playlist stringclasses 160
values | file_name stringlengths 9 102 | content stringlengths 29 329k |
|---|---|---|
Hidden_Figures_Black_History | The_electrifying_speeches_of_Sojourner_Truth_Daina_Ramey_Berry.txt | In early 1828, Sojourner Truth approached the Grand Jury of Kingston, New York. She had no experience with the legal system, no money, and no power in the eyes of the court. Ignoring the jury’s scorn, Truth said she was there to fight for custody of her five-year-old son Peter, who’d been illegally sold to an enslaver in Alabama. As the trial played out over the next several months, Truth raised funds, strategized with lawyers, and held her faith. Finally in the spring of 1828, Peter was returned to her care— but Truth’s work was far from over. She would dedicate the rest of her life to pursuing justice and spiritual understanding. Truth was born into slavery as Isabella Baumfree in the late 18th century in Ulster County, New York. Although New York state had announced the abolition of slavery in 1799, the emancipation act was gradual. Those who were currently enslaved were forced to serve a period of indentured servitude until their mid-20s. Throughout this period, enslavers repeatedly sold Baumfree, tearing her from her loved ones. Often, she was explicitly prevented from pursuing new relationships. Eventually, she married an enslaved man named Thomas, with whom she had three children. She was desperate to keep her new family together— but the slow progress of abolition threatened this hope. Baumfree’s enslaver, John Dumont, had promised to free her by 1826. When he failed to keep his word, Baumfree fled for her safety. During the escape, she was only able to rescue her youngest daughter Sophia, while her other children remained in bondage. It would be two years before she regained custody of Peter. After that, she would wait another two years before she saw any of her other children. During this time, Baumfree found solace in her faith and became increasingly dedicated to religious reflection. After settling in Kingston, New York, she joined a Methodist community that shared her political views. She continued her practice of speaking aloud to God in private, and one night, her evening prayers took on even more sacred significance. Baumfree claimed to hear the voice of God, telling her to leave Kingston, and share her holy message with others. Though she never learned to read or write, Baumfree became known as an electrifying orator, whose speeches drew on Biblical references, spiritual ideals, and her experience of slavery. Her sermons denounced the oppression of African Americans and women in general, and became prominent in campaigns for both abolition and women’s rights. In 1843, she renamed herself Sojourner Truth and embarked on a legendary speaking tour. Truth saw her journey as a mission from God. Her faith often led her to the nation’s most hostile regions, where she spoke to bigoted audiences as the only Black woman in the crowd. Truth was confident God would protect her, but some crowds responded to her bravery with violence. During one of her sermons, a mob of white men threatened to set fire to the tent where she was speaking. In her memoir, Truth recalled steeling herself to confront them: “Have I not faith enough to go out and quell that mob… I felt as if I had three hearts! And that they were so large, my body could hardly hold them!” She placated the men with song and prayer, until they had no desire to harm her. Truth’s speeches impacted thousands of people in communities across the nation, but her activism went far beyond public speaking. During the Civil War, she became involved with the Union Army, recruiting soldiers and organizing supplies for Black troops. Her work was so well regarded that she was invited to meet President Lincoln. She took the occasion to argue that all formerly enslaved people should be granted land by the government. Truth continued to travel and speak well into her 80s. Until her death in 1883, she remained an outspoken critic who fought for her right to be heard in a hostile world. As Truth once said, “I feel safe even in the midst of my enemies; for the truth is powerful and will prevail." |
Hidden_Figures_Black_History | Why_should_you_read_scifi_superstar_Octavia_E_Butler_Ayana_Jamieson_and_Moya_Bailey.txt | Following a devastating nuclear war, Lilith Iyapo awakens after 250 years of stasis to find herself surrounded by a group of aliens called the Oankali. These highly evolved beings want to trade DNA by breeding with humans so that each species’ genes can diversify and fortify the other. The only alternative they offer is sterilization of the entire human race. Should humanity take the leap into the biological unknown, or hold on to its identity and perish? Questions like this haunt Octavia Butler’s "Dawn," the first in her trilogy "Lilith’s Brood." A visionary storyteller who upended science fiction, Butler built stunning worlds throughout her work– and explored dilemmas that keep us awake at night. Born in 1947, Butler grew up shy and introverted in Pasadena, California. She dreamt up stories from an early age, and was soon scribbling these scenarios on paper. At twelve, she begged her mother for a typewriter after enduring a campy science fiction film called "Devil Girl From Mars." Unimpressed with what she saw, Butler knew she could tell a better story. Much science fiction features white male heroes who blast aliens or become saviors of brown people. Butler wanted to write diverse characters for diverse audiences. She brought nuance and depth to the representation of their experiences. For Butler, imagination was not only for planting the seeds of science fiction– but also a strategy for surviving an unjust world on one’s own terms. Her work often takes troubling features of the world such as discrimination on the basis of race, gender, class, or ability, and invites the reader to contemplate them in new contexts. One of her most beloved novels, the "Parable of the Sower," follows this pattern. It tells the story of Lauren Oya Olamina as she makes her way through a near-future California, ruined by corporate greed, inequality, and environmental destruction. As she struggles with hyperempathy, or a condition in the novel that causes her to feel others’ pain, and less often, their pleasure. Lauren embarks on a quest with a group of refugees to find a place to thrive. There, they seek to live in accordance with Lauren’s found religion, Earthseed, which is based on the principle that humans must adapt to an ever-changing world. Lauren’s quest had roots in a real life event– California Prop 187, which attempted to deny undocumented immigrants fundamental human rights, before it was deemed unconstitutional. Butler frequently incorporated contemporary news into her writing. In her 1998 sequel to "The Parable of the Sower," "Parable of the Talents," she wrote of a presidential candidate who controls Americans with virtual reality and “shock collars.” His slogan? “Make America great again.” While people have noted her prescience, Butler was also interested in re-examining history. For instance, "Kindred" tells the story of a woman who is repeatedly pulled back in time to the Maryland plantation of her ancestors. Early on, she learns that her mission is to save the life of the white man who will rape her great grandmother. If she doesn’t save him, she herself will cease to exist. This grim dilemma forces Dana to confront the ongoing trauma of slavery and sexual violence against Black women. With her stories of women founding new societies, time travelers overcoming historical strife, and interspecies bonding, Butler had a profound influence on the growing popularity of Afrofuturism. That’s a cultural movement where Black writers and artists who are inspired by the past, present and future, produce works that incorporate magic, history, technology and much more. As Lauren comes to learn in "Parable of the Sower," "All that you touch you Change. All that you Change Changes you. The only lasting truth is Change.” |
Hidden_Figures_Black_History | The_chaotic_brilliance_of_artist_JeanMichel_Basquiat_Jordana_Moore_Saggese.txt | A sky blue canvas ripped open by an enormous skull. Teeth bared through visceral slashes of oil and spray-paint. In 2017, this untitled artwork was auctioned off for over 110 million dollars. But it’s not the work of some old master. These strokes of genius belong to 21 year old black Brooklynite Jean-Michel Basquiat – one of America’s most charismatic painters, and currently, its highest sold. Born in 1960 to a Haitian father and a Puerto Rican mother, Basquiat spent his childhood making art and mischief in Boerum Hill. While he never attended art school, he learned by wandering through New York galleries, and listening to the music his father played at home. He drew inspiration from unexpected places, scribbling his own versions of cartoons, comic books and biblical scenes on scrap paper from his father’s office. But it was a medical encyclopedia that arguably exerted the most powerful influence on Basquiat. When young Jean-Michael was hit by a car, his mother brought a copy of "Grey’s Anatomy" to his hospital bed. It ignited a lifelong fascination with anatomy that manifested in the skulls, sinew and guts of his later work – which frequently explores both the power and vulnerability of marginalized bodies. By 17, he launched his first foray into the art world with his friend Al Diaz. They spray painted cryptic statements and symbols all over Lower Manhattan, signed with the mysterious moniker SAMO. These humorous, profound, and rebellious declarations were strategically scattered throughout Soho’s art scene. And after revealing himself as the artist, Basquiat leveraged SAMO’s success to enter the scene himself; selling postcards, playing clubs with his avant-garde band, and boldly seeking out his heroes. By 21, he’d turned to painting full time. His process was a sort of calculated improvisation. Like Beat writers who composed their work by shredding and reassembling scraps of writing, Basquiat used similar cut-up techniques to remix his materials. When he couldn't afford canvases, he fashioned them out of discarded wood he found on the street. He used oil stick, crayons, spray paint and pencil and pulled quotes from the menus, comic books and textbooks he kept open on the studio floor. He kept these sources open on his studio floor, often working on multiple projects at once. Pulling in splintered anatomy, reimagined historical scenes, and skulls transplanted from classical still-lives, Basquiat repurposed both present day experiences and art history into an inventive visual language. He worked as if inserting himself into the legacy of artists he borrowed from, producing collages that were just as much in conversation with art history as they were with each other. For instance, "Toussaint L’Overture versus Savonarola" and "Undiscovered Genius of the Mississippi Delta" offer two distinct visions of Basquiat’s historical and contemporary concerns. But they echo each other in the details, such as the reappearing head that also resurfaces in "PPCD." All these pieces form a network that offers physical evidence of Basquiat’s restless and prolific mind. These chaotic canvases won rapid acclaim and attention. But despite his increasingly mainstream audience, Basquiat insisted on depicting challenging themes of identity and oppression. Marginalized figures take center stage, such as prisoners, cooks and janitors. His obsession with bodies, history, and representation can be found in works evoking the Atlantic slave trade and African history, as well as pieces focusing on contemporary race relations. In less than a decade, Basquiat made thousands of paintings and drawings- along with sculpture, fragments of poetry and music. His output accelerated alongside his meteoric rise to fame, but his life and work were cut tragically short when he died from a drug overdose at the age of 27. After his death, Basquiat’s work only increased in value- but the energy and flair of his pieces have impacted much more than their financial worth. Today, his influence swirls around us in music, poetry, fashion and film- and his art retains the power to shock, inspire, and get under our skin. |
Hidden_Figures_Black_History | The_Atlantic_slave_trade_What_too_few_textbooks_told_you_Anthony_Hazard.txt | Slavery, the treatment of human beings as property, deprived of personal rights, has occurred in many forms throughout the world. But one institution stands out for both its global scale and its lasting legacy. The Atlantic slave trade, occurring from the late 15th to the mid 19th century and spanning three continents, forcibly brought more than 10 million Africans to the Americas. The impact it would leave affected not only these slaves and their descendants, but the economies and histories of large parts of the world. There had been centuries of contact between Europe and Africa via the Mediterranean. But the Atlantic slave trade began in the late 1400s with Portuguese colonies in West Africa, and Spanish settlement of the Americas shortly after. The crops grown in the new colonies, sugar cane, tobacco, and cotton, were labor intensive, and there were not enough settlers or indentured servants to cultivate all the new land. American Natives were enslaved, but many died from new diseases, while others effectively resisted. And so to meet the massive demand for labor, the Europeans looked to Africa. African slavery had existed for centuries in various forms. Some slaves were indentured servants, with a limited term and the chance to buy one's freedom. Others were more like European serfs. In some societies, slaves could be part of a master's family, own land, and even rise to positions of power. But when white captains came offering manufactured goods, weapons, and rum for slaves, African kings and merchants had little reason to hesitate. They viewed the people they sold not as fellow Africans but criminals, debtors, or prisoners of war from rival tribes. By selling them, kings enriched their own realms, and strengthened them against neighboring enemies. African kingdoms prospered from the slave trade, but meeting the European's massive demand created intense competition. Slavery replaced other criminal sentences, and capturing slaves became a motivation for war, rather than its result. To defend themselves from slave raids, neighboring kingdoms needed European firearms, which they also bought with slaves. The slave trade had become an arms race, altering societies and economies across the continent. As for the slaves themselves, they faced unimaginable brutality. After being marched to slave forts on the coast, shaved to prevent lice, and branded, they were loaded onto ships bound for the Americas. About 20% of them would never see land again. Most captains of the day were tight packers, cramming as many men as possible below deck. While the lack of sanitation caused many to die of disease, and others were thrown overboard for being sick, or as discipline, the captain's ensured their profits by cutting off slave's ears as proof of purchase. Some captives took matters into their own hands. Many inland Africans had never seen whites before, and thought them to be cannibals, constantly taking people away and returning for more. Afraid of being eaten, or just to avoid further suffering, they committed suicide or starved themselves, believing that in death, their souls would return home. Those who survived were completley dehumanized, treated as mere cargo. Women and children were kept above deck and abused by the crew, while the men were made to perform dances in order to keep them exercised and curb rebellion. What happened to those Africans who reached the New World and how the legacy of slavery still affects their descendants today is fairly well known. But what is not often discussed is the effect that the Atlantic slave trade had on Africa's future. Not only did the continent lose tens of millions of its able-bodied population, but because most of the slaves taken were men, the long-term demographic effect was even greater. When the slave trade was finally outlawed in the Americas and Europe, the African kingdoms whose economies it had come to dominate collapsed, leaving them open to conquest and colonization. And the increased competition and influx of European weapons fueled warfare and instability that continues to this day. The Atlantic slave trade also contributed to the development of racist ideology. Most African slavery had no deeper reason than legal punishment or intertribal warfare, but the Europeans who preached a universal religion, and who had long ago outlawed enslaving fellow Christians, needed justification for a practice so obviously at odds with their ideals of equality. So they claimed that Africans were biologically inferior and destined to be slaves, making great efforts to justify this theory. Thus, slavery in Europe and the Americas acquired a racial basis, making it impossible for slaves and their future descendants to attain equal status in society. In all of these ways, the Atlantic slave trade was an injustice on a massive scale whose impact has continued long after its abolition. |
Hidden_Figures_Black_History | Zora_Neale_Hurstons_Their_Eyes_Were_Watching_God_Tanya_Boucicaut.txt | Baritone thunder. Snarling winds. Consuming downpours. Okeechobee, the disastrous hurricane of 1928, tore through the North Atlantic basin, laying waste to entire communities. In Eatonville, Florida, the storm forced many to flee. But for Janie Crawford, it inspired an unexpected homecoming. Janie’s return begins “Their Eyes Were Watching God,” Zora Neale Hurston’s acclaimed novel about a Black woman’s quest for love and agency in a time that sought to deprive her of both. When Janie arrives back in Eatonville, her arrival is shrouded in mystery. Her neighbors and friends are quick to gossip about her reappearance, her finances, and most importantly, the whereabouts of her missing husband. But only Janie’s friend Pheoby gets to hear the whole story. Over the course of a conversation that spans most of the novel, Hurston untangles Janie’s life story; from her complicated childhood and her life in Eatonville, to her scandalous departure and the shocking events that followed. The specifics of Janie’s story are often larger than life, but many of the book’s details reflect the incredible experiences of its author. Zora Neale Hurston was raised in Eatonville, one of the first planned and incorporated all-Black communities in America. Like Janie, she also left Eatonville abruptly, traveling first to Jacksonville and DC, before eventually moving further north. In New York City, Hurston studied anthropology and became a renowned author in the Harlem Renaissance, a cultural, literary and artistic movement that’s still considered a golden era of Black artistry and creativity. Here, her work garnered enough support to fund research trips through the South, where she collected stories and folktales from Black Americans. By 1937, her fieldwork had taken her all the way to Haiti, where she wrote most of “Their Eyes Were Watching God.” Hurston drew on all these experiences for the novel, incorporating folkloric elements alongside her own family and romantic history to bring readers into the intimate spaces of Black southern life. She uses regional phrases and sayings to capture the dialect of her Floridian characters. And the novel’s omniscient third-person narration allows Hurston to unleash her poetic prose on everything from birdsong, architecture, and fashion, to her characters’ deepest feelings and motivations. Perhaps more than any specific details, Hurston’s experiences of being a Black woman in America at this time are more evident in the novel’s themes. Over the course of one long evening, Janie and Pheoby discuss the nature of family, marriage, spirituality and more. But their conversation always comes back to Janie’s truest desire: to live honestly and be truly loved in return. As a teenager, Janie resents an arranged marriage, despite the safety it offers her and the wishes of her loving grandmother. When her family becomes well-respected in Eatonville, she struggles with the judgmental eyes of strangers and a husband who wants her to be something she’s not. Throughout her life, Janie frequently feels she’s at the whim of natural and spiritual forces that can shift the course of her existence without warning. And when she finally does find true love, these unknowable powers continue to act on her, threatening to destroy the life she's so painstakingly built. The story takes place during a time where women had little to no agency, and Janie’s life is full of complicated characters who demand different kinds of love and submission. But despite the loneliness of her situation, Janie navigates these trials with defiance and curiosity. Her questions and commentary push back in subtle, clever ways. And as the reader follows Janie’s journey from childhood to middle age, her confidence becomes infectious. Just like Hurston, Janie defies the restrictive expectations for a woman in her time. Early in the novel, Hurston writes that “there are years that ask questions and years that answer,” suggesting that life can only truly be understood by living it. But through her empathetic storytelling, Hurston invites us into Janie’s life, her life, and the lives of so many other women. |
Hidden_Figures_Black_History | Debunking_the_myth_of_the_Lost_Cause_A_lie_embedded_in_American_history_Karen_L_Cox.txt | Between 1860 and 1861, 11 southern states withdrew from the United States and formed the Confederate States of America. They left, or seceded, in response to the growing movement for the nationwide abolition of slavery. Mississippi said, “our position is thoroughly identified with the institution of slavery.” South Carolina cited “hostility on the part of the non-slaveholding states to the institution of slavery.” In March 1861, the Vice President of the Confederacy, Alexander Stevens, proclaimed that the cornerstone of the new Confederate government was white supremacy, or as he put it, “slavery” and “subordination” to white people was the “natural and normal condition” of Black people in America and the “immediate cause of the late rupture and present revolution.” Three weeks after the now-infamous Cornerstone Speech, the American Civil War began. The conflict lasted four years, had a death toll of about 750,000, and ended with the Confederacy’s defeat. By 1866, barely a year after the war ended, southern sources began claiming the conflict wasn’t actually about slavery. Meanwhile, Frederick Douglass, a prominent abolitionist and formerly enslaved person, cautioned, “the spirit of secession is stronger today than ever.” From the words of Confederate leaders, the reason for the war could not have been clearer— it was slavery. So how did this revisionist history come about? The answer lies in the Lost Cause— a cultural myth about the Confederacy. The term was coined by Edward Pollard, a pro-Confederate journalist. In 1866, he published “The Lost Cause: A New Southern History of the War of the Confederates.” Pollard pointed out that the U.S. Constitution gave states the right to govern themselves independently in all areas except those explicitly designated to the national government. According to him, the Confederacy wasn’t defending slavery, it was defending each state’s right to choose whether or not to allow slavery. This explanation effectively turned white southerners’ documented defense of slavery and white supremacy into a patriotic defense of the Constitution. The Civil War had devastated the country, leaving those who had supported the Confederacy grasping to justify their actions. Many pro-Confederate writers, political leaders, and others were quick to adopt and spread the narrative of the Lost Cause. One organization, the United Daughters of the Confederacy, played a key role in transmitting the ideas of the Lost Cause to future generations. Founded in Nashville, Tennessee, in 1894, the UDC united thousands of middle and upper class white southern women. The UDC raised thousands of dollars to build monuments to Confederate soldiers. These were often unveiled with large public ceremonies, and given prominent placements, especially on courthouse lawns. The Daughters also placed Confederate portraits in public schools. They monitored textbooks to minimize the horrors of slavery, and its significance in the Civil War, passing revisionist history and racist ideology down through generations. By 1918, the UDC claimed over 100,000 members. As their numbers grew, they increased their influence outside the South. Presidents William Howard Taft and Woodrow Wilson both met with UDC members and enabled them to memorialize the Confederacy in Arlington National Cemetery. The UDC still exists and defends Confederate symbols as part of a noble heritage of sacrifice by their ancestors. Despite the wealth of primary sources showing that slavery was the root cause of the Civil War, the myth about states’ rights persists today. In the aftermath of the war, Frederick Douglass and his abolitionist contemporaries feared this erasure of slavery from the history of the Civil War could contribute to the government’s failure to protect the rights of Black Americans— a fear that has repeatedly been proven valid. In an 1871 address at Arlington Cemetery, Douglass said: “We are sometimes asked in the name of patriotism to forget the merits of this fearful struggle, and to remember with equal admiration those who struck at the nation’s life, and those who struck to save it— those who fought for slavery and those who fought for liberty and justice. [...] if this war is to be forgotten, I ask in the name of all things sacred, what shall men remember?” |
Hidden_Figures_Black_History | Zumbi_The_last_king_of_Palmares_Marc_Adam_Hertzman_Flavio_dos_Santos_Gomes.txt | During the 1600s, an expansive autonomous settlement called Palmares reached its height in northeastern Brazil. It was founded and led by people escaping from slavery, also called maroons. In fact, it was one of the world’s largest maroon communities, its population reaching beyond 10,000. And its citizens were at constant war with colonial forces. The records we have about Palmares mainly come from biased Dutch and Portuguese sources, but historians have managed to piece much of its story together. During the Trans-Atlantic slave trade, which began in the 1500s, nearly half of all enslaved African people were sent to Portugal’s American colony: Brazil. Some escaped and sought shelter in Brazil’s interior regions, where they formed settlements called mocambos or quilombos. Fugitives from slavery probably arrived in the northeast in the late 1500s. By the 1660s, their camps had consolidated into a powerful confederation known today as the Quilombo of Palmares. It consisted of a central capital linking dozens of villages, which specialized in particular agricultural goods or served as military training grounds. Citizens of Palmares, or Palmaristas, were governed by a king and defended by an organized military. African people and Brazilian-born Black and Indigenous people all came seeking refuge. They collectively fished, hunted, raised livestock, planted orchards, and grew crops like cassava, corn, and sugarcane. They also made use of the abundant palm trees for which Palmares was named, turning palm products into butter, wine, and light. Palmaristas crafted palm husks into pipes and leaves into mats and baskets. They traded some of these goods with Portuguese settlers for products like gunpowder and salt. In turn, settlers avoided Palmares’ raids during which they’d seize weapons and take captives. The Portuguese were concerned with other invading imperialists, but regarded Indigenous uprisings and Palmares as their internal threats. Palmares endangered the very institution of slavery— the foundation of Brazil's economy. During the 1670s, the Portuguese escalated their attacks. By this time, Ganga-Zumba was Palmares’ leader. He ruled from the Macaco, which functioned as the capital. His allies and family members governed the other villages— with women playing crucial roles in operation and defense. As they fought the Portuguese, Palmaristas used the landscape to their advantage. Camouflaged and built in high places, their mocambos offered superior lookouts. They constructed hidden ditches lined with sharp stakes that swallowed unsuspecting soldiers and false roads that led to ambushes. They launched counterattacks under the cover of night and were constantly abandoning and building settlements to elude the Portuguese. In 1678, after years of failed attacks, the Portuguese offered to negotiate a peace treaty with Ganga-Zumba. The terms they agreed upon recognized Palmares’ independence and the freedom of anyone born there. However, the treaty demanded that Palmares pledge loyalty to the crown and return all past and future fugitives from slavery. Many Palmaristas dissented, among them Zumbi— Ganga-Zumba’s nephew— a rising leader himself. Before long, Ganga-Zumba was killed, likely by a group affiliated with his nephew. As Palmares’ new leader, Zumbi rejected the treaty and resumed resistance for another 15 years. But in February of 1694, the Portuguese captured the capital after a devastating siege. Zumbi escaped, but they eventually found and ambushed him. And on November 20th, 1695, Portuguese forces killed Zumbi. His death was not the end of Palmares, but it was a crushing blow. After years of warfare, there were far fewer rebels in the area. Those who remained rallied around new leaders and maintained a presence, however small, through the 1760s. Though, Palmares is no more thousands of other quilombos persist to this day. November 20th, the day of Zumbi’s death, is celebrated across Brazil as the Day of Black Consciousness. But Zumbi was just one of many Palmaristas. We only know some of their names, but their fight for freedom echoes centuries later. |
Hidden_Figures_Black_History | Why_are_US_cities_still_so_segregated_Kevin_EhrmanSolberg_and_Kirsten_Delegard.txt | On October 21st, 1909, 125 residents of an affluent Minneapolis neighborhood approached William Simpson, who’d just bought a plot in the area, and told him to leave. The Simpsons would be the second Black family in the otherwise white neighborhood, where they intended to build a home. When the Simpsons refused offers to buy them out, their neighbors tried blocking their home’s construction. They finally moved into their house, but the incident had a ripple effect. Just a few months after the mob harassed the Simpsons, the first racially restrictive covenant was put into place in Minneapolis. Covenants are agreements in property deeds that are intended to regulate how the property is to be used. Beginning in the mid-1800s, people in the United States and elsewhere began employing them in a new way: specifically, to racially restrict properties. They wrote clauses into deeds that were meant to prevent all future owners from selling or leasing to certain racial and ethnic groups, especially Black people. Between 1920 and 1950, these racial covenants spread like wildfire throughout the US, making cities more segregated and the suburbs more restricted. In the county encompassing Minneapolis, racial covenants eventually appeared on the deeds to more than 25,000 homes. Not only was this legal, but the US Federal Housing Administration promoted racial covenants in their underwriting manual. While constructing new homes, real estate developers began racially restricting them from the outset. Developments were planned as dream communities for American families— but for white people only. In 1947, one company began building what became widely recognized as the prototype of the postwar American suburb: Levittown, New York. It was a community of more than 17,000 identical homes. They cost around $7,000 each and were intended to be affordable for returning World War II veterans. But, according to Levittown’s racial covenants, none of the houses could “be used or occupied by any person other than members of the Caucasian race,” with one exception: servants. Between 1950 and 1970, the population of the American suburbs nearly doubled as white people flocked to more racially homogenous areas in a phenomenon known as “white flight.” The suburbs spread, replacing native ecosystems with miles of pavement and water-guzzling lawns. And their diffuse layout necessitated car travel. American automobile production quadrupled between 1946 and 1955, cementing the nation's dependence on cars. Federal programs like the G.I. Bill offered American veterans favorable lending rates for buying homes. But it was difficult for people of color to take advantage of such resources. Racial covenants restricted them from certain neighborhoods. And, at the same time, government programs labelled neighborhoods of color bad investments and often refused to insure mortgages in those areas. Therefore, banks usually wouldn’t lend money to people purchasing property in neighborhoods of color— a practice that became known as redlining. So, instead of owning homes that increased in value over time, creating wealth that could be passed to future generations, many people of color were forced to spend their income on rent. And even when they were able to buy property, their home’s value was less likely to increase. The suburbs boasted cul-de-sacs and dead ends that minimized traffic. Meanwhile, city planners often identified redlined neighborhoods as inexpensive areas for industrial development. So, the massive freeway projects of the mid-20th century disproportionately cut through redlined neighborhoods, accompanied by heavy industry and pollution. As a result, many neighborhoods of color experience higher rates of drinking water contamination, asthma, and other health issues. People targeted by racial covenants increasingly challenged them in court and, in 1968, they were finally banned under the Fair Housing Act. But the damage had been done. Racial covenants concentrated wealth and amenities in white neighborhoods and depressed the conditions and home values in neighborhoods of color. As of 2020, about 74% of white families in the US owned their homes, while about 44% of Black families did. That gap is greatest in Minnesota’s Twin Cities. Across the country, neighborhoods remain segregated and 90% of all suburban counties are predominantly white. Some landlords, real estate agents, and lenders still discriminate against people based on race— rejecting them, steering them to and away from certain neighborhoods, or providing inaccessibly high interest rates. Gentrification and exclusionary zoning practices also still displace and keep people of color out of certain neighborhoods. Racial covenants are now illegal. But they can still be seen on many housing deeds. The legacy of racial covenants is etched across the pristine lawns of the American suburbs. It’s a footnote in the demographic divides of every city. And it’s one of the insidious architects of the hidden inequalities that shape our world. |
Hidden_Figures_Black_History | The_life_legacy_assassination_of_an_African_revolutionary_Lisa_Janae_Bacon.txt | In 1972, Thomas Sankara was swept into a revolution for a country not his own. Hailing from the West African nation of Burkina Faso— then known as Upper Volta— the 22-year-old soldier had travelled to Madagascar to study at their military academy. But upon arriving, he found a nation in conflict. Local revolutionaries sought to wrest control of Madagascar from France’s lingering colonial rule. These protestors inspired Sankara to read works by socialist leaders like Karl Marx and seek wisdom from military strategy. When he returned to Upper Volta in 1973, Sankara was determined to free his country from its colonial legacy. Born in 1949, Sankara was raised in a relatively privileged household as the third of ten children. His parents wanted him to be a priest, but like many of his peers, Sankara saw the military as the perfect institution to rid Upper Volta of corruption. After returning from Madagascar, he became famous for his charisma and transparent oratorial style— but he was less popular with the reigning government. Led by President Jean-Baptiste Ouédraogo, this administration came to power in the 3rd consecutive coup d’état in Upper Volta’s recent history. The administration’s policies were a far cry from the sweeping changes Sankara proposed, but, by 1981, Sankara’s popularity won out, earning him a role in Ouédraogo’s government. Nicknamed “Africa’s Che Guevara," Sankara rapidly rose through the ranks, and within two years, he was appointed Prime Minister. In his new role, he delivered rallying speeches to impoverished communities, women, and young people. He even tried to persuade other governments to form alliances based on their shared colonial legacy. But Ouédraogo and his advisors felt threatened by Sankara’s new position. They thought his communist beliefs would harm alliances with capitalist countries, and just months after becoming Prime Minister, Ouédraogo’s administration forced Sankara from the job and placed him on house arrest. Little did the President know this act would fuel Upper Volta’s 4th coup d’état in 17 years. Civilian protests ensued around the capital, and the government ground to a halt while Sankara tried to negotiate a peaceful transition. During this time, Blaise Compaoré, Sankara’s friend and fellow former soldier, foiled another coup that included an attempt on Sankara’s life. Eventually, Ouédraogo resigned without further violence, and on August 4, 1983, Thomas Sankara became the new President of Upper Volta. Finally in charge, Sankara launched an ambitious program for social and economic change. As one of his first agenda items, he renamed the country from its French colonial title "Upper Volta" to "Burkina Faso," which translates to “Land of Upright Men." Over the next four years he established a nation-wide literacy campaign, ordered the planting of over 10 million trees, and composed a new national anthem— all while cutting down inflated government employee salaries. But perhaps the most unique element of Sankara’s revolution was his dedication to gender equality. He cultivated a movement for women’s liberation, outlawing forced marriages, polygamy and genital mutilation. He was the first African leader to appoint women to key political positions and actively recruit them to the military. However, Sankara’s socialist policies were met with much resistance. Many students and elites believed his economic plans would alienate Burkina Faso from its capitalist peers. His crackdown on the misuse of public funds turned government officials against him as well. After four years, what began as an empowering revolution had isolated many influential Burkinabes. But Sankara was not ready to yield his power. He executed increasingly authoritarian actions, including banning trade unions and the free press. Eventually, his autocratic tendencies turned even his closest friends against him. On October 15, 1987, Sankara was conducting a meeting when a group of assailants swarmed his headquarters. Sankara was assassinated in the attack, and many believe the raid was ordered by his friend Blaise Compaoré. Though his legacy is complicated, many of Sankara’s policies have proven themselves to be ahead of their time. In the past decade, Burkinabe youth have celebrated Sankara’s political philosophy, and nearby countries like Ghana have even adopted Sankara’s economic models. On March 2, 2019 a statue of Sankara was erected in Burkina Faso’s capital, establishing his place as an icon of revolution for his country and throughout the world. |
Hidden_Figures_Black_History | The_secret_society_of_the_Great_Dismal_Swamp_Dan_Sayers.txt | Straddling Virginia and North Carolina is an area that was once described as the “most repulsive of American possessions.” By 1728, it was known as the Great Dismal Swamp. But while many deemed it uninhabitable, recent findings suggest that a hidden society persisted in the Swamp until the mid-1800′s. So, who lived there? And what happened to them? People long suspected that communities had settled in the Swamp, but the historical record was spotty. It wasn’t until 2003 that the first systematic archaeological foray finally launched. But, despite having been extensively drained over the years, the wetland still presented many practical challenges. Researchers had to penetrate thorny thickets, wade through waters studded with sinkholes, and braved the threats of dangerous animals. After several months, they finally found islands in the Swamp’s interior. These formations quickly revealed traces of centuries-old secrets. Archeologists found buried markings that appear to have been left by raised log cabins, fire pits, and basins that may have collected drinking water. They identified what seems to have been a palisade wall and excavated more than 3,000 artifacts, including weaponry, stone tools, and fragments of ceramic pipes and vessels. These discoveries, combined with previous findings, helped tell a story that reaches far back in time. Indigenous American people began regularly inhabiting or visiting the area around 11,000 BCE, before it was even a swampland. A second era of occupation began much later. In the early 1600′s, more Indigenous people came seeking refuge from colonization. And later that century, it seems that Maroons— or people escaping from slavery— began entering the area. In fact, the team’s findings support the theory that the Great Dismal Swamp was home to the largest Maroon settlement in all of North America. Because their success and survival depended on staying hidden from the outside world, these Swamp communities were largely self-sufficient. Based on primary sources, historians believe that people cultivated grains and created homes, furniture, musical instruments, and more from the Swamp’s available resources. These organic materials had probably already decomposed by the time archaeologists came to investigate. But researchers were able to find more durable objects, like ceramic and stone items that were likely left by ancient Indigenous people then reused and modified by others later on. Around the turn of the 19th century, It seems the relationship between the Swamp’s community and the outside world changed. Lumber and manufacturing companies began encroaching on the Swamp’s interior. They brought thousands of free and enslaved workers to live in the Swamp and made them harvest wood, excavate canals, and drain fields. Certain findings suggest that the Swamp’s hidden communities might have switched to a more defensive mode during this period. But researchers also observed more mass-produced objects from this time, indicating that trading was taking place. Researchers think that the secret Swamp communities dispersed during or soon after the American Civil War, by the end of which slavery was abolished in the United States. Some people may have stayed in the Swamp until they passed away or left to settle elsewhere. Most of what we know about these hidden communities has come to light after archeologists excavated sections of a single island. However, there may have been hundreds of habitable islands dotting the Swamp’s interior at the time. Between 1600 and 1860, many people lived in these hidden settlements. Some probably lived their entire lives within the Swamp and never saw a white person or experienced racial persecution in broader American society. Generations of Black Maroons and Indigenous Americans resisted slavery and colonization by creating an independent society in the heart of the Great Dismal Swamp. They fostered a refuge in what might seem like the unlikeliest of places— but one that was more hospitable than what lay outside. Today, this area offers a partial record of that secret, self-reliant world, imagined and built for survival and the preservation of freedom. |
Hidden_Figures_Black_History | Why_did_Phillis_Wheatley_disappear_Charita_Gainey.txt | In late 1775, the newly appointed General George Washington received a poem from one of colonial America’s most famous writers. Its verses praised the burgeoning revolution, invoking the goddess of their new nation to aid the general’s righteous cause. But this ode to liberty wasn’t written by some aloof, aristocratic admirer. Its author was a young Black woman who’d been enslaved for over a decade. The young girl, who’d been renamed Phillis Wheatley, had arrived in the colonies on a slave ship in 1761. The ship landed in Boston, where Susanna and John Wheatley purchased Phillis to work in their house. However, for reasons that remain unclear, they also taught her to read and write. Over the following decade, Wheatley became well versed in poetry and religious texts, eventually beginning to produce her own poems. The family published her work in a local newspaper, and in 1771, her elegy for renowned reverend George Whitefield captured the public’s imagination. The poem’s repetitive rhythms, dramatic religious references, and soaring spiritual language depicted how Whitefield’s sermons “inflame the soul and captivate the mind.” Wheatley ends with an arresting image of life after death, trusting that divine forces “will re-animate his dust.” This moving tribute found an audience in both the US and England. And since the piece was published with a note identifying the author as an enslaved woman, many readers were as fascinated with the poet as they were with the poem. In 1773, Phillis traveled to London, where her collection of “Poems on Various Subjects, Religious and Moral” became the first book of poetry published by an African-American woman. It was filled with profound meditations on life, death, and religion, as well as Biblical and classical references. In “A Hymn to Humanity,” Wheatley linked these themes to her own creative growth, portraying herself as a muse smiled upon by heavenly bodies. Unsurprisingly, Wheatley had her critics. Many white Americans believed Black people were incapable of producing intellectual and creative work. Thomas Jefferson wrote that her writing didn't even deserve to be called poetry, and others dismissed her as a poor imitation of another well-known poet. But many readers of the time were enamored with Wheatley's work, including prominent European writers and politicians. Many modern readers, however, might expect her work to cover a different topic: slavery. Wheatley rarely wrote directly about her experiences as an enslaved person. And her poem addressing the topic has been criticized for suggesting she was grateful that enslavement led her to Christianity. But it’s incredibly unlikely Wheatley would have been able to publicly condemn slavery without serious consequences. And many readers have found a more nuanced critique hidden within her work. For example, Wheatley was a vocal supporter of American independence, writing that her “love of freedom” came from early experiences of being kidnapped into slavery and separated from her parents. When disparaging England’s imperial control, she evokes imagery of an “iron chain.” And by comparing her lack of freedom to America’s lack of independence, Wheatley subtly laments her own circumstances. Thankfully, Wheatley secured her freedom after returning from London. The reasons for her emancipation aren't entirely clear, as there’s no evidence of the Wheatleys freeing other enslaved people. However, since Phillis could have remained free in London, some believe she bargained to make emancipation a condition of her return. It’s difficult to know exactly what happened, both here and throughout the rest of Wheatley’s life. Her proposal for a second book was never published. In 1778, she married a free Black man named John Peters. The two are believed to have had three children, all of whom died in infancy. Their last child is thought to have died around the same time as Wheatley, and the two were buried together in an unmarked grave. While some of Wheatley’s letters survived, she never released an account of her life. So despite her tenure as perhaps the most famous African on the planet, Wheatley’s story has been lost to the ravages of history, like those of countless other enslaved peoples. But her poetry lives on today— celebrating creative growth and offering spiritual sustenance. |
Hidden_Figures_Black_History | One_of_the_most_dangerous_men_in_American_history_Keenan_Norris.txt | In 1830 at a clothing store near the Boston Harbor, David Walker carefully stitched a pamphlet into the lining of a sailor’s coat. The volume was thin enough to be completely hidden, but its content was far from insubstantial. In fact, at the time, many members of the US government considered this pamphlet to be one of the most dangerous documents in American history. So to ensure this volume reached his audience, Walker had to hide his work in the clothing of both willing co-conspirators and unknowing sailors; smuggling the pamphlet throughout the country. But what was this incendiary document? And who exactly was the man who wrote it? Son of an enslaved father and a freedwoman, David Walker was born free in the late 18th century in Wilmington, North Carolina. From a young age, he sought to extend his freedom to all Black Americans, and after moving to Charleston as a young man, he became closely involved with the African Methodist Episcopal Church. At that time, the mainstream movement to end slavery was comprised of societies led by wealthy white men who favored gradual change and avoided confrontation with slaveholders. But the AME Church practiced a more radical brand of abolition. In 1822, AME leader Denmark Vesey planned a major insurrection intended to violently liberate Charleston’s enslaved community and set the city ablaze. It’s unclear if Walker contributed to Vesey’s plan, but he wasn’t among the many AME members who were arrested and executed for this attempted rebellion. In 1825, Walker surfaced in Boston, where he rejoined the fight against slavery. In addition to marrying fellow activist Eliza Butler and opening his clothing store, Walker helped fund America’s first Black-owned newspaper. Is passionate articles and public speeches sought to instill pride and camaraderie into those fighting for Black liberation. But to truly unite free and enslaved Black Americans, Walker would have to go beyond Boston. In 1829, he poured his ideas into the “Appeal to the Colored Citizens of the World.” This treatise was punctuated with furious exclamation marks and emphasized the spiritual righteousness of resistance. He described the suffering of enslaved people in graphic detail to prove that the reality for Black Americans was often “kill or be killed.” And given these circumstances, Walker staunchly defended the right to militant action. This wasn’t his only departure from moderate mainstream abolitionists. To stress the importance of Black solidarity, Walker connected American abolitionism with global movements for Black liberation. He called for an international Black freedom struggle in an early display of what would come to be called Pan-Africanism. But at the same time, he opposed the popular movement for Black Americans to emigrate to Africa. While the “Appeal” criticized the Founding Fathers for their hypocrisy, Walker insisted that Black people were essential to the country’s creation, and had an undeniable right to American citizenship. Walker suspected these incendiary arguments would make him a target for violence. But in spite of the danger, he continued using sailors to smuggle his work. The “Appeal” traveled down the coast into the hands of shopkeepers, church leaders, political organizers, and underground abolitionist networks. For these readers, Walker’s words galvanized militant efforts to overthrow slave owners and its call to arms struck fear into white officials. Police intercepted its delivery, and quarantined Black sailors at Southern ports. The pamphlet inspired Louisiana to ban anti-slavery literature, and both North and South Carolina cracked down on Black education to prevent literacy among enslaved peoples. Southern officials even placed a bounty on Walker’s head worth the modern equivalent of $322,000. But while friends urged him to flee, Walker refused to abandon his cause. Tragically, his bravery couldn’t protect him from the deadliest disease of his time. In August 1830, Walker was found dead. And while his associates declared him the victim of assassination, it’s now widely believed that he died from tuberculosis. Following his death, Walker's message continued to resound. Frederick Douglass credited him as the originator of radical abolitionism, and his “Appeal” inspired some of the most influential members of the 20th century liberation movement. From Malcolm X’s militant approach to Black resistance, to James Cone’s writing on Black spirituality, Walker’s legacy remains crucially important to the history of Black resistance movements— and their visions for the future. |
Hidden_Figures_Black_History | One_of_the_most_banned_books_of_all_time_Mollie_Godfrey.txt | In 1998, a Maryland school district removed one of American literature’s most acclaimed works from its curriculum. Parents pushing for the ban said the book was both “sexually explicit” and “anti-white.” Following an outcry from other parents and teachers, the decision was eventually reversed. But this was neither the first nor the last attack on Maya Angelou’s “I Know Why The Caged Bird Sings.” Few books have been challenged more often than Angelou's memoir. And while book banning decisions typically aren’t made at the state or national level, most of the schools and libraries that have banned Angelou’s book have given similar reasons. Most commonly, they argue that the memoir’s account of sexual assault and the violence of US racism are inappropriate for young readers. But these concerns miss the point of Angelou’s story, which uses these very themes to explore the danger of censorship and silence in the lives of young people. Published in 1969, “I Know Why The Caged Bird Sings” traces the author’s childhood growing up poor, Black and female in the southern US. Central to the narrative is Angelou’s experience of being sexually assaulted when she was seven and a half years old. Surrounded by adults who consider the subject too taboo to discuss, Angelou decides that she is to blame. And when she finally identifies her abuser in court, he is killed by vigilantes. Angelou believes her voice is responsible for his death, and for six years, she stops speaking almost entirely. The book chronicles Angelou’s journey to rediscover her voice, all while exploring the pain and misplaced shame that emerges from avoiding uncomfortable realities. The memoir’s narrative voice expertly blends her childhood confusion with her adult understanding, offering the reader insights Angelou was deprived of as a child. She connects her early experiences of being silenced and shamed to the experience of being poor and Black in the segregated United States. “The Black female,” she writes, “is caught in the tripartite crossfire of masculine prejudice, white illogical hate, and Black lack of power.” Her autobiography was one of the first books to speak openly about child sexual abuse, and especially groundbreaking to do so from the perspective of the abused child. For centuries, Black women writers had been limited by stereotypes characterizing them as hypersexual. Afraid of reinforcing these stereotypes, few were willing to write about their sexuality at all. But Angelou refused to be constrained. She publicly explored her most personal experience, without apology or shame. This spirit of defiance charges her writing with a sense of hope that combats the memoir’s often traumatic subject matter. When recalling how a fellow student defied instructions not to sing the Black National Anthem in the presence of white guests, she writes, “The tears that slipped down many faces were not wiped away in shame. We were on top again... We survived.” Angelou’s memoir was published amidst the Civil Rights and Black Power movements, when activists were calling for school curricula that reflected the diversity of experiences in the US. But almost as soon as the book appeared in schools, it was challenged. Campaigns to control lesson plans surged across America in the 1970s and 80s. On the American Library Association’s list of most frequently banned or challenged books, “I Know Why The Caged Bird Sings” remained near the top for two decades. But parents, students, and educators have consistently fought back in support of the memoir. And by 2013, it had become the second most taught non-fiction text in US high school English classes. When asked how she felt about writing one of the most banned books, Angelou said, “I find that people who want my book banned have never read a paragraph of my writing, but have heard that I write about a rape. They act as if their children are not faced with the same threats. And that’s terrible.” She believed that children who are old enough to be the victims of sexual abuse and racism are old enough to read about these subjects. Because listening and learning are essential to overcoming, and the unspeakable is far more dangerous when left unspoken. |
Hidden_Figures_Black_History | Ugly_History_The_US_Syphilis_Experiment_Susan_M_Reverby.txt | In the 1930s, the United States was ravaged by syphilis. This sexually transmitted infection afflicted nearly 1 in 10 Americans, producing painful sores and rashes that persisted for roughly two years. After these initial symptoms, late-stage syphilis was known to cause organ damage, heart and brain disorders, and even blindness. It was incredibly difficult to slow the disease’s spread. Experts cautioned against unprotected sex, but the infection could also be passed during childbirth. Worse still, existing treatments like mercury and bismuth were considered unreliable at best and potentially harmful at worst. Today these heavy metals are classified as toxic, but at the time, doctors were still uncovering their dangerous side effects. Amidst the uncertainty, health care professionals had two key questions. Did late-stage syphilis warrant the risks of existing treatments? And, did the infected individual’s race change how the disease progressed? Many physicians were convinced syphilis affected the neurological systems of white patients and the cardiovascular systems of Black patients. There was little evidence for this theory, but the U.S. Public Health Service was determined to investigate further. In 1932 they launched a massive experiment in Tuskegee, Alabama. The town had already possessed a small hospital, and the area was home to a large population of potential participants. The PHS collaborated with local doctors and nurses to recruit roughly 400 Black men presumed to have noncontagious late-stage syphilis, as well as 200 non-syphilitic Black men for their control group. But their recruitment plan centered on a lie. While the researchers planned to observe how syphilis would progress with minimal treatment, participants were told they would receive free drugs and care for their condition. At first, researchers gave the men existing treatments, but these were soon replaced with placebos. Under the false pretense of providing a special remedy, researchers performed painful and invasive spinal taps to investigate the disease’s neurological consequences. When patients died, the PHS would swoop in to study the body by funding funerals in exchange for autopsies. In their published studies, they listed the men as volunteers to obscure the circumstances under which they’d been recruited. Outside Alabama, syphilis treatment was advancing. A decade after the study began, clinical trials confirmed that penicillin effectively cured the disease in its early stages. But in Tuskegee, researchers were determined to keep pursuing what they considered vital research. They had yet to confirm their theories about racial difference, and they believed they would never have another opportunity to observe the long-term effects of untreated syphilis. The study’s leadership decided to withhold knowledge of new treatments from their subjects. During World War II, researchers convinced the local draft board to exempt men from their study, preventing them from enlisting and potentially accessing penicillin. The study even continued through the 1950s when penicillin was shown to help manage late-stage syphilis. By today’s bioethical standards, withholding treatment in a research study without a patient’s informed consent is morally abhorrent. But for a large part of the 20th century, this practice was not uncommon. In the 1940s, US led studies in Guatemala infected numerous prisoners, sex workers, soldiers, and mental health patients with sexually transmitted infections to study potential treatments. And other studies throughout the 50s and 60s saw doctors secretly infecting patients with viral hepatitis or even cancer cells. Eventually, researchers began objecting to these unjust experiments. In the late 1960s, an STI contact tracer named Peter Buxtun convinced the PHS to consider ending the study. But after leadership decided against it, Buxtun sent his concerns to the press. In July of 1972, an exposé of the Tuskegee study made headlines across the country. Following public outcry, a federal investigation, and a lawsuit, the study was finally shut down in 1972— 40 years after it began and 30 after a treatment for syphilis had been found. No evidence of any racial difference was discovered. When the study ended, only 74 of the original 600 men were alive. 40 of their wives and 19 of their children had contracted syphilis, presumably from their husbands and fathers. In the wake of this tragedy, and concerns about similar experiments, Congress passed new regulations for ethical research and informed consent. But systemic racism continues to permeate medical care and research throughout the US. To truly address these issues, the need for structural change, better access to care, and transparency in research remains urgent. |
Hidden_Figures_Black_History | Ode_to_the_Only_Black_Kid_in_the_Class_poem_by_Clint_Smith.txt | I'm Clint Smith and this is "Ode to the Only Black Kid in the Class." You, it seems, are the manifestation of several lifetimes of toil. Brown v. Board in flesh. Most days the classroom feels like an antechamber. You are deemed expert on all things Morrison, King, Malcolm, Rosa. Hell, weren’t you sitting on that bus, too? You are every- body’s best friend until you are not. Hip-hop lyricologist. Presumed athlete. Free & Reduced sideshow. Exception and caricature. Too black and too white all at once. If you are successful it is because of affirmative action. If you fail it is because you were destined to. You are invisible until they turn on the Friday night lights. Here you are star before they render you asteroid. Before they watch you turn to dust. |
Hidden_Figures_Black_History | The_dark_history_of_zombies_Christopher_M_Moreman.txt | Animated corpses appear in stories all over the world throughout recorded history. But zombies have a distinct lineage— one that traces back to Equatorial and Central Africa. The first clue is in the word “zombie” itself. Its exact etymological origins are unknown, but there are several candidates. The Mitsogho people of Gabon, for example, use the word “ndzumbi” for corpse. The Kikongo word “nzambi” refers variously to the supreme being, an ancestor with superhuman abilities, or another deity. And, in certain languages spoken in Angola and the Congo, “zumbi” refers to an object inhabited by a spirit, or someone returned from the dead. There are also similarities in certain cultural beliefs. For example, in Kongo tradition, it’s thought that once someone dies, their spirit can be housed in a physical object which might bring protection and good luck. Similar beliefs about what might happen to someone’s soul after death are held in various parts of Africa. Between 1517 and 1804, France and Spain enslaved hundreds of thousands of African people, taking them to the Caribbean island that now contains Haiti and the Dominican Republic. There, the religious beliefs of enslaved African people mixed with the Catholic traditions of colonial authorities and a religion known as “vodou” developed. According to some vodou beliefs, a person’s soul can be captured and stored, becoming a body-less “zombi.” Alternatively, if a body isn’t properly attended to soon after death, a sorcerer called a “bokor” can capture a corpse and turn it into a soulless zombi that will perform their bidding. Historically, these zombis were said to be put to work as laborers who needed neither food nor rest and would enrich their captor’s fortune. In other words, zombification seemed to represent the horrors of enslavement that many Haitian people experienced. It was the worst possible fate: a form of enslavement that not even death could free you from. The zombi was deprived of an afterlife and trapped in eternal subjugation. Because of this, in Haitian culture, zombis are commonly seen as victims deserving of sympathy and care. The zombie underwent a transformation after the US occupation of Haiti began in 1915— this time, through the lens of Western pop culture. During the occupation, US citizens propagated many racist beliefs about Black Haitian people. Among false accounts of devil worship and human sacrifice, zombie stories captured the American imagination. And in 1932, zombies debuted on the big screen in a film called “White Zombie.” Set in Haiti, the film’s protagonist must rescue his fiancée from an evil vodou master who runs a sugar mill using zombi labor. Notably, the film's main object of sympathy isn't the enslaved workforce, but the victimized white woman. Over the following decades, zombies appeared in many American films, usually with loose references to Haitian culture, though some veered off to involve aliens and Nazis. Then came the wildly influential 1968 film “Night of the Living Dead,” in which a group of strangers tries to survive an onslaught of slow-moving, flesh-eating monsters. The film’s director remarked that he never envisioned his living dead as zombies. Instead, it was the audience who recognized them as such. But from then on, zombies became linked to an insatiable craving for flesh— with a particular taste for brains added in 1985′s “The Return of the Living Dead.” In these and many subsequent films, no sorcerer controls the zombies; they’re the monsters. And in many iterations, later fueled by 2002′s “28 Days Later,” zombification became a contagious phenomenon. For decades now, artists around the world have used zombies to shine a light on the social ills and anxieties of their moment— from consumer culture to the global lack of disaster preparedness. But, in effect, American pop culture also initially erased the zombies origins— cannibalizing its original significance and transforming the victim into the monster. |
Hidden_Figures_Black_History | The_breathtaking_courage_of_Harriet_Tubman_Janell_Hobson.txt | Escaping slavery; risking everything to save her family; leading a military raid; championing the cause of women’s suffrage; these are just a handful of the accomplishments of one of America’s most courageous heroes. Harriet Tubman was born Araminta Ross in Dorchester County, Maryland, in the early 1820s. Born into chattel slavery, Araminta, or Minty, was the fifth of nine children. Two of Minty’s older sisters were sold to a chain gang. Even as a small child, Minty was hired out to different owners, who subjected her to whippings and punishment. Young Minty’s life changed forever on an errand to a neighborhood store. There, an overseer threw a two-pound weight at a fugitive enslaved person, missed, and struck Minty instead. Her injury caused her to experience sleeping spells, which we know of today as narcolepsy, for the rest of her life. Minty’s owner tried to sell her, but there were no buyers for an enslaved person who fell into sleeping spells. She was instead put to work with her father, Ben Ross, who taught her how to lumber. Lumbering increased Minty’s physical strength and put her in touch with free black sailors who shipped the wood to the North. From them, Minty learned about the secret communications that occurred along trade routes, information that would prove invaluable later in her life. In this mixed atmosphere of free and enslaved blacks working side by side, Minty met John Tubman, a free black man she married in 1844. After marriage, she renamed herself Harriet, after her mother. Harriet Tubman’s owner died in 1849. When his widow planned to sell off her enslaved human beings, Harriet feared she would be sold away from everyone she loved. She had heard of an “underground railroad," a secret network of safe houses, boat captains, and wagon drivers willing to harbor fugitive enslaved people on their way north. So Tubman fled with two of her brothers, Ben and Harry. They eventually turned back, fearing they were lost. But in one of her sleeping spells, Harriet dreamed that she could fly like a bird. Looking down below, she saw the path to liberation. And in the autumn of 1849, she set out on her own, following the North Star to Pennsylvania, and to freedom. Tubman returned to the South 13 times to free her niece, brothers, parents, and many others. She earned the nickname Black Moses and worked diligently with fellow abolitionists to help enslaved people escape, first to the North, and later to Canada. Harriet Tubman worked as a Union army nurse, scout, and spy during the Civil War. In 1863, she became the first woman in United States history to plan and lead a military raid, liberating nearly 700 enslaved persons in South Carolina. After the war, the 13th Amendment to the U.S. Constitution legally abolished slavery, while the 14th expanded citizenship and the 15th gave voting rights to formerly enslaved black men. But she was undaunted, and she persisted. She raised funds for formerly enslaved persons and helped build schools and a hospital on their behalf. In 1888, Tubman became more active in the fight for women’s right to vote. In 1896, she appeared at the founding convention of the National Association of Colored Women in Washington D.C. and later at a woman’s suffrage meeting in Rochester, New York. There she told the audience: “I was a conductor on the Underground Railroad, and I can say what many others cannot. I never ran my train off the track, and I never lost a passenger.” As her fame grew, various friends and allies helped her in the fight to collect a veteran’s pension for her service in the Union Army. In 1899, she was finally granted $20 a month. In a fitting twist of fate, the United States Treasury announced in 2016 that Tubman’s image will appear on a redesigned twenty dollar bill. Harriet Tubman died on March 10, 1913. Even on her deathbed at age 91, she kept the freedom of her people in mind. Her final words were: "I go away to prepare a place for you.” |
Hidden_Figures_Black_History | The_records_the_British_Empire_didnt_want_you_to_see_Audra_A_Diptée.txt | In 2009, five Kenyan people took a petition to the British Prime Minister’s office. They claimed they endured human rights abuses in the 1950s, while Kenya was under British colonial rule and demanded reparations. They had vivid accounts and physical scars from their experiences— but their testimonies were undermined. They had no documentary evidence that Britain sanctioned systems of torture against Kenyans— at least, not yet. Thousands of secret files were waiting to be discovered. In 2010, a historian joined the trial as an expert witness and attested to having seen references to missing documents. They noted that Kenya had repeatedly requested the return of stolen papers, which the British government had refused. In fact, many historians suspected there were gaps in the archives. As a result, the court ordered the release of any relevant documents. And, days later, British officials acknowledged that 1,500 pertinent files were being held in a high-security archive. It soon became clear that these were just a small sample of documents Britain hid between the 1950s and 70s, while former colonies declared independence, as part of a widespread colonial British policy called Operation Legacy. The policy was for British colonial officers to destroy or remove documentation that might incriminate Britain and be of strategic value to the new governments. They were instructed to destroy, alter, or secretly transport these papers to the UK. Documents slated for destruction were to be burnt to ashes or sunk in weighted crates far from shore. During the trial, between 2010 and 2013, an independent historian revealed they had located more than 20,000 previously hidden Operation Legacy files from 37 former colonies. Finally, an estimated 1.2 million colonial files, sprawling kilometers in the archive’s so-called “Special Collections,” were also exposed. And these were only the documents that British forces kept. How many were destroyed— and what information they contained— remains unknown. About 3.5 tons of colonial documents were slated for incineration in Kenya. Ultimately, Operation Legacy’s objective was to obscure critical aspects of the truth. In the words of Britain’s attorney-general in Kenya, “If we are going to sin, we must sin quietly.” So, what really happened in Kenya? Beginning in 1895, the British administration forcibly removed people from their traditional lands, giving the most fertile areas to European settlers to establish large-scale farms. They mandated forced labor systems, implemented reservations for Indigenous African peoples, and restricted their movement. Kenyan people resisted these incursions from the start and grew increasingly organized over time. One movement, the Kenya Land and Freedom Army, aimed to forcibly remove white settlers and overthrow the colonial government. When the British declared a state of emergency in 1952, they were giving themselves permission to take otherwise illegal special measures to regain control. The newly revealed Operation Legacy documents confirmed that people suspected of participating in the resistance were subjected to horrible abuses. Between 1952 and 1959, the British imprisoned over 80,000 people without trial, sentenced over 1,000 people convicted as terrorists to death, and imposed extreme surveillance and interrogation tactics. Some people were beaten to death. Others were raped or castrated. Many were shackled at the wrist for years. Children were killed. One person was burnt alive. Ndiku Mutwiwa Mutua testified to being castrated while handcuffed and blindfolded. Wambugu Wa Nyingi said he was suspended upside-down, beaten, and had water thrown on his face until he could barely breathe. Jane Muthoni Mara said she was sexually violated with a hot bottle, and imprisoned for years without cause. In response to the new evidence, the British government issued a formal apology, and made an out-of-court financial settlement with the 5,228 Kenyan claimants ultimately involved in the case. The original five claimants had made history— and paved the way for it to be rightfully rewritten. The uncovered files challenge fundamental myths about British colonialism as a benevolent institution that brought freedom and democracy to its subjects, then graciously gave them independence. Instead, the newly exposed evidence confirms what many people knew to be true, because they lived it— and survived to rescue history from the ashes. |
Hidden_Figures_Black_History | The_hidden_life_of_Rosa_Parks_Riché_D_Richardson.txt | In 1944, 11 years before her fateful decision on a Montgomery Bus, Rosa Parks was investigating a vicious crime. As an emissary for the National Association for the Advancement of Colored People, she had traveled to rural Alabama to meet with Recy Taylor, a young woman who had been sexually assaulted by six white men. It would be difficult enough to convince an Alabama court that even one of these men was guilty, but Rosa was undeterred. She formed a committee to defend Recy in court, flooding the media with testimony and sparking protests throughout the South. When a jury failed to indict the attackers, Parks demanded the governor assemble a new grand jury. She wrote, “I know that you will not fail to let the people of Alabama know that there is equal justice for all of our citizens.” Throughout her life, Parks repeatedly challenged racial violence and the prejudiced systems protecting its perpetrators. But this work came at an enormous risk— and a personal price. Born in 1913, Rosa was raised by her mother and grandparents in rural Alabama. But outside this loving home, the fear of racial violence cast a long shadow. The Ku Klux Klan frequently drove past their home, and Jim Crow laws segregated public spaces. At 19 she settled in Montgomery and married Raymond Parks, a barber who shared her growing fury at racial injustice. He was involved with the local chapter of the NAACP; a role many avoided for fear of persecution. At first Raymond was eager to keep Rosa safe from the potential dangers of activism. But as she grew more incensed at the limitations imposed on African Americans, she could no longer stand by. When she officially joined the NAACP in 1943, Parks and Johnnie Rebecca Carr were the only women in the Montgomery chapter. She began keeping minutes for their meetings, and soon found herself elected secretary of the chapter— formally beginning her secret double life. By day, Rosa worked as a seamstress to support her mother and husband. By night, she researched and documented numerous civil rights cases, from local policy disputes to high-profile murder cases and hate crimes. As secretary, she prepared public responses on behalf of the Montgomery chapter, battling the harsh sentencing, false accusation and smear campaigns frequently used against African Americans. In addition to her legal work, Parks was a brilliant local strategist. As advisor to the NAACP youth group council, she helped young people navigate segregated systems including voter registration and whites-only libraries. Through the cover of the NAACP, Parks strived to bring clandestine civil rights activities into the open. She advocated for civil disobedience training and spoke out against racial violence, particularly the murder of Emmet Till. In 1955, her refusal to move to the back of a segregated bus helped ignite the grassroots movement she had hoped for. Parks was arrested and jailed for her one-woman protest, where she was visited by local activists. Together they planned a twenty-four hour bus boycott. It lasted for three hundred and eighty-one days. Park’s simple act had transformed nascent civil rights activism into a national movement. In 1956, the boycott ended when the Supreme Court ruled in favor of desegregating public transport. But this victory for the movement had come at a great cost. Rosa had been receiving vicious death threats throughout the campaign, and was unable to find work in Montgomery because of her political reputation. In 1957, she moved to Detroit to continue working as a seamstress, until being hired by Congressman John Conyers to help support his burgeoning civil rights campaigns. Ever vigilant in the fight against racial inequality, Parks remained active for the next 40 years. She wrote several books, traveled across the country giving talks to support other activists, and established an institute for the education of young people in her late husband’s memory. Today, Rosa Parks is remembered as a radical spirit who railed against the most powerful people and policies. Her call to action continues to resound: “knowing what must be done does away with fear.” |
Hidden_Figures_Black_History | Notes_of_a_native_son_The_world_according_to_James_Baldwin_Christina_Greer.txt | Over the course of the 1960s, the FBI amassed almost two thousand documents in an investigation into one of America’s most celebrated minds. The subject of this inquiry was a writer named James Baldwin. At the time, the FBI investigated many artists and thinkers, but most of their files were a fraction the size of Baldwin’s. During the years when the FBI hounded him, he became one of the best-selling black authors in the world. So what made James Baldwin loom so large in the imaginations of both the public and the authorities? Born in Harlem in 1924, he was the oldest of nine children. At age fourteen, he began to work as a preacher. By delivering sermons, he developed his voice as a writer, but also grew conflicted about the Church’s stance on racial inequality and homosexuality. After high school, he began writing novels and essays while taking a series of odd jobs. But the issues that had driven him away from the Church were still inescapable in his daily life. Constantly confronted with racism and homophobia, he was angry and disillusioned, and yearned for a less restricted life. So in 1948, at the age of 24, he moved to Paris on a writing fellowship. From France, he published his first novel, "Go Tell it on the Mountain," in 1953. Set in Harlem, the book explores the Church as a source of both repression and hope. It was popular with both black and white readers. As he earned acclaim for his fiction, Baldwin gathered his thoughts on race, class, culture and exile in his 1955 extended essay, "Notes of a Native Son." Meanwhile, the Civil Rights movement was gaining momentum in America. Black Americans were making incremental gains at registering to vote and voting, but were still denied basic dignities in schools, on buses, in the work force, and in the armed services. Though he lived primarily in France for the rest of his life, Baldwin was deeply invested in the movement, and keenly aware of his country’s unfulfilled promise. He had seen family, friends, and neighbors spiral into addiction, incarceration and suicide. He believed their fates originated from the constraints of a segregated society. In 1963, he published "The Fire Next Time," an arresting portrait of racial strife in which he held white America accountable, but he also went further, arguing that racism hurt white people too. In his view, everyone was inextricably enmeshed in the same social fabric. He had long believed that: “People are trapped in history and history is trapped in them.” Baldwin’s role in the Civil Rights movement went beyond observing and reporting. He also traveled through the American South attending rallies giving lectures of his own. He debated both white politicians and black activists, including Malcolm X, and served as a liaison between black activists and intellectuals and white establishment leaders like Robert Kennedy. Because of Baldwin’s unique ability to articulate the causes of social turbulence in a way that white audiences were willing to hear, Kennedy and others tended to see him as an ambassador for black Americans — a label Baldwin rejected. And at the same time, his faculty with words led the FBI to view him as a threat. Even within the Civil Rights movement, Baldwin could sometimes feel like an outsider for his choice to live abroad, as well as his sexuality, which he explored openly in his writing at a time when homophobia ran rampant. Throughout his life, Baldwin considered it his role to bear witness. Unlike many of his peers, he lived to see some of the victories of the Civil Rights movement, but the continuing racial inequalities in the United States weighed heavily on him. Though he may have felt trapped in his moment in history, his words have made generations of people feel known, while guiding them toward a more nuanced understanding of society’s most complex issues. |
Hidden_Figures_Black_History | The_exceptional_life_of_Benjamin_Banneker_RoseMargaret_EkengItua.txt | Sometime in the early 1750s, a 22-year-old man named Benjamin Banneker sat industriously carving cogs and gears out of wood. He pieced the parts together to create the complex inner working of a striking clock that would, hopefully, chime every hour. All he had to help him was a pocket watch for inspiration and his own calculations. And yet, his careful engineering worked. Striking clocks had already been around for hundreds of years, but Banneker's may have been the first created in America, and it drew fascinated visitors from across the country. In a show of his brilliance, the clock continued to chime for the rest of Banneker's life. Born in 1731 to freed slaves on a farm in Baltimore, Maryland, from his earliest days, the young Banneker was obsessed with math and science. And his appetite for knowledge only grew as he taught himself astronomy, mathematics, engineering, and the study of the natural world. As an adult, he used astronomy to accurately predict lunar and solar events, like the solar eclipse of 1789, and even applied his mathematical skills to land use planning. These talents caught the eye of a local Baltimore businessman, Andrew Ellicott, who was also the Surveyor General of the United States. Recognizing Banneker's skills in 1791, Ellicott appointed him as an assistant to work on a prestigious new project, planning the layout of the nation's capitol. Meanwhile, Banneker turned his brilliant mind to farming. He used his scientific expertise to pioneer new agricultural methods on his family's tobacco farm. His fascination with the natural world also led to a study on the plague life cycle of locusts. Then in 1792, Banneker began publishing almanacs. These provided detailed annual information on moon and sun cycles, weather forecasts, and planting and tidal time tables. Banneker sent a handwritten copy of his first almanac to Virginia's Secretary of State Thomas Jefferson. This was a decade before Jefferson became president. Banneker included a letter imploring Jefferson to "embrace every opportunity to eradicate that train of absurd and false ideas and opinions" that caused prejudice against black people. Jefferson read the almanac and wrote back in praise of Banneker's work. Banneker's correspondence with the future president is now considered to be one of the first documented examples of a civil rights protest letter in America. For the rest of his life, he fought for this cause, sharing his opposition to slavery through his writing. In 1806 at the age of 75, Banneker died after a lifetime of study and activism. On the day of his funeral, his house mysteriously burned down, and the majority of his life's work, including his striking clock, was destroyed. But still, his legacy lives on. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 71_Introduction_to_the_Doppler_Effect.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we're going to discuss the classical Doppler effect. The Doppler effect was first noted by Christian Doppler, hence the name, in 1842. And what we're discussing as the Doppler effect is a frequency change when moving from one reference frame into another. So this can happen, for example, if the source of a sound wave is moving like we experience when a car is passing by. The application of the Doppler effect are numerous. We can use the Doppler effect to measure the velocity of a speeding car. We can use the Doppler effect to measure the distance of distant galaxies. As a reminder, the velocity of a wave, so the velocity in which the wave is propagating is equal to the frequency times the wavelength. So when we move relative to the source of the wave, we see this apparent change in frequency. But let's introduce this topic not in my words, but in the ones of Sheldon Cooper from The Big Bang Theory. [VIDEO PLAYBACK] - We go to the party. - I don't care if anybody gets it, I'm going as the Doppler effect. - No, it's not-- - If I have to I can demonstrate. [IMITATES DOPPER EFFECT] PROFESSOR: That's the sound wave of a moving car. - So what time does the costume parade start? - The parade? - Yeah, so the judges can give out the prizes for best costume. Most frightening. Most authentic. Most accurate visualization of a scientific principle. - Sheldon, I'm sorry but there aren't going to be any parades or judges or prizes. - This party is just going to suck. - No. Come on, it's going to be fun, and you all look great. I mean, look at you, Thor, and oh, Peter Pan, that's so cute. - Actually, Penny, he's-- - I'm Peter Pan. And I got a handful of pixie dust with your name on it. - No, you don't. Hey, what's Sheldon supposed to be? - Oh, he's the Doppler effect. - Yes. It's the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer. - Oh, sure. I see it now, the Doppler effect. All right. [END PLAYBACK] MARKUS KLUTE: The Doppler effect, so now we have it. All right, so let's look at a specific situation here, in which we have a sound wave emitted by a moving car. The car is moving with the velocity u, and it's emitting sound, via the engine or the horn of the car. If you then look at the wavelength as we observe, so we have an observer here. As we observe the wavelengths, the wavelength is actually modified because the crests of the moving car, the horn, the engine, they are moving themselves with the velocity u times to 0, which is a period, which is equal to u times 1 over the frequency as the frequency emitted by the car addressed. So the wavelength is modified to lambda minus u times 1 over f0. So then we can use this equation here to calculate the wavelength at the frequency as observed by this person here, which is the velocity of the moving wave divided by the modified or the observe wavelengths. And so that's v over lambda minus u times 1 over f0 or f0 times 1 over 1 minus uv. All right, so we see that the frequency is modified when we observe the moving car. And you have all experiences in life already, so let's look at a few more examples of this. I'll just share this little video. It's a very short video. [VIDEO PLAYBACK] [HONKING INCREASES AND FADES] [END PLAYBACK] So you're able to hear the change in frequency in this example. Here's another one. [VIDEO PLAYBACK] [SIREN INCREASES AND FADES] [END PLAYBACK] So this is interesting now. So you've heard exactly what I was describing before, the sound of a moving vehicle. In the first case, a car honking, in the second case, a European fire truck passing by. And so there is an additional part of the story here. If I would ask you, please do make the sound of a fire truck, I think not many of you would say nina, nina, nina, which is the sound of this fire truck for European. There's a few more of those examples, which are really weird. If I ask you to make the sound of a rooster, you and I will probably disagree on the sound of a rooster. I would say [NON-ENGLISH] and you would make a sound which is rather unfamiliar to me. So that's just a fun fact on the side, but this was a short introduction to the classical Doppler effect. And in the next video, we will look at the relativistic Doppler effect. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 34_Stellar_Aberration.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. So we're going to continue the discussion that makes a case against ether when discussing stellar aberration. So the problem we have in mind here is the ones where we use a telescope to look at a distant star. When we do this, the fact of ether might change the way we have to orient our telescope. Let's have a look at this. So we are in a situation that our hypothesis is that the ether exists. This carries-- is the medium in which light travels. But it's dragged with earth. So this is our case number one. In that situation, the telescope looks at the star. The light of the star enters our telescope at the top and then tries to go to the bottom. Because the medium in which the light travels and the telescope move with the same velocity-- remember, the ether is dragged with the earth-- the light appears to go straight down the telescope. Without the-- [INAUDIBLE] case 2, without the ether being dragged or even exist, we actually do have to slightly tilt our telescope. Why? Because after the light hits the top of our telescope, the telescope keeps moving because it's in a different reference frame. And therefore, we have to have this slight tilting. The value of the tilt is equal to velocity of the Earth over C. This is a well-known effect in astrophysics and, by the way, was already studied way earlier by James Bradley in the 1720s. He actually developed the-- not really supported theory of light where he was talking about particle nature of light. So his idea was very similar to case 2, that the Earth is moving in a different reference frame with respect to the star and the sun, moving around the sun. And therefore, the tilting angle needs to be the tangent of the tilting and needs to be equal to the distance of Earth to the Sun. And the distance is [INAUDIBLE] this angle is very, very small because the nearest star [INAUDIBLE] to Earth is about four light years away. So with this now, we can again do our ether versus particle nature model comparison. So in our first discussion, we concluded clearly that the wave and ether hypothesis dominates or wins. But now we have studied at least two further experiments. So, again, I invite you to stop here and think about how stellar aberration and Michelson-Morley would be answers to the particle model or wave model [INAUDIBLE] explanation. As a thought experiment, the Fizeau and Airy experiment that we haven't and did not discuss this here. All right. So for stellar aberration, the particle model actually seems to work, as it was proposed apparently to solve this very problem. And the Michelson-Morey experiment also is consistent with the particle model. So now we're in this dilemma that some experiments or experimental evidence for particle ether nature and some for the particle nature of light. So the question of how do we get out of the dilemma-- and we'll discuss this in [INAUDIBLE] the next section. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 107_Compton_Effect.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this last example of relativistic kinematics, we want to investigate scattering-- in this specific case, a scattering of a photon on an electron at rest. So we have as an initial state a photon, an electron at rest, and then the photon is scattered and we also observe a scattered electron. There's one important piece of physics here, which we add without further explanation, which is the Planck-Einstein relation, which relates the energy of the photon to the frequency of the photon or the wavelength. This is fundamentally important in quantum physics, and can be explained or tested with the photoelectric effect for which Einstein received the Nobel Prize. So what we want to do here is find the wavelength shift, so delta lambda, which is the wavelength of the incoming photon minus the wavelength of the outgoing photon, as a function of the scattering angle zeta, as shown in this picture here. OK, so again, this is an activity I want you to work on and try to find out this. The algebra here is not trivial, but knowing how to set up a problem like this is important. So let's try. So the way to set this up is to write this four vector relation, or you could just simply write down energy conservation and momentum conservation. So you have an initial state, the before, and the final state, the after, where you simply add the four vectors of the initial electron and photon and set this equal to the scattered electron and scattered photon. Now, we are interested in a quantity delta lambda, which is related to the change in energy of the photon. So therefore, it brings the four vector of the four scattered photon over here to this side, and builds a square, which allows us then to use our invariant information in the scattering process. When we explore the squared here, we find the photon four vectors squared for the scattered and the unscattered photon, minus 2 times the product of the two four vectors. Now, the mass of the photon is 0, and then hence the invariant mass is 0, too, so this invariant four vector is 0. So this cancels and this cancels. And then we know that the mass of the electron is the mass of the electron, the initial momentum of the electron is 0, and we just for the further, not to get confused, we said C equal to 1. So then we just go through a sequence of algebra here, making use of information that those guys here are simply the mass of the electron. And we move things around a little bit and then find this equation here, which relates the energies of the two photons [INAUDIBLE] the scattering angle, which we get from the scattered product of the [? three ?] momentum of the photon to the change in electron energy, which is the energy of the electron minus the mass. OK. And then we start using the Einstein relation here. And again, a little bit of algebra then brings us to delta lambda equal h over me times 1 minus cosine theta. So this relates the shift in wavelengths to the scattering angle of the photon. Important. If you want to recall this, the most important part through this problem is setting up this first equation here, which relates the energy and the momentum, or the four vector of those particles, before and after the collision. And again, then it takes a little bit of practice. But the way to approach most of this problem is to make use of the invariant four vector squared, or the invariant mass of the objects involved, if we know the masses of the object involved. OK. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 106_Creation_of_Particles.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. In the previous section, we have seen how we can look at energy and momentum of particles in a decay. Here we now want to, in collisions of particles, create new particles. The example, the first example here, is the collision of two protons to create a proton, a neutron, and a charged pion. The masses are given there. So the question now is, what is the minimal energy needed in order for this process to occur in a fixed-target experiment? Fixed-target experiment is we have an accelerated proton and another proton at rest. This might just be a hydrogen target just sitting there. So the question is, how much energy-- how much do we have to accelerate the proton for this process to be possible? Now, again, stop the video, and try to work this out. The important part here is to realize that minimal energy here means that, after the decay or the decay after the process occurred, all the new particles need to be addressed. That is when the process requires minimal energy. So, instead of analyzing this in the laboratory frame, we want to analyze this in the center-of-mass frame. All right, the momentum has to be conserved in this discussion. So there needs to be some sort of momentum. But, in the center-of-mass frame, that's not required. So, in that frame, the momentum of all outgoing particles can be 0. And that's how we start the discussion here. So, in this S prime frame-- here S prime is the center-of-mass frame-- the energy, the minimal energy required, is 2 times the mass of the proton times gamma. So here, two protons are colliding with the same velocity. And that's then equal to the energy after this process, c squared times the sum of the masses, the sum of the mass of the proton, the neutron, and the charged pion. And then you just have to solve this for gamma to find gamma equal to 1.08 or beta in this frame of 0.37. Note, this is the gamma, relativistic gamma, or the velocity beta of the protons, two protons in the center-of-mass frame. So we're not quite there yet with our answer. The answer then needs to be boosted back into the laboratory frame. And we have seen how we can do this for beta or velocities in general. We find beta in the laboratory frame is 2 times-- or just result, 0.37, over 1 plus 0.37 squared, which is 0.65. That velocity, we can then take and calculate the gamma factor of the proton in the fixed-target experiment. All right, so we analyzed this situation in the center-of-mass frame and then did a Lorentz transformation by just looking at the velocity into the fixed-target frame. So this means now, numerically, that the proton colliding with the proton at rest has a total energy of this one proton of gamma m0 c squared, which is 0.32 times 938 MeV over c-- MeV. And so that results in 1.238 GeV. But we're interested in the kinetic energy. So the kinetic energy here is given by gamma minus 1 m0 c squared, which is 300 MeV. So we have to accelerate a proton to 300 MeV in order to be able to have this process to occur. All right, very similar problem now, but here we want to produce anti-matter. So we have a process of proton plus proton into three protons and an antiproton. Charge is conserved. In the initial state, the charge was 2. In the final stage, the charge was plus 2 as well. OK, this works very similar as in the previous problem. But what we want to do here is compare the fixed target with symmetric collisions. OK, so, again, the question is, what is the minimal energy needed in order to produce antiprotons in proton-proton collisions? OK, so, exactly following the same procedure as before, in the center-of-mass energy, the energy is 2 times the mass of the protons times gamma times c squared. And that's 4 times the mass of the proton. OK, gamma prime, so the gamma factor in the center-of-mass frame is 2. Beta is 0.75. And then we just do the very same thing again. We calculate the velocity in the fixed-target frame. And we find the velocity of beta of 0.96 and gamma of 3.57. So, if we compare this now, we need a pair of 1 GeV-- remember, gamma minus 1 is the kinetic energy-- protons in a collider experiment or 2.57 GeV protons in a fixed-target experiment. OK, so you see that, in fixed-target experiment, in order to produce new particles, the energy has to be much larger, a factor of 2.5 here in this example, than a colliding experiment. And that explains why we use collider experiments in order to test the energy frontier, in order to produce the largest possible energies. And the LHC is one example where we have proton-proton collisions in a circular ring where those protons are brought together in symmetrical collisions. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 42_Muons.txt | PROFESSOR: Welcome back to 8.20, special relativity. In the last section, we discussed that moving clocks tick differently than those which are at rest. And here, I would like to discuss real life example of this. The muon is an elementary particle very similar to the electron. It's mass is about 200 times as heavy. The muon was discovered in the 1930s by Anderson and Neddermeyer at Caltech. And it's really one of my favorite particles because you can-- they are abundant. There's many of them in cosmic air showers. You can study them, you can study their lifetime. You can even calculate the lifetime on a piece of paper. So what Anderson and Neddermeyer did is they just basically went outside and discovered a particle which comes from the sky. And so they studied cosmic radiation. Muons are produced in cosmic air showers, and we look at one of those a little later. Basically, protons hits the upper atmosphere, and in a shower of various particles, muons are being produced. And then those muons are not stable particles, but they are stable enough to reach us. On average, if you hold out your hand right now, about one muon travels through your hand every second. How is this possible? So if you look at this muon, it gives you a little bit of particle physics explanation here. Again, the muon is not a stable particle. They decay via the weak interaction. For those who are interested, this is a Feynman diagram for this decay. The muon couples to the w, and as a result of the decay, you find an electron, an anti-electron neutrino, and the muon [INAUDIBLE]. The lifetime is about 2.2 microseconds-- 2.2 times 10 to the minus 6 seconds. And I just taught 8.701 which is introductory class into particle and nuclear physics, and the students calculated the lifetime of a muon in that class. So you can calculate this. And you need a few tools, but it's not that hard after all. The average velocity of the muons when they're being produced is close to the speed of light, or 0.998 times the speed of light. And if you do a classical calculation, and you want to figure out how long do the muons on average live-- fly, you find that this is about 660 meters. Now they are produced in the upper atmosphere, and nevertheless, we can find them down here on Earth. So something is not quite right. What is not quite right-- you can already assume-- is that the clock in the muon as observed by us ticks much, much slower than for the muon at rest. And so the lifetime of the muon of 2.2 microseconds is basically extended. If you calculate this-- this is average velocity-- we find gamma factor of 15. Using the equation we-- [? of ?] time dilation, you just simply multiply 15 times 2.2 microseconds, and you find that muons, indeed, reach our hand on the surface of Earth. This is a really fun example. Again, you can study those cosmic showers, those muons, and learn about the muons in very simple experiments. This picture here shows you one of those air shower formations. So the story is a little bit more complex. As I explain, this is a spectacular air shower, or a picture of one, where you have an [INAUDIBLE] coming in at an energy of 10 to 15 electronvolts. And so even its lower energies show us look like the one here. It produces, in collision with the atmosphere, many, many particles-- pions, protons, additional protons, neutrons, and pions again. And those pions then, they decay into muons. And this all happens in the upper atmosphere, but also in some cases, further down. So here, we have seen now an example which you can actually see and observe in nature, where particles travel with high speed. And there are relativistic effects we can measure and observe. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 45_Lorentz_Transformation.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. In this section, we want to discuss Lorentz transformation. Or, in other words, given an event observed by Bob, we want to express that event as observed by Alice. We want to find the translation between the observations in Bob's reference frames to the observation in Alice's reference frames. We have already done this for the classic case as Galilean transformation. Now, we want to do this in the framework of special relativity. In order to simplify the discussion, we don't worry about the y- and z-component here. Those dimensions can be neglected if we assume that the relative motion between the two reference frames only in x-direction. We also know from the previous discussion that you can use the invariant interval. ct squared minus x squared is the same observed in Bob's and in Alice's reference frame. We'll make use of this fact. And, lastly, we can assume that this transformation has to be linear. Why? Because we transform something like a measurement of distance into a measurement of distance. It has to be linear. If not, we find something like a length squared or the same for time. And we might end up on time squared if we don't do this correctly. All right, so we can write this down as a linear equation, which is a multiplication of a matrix with a vector, ct, x, into a vector, ct [? x ?] [? prime, ?] x prime. OK, so the goal here now is to find the parameters or the coefficients of this matrix, OK? I invite you to stop the video here and try to work it out. It's an interesting exercise. It tests your algebra knowledge. There's not much physics in here, but it's still useful to go along and try to work this out. So the first thing we want to do is assume that the origins coincide at t equals 0. And then we can follow along the trajectory of the origin of S prime in the S frame. So this is just ct, vt. OK, great. This already gives us a constraint on the coefficients a1,0 over a1,1, which is equal to minus v/c, OK? And then we can use the invariant interval, which is another constraint. And we can use this to obtain the set of equations here. I will not read this for you. And that's already enough in order to solve the set of equations. So, if you do this and follow along, you find answers for all four coefficients given gamma and beta as we defined them before. This then simplifies to our Lorentz transformation. So the only thing we did here is we simplified a little bit. We assumed that this is a linear transformation. We used the invariant interval in order to set the constraints. And we find Lorentz transformation. If I summarize this, we find this matrix here with coefficients gamma, minus gamma beta, minus gamma beta, and gamma. Great. Or, if you want, you can write this as an equation for the spatial component and the time component. So does this make sense? There's always a chance that we make a mistake in this kind of calculation. So we want to make sure that the answers we developed in previous sections actually are reflected by this transformation. So let's go one by one. The first thing we can do is check units. If we do that, we see that this first equation here is of unit meter, and then we can analyze the second part of the equation. OK, so gamma is unitless. x is of unit meter. And then we have beta ct. Beta is unitless. c is meter per second times second, also of unit meter. So this checks out. The second equation is very similar. c times t is of unit meter. Meter per second times second is of unit meter. Gamma is unitless. Beta is unitless. And then we have an x, unit [? meter, ?] plus ct, c, meter per second times second, also meter. So this checks out. So this is great. At least we find that we have a linear transformation by design, and the units work out. So now we can see, what happens now if we use this for velocities which are much, much smaller than the speed of light? In this case, gamma is equal to 0, and beta is very close to 0. If we put this in our equations, you find x prime is equal x minus vt. And t prime is equal to t. OK, this checks out because this is our Galilean transformation. So, for systems which move relative with very low difference in velocities, we can use Galilean transformation as an approximation of Lorentz transformation. OK, at a third part, now we can investigate a little bit further. For example, what happens now to a distance, just a measure of distance or a measure of length, which we obtain by making this measurement simultaneously at t2 equal to t2? We find delta x prime is equal to gamma delta x. All right, that's length contraction. If we do the same thing for delta t, for doing the measurement of time at x equals-- x2 equals x1, we find time dilation. All right, this is exactly what we expect. And then we can look at two events which happen at the same time in frame S and see what happens to the time, as measured in system S prime. Delta t prime is equal to gamma delta t. Well, in this example, we set this to 0. And then we have the second term, which is minus beta over c gamma delta x. So we find that, while this event happened simultaneously in our frame S1-- or in S, it does not happen simultaneously in our frame S prime. There's an extra term, which is not 0 unless you actually measure at the very same point, l is equal to 0. So this is the relativity of simultaneity. Again, this checks out. And I think we're good with our Lorentz transformation. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 22_Galilean_Transformation.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. In this video, we talk about Galilean transformation. So what is it you're going to do? We want to describe our event P, maybe Professor Klute exploding, with two different reference frames. We can call one of the reference frames our laboratory frame. Maybe that's the frame in which Professor Klute was stationary. It has an origin and has axis x, y, and z. And then we have a moving frame, which is moving with a constant velocity with respect to the laboratory frame. The origin is o prime. The axes are x prime, y prime, and z prime. All right, so now what do we learn from this? Let's think about an example. An example most of you have experienced before is the one where you sit in a train car. And, if it's not a train car, it can be a car or a plane, something which is moving with respect to Earth. If you look out of the window and the acceleration is very, very minor, it's often not clear whether or not the Earth, the train station, or the train car is moving. And so you have this kind of weird feeling that, you know, I don't know if maybe the neighboring train started moving or I'm moving. But what we're going to do here is describe you sitting in the train car, reading a newspaper, once within your laboratory frame, within the frame of the train, and then we want to describe the very same events or sequence of events in the frame of the stationary train station. All right, let's look at a specific example. Here again our professor is exploding at a time tP at xP, yP, and zP. To make this a little bit easier, we define, at time t equals 0, the origin of the two frames coincide. That just means that, at the origin, we have two clocks, two watches. And we make sure that they're synchronized. And then my watch stays with me, and then the second watch may be with you, which moves along. And those are great watches. They are synchronized. We also want to simplify-- we can always define the direction of our coordinate system such that the velocity, the relative velocity between the two reference frames, is in one specific direction. And here I decided to use x. I could have used y and z, and I could rotate the coordinate systems or the relative movement of the coordinate systems in any way. It's just a simplification here. When I do that, I can rewrite this event P in the S prime frame through the S frame in the following way. So, for x, we find that xP is given by-- x prime P is given by xP minus v, the velocity, relative velocity-- I could put a little label x here-- times tP. And then, for the y-coordinates and the z-coordinates, there's no change. For the time, intuitively, you say that those two watches are run with the same speed, meaning that the time in both frames for the same events are the same. So now I'm asking you, if you're watching this video, to find the velocity and the acceleration. It might be good to stop and just write this down. So find the velocity and acceleration of S prime, of S prime expressed by the S-coordinates. So let's try to do that. So, first, we build the derivative dx prime dt prime, which is our velocity. That's the velocity of an object in my prime frame. And that's given by d dt-- I can just do that here because the times are the same-- times x minus vt. That's [? x, ?] ux, the velocity in my S frame, minus v. That's really what you expect. You just subtract or add the velocities. If I then build the acceleration, I have to build the derivative of u x prime, which is our acceleration, in the prime frame. Here again I just do this in x-direction because the solution for the y-direction and z-direction are trivial. So now I find d ux dt minus dv dt. Now, the velocity, as we defined, between the two reference frames is constant. Therefore, this is 0, meaning that the velocity-- the accelerations in the two frames are the same. If the accelerations are the same, that means that the forces in the two frames are the same. And that means that the forces or the accelerations are invariant. They do not change based on the reference frame that I use. So now, coming back to the example we discussed in our very first lecture, you are experiment in the train car. S, the velocity between those two frames, the frame in your car or a second frame, are constant. There's no way to tell whether or not your train car is moving or not. That is only true as long as the velocities are constant and unchanged, constant. So, in summary, in Newton's mechanics, time and accelerations are invariant and, therefore, also the forces. There is no inertial frame which is above another one. So you can pick one or can pick another one. There's no difference in the descriptions of the physics between those two frames. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 31_Light.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: Welcome back to 8.20. In this section, you want to look at light, what is it, and how does it propagate. In this video, specifically, I give you a little bit of a preview of 8.02. And I don't do this in a very topological way. I just give you some information. So if we study 8.02, we'll see Maxwell equations are being developed in there. We look at Maxwell equations for electric and magnetic field E and B in vacuum. We can rewrite the Maxwell equations and define wave equations. The solutions of the wave equation, as the name tells you, are waves. So what we are looking at here is you want to describe the propagation of electric and magnetic fields in vacuum. In this situation, this is maybe at some time, t equal to 0, we have an electric field in this point here, and a magnetic field-- electric field points into the y direction, the magnetic field into the z direction. And what the equations now describe is how the wave propagates in space and in time. And you can already tell from the name, wave equation, the solutions of this equation-- these differential sines and cosines. So one solution here is are Ey equal to E0, times cosine, kx minus omega t. We find then that the speed in which the wave propagates-- you pick one peak of a wave, and you see how it propagates-- one point of variance here propagates. The speed in which it propagates is the speed of light, c. And you can find c here through those constants in the Maxwell equations and wave equations. Find c is 1 over square root epsilon 0 and mu 0. The permeativity and the permeability, the product of the two gives you the speed of light. So if you look at this some more, and connect the Maxwell equation to the Lorentz force, again, as a reminder, for those who had had 8.02 already, the force on the charged particle in an electromagnetic field is given by 2 times E, plus 2 times V cross B. If you have two charges, the force between those two charges is the product of the 2 divided by r squared, times 1 over 4 pi x mu 0. Again, [INAUDIBLE]. And the force between two wires-- this current-- current i1 and current i2-- is equal to the product of the two currents, divided by r, times l-- the length of the wires-- times mu 0 over 2 pi. So this is fantastic, because now you can calculate the speed of light by just measuring the forces between charges and current in wire's centers. The value of c is also very interesting. It's large-- very large. 3 times 10 to the 8 meters per second. So just let that sink in. We, as humans, move with a few meters per second. Light travels-- a few nanoseconds is needed for light to travel about 1 meter. It takes just nanosecond. Let's stop the video here. The next thing I want to do is an exercise. I want to have you play with this differential equation, and there's a solution of the differential equation. But the challenge or the exercise is to show that if you have a function which you can write as f0, which is an arbitrary function, which is a function of x minus ct, those functions, regardless in how they look like, are solutions of this differential equation. Note that I replaced our constant epsilon 0 and mu 0 now with 1 over c squared. So f0 can really be an arbitrary function. You need to be able to build the derivative, though. So I do the function here as a function of x for some time equal t0. And then I drew the same function 4 times equal to 1. And so you can, from this picture, see that the delta x over delta t is minus c in this case. So my function-- my wave is moving with the speed of light in minus direction. So I want you to show that this kind of equation [INAUDIBLE] wave equation. So I would like you to do this, and stop the video, and show you the solution next. So the way to approach this is simply applying the chain rule. And that might be something you want to remind yourself of. So after I do this, I'll define this little helper function here u is equal to x minus ct. And this makes our function a function of u, which is itself a function of x and t. So if I built a derivative with x, I have this df of u, du times du dx. If I build a second derivative, there is a product you have to take care of. So I find that d is the second derivative of f of u here, times du dx squared. And then I have to add df du times second derivative of u. This follows very similar for the derivative of t. And then I can investigate what we find. So my du dx is equal to 1-- du dx. If I build the derivative of x minus et, this x, I find 1. I do the same with t-- I find minus c. I will use this second derivatives of u. x and t are all 0. If I put this now in my equation, I find second derivative of f with u is of 1 minus c-- sorry, 1 minus 1 over c squared times c squared is equal to 0. And since this is always 0, we have just proven that any sort of function which I can build the derivative of which is of the from x minus ct solves that equation. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 54_Regions_in_Spacetime_Diagrams.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we want to study space-time diagrams a little bit more in detail, and also define certain regions in space-time diagrams. So let's start again with Alice's space-time diagram here in which we plot or draw Bob's space-time diagram. The relative velocity difference is 0.5 times the speed of light, and that leads to a gamma effect of 1.2. We also plotted the world line of light in here in yellow. Light is the speed of light equal to c. I want to discuss two specific events. The first one here, event number 1, is the one where tA, the time for Alice's [INAUDIBLE],, and xB, the space for Bob, is equal to 0. So this event lies on Bob's timeline. If we read off the time on Bob's clock, we see it's 0.83, 1 over gamma. And here, we can immediately [? read of ?] time dilation for this event. Note that while xB Bob is equal to 0, xA for Alice is not 0. Similarly, we can look at the second event here, where we read off xA equals 1. [INAUDIBLE] 1, in this case, light year for Alice. And so now we want to investigate this length in Bob's reference frame. For him, time is equal to 0. So tB equals 0. We can immediately again read off xB equals 0.83, and that indicates length contraction as of the [INAUDIBLE]. Important to note here is that those two, Alice and Bob, will not agree when the time and the measurement was made. All right. So let's zoom out here a little bit and look at another space-time diagram. So in this space-time diagram, again, I drew light-- blurred lines, or blurred lines for light-- in yellow. And I [? wrote ?] a total of 12 different events. Now we want to characterize those events. And we want to characterize them based on whether or not they are time-like, light-like, or space-like. As time-like, we define those events. We have c squared, t squared, minus x squared, is greater than 0. Light-like are those which are like light in a blurred line. [? ct, ?] c squared, t squared, minus x squared, equal to 0. And space-like, those were c squared, t squared, minus x squared, smaller than 0. The first task is now to find to which of those regions the individual events correspond. And so again, stop the video, and try to figure out whether or not you can find the solutions. Because the solutions are given here. Time-like are events 2, 5, and 6. Light-like are the ones which lay on the yellow lines. 1, 7, 4, and 9. And space-like are 8, 12, 11, 10, and 2. One of the things you can find, if you are starting here in the origin, and you're going to correspond to any event in the future, you can only do that if the events are time-like. If the events are space-like, you will not be able to correspond between those two events. That's one of the ways to read this kind of space-time diagram. And in the next section, in the next video, we'll talk about causality, meaning can a specific event cause something to happen, another event? Again, this can only happen if the events are actually time-like. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 33_MichelsonMorley_Experiment.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section and also the next one, we're going to make a case against aether. Aether was presumed to be the medium in which light traveled. We do this-- we make this case by studying the Michelson-Morley experiment. Michelson and Morley went out to detect the motion of the Earth relative to aether. And by not finding the result, we can conclude that, at least in this form, aether doesn't exist. So what is the experimental setup? Also, it's shown in the pictures above. We have a light source here, through which the light travel to some mirror which reflects about half the light. About half the light goes through reflected on the second mirror and then comes to a screen. The other part of the light is up and back down. So the light either travels against the aether wind or perpendicular to the aether wind. If aether is the medium in which light travels, then the velocity needs to change. The velocity of light in this medium will change. If there is no aether wind, there is no change in the velocity of the light. The result of the experiment is an interference pattern on the screen. The experiment is called an interferometer. So there are some systematic uncertainties here. There are some unknowns. For example, if you're building your table, it's not really clear in which direction aether actually-- which direction is actually the direction of the aether flow. But you can get by this by rotating the table and making various sets of experiments. And that's what Michelson and Morley did. I want you to show that the effect they're about to observe-- or they will not observe-- is of the order of v over c squared. And what you want to do is compare the light as it travels on this path number 1 to one mirror and on the second path to the second mirror. So if you're following the video, I ask you to just stop here and figure out how long does it take for the light to travel on this path here-- on the lower path, path number 1, or path number 2. Well, obviously, I've worked this out already. So we want to calculate the time it takes light to travel path 1 that is going up and coming down-- going left and going right. So in the first case, the velocity is reduced by the velocity of aether. So the time it takes is l1, the leg, divided by c minus v plus the return leg, l1 over c plus v. And you work this out, and you simplify a little bit, you get this first equation. For the second part, the one up and down, you want to draw this triangle here to figure out that t is equal-- the time it takes is equal to the square root of l square over c square minus v square. And then you simplify again, you get this [INAUDIBLE] square root [INAUDIBLE]. What's important is the time difference for the light traveling with two legs. And we find that the time difference t1 or t2 minus t1 is given by this bit complicated looking equation. So now in the experiment, you want to set up l1 equal to l2. To simplify, you might not be able to do it very precisely. What you can do, however, is rotate the table. If you rotate and then compare the difference of the differences, you find that here there's this v square over c square dependence. I used a little trick to simplify the square root as I use this 1 plus or minus x to the nth power. It's about 1 plus minus n times x for x plus 1. And since the velocity of aether is presumed to be small compared to the speed of light, that's a good approximation. The result of the experiment was there was no effect. They tried-- they improved the experiment. They tried to find smaller effects. They didn't find any. The reason for this is that there is no aether. But we can use this now to make a case against the aether. But you could say that maybe they didn't observe anything because the aether is dragged with the Earth. So instead of the Earth going through this aether, and therefore, we experience aether winds, you could argue that maybe aether is localized around the Earth, moving with the Earth or maybe moving with the universe. And so in the next video, we'll look at a case against that scenario as well. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 108_Global_Positioning_System.txt | PROFESSOR: Welcome back to A20 special relativity. In this last section of this chapter, we talked about applications and implications of special relativity. We talked about the global positioning system. You all have used GPS before, be it in your car when you're trying to find your way through town or when you go for a run and you want to measure how long and how fast you actually are running. The global positioning system is a set of satellites all equipped with atomic clocks. And they're used for navigation and have been developed first under the name of Napster GPS. And GPS is one of many global or few global navigation system. Others are GLONASS, BDS, Galileo Systems developed in Europe, in Russia, in Japan, and so. This picture here shows you how you want to view this view of Earth and then at around 20,000 kilometer, GPS has about 31 satellites, which zooms around the Earth. And as soon as you have three satellites, which you can see from your viewpoint, you can actually figure out where that viewpoint is. Think about each satellite gives you an information of a sphere. The cross section between two spheres is a line. And if you have three spheres crossing, that gives you three-dimensional information about location on which you're looking at this. GPS was-- the first satellites were launched in 1978. Again, 31 satellites in orbit at the time right now. So how does this work? So the satellite transmits information about position in time in regular intervals. It's does have a clock. They use a specific frequency or multiple frequency to send signals. And then the receiver calculates how far away the satellite is and how long it took the message to arrive at the place of the receiver. Again, three satellites are used to extract the exact location. However, as you can imagine, the reason why I bring this up in this class is that we have to consider effects of special relativity and also, general relativity, as we will discuss later in this class. So you have clocks on the satellites. And you have identical clocks on Earth. And they do tick differently as we discussed many times in this class. If you were to calculate the effect of special relativity, the prediction is that the clocks take about 7 microseconds a day slower on the satellite then compared to the one on Earth. However, the satellite at a higher orbit. And the gravitational pull at a higher orbit is less strong as on Earth. And that then, as a result, is the clocks on the satellites actually run faster than the ones on Earth. And that effect is larger, very larger than the effect of special relativity, which is effect from then just moving relative to the observer. The fact is 47 microseconds a day. And then you can just calculate the net correction, which is 38 microseconds a day, which corresponds to about 11 kilometers. Now imagine on your run and there's 11 kilometer difference a day. I don't expect you to run for a day, but you might run for an hour. Still the effect is rather large. So you want to have a more precise system. And therefore, those effects need to be corrected, and they are. They're typically built in into the electronics and automatically corrected. The precision achieved with GPS system ranges a little bit based on how you use the information, what kind of transceiver you use. It's in the order of 505 meters to 30 centimeters, so about this of a distance. So anywhere on this planet you can pinpoint your location by about this with this precision. Well, this concludes the discussion of implications and applications in special relativity. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 82_Introduction_to_4Vector_Notation.txt | MARKUS KLUTE: Welcome back to 8.20 Special Relativity. In this short section, we want to introduce a new notation, four-vectors. And if you look at previous discussions, this is actually not that new. We have seen that we need to treat time and space in a consistent manner. And you have often applied Lorentz's transformation, for example, to a vector of time and the next component of space. Now you just want to do this with x, y, and z here and not treat the y component and z component as 0. So as a starting point, you can just simply say, OK, we have this new four-vector. And the 0's component is the time or time times the speed of light. And then the first component, second and third component are the spatial component, x, y, and z. Now I wrote a vector Xi mew here, with the mew being the upper index. I can also introduce Xi with a lower index. And you see little y and little y is useful. Where the 0's component is not t but minus ct-- but minus ct. As a reminder for three-vectors, you learned about the dot product, which is just a multiplication of two, three-vectors where all vectors with n components, where you multiply the same component of each vector and add those results together. So the dot product of vector a and vector b is the sum of all indices for ai and bi. Now for our four-vector, we do the very same thing. We just sum over all four components. And we treat the vectors as a product of the vector with the lower index and the upper index. And you find here then we get minus c squared t squared plus x squared, y squared, and z squared. More generally, this is for two vectors of the same-- two of the same vectors. More generally for two different vectors, you can write in this way. Or in short, you can define a new notation in which you basically sum over all indices which are equal. So here we have an upper and lower indices together. So you sum over this case here where there's the same index, mew, for both vectors. And one is lower and one is upper. And we can continue the introduction and just introduce a few tools to work with those vectors. For example, if you wanted to bring the component mew from the bottom to the top, you can do this with multiplying the vector with a matrix. And the matrix here is also called a metric. And simply what you have to do is multiply the first component with the minus 1 and the rest with 1. You see this here on the diagonal and on other components later on. What this does-- you can check this if you want-- is bringing the index of the vector from a lower to an upper one. An interesting example is the product of a four-vector with itself. And we have already seen this because we saw this as our invariant interval. Here, the four-vector is the distance in space and time between two events. So we looked at delta Xi mew times delta Xi mew. And delta Xi mew is the difference between event A and B. And so we have seen this already and calculated the invariant and showed that this squared over a distance of two events is actually invariant in the Lorentz transformation. But there's other examples for vectors. The first one we'll investigate some more in the next sections to come. It's the energy momentum four-vector, where we place in the first component the energy-- in the 0's component the energy, and then the first, second, and third components the three-vector of the momentum. But there's others, for example, the four-potential, where in the 0's component, you have the potential-- the electric potential. And then the first, second, and third component, you have this new field A, which is related to the magnetic and electric field. So E and M is not part of this course, but we'll come back to this in the last week and discuss the consequences and ideas a little bit more. But if you then look at the invariant four-vector, which is a product of the energy momentum vector, you find that the first component, the energy square or minus the energy square over c square plus the three-component vector of the momentum squared. And that's constant, we can just here name this mass or minus mass square times c square. So if you write this, you find this energy momentum mass relation E squared is equal to p squared c squared, plus m squared c to the fourth power. And if you look at this four particles of 0 momentum, in which case this component here is 0, you find the equation E is equal to mc square. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 73_Redshift.txt | MARKUS KLUTE: Welcome back to 8.20. In the previous section, we have seen the relativistic Doppler effect, and now we want to study how light-- in this case, a monochromatic plane wave-- transforms in the Lorentz transformation. In other words, we have, for example, a distant star emitting light at a specific frequency. The question now is, how do we observe this light when the star is traveling away from us or towards us? So here you see our monochromatic plane wave. We have an amplitude A and then just a simple cosine, which is a function of x, y, and t, time. This is the solution of the wave equation, and we have already seen this as part of the p sets, but also discussed in class. So the wave is characterized by so-called wave numbers in x direction and y direction. The squared sum-- the square root of the squared sum is the so-called wave number. The frequency omega is equal to 2 pi f, where f is the frequency and omega is the angular frequency. And if you divide the angular frequency by the wave number, you get the speed of light. Similarly, you can multiply the frequency and the wavelength. OK, so as a first activity, I will ask you to see that how does this solution, how does this specific wave, transform on the Lorentz transformation? As a reminder, we have seen that the equation which governs how this light propagates is [INAUDIBLE] the Lorentz transformation. But now we want to investigate what happens to the wave itself. OK? So we have to investigate this specific solution and Lorentz transform x and t. And I just do this here in this equation. So you see that now we have, as part of the cosine, Kx gamma x prime plus beta ct prime plus no change in y direction as we look at the Lorentz transformation in x direction, and then we have the transformation of the time axis. OK? So now this looks very cumbersome or complicated, but we can try to refind the very same characterization of the wave as we had before. How does now the transformed wave number look like or does the frequency look like after Lorentz transformation? And so we want to identify the individual terms Kx prime, where we label Kx prime as the parameter you find here in the solution, in this Lorentz transform solution, and we do the same for omega prime, and you find a solution here. So now there's this angle cosine I defined as the angle with respect to the line of motion. Omega prime is now the baseline frequency and omega the one which is detected. That's just a matter of changing the direction of beta with the plus and minus sign. But if we use that definition, we can now discuss the result. So as part of the discussion, we can look at the specific case where the wave is moving towards us. OK? So zeta is equal to 0 and beta is positive. In this case, omega is larger than omega prime. And so the frequency is going to be higher. So the detected frequency is going to be higher blueshifted. So if you have a situation that a star is moving up towards us and emitting light, the light is detected by us, maybe by our eyes or by a telescope, that light is going to be blueshifted. It's going to go to higher frequencies. The opposite scenario is where zeta is equal to 180 degrees and beta equal to-- greater than 0, or the other way around. We could have defined this also as zeta equals 0 and beta negative. In this case, omega is smaller than omega prime. So the frequency is lower, meaning that the light we observe is redshifted. And therefore, this term redshift is a measure of whether or not the source of light is moving towards us or away from us. And the larger the redshift, the higher the velocity is of this object moving away from us. So we can define this redshift as the relative change in frequency omega prime minus omega over omega, or we can define 1 plus Z, 1 plus the redshift is omega prime over omega, which is square root of 1 plus beta over 1 minus beta. All right? So if you now observe the stars in our galaxy, and you can do this, for example, by its specific spectral form. There are the specific spectral lines, lines of specific frequency, which we can observe from stars as they are in certain distance from our solar system. And if we do this, we basically see all stars being redshifted, meaning all stars are actually moving away from us, which is a measure of the fact that the universe is expanding. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 13_History_of_Special_Relativity.txt | MARKUS KLUTE: Welcome back to 8.20-- Special Relativity. In this short video, we look at the historic backdrop, the time in which Einstein was able to develop the theory of special relativity. What were people thinking? What was the physics landscape of the time? How was technology developed? And how did all those things come together for Einstein to thrive and come forward with those important discoveries in physics? We have to go back to 1900, around that time in which Einstein was able to break through, break out, and come up with completely new ideas in physics. Before we go into the discussion of the timelines, I'd like you to be invited to come to Geneva, to Switzerland, to Bern, to Zurich-- to the places in which this all happened. Especially Bern is a historic town, wasn't destroyed in the Second World War. And if you walk down the streets, here is a picture of me two years ago in front of the Einstein house. The streets are basically unchanged for the last 150 years, so you look up into the window, and you see Einstein looking down at you. So if you go back 200 years, in Italy, people like Volta and others started to make use of electromagnetic effects. The first batteries were developed. And a theory, a wave theory of light was developed. That then let-- slowly developed into the theory of electromagnetism and the understanding of those phenomena. And the very same time, in Europe, the first railroad systems were developed. Electromagnetic induction was understood. And then in the 1860s, Maxwell was able to put all of those concepts together in his famous Maxwell equations, which are discussed at length in 8.02. People were then, because of the railroads but also because of telegraphic cables, able to communicate and move over a larger distance in a shorter amount of time. And that led to the need of synchronizing clocks. To that point, in each town had one or a few clock towers. And you just read off the time, and it was the time of day. There was no need to be able to tell what time is it in London when you have your breakfast in Munich or what time the store opens in a different city. There was no need for this kind of synchronization. But with the expansion of railroads, specifically, there was a need to understand when a train when a train is on a track and such that it doesn't intersect with another train, avoiding collisions. You also want to know when you have to be at a train station in order to catch the train. Those things became important, and they led to patents filed around this time and also later on. On the theory of light, people developed all kinds of ideas and conflicting ideas of, for example, mechanical models of light, which relied on the existence of a medium in which light travels. And we'll discuss this at length. In the 1880s, there's a new phenomena in the structure of physics. Up to this point, there was a professor of physics at an institute. He had a chair, and everybody was working for them. But then there was a change in the way physics research was conducted, such that there started to be a division of labor between experimental physicists and theoretical physicists. Around this time, Thomas Edison developed light bulbs, meaning that light and electricity, those two things became more woven together. Electromagnetic waves were discovered. And Michelson and Morley conducted the experiment. And they did not find a medium which carried light waves. And we'll discuss the Michelson-Morley experiment in much more depth. But this was basically the backdrop. Michelson and Morley-- the idea of electromagnetic waves, how do they actually travel. What is the speed in which they travel? What are the medium? And how does this all interact? So Einstein was born into this. He was born in Ulm, a German town, a small German town, on March 14, 1879, on Pi Day. He spent his youth in Munich, where his father-- he had an electrical company, a company which provided electrical services-- for example, for the Oktoberfest in Munich. And his business went up and down and had to be relocated later into Italy. It was a difficult and interesting time. So Einstein went to school in Munich, in a gymnasium in Munich. And at some point, it was time for him to go to military service. And he avoided this by becoming stateless. He then wanted to enroll in a university in Switzerland, Polytechnic of the Eidgenossische Technische Hochschule in Zurich. But he wasn't quite admitted, mainly because his French wasn't quite sufficient. So he had to go an extra round of studying in order to be then admitted to the Polytechnic. One of his mentors at the time was Herr Professor Weber. He was a leading physicist there. And they had an interesting relationship, which I come back to in a little while. While studying in Zurich, Einstein met his wife, Mileva Maric. She was also a physics student, one of the few female physics students there. And they fell in love, and they married. And they had a child together, or children together. In 1900, Einstein graduated. He wasn't the best student in class. Neither was Mileva Maric. And he had a hard time finding a career. So he wanted to stay at the university and enter an academic career, and that required the mentors to be in favor of this. And since Einstein didn't quite develop a good relationship with Herr Weber and the other professors around, they didn't want him there. They didn't promote his career. So he didn't know quite what to do-- went back to his parents, to Italy, and didn't get any traction from where he was. And he was quite anxious about this, which is visible in letters he wrote. But he had good friends there. He had a good friend, Marcel Grossmann, specifically, who helped him get into, after some back and forth, a position as a lower level patent clerk in a patent office in Bern. And so this was the starting point for a career. So he had this-- he was a patent clerk. He had a lot of time on his hands beyond that. And that started something which was called the Olympia Academy, which consisted of a group of people who studied with and friends of Einstein, including Marcel Grossmann. And they talked and spent time, took long walks, spend time drinking, partying, if you want, but specifically talking about physics. And in this framework, Einstein was able to develop this ideas. And then 1905 came-- not in a sudden, but in 1905, he was able, out of this context, to develop five very, very important papers. The first one was on light quantum. He was able to describe the emission and absorption of light. In April, then, which was part of his PhD thesis, he was able to characterize the size of molecules. And in May, he was able to show the existence of atoms. He was basically demonstrating this by following Brownian random motion of atoms. And then in June, he wrote a paper named electrodynamics of moving bodies. And that is the paper in which he discovers or describes special relativity. On further review, he discovered that there's a consequence of special relativity that energy and mass are equivalent in his famous equation E equals mc squared. And we'll come to this, as well. So this is basically the framework of this lecture. If you then move forward into general relativity, which was developed by Einstein in 1915, you can then, again, follow the development of signs of quantum mechanics of general relativity and the conundrum of describing those two concepts at the very same time, in which, then, Einstein lived and moved. His career developed then from there on, not instantly but systematically. He had a call as a professor in Berlin later on and, after the Nazis took over in Germany, decided to flee Germany. And he moved to the United States, where he took on a position in Princeton. The rest is history, as you want, and we'll hear more about this tomorrow. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 83_Proper_Velocity.txt | MARKUS KLUTE: Welcome back to Special Relativity, 8.20. In this section, we're going to talk about proper velocity. We have seen already concepts of proper time and proper length as the time and the space as seen in the object's own reference frame. So now we want to try to find something similar for velocities, as we have seen the Lorentz transformation applied to velocities of a quite difficult form. As a reminder, velocity is given as a change in space of a change in time. We have seen the Lorentz transformation of a velocity x in a reference frame which is boosted in the same direction x. And you see that this new velocity x prime is given by new x minus v. So there's a velocity addition going on, which is corrected then by this factor 1 minus uxv over c squared. You've also seen that even so, the boost is in x direction. There's also modification of the velocity in y direction and in z direction. OK? So you see that basically, there is this-- the velocity itself is corrected with this new factor. Note here that there is a special case in which the direct-- the velocity in x direction is equal to 0. Think about this object being in its own rest frame, again, where the velocity in the booth direction is 0. You see that both equations simplify for ux prime that would be simply equal to minus v, where uy prime would be uy prime and uv prime would be uv. So let's try to get at it. Let's try to express velocities in terms of the proper time, at the time as it ticks in the object reference frame. So we have seen that the time is given by gamma times the time in the rest frame or time in the proper-- times gamma times the proper time. Note here that we have two different gamma factors to play with. One is a gamma factor of the Lorentz transformation. And this gamma here is the gamma using the speed of the object in a specific reference frame. So this is the gamma which is a gamma factor which is a gamma of v of the velocity or the speed of the object. And now we can just simply define proper velocity if I use this vector eta here, which is a four vector, which is the derivative of the spatial component with the proper time. And when we do this, you find this relatively simple solution of gamma times c for the 0's component, gamma times ux, gamma times uy, and gamma times uz for the last component. So the question now is, we defined this new velocity of an object where the time of the object ticks in its own reference frame using this property as proper time. OK? If we now try a Lorentz transformation on this, we can simply apply the matrix for Lorentz transformation on this four vector and find the solution here. You can see that this is consistent by doing this with the original components in this proper time as well. You see that those actually are a consistent answer. It makes a lot of sense. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 111_Charge_and_Current.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: Welcome back to 8.20, special relativity. In this section, and also the next time, we talk about electromagnetism. Electromagnetism is not part of the core of 8.20. We are not requiring electromagnetism as part of the prerequisite. I will not test or include electromagnetism in the final. But nevertheless, it's interesting to discuss electromagnetic effects in the context of special relativity, as they led to the development of special relativity in the first. After all, the paper which describes the theory of special relativity is about the electrodynamics of moving bodies. And so let's have a look at the natural dynamics of moving bodies. And so here, we have a source-- a charge which is moving with a velocity V. And moving charges, or currents, create magnetic fields. You can use your right-hand and see particles are moving in this direction here. The electric field lines are curling around you. Good. So now I have a second charge-- a test charge which is moving with the velocity u. That test charge will experience the magnetic force. The force is equal to the test charge itself, qt, times u cross B, the velocity cross the magnetic field. So there's clearly a magnetic force acting on this [INAUDIBLE] charge in this reference frame. However, if I now move into a reference frame where the source charge is stationary, a stationary charge is not creating magnetic field, so the magnetic field is 0. Hence the magnetic force on the test charge is 0. I can, alternatively, look at the reference frame in which the test charge is stationary. And so also here, because u is equal to 0, the magnetic force is 0. So clearly, we do have to treat magnetic and electric fields in sort of a consistent manner as we treated time and space, and energy momentum in a consistent manner. But we need a concept of electromagnetic fields. And previously in this class, we looked at electromagnetic field light before. The difference here is-- and that's the context of the next section-- is that you want to understand how we create those fields-- how we can create electric and magnetic fields, and how that all works together. Now the first activity, I want you to consider a cube of length L with n electrons. There's n charges inside, and everything is at rest. And what I want you to figure out is what is the charge and current density [INAUDIBLE] and j0 of this cube? There's a second step. As you can imagine, I ask you to consider the very same thing from a moving reference frame, S prime, with some velocity u which is the velocity of S prime with respect to S. What is the charge in the current density now in this new reference frame? I'd like you to figure this out. So now we look at this cube, and the total number of charges is N. The length-- or the volume is l0 cubed. And so the density is N divided by l0 cubed. The current 0. Nothing is moving. There's no moving charges [INAUDIBLE].. All right, this one is more straightforward. But now in our moving reference frame, the situation changes. Here, one of the directions-- the direction in which we are moving-- is Lorentz contracted. So we have lx equal to lx 0 or l0, times-- divided by gamma, or times square root 1 minus u squared over c squared. The volume then of the same cube in the S prime reference frame is l0 cubed divided by gamma. The number of electrons, however, is unchanged, so the charge density is the previous charge density divided by the volume. And if you compare-- sorry, it's the charge divided by the new volume which is the density times gamma. For the new charge density, we simply have to multiply the current density. For the current density, we have to multiply the charge density times the velocity. And again, we find ro 0 times 0, times gamma. Good. So if you look at those relations, they look very much like the relationship between the current and the charge density. They look very much like the relations we had between momentum and energy, and time and space. And Lorentz transformation looks very similar. So we can, motivated by this, write a 4 vector, which is the first component c times the density. And as the first, second, and third component, the current. And you find that the invariant here is very similar to time and space, energy and momentum given as the density where the charges are addressed. And that's the invariant, and you can calculate this from multiplying the 4 vector [INAUDIBLE] square [INAUDIBLE].. So concept questions. Is the electric charge conserved in the Lorentz transformation, or did we actually change the charge? So we have the charge density here and the current here, but did we actually change the charge on the Lorentz [INAUDIBLE]? The answer is, no, we did not change them, so the charge is conserved. The charge is invariant in the Lorentz transformation. Is electric current conserved in Lorentz transformation, or invariant on the Lorentz transformation? Should rather use invariant [INAUDIBLE].. And the answer is, no, it's not. So we have seen from the very first example that you start with the current which is 0, and then in the S prime frame, the current is of a certain value. So certainly, the current was seen from two different observers is changed. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 63_SpacecraftonaRope_Paradox.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. In this section, we want to talk and investigate a bit more length contraction. We have seen length contraction a few times already in this class. We have derived it. We have seen it in application. But we want to get some sort of feeling to how can we actually understand what's happening to the objects. Later in this section, in this video, we'll talk about another paradox, spacecraft on a rope. So let's get to it. So the situation here is as we have seen a few times already. We have Alice being at rest, and Bob is moving with a velocity v. And what we are interested in is this object here, which might be a rod of some sort. You can think about a spacecraft if you want, but a specific object, which, at rest, has a length LB. For Alice, this object is Lorentz contracted, and it appears shorter. So now what happens now if Bob accelerates from his velocity v to a velocity v plus delta v with respect to Alice? How does the acceleration occur? And how can we understand then the further shrinking of the spacecraft? Bob tries really, really hard to accelerate such that all elements of this rod or spacecraft are being accelerated simultaneously in his framework. You can think about splitting up the spacecraft into small elements. They're all getting a little bit of a kick, a little bit of an extra momentum at the very same time. So, if now Alice observes the same situation, we find that she looks at the spacecraft. And, because the leading clock in the spacecraft, in Bob's spacecraft, lags, she observes that the spacecraft's back is being accelerated first. And, because it's accelerated first, she observes that the spacecraft shrinks just a little bit because of the additional velocity. Well, that's kind of an interesting picture to think about how we can understand length contraction and how we can understand length contraction once there's acceleration involved. OK, so the next question now or the next topic here in this video is the spacecraft on a rope paradox. This was phrased by Bell in the 1950s and '60s. He was working at CERN at the time and roaming the corridors, discussing with his colleagues. The situation here is related to the one we just discussed, but slightly different. So let me explain. So again we have Alice as an observer, observer in a reference frame A, observing two spacecrafts. They are identical spacecrafts. They have the same engines. And they are separated by distance D. So now Alice gives a signal to both spacecrafts simultaneously in her reference frame to accelerate at the same time such that the distance between B and C remains constant. So they're asked to accelerate such that the distance remains constant. Well, the question now is, when those two spacecrafts are connected with a rope, will this rope break? So I'll let you think about this a bit and come up with your own answer. In the meantime, this is not such a hard problem actually. In order to keep the distance constant for A, the distance in the BC reference frame, in the reference frame of the two spacecrafts, needs to expand. So LA, so the distance as observed by Alice or reference frame A, is equal to 1 over gamma, the distance between the two [? planes. ?] And for this to stay-- for LA to stay constant while there's acceleration going on, LBC needs to increase. That's why the rope will break. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 32_Waves.txt | MARKUS KLUTE: Welcome back to 8.20 Special Relativity. In this section, we'll talk about waves. You have all seen waves. You know what a wave is in principle. You might have had an opportunity to surf on a wave just like the one behind me. What we want to do here is be more quantitative and more precise in the definitions, and also look at different sorts of waves. I find this Wikipedia article here quite interesting. It starts with saying that, in physics, a wave is an oscillation accompanied by the transfer of energy that travels through space or mass. It continues then talking about the various types of waves, and it makes a distinction between mechanical waves, which travels through a medium or substance, and the deformation of the substance is reversed by restoring force. In contrast, there's electromagnetic waves, which we have just seen in the previous section. They do not require a medium, and that is topic of the next section as we continue the discussion of special relativity. Here, the electromagnetic waves consist of an oscillation of electrical and magnetic fields, which are generated through charged particles. Things that don't require a medium, they can travel through a vacuum, but an electromagnetic wave can also travel through a medium like water or anything else you come up with. It doesn't stop there really, because the concept of waves are really everywhere in physics. Specifically, when you start studying quantum mechanics and the behavior of particles, those are described by waves. Really exciting recent results are those of the discovery of gravitational waves, which are vibrations or movement of a gravitational field. Also those don't require a medium. They travel through vacuum. When we look at waves, we can start characterizing them, and one primary sort of characterization is the polarization of a wave, meaning whether or not the oscillation itself happens in a transverse way with respect to the direction of movement, or longitudinal to the direction of movement. Mechanical waves can be transverse and longitudinal, polarized, or have transverse and longitudinal components. Electromagnetic waves in free space are transverse only. So here's a picture of a wave, sine or cosine, and we can start with the characterization. One aspect is the amplitude. How big is, for example, the water wave? How big is the maximum strength of the electric or magnetic field? That's the amplitude. Waves propagate and they have a velocity. That is a characterization. The length of the wave, the wavelength, is another way to characterize them. In physics, it's always important to understand the units of the object we discuss. Here, just as a reminder, the velocity given in meter per second. The frequency of your waves. So how often do we find a trough, for a wave for example, per second. The frequency 1 over second. The wavelength is in meters. We can continue with the characterization and be a little bit more complete. We can start from the medium, the period, the polarization, transverse and longitudinal, the wavelength, frequency, velocity, or even how much energy is being carried by the wave. When we compare waves and look at their properties, we have to consider the phase of the waves. So where do, for example, two waves line up, the difference in phase between two waves, and some waves can interfere. So if you have two waves which interfere and they are out of phase, like the one drawn in this picture, the resulting wave has amplitude 0. This is called destructive interference. You can have constructive interference. For example, when those two waves are aligned, there's no phase difference, and then the amplitudes you simply add. The situation can be more complicated. When we study the speed of a wave, there are a number of considerations. The first one is that the speed of a wave depends on the medium in which it travels. And the study of the speed or the velocity dependent on the medium is part of what we will discuss in the discussion of whether or not there is a medium responsible for carrying electromagnetic waves. The source. So for electromagnetic waves, you have a charged particle. Does the speed of the source change the speed of the wave? The answer is no. It changes the wavelength or the frequency, but the velocity is not changed, and you find this, for example, in sound waves in the Doppler effect. When you listen to a police car, you hear that the frequency changes depending on whether or not the police car is coming towards you or driving away from you. That is called the Doppler effect. That does not change the velocity of sounds in air. That's independent. When your medium is moving, that changes the velocity, and so here you have to add the velocities of the medium. So for light, as a summary, light is an electromagnetic wave which is moving in vacuum with speed C, and that is independent of the source. But you can ask, in which frame. In which frame is that the velocity of light, and what is the medium, and that is really the discussion we want to carry on from here. At the time, when Einstein developed special relativity, there was still a discussion going on whether or not electromagnetic waves are of the nature of a particle or of the nature of a wave and whether or not that wave moves in a medium, which was called ether. So we can then experimentally determine this. We can look at various properties of our waves and ask whether or not this is consistent with the hypothesis that this is a particle, this is a wave in ether, maybe both, maybe neither. And we can then fill a table like this one here and answer the question. So this is, again, an opportunity for you to stop the video and think through this and try to answer the individual questions. I do this here for you. One characteristic of light, at least when there is no heavy masses involved, is that it travels in a straight line. That is certainly consistent with light being a particle, but it's also consistent with it being a wave. So the answer is both are correct. Interference and diffraction pattern. That's rather difficult to describe for a particle model, but waves, as we just saw, can interfere, and there can be diffraction. Polarization. What does it mean for a particle to be polarized, but waves can be polarized. We have just seen transverse and longitudinal polarization. Light velocity depend on the velocity of the source velocity. For particles, it doesn't seem to hold. For waves, this does. And then, the last, is the speed of light greater in air than in water. That's true for a wave. For particles, you might argue this one, but I put a no in this table here, which means that in our discussion up to this point, clearly the hypothesis of a wave for light and ether holds. You'll see in the next sections that there is aspects of light where the discussion will not hold like this, especially for the ether hypothesis. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 11_Course_Organization_820_Introduction_to_Special_Relativity.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome to 8.20. Welcome to Special Relativity. And let me start by wishing you all a happy New Year, happy New Year 2021. I'm pretty sure this is going to be an exciting year with a lot of changes ahead and a lot of exciting events. My name is Markus Klute, and I will guide you through this IAP lecture on special relativity. This is very likely my favorite class at MIT, A, because it's IAP, and we start a new year. There's a lot of excitement in the air. And we have a chance to focus for this one month of January on this specific subject. B. I have a chance to introduce a man, Albert Einstein, through a discussion of his physics, through discussion of him as a person, and also through a discussion of the historic context in which he developed special relativity. And C, if this is the first time you encountered special relativity, it will blow your mind. And it actually is quite fun as an instructor to do a little bit of a transformation in your understanding on physics. So let's get started with a quote, with an Einstein quote. I have a number of those as we go through the class. This one here is really kind of the theme of the class. Let me just read this to you. "It is not the result of scientific research that ennobles humans and enriches their nature, but the struggle to understand while performing creative and open-minded intellectual work." It is really the struggle with the ideas, really the trying to get understanding which ennobles humans and enriches their natures. And let me add here that, through your own work, through your own mind, you can transform yourself and your understanding of physics in general. This first video and this first lecture will mostly be concerned with organizational topics. So I'll lay out the schedule of the class and also how you get a grade, of what kind of PSETs we have in mind and so on. Let's get started. So I introduced myself already-- again, Markus Klute. You can reach me by email. It's klute@mit.edu. We have a graduate TA, Justin. His email is given on the slide, as well. And then we have, too, graders and TAs-- Mohit. Zach, YuQing, and Stephen. Those four will guide us through the class, as well. And especially when we go into breakout rooms in the live class, they will help you in the discussion. The class hours are from 3:00 to 4:30 PM on Zoom. And you can-- I encourage you to join the class, to participate, to be active in the class. But you will also be able to get through the material through recordings. I will not record the live class, because I want to encourage you to be just as open as possible. But I pre-record the class, and after the class time is over, I'll upload those videos for your reference. And also, if you live in a different time zone, you have a chance to listen to the class. Office hours will be Tuesday at 9:00 AM and Friday at 5:00 PM. Tuesday, you reach me, and Friday you reach Justin. Will set up a Slack workspace which we use for our internal discussions with a teaching staff. But there's also going to be channels for you to discuss PSETs as channels we discussed for physics. And there's going to be a channel, which will be important when we go to the exam discussion. The expectation is that you spend about 30 hours a week during IAP. Some will find the PSETs very straightforward and quickly to do. Others will need more time. So not a-- don't set your stopwatch for 30 hours. That's just an average guide. I will evaluate you at the end of the term, and I will use this metric here. 50% of your grade will come from the homework. There's going to be five different homework assignments. There's going to be two midterms with 15% each and one final that's 20%. If I do the math correctly, this should lead us to 100%. You earn an A with 90%, a B with 75%, a C with 60% or higher, a D with 50% or higher, and then a failing grade with less than 50%. So all lectures will be on Zoom. You'll find the Zoom links on the Canvas page. And you probably found this already. Otherwise, you wouldn't be able to find this video. And again, as I was saying, the recordings will be available after class. This is the schedule. Let me start by explaining this picture. That's Little Thomas here and then a German ICE train. This class-- the way I look at this class is that it's a train which is just leaving the station right now. And it will very slowly pick up pace. But you want to make sure that you stay on the train and you don't jump off, because this will, once it's picked up its pace, will be a fast moving train. And missing a few days in IAP will cause you to have trouble following along with the material. We'll start today with this introduction. Tomorrow, we have a very interesting lecture by David Kaiser. This is the only one where we actually record the live event. So we record the Zoom session. And then the rest will unfold, I was about to say. You find this red block here. Those are our midterms and the final. On those days, there's not going to be a live class. But I will make the final-- sorry, the exam available for about 24 hours. You should not spend more than an hour, maybe an hour and a half on the exam. And then submit once you are done. The exams are open book in a sense that you can look at the textbooks, and you can look at the course material. But you are not supposed to do an online search for the solutions, and you're also not supposed to discuss with your peers. You can ask clarifying questions on one of the Slack channels, and we will be happy to answer as quickly as we see them. But again, this is an evaluation of your own performance, and you should submit your own performance. The PSETs are different in a sense that I encourage that you guys build small groups and work on them together. Each PSET then has to be submitted by you as an individual, meaning that I don't want to see copies of PSETs of other folks. I want your own answers in your own words to the questions we pose. Very important day in January is Martin Luther King Day, and IAP gives us all a little bit of a breathing room and extra free day. And you see that there's going to be, after the discussion at the very end, after the final, a special lecture where I introduce the topic of general relativity, which is not part of the core content, especially of this special relativity class. Textbooks-- I will not really follow the textbooks line by line or chapter by chapter. But I encourage you-- and I give you an outline-- to read them, especially the first two ones. The first one is by French. The second one is by Resnick. The first one is the former MIT professor. You are supposed to read this book cover to cover, and I give reading assignments on this. Resnick has a lot of good examples and problems, and we will focus on the first four chapters of that book. There is a lot of literature out there on special relativity, as a slew of textbooks available. If you have one at hand, you can also use that and find the relevant chapter in there. I give you one extra book here, which is nice because it is an excellent explanation of the mathematics involved. If you're more inclined to course 18, that might be an interesting textbook for you. Again, there's many, many resources on the topic of special relativity, and I'll point out a few as we go. In addition to the textbooks, I give you a number of papers-- for example, the paper in which Einstein explains or describes the concept of the theory of special relativity. And there's a few others which I'd like you to read. They're interesting because they introduce the physics, but they're also interesting as they have been written more than 100 years ago in a slightly different language than we would be using today. And Einstein, specifically, doesn't even write papers the way other people wrote papers at the time. He had his own style in writing papers. And we will see that. So here is the reading assignment week by week. We start in week one by Resnick Chapter 1, French Chapter 2. There's two papers, the Michelson Morley paper and the Einstein paper on special relativity. And then we go on as we go through IAP. Your homework schedule is here. I hand out PSET number 0 today, and I'll explain it on the next page. This is a PSET which allows-- I mean, this is a PSET which will keep you busy for the entire month, or most of it. The other PSETs are shorter, and they are more pointed. And you get the PSET, you work on it, and your hand in the solution. And by the time you have done that, the next PSET rolls around. I would use the PSET load a little bit, because I want to acknowledge the fact that it will be harder for you to work in teams. I still encourage this. You find yourself on Slack. You find yourself on Zoom channels. Some of you might have housing together. I really encourage you guys to discuss the physics. There are some really difficult concepts which just need you to think through and to talk through. And the talking through best happens with your peers. And you will see that that will be very beneficial, especially when it comes to the PSET. But here is homework number 1. I've done this now a few times, and I really, really enjoy the solutions given by the students. So this is a creative project. You have basically almost all of IAP to hand it in. And the idea is to be creative around a topic relating Einstein, or/and the theory of special relativity. The project might be a video. It might be a poem. It might be a musical piece, artwork, animations, a game, a structure-- you name it. What I really want is that you take something you do very well and combine it with the topic of this class. I want you to take something where you are very familiar with where you are feeling comfortable and then enter this new topic, this new topic of special relativity. I will rate it based on your creativity, on the quality, and on your effort. And again, please hand this in by January 22. I really encourage you to not wait to the last minute to start working on those topics, those creative projects. They always take a little bit more time than you might expect. This class will have an interactive component. And we will use breakout rooms for discussions and smaller problems as you would be doing in the classroom, as well. Here is the very first one. People who are taking this class asynchronously, I encourage you to stop here and work through the problem. And then look at the solution afterwards, or look at the discussion of the solution afterwards. So this first Gedanken experiment, thought experiment, is about the topic of relativity. And you want to first understand what that is, what relativity is, before you discuss what special relativity is. So imagine you wake up inside a room with no windows and one locked door. You checked. You make sure that the door is locked. You cannot get out. You cannot see outside. Looking out around, you see a table with a number of items on it. There's a desk lamp plugged in and turn on. There's a tennis ball. There's a bunch of string. There's a pitcher of water. There's a cup. There's a candle. There's a box of matches. And there's a music player with headphones. A skateboard and a wooden stool are also in the room. The music player has a sign on it saying turn it on for instructions. So you do. You're told that you are in a specially designed vibration-proof and noise-proof train car [COUGH]---- sorry-- on a set of straight and level train tracks. Your task is to use one or more of the items in the room, perhaps in combination with each other, to determine whether the train car is stationary or is moving on the tracks. There's a 30 minute time limit on your test or tests. And you cannot destroy or you cannot modify the walls or the floors or the ceiling of the car. That is not permitted. So the question to you is, can you think of some creative way to use the items that might indicate whether or not you are moving. Think about this and discuss in your breakout rooms-- that is, for the live session here. Again, I want to encourage you to stop and to think about a solution. Can you think about a creative way to work with this? The typical time is about 10 minutes. You might think about it. You might want to take some notes. All right, the answer is you cannot figure it out. Every reference frame by itself is inert. You cannot figure out whether or not you are in a moving or in a stationary reference frame. And as long as you cannot feel vibrations, or you hear the sound, or you have some other internal, external indication, you will not be able to figure out whether or not the room is moving with a constant velocity. You will be able to feel or measure accelerations. And so now you could argue that since the train track is on Earth, and the Earth is actually-- since you are on Earth, and if you're moving, you would be able to feel the rotation of the Earth, there might be a way to figure this out. But that will be very, very hard to do. So the answer is you cannot. You cannot distinguish one reference frame from another. You don't know what an absolute velocity is. So another way we will interact in this class is through concept questions. So here, concept questions are meant to give you a little bit of a checkpoint to encourage you to think about what I just explained in a video or in the class and to stimulate some sort of discussion. Sometimes you will use contact questions in breakout rooms for the discussion part sometimes you just use Zoom features in order to figure out what the answer is. But besides the technology, those concept questions are really for you to figure out whether or not you just understand the concept. So again, I encourage you to pause the video to think about the concept question and then move on with the answer. Here's an example. A new Star Wars movie came out, 2019. And I ask you to pick the answer closest to your level of Star Wars expertise. I saw the movie, and I consider myself a Star Wars fan. I saw the new movie, but I do not know much about Star Wars. I heard about Star Wars, but I didn't see the movie yet. Always liked Star Trek better. What? Star Wars? OK, so very straightforward-- here I would give you an option-- 1, 2, 3, or 4. You collect the answers, and we get some feedback. This is obviously-- there's no right answer here. There can be concept questions where there is a right and a wrong answer. And again, for you, this is supposed to give you feedback on how well you understood the concept. All right, before we close this first video, I want to introduce players, Alice and Bob. We will have, often, discussions of reference frames. Typically Alice or Bob are in one reference frame. And then the other person is in the other reference frame. I want you to think about those Star Trek figures. We will use spacecrafts. We will use light-- so phasers or light pistols-- in order to demonstrate the impact of special relativity and understand the concepts. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 92_Energy_Conservation.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we're going to further investigate the energy momentum for vector, which we introduced in the previous sections. But here, we focus on the zeroth component, the first component of this vector, where we find mass in A for particle A times the proper velocity of the zeroth component is equal to mA times c times 1 over 1 minus uA square over c square, which is the energy of this particle A over c. Or in other words, the energy is equal to the mass times c square over 1 square root of 1 minus uA square c square. So let's discuss or look at-- let's have a look at this a little bit more. The first question we can ask-- how does this now look like for particles which travel with reasonably low velocity? So uA-- much smaller than c. So we can Taylor expand this following this equation here, which we discussed earlier. And we find that the energy is equal to mA c square. That's the first term, which we call rest mass, the energy given-- just the rest mass by the mass of the particle times c square, plus 1/2 mA c square times uA square over c square. The c squares cancel. And we find what we know as the kinetic energy, 1/2 m v square, or in this case, 1/2 mA uA square. That looks very familiar. So the energy of a particle is given by its rest mass plus its kinetic energy. All right, now investigating this for vector, then we can ask, how does the invariant interval look like? How does this property, which is invariant, and the Lorentz transformation look like when we multiply the vector with itself? Here, we find minus E square over c square plus the 3 momentum squared is equal to minus m0 c square. Or in other words, we find this energy momentum, energy mass relation, energy momentum mass relation, where the energy is given by the momentum square times c square plus the rest mass square times c to the 4th power. Again, we can unroll this now and ask, how does this look for a particle at rest? And again, we find the energy is equal to mc square. No surprise. That's how we started the definition of this. In general, we can find that the energy is equal to a relativistic mass times c square, which is equal to the rest mass times gamma times c square. And that's equal to the rest mass times c square plus k, the kinetic energy, square. All right, does this definition-- I can tell you that this confused me as a student quite a bit. This understanding that the mass becomes heavier for a part of this-- really, one I didn't quite like. I just like to think about the fact that the kinetic energy-- there's a relativistic component to the kinetic energy, which is owned by the particle in addition to the rest mass of this particle times c square. Also interesting to note is that for particles at rest-- particles which are massless, like a photon, the energy is equal to the momentum times c. If you want to know what the energy is of a photon, you need to know what the momentum is of the photon multiplied by c. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 122_Time_Dilation_Effect_on_Earth.txt | MARKUS KLUTE: Welcome back to [INAUDIBLE],, special relativity. So moving from special relativity to general relativity, we relax the requirement that reference frames can only have a constant relative velocity to each other. So we are looking at what happens now if you have accelerating reference frame, and the first example is that of rotational time dilation. So I mentioned you have a device which has two clocks. And so the first block sends it ticks up in terms of light beams, beams with some specific frequency. So there is a transmission going on, and there is a receiver on the other side. And the clock, the upper clock, u clock, submits the same things. They both have the same frequency, let's say 10 pulses per tick. We also assume now that we don't have to consider special relativity effects, yes, so the velocity of the thing is very small. OK so relativistic effects are small. But when I now accelerate this device, there is an interesting effect. The lower clock will receive more than 10 pulses per tick, so it is perceived as running slow from the upper clock, right. So the lower clock is expecting 10 ticks, but it's seeing 12, so you think that your clock is running slow. Similarly you can discuss the upper clock, which because it's accelerating upwards, it only receives 8 pulses per tick. So it's expecting 10, so it's perceived to be faster. Now, this acceleration as in this example, as I've discussed previously, is equivalent to being in a gravitational field. So we have a time dilation effect on us. We already discussed the consequence in the discussion of GPS systems. Here is a summary of this plot, and just showing you the effect of changing gravitational fields as a function of the distance of the center of the Earth. And you see that the rotational speed-up, 0 on the Earth's surface, becomes rather large the further away you get from Earth, while if you consider that the object, a a satellite, in an orbital motion, the speed of this object is kind of compensating partially. But there is this net effect, as we discussed earlier, on the GPS system. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 14_Guest_Lecture_Space_Time_and_Spacetime.txt | so welcome to age 20 special relativity it's a great pleasure to introduce david kaiser david is a faculty in the physics department um he's also historian of science so he's in a super great position to talk about einstein and special relativity and give kind of the frame for this class so please welcome david um and you know take it along great well hello everyone happy new year uh welcome back at least virtually to mit it's really a great pleasure um to talk with you today about this material i love this material so hopefully you can all see that first slide smiling einstein on the bicycle okay so i want to talk about three main um parts today for for the for the material i will talk about how all the of of the kind of most accomplished physicists were thinking about motion of bodies through space and time during the middle and late years of the 19th century roughly say 100 to 150 years ago give or take because that really sets up a pretty sharp contrast with how at the time a rather young and very a little known person named albert einstein began asking similar questions but often in very different ways so we'll start with some of this context of what was happening before einstein even came on the scene to help us better make sense of why his approach seemed really so unfamiliar and so surprising at the time and then for the last part it's i find it really fascinating drawing on work from some of my friends and colleagues other historians will try to ask what was going on with einstein why was einstein's approach so different than can we make sense of it since it wasn't just the ordinary routine that we might have otherwise expected that's the last part about coordinating clocks let's jump right in let's start not with albert einstein with another very familiar name uh james clark maxwell i thought about growing a beard like that um during covet i haven't made much progress but that's a typical 19th century fine cambridge beard cambridge england so we all know maxwell's name many of you probably own a t-shirt with uh maxwell's equations on it if you do i'm very jealous i don't have it anymore anyway so we we still use um his approach to electromagnetism as of course you all know very well and we've been able to boil it down to basically a tweet or a single t-shirt it turns out as you may know what we now call maxwell's equations were were hammered out by maxwell and actually some other colleagues during the 1860s and early 1870s so approximately 150 years ago and in fact when he first wrote them down they weren't in such a nice compact form it was in a 900-page two-volume treatise that was first published in 1873 two volumes a total of 900 pages was a weapon it was you could like hurt someone with these books they were so fat even though now we can boil it down to a simple t-shirt and so what's even more interesting to me is that even though we still use maxwell's equations what we think they say about the world is really really different than what maxwell and his immediate circle thought and the biggest difference of all is that for maxwell and really for all of his contemporaries and his students the equations of electricity and magnetism as far as they were concerned had everything to do with some physical substance some medium that they called the ether that they assumed must have spread evenly throughout the entire universe filling every nook and cranny of space so all of the phenomena that we would associate with electricity uh or mechanism the flow of currents um the the splaying of iron filings around a bar magnet and even much more complicated things to maxwell and his group these were all just sort of evidence of the state of stress of this underlying kind of physical medium was like a springy uh substantial resistive medium that they called the ether and he says that in the opening pages for example of this now very famous treatise and throughout that the whole point of of studying this field as far as he was concerned was to study the behavior of the distribution of stress in this medium extending continuously throughout the universe the ether it wasn't just maxwell uh his uh a little bit more senior colleague william thompson who went on to be known as lord kelvin like the kelvin temperature scale and many things we we uh still use from from uh thompson's work uh he wanted to give a sense to uh non-scientists a few years later in a popular lecture what all the excitement was about he said here's what we're doing we want to get a sense of what what we're doing we're studying the physics of this elastic physical medium called the ether and he instructed his very elite very fancy audience who went to one of these popular lectures stick your hand in a bowl of jelly he tells them and see how it wiggles and vibrates as you move your hand around that's the highest of high-end physics in the 1860s and 70s it was all about understanding that the behavior of this elastic medium called the ether and that had very specific um follow-on implications for this group so one of the first things that you all know already one of the most exciting features of maxwell's work that really got himself excited and solidified his reputation was this unexpected unification that he put together in the 1860s not only was there a deep relationship between electricity and magnetism that had been wondered about but never really formalized but even more surprising that these two areas electricity magnetism were also deeply associated with optics with the propagation of light so as maxwell as i'm sure you know and probably have already done this on problem sets by now yourselves using maxwell's equations you can derive the quantitative behavior of light as it travels through space light as maxwell um intuited was nothing but a certain pattern of propagating electric and magnetic and magnetic fields again to maxwell these were these were propagating in this material physical ether they were disturbances propagating skittering through this physical stuff that filled all of space and in fact by trying to calculate the speed with which disturbances would travel through space based on other properties of the ether that he and his colleagues had already measured basically sets of like spring constants he found that the speed of propagation would be equal to the value that had already been calculated many years before for the speed with which light travels and he says in his own words in this lovely old-fashioned phrasing the velocity of transverse undulations a certain kind of wave in our hypothetical medium agrees so exactly with the velocity of light calculated from optical experiments mostly astronomy at the time that we can scarcely avoid the inference that light consists of the transverse undulations of the same medium which is the cause of electric and magnetic phenomena that's why his book was 900 pages right today we put on a t-shirt he was really just walking through this very old-fashioned language to come to the point saying light was nothing but a certain kind of pattern in the ether and so in fact they often called this not just the ether but an even fancier name they called it the luminiferous ether and that stands that's uh from the latin it means light bearing or light carrying so lumen like lumos for you harry potter fans that part you probably recognize and the ferris is like a like a ferris wheel that means like to carry or to to to kind of um to move through space so this is the light bearing or light carrying ether and that's what maxwell and really all of his contemporaries were convinced was most exciting about maxwell's work uh that this was a way of characterizing the state of the ether including these optical phenomena okay so that's where things stood in the 1860s and 1870s but that led to a whole new set of questions and the next generation quite understandably wanted to kind of follow up on that and ask kind of ways to generalize that framework maxwell's work had really assumed that there was no relative motion between either the source of light or the receiver of light that everything was analyzed in what we would now call the same kind of rest frame or frame of reference as the ether itself but of course that wasn't the most general situation as as maxwell's own contemporaries was sort of the next younger generation began to wonder about one of the most influential of that next wave is this gentleman shown here hendrick lorenz from the netherlands from leiden so laurence wanted to ask about the electrodynamics of moving bodies what if either the emitter of light or the receiver or both were moving with respect to this all-pervasive medium and so lorenz as we'll see had really two kind of distinct motivations in mind when he when he tried to think about the electrodynamics of moving bodies or let's say optics for moving sources or receivers he had a kind of mathematical set of quandaries we'll talk about those first but he also had a series of very puzzling experimental results so he had a kind of empirical set of ideas we'll come to in a moment and also these more formal or mathematical ones we'll start with the math uh lorenz was really one of the the best trained leading mathematical physicists of his age so he knew very well how to handle relative motion galileo had derived that in the 1610s that was hardly news in the 1890s we would do what we still call the galileo newton transformation that if you have for example that galileo wrote very famously in his charming dialogues in the 17th century if you watch a friend float down a river on a boat that moves at a constant speed neither speeding up or slowing down then you know very well how to compare the coordinates between you sitting at rest on the on the back of that river and your friend as she floats down on her boat that what you call the origin the spatial origin of your coordinate system you can fix at say x equals zero but your friend could call the center of her boat the origin for her coordinate system and then you can see how your system or how her system coordinates will will um will move with respect to yours if the boat moves at a constant speed little v then you just have to calculate the offset between where the origin the center of her boat is as it floats further and further down the river so what she calls the origin of her spatial coordinates will be offset from yours because it's relative emotions relative drift meanwhile as newton himself famously said in the beginning of his um principia where we where we learn about uh newton's laws of mechanics the time is time is time there's some absolute time he actually referred to the censorium of god he thought there was only one possible time to consider and so formally there'd be no change to the rate at which your clock ticked versus the rate at which your friends clocked ticked on her boat this is called the galileo newton transformation i made a big deal out of it you probably would have done this in your sleep lorenz did this in his sleep however it drove him to nightmares because when he then applied this to maxwell's really beautiful and much newer set of equations for things like optics or the propagation of electric and magnetic disturbances in the ether he found that in that transformed reference frame if you take into account that relative motion between either emitter or receiver of light then maxwell's beautiful description of light this quantitative description looked very ugly and in particular it would no longer suggest that light should behave as kind of oscillating sines and cosines it works beautifully with maxwell's equations if you look only at both sources and receivers of light that are that are fixed in place and not moving with respect to the ether once you apply the the very very well known transformation to take into account relative motion then your description of light gets all literally out of whack and this was a mathematical conundrum for lorenz it didn't seem to make any sense because on on the earth on the moving earth presumably moving through the ether he and his colleagues could measure light to behave like beautiful signs and cosines all the time so we have an empirical kind of um reminder that maxwell's description seems really really accurate for describing optics even on our own moving ship of the earth so this is a real conundrum for him he was a very very you know clever mathematically gifted physicist so he introduced a mathematical clue she introduced a mathematical um stop gap to address this he called it local time and he was actually very clear in his writings he said it was a mathematical fiction his word it was literally just a mathematical trick if he introduced a new transformation for the time variable so when you compare your coordinates with those of your friend on the boat what if you actually did make a transformation of t t prime not just of x prime and the transformation he kind of worked out he kind of reverse engineered the form it would need to have so that maxwell's description of light waves would be restored to the very simple form uh that maxwell himself had written down a few years earlier so this is where we still call this the wrench transformation many of you might have seen it if not you'll have plenty of time to learn about it and practice it during iap we call it lorentz not einstein because lorenz derived these equations first in a series of papers in the 1890s and he was responding to this question of the electrodynamics of moving bodies how do you preserve this beautiful description of light from maxwell even if there's relative motion with respect to the ether he said it was a trick it was merely mathematics but it would at least get the right form back so that was the kind of mathematical motivation lorenz as we know was also concerned about an empirical or experimental curiosity he was following very closely the work of the u.s based physicist named albert michaelson michaelson's story is really fascinating i won't take too much time now but um super interesting he was actually an immigrant from what's now germany was often part of poland kind of central europe he moved to the us when he was about two years old there young and was raised right around the time of the gold rush in california and before he get carried away i'll stop talking about michaels and although he's super interesting so what michaelson wound up doing was he was fascinated by this question of maxwell waves of waves of electricity and magnetism in the ether and he was convinced that if we are on this moving ship of the earth moving through the ether that should have a measurable impact on the light waves that we can produce and measure here on earth so he set really his life's goals starting from very very early trying to build devices extremely sensitive instruments with which he could measure the fact that the earth was moving through this ether like all his contemporaries michaelson believed without any hesitation that there must exist this the material the ether uh what else could light waves be but disturbances in the ether and then he figured that much like if a bicyclist on a on a going for a bike ride we should be able to tell that we're moving through that medium so before i describe his device called the interferometer let me just give a little more kind of analogy for why michaelson was so convinced there should be a measurable effect if we walk outside our houses which is now a luxury we can still do it with masks on walk outside on a still day when there's no wind or no breeze we won't feel any particular wind on our face if we're standing still so we're outside in a physical medium there's a physical atmosphere but if we're at rest with respect to that atmosphere if there's no wind we don't feel it on our face on the other hand if it's still a still day there's no breeze when we get on our bicycle and pedal really quickly we'll now feel a breeze on our face because we're moving through that physical medium we're moving through the earth's atmosphere even if the atmosphere seems to be perfectly at rest no strong breeze no hurricane whatever if we're going quickly enough through it we'll feel a breeze on our face and michaelson thought the same thing should be happening for light waves as they are carried by the earth through this medium not the earth's atmosphere but this all-pervasive light bearing ether so we really thought it'd be like the bicyclist now his his real genius was devising this device we now call it the interferometer with which he could try to measure the impact of that motion of the earth the entire earth through the medium so we took a source of light today we use lasers because they're awesome and they're also monochromatic their light shines at basically one frequency or very dominantly one frequency lasers of course weren't available to michelson so he used um very bright sodium arc lamps sodium lamps will shine mostly at one dominant color not nearly so monochromatic as modern lasers but they were the kind of industry standard of the time so we basically took sodium lamps that mostly shine as a particularly yellow color shine them at what we would now call a beam splitter it was a half silvered mirror as its name implies that would let half the light through it would act like a window roughly half the time but act like a mirror the other half the time so half of this incident beam will go right through this as if it were just a plate of glass it would be transmitted it will then make its way to a fully reflecting mirror sorry a fully reflecting mirror at the end of the path it'll be reflected to come back and then again half of that return light will be reflected by the mirror and come out to a screen meanwhile you can run the same kind of story for light that gets deflected towards path two from that same single incident source half of the incident being will be reflected by that beam splitter half silver mirror it'll travel a path down to a fully reflecting mirror come back and then half of that returning light will be transmitted so you have light coming together on some detector screen that started out as a single light wave from a single nearly monochromatic source but you split it into two beams so the beams travel different paths and now the light starts out fully in phase with itself it's one light beam so crests are with crests and troughs with troughs and the idea was that you then split it to set up a race to see if there's any difference in the travel for light that traveled through path one and out to the screen versus like the traveled path to and back to the screen now all the light starts uh as a single light wave in sync or in phase of itself it was if there was any difference in the travels of that light between path one and part two you should see a shift in the interference patterns crests should no longer arrive lining up with crests or troughs with troughs you should see a very characteristic set of interference fringes what's called an interferometer extremely sensitive now his idea was if if like the bicyclist we're on the moving earth then the path that's heading directly into the ether the that is a the path that's along the earth's motion through the ether should be just like the bicyclist feeling the direct kind of headwind of that of that breeze directly on on her or his face that means the effective speed let's say this is the direction of the earth's motion the effective speed with which this light beam could travel toward its mirror should be experienced a direct headwind for its upward uh path and a direct tailwind for its downward path it should have a different speed than this or the light that follows some orthogonal path so that even the light starts out perfectly in phase you should find a shift in the interference fringes because the light's effective speed through the medium uh is different for path one than path two again an analogy to the ether wind of a bicyclist through the atmosphere not only that is a little uh foreshadowing for what's to come later this device is so sensitive it's sensitive to second order in a very small quantity so if there was any um it could measure differences to the second order in the relative speed of say the earth through the ether compared to the the speed of light so that's called a second order because it goes like this small quantity squared that's how sensitive is really ingenious device was and as many of you might know we use interferometers all the time now in all kinds of industrial applications even more my favorite example is from our friends who detect gravitational waves with ligo that's essentially a super-sized interferometer with very very similar principles of design so michelson was doing this really cool work it was among the earliest work that any of the super fancy elite european scientists paid any attention to coming out of the u.s michaelson was among the first who made even the people like hendrick lorenz sit up and pay attention at a time when the u.s was still otherwise pretty much a scientific backwater so it's a really cool idea it's a remarkably clever design for device michaelson built a small prototype then he supersized it with his colleague edward morley they built a device 11 meters long for each arm so you know roughly 33 or 34 feet long huge huge for this day very long arms they floated the whole thing in a vat of mercury which i don't recommend but they're trying to tamp down any vibrations from nearby cable cars they want to have a clean laboratory environment then they did this for years and years and years months and months and years and years because of course you never know what the what the ether's actual rest frame is who's to say that at this moment this is the direction of the earth's motion to the ether maybe it's this direction or some other angle in between they would look for day-to-night variations they would look for annual variations as the earth went around its orbit around the sun they would try to find any time day or night winter versus spring year after year when they could measure some measurable offset uh or change in the in those interference fringes because at some point this direction should have lined up with the earth's direction of motion even it wouldn't be every time that's why they kept doing this over and over and over again and as you might know they found nothing so after years of painstaking data collection they found no no compelling empirical evidence of the earth's motion through the ether michelson won the nobel prize for this and related work he was a astoundingly accomplished experimentalist he was the first u.s based physicist to win the nobel prize he lived 20 more years after that and died considering himself a failure so i hope you all win the nobel prize and i hope you think better of yourselves and give yourselves more credit uh michelson knew for certain the ether must exist and yet he had failed to find it so lorenz was following this back in europe and this was really his second main motivation to think about these funny ways of handling coordinates and what we now call the lorenz transformation so i mentioned his mathematical concerns about the transformation properties of of maxwell's equations he was also following um uh the michaels and morley work with great interest in citing it and we know he was really following each update so he was concerned about this failure to measure a shift in the interference fringes but lorenz said oh but actually maybe there's a physical reason to account for that if there really is as he knew there must be this physical resistive medium a kind of elastic medium through which uh the earth and everything else is moving then there must be a force exerted by that kind of viscous medium on every single atom and molecule making up all the stuff in michaelson's device as well as in everything else and so it'd be like picturing a beach ball well if you try to drag a beach ball under water in a resistive medium the shape will be deformed along the direction of motion if you drag that beach ball fast enough underwater it will shrink in the direction of motion be deformed like this picture here and so lorenz said that must be happening for every bit of matter in the arm of michelson's interferometer that's that's most subject to that resistive force of the ether there should be a contraction along the direction of motion so then again because he's a very gifted mathematical physicist he calculated exactly how much shift must there be for the resultant path to be shortened just enough to make it a tie race after all so the leg of the or the arm and the infrared that's experiencing this shrinking this extra force due to the resistive medium would have to shrink by a specific calculable amount we now call it use the greek letter gamma if you don't haven't seen that already you'll see it throughout iap and the rest of your life it takes this somewhat simple looking form 1 over the square root of 1 over 1 minus v over c squared if you plot it you can see it diverges at when when the relative speed gets close to the speed of light you have 1 minus 1 in the denominator you divide by zero your eyes bug out cats and dogs live together everything goes crazy so it diverges at v equals c on the head for small speed small compared the speed of light this factor is indistinguishable from one so here's a quick plot for ordinary speeds that we encounter on the highway let alone on a bicycle our speed compared to the speed of light is is vanishingly small so this factor gamma is that is very very close to one only when you get to speeds approaching the speed of light would you expect to to measure any shift from that nonetheless lorenz said in principle this happens at any speed and they shrink the amount of contraction is governed by that relative speed given by this this greek letter gamma so gamma is always greater than or equal to one that means one over gamma must be um smaller than one and that's saying that the contracted length of one of those arms of the interferometer got shorter by a certain amount given by or depending on the relative speed that would be enough to account for this null result lorentz argued it turned out that exact form of gamma was the same form he'd found for his mathematical kind of um local time manipulations so here's how we can live in a physical ether even though we don't measure its effects okay let me now shift to how uh a different person began thinking about very familiar questions by that point but coming at them quite differently that's young albert einstein so einstein was born in the midst of all this he was born in 1879 in kind of rural roughly speaking nowheresville germany not near any of the big cities so laurence was already a practicing physicist when einstein was born he was much younger einstein's main ambition was actually to become an electrical engineer at a time when that term itself was actually brand new uh einstein's father and uncle had gone into business together in this really brand new field of electrical engineering this is the age of electrification think about extended street lighting uh trolley cars with with the shared electric lines above this was transforming the face of everyday life in cities and eventually even in more rural areas and einstein's father and uncle were were in on that they were early professionals in this new field of electric engineering and einstein loved it he loved to tinker with electro technical gadgets his he had many strong feelings about many things among them he hated what he considered the overly militaristic german high school that he attended he would argue and insult his teachers they didn't like him any better they were delighted when he dropped out so they didn't have to kick him out he was really very obnoxious it turns out so he dropped he was a high school dropout he dreamed of entering uh the swiss federal uh federal um technical institute it was kind of like the mit of switzerland except to be more proper mit is the is the uh massachusetts version of their school zurich was was founded first it's often called the eth or the ignorance technician it was really a very elite technical university in zurich switzerland einstein by this point had renounced his german citizenship he dropped out of high school and one of the best things about this technical university in zurich was that you didn't have to have a high school diploma he said perfect that's the place for me so they had an entrance exam which he very dutifully studied for and then failed because after all he ignored the topics that weren't of interest for him he he did well enough on the physics and math portions of the overall exam that a kind of kindly physics professor took him aside and said you know if you go to basically a kind of regional something like almost a community college go to a local regional school for a year study up retake the exam you might do better and that's what einstein did so he went to a little regional school in um again in rural switzerland for a year took the exam again and passed so now he's able to enter his dream school the eth in zurich once he then worked so hard to get in he proceeded to cut classes this guy was a horrible student whatever you do don't be like albert einstein at least while you're in university he works so hard to get in and then he again would insult his professor as much as he'd insulted his high school teachers he thought everything that they did was boring they didn't know what was really interesting so he would cut classes and read on his own and then borrow notes from his girlfriend his long-suffering very patient girlfriend maleva marich who was doing very well in her own physics and math courses and likewise he borrowed notes from another friend of his martha grossman so he just cram and scrape through for the exams not an ideal student for some reason einstein couldn't possibly fathom because he actually was adult frankly he impressed none of his professors so when it came time to graduate none of them would write him a strong letter of recommendation again his life bears many important lessons for us today uh go attend class and be um slightly less obnoxious to your instructors and maybe things will turn out better so none of his teachers would basically support him because he had middling grades and he was rude to them so he couldn't get a job after graduation so finally uh of one of his close friends marcel grossman one of the folks from whom he'd borrow notes his father had connections and so the basically through connections he was able to get einstein an entry-level civil service job at the patent office in barron switzerland so still in switzerland he was a patent officer third class as i'm always fond of saying there was no fourth class that was literally the lowest entry-level gig so even for you einstein it wasn't what you know it was who you know what he then proceeded to do was he had a day job at what they called the electro technical desk much like what his father and uncle were doing he was a patent examiner for a lot of these cool new electrical gadgets and then he would go hang out with a bunch of friends and drink beer a lot at the at the pubs so they formed what they called the olympia academy and this was very ironic they gave themselves the most elite sounding name because they're basically three semi-bums hanging out reading and blowing off their families so they gave this was a very very prestigious name even though it was literally three recent college graduates hanging out it was um maurice sullivan conrad had peach and young albert einstein and they would go and sometimes drink coffee often drink beer and talk about stuff they'd read and they would read a lot of physics and read a lot of philosophy and talk about it and one of the books that we know from their correspondence at the time that they were really interested in was this fascinating book by an austrian polymath named aarons mock here's the book translated into english known as the science of mechanics it had come out in the 1880s mock was really remarkable he was both a mathematical physicist and experimental physicist you might know the terminology for mach number like speed of sound mach 1 mach 2 mach 3 seymour he did a lot of studies of acoustics and optics but also of what we would now call psychology like sensory experiments and so on and eventually a medical surgery and then at the end of his career he was a professor of the history and philosophy of science this guy did it all he wrote these very very dense um conceptual critiques of newtonian physics among other things and he was mark was convinced that newton was really getting getting himself into a horrible muddle because he had not paid sufficient attention philosophically and this is what the young members of the olympia academy got really excited about we know from their letters and their notes at the time mark was was advocating a position that came to be known as positivism according to ants mock unlike isaac newton or for that matter even some folks like maxwell mach argued that only quantities that could become objects of positive experience let's say things we could actively measure or sense or feel or touch only those things belonged in our scientific theories anything else was just like mere speculation like you know counting the number of angels that could dance on the head of the pin he was scathingly critical of newton and in particular mock wrote in detail that newton's notions of absolute space and absolute time had no meaning because how could you ever measure absolute time show me the clock that could measure absolute time that was the kind of classic machine response i can measure the passage of time by using a clock but who says that's what newton wrote absolute time was this had an amazing impact as we as we know on the young einstein so it's in this milieu he's now not in the university setting he'd scrape through his undergraduate studies he has a nice day job but he's not doing particularly well professionally he's hanging out with his buddies in reading some interesting and hard obscure kind of philosophy science uh in the evenings where he has what's now commonly called his miraculous year his honest mirabilis in 1905. in fact he only took about six months it was like half a year during which he submitted four really astonishing original papers to the leading journal of physics the honolulu physique in germany at the time they've since been published and translated there's a particularly nice addition you can find edited by uh john stachel uh with very nice um essays to accompany them you can find them online so for for iap and for the rest of today and even throughout the month we're really interested in what was the third of these articles that he submitted he submitted to the journal in june of 1905 it's on what we would now call special relativity its title as you can see up top was actually in translation on the electrodynamics of moving bodies a thoroughly familiar title exactly what lorenz and all of laurence's colleagues have been talking about for for decades by that point so the title of einstein's paper in 1905 was not surprising even though as we'll see his approach was quite distinct he begins this now very well known paper not by saying i found an error in lorenz's calculation there's a missing factor of two pi or you know whatever he doesn't say i conducted my own measurements and i found these results within experimental error he starts out by saying there's an asymmetry in the explanation which is not present in the phenomena it sounds very philosophical he goes on in this very opening paragraph to say that when we use maxwell's equations we come up with very different kinds of accounts for what should be the same phenomenon very simple nothing super fancy like an interferometer just take a bar magnet and a coil through which current could flow and make sure you have a current meter an ammeter attached to the coil so if current flows electric current flows through the coil it will be measured the ammeter needle will be deflected and we are free to move either the magnet or the coil in this opening paragraph einstein says physicists had treated this situation completely orthogonally even though einstein was concerned as far as that's concerned there's only one explanation needed and so with case one if you assume the bar magnet is moving you're shaking the magnet back and forth and keeping the coil fixed in location then you would appeal to one set of maxwell's equations and again on on the t-shirt you'll be this one you have a time-varying magnetic field which will induce a spatially varying electric field and that will exert a a push a force on the little ions little electric charges within that coil they'll feel a push they'll move along the coil electric current as maxwell himself had argued was nothing but the motion of these electric um electric charge bearers so because of a time varying magnetic field the the little bits of matter in the coil will be pushed along you'll induce a current that's case one moving magnet stationary coil but if you want to analyze the other situation hold the magnet rigidly fixed in place and shift the coil back and forth then physicists would give an entirely separate explanation they would appeal to a different one of maxwell's equations now they'd say that there's a static magnetic field varied in space its spatial gradients were non-vanishing and that would exert a a force like a kind of lorentz force law on those charges so they're moving they have some velocity in a magnetic field they'll be pushed along the wire then you'll measure a current einstein said that's one explanation too many all that we could ever measure again you can almost hear the kind of mock coming through all we can ever see as an element of positive experience is some relative motion between the magnet and the coil which induces an electric current who's to say one was actually still while the other was moving so einstein begins this paper by saying that there's something wrong with with the stories we tell about the equations he doesn't argue about the equations he argues about our interpretation of equations very striking he then goes on just in like paragraph three still very early in the introduction to posit two postulates he doesn't prove them he doesn't derive them he doesn't say i have uh demonstrated these by doing experiments he just pop he says let me hypothesize these and see what follows and these are the two in in rough paraphrase the first one was actually already called the principle of relativity or sometimes simply called a galilean relativity going back again to the 17th century galileo had argued by thinking about that that boat floating down the river that to a person on the boat as long as the boat is moving at a constant speed neither speeding up or slowing down that all the laws of mechanics should work perfectly well for that observer as if the same as they would for us at rest on on the shore if you toss a boat it'll land back in your hand as you expect even though to us we see the ball uh trace a complicated parabola to the person on the boat she's a perfect entitled to say i'm sitting at rest the ball went straight up and let it straight back in my hand all the laws of mechanics should work equally well in any reference frame that's moving at a constant speed einstein just takes that existing principle of relativity from mechanics and just assumes just hypothesizes that that should apply not only to mechanics but to every physical phenomenon electricity magnetism optics thermodynamics and that's just a leap he just says what if what if this applies to any kind of physical phenomenon not just mechanics and then he introduces the second postulate which seems actually to be in tension a bit with the first his second passage is the speed of light is a constant independent of the motion of the source what that doesn't sound right if you go back to galileo and watching his friend on the boat if she you know fires a cannon the speed with which we measure the cannonball is different than speed she measures it right for if if you're if you're if you watch her lobbing a tennis ball our measurement of the speed of an object on her reference frame is not the same as ours and einstein here is saying light is special that unlike tennis balls ping pong balls cannonballs or trains or anything else the speed of light as a postulate will be constant for any mode any observer as long as they're in an inertial frame of reference and here's an excerpt from the english translation i won't read all this out but he says what follows from these postulates is that the luminiferous ether will prove to be superfluous uber he doesn't say i've just proven the ether he says it's just irrelevant 50 years worth by that point 100 years worth of study by all of europe's most prestigious physicists was irrelevant that's probably why he didn't have many friends in physics at the time so he doesn't disprove the either he says we just won't even need to refer to anymore it makes the entire set of questions that are driven people like lorenz really kind of fall away at least at least as far as einstein himself is concerned so why does einstein introduce that second possum again historians and physicists uh philosophers have studied this question a lot and we have some good documentary evidence because einstein was writing letters and diaries and stuff all the time and a lot of those have survived so we have some contemporaneous documentation as well as his own later recollections and so on we know that actually 10 years before this paper back when islam was was a mere teenager he'd like to pose these kind of thought experiments or questions to himself and one of the questions he would return to really over the course of a full decade was what would it look like what would you experience if you could catch up to a light wave any reason it would be like a surfer riding along an ocean wave that to us on the shore we'd see both these things moving we would see a dynamical wave moving over time not just not just a frozen waveform in space but to the surfer if she's really moving at the same speed as the wave then she would see the wave frozen in space it should be frozen in time it would be a crest here a trough there but the surfer wouldn't see at least for some extent a period of time the surfer would see the wave frozen not dynamically a frozen wave form a varying in space but frozen in time so einstein said that doesn't make any sense if we think about maxwell's equations when he got a little more sophisticated and learn more about maxwell he said there's no solution to maxwell's equations if you're in a source-free region if there's no clump of electric charge around no electric currents around if both rho and j vanish then there's no way to have spatially varying electric and magnetic fields that are nonetheless static right that seems to be a contradiction so how do you get out of this contradiction of thinking you would have a static waveform of light if you could catch up and move at the same speed as that light wave how do you avoid that you just make sure you can never catch up to the wave how can you never ever catch up to the wave if the wave is always traveling at the speed c even if you're traveling on a very fast train or now you know a hypersonic jet or a spaceship of your imagination that second postulate for einstein we now know was was really the end point of a 10-year series of thought experiments about what would it look like to catch up with a light wave he said that would be that would make so many other things mutually inconsistent let's make sure no one can ever catch a light wave and that's what starts driving much of the rest of his thinking not worrying about you know interferometers and all the rest so what he does unlike lorentz and really all the all the masters in the field at the time uh einstein begins with kinematics with the most with the force-free motion of bodies through space and time which is a very mocking thing to do what can we see we measure objects moving through space and time whereas people like lorenz and maxwell for that matter have been starting with forces with dynamics remember lorenz has this great idea that there's a force from the ether on the matter in the in michaelson's instrument it's all about dynamics forces einstein inverts that order because force is really important but first let's make sure we understand force-free motion kinematics first after all that's that's what could become objects of positive experience and so he has these wonderful quotations is now from from the introduction of the paper it's on the bottom of page two he's really redescribing how to lay out coordinates it sounds ultimately it sounds childish it sounds thoroughly uh unprofessional to the to the folks of the time hendrick lorette says don't you don't bother with with what do we mean by coordinates but einstein was convinced we have to think through how do we describe motion to space and time first and i won't bother reading it out i'll share this slide you can see it but that's really what he's doing in the very opening paragraphs of this paper and that leads him to other follow-up conclusions one of which very famously becomes known as the relativity of simultaneity you might have heard of that before but you'll spend more time at this iap i'm sure if there's no such thing as absolute time if time is what we measure with time measuring devices like clocks if that's all time is is what can be measured by an actual instrument like a clock then how can we measure how can we compare the times in different places after i'm sitting here with my clock which is one thing we could do is trade light signals because by at least according to his postulate number two light is special if i throw ping pong balls that's not so special because we will disagree on the speed with which those ping-pong balls travel if i send a light wave then we could better agree on the speed with which that light moved from point a to point b because of postulate 2. so we start thinking about one of his favorite things in the universe trains he loved trains they come up all the time so he imagines a train moving along a platform at a constant speed a nice inertial frame of reference einstein is standing here on the embankment the train platform he has two friends we can call them alice and bob a and b uh he ellenstein has marked himself out to be in the perfect midpoint of where alice and bob are standing so they mark this all out ahead of time he marks himself perfectly in the middle alice and bob are each equipped with a lantern and a watch and by prior arrangement they say at 12 noon on the dot turn on your lanterns so the light will will travel from both a and b toward the midpoint point your lanterns toward einstein at point m turn them on at the same time meanwhile on this zooming back train there's another uh friend of einstein sitting at the midpoint of the train at the point there's march m prime here she knows that she's the exact midpoint of the train she knows by prior arrangement that alice and bob are standing one train length apart from each other and again she's expecting that they will turn their lanterns on at the same time so how do these different folks describe the series of events that follow let's focus first on what einstein sees when he's standing still on the train platform the embankment watching the train go by the observer who's standing still on the platform at point m was by prior arrangement an equal distance from uh alice and bob i guess i changed it now it's a she so um maleva his wife uh he's staying there maybe m stands for millennial so she receives a light weight from points a and b at the same time so she can conclude that the light flashes were emitted at the same time the the event of person a and person b shining their lights those events were simultaneous how do you know because the light had equal distances to travel and the light could only travel by at but one speed and they were arrived at the same time they must have started the journeys at the same time the act of shining those lanterns must have been simultaneous but what about the person who's move who's on the moving train she sees the following uh as she's zooming along at point m prime she sees the light from point b arrive at her location first now we malevo or i say would say oh that's because you're racing toward it she says not so fast i'm in a perfectly self-consistent mechanical system all the laws of physics work as well for me is for you and the light couldn't have sped up or slowed down by your own postulate she would say she would be entitled to say so if she's equal distance between points a and b and she receives the light from b first the only possible explanation is that the lights were not shown simultaneously that a and b did not turn their lanterns on at the same time so the person of the on the train platform and the person on the moving train disagree about what happens simultaneously this becomes known as the relativity of simultaneity so who's correct i've already said according to einstein no one or both of them because of posture one they're both entitled to work out a perfectly self-consistent set of uh of of laws involving electromagnetism optics and mechanics they agree they are at the midpoint they agree that light didn't speed up or slow down so they are both right which is to say there is no right answer to the question what was really simultaneous simultaneity becomes relative to one's frame of reference that's a pretty deep change from the newtonian system and as as einstein says against bottom of page two very early in the paper we see that we can attribute no absolute meaning to the concept of simultaneity but the two events which examined from a coordinate system are simultaneous like him on the embankment can no longer be interpreted as simultaneous events when examined from a system which is in motion relative to that system okay that's about kinematics he's not about forces or resisted medium or uh or dynamics so now he goes on again very early on in the paper once you get to the relativity of simultaneity again as you'll have a chance to unpack with more patience in the coming days and weeks other strange phenomena seem to follow from that argues einstein one of the first he talks about is length contraction so this had been found and published by lorenz einstein's about to give an entirely different derivation even though quantitatively it's the same form of the equation for einstein has nothing to do with forces it's all about um kinematics how to in general like a good machine he asks how do we measure the length of an object how do we make length an object a positive experience well at the same time measure the location the front of the object and the back of the object and take the difference so if you want to measure the length of the train that's fine just measure where the front is and the back is at the same time and then mark off the difference between those two locations in space our friends alice and bob can do that with the train and they uh find the answer capital l the train is length l long meanwhile uh alice and bob uh that's i should i'm sure that's when the train is at rest when they do this with a moving train alice and bob measure the length of the moving train they measure some some length shorter than what the person on the train was expecting the person on the train expected the full length l the person who's moving with the object either at rest with it or moving together with it says the train is l units long but when alice and bob measure the train by measuring the front and back at the same time they find a shorter answer they found it shrunk along the direction of motion by some specific amount now how could that be remember we disagree about at the same time so if we disagree about simultaneity we will per force we must start to disagree about lengths so the person on the train says you did your measurement wrong you measured point the front of the train first we know that event b happened first she says because we got the light from from event b first so you measured the front of the train and then waited while the train slid and then later measured the position of the rear of the train so of course you have a shorter distance because you performed a bad measurement because you didn't measure the front and back at the same time we say don't be silly they were at the same time we checked by uh trading light beams we know they were simultaneous and we got the answer l prime your train is short who's right sort of both and neither that because we disagree on simultaneity we just agree on on the outcomes of simultaneous events like measurements of length and so he goes on with just a few lines of algebra to find the exact form of what was already known as the lorenz contraction the same factor gamma that you'll be using all iap that the the measurement of an object in motion shrinks along the direction of motion whereas if we were at rest with an object we find a longer length and the amount it shrinks depends on that relative speed v over c depends in particular on this form gamma this has nothing for einstein to do with forces or dynamics it's a totally different derivation to lorenz's it's simply a consequence of kinematics and in particular about simultaneity uh it's a very simple exercise you'll probably end up doing it very soon to find a similar consequence for the rate at which clocks will tick this is now became known as time dilation imagine a very simple kind of clock the simplest one two highly polished mirrors with a light beam that just bounces straight back and forth between them if the clock is at rest with respect to us we hold the clock perfectly still at a fixed height then the time it takes for that light to travel from the bottom mirror to the top mirror will count as one tick of our clock and we know that light can only travel at a fixed speed so it's a really great uniform clock as long as we can hold that height really fixed this is a great way of defining a unit of time and therefore our clock rate what happens if our friend has an identical clock same mirrors held equally rigidly at the same height apart but zooming past us on that moving train then we watch the light have to travel this longer path because but after the light beam leaves the bottom mirror in order to reach the top mirror the whole assembly has moved with the train to the right some distance v times t in the in the time during which that light beam was was in transit so it has to travel the hypotenuse of this right triangle instead of just a simple up and down it travels per force a longer distance and likewise back down so our observation of the moving clock is that it takes longer between ticks that becomes known as time dilation the time between ticks has stretched which is to say the time the clock is running slowly as we measure we measure the moving clock to run slowly by that same factor gamma the time between ticks gets stretched remember gamma is is greater than one so the time between ticks gets stretched and that's the thing that the clock is running slowly meanwhile the person on the train says you're ridiculous i had the clock with me the whole time it kept perfect time but your clock ran slowly i watched as i ran by the station perfectly symmetrical conclusions so einstein was working those all this out very much inspired by uh airsmach by this notion of positivism of what can we hope to measure uh not starting with forces but with with the force-free motion of objects through space and time and really trying to bring it back to what would be measurable what would i see if i were in this situation one of his teachers at the eth a mathematician named heman minkowski had formed a deservedly low opinion of student einstein i used to cut class i was very rude one of their mutual friends after einstein's paper came out sent the paper to minkowski saying you know it's actually kind of interesting to take a look to paraphrase and mankowski said well despite the fact that i knew einstein was up was was never going to amount to anything i read the paper and i was right mccaskey says he really made a mess of this as well so mankowski is just interested enough to redo einstein's work in a form that made much more sense to minkowski who was after all a professional geometer he wasn't just a mathematician he loved geometry most of all so mankowski not einstein is the one who actually brings these pieces together a few years later in it was published in 1908 instead of 1905 and he says all this talk about moving trains and and polished mirrors all that's just a distraction to a properly trained mathematician which einstein was not all we're doing is performing rotations or projections in a certain kind of space in this case a space time in which there's say one direction of space running along the horizontal axis and one dimension of time running up the page we now call these minkowski diagrams in his honor or simply space-time diagrams none of this appears in einstein's early work this was done by his former teacher in response to what mckowski's considered to be einstein's continuing confusion and sloppiness and there's a benefit from doing that minkowski finds he agrees with einstein that we will that we will disagree on certain kinds of measurements for example lengths and durations of time because we disagree on simultaneity but minkowski finds something that had not been so clear to einstein himself there are combinations of those kinds of intervals on which we will all agree there's what we would now call a space-time invariant put together a combination of the time interval minus the space interval a relative minus sign that that comma and each of those quantity squared that combination we will all agree on as long as we're each in states of um of non-accelerating motion even though we disagree separately on the duration between two events delta t or the lengths between them delta x and that for minkowski is what any geometer should do some relationships remain invariant even under changes of coordinates and for mikowski that was what any undergraduate geometry student should know to do first so we actually have this notion of one kind of thing called space time coming from minkowski kind of redoing isis work and as mikowski famously writes in his own article was published posthumously he died quite young and he gave a lecture and it was published soon after he died and he wrote upon introducing this new work henceforth space by itself and time by itself are doomed to fade away into mere shadows and only a kind of union of the two will preserve independence by shadows he really meant projections just drop a perpendicular to the appropriate axis and then think about uh coordinate transformations in that space time of x and t so that's again something you'll get much more practice with uh if you haven't seen it already but that's a preview that comes in response to einstein's work it wasn't denied science owned originally although just to jump ahead very briefly einstein first thought that was horrible because he didn't like mizkowski but then over the next several years the better part of a decade einstein himself became more impressed by that way of thinking and really adopted it for his own later work on what became known as the general theory of relativity that's that's for later okay very briefly here's the third part of the material i want to share with you and then hopefully some time for discussion as well so this this last part is much more brief but how do we make sense of this why was einstein doing all this seemingly unusual maybe even crazy stuff certainly out of step with lorenz with his own teachers like minkowski what how can we account for einstein's quite idiosyncratic approach to what was otherwise a common set of questions about the electronics moving bodies and here's some some of my favorite work by other colleagues by other historians and physicists who really looked at this um in great detail so for a long long time i mean literally for for the better part of 100 years since people began taking relativity seriously which is a long time by now it had become very common to say well einstein must have been motivated much like hendrick lorenz was really like that much of that generation was einstein must have been kind of responding to the null result from michaelson and morley and their interferometer but einstein must have been trying to explain why they couldn't measure our motion to the ether much as lorentz we know very explicitly was trying to do and there's a fascinating one of the first really careful examinations that claim uh was by gerald holton who's a real kind of hero of mine uh holton uh did his first phd in uh in physics low temperature physics and then retooled many years later in the history science and he wrote a series of really remarkable essays about einstein uh once a lot of einstein's papers and letters and notes became available holton was among the first to really dig through these with great care uh and and uh almost like a detective going through the evidence and what what holton concluded was it's not clear that einstein even knew about the michael simoerly interferometer lorenz knew about it and followed it carefully there's very little trace that einstein paid it any attention at all if he did know about it it was kind of in passing it was certainly not sort of front of mind in the months and years leading up to the 1905 paper he might have known about it secondhand from reading through all the footnotes of loretz but it's not something he was he was seems to have been paying much attention to at all and so there's nice textual evidence even in einstein's paper that 1905 paper that i keep showing some some brief excerpts from again this is on on the first or second page when he's trying to to um to reason why we should think about maybe giving up on the ether he mentions without citing any of them he mentioned several unsuccessful attempts to discover any motion of the earth relative to the light medium the ether he doesn't say which ones he means but he actually then goes on to say the same paragraph none of these have found any deviation to first order in v over c and what's so striking as halton reminds us is michael simorley was actually second order einstein seems to be talking about a whole different class of earlier experiments even before michaelson got in the game uh he seems to not to be he's not obsessing over the michael simori interferometer the way other people were and so some people then read holton's analysis to say a kind of over-correction does that mean einstein didn't care about any kinds of experiments at all if he wasn't trying to respond directly to the michael symond experiment was he just often a kind of philosophical dream world and that's where i think we again can now say quite firmly no in fact it makes sense to go back to einstein's own early years his his early fascination with electrotechnical gadgets and remind ourselves what his day job was in the early 1900s including 1905 put it back in the patent office in central europe at this very specific moment so let's think about einstein's favorite technology trains trains and railroads themselves were actually fairly new in the 19th century they were introduced to widespread commercial usage for kind of commercial transportation uh really only in the 1830s and 40s remember einstein was born in 1879 they were relatively new even in his childhood during this whole time until very late in the 19th century there were no coordinated time zones there was nothing like eastern standard time or pacific time or you know central european time each town set its own local time and all the residents could coordinate by agreeing that we'll use that clock tower either you know some municipal building or often a tall church in the town square so that's our local time we'll set our local watches to when that clock you know chimes 12 noon so each town kept its own local time and before railway that was who would ever need anything else because one was rarely encountering more than one town in a given day after the advent of railroads you still had this local time but that became more and more of a problem it was true in north america as in europe and other parts of the world as well so here in closer to to to boston passengers riding on the train between boston and new york during this time would have to change their watches by an average of 37 minutes you know today we're all in the same time zone but that's how different the sort of local variations tended to be just up and down the u.s portion of kind of mid-atlantic sea coast and likewise throughout europe that was becoming more and more of a problem when you have to coordinate lots of trains traveling through across large distances it wasn't only a problem for commercial railway or for shipping it became a very potent added challenge in germany during einstein's youth so again as many of you may know there was there only became one single country of germany a unified germany in 1871 following yet another war with france that was basically the story of the previous 500 years german-speaking lands fighting with french-speaking ones there was another one the franco-prussian war throughout 1870 the german-speaking folks prevailed and it was at that moment in the wake of that war there was then a first unification of a country of germany now a much larger expanse of what of what had otherwise been independent german-speaking territories so now there's an added reason to worry about trains that have to be coordinated across distances there's now a new country's worth that has to defend its borders all the way from basically poland or russia on on the east and france on uh the west and there's a famous quotation from a leading uh german uh general counca who in in the 1890s you know 20 years into german unification is basically saying we have a problem on our hands we have to coordinate trains for military purposes as well as commercial um uh you know passenger rail and merchant shipping we have a problem with the fact that we don't have unified time zones this is all happening in einstein's childhood remember eisen was born in 1879 very early in this period of unified germany and a new focus on coordinating clocks at a distance so one of the main ideas that many clever people inventors came up with throughout europe and other parts of the world as well was to try to coordinate clocks and therefore help coordinate train stations in these different cities that you know europe much of continental europe was now being connected by rail if the train if the major hubs of the major train stations could agree on the time then you could set your watch then and and uh and the trains could be better coordinated throughout even into the hinterlands and the main idea that many inventors and entrepreneurs kind of zeroed in on was to install these so-called motor clocks mother clocks central clocks that you would say were like the standard ones and then connect those by electromagnetic signal either telegraph or increasingly radio waves literally traveling electromagnetic waves in the ether some of you might know that the eiffel tower which is under construction at just this period was originally built to aid radio communication it was a radio beacon before it was a reason for all of us to go visit paris there were other reasons for that the eiffel tower was built really in this period in large part to become a radio beacon to broadcast standardized time signals and then the idea was you would know how far you were from paris you know there'd be some delay when you'd expect to receive the radio beacon if it's 12 noon in paris when that beam was sent you have an offset based on how far away you are in the speed of light that it would take to get there so then these very clever folks who make all these kind of gadgets to both receive the time either telegraph or a radio wave and then kind of um implement the offset and then reset the clock in your next train station clock and down the line so imagine these coordinated sets of coordinates where you have local clocks at every at every uh important location that are sending and receiving electromagnetic signals to coordinate their their local time that's exactly the scenario that einstein mentions in the abstract in the opening pages of his 1905 paper he doesn't talk about these patent applications but or about trains in this more um kind of practical way but it turns out as as my friend uh peter galison uh shows you in this beautiful book einstein's clocks point grace maps this was the subject of a kind of um enormous tech spree this is like the tech challenge of the day for all these very smart young electrical engineers to build up and patent every bit every switch in gear for this way of coordinating clocks at a distance often using uh electromagnetic signals not only that einstein was at the electrotechnical part of the barren patent office where a lot of these were flowing through so he was an examiner on many of the little widgets and gadgets as part of this clock coordination sequence even better my favorite part of peter's otherwise quite fascinating book was it peter and his research assistants pieced together the walk that einstein used to take from his apartment to the office he had a lovely stroll through baron during this time of his life and then finding out when each of the clocks along his route were wired up to this now um coordinated system so literally the clocks that i said would walk past between his apartment and his office were in this exact moment being wired up in this new explicit electro-technical clock coordination system not just across distant train stations but soon even across um the kind of semi-urban regions where he lived it was literally his day job and and the path that he worked and walked to get to work so this notion of using electromagnetic signals to coordinate clocks in a kind of measurable repeatable way was of philosophical interest it was certainly inspired in part by uh the writing of air smock but einstein was also immersed in a different set of realities than someone like uh hendrick lorenz or the other maxwellians and so when it comes to the question of the electronics electrodynamics of moving bodies einstein's immersed in different kind of conversations and day-to-day questions compared to those of um the kind of physics elite of his day so we come back to this very um iconic but kind of unusual paper from 1905 on the electrodynamics of moving bodies i think we can make a bit more sense of it informed by work by uh people like peter gallison the paper that i say is paper has almost no references much like many uh patent applications you want to sort of emphasize priority and downplay precedence default outside focuses on the operational details how would you actually perform these measurements and compare the answers at some distance it starts to look sort of like a patent application not like a fancy exercise in mathematical physics so we can come back to this question of why was einstein doing things differently what was he up to we see him really enmeshed in just a different set of of ideas of philosophical conversations of mathematical techniques and also of actual gadgets compared to some of the other experts of his day you |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 74_Galaxy_Travel.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we're going to build on-- we just learned about the relativistic Doppler effect and redshift. So we take on traveling through the galaxy from here, from Earth, towards the center of the galaxy. The situation is as follows. We have Bob, who's stationary on our planet, Earth, and Alice, who makes use of a new spacecraft. This spacecraft is able to travel with a velocity of 0.99999998 times the speed of light. So that's really fast. It corresponds to a gamma factor of 15,000. Now, the center of the galaxy is about 30,000 light years away. And in Bob's reference frame, this journey will take about 30,000 years, because velocity is about the speed of light. For Alice, however, the journey will only take two years. So it's quite doable. The question, now, is what does Alice see? Literally, what is she going to see while she is looking out of the windows of the spacecraft? Is the picture similar to the one we see in some of the movies, where on the horizon, there's lots of stars, and once the spacecraft accelerates, you see those dots kind of blurry, coming towards us, right? Or is the situation somehow different? The starlight has a wavelength of about 600 nanometers, and the cosmic microwave background a wavelength of 1.06 millimeters. So how is Alice going to observe those two light sources in her travel? So I invite you to work this out, but also think about the next question. How long does it take for Alice to accelerate from 0 to her velocity with an acceleration of 10 meters per second squared, which is 1g, which is very, very doable? OK. So I invite you to stop the video here and work out those numbers to get a feel and speculate a little bit about how this journey is actually going to look like. So here's the solution. So the light's moving towards us, so it's going to be blueshifted. The velocity is given here. And with beta, we have seen that redshift or 1 plus redshift is equal to the emitted wavelength divided by the observed wavelength. And you find that that factor is 10,000. So we just have to divide our emitted wavelength by 10,000 and find that the observed starlight has a wavelength of 0.06 nanometers, which is X-ray. So she's going to be flooded by X-rays of light coming from the stars. And similarly, the observed cosmic microwave background is going to be about 106 nanometers, which is ultraviolet light. The ultraviolet light-- there's a spectrum to this. So what she's going to see is X-rays, which she can actually not see with her eyes. But she will be able to see the ultraviolet, or some part of the spectrum, as kind of a blurry, fuzzy kind of background all over the place. So the situation is actually different from what we just saw in this picture. A few more fun facts about the cosmic microwave background. It's actually at a temperature. So the spectrum of cosmic microwave background, those photons, they correspond to a spectrum emitted by [INAUDIBLE] which corresponds to a specific temperature of 2.7 kelvin. That is the temperature of our universe. This temperature was about 3,000 kelvin about 380,000 years after the Big Bang, the age of the universe at the time. And so then at that time, this corresponds to visible light. But at that moment, the light stopped interacting-- well, stopped-- The likelihood for the light to interact with something out there in the universe became so low that it just stopped interacting. And then the frequency changed, because the universe was expanding. So what we are seeing today is kind of a relic of the universe at that time, at 380,000 years after the Big Bang. And if you study the cosmic microwave background with some more precision, you see that there are actually fluctuations which can be analyzed. It turns out that you can correlate those fluctuations-- the fluctuations of the energy density 380,000 years after the Big Bang-- to this, the present of today's stars and galaxies and galaxy clusters. So those energy fluctuations, they served as seeds for the formation of galaxies and galaxy clusters. Quite interesting. Today, we have about 400 of those photons-- microwave photons-- per square cubic centimeter. So there's quite a busy environment around here. So like, this little cube has about 400 of those photons. Now, this is a spectrum as well. It's not just a monochromatic background, but it's a spectrum which corresponds to these temperature [INAUDIBLE] All right. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 105_Decay_of_a_Pion.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we want to study the decay of a particle, in this case, the decay of a pion. The pion is a particle which consists of quarks and antiquarks, which are bound together by gluons. They're part of a family of particles which are called mesons. And they can be charged and neutral. So we have positively-charged, negatively-charged, and neutral pions. In this example, we look at a neutral pion decay and the specific decay into an electron and a positron. Mostly, neutral pions and decay into a pair of photons. But we study this effect here because it's more fun. The lifetime or the mean lifetime over pi 0 is 8 times 10 to the minus 17 seconds. So when we produce neutral pions in our detector, they immediately decay, as I said, mostly into a pair of photons. That we discovered in the 1940s. So let's get to it. So the mass of a pion-- of neutral pion is 135 MeV. So it's much heavier than an electron, which is 500-- has a mass of 511 KeV. So we're looking at this decay here. We have a pion a rest into an electron and a positron. And so the charge for you now is to find the gamma factor of the electron or the positron or, [? with ?] that, the velocity of those particles and the decay of a pion at rest. So again, as usual, stop the video, and try to work this out. And again, what we want to do is just write down the four-vector of the particles involved. We start with the pion, which has an energy of the pion mass times c square. And it's at rest in this example here, which means that the momentum is 0. And so then the outgoing particles are the electron and positron with their energy and their momentum. And then just from this first line here from the energy relation, you can-- by knowing that the mass of the electron and a positron are the same, by knowing that the gamma factor is the same that comes out of the momentum relation here, that they have the same velocity, you can just simply find gamma as equal to the pion mass divided by 2 times the electron mass. And that is about 132. So again, we studied the decay of a pion. And as a general example of the decay of a particle into two new particles, we used the energy momentum relation. We make sure use of the [? effect, ?] in this case, that we are in the rest frame of the pion. And then we are able to get to the velocity or the gamma factor of the outgoing particles. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 125_General_Relativity.txt | MARKUS KLUTE: Welcome back to 8.20 Special Relativity. In our quest to understand how we get to general relativity, there is two things to consider. The first one, this lecture is not meant to give you a full description of general relativity, but just a view into where this might lead, where this discussion might lead. So in this quest, we can understand the theory of general relativity as a theory on how to patch together the different reference frames which each can be described in special relativity, in the framework we discussed up to now, and it's valid in short intervals in spacetime. Consequences of general relativity are that spacetime is curved. So we have modified geometries. We learned that, because of gravitational effects, matter curves spacetime. As a consequence of that, there must be modification of gravity based on matter distributions, and so there must also be gravitational waves, gravitational lenses which bend light, black holes, and there's cosmological predictions coming out of general relativity. So let's have a discussion first. What does it mean to have a changed or modified geometry? What could that mean? So you are all used to Euclidean geometry, where, when you draw a triangle, you add up all the angles to 180 degrees. If you try to parallel lines that never cross, they also don't diverge. But if you have a modified geometry-- for example, the geometry on a sphere, like on our globe-- the angles do not add up to 180 degrees. Actually, the sum is larger than 180 degrees, and parallel lines will cross. We will call this kind of space positively curved, but you can have the opposite example, like on a saddle. So you can have other spaces and other curved spaces, and they can be negatively curved. In this example, if you add up all angles, you find they add up to less than 180 degrees. Parallel lines do not cross, but they will diverge. Mass changes the geometry of spacetime. We just talked about light bending, and because of the change in geometry, light will not go on a straight line anymore, but will bend around massive objects. Spacetime is curved. Geometry of spacetime tells us how the mass is moved. You can think about a trampoline. When you put a heavy object on a trampoline, all the other objects on the trampoline will gravitate towards the heavier object, and that's kind of a picture on how spacetime actually looks like. Einstein used those findings in order to redefine Newton's first law and found the so-called Einstein field equation. So on one side of the equation, there's a description of spacetime and its curvature, and on the other side of the equation is the energy momentum tensor, the description of how energy and momentum of object is distributed. And those two things, spacetime and energy and momentum, they're kind of interlinked in this equation. So if you read this description, you can read it from one side to the next. Spacetime tells matter how to move. Or you read from the other direction, say matter tells spacetime how to curve. That is an equation, and you can just read it from the left to the right or from the right to the left. Our understanding here. It says space and time are not fixed things through which matter and energy moves through. The matter and energy themselves define spacetime. And matter, because of spacetime, is dynamical. It's changing. It's interacting with the matter and with the energy. This is a super exciting picture from Hubble, the Hubble Space Telescope. And you see galaxies, but what you also see is those structures which looks like the light has come through lenses. Those lenses are actually matter distributions, galaxies, which actually lead to the bending of the light and those lensing effects. OK. If you want to summarize general relativity, you can first say that spacetime is curved and it follows the pseudo-Riemannian manifold with a specific metric. We have seen the metric before. It's minus, plus, plus, plus. And the relationship between matter and curvature is given by the Einstein equation, and here I give you a slightly different form where there is the dynamics, again, on one side and the energy momentum on the other side. Let's just look at one example here. So we discussed, in special relativity, invariant intervals, and we have this delta squared, or we have a different name for it. I given by minus dt squared plus dx squared plus dy squared plus dz squared, and we could have just written this in polar coordinates as well, where you find that dr squared and r squared d theta squared, and then r squared sine squared theta d phi squared. OK. Same thing. It's just a different coordinate system. So as a solution to Einstein equation, we find something which looks very, very similar. That's not a surprise, as we find general relativity as a patchwork of small spaces of special relativity. So the solutions might be very similar. And the solution found here, the so-called Schwarzschild solution, which is a unique solution in vacuum with spherical symmetry of a matter distribution. So you have a spherical matter distribution like our sun, and this is a solution which describes spacetime around this. You find this invariant interval here has two interesting features. There's two singularities in here. This should be a minus 1. You find those two singularities. One is at r equals 0. That's kind of expected. In the middle of the mass distribution, this thing is not defined anymore. There's no mass left. But there's also a second singularity at 2GM. This is called the so-called Schwarzschild radius, and if you get to the singularity, you basically don't define anymore this invariant interval. You can think about the surface of a black hole as this singularity. At this r value, at the singularities, everything becomes timelike, or everything within the radius becomes timelike. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 91_Momentum_Conservation.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20, Special Relativity. So in this section, we want to look again at collisions and study momentum conservation. And so we have this scenario here in which we have two balls colliding with velocities uA and uB, mass mA and mB. And after the collisions there, the mass is changed to mC and mD. And the velocities are uC and uD. Momentum conservation tells you that the product of the mass and velocities and the sum of the two particles before and after the collisions is the same. But now what happens if you boost the system? If we look at the very same system with the boosted reference frame, and we can just simplify the case here by just considering the x direction. So the question is if momentum is conserved in a frame s, like the one we're looking here in this picture, is the momentum also conserved in a moving reference frame with relative velocity v? And so you can show this quite easily, that this is actually not the case, right? So you write down the velocities. The momentum equation is the same as before with boosted velocities. And you find that this is not the case. You can easily show this by, for example, setting the right part of this equation to 0, and see whether or not this equation will hold true in general. And it doesn't. But in the last section, we introduced the new concept of proper velocity. So how about redefining momentum through proper velocity and just saying mA times proper velocity A plus mB times proper velocity B is equal to mC times proper velocity C, and so on, and see whether or not we can learn something from this equation? So why don't we write down this very same equation and with proper velocity and see whether or not it's invariant in the Lorentz transformation? OK. Again, a good moment to stop the video and just work out the math by yourself, on your own. So I did this here. And I'm just doing this for the x component. So we have our proper velocity vector, which is gamma times C, gamma times uX, gamma times uY, and sometimes uZ. And so the x component is the first component. And in the boosted reference frame, the proper velocity of the x component in our boosted frame is gamma times the proper velocity of the particle A, first component, minus beta, upper velocity of A0's component. And then you can write your equation. And this should be our momentum energy, momentum conservation equation. OK. And the Lorentz transformation? We find this one here. And so all we need to do now in order to show that this is always true, or true in general, is to reassign, relabel, reorder the individual terms. And I did this here to make this visible. You see everything behind, in this bracket behind the beta, is equal to the equation we had before with the minus sign in between, which means it's 0. And everything we see on the top of the first component, it also be 0. That's for the boosted reference frame. So we see here, and it just shows this for the x component. You can show this for all components that momentum is conserved. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 121_Equivalence_Principle.txt | PROFESSOR: Welcome back to [? A20, ?] special relativity. Very early in this lecture, we discussed what mass is and how we can understand the mass of a proton and the mass of an electron as it interacts with the Higgs background field. But here, the question is slightly different. You want to understand the difference or the lack of difference between the mass in a gravitational field compared to the inertial mass, a mass being pushed through some force. And it's probably Einstein's biggest realization that those two things are the same. And that finding is called the equivalence principle, the equivalence of the gravitational and inertial mass. So if you are a freely falling person, you will not feel your own weight in your gravitational field. You will just drop. And the gravitational field provides an acceleration. And that acceleration, if you are standing in an elevator or just sitting on a chair like I'm doing right now, then sets up an accelerating reference frame. So there is an equivalence between being accelerated or being stationary in a gravitational field. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 123_Bending_of_Light.txt | MARKUS KLUTE: Welcome back to special relativity. Bending of light is one of the spectacular consequences of general relativity, a prediction, if you want, [? of ?] [? general ?] relativity, which was experimentally confirmed. But let's analyze the situation first before we look at the actual evidence. They use Alice and Bob again, where Alice is an elevator operator. She's stationary and operating this elevator. Bob is a passenger and riding within the elevator. So Alice is injecting a light beam and watching this light beam in the elevator from outside. And the light beam, as you would expect, is going in a straight line. We'll see pictures here show the elevator at three different times-- [? t1, ?] [? t2, ?] and t3. For Bob, the very same situation looks completely different. He sees the light entering the elevator. And then after some time, the light pulse is maybe on the height of his head. And after some additional time, he sees the light at the end-- or at the bottom part of the elevator. So if you would draw a line on the back of the elevator where he was able to observe the light pulses, you could draw this line. So for Bob, who's stationary in this accelerating reference frame, light is bending. As we discussed in the first section in this chapter, there's an equivalence between accelerating and the gravitational field causing an acceleration. And so you can use heavy objects, like the Sun, in order to bend light. And this led to the first observation of the rotational bending of light. And so in order to do this, we want to have a star very close, or starlight passing very close to the Sun. And because the Sun is very bright, you want to use solar-- a total solar eclipse in order to test this effect. And this was first achieved in 1919 with a solar eclipse in Brazil, also at the West Coast of Africa, by Arthur Eddington. And so this then led to Einstein really becoming famous. I talk about this later. You might remember the last partial eclipse here in Massachusetts in 2017. I have a vivid memory of this, showing this to my kids up into the sky, obviously, with proper eye protection. So the idea is, again, that you have a star, and light passes by the Sun. And because of the gravitational bending of the Sun, there's a bending effect. So there seems to be an offset of the actual start position due to the bending. And this effect, again, was discovered-- or was measured in 1919 by Arthur Eddington, and let's-- maybe because of an article in The New York Times to the fame of Albert Einstein. So at the time, there was no science reporters for newspapers. So a former sports reporter reported on the scientific endeavor. And the way he writes about it is rather interesting. You might want to read the entire article. But here's just the headlines. And it read like, "Lights All Askew in the Heavens," "Men of Science More or Less Agog Over Results Of Eclipse Observations," "Einstein's Theory Trumphs," Stars not Where They Seemed or Were Calculated to Be, But Nobody Need to Worry," "A Book for 12 Wise Men," "No more in all the world could comprehend it," said Einstein, when his daring publisher accepted it. So, again, this populous writing made Einstein very popular. He was then later invited to come and visit to the United States and made a tour, which he also used for political reasons. But, really, it made him very, very famous-- made him a pop star of the time. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 81_Algebra_of_Lorentz_transformations.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20 Special Relativity. So we're starting a new chapter. In this chapter, we talk about some aspects of special relativity, which are not crucially important to understand the concepts, but they help you to go a little bit deeper in your understanding. I hope this is going to be useful. So we want to talk about the algebra of known transformations. So we have seen that our gamma factor is 1 over square root 1 minus beta squared with beta equal the relativistic velocity v/c. And thus, this we can rewrite it as gamma squared minus beta squared gamma squared is equal to 1, OK? So now, I would like you to recall hyperbolic functions sinh and cosh and cosh squared minus sinh squared is equal to 1. So the form here and here are pretty much the same. And something squared minus something else squared equal to 1, OK? Good. So let's see how this looks like. As a reminder for us, the hyperbolic functions as defined as 1/2 e to the x power minus e to minus x and cosh equal to 1/2 e to x plus e to minus x. OK? The tangent is then defined as a ratio. And you can plot those functions, and you can see the functional form as given in these two diagrams. Well, we want to come back to those two equations looking very much the same. So we can define now eta, the rapidity, as gamma equal to hyperbolic function cosh eta and beta gamma equals sinh eta. So basically, we have this rapidity, which is a measure on how much the system is boosted as being equal to this kind of hyperbolic angle, right? You can then process again, where beta is equal to the tangents of this hyperbolic angle. And just remember that the slope in our space time diagram is 1 over the velocity. We find that angle again now is being called rapidity, OK? And just as a reminder, beta goes from minus 1 to 1, depending on the direction and the speed of it is less than the speed of light. And then eta goes from minus infinity to infinity. OK, so then we can rewrite our Lorentz transformation. Instead of writing gamma and beta gamma and minus beta gamma and so on, we can write this through the hyperbolic angle. OK? So you should always ask why is this useful. The first part is that when we add velocities, we found this complicated transformation where the new velocity is equal to the first velocity times the second velocity over 1 plus the product of the two velocities. And this is much easier now as we can just add the velocities. So the third velocity is equal to the first plus the second. This is much, much easier to actually calculate. And the proof of this is coming directly from the proof of those hyperbolic functions here. The second part where this becomes useful is when you think about the angle in your space time diagram. How does this now compare to a normal rotation? So let's start here. So we have a normal rotation. We have a rotation at a angle, and our coordinate system just rotates by specific angle. Let's call it phi here. And what we do now, we have a similar but a hyperbolic rotation in which the coordinate system in our space time diagram rotates. All right? In the normal rotational case, x squared plus y squared is invariant. And in our Lorentz transformation, c squared t squared minus x squared. All right? If you then have a more general transformation, a rotation, and Lorentz transformation, you find x squared plus y squared plus z squared minus c squared t squared [INAUDIBLE]. OK, so we have just relabeled things, but now we can make use of everything we know about hyperbolic functions when we think about adding velocities. Because the rapidity-- the relative distance and speed between two reference frames is basically the angle of the hyperbolic angle. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 103_Deuteron_Production.txt | MARKUS KLUTE: Welcome back to A20 special relativity. In this section, I am going to talk about the deuteron, which is one of the two stable isotopes of hydrogen. The atom is called deuterium. The nucleus actually contains one proton and one neutron. So it's quite simple. And so we can use this as an example to understand the concept of binding energy and how we can create objects of heavier mass or nuclei of heavier mass. We can do this by bringing protons and neutrons together. And when we do that, they have a likelihood of binding together to deuterium. And they release an energy in form of a photon of 2.3 mev So the way to understand is that the combination, the bound combination of the protons and neutrons have more favorable energy state, which is visible in this nuclear potential here. So the deuteron lives as a combination of the proton and the neutron here. In order to now split up the deuteron, which is stable, we have to add energy. We have to add at least 2.3 mev of energy in order to release the proton and the neutron and make them free. The natural abundance of deuterium in the Earth is rather limited with 0.0115% in the Earth. But, nevertheless, deuterium is extremely useful and extremely important in the evolution of the universe, in the synthesis of more heavier elements, as it's kind of a part of the chain, which allows the creation of helium. And then helium can be used in order to create even heavier nuclei. To give you a sense in the numbers, the mass of the neutron is 2.01355 in atomic units. The mass of the proton is about one. The mass of the neutron is a little bit bigger, but also about one. So in order to now get the binding energy-- so the energy, which is kind of stored in the deuteron, when it's binding together, the proton and the neutron can be accessed by adding the proton, and the neutron mass, and subtracting the mass of a deuteron. Again, that's a minimal energy needed in order to free up the proton and the neutron. If you add more energy, then you actually also give kinetic energy to the proton and the neutron in this reaction. The deuteron was discovered in 1934. And it's fundamental also in production of hydrogen bonds, which were produced about 20 years later. So the deuteron itself is a rather important nuclei. Again, as I said before, it's fundamental. It's stability is fundamentally important in nuclear synthesis after the Big Bang. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 44_Invariance.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. Let's start here with a short summary. We have seen through experimental measurements that there is no ether, that electromagnetic waves travels through vacuum. We have discussed the concept of the relativity of simultaneity, meaning that two events might occur simultaneous to one reference-- in one reference frame, to one observer, while they're not to another. In the last two videos, we looked at clocks. And we've seen that moving clocks run slow. We've also seen that moving objects appear smaller. They're lengths contract. We found that the time of a moving clock is related to the time in the clock at rest with a gamma factor because it's time dilation. And, for the length, we have seen that there is a 1 over gamma dependency, length contraction. The entire discussion was based on Einstein's postulates. We simply used Einstein's postulates. And then we looked at experiments of clocks. Now, you might argue that the setup of the clock is actually what's tricking us here, but I can tell you that is not the case. As you have seen for the muon, the muon doesn't know about optical clocks. It just decays based on its own properties. I think it's fair to say that Poincaré and Lorentz came to similar conclusions about the same time as Einstein did. Delta t at rest and delta L at rest, the time and the length, are also sometimes called the proper time and length. In German-- and I actually prefer this a bit-- we use the word eigen, which means own. So it's basically the time, the own time of the object, the object being addressed. So we can now ask, [INAUDIBLE] seen time is suspect. What is not suspect? What are the observables which are invariant? And, by this, I mean the observables, when we have two different reference frames and we have a conversation, we do actually agree in a conversation about an observation we have. We have seen we cannot agree on time, and we cannot agree on the length in the direction in which we are moving. So can I ask, for example, what happens to the width or the height? If I put a train on a train track and it's fast moving, is that contracted or even expanded or changing at all? The answer is no. Similarly, if I put a train track on a track and I go very fast into a tunnel, does the height of the tunnel change? Also here the answer is no. And we can verify this later quantitatively. In summary, transverse dimensions are not affected. They are not suspect. We can agree in a conversation of two people in different reference frames about the height and the width of a train. That's good. But what else is invariant? So here I want you to consider the time and distance between two events or maybe even three events observed from different reference frames. And we introduce or reintroduce our characters Alice and Bob and add Carol to this. So we're going to have three reference frames of three observers in this discussion. So I want you to look at this one here. So assume that Bob has a clock, and it's moving with a velocity v. And Alice is observing Bob's clock. We have done this before. Now, we want to also add Carol to this. And Carol is moving with three times the velocity and is also observing Bob's clock. What I want you to do is look at this property here. So we have seen that the height is invariant. So let's look at what happens if I calculate 2 times the height squared, and I use x and t, time and space, in order to express the height, all right? Again, this is an opportunity to stop the clock, stop the video, and work this out on a piece of paper. So it turns out it's not that hard. We basically find that 4 times the height squared is equal to-- maybe it is hard-- c squared times t squared minus x squared. And, since the height is an invariant, this property, c squared, the speed of light squared, times the time squared minus x squared, is invariant. So we can think about them as delta t and delta x. When we look at the difference in time and the difference in space between two events-- Professor Klute entered the class. And, at the other end of the class, Professor Klute exploded. If I do the delta t and the delta x between the two-- we square them and subtract them-- that is an observation we can all agree on. Whether or not you're stationary in the classroom or you're passing by really quickly with your spacecraft, that observation is something we can agree on. And it's invariant. This property is called the invariant interval. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 51_Voyager_Program.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, you're going to take a journey through our solar system and beyond. 1977 marks the first flight of the Voyager program. Voyager is an MIT-led program with the original mission to study planetary systems. Now, Voyager 1 and Voyager 2, two spacecrafts, are studying the interstellar space. Those two probes are still sending back now, more than 40 years later, information about measurement of their surroundings back to Earth. Those are the two furthest-away objects we were able to send. What we have here in 8.20-- and it's IAP, so we are a little bit more ambitious-- our mission is to bring us to Alpha Centauri. Alpha Centauri is the closest star and planetary system to our solar system. It's actually not just one star, but multiple stars. And our journey brings us to Proxima B, which is a planet in the Alpha Centauri system. So we have a very fancy spacecraft. It's able to travel with a constant velocity of 0.943 c, speed of light. The gamma factor is 3, and the path which it's going to take is 5 light years long. Alice is our ground control. She is going to stay on Earth. And Bob is going to take this journey with Virgin Galactic. After some time, Bob is going to arrive on Proxima B. So your first challenge is to now calculate how long does this trip take from Alice's and Bob's perspective? So I ask you to stop the video here and just work this out. So for Alice, the time needed is 5 light years divided by 0.9, by the velocity of 0.943, times the speed of light. And that gives us 5.3 years. So after 5.3 years, Alice is observing Bob's arrival on Proxima B. For Bob, the length of this path is Lorentz contracted-- 5 light years divided by 3, which is the gamma factor, which results to 1.67 light years. So for him, the journey takes only 1.77 years-- 1.67 light years divided by 0.943 times the speed of light. So apparently, the time experienced by Alice and Bob for the very same journey is different. So this now is the first part, where we enter discussions of paradoxes in special relativity. And this will bring us, ultimately, to an understanding of the twin paradox later in this course. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 132_Course_Review.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20, Special Relativity. So in this section, we're going to review the content of the material, but underlined with a few questions and examples, very similar to the previous section. It's just this one is interleaved with activities. So let me start by bringing back two of those Einstein quotes, which nicely relate to each other in a sense that they feed off each other. The first one is, "I have no special talent. I'm only passionately curious." Albert Einstein. And the second one, "It is a miracle that curiosity survives formal education." And I [LAUGHS] sincerely hope that I didn't stop your curiosity with this lecture-- quite the opposite. Like we discussed here, it's just the starting point of a wider discussion of general relativity in your education at MIT in the Physics Department. You could learn about quantum mechanics, quantum field theories. And a lot of physics is out there, which is super exciting and interesting. So it basically needs your curiosity in order to tackle challenging questions in physics today. We started the discussion looking at the background which led Einstein to make his discoveries. And specifically the year 1905, in which he was able to come out with five papers, all breakthrough papers, including the theory of special relativity. His career didn't stop there. He developed the general theory of relativity and published a paper on this in 1915. And his fame as a physicist really comes out of the predictions he made at that time. So we set the context of this class, and we started with a question of Galilean transformation and whether or not you can tell in a moving train car whether or not this is actually moving or stationary. And we demonstrated that time and acceleration is invariant in the Galilean transformation. And then, therefore, since you cannot distinguish the strength of a force in two reference frames which move to each other with moderate velocity, you cannot tell whether or not the train car is moving or not. Very important are the development of clocks and times and signal processing of the time. And we discussed this in the context of trains and train lines, but also in Einstein living in the city of Bern, with a large number of clock towers which needed synchronization. And even working in the patent office certainly confronted him with those questions all the time. So time is suspect. That is really the key to moving out from Galilean transformation into Newtonian mechanics to special relativity. When we classify or when we look at specific series, we have to understand that they live within a context, within a range of validity. And so classical mechanics is not wrong because it breaks down at large velocities. It's just only correct in the frame of slow velocities. And special relativity also has its limitations, as in it only describes reference frames or scenarios in which there is no acceleration between two reference frames. Going back to the question of the time, Michelson-- and a number of other experiments leading to the very same direction-- was trying to establish that there is an ether wind, a medium in which light is moving. And his experiment, at the time, failed to demonstrate that ether actually exists. His experiment used a light source and a couple of mirrors in order to show interference patterns. And those interference patterns didn't manifest themselves. So he thought for a long time that his experiment is limited or that he has made a mistake. But it turns out that ether indeed doesn't exist. Einstein tackled this problem by making two postulates. The first one is the principle of relativity, that there is no preferred reference frame if you want; and the second, that the speed of light is constant-- and constant and the same in all reference frames. And in this class we use those two postulates in order to derive everything we know about special relativity. So we looked at the implications. And the implications started from time dilation, length contractions. We were able to derive the Lorentz factor and Lorentz transformation. And we did this by showing this light clock here, where you observe a ticking clock in which there is two mirrors and light bouncing back. And each time there is a bounce, we count this as one tick of the clock. So if the clock is moving, the light has to travel a longer distance. And hence time is delayed. And from just the geometry of this problem, we were able to derive this gamma factor here-- 1 over square root of 1 minus beta squared. The beta is velocity over the speed of light. So as the first activity in today's class, I want you to think about a clock which is moving with a photon, a clock which is moving with the speed of light. And also discuss why isn't it possible to go faster than the speed of light. And why can you not just keep accelerating? So think about this a little bit. In the live class we will have a discussion. But just come up with some sort of answer of why this is the case. So here we have, in the class, actually showed that there is a real speed limit, that if you try to go faster, past [INAUDIBLE] velocity, you run against a boundary, a real speed limit. If you think about keeping accelerating, giving more energy, you find that the amount of energy you need in order to go faster and faster, faster, doesn't get you to velocities which are faster and faster. And again, you enter a speed limit-- the speed of light. An important topic in understanding some of the paradoxes in special relativity, and some of the confusion, is the concept of the relativity of simultaneity. Now, it can be illustrated quite nicely in this example here, where you have a carriage train car with light being emitted and clocks which record those events at each end of the train car. For the stationary person, those clocks will tick in sync. They will always show the same tick and the same time. But for somebody who's observing this train from a platform, or somebody who's moving with a relative velocity towards this train car, you will see those clocks not ticking at the same time. So the clear evidence is given in this picture again, where you see that the light's being emitted in the center, but one side of the train car is hit first and the second side is hit afterwards. So you see that the leading clock lags. The leading clock in this example lags behind. And so what you find here is that events which are observed simultaneously for one observer-- in this case, the person inside the carriage-- they will not be simultaneous for an observer who's moving with a relative velocity. And that led us to the understanding of the pole in the barn paradox, where, in one example, the event of the front of the pole hitting the back of the barn and the event of the back of the pole hitting the front of the barn, they are simultaneously for the barn owner. But they're not simultaneously happening for the person who's carrying the pole. In this case, the event of hitting the back of the barn is simultaneous to an event where the back of the pole is still sticking out of the barn. So there is a clear disagreement. But the disagreement can be resolved by understanding that simultaneous events are not necessarily simultaneous to two observers. Then we moved on to a variety of other paradoxes in special relativity. And the most famous likely is the twin paradox, where we discussed that a person moving away and then returning is younger than the person who actually stayed at rest. And we discussed that we were able to use time dilation or length contraction in order to quantitatively figure out the difference in time. But we also discussed that the person who is moving away and then coming back needs to describe the journey in two different reference frames. And from the fact that you don't consistently can describe this sequence of events as two reference frames can see the paradox and the extra confusion. So here we have another activity-- an asymmetric travel. So we discussed also the example where two people move away and then they come back in a symmetric fashion. But here we want to discuss the case where there are three trends. Carol stays on Earth. Bob moves to Star 1. And Alice moves to Star 2. The distance to Star 1 is longer than the distance to Star 2. So the question is, in this journey, they both start and they both return at the same time for Carol. But which of the twins is the youngest? So again, I invite you to just work this out. You can use some numbers if you want a quantitative answer. Or you can just reason about [INAUDIBLE].. The answer here is that Bob is the youngest of the three once they return to Earth. And the reason for this is the distance he has to travel is the longest. Hence the velocity he has to travel in is the largest. And hence the effect of time dilation for him is the biggest. And therefore he's going to be the youngest of the three. All right. We had a rather long discussion about waves and light, Doppler effect, and relativistic Doppler effect. Here, just as a reminder, the wave equation for an electric field in a vacuum. And the solution to the wave equation is use the second derivative with space and time. And the solution simply can be expressed as a cosine, which is a function of space and time. We have talked about light quite a bit. And just as reminder, [INAUDIBLE] the energy of photon is related via the Planck-Einstein relation to the frequency. So the higher the frequency, the higher the energy. The higher the frequency, the higher the energy. And here, in this picture, you can see the effect of the Doppler effect, where, when you have a moving source, the observer sees the waveline modified. Objects which move towards us are blueshifted, starting from white light, or green light in this example. And objects that move away from us are redshifted. The effect can be used, for example, in speed measurements of cars. It can also be used in order to measure speed or distances of stars moving away from us. And so that defines, then, the concept of redshift, which is simply the ratio of the difference in wavelength divided by the wavelength as observed by the observer. So here there's two concept questions. The first one is, is the wave equation, which you can see there as an example, invariant under Lorentz transformation? And the second question is, how about the solutions? Are the solutions to the wave equation invariant under Lorentz transformation? So I'll have you work this out again. And the answers are yes and no. The wave equation is invariant. The wave equation describes the physics. It explains how electric and magnetic fields change. And the laws of physics need to be invariant under Lorentz transformation. Otherwise they will not be valid. They will violate the postulate we just made that all reference frames are equal to each other. However, the solutions of the wave equation-- light itself-- is not invariant under Lorentz transformation. We've just discussed redshift and blueshift, which means that the wavelength and frequency of light changes with respect to the observer, or for each observer. So the solutions-- light-- are not invariant under Lorentz transformation. And then we went a little bit into particle physics. And I have to apologize for my own preference. But elementary particles, as they have been reproduced or observed, are typically moving at rather large velocities. So they are very good examples to study effects of special relativity. We looked at energy, the total energy m0 gamma c squared, which also can be expressed as the energy of the rest energy of the particle, m0 c squared, plus the kinetic energy. And we looked at the total energy as being invariant, one of those invariants, as equal to the total momenta squared. The total energy squared is equal to the total momentum squared times c squared plus the rest mass squared times c to the fourth power. And then we went through a larger number of examples, from accelerating electrons to composite particles. We talked about deuteron photon absorption and emission, the creation of particle, the creation of antiparticles, and the scattering of particles. So here we had another example. Oops-- without the solution. In 1995, at Fermilab, a proton-antiproton collider, the Tevatron, top quarks were discovered. And we measured the top quark mass to 175 GeV. The center of mass energy at the Tevatron was 1.8, and later almost 2 tera-electronvolt, and clearly sufficient for the production of top and antitop. But what is the minimal energy in order for this process to occur? And here we went through a number of examples. The minimal energy-- sorry. I have to work this out again. The minimal energy required can be derived or extracted in the center of mass frame, where the top and antitop are produced at rest. And if you do this-- in this example, the proton in this collider experiment-- the experiment is already conducted in the center of mass frame. So the minimal energy is simply 2 times the top mass times c square, or 2 times gamma times the mass of the proton times c square, which gives you a gamma factor of 175. But the likelihood to actually observe a top quark and a antitop quark at that energy, 175 GeV proton or antiproton energy, is rather 0. And the reason for this has to do with the structure of the proton. The actual interaction between the proton and the antiproton is such that the quarks and antiquarks inside the proton, and also the gluons, interact. And they only carry a fraction of the momentum and the energy of the proton. And hence this minimal calculation is insufficient to get a sufficient cross-section likelihood for top quarks and antiquarks to be produced. But that is particle physics and goes beyond the scope of this lecture. One last point, which leads sometimes to confusion, is the concept of conserved and invariant properties. When we look at the meaning of the word, invariant means never-changing. And in the concept of special relativity, properties are invariant when they do not change under Lorentz transformation or Galilean transformation, as we discussed earlier in the class. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 72_Relativistic_Doppler_Effect.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we're going to talk about the relativistic Doppler effect. And we make good use of our space-time diagrams, which we discussed earlier. So the situation is as follows-- to simplify this, we have a source which is emitting pulses. So the waves are pulses. Every now and then there is a beep, and another beep, and another beep. And those pulses travel with their velocity-- with their wave velocity. And they have a world line represented here in the space-time diagram. This is pulse number one, and this is pulse number two. The distance between those two pulses is our period, the period of our wave, which we call tau. The question now is, how is this being observed by an observer which is moving with a relative velocity v with respect to the source? So let's analyze this. So if we want to characterize or find our position x1 and x2, we can do this by saying x1 is equal to ct1 or equal to x0, which is the distance of the observer to the source plus c times t1. v is the velocity in which the source is moving. And similarly for t x2, we find c times t2 minus tau. And that's also equal to x0 plus v times t2. So the distance in time-- we're still in the reference frame as of the source-- is given by c times tau over c minus v. And the distance in space is given by v times c times tau over c minus v. So the question is not how this observed-- how this is seen by the source but how this is being seen by the observer. So we have to apply Lorentz transformation. So in the s prime frame, which is the observer frame, we find delta t prime is equal to gamma delta t minus v over c squared delta x. And then we just fill in the information as we discussed before. Tau prime is then gamma times c tau over c minus v times 1 minus v square over c square. And then you make use of delta equal v of over c. And we make use of gamma equals 1 over square root of 1 minus beta square. And we find then-- this is a little bit of an algebra exercise here-- that the period now is given by 1 plus beta over 1 minus beta square root of that times tau. And the frequency is the inverse. We'll have 1 minus beta over 1 plus beta square root of that [? times ?] the frequency. So we just calculated relativistically how the period and the frequency of a wave is Lorentz transformed. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 126_Experimental_Evidence.txt | MARKUS KLUTE: Welcome back to the last section of 8.20 Special Relativity. So as we discussed for special relativity itself, also general relativity is a theory which requires experimental evidence to be confirmed. And there's plenty of experimental evidence for general relativity. We talked about a few examples of them. But let's go through this, one by one. Also, a little bit with an historical context. So one of the first experimental pieces of evidence pointed out by Einstein was the procession of mercury and also of other planets. This was always this problem that's a procession of mercury, deviates from Newton's prediction was well known and first recognized already in 1859. And it turned out that attempts to correct this failed. You can think about, maybe there's other objects in the solar system which modify the trajectory of mercury around the sun. But nothing really added up correctly. And then when Einstein calculated the effect of the procession, he found that it's in good agreement with the observation. This was already very strong evidence for general relativity effects. And then there's gravitational lensing. And here, this was first measured by Dyson and Eddington, 1919, of light passing the sun in a total eclipse. The observation was in Brazil but also at the West Coast of Africa. This wasn't the first attempt to measure this. It was an eclipse, a total eclipse in Argentina in 1912. But unfortunately, this expedition didn't lead to a result, because it rained out. There was eclipse shortly after in 1914, but that happened during the Second World War, and there's long stories and accounts of how this failed. But basically, one of the expeditions wanted to travel to Crimea in Russia. And because Russia was in war with Germany, material was confiscated, and people were imprisoned. So this was canceled, if you want, due to the second and first World War. But then in 1919, this led to the observation the data was not as clear. I think there was a little bit more hope than science in the interpretation. So there was-- there was not a strong evidence, a strong significance of the results. But the evidence, nevertheless, was there. And as I was explaining earlier, that led to the fame or the triumph of Einstein, where really, his fame resulted out of the reporting of those events. There's more experimental evidence. Light travel time but around or close to massive object is modified. We talked about gravitational time dilation, which can be measured or has been measured. Other tests of the equivalent principle, but also the observation of gravitational waves. Gravitational waves were predicted by Einstein by the theory of general relativity. And only very recently, we were able to observe those. And then, in addition, there is plenty cosmological tests, which require a precise understanding of general relativity in order to get to agreement between the observations and the theoretical predictions. But let's talk about-- let's talk about gravitational waves. So those we predicted, as I was saying, but they're very, very difficult to measure. First, indirect measurement was performed by Hulse and Taylor. They were able to study binary neutron star system. And because the orbits of those two decayed required lots of energy, and that lots of energy needs to happen somehow. And it was theorized or predicted by general relativity that that loss of energy is due to the fact that gravitational waves are emitted. And they received for their findings the Nobel Prize in physics in 1993. So how are gravitational waves generated? You can ask-- I have a spinning sphere, like our sun. Would that generate a gravitational wave? The answer is no. It's a symmetrical situation. There's no change of the metal distribution. And therefore, spacetime is not modified. But if you have a sphere with a little bump, that would create gravitational waves. If you have a mass which is moving by, maybe two passing galaxies, that would not directly create gravitational waves. But if you have those galaxies rotating, or two stars rotating, or neutron stars rotating, or black holes rotating around each other, those generate gravitational waves. And the closer the object, the higher the masses of the objects, the stronger the gravitational waves are. So how can you measure gravitational waves? Very similar to the Michael Smalley experiment. What you want to do is measure differences in arms of your interferometer. And you do this with very powerful lasers and with very precise mirrors. So it's very clear the very same experiment as Michael Smalley, just much, much bigger. So we're talking about multiple miles of arms and very powerful lasers in order to conduct those experiments. LIGO, which is one of those measurement, of those devices, experiments, measures the change in the length of one arm with a precision smaller than the diameter of a proton. So that's just really-- it's mind-blowing, the level of precision, the level of understanding needed in order to measure gravitational waves. But nevertheless, they succeeded. So here, you see two experiments. LIGO has actually two experiments, two of those devices in the United States. And there's other experiments similar worldwide. You see also highlighted here, Caltech and MIT. Those are the leading communities of the leading universities in this endeavor. And then the first observation of gravitational waves happened in September 14, 2015. And this first observation was rather spectacular because it was not just any observation, but it was the observation of two collapsing black holes. So you have two black holes. They get close to each other, than they circle each other and create a new, heavier black hole. So the collision of those two black holes with masses around 30 times the mass of the sun, it actually took place 1.3 billion years ago. So the gravitational wave was traveling towards us for 1.3 billion years. The energy of about three times the mass of the sun was emitted as gravitational waves in fractions of seconds. So the huge amount of energy released in form of gravitational waves. The collision happens with both black holes moving with half the speed of light. So this is just a catastrophic kind of event, in our universe. And researchers or faculty at MIT and Caltech received the Nobel Prize in physics in 2017 for this discovery, only two years after the discovery actually happened-- and very deserved, very deserved. Let me close this lecture by just reminding you of a quote of Einstein, which I use in order to start this very same lecture. It is true that we are living through some difficult times, some turbulent times. But if you think about the bigger picture, I think we're making a lot of progress scientifically but also as humanity. And I like this quote from Albert Einstein a lot. "It is not the result of scientific research that ennobles humans and enriches their nature, but it's the struggle to understand while performing creative and open-minded intellectual work." I think if there's one thing I want you to take away from this lecture, it is this quote. I want you to be encouraged to be creative, to be open-minded, to question, and to perform high-level intellectual work. Thank you. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 12_Prof_Klutes_Research.txt | PROFESSOR: Hello, and welcome back to 8.20, special relativity. In this little video, I'm going to continue with my introduction, and talk about the research I'm interested in. So this is not strictly on the topic of special relativity, but you will see some of the influences of my research in the class as well as we move along. So what am I interested in, and what am I working on? I work on the Large Hadron Collider. You see behind me here a picture of the CMS detector. CMS detector is one of two omni-purpose detectors at the Large Hadron Collider. There's also LHCb and ALICE, two more dedicated experiments. The Large Hadron Collider collides protons at the highest possible energies. In some units, 13 TeV-- tera-electronvolts collision energy. Collisions happen around 40 million times per second in this machine when it's operational. And we have made great progress in understanding nature using this machine in the last about decade. The Large Hadron Collider started operating in 2009. They are currently in a shutdown phase, but we hope to restart next year with even higher center of mass energies available for our studies. So why do we need a machine like this? Colliding particles at high energies allows us to probe the structure of matter like with a big microscope. And so we can look very deeply into the structure of the proton. At the very same time, we can-- this high center of mass energies and collisions produce perhaps new particles-- unexpected particles. We will later see E equals mc squared as a result of special relativity. And when you have enough energy, you might be able to produce a new particle of high mass. And so that's kind of the Holy Grail, and what we're trying to do. And the other thing we do here is by colliding protons and sometimes even lead ions, we are able to create a very hot and dense form of matter similar to the environment after the Big Bang, and be able to study this new form of matter. Let's see how mass and matter are being built. If you take the table in front of you, and you start looking in detail, you start seeing molecules and atoms. The atoms are built of electrons and the nuclei. The nuclei itself is built of protons and neutrons. And if you look more precisely-- drill deeply into the structure-- you see that a proton on the surface is built out of quarks-- up quarks-- two up quarks and a down quark. If you further investigate the structure of the proton, you see that there's much more going on. There's gluons-- particles holding the quarks together. And there's also bunches of quarks and anti-quarks. This is by now well understood. If you ask what is the mass of the proton, it's about one giga-electronvolts, or 938 mega-electronvolts. But where does the mass come from? The mass of the proton comes, in part, of the mass-- from the mass of the quarks. But in most parts from the gluons, or the field which holds the quarks together. That's kind of surprising, but if you had 8.02 already, you know that there's energy stored in a field, and that energy, again, is equivalent to the mass. So the energy stored in the gluon field holding the quarks together gives mass to the proton. And this was quite well. There's a theory which describes all of this. It's called QCD-- quantum chromodynamics. And if you-- with some assumption, you can calculate the mass of a bunch of particles. So this plot here shows the light hadron spectrum which can be calculated using just [INAUDIBLE].. What I'm actually interested in is the mass of elementary particles. So this discussion so far was a brief overview in how composite particles like your table becomes massive. But how does the quark itself acquire mass? How does an electron acquire mass, or a muon and a tau. This picture here shows you all known elementary particles. We can put them in three boxes-- quarks-- those are the particles-- the up quarks and the down quarks we found in the proton. The electron makes-- together with the proton makes the hydrogen atom. And there's neutrinos. Those are core electrons. And then there's force carrier. And we just met the gluons, but there's also the photon, the W and the Z boson And the WZ boson, they are themselves also massive particles. How do they acquire mass? The answer was found by us about 8 years ago with the discovery of the Higgs boson, a new particle. And the underlying theory explains how particles acquire mass. And so basically solved, right? Not quite. So this is really mysterious to see how different the masses of those elementary particles actually are. You see on this logarithmic table here. Again, here are our friends the down quark, the up quark, and the electron. And if you compare this, for example, with the heaviest known elementary particle, the top quark, you see many, many orders of magnitude difference. So how does this actually work? And then you see some of the bosons-- the force carriers are massive. Others, like the photons and gluons, are massless. The answer to this was the Higgs mechanism. And a very simple explanation how the Higgs mechanism actually works for fermions for those quarks-- so the electron, for example-- is given in this cartoon. So the idea is that a field fills all of space. It's basically a property of the vacuum. And when you travel an elementary particle through this vacuum, you interact with this field. And the stronger you interact, the more drag you kind of get. There's some sort of-- you feel an inertia. And this inertia is what we know as the mass of the elementary particle. So there is an equivalence between how strongly you're coupled to the vacuum-- to the Higgs field, and your mass. And so a top quark couples strongly to this Higgs field, while an electron only slightly. Great. So we have understood everything. So the question is why do we still collide protons and bosons at the LHC? Is there anything else to be discovered? So it turns out that we have a very sophisticated theory describes those particles and their interactions, but this theory fails to explain all of the observations we have in nature. And so that is kind of the driving force behind the experiment I'm conducting right now. And so for example, we know that there is dark matter. When we look at the rotation of stars and galaxies, we find that they don't behave as you would expect simply based on the distribution of matter in those galaxies. There must be something else out there, and that's what-- since it's not visible-- is called dark matter. And those dark matter-- dark matter could be a particle we might be able to produce at the LHC. So that's an interesting question. Also when you look out into the universe, we see a lot of matter. You don't see a lot of antimatter. So there must be an asymmetry between how matter and antimatter is being produced. And so that is also not fully understood yet. And then there's more question. For example, those neutrinos, they are really, really light. On this logarithmic scale, I had a cut off, and then the neutrino masses. Do neutrinos acquire mass as an electron does, as a top quark does, or is it a different mechanism? We don't know. Gravity is not even included in the standard model. And the fact that the Higgs boson was discovered at a specific mass which is rather small it's also a little bit unnatural. And so there is this entire list of questions and unresolved mysteries which we're trying to answer. And the way we do this is with big cameras. So this is CMS detector. There's a similar picture that's behind me. You can think about it as a big camera looking at the interaction of the collision of two protons. And it starts off with around this interaction region with pieces of silicon, which we use to track charged particles going through. We put all of this in a magnetic field, and if you listen to 8.02 already, you know the charged particle in magnetic field, they follow a curvature. And from the radius of the curvature, we can calculate the momentum of those particles. And then we stop the particles in order to measure their energy. So we do this in kilometers. And what we use here is the lead tungstate electromagnetic calorimeter, and a second calorimeter for particles which are harder to stop. So those are called atomic calorimeters. And then the silver part in the middle here gives the CMS detector its name. It's the solenoid. It's a 2-- a 3.8 Tesla superconducting magnet. And then we have more detectors there to see whether or not some particles might escape, and we try to measure those as well. There's another very nice picture. After opening the detector, you see this silver thing in the middle here. It's the pipe in which the protons zoom through the detector, and broaden into collision in the very center part of it. And then we take those pictures. Here's one, and this is a very famous one. It's a Higgs candidate event, where the Higgs boson might have decayed into two Z bosons, and then those Z bosons themselves decayed again into a pair of electrons shown here, and a pair of muons here. And then we can use those individual particles to reconstruct the property of the Higgs boson. Here's another candidate where there's two photons being reconstructed. And again, those two photons can then be used in order to reconstruct, for example, the mass of the particle, which is the original first two protons. So we have done this for the last years, and collected quite some data. And if we look at the entirety of the data, we can make this plot here. And what this plot here shows is the mass of the particle and the coupling of the particle to the Higgs field. And what you see, there is a linear relationship in this log-log plot between those two, and that gives us some confidence that the elementary particle-- like a muon here, like a tau lepton, and a bottom quark here, like a top quark here-- they acquire mass through the coupling of the Higgs field. And so there's this linear relationship-- the correspondence between mass and coupling to the Higgs field. Great. So we have this all together, and it gives us a complete theory. Again, there are a large number of open mysteries and questions we'd like to answer. And the way I look at it is it's a little similar to the exploration of Christopher Columbus. So what we're trying to do is to go to higher and higher energies, to higher and higher intensities to find out whether or not we find first hints of something new and unexplored. So we made this discovery. We made the discovery of the Higgs boson, but whether or not this particle is really the Higgs boson is still out there. We're trying to measure it with more and more precision. Maybe we find deviations from its expected properties to the ones we observe. Similarly, Christopher Columbus, when he sailed off from Spain, he tried to reach the Indies or Asia, and in his lifetime, he never figured out-- they didn't actually accomplish this. And similarly, maybe we have discovered a new particle which helps us to understand more about the inner structure of particle [INAUDIBLE].. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 101_Tests_of_Special_Relativity.txt | [SQUEAKING] [RUSTLING] [CLICKING] MARKUS KLUTE: Welcome back to 8.20, special relativity. So we're starting a new chapter in which we look at tests and implications of special relativity. We started the entire discussion and evaluation of Lorentz transformations and the description of the paradoxes based on Einstein's postulates. Those are not axioms, which means that we do actually have to verify. They are a prediction of how nature functions, and experimental verification is needed to gain confidence that those postulates are actually correct or realized in nature. And we have studied some of those tests already. And this serves as a little bit of a review of the discussions we had to this point. One experimental test is stellar aberration, which we discussed can be explained by special relativity and by velocity addition. So this gives us some idea about what light is. Light isotropy is being tested in a variety of different experiments, starting from Michelson-Morley, which basically tests that the speed of light is independent of the orientation of the apparatus. We have not discussed in detail another experiment, which is very similar. It's the Kennedy and Thorndike experiment, which tests that the speed of light is also independent of the velocity of the apparatus itself. As one of the homework assignments, we looked at de Sitter who tested that the speed of light is independent of the speed of the source. And that has been tested, for example, with the motion of binary stars, as we did in our pset, in our homework assignment. We can look in particle physics at the decay of a pion into two photons. And those pions, they can have a lot of energy, for example, with a beta of 0.999 times the speed of light. And still the photons are of this decay. They behave like any other photon. They move with the speed of light. Let me discuss the Doppler effect and relativistic Doppler effect and analyze light with various frequencies. And one of the hypotheses you could have is that the photon actually is a massive particle. This would directly modify Coulomb's laws, which are tested experimentally. And those results would be dependent on the frequency. There is weird electromagnetic effect if you introduce a mass to the photons like torque on a magnetic ring Again, here precision measurements have been performed. And they're all in agreement with the hypothesis that the photon is massless. And we talked about the Doppler shift and redshift for light. Another class of experiment is where we look directly at time dilation, for example, in the decay of the muon. As we discussed, we have cosmic muons to study, or we can produce muons in the laboratory as well and study them. Or we can put very precise clocks on planes, fly them around the globe, and compare them with stationary and just simply measure the effect of special relativity on those clocks. In all of this experimentation and experimental verification, it's important to understand the importance of uncertainties in the scientific process overall. I think that-- remember, when one has the historic perspective on science, one often forgets that a specific measurement comes with an uncertainty, often, a statistical uncertainty or systematic effects, which are quite important to quantify the level of verification of the theoretical hypothesis. One example here is-- and experiments can have biases as well. And one example of a biased experiment is here one by Walter Kaufmann who tried to measure e/m, the electric charge over the mass of the electron. But he had a rather strong theoretical bias. And he conducted the experiment at the time Einstein was proposing his theory. The bias really came from the model of an electron at the time. And, as experiments, they were inconsistent with Einstein. So he said Einstein is wrong. Einstein and Lorentz are wrong. Planck looked at this and said maybe, but Einstein's conclusion immediately by looking at this was, no, this cannot be. And it took a little bit of time to tee up those experiments until 1940. You can read more about this in this Wikipedia article about Walter Kaufmann's experiments, but let me just read this to you. "The prevalent results decidedly speak against the correctness of Lorentz's assumption, as well as Einstein's. If, on account of that, one considers this basic assumption refuted, then one would be forced to consider it a failure to attempt to base an entire field of physics, including electrodynamics and optics, upon the principle of relative movement." We now know this was wrong, but scientific process happens in scientific environment. And I started this class by explaining that one needs to be open minded to learn and to study and to grow scientifically. And one has to question the assumptions, understand the assumptions when then go into measurements. So this is the first part of this chapter where we talk about tests. We will have a discussion of implications of special relativity as we move on from here. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 53_Spacetime_Diagrams.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we talk about spacetime diagrams. They turn out to be very useful tools to describe events or sequences of events, in particular when observed by multiple observers. So what is a spacetime diagram? Here's an example. You have an x-coordinate and a t-coordinate for space and time. I plotted an event in blue here. And the world line of events. A world line is just a sequence of events as they occur. In this case, something seemed to be moving with constant velocity. The world line is just a continuous line of movement. The velocity of this event is delta x over delta t, which is 1 over the slope. Let's have a look here. So the time axis is defined as those events which all occur at the same space, x equals 0, whereas the x-axis is defined as those events which all occur simultaneously at the same time. And then you can draw additional lines into the spacetime diagram where, for example, all times are equal to 1. You might want to add a unit. I omitted this here. Time might be given in seconds, in days, in hours, in years-- whatever you like-- similar to space in meters or light-years. So lines here in green of the same time, meaning time is constant. All events on that line happen simultaneously, while in blue are those lines where events happened at the same location, so x is equal to constant some specific values. Here in red, I add one additional caveat, which you're typically not aware or considering very much in diagrams is the role of tick marks. Here in this spacetime diagram, our tick marks are perpendicular to the axis. It's also OK or correct here to say that they are parallel to the second axis, and we'll come back to this point later on. An example of a world line is simply drawing all events which correspond to me, right? Professor Klute is pacing in his office. You know, maybe on a line, just the x-coordinate is plotted here, time passes, and I'm just pacing the long-changing direction. Each time, each little segment is constant velocity. That's the world line of me from some time t for minus t to term time equals 40. OK, so now our first concept question. Let's consider the set of world lines, 1, 2, 3, 4, and the question is, which of the objects which correspond to the world line is moving the slowest? Let's consider this for a second, and then we look at this. As the velocity is 1 over the slope, the object with the steepest slope, the largest value of the slope, is moving the slowest. And in this case, it's object number 2. All right, now we want to actually make them useful. Yes, they can be used in order to describe certain event lines, but they're really useful when you describe events happening for different observer. So in this activity, I invite you to draw Bob's spacetime diagram into Alice's, and then as a second step, draw Alice's spacetime diagram into Bob's. The situation is very similar to previous ones discussed in this lecture. Alice is stationary and Bob moving in this rocket with a velocity of half the speed of light, or a gamma factor of 1.2. All right, try. Go ahead. Try to show where is the time axis for Bob and where is the spatial coordinate for Bob. OK, so the way to approach this is the following. We want to use Lorentz transformations in order to figure out what is the value of Bob's time axis and space axis for different values of Alice's spacetime diagram. So we start with drawing Alice's spacetime diagram. And then if you want to find the x-axis as seen by Bob, we have to set the time for Bob to equal 0 and then find the corresponding elements or tick marks on the axis. So the first point we're going to find is tB equals 0 and xB equal 1. So with the Lorentz transformation, we find that xA 1, so this point corresponds in Alice's spacetime diagram to xA equal gamma equal 1.2 and tA equal gamma times v over C squared equal 0.6. So we can make this-- find this first point and plot it in our diagram. It's right here. OK, and then we go move around and find the second point and the third point, and we do the same for the time axis, where xB is equal to 0 and tB equal 1, then corresponds to 4.6 in xA 1 and 1.2 in xA 2, where we find these points here. I failed to say that the origin of those two spacetime diagrams [INAUDIBLE]. OK? So this is already it. So we found Bob's time axis, where xB is equal to 0, and Bob's x-axis, where tB is equal to 0. And I did draw those tick marks parallel to the second axis. So if I want to now find out the time axis for xB equal 1, I just have to follow along and draw a parallel in the picture here. All right, so the second question then is, where Alice's axis in Bob's spacetime diagram? So the procedure is very similar as before. We draw Alice's axis-- sorry, we draw Bob's axis, and we find Alice's x-axis by setting tA equals 0, and then we find the number of points and connect those. And the time axis is found by setting xA equals 0 and finding them points for various values of time. And you see here, this looks a little different than before. I just zoom in here a little bit. What you find specifically, because relative the direction of motion changes, the positive values of x are in positive value-- and negative-- so the positive values of the x-axis are in the negative time direction, while the negative values of x are in the positive time direction, so would be across down here. So this needs a little bit to get used to, but you will later see when I draw in any of the two spacetime diagrams specific sequence of events, I can immediately read off how this event is perceived from Bob's and from Alice's perspective. And this makes our space diagram very, very useful tools. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 15_Categories_of_Physics.txt | MARKUS KLUTE: Welcome back to 8.20 Special Relativity. In this short video, you're going to see how special relativity, here shown as SR, fits into the general landscape of physics theories. Very likely you have listened to 8.01 last semester or some previous time in your career at MIT. And once we enter the discussion of special relativity, it might seem to you that what you learned there was simply incorrect. A better way to look at this is that each of those physics theories, they have a range of validity, a range in which the theory works and describes physical phenomena. And so here in this picture, I'd like to point out that even special relativity has its own range of validity. But let's start with classical mechanics. So classical mechanics describes the motion of objects, a collision of objects, a rotation of objects. In 8.02, you are studying electrodynamics. But you do that typically at MIT in those courses at velocities which are modest-- small velocities, small velocities compared to the speed of light. Once you accelerate, once you have velocities which are really, really large, then the description breaks. The validity of those theories breaks. And that's where special relativity comes in. And it's especially important when you discuss very fast moving bodies, fast moving reference frames, or behavior of light. And so this is what special relativity does. The range of validity is basically fast-moving phenomena with high speed. But special relativity by itself doesn't help you to understand, for example, object of small scales. In order to have a proper description here, you need to study quantum mechanics. If you have object of very mass large or with large accelerations, you need to enter a generalized view, general relativity. And if you, for example, combine high velocities and small scales, you have to study quantum field theories. So the point of this slide is really to point out that special relativity has a specific use case. It's limited. It describes fast-moving objects. It doesn't describe large accelerations. So that is the generalization you get when you study general relativity. So in summary, special relativity is a complete description of physical phenomena at high speed. It is consistent with 8.01 and 8.02. And you will later see that whenever we find integration of motion for object at high speed, if we use smaller velocities, we'll find the solutions of 8.01 and 8.02. When we combine special relativity with quantum mechanics, it describes quantum field theories. And as I was saying, the generalization of special relativity, SR, is general relativity, GR. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 104_Absorption_and_Emission_of_Photons.txt | MARKUS KLUTE: Welcome back to 8.20 Special Relativity. In this section, we're going to talk about the emission and the absorption of photons. So you can think about a scenario where you have an atom which emits a photon, like in this picture here. And to create a new atom, which we call the atom prime, in the absorption process, you have a collision between a photon and the atom. And you make a new atom, again, indicated by a prime. So I want you to actually work this out in two activities. The first one is on absorption. So here we have a stationary particle, the atom, with a mass M0, and it's struck by a photon. We have this scenario here. And the photon has an energy, Q, and becomes-- and there's a new particle coming out with the rest mass M0 prime and recoiling velocity, it has to have some sort of momentum, v. And so the question is, how can we now find the mass of this new particle and the velocity? So please stop the video, and try to work this out. So the way to work this out is by writing energy and momentum equation. You can do this just with a four-vector. even though we are not performing any Lorentz transformation in this part, you can just write on the four-vector. And this needs to be conserved energy. And momentum need to be conserved in this correlation. All right, so we have the mass of this initial particle at rest. So the energy is M 0 C square. The energy of the photon is Q. The momentum of the photon is Q over c. And so then the final particle-- the outcoming new particle has its own mass, but it has a boost. So the total energy is M 0 prime gamma using the velocity of this particle, C square. And the momentum is M 0 prime gamma times v. So that's after the correlation. And so then you can use the energy relation in order to get to the mass. This is a function of the velocity here. And you can get the velocity by looking at the momentum. So you basically solve for this v or v over c in this case. And then if you want, you can add this back in here in order to get the mass as a function of the mass of the initial particle and the energy of the photon. The second example now is the emission. So here we have a stationary atom with a rest mass M0. And it emits the photon, this energy Q. And it becomes the new atom with rest mass M0 prime and recoiling with the velocity. Now given the two masses, what is the energy of the photon? Q. So we set this up in the same way as before. Now we start with this particle at rest. And the energy is M 0 c squared. Momentum is 0. And that's equal to the energy and momentum of the new particle and the photon, in the very same way as we set this up before. So now what you want to do here is use our invariants of the four-vector using the energy and momentum relation. So the mass of this new particle times c square squared is equal to the energy squared . The energy squared-- you just bring this over here-- is M0 c squared minus Q squared. And then the momentum is simply of this new particle, is simply Q square. This you cannot solve for Q. And you find this relation here. So that's the energy of the outgoing photon. If you now define Q0 as the difference in masses or the difference of the masses times c square, then you can write the energy of the photon as Q0 times 1 minus Q0 over 2M c square. And that's always smaller than the difference in masses. So if you now analyze this and look at this some more and try to understand what it means, what we find is that the emission and the absorption, they only occur at a very precise value of the energy of the photon. It also means that an emitted photon can only be reabsorbed if the particles are moving with just the right velocity. So if you're an atom, and the photon goes out, and your neighboring atom is just sitting there next to you at rest as you were at rest before, it is not able to reabsorb the photon. So we learned just by using special relativity quite a bit about the physics involved in absorption and emission of photons. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 124_Redshift_Tests.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. So if there is a time dilation effect due to gravitational fields, then there's also a redshift which is of gravitational fields. So a consequence of time dilation is a change in the light frequency. And I asked you to estimate the magnitude of this effect. And the example you want to use is the one shown here. So you have a tower just-- not randomly, but 22 and 1/2 meters tall. And a light beam is sent down. Basically, the tower is built on this planet, and there's gravity that's acting. So this is basically an accelerating reference frame. The length of the tower is, again, 22 meters. And I would like you to just get a feeling. How big can this effect be, the effect of redshift here? So please, try to work this out. The way to think about this is first to say, OK, now the light-- the delta t equals the light to travel-- is l divided by c. The speed of light is c. The length is l. The change in velocity is g, acceleration, times l divided by c. So the Doppler shift then is the frequency, the new frequency, divided by the initial frequency. And that can be approximated by 1 plus delta v over c. So we find that it's 1 plus g times l over c square. Now the speed of light is pretty fast, 3 times 10 to the 9 meter per second. And this distance is only 22 and 1/2 meters. So we find that this is a tiny, tiny, tiny effect. But nevertheless, experimentalists at Harvard tested this effect. So Pound, Rebka, and Snider in the 1950s and '60s were able to show this very tiny effect. You want to know more about this, you can, for example, look up a small description in Wikipedia here. But there's quite some literature on those experimental tests [INAUDIBLE] |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 94_Forces_and_Kinetic_Energy.txt | MARKUS KLUTE: Welcome back to special relativity, 8.20. After discussing energy and momentum and examples with collisions, we now want to talk about forces. And we get back to the example of Alice traveling to the center of the galaxy and asking what does it mean in terms of acceleration. So we start from Newton's second law. We know that a force is a change in momentum. We can write this down as d dt m0 times u over the square root of 1 minus u squared over c squared, or just with a gamma factor. And the thing to consider here is that now the gamma factor and the velocity are actually time dependent. So there's two components to this. We'll come back to this. The kinetic energy is the work done by an external force. And you can get to the kinetic energy by just integrating, let's say, for a particle which is accelerated by an external force from the velocity 0 to some velocity v. That's the integral over the path of the particle, over the path of this object, times the force. If you just assume here uniform motion in x-direction, then this simplifies to just an f dx. So, as the first activity, I want you to find the kinetic energy of an object with velocity v and the mass and mass m0. And, as the second part, I want you to test this result for velocities much, much smaller than the speed of light where you're used to doing this kind of calculation, and you're familiar with the outcome. So we just have to integrate, just have to integrate. So this is a little bit involved here. So we have to integrate from 0 to v m0-- that's our constant; we can take that out-- d dt of this u times gamma dx. OK, so you find there's two components here. And then we do a trick where we introduce this du dx dx. And then the integral becomes an m times u du over 1 minus u squared over c squared to the third, to the third power half. OK, and then you can just look up the integral or work it out. It's not that difficult actually, but you find that this is equal to m0 times c squared over square root 1 minus u squared over c squared, which you have to evaluate for velocities v and 0. And, when you do that, you find those two components here. OK, the first one is m0 c squared times gamma. And the second one is m0 c squared. So that result is actually not too surprising, as we saw that we can write the energy equal to m0 c squared plus k. And what we just calculated here from this example is k is equal to energy minus m0 c squared, OK? So that result already makes sense with respect to the discussion we had to this point. Or you can simplify this by saying, the kinetic energy is gamma minus 1 times m0 c squared, OK? So, if we now evaluate this for small values of v, as we did before, we find that the kinetic energy is 1/2 m0 v squared. And I find it interesting, illustrative, to plot what this means now. So, if we plot the kinetic energy of a particle as a function of its velocity, we find that, for small values, those two curves basically overlap. For small values, m0 c squared times gamma minus 1 is basically the same as 1/2 mv squared, which we just derived here from the Taylor expansion. But, for larger values, this diverges and especially when you get closer to the speed of light. Just to get a quantitative example, I asked you to do another calculation here. I want you to reinvestigate Alice's journey to the center of the galaxy where she has a spacecraft, which moves with a gamma factor of 15,000, an acceleration of 10 meters per second squared, and the mass of the spacecraft, let's say, is 10,000 metric tons or 10,000-- sorry, 100 metric tons or 100,000 kilograms. So compare the kinetic energy using Newtonian mechanics or special relativity. And you find that the difference is astonishingly large. So, if you just work this out, 1/2 mc squared-- we can just use c squared here because the velocity is basically c-- 5 times 10 to the 22 kilograms meter squared over seconds squared. And, in relativistic terms, the answer is 30,000 times larger, so 30,000 times larger than the classical case. So there's a huge difference between the classical evaluation and the evaluation with special relativity. One more word on F equal ma, the question is, how does this transform under Lorentz transformation? It's something we already halfway figured out. So here you basically want to see how a transforms under Lorentz transformation. We have started the discussion by saying, you know, in Galilean transformation, the acceleration is invariant, while, in Lorentz transformation, that's not the case. But, if you investigate again the second law of physics, the force as a change in momentum, you find that you get those two components here. One is parallel to the acceleration, so m times gamma a. But the second one is not. The second one is m0 times u times the change in time of the gamma factor. And that's not parallel to F or to a. And so you find that there's two-- the new vector or the new force of a particle is not parallel to the acceleration anymore. That's kind of counterintuitive. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 21_Events.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: Welcome back to 820. So in this section, we want to define events and frames. The goal is to define the principle of relativity. So let's think first about what an event actually is. An event is something that happens, and it happens independently of the frame of reference, independently in how we want to describe the event itself. We might use the frame, the reference frame, to define the event. But again, the event is an independent thing which happens. Let's look at an example, probably one of my favorite example in this class. Guess what happens? What? Professor Kloots exploded. Really? And then you start asking the questions, when and where. When did it happen, and where did it happen? But the event happens without a description of the when and where. We can then draw a reference frame, as I did here, with an origin, with an x-coordinate, and a y-coordinate, and z-coordinate. And in this picture, we have a spot defined for the event. And we need to also have a fourth coordinate for this, the time, in order to describe when the event actually happened. And I did this here. And some coordinates I give you. It's 42 degrees north, 75 degrees west, Cambridge, Massachusetts at MIT. You see, MIT, the dome behind me, at 3:30 PM on January 6, 2021. So I give you the when and the where for this event. I didn't give you the altitude, for example. This is, in this picture, maybe the y-coordinate or the z-coordinate. But that would be needed as well. I just tell you, maybe it's my office on the fourth floor in building 24. So when we want to consider an event describing the event, we have to consider the space, so where, and the time, the when. And we can do this as a four dimensional vector or a four vector data set. And so our event, Professor Kloots exploded, can be described with this vector P, which has an x-coordinate, a y-coordinate, a z-coordinate, and a time coordinate. So the important part here is that events happen independent on the frame of reference. If you are sitting in your room somewhere in Europe right now, you can describe this event with your time frame. There's a six hour time difference. So this 3:30 PM would be 9:30 PM in the evening. You can define your little room as the origin and then draw an x-coordinate and a y-coordinate and z-coordinate that describes this specific event. This description will look different from the description I just gave you. But the event is unchanged. It's independent of the choice of reference. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 93_Collisions.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section we're going to talk a bit more about collisions. I've already seen collisions in study of momentum conservation in previous sections. So here we can have a collision. Then we can describe them in the center of mass frame, for example, where the total momentum is equal to 0. So in the case of the collision of two particles, the momentum of particle one plus the momentum of particle two is equal to 0. We can then describe the energy and the momentum of the particles before and after the collision. In the lab frame, the situation is different. Here typically we have one particle with some momentum hitting another particle, which is at rest. But we can also have different types of collisions. We describe or characterize elastic collisions where the kinetic energy is conserved and so is the mass. So you can think about two billiard balls colliding without any friction, in which case they don't change their appearance, their mass. Everything is unchanged, so you must change the direction. The total kinetic energy in these collisions are typically conserved. But we can also have inelastic collisions. And there's two different kinds. There's sticky kinds, where the mass after the collision is greater. So you have two particles, for example, maybe they stick together-- they're some, like, Play Dough balls-- and the kinetic energy after the collision is smaller. Or you can have explosive collisions, where the mass afterwards is smaller. Maybe you start from one heavy, big object and then which explodes into many smaller ones. But the kinetic energy after the collisions is much smaller. Those are also collisions. So here we want to do an activity and study an inelastic collision. So before we have two particles there, or billiard balls. They're exactly the same and have a velocity u. And after the collision their mass is capital M, big mass. And you're going to describe this collision once in the center of mass frame and one in the laboratory frame. And so the question now is, are the masses and is energy conserved in those collisions? And you're going to just described this in both reference forms. So again, stop the video here and try to work this out. I already did this, so I discussed before, in those collision problems it's always important to be really clear. The situation before the collision was A. The situation after the collision was B. So I'm describing this here. First in the center of mass frame where the x-- and I'm just talking about x component here-- the x momentum is 0, which is equal to the mass times u times gamma minus the mass times u terms gamma. That's the 0. The energy before is 2 times the mass times gamma times c squared. After the collision, the particle is at rest. The new one particular is at rest and has an energy, large M over-- times c squared. In the laboratory frame situations, different case. X momentum 0 minus m times u prime-- this is a different velocity-- times gamma of u prime. So here I'm trying to indicate that this gamma is not the same gamma as over here. This is a gamma, but it's the velocity of u prime. And the energy is the rest mass of the particle addressed plus the mass times gamma times c squared off the second particle. After the collision, the particle has some velocity u. And so the momentum in x direction is minus large M times u times gamma of u again. And the energy is large M times gamma u times c squared. OK, good. So now we can use momentum conservation and find this equation here. And from which we can then calculate that the large mass is equal to 2 times the smaller mass. So what you find, and this is the relativistic math, you find that at the conclusion that the rest mass is not conserved. The mass of this big ball is not simply the mass of the two rest masses, or 2 times the mass of the rest mass. You have to consider this gamma factor here. It's 2 times the relativistic math, if you want. But you also find that the total energy is conserved in circulation so that the sum of m0 gamma times c squared is conserved in the collision, irrespectively in how you actually reference it when you discuss the problem. I want to close this part of collisions with a small discussion of units. And that will become interesting and important later on when we look at particle physics examples. So in particle physics, we often talk about units of electronvolt in collision experiments, or mega electronvolts, or kilo electronvolts, tera electronvolts. So 1 electronvolt is the kinetic energy of the particle with charge e, which is accelerated in a potential of 1 volts. So that corresponds-- that's a unit of energy and it corresponds to 1.6 times 10 to the minus 19 joules or 1.6 times 10 to the minus 90 kilograms meter squared over 2nd square. But the mass of an electron is really, really small. And those units here are introduced because the mass is small and you want to have reasonable numbers to work with. So the mass of the electron is 9.11 times 10 to the minus 31 kilogram. So if you just rewrite an m0 as equal to m0 c squared times 1 over c squared you find that, huh, now we rewrite this and find that the masses 8 times 10 to the minus 14 joules over c squared. Or in units of electronvolts, 5 times 10 to the 5 electronvolts over c squared, which is 0.511 mega electronvolts over c squared or 511 kilo electrons over c squared. So when we talk about the mass of an electron, we sometimes approach this with natural units, in which c squared is equal to 1. And that just simply says that the mass of an electron is 511 kilo electronvolts. The math of a neuron is mega electronvolts, and so on, and so on. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 112_Electric_and_Magnetic_Fields.txt | MARKUS KLUTE: Welcome back to 8.20. So this section is a preview or maybe a review of 8.02 depending on whether or not you have listened to electromagnetism [? already. ?] So first I just want to remind you how we relate electric and magnetic fields and how we can describe them from a different moving reference [? point. ?] So if the reference point moving in x direction, then the x component of the fields do not change. But the transverse components do change, as we have discussed before. The core of this section is about how the fields-- the electric and magnetic fields are actually generated by charges and their distributions. And this relation is described by Maxwell's equation. The entirety of 8.02 are classes on electromagnetism is about how to understand Maxwell's equations. So I'll do this here in a very short and brief manner. So you can write Maxwell equations in four different equations. The first one is called Gauss's law. And if you read the equation, it just says that the divergence of an electric field gives the density of the source or the charge density of the source. You can also read this equation by saying a charge density generates an electric field. So charges generate electric fields. Similarly, Gauss's law for magnetism can be read as magnetic charges generate magnetic fields. Or the diversion of the magnetic field is the density of the magnetic source. However in nature, we haven't observed magnetic monopoles or at least not yet. And so therefore, there's no such thing. There's no magnetic density. You can read this equation also saying that all magnetic field lines need to be closed. And so that's another way to look at this. And we have Faraday's law, which means that we can induce electric fields in a coil equal to negative change of the magnetic field. In other way, if you want to create an electric field, you can do this with a charge. Or you can do this by changing, as a function of time, the magnetic field. Changing magnetic fields generate electric fields. And very similarly, we can look at Ampere's law and saying that changing electric fields generate magnetic fields. And you can also generate magnetic fields with a current, as we have seen in the previous [? section. ?] So this is how we can understand [INAUDIBLE] Maxwell's equations. The difficulty now, 8.02, is often to understand the concept of fields, the fact that there is a [? vector ?] describing the strengths of this abstract thing, of an electric or magnetic field somewhere in space or [INAUDIBLE] are changing this time. That's [? complicated. ?] And then there is also a little bit of functional analysis needed in order to understand and how to apply the electric field by a specific charge. Those cases can often be simplified by having symmetric configurations, like a charged atmosphere, or a point charge, or a cylinder, or charges along the line. In those cases, those integrals or those divergences can be calculated in a straightforward manner. OK, so then there's another aspect which is relating [? charge/discharge ?] distributions or fields to forces. And that's done by Lorentz force. So the force of the charged particles which is moving in electromagnetic field is given by the strength of the charge itself times the electric field, plus the velocity of the charge, plus the strength of the magnetic field. OK, what that means is I can-- if I put a charge in an electric field, it's being pulled. It's being accelerated. If I have a moving charge in a magnetic field, it's being bend around or the force bending it around. And then I can have this relativistic equation of motion, which uses our relativistic equation of motion and sets it equal to our Lorentz force. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 61_PoleintheBarn_Paradox.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: Welcome back to 8.20. In this section and the following ones, we talk about paradoxes in special relativity. A paradox is something which is absurd or self-contradictory. So we have statements which don't really make any sense when put together. The pole in the barn paradox is rather interesting. And we will analyze this, and at the end of the discussion, we hopefully agree that there is no paradox here. It's a pseudo paradox. So the situation is as follows. We have Alice. She has a pole. The pole is 10 meters long in her reference frame. And Bob is very proud of his New England barn which is, in his reference frame, 8 meters long. Alice, however, is moving with a velocity of 0.6 times the speed of light, which gives us a gamma factor of 1.25. Does the pole fit into the barn is the question, or not? So stop here and think about this for a second, and we will continue with an analysis of this. So here is the analysis for Alice's frame and the analysis for Bob's frame. For Alice, the barn in her reference frame is Lorentz contracted. It's 6.4 meters long, but her pole is 10 meters long. So we should clearly answer this question by saying it doesn't fit. In Bob's frame, the barn is 8 meters long, and the pole is Lorentz contracted-- also 8 meters long. So Bob will say, yeah, it fits-- it just barely fits. They're exactly the same size, so yes, it all fits into the barn. And here is where you might think this is an absurd statement. They cannot be both right. We will see they can. They can both be right. They just disagreed on the fact that events happen simultaneously. What are the crucial events here? When does the barn hit the end-- does the pole hit the end of the barn, and when does the back of the pole hit the front of the barn? Those are the two things we have to study in detail. But let's get to it. How can they, or why can they disagree? So the idea is that you draw space-time diagrams for the pole in the barn, and show that there's no paradox by using the world lines of the pole. Before we do this, we're going to analyze this a little bit more. So assume that the front of the pole enters the barn at time equals 0 for both Bob and Alice. Then Bob observes the pole entering his barn, and it takes 44.4 nanoseconds-- 8 meters divided by 0.6 times the speed of light-- for the front of the pole to reach the back of the barn, and the back of the pole to reach the front of the barn. So after 44 nanoseconds, in Bob's reference frame, the pole is in the barn. Alice, however, sees the barn Lorentz contracted. It's 6.4 meters long. She moves this 0.6 times the speed of light. So for her, she reaches the back after 35.6 nanoseconds, in which case, Bob's clock only shows 28.4 nanoseconds, because Alice's clock time is Lorentz contracted. So we can clearly conclude here that that's not enough time for Bob such that the pole actually entered the barn for the full length. So the back of the pole is still outside. So we want to consider three different events. The first event is after 44 nanoseconds, and in the space of 8 meters in Bob's reference frame. For Alice-- this is the situation we just analyzed-- 36.6-- 35.6 nanoseconds passed. And in her reference frame, the front of the pole is at 0 meters. The second event is then the other side of the barn in Bob's reference frame after 44.4 nanoseconds 0 meters. He sees-- or she sees that 55 nanoseconds have passed. We use Lorentz transformation here, but the position is minus 10 meters. And the last point is 28.49 nanoseconds and 0 meters. That is the observation when Alice sees the end of the-- front of the barn-- the front of the pole at the end of the barn. That translates into Alice's frame a 35.6 nanoseconds, and minus 6.4 meters. So the minus 6.4 meters tells you very clearly what we just already said. The back of the pole is still outside. So that's the quantitative or numerical kind of evaluation. And we can also show the very same thing in the space-time diagram. So we show the space-time diagram here, and this is Bob's reference frame. So the pole just touched the front of his barn, and the barn is located at 8 meters-- the end of the barn is located 8 meters. The front of the barn is located at 0 meters. After 44 nanoseconds, there's event number one and event number two. The pole is fully in the barn. But we can also show the pole in event number three. So one-- where is the end of the pole? We look at this diagram here. Where is the end of the pole when the front of the pole hits the end of the barn? You see clearly there's a piece sticking out. We saw that there's-- in this event here, 6.6 meters in Alice's frame still seeking out. So we see that event number three is located here, and not all of the pole is actually contained within the. So Bob and Alice disagree on whether the front and the back of the pole are in the barn simultaneously. That's where the situation becomes contradictory. They don't agree that two events which happened at the same time in their reference frame-- in Bob's reference frame occurs at the same time in Alice's reference. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 41_Time_Dilation.txt | [SQUEAKING] [RUSTLE] [CLICKING] PROFESSOR: Welcome back to 8.20. In this section, we're going to talk about time, timekeeping, and how to relate time between two different reference point. Now, let me start with a quote by Albert Einstein. "Everything should be as simple as possible, but not simpler." So let's start with this in mind. And recall that we ended the last section by finding that the wave aether model doesn't really describe electromagnetic waves very well. We see that there is a problem between the experiments, specifically the one by Michelson-Morley, and the theoretical picture people had in mind. So Einstein approached this in an interesting way. He simply postulated the things he thought need to be true. He said, "The same law of electrodynamics will be valid for all reference frames where all laws of mechanics hold good. This is the principle of relativity." The second postulate is that "Light is always propagated in empty space at the velocity 'c,' independent of the state of motion of the emitting body." So with these two postulates, we will now derive the theory of special relativity. And again, we'll start by talking about time. So time is suspect. And I alluded to this already when we looked at Galilean transformation, where it simply, out of our intuition, assumed that time is invariant. Now, when we now talk about time, the viewpoint I would like you to have is that we want to look at clocks from different reference frames. We want to investigate whether or not events happen simultaneously or not. What does it mean? When we make a statement like a train arrives at 7 o'clock, what we mean is that there is a simultaneous-- two simultaneous events happen. One is that this little clock here shows to point at seven and 12, meaning that it indicates to us that-- this event indicates to us that it's 7 o'clock. And the second event is that the train actually arrives at the station. So those two events happen simultaneously. The question now is whether or not two observers, one stationary and one moving, agree with this observation. And I take it away-- the answer is no. There is a relativity of simultaneity, meaning that two observers can very much agree on the description of two events, but not necessarily that those two events happen simultaneously. So let's investigate. And we use our two friends, Alice and Bob, in order to have this discussion. All right, so we start from a situation where Alice and Bob are both stationary. Alice is on her spacecraft, and she has a device on her spacecraft which shoots light or paint balls towards two clocks. And each time this happens, the clock ticks, right? And we just look at one situation. So she has a clock on the left and a clock on the right. Bob observes Alice's clock. And he can compare this observation of Alice's clock with his own. So in this station, as to duration, there's a TA and a TB. Those are the times of Alice and Bob. Both are 0. This is when the situation starts. And the capital T indicates for Alice and for Bob when they observe that the clock has been hit. I should add here that when we talk about observation in this entire class, unless I make a very explicit exception to this, we don't consider the fact that observing actually means that light has to be emitted from the clock and enters Bob's eye in order for him to conclude that there was something happening. The observation is like taking an instantaneous picture. OK, so we have to keep this in mind. But in this simple situation, nothing is moving. We can hopefully agree that the times being read for Alice and Bob on the left and the right clock are all the same. Now, we go in the second situation, where we use the same device but with a paintball. So now, Alice moves and Bob is observing her. She moves with a relative velocity, v, and shoots off the paint balls with a velocity, u. The velocities will add, meaning that the answers to clocks are initially synchronized. So there is a small tA equals small tB equals 0. Once the clock hits, you can hopefully agree that Alice and Bob will agree that the times when the left clock and the right clock hit are the same, right? But now, we want to enter the situation where we use light. So we use a phaser in order to do the very same. So Einstein just postulated that the speed of light is constant, is c. And it's the same in all reference frames. And it's independent of the emitter, which means that we cannot add the velocities anymore. So the velocity, as seen by Alice, of light is c. The velocity of the same light by the moving observer, Bob, is also c. So here, we can conclude that the times for Alice for where the situation is stationary, both clocks will hit at the very same time. Those two events, clock one and clock two are hit are simultaneous. Why, for Bob, this is clearly not the case. You can see here that this lagging clock is being hit first, while the leading clock is hit a little while after. So if Bob and Alice now meet and they discuss whether or not those two events happened simultaneously, they will disagree. For Alice, those two clocks were hit simultaneously-- at the same time for her. But for Bob, the first clock was hit first and the leading clock was hit second. All right, we can conclude the two events can be simultaneously to one observer but not to another one. This is rather confusing. And we will see and use this fact a few times later on when we discuss the famous paradoxes of special relativity. So let's look at this in a concept question to just make sure that we're all on the same page. Again, we discuss your diagram three. Alice move to the right. Bob is the observer. Alice's fires her phaser at times equal 0. Then the situation unfolds. At time TA, capital TA, Alice observes that both blocks are hit. At time TB1, Bob observes that the left clock is hit. At time TB2, it's the right clock. Which of the following answers is correct? So here, you want to stop the video and think about which of the answers is correct. So moving forward, the correct answer is number three, where TB1 is smaller than TA is smaller than TB2. So again, the leading clock lags. The leading clock has a larger TB, which means that clock ticks a little slower. And again, the two events, they can be simultaneously to one observer-- Alice, in this case-- but not to another, Bob. All right, let's look at clocks a little bit more and design an optical clock. So here, the situation is as follows. We have two mirrors in which we inject light. The light travels up and travels down. And that's what we call one clock tick of this optical clock. The length between the two mirrors is L. So for Alice, she has this clock in her hand. And she can happily observe the ticking of the clock. OK, Bob observes Alice's clock and compares it with his own identical clock. There's a relative speed between Alice and Bob, and that's v in x direction. Now, the task for you is to relate the clock ticks which are observed by Bob and the ones which are observed by Alice in Alice's clock. So again, stop the video and work out the algebra. The answer is going to be, again, surprising. So if you do this now, we find this picture. So we calculate how long does a clock tick take. The light has to travel to L with a velocity c. So the clock tick is 2L over c. The length can be expressed as c times t delta tA over 2. For Bob's, the situation is a little bit more complicated. And we have to use Pythagoras in order to calculate the length. So we define that the length the light has to travel is D, then the delta tB as Bob observes this is 2 times D over c. Again, for Bob, the light travels with the speed of light. Einstein just postulated it. And then we find the length as expressed to the time as c times delta tB over 2. The length in x is simply given by the relative velocity, v, times the time it takes for the clock to tick-- v times delta tB. So then we can express D square via L squared plus x squared over 4 and use those expressions here. So we just use this for L, this for D, and this 4x, we find this expression here. All right. And then we solve this for delta tB. And we find the relation between delta tB and delta tA and can find that it's 1 over square root 1 minus v squared over c squared, which is the Lorentz factor. So we just used a simple clock and Einstein's postulate derived time dilation. We find that for Bob, Alice's moving clock moves slower. Great. So again, gamma is 1 over square root 1 minus v squared over c squared. We often use, in short, beta as a relativistic velocity. It's unitless and defined as v over c. gamma is always greater or equal to 1. And it's mostly one for everything we observe in nature. So in one of the p sets and also here, I invite you to simply calculate values for gamma for things you might think are fast-moving objects. So we start with a fighter jet. We look at the International Space Station, the Earth around the sun, the particle which almost moves with the speed of light, and the proton at the Large Hadron Collider, which is only 3 meters per second slower than the speed of light. So again, stop the video and work out those numbers. You will need a calculator for that. So if I do this, I find for this very, very fast F15 fighter jet, which moves with speeds of 2,680 kilometers per hour, that the number for gamma is 1.00000000000, which is 11 zeros, 3. So we find this very, very small number or number which is very, very close to 1. The duration for the International Space Station changes a bit-- only 9 zeros. For the Earth around the sun, the Earth is really, really fast, travels a long distance. Every year, we travel once around the sun. And you know, every year you get older. You have a lot of mileage on your back. Here, you have eight zeros. Particle which moves with 0.9 times the speed of light, here the gamma factor is very different from one. It's 2.3. And the protons we have at the LHC, they have a gamma factor of 7,000. So you see, once you get close to the speed of light, the gamma factor approaches large numbers. And that's where our relativistic effects really are visible. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 102_The_Large_ElectronPositron_Collider.txt | welcome back to 820 special relativity in this section we want to talk about particles in accelerators and we use the large electron position collider at cern as an example lab the large electron position collider was a collider which was operated in the late 1980s up to the year 2000 at center of mass energies from 91 gev to up to 209 gev this energies were used to probe the z-boson to produce pairs of w boson and also to look for the higgs boson which didn't quite work because energy wasn't quite sufficient the 27 kilometer or about 16 mile circumference collider is now the home of the large utron collider etc and it's being used there to collide protons with protons um lab had four experiments alef delf delphi l3 and opal and i was happy to you know study data from the opal detector as part of my undergraduate undergraduate thesis so this was very good memories there but that's not the topic today so the topic of the day is to find out how quickly or fast the particles in the collider moving when we know the center of mass energy so if you know the center of mass energy to be 209 geb we know that the energy in the beam is half of that 104.5 gb um so now to figure out how fast the electrons are moving the electrons and the positrons we can use our relativistic total energy for the electron or for the proton which is the kinetic energy plus the rest mass of the electron or positron which is m naught gamma times c square so with a mass of 511 kv over c square and this energy of 104.5 gb this results in a gamma factor of 200 000 and if you do the mass you'll find that electrons and positrons are almost moving with the speed of light in where moving this is speed of light um as a in a fun additional fact is we are working right now on proposing a even larger collider which will then allow it to efficiently go up to energies of 350 geb and that collider then will be able to you know study the higgs boson with precision but also look at top pair production top crop hair production in this kind of color another fun fact is that you know cern the collider which is about 100 meter under the surface um is spanning two countries switzerland and france and so each particle lapse lab i mean makes a lab makes about 11 200 laps per second that means that there is about 44 800 border crossings for an electron position so they all constantly have to show their passport when they're moving around in that to look at its second example here with a small accelerate we can accelerate electrons through a electrostatic potential of 511 kilo volts or 0.5 11 mega volts um the total energy then is again the kinetic energy plus the rest energy which is in this example 4.5 11 mev plus 0.511 mev and so here we find the gamma factor of 2 and a velocity of 0.866 so even in this smaller electrostatic potential of 500 kilovolts we find that electrons are moving with very very high velocities with velocities very close to the speed of light you |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 62_Twin_Paradox.txt | MARKUS KLUTE: Welcome back to 8.20, Special Relativity. In this section, we talk about the famous twin paradox. It's probably the most famous paradox in special relativity. I want to get to the bottom of this, and understand really where there is a conflicting or contradictory statement in this story. Let me just first say that this is personal to me. I do have a twin brother, and you can see three pictures of myself and my twin brother here. We were very little on our first day of school in Germany. You get a little box of candy when you go to schools to make it more attractive to actually learn and study. And then a picture, which is probably already about 10 years old. What you can take away from here is clearly moving clocks are slow. It turns out that my twin brother lives in the very same village in Germany where I grew up, where we both grew up, while I traveled the world constantly and constantly on the road between France and Geneva and Switzerland and the United States. And again, I think there is no dispute here. It can be seen from this picture that your professor looks much younger. I even have a more recent picture. This is two years ago. The German Kris Kringle Market, where I asked my brother to take this picture for this class, for 8.20. And again, I think the answer to the question is clear. Professor Klute has aged less. All right. On a more serious note, we're going to quantitatively understand and analyze the situation. And we use Bob and Alice again. In this situation, here Bob stays local. Alice has a spacecraft, and she moves with the velocity of 0.6, six times the speed of light, a gamma factor of 1.25. The travel takes her to a distant star, which is in this example It's three light years away from Bob, measured by Bob. The journey takes her, on Bob's clock, five years, and the return takes another five years. She doesn't spend much time. She wants to go home as quickly as possible. If you analyze this, from Alice's perspective, we see that for Alice, the journey takes four years, and the distance traveled for her in a spacecraft is 0. From Bob's perspective, the journey, as seen by Alice, is only 3.2 years long. And so we find that there's already a conflict. If you add the times together, both ways, the inbound and the outbound ways, 6.4 years is not equal to 10 years. So there's already a contradictory statement in this story. But the key to the understanding of this problem is that Alice, in order to return, has to change reference points. And there, we do have to resynchronize the clocks, if you want, or add a specific extra factor. And we'll go back to this when we look at space and [INAUDIBLE]. So the time, as seen by Bob, is 3.2 years for the outbound journey, and then 3.6 years in order to resynchronize the clocks on the return, on the turning around. And then 3.2 years on the return, which makes 10 years. And so that observation of Bob, of Alice, is in agreement with Bob's own clock. All right. So we saved the day here. Let's look at space-time diagrams. The outbound journey is shown here. You see I've got it-- in addition to Bob's reference frame, I've plotted Alice's reference, and it makes it easier to understand what's going on. So we see in Alice's reference frame, the journey takes four years. If you then go back to the position in which Bob is, 3.2 years have passed. So this is iffy. At the time when Alice arrives we go back to the position of Bob, 3.2 years have passed. We then turn around and ask the very same question. At that time-- it's still four years-- we go back the other direction now to Bob, we are already much further ahead, 3.2 years plus 3.6 years. And then the journey continues, and we add another 3.2 years to the journey. When Alice and Bob reunite, Alice aged by eight years, 2 times 4, and Bob aged by 10 years. So the question now is, there may be a paradox here. Is it possible that we missed somehow that by-- and try to understand why this is a probably not symmetric. Why can I not just use the other reference frame, and just declare that Alice stayed stationary in her spacecraft while Bob moved away with Earth and then came back? Why are those two things not consistent? The answer is that it's not Bob who has to change reference frame, but Alice. It's Alice who has to do this. There is where the asymmetry is. You can argue if you want that, in order for Alice to do that, she actually has to accelerate. But we don't have any sort of discussion of how the acceleration actually went about. It's really the change in reference frame which is crucial in this discussion, and causes the asymmetry between Bob and Alice. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 52_Velocity_Addition.txt | MARKUS KLUTE: Welcome back to 8.20, special relativity. We're going to continue our discussion of galactic space travel. Here the situation is slightly modified from the previous one. We still have Alice being our ground control and Bob riding on a spacecraft in order to explore planetary systems and solar systems. This situation is different in the sense that the spacecraft has an escape rod. So it's able to send probes to planets in order to study them. And so, in this specific case, the velocity of this escape rod is uB delta xB over delta tB, as measured in Bob's reference frame. The direction of this velocity is the same as the direction of Bob's spacecraft when looking in longitudinal direction. Again, as a reminder, velocity is distance over time or delta x over delta t. So the question now is, what does Alice observe? Because this is an activity you should try to work out yourself, stop the video. I'll just continue here. So, if you calculate now the velocity as seen by Alice, delta xA over delta tA, that's given by gamma times delta. We just use Lorentz transformation-- gamma times delta xB plus v times delta tB over gamma times delta tB plus v over c squared delta xB. Now, we can cancel the gammas and take out delta tB out of the brackets. And then we find uB plus v over 1 plus v over c squared uB. So this looks like an addition of velocities with a correction factor of 1 plus v over c squared uB. Good. So does this make sense? So, whenever we have a calculation like this, we should check that it actually works out, that extreme cases are preserved, and that units work out. So, again, the units work out here. On both sides, you find meters per second, the unit of velocity. If you check now what happens if we set uB equal to 0, we know that the escape rod is at rest. There's no velocity with respect to Bob's reference frame. In that case, we find that uA is equal to v, exactly the velocity difference, the relative velocity between those two reference frames. If that velocity is 0, we find uA equal to uB. Again, that's expected. If Bob and Alice are in the same reference frame and they observe the same escape rod, they better measure the same velocity. And, lastly, if we now, instead of having an escape rod, we send a beam of light out, which has a speed-- a beam of light has the speed of light, uB equal to c, we find that the velocity observed by Alice is also c, which brings us to an interesting point here. Yes, we still add velocities with a little bit of a relativistic correction, but we will never get larger velocities of the speed of light. So the speed of light is an absolute speed limit. Let's analyze this a little bit more in the context of our light clocks. So what now happens if the velocity is equal to c is that gamma goes to infinite? And, in the context of the light clock, you can notice that the upper mirror can never be reached. It's moving with the speed of light, the same velocity as the light itself. So light is never able to reach this. The clock will stop. All right, so there's an absolute limit of velocity at the speed of light. OK, so now, so far, we discussed only velocities in the direction in which the two reference frames move or the second reference frame moves with respect to the first one. What now happens if we consider perpendicular velocities? So, in this case, Bob's spacecraft, this escape rod goes up. Maybe he's circling the planet, or he's just approaching the planet. And [INAUDIBLE] when it [? pops at ?] that planet specifically. So here we want to work out the example in which the perpendicular velocity is not 0, but the longitudinal velocity is 0. So what does Alice observe? So we do this as a concept question. Which of the four answers is correct? Is the velocity unchanged because we are studying perpendicular velocity? Is the velocity smaller, larger, or you don't know because you actually have to figure it out, work it out? OK, so the velocity, as observed by Alice, is actually the absolute value is smaller than the one observed by Bob. We can do the very same calculation. So we have uB y is delta yB over delta tB. And then, for Alice, this is uA y delta yA over delta tB. So the y-component, the length measured in y-direction between Bob and Alice, is invariant, as we saw in the previous section, but the time is not. So we do have to do the Lorentz transformation of delta tA and find that, in the case where uB x is equal to 0, that we just have to divide uB y over gamma. Situation is a little bit more involved when there's also a longitudinal velocity, but you see here how this would unfold. So 2 was the correct answer here. So, while the length in longitudinal directions are invariant, the velocities are not. And that's because time is suspect. Time needs to be corrected in the two reference frames. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 55_Causality.txt | MARKUS KLUTE: Welcome back to 8.20. In this section, we're going to talk about cause and effect and causality. And we do this with an example which talks about the good guys and the bad guys. There's always good guys and bad guys in the universe. In this example, those two groups, they were fighting a war. And after some very long and grueling time, they were able to sign a peace treaty. So this happens in this story at year number 0. So in year 0, a peace treaty is signed. And the good guys, they go back to their families. They go back home to their planet. There's a spacecraft, which is able to go with a velocity of 0.6 times the speed of light. The bad guys. That's the bad guys. And so they develop a faster-than-the-speed-of-light spacecraft to follow the good guys. And they succeed with the development of the spacecraft after four years. So they do. They follow the good guys. And in year number 5, the bad guys attack and destroy the good guys. The challenge for you now is to explain the story and answer the following question. When and where does the attack happen, from the good guys' perspective? And when did the bad guys invent the spacecraft, from the good guys' perspective? You will see some surprising results in this analysis. So please stop the video and try to work this out. I recommend drawing a space-time diagram in order to get to a good picture of what is actually happening here. If you do this, you find the space-time diagram here for the bad guys. So again, there's a number of important events. In year number 0, the peace treaty is signed. And then the good guys, they start traveling away at 0.6 times the speed of light. So what you see here is a world line of the good guys. In year number 4, in the reference frame of the bad guys, they developed a spacecraft, and they keep following the good guys. And here in year number 5, the attack actually happens. So if you analyze this for the good guys, you find that the exposition, the position of the attack, is in their spacecraft. So let's figure out whether or not this is correct. You find gamma and xB minus v times tB. If you put in the numbers, you find, luckily, the exposition is equal to 0. At what time did this event happen? Here, for the good guys, you find gamma times tB minus v over c xB. And again, if you put the numbers in, the attack for the good guys happens in year number 4. But then the question is, when did the bad guys invent the spacecraft in the reference frame of the good guys? And if you put in the numbers, you find the invention actually happens in year number 5. So the attack happens before the spacecraft is invented in the reference frame of the good guys. And this clearly violates causality. You cannot use a device which is not invented yet unless you travel faster than the speed of light. You travel in time. All right. I find this a really funny and geeky example. And the reason why this doesn't work out, why it's geeky, is that they travel faster than the speed of light. So let me conclude this part of the section here with a concept question. We have an event A, and ask ourselves, can event A cause other events? So event A might be the invention of a spacecraft. Can event A cause event C, event D, or more than one event, like B and C or B and D? Note that the time access here has units years and the x-axis has units light years. Again, stop your video here. Think about your answer. Maybe you draw in the space-time diagram in order to find the answer. And the answer, the correct answer, is C. Only event C can be caused, because only C lies within what we call the light cone of event A. In order to impact, in order to cause an effect on any other event, the message needs to be traveling with speed smaller than the speed of light. So that means that, in order to reach event C, we can design a spacecraft, or we can send a light beam to this event or in this direction. And then just wait a little bit, in order to make any sort of impact on event C. We cannot travel faster than the speed of light, which would be necessary in order to reach event D. And you can also not go backwards in time, this message is, in order to reach event D. |
MIT_820_Introduction_to_Special_Relativity_January_IAP_2021 | 43_Length_Contraction.txt | MARKUS KLUTE: Welcome back to 8.20. In this short video, we want to discuss length contraction. We're going to actually derive length contraction. And we do this with an experiment. The question is how long is Bob's spacecraft? The experiment is conducted by taking two pictures. So let's [? read ?] it here. The first picture is when Alice's and Bob's spacecrafts just start to meet. Bob's relative to Alice's, is moving with a velocity, v. And we take the second picture when the back of Bob's spacecraft is meeting the front of Alice's, just like it's shown here. And the pictures-- take pictures of clocks. So in the first picture, we see Alice's clock showed a T A1 and Bob's T B1. And for the second equivalently we see T A2 and T B2. So your task is now to express the length of Bob's spacecraft as Alice sees this from these pictures and as Bob now see it from his own pictures by comparing the time and the velocity. Now pause the video and try to work this out. So I did this here for you. We calculate the length. We can do this with the velocity and simply the product of the velocity, and the time difference, and the pictures as shown [? of ?] those two clocks. We see this for Alice, and we see this for Bob. And now we can start to compare. For Bob, Alice now is moving. So the time difference in Alice's clock will be smaller by 1 over gamma compared to what Bob sees on his own clocks. Bob says, "Your clock is slow." So we can use this. We can then calculate the length of the spacecraft as Alice sees it equal to v times delta T A. And we'll just use the time dilation here. In this equation, we find that the lengths are actually not the same. The length as Alice sees this is 1 over gamma times the length as Bob sees this of the very same spacecraft. So the length of the spacecrafts are not the same as seen by Alice and seen by Bob. Now here you have to see that, in this example, what I just did is I changed around who's moving and who's resting. So here the observation of Alice of Bob's spacecraft is that of a moving spacecraft. So Alice sees a moving spacecraft, which is shorter than the spacecraft itself at rest. So how can we resolve this? Alice will argue that the two clocks Bob used are actually not synchronized. And if you paid a lot of attention, you'll see that I'm actually not looking at the same clock. I'm [? having ?] a look at the clock at the beginning of the spacecraft and at the end of the spacecraft. And while they're synchronized for Bob, they're not synchronized for Alice. Who's right? Both are right. They're just observing events or sequences of events as two different [? reference ?] points. |
Lecture_Collection_Introduction_to_Robotics | Lecture_15_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started well today we have uh a very nice video actually uh this is one of uh my favorite videos and um I think you let's see if we could have the light off please the projectors [Music] [Music] [Music] this video was made in 93 by Mark [Music] oh [Music] well it's not finished continue keep the light up so simulations does not always work in robotics it's very important to remember that experimental validation is very important yeah [Music] y yes okay that was quite amazing uh performance of this robot let's go back to control so I'm sorry for the figure on the left it's a two degree of Freedom manipulator um revolute revolute and um last time we saw how we can develop a p controller for a one degree of freedom and then deal with the control of that uh robot if we add one more degrees of freedom that is if we have a a robot with some Dynamic coupling we are going to see that the performance of a p ID controller is not going to be satisfactory especially if we are tracking a trajectory so this equation of motion you're familiar with I'm going to rewrite it in two equations so here is the equation of motion and you have two joints so the vectors are 2x one and the mass matx is a 2X two so if we split rewrite this equation in two equations now uh you're probably more familiar with this form so what do we see we see here sort of mass acceleration or inertia rotational acceleration equal torque but there are all these additional terms that are appearing so essentially the acceleration of joint two or link two is appearing in the first equation and is affecting the dynamic behavior of the first link so as you accelerate you have a coupling through M12 and that is producing uh disturbance forces or inertial forces on the first joint we have opposing forces coming from the acceleration of joint one on joint two we have on joint two what kind of forces here come on you have immediately to recognize this form what what kind of forces are these centrifugal forces on the top coras so product of velocities coris square of velocities centrifugal and these gravity gravity forces so so all of these are going to have an impact on the robot and it's really difficult to imagine that these are zero or or treat these as disturbances they are going to be there and as you accelerate one joint you have an effect on the other joint so really the system that we talked about this controller that is controlling a joint a link uh is going to receive disturbances coming from the gravity and coming from from the motion of the other link you see that which means that well to ignore it we me we need to really uh I mean treat these as disturbance forces but these are large disturbance forces and the only way you can deal with the with the rejection as you know is increasing KPS to reject your disturbances so treating the system as an independent joint is not a valid assumption and in fact most robots today in Industry still use this scheme they say okay I'm not going to care about uh the motion of joint two I'm going to uh put a controller with large gains so I can control each of the joints separately whereas in reality the end degree of Freedom system each of those joint is receiving those coupling forces and these coupling forces are interacting so there are something I mean not not not only disturbances but the fact that they are interacting there is something about the stability how can we say that this system is stable if we are controlling it with a PD controller right because you have this interaction taking place between The Links at the move so the stability of a system that is highly non nonlinear and interactive uh in the way we saw is not evident how can we how can we prove that this system with a PD controller with all these disturbance forces coming from the interaction is stable anyone Yes Energy so in fact what we're going to do to prove this is to go and do an abstraction to see what kind of energy we are putting in the system and how we can show that the system is stable in fact in the uh late 7s there were a lot of Papers written about the PD control stability to prove that yes the system can be stable just with PD controller so instead of looking at the equation in this form a torque equation with a controller that is with this is a vector uh PD it's not just a a component it is applied to all the joints so what we're going to do we're going to go to uh an equation you are very familiar with which is is which equation of energy we're going to use the lrange equation so instead of analyzing this component we will go and write this equation in this form so now we are doing an abstraction to the a scalar quantity which is the energy the kinetic energy and the potential energy in this form we can analyze the impact of those controls we are applying to the system and immediately we will see their effect so when we apply the torque torque equal minus KP Q minus Q desired essentially we are putting a first term this proportional term is a gradient right it's a conservative Force it's a gradient and this gradient is coming from the potential energy of the springness KP which is varying on different joint KP is a a matrix diagonal matrix you can imagine you have different KPS for different joints but still it's a gradient of some energy so now how can we prove the stability of the system it's now much easier yes isre so what we are going to decreasing or equal to zero or basically what what do we have here you see a v here and a v there so what you are doing is basically by adding this if you move this to here you can see the effect of your control your controller is essentially here is the inertial forces and your control is modifying the initial potential energy of the system which is the gravity forces to change it to VD this potential energy desire so instead of having your energy uh making you fall down now you have some energy that is going to help you go to those desired goal positions in addition on the right hand side we have a term that is minus K VQ do so this this term is going to bring additional stability that is it's going to bring uh atic stability because it will dampen and oppose the Motions along the Q dots so you remember this condition that these forces are acting against the velocity so as long as your KV is positive this condition is satisfied and we can show that uh essentially this system is going to achieve this motion so you will go to the minimum of that potential energy and because of this term you are going to be able because this damping term you are able to asymptotically stabilize your system so yes a PD controller is stable but this doesn't tell you anything about the performance it tells you yes you're going to move and reach that configuration but along the motion you have all the coupling so in here whatever coupling you have it is going to appear in the kinetic energy derivatives and that will produce the inertial forces so if you're moving to a goal position there is no problem you can can move and reach that goal position and you will have a small offset because of the gravity you can in fact put in VD some compensation of the gravity and take it out so you can reach that goal position but the performance along the trajectory are not going to be if you are tracking a trajectory are not going to be good so how do we evaluate our performance what are the parameters in this controller that affect the performance yes the KP and the KV the KV allow us to uh dampen the system and KV KP allow us to reject disturbances so what would you like to do in order to reject the disturbances very very large as large as possible so you need High gains right so if you have uh if you're able to implement High gains then you get better disturbance rejection however your gains are limited in practice you start cranking up your gains and you start we we had the the the simulation and I showed you as we increase the gains we reach a point when we start vibrating and this was only a simulation in the real system what do we have in the real system we have structural flexibilities we have flexibilities in the links we have flexibilities in the drives so if you take the Puma the Puma has those Motors placed inside and they are connected to the joints through rods and they have flexibilities and when you start cranking up your gains what's going to happen higher gains you start to to have a Clos Loop frequency closer to your flexible structural frequencies and that leads to un stabilities you excite those modes so if you have flexibilities you have to make sure sure that your gains result into Clos L frequency below your your structural flexibilities and not only that you have time delays in the actuation So you you're sensing you're measuring the encoders and then you are going to compute your controller and then you are sending this controller to the motor and the motor has some time constant so you have some delays and you have also the sembling rate the servo Loop is running with some sampling rate that needs to be quite high and this is not anymore a problem it used to be a problem when we had slower computers but now the CPUs are so fast that this is not anymore an issue so here is a rough idea about the limitations you have if we take the resonant mode the lowest resonant mode on your system you have to make sure that your Omega n is lower than half of that resonant mode in order to be able to avoid uh catching with those uh structural flexibilities resonant mode with term in term of delays you are limited also by 1/3 of the frequency of the delays well as I said this is not anymore a problem so the the key issue is this lowest structural flexibility in your system and it is there and believe me you start cranking these gains up tuning your gain and you start to hear the vibrations so it's very difficult to achieve High gains and remember the delay is in time it's 2 pi divided by the the I mean the frequency is 2 pi divided by the time of the delay this toe is time okay so what is the solution if we cannot really crank up those gains so high that we will cancel all disturbances Dynamic disturbances what is the what is the solution so if you remember the problem is like this at each of the joint you are getting this disturbance Force so what should we do I have the model I have good estimate of M JS good estimate of the gravity because you you you just uh did a lot of homework with Dynamics so you know how to to find the structure you can uh eventually identify the constant involve the masses the center of mass you can measure all of these and then what can you do oh I'm sorry you're sitting so far on the left see if you use a model of your plant of the spec especially the nonlinear part of your plant in your control LW and effectively you can turn it if you and also if you diagonalize the mass Matrix you turn it into a decoupled uh simp a very simple one over s squ plant yeah well I mean the idea is Right basically let's use this model if we if we know the Dynamics in advance we can use this model and essentially what what do we need to do we need somehow to counter this Force so how we counter this Force we computed and and and and like injected back in the system to counter now the diagonalizing diagonalizing the metrix is not really the issue because if you uh essentially you want uh to compensate for the coupling forces so what you want to do is is essentially for joint two somehow to compensate for those terms these coupling terms so this brings us to something we discussed earlier you remember this unit Mass system that we can achieve by compensating by scaling our controller with a gain that is proportional to the mass m alpha plus beta or Alpha fime plus beta you remember that well now we are going to extend it there and what we're going to do is to go to that first equation in The Matrix form and we are going to say we are going to take a torque that is proportional to the mass Matrix so our torque will be some estimate of the mass Matrix times torque prime plus a beta that compensate for V and the gravity and this is what we're going to do oh nice what was this so what we're going to do is we take this model and we imagine that somehow after a lot of identification work we identified some estimate of the mass metrix so the mass metrix actually you know its structure you know the structure but it has in it parameters inertial parameters distances that you don't know exactly but you know approximatively within Epsilon so you take an M hat of the mass Matrix and you select a controller torque Prime so this is Alpha F prime plus beta so think about just taking a controller that compensates for the gravity estimate so essentially what you're going to say if this is perfect I take this out so this is not anymore in the equation right if this estimate is perfectly equal to this this will drop your controller will be M acceleration equal m torque Prime if m is exactly equal to M your unit acceleration is equal to torque Prime so basically you are able now to drive the acceleration Thea a double dot with Theta Prime with torque Prime and your torque Prime is now the input the control input of the decoupled system the unit Mass system so if we select this structure and if we apply this structure to the equation and if we multiply this whole equation by m minus one to bring Theta double dot out what you obtain is m minus1 m hat torque prime plus m minus1 v minus V hat G minus G okay suppose that g hat is identical to G just suppose for a second and V hat is identical to V the second term is going to disappear will be zero M minus one and M hat if M hat is identical to m m minus one m is identity so with perfect estimate this is your behavior magical so essentially the nonlinear Dynamic decoupling of a multibody system relies on this structure if you use this structure to take a controller any controller not a PD controller any controller you want and this controller is going to to be scaled by the mass Matrix and add to it nonlinear Dynamic compensation to compensate for the terms nonlinear terms in your controller then you achieve Dynamic decoupling with a behavior that is controlled by torque Prime and torque Prime now is controlling the unit mass unit inertia of all your degrees of freedom so if you select your torque Prime in this way like this is a trajectory tracking to track Theta desired Theta do desired and Theta double do desired then you are going to be able to achieve a closed loop in this form if you have perfect estimates now in reality do you think we will have perfect estimate no but if we don't have perfect estimate we have to have good estimates in order for this to work so you will have a difference between V and vad a difference between G and Gad and some difference between there and that will bring some Epsilon error disturbance and this disturbance will appear here right it's not going to be equal to zero you will have some disturbances so what is then going to be important is that your KP Prime and KV Prime to be selected so that you reject those disturbances but those disturbances are much much smaller than the the initial Dynamic coupling that we saw before the compensation and uh the decoupling that we use in this structure so this controller as long as your estimates are reasonable is going to give you decoupling with some errors and you still need relatively large gains but much much smaller gains than the one one will be needed without any Dynamic compensation okay now it turned out that you should be able to identify your Dynamic the rigid part of the the systems Dynamics to a large extent to a small Epsilon so those epsilons are small however in the initial equation we are neglecting what are we neglecting on this first equation we are not showing here this is the system the robot what is missing there no friction so there is friction there is actually nonlinear friction that is difficult to model and there are a lot of other nonlinearities and then you have the higher order dynamics that we are not modeling we are modeling just second order Dynamics the flexibilities are not modeled so you are going to find limitations here you can try to compensate for part of the uh fraction forces but it's not going to be perfectly compensated for so higher gains are very important here in practice compensation for the gravity suppose we do not we we we made a big mistake with the gravity the gravity is we estimated the gravity and we we were wrong and we we we selected Gad to be zero what is the effect of that on the robot so you will have just like a sort of constant torque offset that is will be divided by your your KP Prime Times m so that's not going to affect the stability you will you will just Reach the the final goal position with little error so what is the effect of M minus one and M hot well if you make a mistake in the estimate of the mass matrix it's going to sort of uh scale up or down your gain your KP so your KP will be with this structure M Hat Time torque Prime M Hat Time torque Prime means KP Prime is multiplied by m hat and if that estimate is little wrong you are changing your gains up or down so it is a sort of like a gain that you will have what about V and V hat so estimate of M hat if you make an mistake in the estimate of mut as long as mut is positive definite from your structure you shouldn't make a mistake if you make a an estimate the effect of it is just scaling up and down your gains it's not going to affect too much the stability as long as you have positive definite Matrix what about V if you make a mistake in V hat so what is v v is function of theta and Theta do you make a mistake in V hat which is coming from your estimate of theta dot theta. hat you're estimating to compute V hat and that might be really dangerous because V is function of velocities and if you make a mistake in your estimate of centrifugal corus forces you might create sort of negative feedback in velocity which could destabilize the system so in fact if you do not know vat well it's better just to treat V hat equal to zero rather than making a wrong estimate so beside V hat errors in G or in M are not a problem for the stability it's V hat and if you don't have a good estimate you you can just use a zero estimate of VAP okay so this is the overall control system basically what we are doing is we are doing a feedback that is compensating for for centrifugal coriolis gravity and all these forces and then we are scal in our input so this is torque Prime it's scaled by m hat and we are adding these to produce the torque so the torque is M Hat Time torque prime plus those estimates and then we are able now to compensate for centrifugal corus gravity forces and scale our gains and decouple the system and the overall behavior of the system will be decoupled so now we are able to control our robot but what we did here we controlled our robot in term of its joint motions where do you get these trajectories so you have your camera you have your hand and you want to move to the goal position remember our initial problem you want to move there how how do you compute all these trajectories for the joints and this brings the problem of inverse kinematics uh that you need to use to transform your task description in term of joint trajectories well this brings us to task oriented control where instead of thinking about the inverse kinematics as a way to transform the trajectories we go directly to controlling the task and I talked about this earlier but let's take a simple example to illustrate this problem so here is a robot a mobile manipulation platform and our goal is to move this robot and aor to move it to this goal position to grasp this object so how do you do do that en join space control quickly I have a lot of things to talk about so quickly think about it what what do we need to do in order to go to that goal position maybe translate uh positions to uh Ori to Joint orientation well I mean essentially if you if you go like little bit uh uh to a higher abstraction what you can say well I need to imagine the final configuration where I'm going to to be and in fact I need to imagine all the configurations of the robot I need to find this configuration how many configurations like this you can have in this case to reach this configuration you can be here you can be there you can be in infinite ways so the question is you have infinite ways from starting from here to go to there because of the redundancy and it's really difficult a prior to say which one is better it's really difficult when you have redundancy to resolve all that uh U redundancy and decide about it ahead of time and if you do that actually you lose a lot of inertial properties that you can use in the control of the road robot as we will see a little bit now and uh much more later when we analyze uh the the fact that the this base is heavy this arm is lighter and if you are able to use this fact of macro structure and mini structure and combine them you are going to be able to better use the uh reduced effective inertias that are coming from macro mini uh uh uh structures and controlling the robot in a much better way that is we will see that we will be able to really reach rapidly and then stabilize the micro structure reach with the fast dynamics of the system and then stabilize the system so how can we control this differently how can we move from here to the goal position without without deciding a prior what final configuration we're going to have okay what the structure does exactly and this was the idea I found many many years back which was to say I mean I don't need to know exactly how the structure is going to move at this time let's just move the end of factor so pull on the ector and pull on the end factor is very easy you just apply the gradient of a force that is pulling on the end Factor right so if you you virtually apply a force at the end of factor that is the gradient of this Force which has its minimum at the goal position essentially this end factor is going to fall in this potential energy and it will move towards the bottom right well you don't know exactly how it's going to move it might like because this is not a point Mass you're dropping in a potential energy it's a multibody system and U this is what we call task oriented or operational space control operational space control relates to what we are trying to do with the robot it could be the end effector if we have multiple arms it could be both effectors or or it could be any set of descriptions related to the task you have a question yeah do you have to give priority to different joints and you do this or is ited into the equ so let's see it little by little and then you see how you can put priorities how you put constraints uh on Wednesday I will describe how this simple concept of a combination of an attractive potential energy that pulls you to the goal combine and now this is what is interesting is I'm going to move to this goal right and this is the goal but as I move there are obstacles on the way now I can apply repulsive forces on those different links that are interacting with the environment to move them away which means that if I was doing this in joint space I need to do the whole motion planning in ahead of time to make the decision about how to move but suddenly an obstacle comes and move or I discover that this obstacle is not there how can I readjust and recompute or the inverse kinematics I would need to do replanning in real time whereas in here you are able directly to affect those constraints so we will see uh a lot of uh uh issues that relates to how in real time we can deal with many other criterias related to how far away I would like to keep the body of the robot from there what type of angles orientation I would like you can add all of these as well but this will bring us to how we control a redundant manipulator and these are issues that will come in advanced robotics next quarter but essentially the basic concept is very simple the basic concept is to say I want to move to those goal positions you can have goal positions for the end effect factor for the base for any part of the robot and then I need to protect the robot from touching anything so I apply these repulsive forces and you drop the system in that field and it will just move and go there now I didn't talk anything about the Dynamics but it's entally this is the the basic concept now how can we apply this Force so the gradient of a v is a force how do you apply this Force to the motors so you want to apply a force here F what would be the corresponding torque so it will be torque equal F we need some something to relate this Force to the torque J transpose F equal torque that's it so if you just apply a j transpose F where f is this gradient of your goal position in X you apply this to the system and the result is going to be that you will will move there and it works again it doesn't deal with the Dynamics your motion is not guaranteed to be decoupled you are just like pulling all these links and moving so what we need to do is really to look at the dynamic properties of the system and evaluate how the system is behaving when we are moving in those ways the other interest in thing about this controller is and this is really the the fundamental property of those controller is the fact that now you have a way to deal with contact because when we are going to move this object so I grasp the object and suppose the task is to move this object over the table while it's in contact and I need to push on the object I'm doing some assembly and I'm touching the object and applying a force to apply a force in some direction it's very easy if you know your contact forces you you do J transpose F contact but now to move we are translating the control for motion in term of f motion so essentially we are unifying the two problems in one problem you just add them and you take the sum of the two and now you have a unified way to control both motion and forces you have to make sure that your contact forces are in the proper space that is uh in in this case it's the vertical directions and your motion control are in that space and when you go to uh more complex contact you have to do some selection and project your controllers in those spaces so the dynamic problem that comes with this is quite interesting it is a problem that comes from the fact that when you apply this Force to the end of factor and as I said this is not appointments this is a multibody system you are going to obtain an acceleration so if you have a point mass and if you apply a force in the X Direction This Is The X Direction your acceleration will be along the same direction in this case your acceleration is going to be somewhere else because of the coupling there is a coupling between F and acceleration so you need to find the relationship between forces and acceleration at this point there is a mass Matrix associated with this x that allows you to evaluate the Mass properties in the different directions and the coupling between them so if I'm here and if I push in this direction I will feel sort of an effective mass of like say 5 kilg but at the same configuration if I push in this direction the mass is different 2 kilg and if I rotate about this axis I will feel some inertial force that is different from this one and this is your mass Matrix basically this is different from the mass Matrix that we were talking about the inertial properties in joint space the effective inertia about this axis that is varying the effective inertia about this axis and the coupling between them now if I push in this direction I will see this motion because there is coupling between the y direction and the Z Direction and The X Direction and the rotation between them but if I am able to model this then well there is centrifugal coriolis gravity forces as well so we are just going to be able to model the Dynamics in that space for this point and if we obtain this model which is simply a relationship between forces and acceleration velocities and positions at different configuration you have different properties once you have this model you can do the same thing we did with controlling the robot in joint space by decoupling making estimate of this decoupling and making an fstar or F Prime now we we select an F that is proportional to the mass the estimate of the mass Matrix with compensations of centrifugal coris and gravity forces and then you can control it you find your F you multiply it by J transpose and transform it into term of torque and suddenly now you are directly control calling your robot in operational space in the space of the task and that would allow you to uh control directly your errors in that space so your Force control will be function of those estimates in the same way with the same structure that we saw just earlier okay so to discuss this model I'm not going to go to the redundancy we can do that in the Redundant case and as I said we will discuss this more in advanced robotics but essentially what you are doing you are making an abstraction to the robot focusing on on the task which is in this case this Frame at the end of factor and in the non-redundant case you select a set of generalized coordinates which are going to be your XY y z alpha beta gamma if you have a six degrees of freedom and an factor with six degrees of freedom basically what you're saying instead of selecting q1 Q2 to q6 as generalized coordinate I'm going to select X1 to X6 as generalized coordinate and then you write your equation of motion with respect to those coordinate so here is your generaliz coordinates now you rewrite your equations of motion with respect to those coordinate we're taking the partial derivative with respect to X we're talking about x dot we're talking about not torqus but forces acting along x y z and the different components and if you do this competition you end up with the similar model identical model I mean I'm changing Q in X now you have a similar model but now captur the properties at your end of factor so these are the definitions basically everything is the same except that now it is related to your task okay so the relationship between these Dynamics the mass Matrix M of inertial properties are the joints and the mass mat matx are the end of factor how can we establish those relationships again energy yeah so what is the kinetic energy of a manipulator described with respect to joint space coordinates with joint joint velocities 1/2 Q do transpose m q dot you agree all right I just changed my coordinates what is the kinetic energy described with respect to x dot supposing the robot is non redundant and these are geniz coordinates so they capture all the motion so it's in X so what would be the kinetic energy 12 x do transpose m x x dot you agree is it the same kinetic energy yeah so we have an identity between the two so if we write this identity between the two see how how how beautiful this is once you you do you go to the right abstraction it's very easy to extract properties if I say the kinetic energy written in this space should be equal to the kinetic energy written in this space I have already resolved the relationship between what is MX related to M do you see it what should we do now to resolve it you see this x dot and Q dot what is the relationship between the two the Jacobian so if I substitute X do with JQ dot we have the answer so I I'm just rewriting the same identity between the two and the mass Matrix we computed in joint space is related to the mass Matrix in operational Space by just m is equal J transpose M JX which now you can compute this as a function of this by multiplying from the left with J minus and from the right by J minus transpose which gives you MX = J minus transpose M Jus one okay that was easy do you agree it's was it easy so in fact once you computed your MX then you can see that in fact your centrifugal coriolus forces they are not J minus transpose they are not just the the these forces are not J minus transpose V they are J minus transpose V minus this quantity so actually in fact when you look at this as a torque and compensate and decouple centrifugal forces in joint space you are not really decoupling the end effector centrifugal corus forces because the relationship between the two is different by the way H is Just J do Q dot so this H appearing there is this so this is the relationship between the two the gravity in joint space and gravity in operational space are related just by J transpose minus so the gravity is a a torque and you compute the force so if you multiply by J transpose J transpose JX equal to J basically minus transpose means it is a minus it is J transpose minus one or J minus one transpose okay so the relationships between the two are very simple well they were not simple to find but they are simple okay here is an example I think we we can quickly go over this example you we we saw this example before remember uh we saw it in joint space and we computed the Dynamics through this and all what I'm doing now is going and finding the Jacobian so this is your Jacobian which I'm putting in frame one and I'm writing now so this is the Jacobian in frame one of this robot and now if you if you remember from uh joint space Dynamic of this robot the mass Matrix was diagonal and decoupled for this robot now if I I'm Computing the mass Matrix in frame one for the end Factor position uh I need to multiply by J minus transpose and J minus one and that turns out to be again a diagonal matrix like this so in frame one the Mass properties so now let's think about if this Mass Matrix in for the end of factor makes sense so this robot has the following properties we said it has a a mass of Link one M1 a mass of Link 2 M2 at this location and now I'm looking at the Mass properties that is what is the effective Mass when I move in this direction because I'm representing this in frame one which is this so if I move in this direction what would be the effective Mass I would see when I'm moving in this direction and what is the effective Mass when I'm moving in this direction so let's see if this Mak sense so if I'm moving in the X1 Direction this direction what do you expect the mass to be this is the end of factor I'm moving in this direction what do you feel when you move in this Direction M2 well this is what you see here correct if you move in this direction what do you what do you push what do you carry what masses are moving with you M2 plus something else because you are pulling you have to move this so you get another term M2 plus something what is this something this something is sort of this M1 reflected there and that depends on the distance right so essentially what you are doing actually M M2 Prime is taking this distance dividing it by two and Computing all the inertias related to M1 the inertias of Link uh two and Link one that that appears here and all as if they were just a mass so this is what you're doing when you go to uh the mass Matrix at a given point you are looking for this mass that produces all these effect that are equivalent to those effect that you see if you had all these masses make sense good so let's now go from this diagonal Mass Matrix in frame one if I go to frame zero I need to multiply it by the rotation Matrix right rotation Matrix look what happens in frame zero the mass Matrix is not diagonal so if I'm moving this structure along the X direction or y direction I'm getting like the aen vectors but if I move in any other direction there is coupling and this is the coupling so the properties that you see in joint space they could be decoupled even for some robots when you go to uh the real space where you are doing the task the properties are different and essentially here what we see is in the Direction X we have M2 so if you draw the ellipso you you see square root of M2 and in the other direction you have M2 + M2 Prime but in any other direction you are basically getting the square root of the magnitude in that direction like this okay so how do we control this robot in fact uh uh well on Wednesday I will show you little bit uh how those properties are through the ellipsoid of inertia and the limitations of the different characteristics but let's do the factor control now and see how we're going to uh show that just applying this controller with a sort of uh a force is going to lead to a stable behavior and then see uh the problems how we can go to Dynamic control so when we apply F to the system is essentially what we are doing is we are saying I'm going to apply a force F and this Force F could be in many different ways we are saying let's apply a gradient of a potential energy so the gradient of the potential energy that depends on the goal is going to be 1 12 KP this Vector x - x goal transpose x- X goal so if you take this force and say this this is my Force it's a gradient of this potential energy the result is this essentially uh what I did in addition I made an estimate of the gravity V so I I put in my gradient some estimate of V heart of the gravity so I'm compensating for the gravity so what happens is the initial system that was K minus V is now transformed into K minus V go it's sort of like I I have my arm I would like to move to this goal position instead of having the robot without any control if the robot you you drop it in the gravity field it will fall like this right now I'm changing the gravity it's going to fall there that's it so so this system is stable it will oscillate about V go it will osculate but stable right instead of ulating about the minimum of the gravity field it will oscillate about the minimum of the uh artificial potential field you are putting about your goal position so how do we stabilize it quickly damping how do we select our damping to oppose x dot and to oppose x dot we put forces that satisfies this condition again so it's very simple you just select FS to be minus kvx dot with KV positive you have your control so this is my estimate of the gravity to remove the gravity and this is the proportional term in X directly you take the proportional term and you have asymptotic stability uh no you have stability here and when you put this additional damping you have asmic stability all right so let's take uh our two degrees of freedom and now treat them with this controller this is the Dynamics someone get a phone call Let's uh write the controller so are you following so far here is the Dynamics in operational space this is your controller let's look at the behavior so I'm just splitting this first equation in two equation X and Y so this is the behavior in X and the second equation is the behavior of Y and what you see is that I'm just copying this from uh the real uh Mass properties of uh the mass Matrix in operational space for the two degrees of freedom so what you see is that you have M1 cine squar of theta 1+ Theta 2 M2 acceleration M1 Star multip by the acceleration so what do we see we see that essentially when I'm accelerating when I'm studying in the behavior along the X Direction I have coupling coming from the y direction and my controller is like this so so what is the Clos Loop Behavior well the Clos Loop behavior is like this it's like I have a mass acceleration plus KV X do kpx and and instead of being equal to zero it's equal to the acceleration of the second joint and the nonlinear terms coming from the errors uh associated with centrafugal corus forces and the same thing for the second joint I mean not second joint second Direction in addition what about this suppose this was Zero these are disturb forces right but what is this it's telling you that as you move your mass effective mass is changing you know your effective mass is very small here and when you reach this location do you know what is your effective Mass I'm pulling too much very small When You Reach here what is your effective mass in this direction ction infinite yeah basically when you reach the singularity this will become so high that you cannot move anymore it will become infinite so so this is a nonlinear system because also the Mass properties are changing still it's asthetically stable it works but you have a lot of disturbances you need High gains we cannot achieve High gains we know why right so what should we do come on we don't have too much time what should we do decision under constraints what should we do use the model to decouple and linearize the system so nonlinear Dynamic decoupling it's very simple when you have a model like this immediately you have a unique structure that will give you the answer this is your structure compound say take out the gravity with an estimate of the gravity take out your centrifugal cor corus forces with an estimate of it and take your controller to be proportional to an estimate of your mass Matrix that's it okay you have a structure like this x could be Q so you will be in join space or in any other task space this is your structure this is your system on the right hand side you have your controller on the right hand side this red thing can be selected in this structure and once you selected this structure you apply and and if your estimates are good you apply it to the system your behavior is going to be unit Mass acceleration equal F Prime okay unit Mass acceleration equal F Prime now you select F Prime whatever you have a goal position you have a a motion tracking you select your fime the way you want and because now you have your F there with that structure at the end so you compute your F Prime for the different uh task you have and then you apply this F over there not this one F Prime is in here you apply J transpose f as a torque and now you are controlling the torque needed to decouple the system and track the desired Behavior you're imposing in fpr okay if your estimates are mass hatot identity V x equal Z basically we are back to the non dnamic control system to the PD controller with potential energy if you are applying the potential energy at this level you are reshaping it and adding the appropriate forces to decouple everything okay good so if I have goal position we saw this controller but if I have tracking I know x x dot and X double dot then so for the for the uh goal position controller your behavior is linear and decoupled and everything is going to let you move to the goal position following this second order Behavior if you have a all these descriptions then your controller will anticipate the acceleration so you had feet forward acceleration like we did in joint space you have feet forward acceleration you have tracking of the Velocity the desired velocity and tracking of your position error and the closed loop is like this which means now you are controlling directly the error in the task space that if is you have your camera you are doing visual servoing you get your ex desired in real time ex do desire acceleration whatever from your sensory information and you are able to track and control the your robot to follow the trajectory specified by your feedback and in real time you are achieving this decoupled Behavior whereas if you do it in joint space actually you are controlling ATT tracking in an error not ex you are controlling EQ and dynamically this is very different because the relationship between ex and EQ is nonlinear so this is what you are controlling now whereas in joint space you were controlling this when you were tracking okay so this is the overall controller the overall controller is now compensating it's the same structure that we saw before but instead of looking at the system in joint space we are using this J transpose to transform this into the end effector behavior and now we are compensating for centrifugal coriolis gravity forces that are modeled for your end effector and we are scaling with an MX with the mass Matrix of the end of factor and we are controlling the trajectory so here we control an F Prime here we get F which is the sum of scaling fime by m and adding the nonlinear compensation and your Dynamics now is controlled in task space right and that's what you want good so on Wednesday we will talk about about uh some additional uh aspect of the control dealing with compliance and uh uh some ideas about Force control and then I'm going to uh give you a a preview of a lot of uh the research going on in humanoid Robotics and in uh design of uh robots uh to work and interact with a human uh in their environments so they have to be safe and we will talk about those issues on Wednesday in the meantime on Tuesday I will see uh half of the group I hope uh for uh the review session and the other half on Wednesday see you on Wednesday |
Lecture_Collection_Introduction_to_Robotics | Lecture_1_Introduction_to_Robotics.txt | (music) >> This presentation is delivered by the Stanford Center for Professional Development. >> Okay, let's get started. Welcome to Intro to Robotics, 2008. Well, Happy New Year to everyone. So in Introduction to Robotics, we are going to really cover the foundations of robotics--that is, we are going to look at mathematical models that represents [sic] robotic systems in many different ways. And in fact, you just saw those in class. You saw a [sic] assimilation of a humanoid robotic system that we are controlling at the same time. And if you think about a model that you are going to use for the assimilation, you need really to represent the kinematics of the system. You need also to be able to actuate the system by going to the motors and finding the right torques to make the robot move. So let's go back to this-- I think it is quite interesting. So here's a robot you would like to control. And the question is: How can we really come up with a way of controlling the hands to move from one location to another? And if you think about this problem, there are many different ways of, in fact, controlling the robot. First of all, you need to know where the robot is, and to know where the robot is, you need some sensors. So, what kind of sensors you would have [sic] on the robot to know where the robot is? Any idea? >> GPS. >> GPS? Okay. Well, all right, how many parameters you can measure with GPS? That's fine. I mean, we can try that. How many parameters you can-- What can you determine with GPS? >> Probably X and Y coordinates. >> Yeah, you will locate X and Y for the location of the GPS, right? But how many degrees of freedom? How many bodies are moving here? When I'm moving this--like here--how many bodies are moving? How many GPS you want [sic] to put on the robot? (laughter) You will need about 47 if you have 47 degrees of freedom, and that won't work. It will be too expensive. Another idea. We need something else. >> Try encoders. >> Encoders, yeah, encoders. So, encoders measures [sic] one degree of freedom, just the angle. And how many encoders we need [sic] for 47 degrees of freedom? Forty-seven. Now that will give you the relative position, but we will not know whether this configuration is here or here, right? So you need the GPS to maybe locate one object and then locate everything with respect to it if you-- Any other idea to locate-- >> Differential navigation. >> Yeah, by integrating from an initial known position or using >> Vision systems. >> vision systems to locate at least one or two objects, then you know where the robot is, and then the relative position, the velocities could be determined as we move. So once we located the robot, then we need to somehow find a way to describe where things are. So where is the right hand? Where the left hand? [sic] Where-- So you need-- What do you need there? You need to find the relationship between all these rigid bodies so that once the robot is standing, you know where to position--where the arm is positioned, where the hand is positioned, where the head is positioned. So you need something that comes from the science of-- Well, I am not talking now about sensors. We know the information, but we need to determine-- >> A model. >> A model, the kinematic model. Basically, we need the kinematics. And when the thing is moving, it generates dynamics, right? So you need to find the inertial forces. You need to know-- So if you move the right hand, suddenly everything is moving, right? You have coupling between these rigid bodies that are connected. So we need to find the dynamics. And once you have all these models, then you need to think about a way to control the robot. So how do we control a robot like this? So let's say I would like to move this to here. How can we do that? The hand--I would like to move it to this location. I'm sorry? >> Forward, inverse kinematics. >> Oh, very good. Well, the forward kinematics gives you the location of the hand. The inverse kinematics give you--given [sic] a position for the hand that you desire. You need to-- You will be able to solve what joint angles-- Yeah. And if you do that, then you know your goal position angle for each of the joints. Then you can control these joints to move to the appropriate joint positions, and the arm will move to that configuration. Well, can we do inverse kinematics for this robot? It's not easy. It's already difficult for six-degree-of-freedom robot like an arm, but for a robot with many degrees of freedom-- So suppose I would like to move to this location--this location here. There are infinite ways I can move there. And there are many, many different solutions to this problem. In addition, a human do not [sic] really do it this way. I mean, when you're moving your hand, do you do inverse kinematics? Anyone? No. So we will see different ways of-- Oh, I will come back to this a little later, but let's-- I'm not sure, but the idea about robots is basically was captured [sic] by this image--that is, you have a robot working in an isolated environment in a manufacturing plant, doing things, picking, pick and place, moving from one location to another without any interaction with humans. But robotics, over the years, evolved. And today, robotics is in many different areas of application: from robots working with a surgeon to operate a human [sic], to robot assisting a worker to carry a heavy load, to robots in entertainment, to robots in many, many different fields. And this is what is really exciting about robotics: the fact that robotics is getting closer and closer to the human-- that is we are using the robot now to carry, to lift, to work, to extend the hands of the human through haptic interaction. You can feel a virtual environment or a real environment. I'm not sure if everyone knows what is haptics. [sic] Haptics comes from the sense--a Greek word that describe [sic] the sense of touch. And from haptics-- So here is the hands [sic] of the surgeon, and the surgeon is still operating. So he is operating from outside, but essentially the robot is inserted, and instead of opening the body, we have a small incisions [sic] through which we introduce the robot, and then we do the operation. And the recovery is amazing. A few days of recovery, and the patient is out of the hospital. Teleoperation through haptics or through master devices allow us to control-- So here is the surgeon working far away, operating, or operating underwater, or interacting with a physical environment in homes or in the factory. Another interesting thing about robotics is that because robotics focuses on articulated body systems, we are able now to use all these models, all these techniques we developed in robotics, to model a human and to create sort of a digital model of the human that can, as we will see later, that can be assimilated and controlled to reproduce actual behavior captured from motion capture devices about the human behavior. Also, with this interaction that we are creating with the physical world, we are going to be able to use haptic devices to explore physical world that cannot be touched in reality--that is, we cannot, for instance, go to the atom level, but we can simulate the atom level, and through haptic devices, we can explore those world. [sic] Maybe the most exciting area in robotics is reproducing devices, robots that look like the human and behave like life, animals or humans. And a few years ago, I was in Japan. Anyone recognize where this photo is? >> Osaka. >> He said Osaka. >> Yokohama. >> Very good, but you are cheating because you were there. (laughter) So this is from Yokohama, and in Yokohama, there is Robodex. Robodex brings thousand and thousand [sic] of people to see all the latest in robotics. This was a few years ago. And you could see ASIMO here--ASIMO which is really the latest in a series of development [sic] at Honda following P2 and P3 robots. And in addition, you could see, well, most of the major players in robotics, in humanoid robotics. Anyone have seen [sic] this one? Do you know this one? This is the Sony robot that-- Actually, I think I have a video. Let's see if it works. The Sony is balancing on a moving bar, and this is not an easy task. And you can imagine the requirements in real-time control and dynamic modeling and all the aspect [sic] of this. And this was accomplished a few years ago. Well, actually, we brought this robot here to Stanford a few years ago, and they did a performance here, and it was quite exciting to see this robot dancing and performing. There are a lot of different robots, especially in Asia-- Japan and Korea--humanoid robots. AIST built a series of robots: HRP, HRP-1 and 2. And they are building and developing more capabilities for those robots. One of the interesting show [sic] that we had recently was near Nagoya during the World Expo in Aichi, and they demonstrated a number of projects. Some of them came from research laboratories that collaborated with the industry to build those machines. This is a dancing robot. Let's see This is HRP. So HRP is walking. Walking is now well-mastered. But the problem is: How can you move to a position, take an object and control the interaction with the physical world? This is more challenging. You see that sliding and touching is not completely mastered yet, but this is the direction of research in those areas. This is an interesting device that come [sic] from Waseda University. This robot has additional degrees of freedom that-- Okay, another problem. So you have additional degrees of freedom in the hip joints to allow it to move a little bit more like a human. Let's see This is one of my favorite. This is a humanlike, and humanlike actuation in it, so artificial muscles that are used to create the motion. But obviously, you have a lot of problems with artificial muscles because dynamic response is very slow and the power that you can bring is not yet-- But we will talk about those issues, as well. Okay, let me know what you think about this one. So? So what do you think? Do we need robots to really have the perfect appearance of a human? Or, like, we need the functionalities of the environment? Like if we are working with the trees, we specialize the robot to cut trees. If we are working in the human environment, then we will have a robot that has the functionalities of two arms, the mobility, the vision capabilities. So these are really interesting issues to think about: whether we need to have the robot biologically based or functionally based, and how we can create those interactions in an effective way. Last one, I think is-- Yeah, this is an interesting example of how we can extend the capabilities of human with an exoskeleton system. So you wear it, and you become a superman or a superwoman, and you can carry a heavy load. They will demonstrate here carrying, I believe, 60 kilograms without feeling any weight because everything is taken by the structure of the exoskeletal system you are wearing. Another interesting one is this one from Tokyo Institute of Technology, a swimming robot. So make sure no water gets into the motors. Anyway, the thing is robotics is getting closer and closer to the human. And as we see, robots are getting closer to the human. We are facing a lot of challenges in really making these machines work in the unstructured, messy environment of the human. When we were working with robots in structured manufacturing plants, the problems were much simpler. Now you need to deal with many issues, including the fact that you need safety. You need safety to create that interaction. And this distance between the human and the robot is very well justified. You don't want yet to bring the robot very close to the human because these machines are not yet quite safe. Well, development in robotics has many aspects and many forms. And really at Stanford, we are fortunate to have a large number of classes, courses offered in different areas of robotics, graphics and computational geometry, haptics and all of these things. And you have a list of the different courses offered all along the year. And in fact, in my-- This is the Intro to Robotics. In spring, I will be offering two additional courses that would deal with Experimental Robotics--that is, applying everything you have learned during this class to a real robot and experimenting with the robot, as well as exploring advanced topics in research, and this is in Advanced Robotics. So, I'm Oussama Khatib, your instructor. And you have-- This year, we are lucky. We have three TAs helping with the class: Pete, Christina and Channing. So let's-- They are over here. Please stand up, or just turn your faces so they will recognize you. And the office hours are listed. So we will have office hours for me on Monday and Wednesday, and Monday, Tuesday and Thursday for the TAs. The lecture notes are here, and they are available at the bookstore. This is the 2008 edition. So we keep improving it. It's not yet a textbook, but it is quite complete in term [sic] of the requirements and the things you need to have for the class. So, um, let's see The schedule-- So we are today on Wednesday the 9th, and we will go to the final examination on March the 21st. There are few changes in the schedule from the handout you have, and we will update these later. There is-- These changes happened just in this area here around the dynamics and control schedule. But essentially, what we're going to do starting next week is to start covering the models, so we will start with the spatial descriptions. We go to the forward kinematics, and we will do the Jacobian. And I will discuss these little by little. That will take us to the midterm. One important thing about the midterm and the final is that we will have review sessions. And the class is quite large, so we will split the class in two. And we will have two groups that will attend these review sessions, which will take place in the evening. And they will take place in the lab, in the robotics lab. And during those sessions, we will cover the midterm of past years and the finals of past years. And what is nice about those sessions is that you will have a chance to see some demonstrations of robots while eating pizza and drinking some So that will happen between 7:00 and 9:00. Sometimes it goes to 10:00 because we have a lot of questions and discussions. But these sessions are really, really important, and I encourage you and I encourage also the remote students to be present for the sessions. They are very, very helpful in preparing you for the midterm and the final. So as I said, this class covers mathematical models that are essential. I know some of you might not really like, well, getting too much into the details of mathematical models, but we are going to really have to do it if we are going to try to control these machines or build these machines, design these machines. We need to understand the mathematical models, the foundations in kinematics and dynamics. And we will then use these models to create controllers, and we are going to control motions, so we need to plan these motions. We need to plan motion that are [sic] safe, and we need to generate trajectories that are smooth. So these are the issues that we need to address in the planning and control, in addition to the fact that we need to touch, feel, interact with the world. So we need to create compliant motions, which rely on force control. So force control is critical in creating those interaction. [sic] And we will see how we can control the robot to move in free space or in contact space as the robot is interacting with the world. And then we will have some time to discuss some advanced topics, just introduce those advanced topics, so that those of you who are interested in pursuing research in robotics could make maybe plans to take the more advanced courses that will be offered in spring. So let's go back to the problem I talked about in the beginning: the problem of moving this robot from one location to another. Suppose you would like to move this platform. This is a mobile manipulator platform. You would like to move it from here to here. How do we do that? Well, we said-- Essentially, what we need to do is somehow find a way of discovering a configuration through which the robot reaches that final goal position. And this is one of them. You can imagine the robot is going to move to that configuration. But the problem with this is the fact that if you have redundancy. So what is redundancy? Redundancy is the fact that you can reach that position with many different configuration [sic] because you have more degrees of freedom in the system. And when you have redundancy, this problem of inverse kinematics becomes pretty difficult problem. But if you solve it, then you will be able to say I would like to move each of those joints from this current position, this joint position to this joint position. So you can control the robot by controlling its joint positions and by creating trajectories for the joints to move, and then you will then be able to reach that goal position. Well, this is not the most natural way of controlling robots, and we will see that there will be different ways of approaching the problem that are much more natural. So to control the robot, first you need to find all these position and orientation [sic] of the mechanism itself, and that requires us to find descriptions of position and orientation of object in space. Then we need to deal with the transformation between frames attached to these different objects because here, to know where this end effector is, you need to know how-- If you know this position, this position of those different objects, how you transform the descriptions to find, finally, the position of your end effector. So you need transformations between different frames attached to both objects. So the mechanism, that is the arm in this case, is defined by a rigid object that is fixed, which is the base, and another rigid object that is moving, which we call the end effector. And between these two objects, you have all the links that are going to carry the end effector to move it to some location. And the question is: How can we describe this mechanism? So we will see that we are raising joints, different kinds, joints that are revolute, prismatic. And through those descriptions, we can describe the link and then we can describe the chain of links connected through a set of parameters. Don't worry-- Denavit and Hartenberg were two PhD students here at Stanford in the early ???70s, and they thought about this problem, and they came up with a set of parameters, minimal set of parameters, to represent the relationship between two successive links on a chain. And their notation now is basically used everywhere in robotics. And through this notation and those parameters, we will be able to come up with a description of the forward kinematics. The forward kinematics is the relationship between these joint angles and the position of the end effector, so through forward kinematics, you can compute where the end effector position and orientation is. So these parameters are describing the common normal distance between two axes of rotation-- So this distance, and also the orientation between these axes, and through this, we can go through the chain and then attach frames to the different joints and then find the transformation between the joints in order to find the relationship between the base frame and the end effector frame. So once we have those transformations, then we can compute the total transformation. So we have local transformation between successive frames, and we can find the local transformation. Now once we know the geometry--that is, we know where the end effector is, where each link is with respect to the others, then we can use this information to come up with a description of the second important characteristic in kinematics, and this is the velocities: how fast things are moving with respect to each other. And we need to consider two things: not only the linear velocity of the end effector, but also the angular velocity at its rotate. [sic] And we will examine the different velocities--linear velocities, angular velocities--with which we will see a duality with the relationships between torques applied at the joints and forces resulting at the end effector. Forces, this is the linear-- Forces are associated with linear motion. Movement, torques associated with angular motion. And there is a duality that brings this Jacobian, the model that relates velocities, to be playing two roles: one to find the relationships between joint velocities with end effector velocities, and one to find the relationship between forces applied to the environment and torque applied to the motors. The Jacobian plays a very, very important role, and we will spend some time discussing the Jacobian and finding ways of obtaining the Jacobian. So the Jacobian, as I said, describes this V vector, the linear velocity, and the omega vector, the angular velocity, and it relates those velocities to the joint velocities. So the Jacobian, through that, gives you the linear and angular velocities. And we will see that essentially this Jacobian is really related to the way the axes of this robot are designed. And once you understood this model, you are going to be able to look at a robot and see the Jacobian automatically. You look at the machine, and you see the model automatically through this explicit form that we will develop to compute those linear velocities and angular velocities through the analysis of the contribution of each axis to the final resulting velocities. So we will also discuss inverse kinematics, although we are not going to use it extensively as it has been done in industrial robotics. We will use-- We will examine inverse kinematics and look at the difficulties in term [sic] of the multiplicity of solutions and the existence of those solutions and examine different techniques for finding those solutions. So, again, the inverse kinematics is how I can find this configuration that correspond [sic] to the desired end effector position and orientation. And then using those solutions, we can then do this interpolation between where the robot is at a given point and then how to move the robot to the final configuration through trajectory that are smooth both in velocity and acceleration and other constraints that we might impose through the generation of trajectories, both in joint space and in Cartesian space. So this-- Oh, I'm going backwards. So this will result in those smooth trajectories that could have via points that could impose upper bound on the velocities or the accelerations and resolving all of these by finding this interpolation between the different points. And that will bring us to the midterm, which will be on Wednesday, February the 13th. It's not a Friday 13th. It's Wednesday. So no worries. And it will be in class, and it will be during the same schedule. Now for the midterm, the time of the class is short, and you'll have really to be ready not really to, like to discover how to solve the problem but really immediately to work on the problem. So that's why the review sessions are very important to prepare you for the midterm to make sure that you will be able to solve all the problems, although we will make sure that the size of the problem fits with the time constraints that we have in the midterm. After the midterm, we will start looking at dynamics, control and other topics. And first, what we need to do is to-- Well, I'm not assuming-- I'm not sure how many of you are mechanical engineers. Let's see, how many are mechanical engineers in the class? Good. And how many are CS? Wow! That is about right. We have half of the class who's familiar with some of the physical models that we are going to develop, and some others who are not. But I'm going to assume that really everyone has no knowledge of dynamics or control or kinematics, and I will start with really the basic foundation. So you shouldn't worry about the fact that you don't have strong background in those areas. We will cover them from the start. We will go to: What is inertia? What is-- How do we describe accelerations? And then we will establish the dynamics, which is quite simple. Anyone recalls the Newton equation? So, let's see. What is the relationship between forces and accelerations? You need to know that, everyone. (laughter) Okay, I need to hear it. Someone tell me. Okay, good. Mass, acceleration equal force. Well, this is all what you need to know. [sic] If you know how one particle can move under the application of a force, then we will be able to generalize to many particles attached in a rigid body, and then we will put them into a structure that will take us to multi-body system, articulated multi-body system. So we will cover these without difficulty, hopefully. The result is quite interesting. So this is a robot. This is a robot that is controlled not by motors on the joints but by cables. So really, the active part of the robot is from here to there, and here, you'll see all the motors and cables-driven system that is on the right. Now if you think about the dynamics of this robot, it gets to be really complicated. So you see on the right here-- So this is the robot, and here you have some of the descriptions of-- Wait, you cannot see anything probably. But you have all the descriptions of-- For instance, what is the inertia view from the first joint when you move? So this inertia is changing as you move. So imagine, if I'm considering the inertia above this axis, right? If I'm deploying the whole arm, the inertia will increase. If I'm putting the arm like this, I will have smaller inertia above this axis. Bigger inertia, smaller inertia. So the configuration-- The inertia view from a joint is going to depend on the structure following that joint. And we will see that essentially all of this will come very naturally from the equations that will be generated from the multi-body system. But what we are going to use for this is a very simple description that again will allow you to take a look at this robot and say, Oh, this is the characteristics, the dynamic characteristics of this joint. And you can almost see the coupling forces between the different joints in a visual form that all depend on those axes of rotation and all translation of the robot. And this comes through the explicit form of dynamics that we will develop. This representation is an abstract, abstraction of the description that we will do with the Jacobian. So I said in the Jacobian case, we will take a description that is based on the contribution of each joint to the total velocity, and we will do the same thing. What is the contribution of each link to the resulting inertial forces? So when we do this, we will look at what is the contribution of this joint and the attached link and the contribution of the others. And we just add them all, and you will see this structure coming all together. So that is a very different way than the way Newton and Euler formalized the dynamics, which relies on the fact that we take each of these rigid bodies and connect them through reaction forces. So if you take all the links and if you remove the joints, you get one link. But when you remove the joint, you substitute the removal of the joint with reaction forces, and then you can study all these reaction forces and try to find the relationship between forces and acceleration. Well, this way, which is called the Recursive Newton-Euler formulation, is going to require elimination of these internal forces and elimination of the forces of contact between the different rigid bodies. And what we will do instead--we will go to the velocities, and we will consider the energy associated with the motion of these rigid bodies. So if you have a velocity V and omega at the center of mass, and you can write the energy, the kinetic energy, associated with this moving mass and inertia associated with the rigid body. And simply by adding the kinetic energy of these different links, you have the total kinetic energy of the system. And by then taking these velocities and taking the Jacobian relationship between velocities to connect them to joint velocities, you will be able to extract the mass properties of the robot. So the mass metrics will become a very simple form of the Jacobian. So that's why I'm going to insist on your understanding of the Jacobian. Once you understand the Jacobian, you can scale the Jacobian with the masses and the inertias and get your dynamics. So going to dynamics is going to be very simple if after the midterm, you really understood what is the Jacobian. The dynamics-- This mass metrics associated with the dynamics of the system comes simply by looking at the sum of the contributions of the center of mass velocities and the Jacobian associated with the center of masses. In control, we will examine-- Oh, I'm going to assume also a little background in control. So we will go over just a single mass-spring system and analyze it, and then we will examine controllers such as PD controllers or PID controllers, proportional derivative or proportional integral derivative, and then we apply these in joint space and in task space by augmenting the controllers with the dynamic structure so that we account for the dynamics when we are controlling the robot. And that is going to lead to a very interesting analysis of the dynamics and how dynamics affect the behavior of the robot. And you can see that the equation of motion for two degrees of freedom comes to be sort of two equations involving not only the acceleration of the joint but the acceleration of the second joint, the velocities, centrifugal, Coriolis forces and gravity forces. And through this, all of these will have an effect, dynamic effect, and disturbances on the behavior. But we will analyze a structure that would allow us to design torque one and torque two, the torques applied to the motor, to create the behavior that is going to allow us to compensate for those effects. So all of these are descriptions in joint space--that is, descriptions of what torque and what motion at the joint. [sic] And what we will see is that in controlling robots, we can really simplify much further the problem by considering the behavior of the robot in term [sic] of its motion when it's performing a task--that is, we can go to the task itself, the task, in the case of the example I described, is how to move the hand to this location, without really focusing on how each of the joint is going to move. And this concept can be captured by simply thinking about this robot, this total robot, as if the robot was attracted to move to the goal position. This is similar to the way a human operate. [sic] When you are controlling your hand to move to a goal position, essentially you are visually surveying your hand to the goal. You are not thinking about how the joints are moving. You are just moving the hand by applying these forces to move the hand to the goal position. So it's like holding the hand and pulling it down to the goal. And at the initial configuration, you have no commitment about the final configuration of the arm. You are just applying the force towards the goal, and you are moving towards the goal. So simply by creating a gradient of a potential energy, you will be able to move to that configuration. And this is precisely what we saw in this example, in the example of this robot here. So this motion that we are creating-- So if we are going to move the hand to this location, we are going to generate a force that pulls like a magnet. It will pull the hand to this configuration. But at the same time, you have-- In this complex case, you have a robot that is standing, and it has to balance. So there are other things that needs [sic] to be taken into account. And what we are doing is we are also applying other potential energies to the rest of the body to balance. So when we apply this force, you see it's just following. It's like a magnet. It's following this configuration. There is no computation of the joint positions. Simply we are applying these attractive forces to the goal. We can apply it here, apply it there, or apply it to both. Now obviously, if you cut the motors, it's going to fall. And it behaves a little bit like a human, actually. When you cut the muscle (laughter) In fact, this environment, we developed-- It's quite interesting. You can not only interact with it by moving the goal, but you can go and pull the hair. (laughter) Ouch. You can pull anywhere. When I click here, I'm computing the forward kinematics and the Jacobian. And I'm applying a force that is immediately going to produce that force computed by the Jacobian on the motors, and everything will react in that way. So we are able to create those interaction [sic] between the graphics, the kinematics and apply it to the dynamic system. And everything actually is simulated on the laptop here. So this is an environment that allow us [sic] to do a lot of interesting simulations of humanlike structures. So you apply the force and you transform it. As I said, the relationship between forces and torques is also the Jacobian, so the Jacobian plays a very important role. And then the computer dynamics--all that we need to do is to understand the relationship between forces applied at the end of factor and the resulting acceleration. Now when we talked earlier about Newton law, we said force-- mass, acceleration equal force. And the mass was scalar. But this is a multi-value system. And the mass is going to be a big M, mass metrics. So the relationship between forces and acceleration is not linear--that is, forces and acceleration are not aligned because of the fact that you have a metrics. And because of that, you need to establish the relationship between the two. And once you have this model, you can account for the dynamics in your forces, and then you can align the forces to move, to be in the direction that produces the right acceleration. Finally, we need to deal with the problem of controlling contact. So when you are moving in space, it's one thing, but when we are going to move in contact space, it's a different thing. Applying this force put [sic] the whole structure under a constraint, and you have to account for these constraints and compute the normals to find reaction forces in order to control the forces being applied to the environment. So we need to deal with force control, and we need to stabilize the transition from free space to contact space-- so that is, we need to be able to control these contact forces while moving. And what is nice-- If you do this in the Cartesian space or in the task space, you will be able to just merge the two forces together to control the robot directly to produce motion and contact. I mentioned that we will discuss some other topics. There will be a guest lecturer that will talk about vision in robotics, and we will also discuss issues about design. I would like to discuss a little bit some issues related to safety and the issues related to making robots lighter with structures that become safer and flexible to work in a human environment. Also, we need to discuss a little bit about motion planning, and especially if we are going to insert those robots in the human environment, we need reactive planning. And there is-- In this video, you can see how a complex robotic system is reacting here to obstacles that are coming at it. It's moving away from those obstacles. And this is simply done by using the same type of concept that I described for moving to a goal position. I said we can create an attractive potential energy. In here, to create this motion, we are creating a repulsive potential energy. So if you put two magnets north-north, they will repel, and this is exactly what is happening. We are creating artificially those forces and making the robot move away. But if you have a global plan, you need to deal with the full plan so that you will not reach a local minima, and we then apply this technique to modify all the intermediate configurations so that a robot like this would be moving to a goal position through this plan. And when an obstacle or when the world is changed, the trajectory is moving, the hand is moving, and all of this is happening in real time, which is amazing for a robot with this number of degrees of freedom. The reason is-- I'm not sure if you're familiar with the problem. Oh, sorry, let me just-- The problem of motion planning in robotics is exponential in the number of degrees of freedom. So usually, if you want to replan a motion when one obstacle has moved, it would take hours to do for a large number of degrees of freedom. And here we are able to do this quite quickly because we are using the structure and we are using this concept of repulsive forces that modifies future configurations and integrate-- So this is an example showing Indiana Jones going through the obstacles modified by--in real time, actually, modified all these configurations. And all these computations are taking place in real time because we are using this initial structure and incrementally modifying all the configurations. Another topic that I mentioned slightly earlier is the implication on digital modeling of human. [sic] And learning from the human [sic] is very interesting and very attractive to create good controls for robots, and also understanding the human motion. In fact, currently, we are modeling Tai Chi motion and trying to analyze and learn from those motions. So you can go from motion capture to copying that motion to the robot. But in fact, you will end up with just one example of motion. The question really is how you can generalize, not just one specific motion. And to do that, if you want to generalize, you need to take the motion of the human from motion capture and map it not to the robot but to a model of the human. So you need to model the human, and modeling the human involves modeling the skeletal system. So we worked on this problem, so now you have-- This is a new kind of robot system with many degrees of freedom, about 79 degrees of freedom. And all of this is modeled through the same model of kinematics, dynamics. And then you can model the actuation, which is muscles now, and from this, you can learn a lot of things about the model. And then now you can control it. You can control-- This is synthesized motion. And you understand how this is working. You just guide the task, and then you have the balance taking place through other minimization of the reminder of the degrees of freedom. And then you can take those characteristics and map them to the robot, scale them to the robot--not copying trajectories but copying the characteristics of the motion. It's quite interesting. We'll discuss also a little bit about haptics. This will be more developed in Advanced Robotics later in the spring, but haptics is very important, especially in the interaction with the environment, the real physical environment. So you go and touch-- And now you have information that allows you to reconstruct the surface and move over now more descriptions of what you are touching and what normals you have. Well, contact. (laughter) Quite amazing. What is amazing about this is this is done in real time. So someone from the automotive industry was visiting us and said, ?Now you have model of skeletal systems and good models for resolving contact. Why don't you use them for crashes instead of using dummies, right? So-- Ouch. But it's only in the model. Well, there is a lot that will come later, but I will mention a few things about the interactivity also with obstacles and how we can deal with those issues and then combining locomotion--walking with manipulation and dynamic skills like jumping, landing and all these different things. Okay, so what is happening here? Okay, this is a different planet. I'm going to just-- Okay, and that will take us to the final, which will be on Friday, the 21st of March. And the time is different. It will be at 12:15. We will announce it, and hopefully we will have again a review session before that. It is on the schedule. In that review session, we'll review previous finals, and here you will have enough time to solve some good problems. So, by the way, not everything that you see in simulation is valid for the real world. And let's see How many skiers do we have here? Skiers. That's all? I would have thought-- Okay. Okay. Does it ski? Let's see the ski. Don't do that. (laughter) All right. I will see some of you on Monday. Okay. |
Lecture_Collection_Introduction_to_Robotics | Lecture_16_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay well we have a special today uh this robot is going to show you a video I'm not sure if you're familiar with those uh cassettes anymore do they exist still so the Puma here is uh uh is going to start the VCR well starting and there is a video inside so this video is about uh a very important aspect of Robotics which is uh compliant motion you see the sponge is pushing up and you you see no deflection on the sponge right which means that there is no Force applied here we are coming to a surface that is unknown and the robot is sliding over the surface so it's making contact at different point if even we remove the whole object now here is a wavy surface that is being followed B just by saying press down and move to the right cleaning a window without breaking it it's very important no and uh well all of this cannot be done without Force control and compliant motion control so you put an error and still you are doing the same task dealing with uncertainties requires you to be able to control the compliance this is linear compliance but we can also do rotational compliance so zero moment when you push to the left or to the right you will have a zero moment so if we could see the video back so here we are creating a strategy to do face-to-face assembly where the contact forces will drive the robot to move about a rotation Center here and that will result into uh a robust strategy to do phas to face without any planning of the trajectory so from any contact point you rotate minimizing these moments of contact this is another example of a pigan hole where the forces of contact are driving the motion to rotate and to insert the peg into the hole these were developed in the late 80s I mean here is a very nice example we following this racket without any specification just the contact forces are driving this motion so first control is very important not only for compliant motion but when you do cooperation between multi robot it is also the same requirement you need to be able to control internal forces otherwise you will break the object and here is an example of two Pumas cooperating to manipulate a pipe or I believe we have three Pumas oh actually still we two Pumas we are moving a large object and you can imagine now you are doing internal Force control and resulting Force control to create the compliance uh to produce that assembly so you need to control internal forces and there is a model called the virtual linkage model that we developed to allow us to capture those variables that you have internally and for the first time we were able to produce uh four arm manipulation you have and uh you have the driver and you can see resulting compliant motion and also uh internal Force control to maintain those contact point and to manipulate this object that we call augmented object all of these issues we will cover in advanced robotics later so I'm going to stop here uh the tape is very long but let's just uh give you an idea about the issue of compliance so let's just take a case of just one degree of Freedom you have uh an inertia some uh uh displacement and you have a force and we come to this all the time by decoupling the system so if we think about the controller F Prime to be just a proportional controller so you have uh uh you're controlling X Y and Z and you have a KP Prime KP uh y Prime and kpz Prime right so what is the behavior you you you you push here and you're going to feel some stiffness right and this stiffness is going to come from your mass times KP Prime it's the overall stiffness now if I want to do compliance in the Z Direction so I would like to push up the robot and I want the robot to just uh move with with the force I'm pushing with what can we do so if I have a a very large KP Z it's going to be very stiff right if I reduce KP what is going to happen if I push it will restore itself I push I reduce it I reduce it it will be a a stiffness that is lower and lower and you will you will basically move in that direction right if you make it zero what's going to happen I'm sorry well if it is zero it is basically it is free now that spring is cut so you move it it's going to move so from the control we developed for the position because we have a control in x y and z in task space at the end of factor end of factor is stiff in this direction stiff in that direction in this direction if you change KP and make it zero now it's free and just by making KP equal to zero you are going to have those relations so the first one will be stiff in the y direction stiff and here you will just feel the Z Prime I mean the Z dot the damping okay so can can we create compliance this way just compliance now it is free but actually what we want to do is not only to create a compliance we want to control the force the contact force so if I'm controlling uh pushing the robot I would like it to maintain zero Force so when it feels there is any force it moves away which means we need feedback we need a force sensor and this is really the requirement in order to interact with the environment most of the time actually you want to apply a specific Force on the environment and that requires control of forces here we have only we just removed the position control from one degree of Freedom One Direction so this compliance along the Z direction is the first step we substitute actually this controller in the Z Direction with a force controller later and then we are able to control forces in the Z direction as we saw earlier and position is in the other direction so this is what we call uh both motion control together with Force control so this KP Prime has a very important role in determining the the the stiffness but really the closed liop stiffness remember you have multiplied KP Prime by m so this is your closed liop stiffness and if you think about it in term of uh just the spring aspect you can see that KX is your stiffness but now what is the corresponding stiffness in joint space you can compute it because this KX is displacing Delta X and Delta X is J Delta Theta so your K Theta the stiffness you have in joint space is your KX with this transformation so you can even evaluate your corresponding stiffnesses in joint space now for really dealing with the problem of force control you need to be able to sense the force and the reason is if you think about it let's say I would like to control some f desire so let's uh if we set just the desired Force to the to uh control Force to the desired Force we will we will see that actually the robot is not even moving and the reason for that is the friction so if you think about the friction as long as your Force desired is within the uh Breakaway friction you're not going to to move your your your your your force will not get the robot to move beyond that uh friction then you have also continuous column friction and you have the viscous friction so it is really difficult to do accurate Force control you all the time you are feeling those forces that are coming so you cannot do it in open loop so what you need to do you need to measure to measure your uh resulting Force at the output so if you apply one Newton meter in this case the output will be zero and you really need to to be able to measure the actual F and form an error between your desired one Newton and your Force actual force and then you can make feedback control and with again then you will be able to do a if you have a sensor then your sensor is going to give you this sensed Force which is the displacement of the sensor a sensor is essentially a stiffness that you're deforming it's very stiff but and uh the Delta X is very tiny so your sensor will give you the information and now you can you can see that uh what you want is to achieve F equal F desired which means that you can select a controller that feed for Ward the desired force and deals with the error and the feedback through a controller that allows you to control this mass that is moving but in term of forces now we can then use this relation between X and F and rewrite this equation in term of force control so we we can take the second derivative of this in term of forces and then we will have this stiffness appearing in the equation and then we close the loop and once we close the loop we are going to have the responses we saw on the video that is when we push we will see not only the feet forward but there will be an error coming from the sensor and this error will produce torqus to move the robot in a way to reduce this error so so if you think about how we can do this basically you have the relation between forces and displacement you take the derivatives and then you design a controller like this with a proportional term derivative term and then you are able to have a closed loop that minimizes this and the response now is in Force Space for the robot I'm I'm not expecting you to understand all these details but essentially Force control is going to be very very important aspect of uh robot control we just dealt with the motion control the question is how we combine the two how we get the robot to apply a force in some direction while moving if you are cleaning a surface you need to be able to move to control directions and also to control the pressure so you have to combine these two and the result is a sort of uh unification of the task control in term of motion control Force control mer together so there are direction of force control and there are direction of motion control and these depend on I'm going to skip this these depend on the relationship between the object you are uh assembling so if you have a sphere and if you want to put the sphere on a contact you lost a degree of Freedom right you cannot move in the Z Direction you can not move in that direction you can still slide on the XY plane right so how many degrees of freedom of motion you have now imagine the end effector is holding the sphere on the surface how many degrees of freedom you have two five you you L yes two in position and then you still the rotations you still have those rotations so you lost one degree of Freedom by this constraint contact constraint how many degrees of freedom we lose here more than one two we we lost the z-axis and we lost the rotation about the Y AIS this axis you cannot rotate about this axis okay how many we lost here so you see every time you have a different shape you are going to lose different number of degrees of freedom here you lose three so what does it mean it means that we really need to uh evaluate that space where we can move but at the same time did we really lose that degrees of freedom because if I'm pushing here or controlling this moment I still can do it I can push with 10 Newton 20 Newton for so the degree of Freedom just went to the space of force control so in those contact if you are talking about controlling that variable the force then you didn't lose the degree of Freedom it went just into the constraint space and you can control not now that force and we can now describe the directions and the relationship between the two and separate the spaces in here it's very easy you can say in X and Y there are no rotation about axis X and Y you cannot rotate this about X and Y you can rotate it about Z but in the Z Direction you cannot move but you can move in X and Y so now the space is split into two parts and to split the space we go to a a a description of the space and we we split it through these Omega matrices that uh allow us to project the motion control in some sub Subspace and the force control in another Subspace and in this case it's very simple what directions you can control forces here the Z Direction so essentially the motion control is in the X and Y and the Z direction is the complement if we call Omega bar is the Subspace for uh Force control so the result is a unified framework what we studied so far was uh sort of this Loop if I have a motion I can control it and I can go to J transpose and I can compensate for centrifugal coris and do this Loop now what we are doing we are projecting those controls in their spaces and we are doing unified control of motion and forces and because we are controlling everything in the task space so we have F motion F forces we just add them together we have a total force that we transform into a torque through Jacobian transpose and that is that is what is nice now you you unify all the characteristics so you can see some lambdas and some other things the Lambda is essentially the Mass properties it is your mass Matrix MX and this is the Mass properties that you are using to scale your controller to decouple the system and this is done also in the force control Loop the result is very interesting because you can decouple the two systems in their own subspaces and then you can control them properly so so this is a little bit of uh maybe the the beginning of the introduction of advanced topics that uh we will see uh I'm going to continue with this uh uh discussion but uh you have really to deal with this very important problem if you want your robot to interact with the world you need absolutely to be able to control the contact forces and contact forces doesn't mean only linear forces but it also means you control moments and with moment control you are going to be able to achieve a lot of things because in any assembly you're going to have some error in the in the Rel reltionship between the two how do you deal with this how do you deal with the fact that when you are pushing down you are going to have a moment generated well you can use it to drive well I'm not going to drive it too far but spill the water but you can use it to drive this rotation and this reaction force here will rotate if you select this point as a center of your as your operational point if you push from here it rotate this will be fixed you can fix it by controlling this point and now you are rotating about this point and we use this a lot in uh doing assemblies between objects because we are able then to cope with the errors and uncertainties that we have we we know to some extent the relationships but we don't know exactly where things are so I'm going to to continue with the discussion of advanced topics and this is part of uh um a ke talk I gave uh recently uh it starts with the with the some really old robots I don't know if you you've seen have you seen this robot Fardo okay so tell me if you have seen this one can you believe this you have a robot that can draw 1773 unbelievable actually there were a whole family of robots that draw that play music that can write at that time how can you make robots like this well she is able to play five different Melodies to do this how do you program this robot with the technology of the time so so basically you're going to use mechanical devices and if you want to see the the computer that was used here is the computer so you have Springs and then you have to drive the motion and you can see all all these sets of designed trajectories on which you are moving the different parts so the robot is going to write ja aat so well robotics was around for a long time right and and really the question was just waiting it was in our minds we wanted to build those machines and and the thing was just we didn't have the right technology and the technology really came much later and brought the first robots that were almost completely industrial robots that were used in this uh U structured plant to uh perform repeated uh tasks uh such in auto industry and uh others now today robotics has moved and we uh saw at the beginning of the class that robot robots are now used or the the application of uh robots is conceived in many domains and especially we we see a lot of Robotics today in medical applications where robots are coming very close to the human in fact inside the body of the human so this is bringing a lot of challenges as we bring robots closer and closer to the human this is bringing uh really the real challenge of Robotics because robotics in in uh the industrial setting uh requires really little from the robot once we understood uh the requirement uh Precision uh repeatability and the performance and speed and robustness we can engineer the machine to do that here the problem is we need much more intelligence in the robot to deal with many things that are uh not known in advance you need to perceive the environment quickly respond and react to everything that is happening and you need to be also moving safely so those challenges bring the perception and sensing issues that we need to deal with in an environment that is unstructured and this is really a big Challenge and also uh a good think for researchers I mean without this challenge in fact robotics would have become just automation but having the challenge of unstructural environment is bringing a lot of interesting issues to the research uh in sensing and perception in planning and control why planning and control is hard here when you have those humanlike robots what is the problem well I mean in the planning the world is changing all the time and uh maybe something else you you wanted to mention multiple inverse kinematic Solutions well the the number of degrees of freedom you have a robot with not 65 anymore you're talking about many degrees of freedom and you need to resolve and respond in real time and you need in those machines to have not just move to a position you need skills almost human-like skills so you are demanding much more you need to deal with the human robot interaction which is uh going to involve both physical interaction touching uh the human and working with the human but also the communication the interaction the cognitive aspect of it and you need to build those robots and you need to make sure that they are safe you need to make sure that they are capable but safe so a very uh important theme in this area is the interactivity of the robot and also the human friendly design of the robot so that the robot can really be integrated and working with humans so if we think about this problem and look at the challenges you can see on the left you can see a robot you don't want to be next to too I mean it shows it is it is a problem on the right you can see a robot you might think well it's smaller I it might be safe but actually the danger in this robot is hidden inside it's hidden inside and we talk about it in class you remember this you you remember this n Square the inertia of the motor this is going to be reflected and when you're going to have an impact Force you're going to really produce a large impact Force because you are going to see the inertia of the rotor of the motor so there were a lot of development uh in the area of making robots lighter safer and uh one of the most beautiful one is you can see it on the top this is the light arm from DLR uh sort of the NASA in Germany they they have been working in fact we had a lot of collaboration with them in designing uh uh torque sensors so that we close the loop at each of the joints but they developed a very nice mechatronic technology that resulted in this very beautiful robot and now you can have compliance you can control the robot to be compliant at all the joints there is still a problem the fact that the open loop characteristics doesn't have time makes it that you do not have time to close the loop and there are still open loop characteristics that will be reflected during impact which might make the robot dangerous so perhaps one of the safest way is to use elasticity with the actuator then when you have an impact the elasticity take the energy well this is not going to uh to give you the performance and that is really the problem and the challenge so if we think about the problem of collision we see that essentially you have a robot moving at some speed it has some stiffness the environment has some stiffness and when you have a collision you are going to have an impact Force this typical problem was studied in the automobile industry and they came up with a criteria called the head injury criteria the head injury criteria measures that uh uh the risk of injury given the impact forces in fact if you think about it in this plane where are looking at the effective inertias and the stiffness we can see that a puma robot is sitting here at almost 90% of risk of serious injury and the only way you can make it safe is by like covering it with almost 20 cm of compliant material so I don't know I mean it will be really really big so what is the problem I know you have a lot of mechanical engineers so they are going to help me now to solve this problem how can we deal with this problem yeah do you need need sensors that are more anticipatory well I less reactive or more well actually the sensor idea is is really really good but I I think there is no way you can guarantee I mean if you have a robot you can never guarantee safety that a passive safety that is not dependent on a computer or dependent on a controller or a sensor anything can break how can I guarantee that whatever happen this robot is safe so I I need really to go further not only to to think about uh how I can improve the feedback how I can all what we can do uh with the sensors skin and protection of the robot is good because that is most of the operation but I want also to make sure that if I have an impact for some reason something broke in the robot the controller the computer something happened I need to guarantee that the impact force is below uh some acceptable uh amount so it turned out that a large part of the design comes from the actuation and you can see it every time I mean you are trying to manipulate an object with this end effector how do you move this end effector you need links and everything we were studying was the carrier of this end effector right and to actuate your links you need motors and because of the motors your links becomes bigger and bigger because you put this motor here to carry this and a factor then you need to carry both with the link with another motor here and you propagate the weight so every time you design a robot you you're hitting this problem how can I squeeze a lighter actu here so you put higher gear ratio but you reflect larger inertia but essentially the the torque you need here is very critical so you come up with some specification you see you say I need the 3.7 Newton meter motor with continuous torque a torque that you can sustain for a long time because a motor has a big torque that could be produced very in short time and if you keep applying it you will burn your motor so once you designed your motor and now you can lift the gravity of the link and the effector and the load and all of that what do you say well but my robot has to accelerate has to give me those performances and now my motor I will pick my motor to have those characteristics so I need strength to carry and I need also the responses Dynamic responses well actually this is what you need you need performances you need but what you are doing here is you pick a motor and then you say this motor need to be as performant as the acceleration I'm going to produce and this is really not necessary because if you think about the problem in the domain of control and the performance you need the magnitude that you will need at high frequency is much reduced so in fact your requirement are you need large torqus at low frequency and smaller torque at higher frequency which means you can design your robot differently we we never think about it but we can design the robot not with one motor but with multiple motor for each joint like human human have multiple muscles actuating and actually different type of muscles so you can place a small motor the structure will be light very safe and you put a big motor at the base you remove the weight but if you do this actually this motor will be reflected at the impact Force but I said this motor is only needed at low frequencies so now I can isolate the motor when I have an impact force it will go through this elasticity so this is the concept we developed in dm2 it's called distributed micro mini actuation that really uh is interesting in the sense that you build a robot that has the capacity of a large robot but the safety and performance of a smaller one well this concept since went much further and uh seeing the performance you can achieve I mean it's amazing you can take the Puma and build a robot like the Puma with 10 times less effective inertia that is you reduce your inertia from 35 to 3.5 and that leads to a large reduction in the inertial properties that you see so I I think you don't understand this let me show you what I mean with this uh where is my simulator anyone saw the Puma simulator somewhere ah right here okay so if we look at the Puma I'm sorry if we look at the Puma we can display the the inertial property of the Puma and you can see like you have small inertia in this direction and large inertia in this direction in fact if I move the Puma you can see that this inertia is changing it become here is we have Singularity you remember the singularity here and if we look at it in this direction you cannot move when you come to overhead Singularity and the inertia becomes very very big the inertia goes back to a reasonable value in here so we can take this Mass Matrix and represent it as the inertial property and describe uh those properties on the robot so let's go back to so you understand those inertial property the green one is what you have now you reduce your effective inertia by 10 times and immediately you improve your control performance so this concept is quite complicated because it requires this elasticity requires the the big motor require Transmissions to to coup it and we came up with another idea which is if we want an an elastic actuator why don't we go directly for muscle like actuat like in this nice concept so we started building a robot like this and the idea in this robot is that you use bones muscles and air pressure so you are bringing the energy through the air and now you can lift the robot you can produce the large magnitude of torqu needed and then you add a small motor on the joint and this motor will allow you to get the Dynamics so combining uh the the the two you get hybrid actuation that essentially you leads to uh a safe uh robot design now the problem with uh this is how can you manage all these uh tubes of air that are going to go to the joints and control every joint will need two uh muscles well because I mean you realize you you have a large uh amplifiers pressure regulators and you cannot think just I'm going to take all these tubes you will have a lot of them and and your your arm will be very heavy so the solution to this problem is what is we don't want the tubes so we want to put this on the arm big so what do you do well make it smaller now we identified the right problem make it smaller we can have a solution to this problem and once uh a graduate student know the the problem they solve it in fact it was a piece of cake just making it smaller no problem so making it smaller now you can distribute it and by Distributing it on the links you essentially uh take one line of pressure running through the whole robot and at each of the joint you are Distributing the pressure press regulated to the value you need at that joint so here is the arm they built and uh in fact this arm was uh almost like measured after my own arm so it is really human like arm with the constraints of a human and not too big uh here are the dimensions um well I'm not sure about the the torque and here is the the arm you can hear the air pressure switching and control so now you can do Force control with this you control the contact forces and you produce motions with an arm that is yet lighter from from the arm that we had before 3.5 here we go to 1.5 kilg so let's see it you go from 3.5 to 1.5 kg maximum and you can see on the red response you have the macro robot just the macro the muscle response is very slow you get almost 0.5 Hertz in the closed loop whereas with the macro mini you can go to 35 Hertz huge huge Improvement the other thing is by adding this tiny motor you you are improving uh the control not only of the overall system but also by thinking about the macro differently and looking at what you are controlling you can also do a lot of improvement what we did we added sensors to measure the tension the actuation using muscles brings a lot of nonlinearities in the model but if you use a sensor you can close a loop and you can improve your control so just the macro with a sensor goes to 7 Herz so this is really really important always to use sensors where you can well with the the group of uh Mark katuski we are working now on the next prototype that will come probably in few month and this prototype is bringing an integration of all these tubes directly inside the structure so this is going to be a cool arm it this is some of the built parts and you can see the overall design so everything is integrated now inside and outside we have skin to protect uh in the first impact and also to produce feedback so you get uh tactile information about the contact location so you can control better your robot okay so how many orders we we have today for this one okay well please contact us and yes sensors are in The Joint but the extremity or both well there are uh uh sensors on the inside the structure to measure tension on the on the muscles but also we have sensors distributed on the skin outside and that will measure contact with the environment so when you hit first you dampen so you reduce the impact force and also you measure where you are touching okay let's move to another aspect of uh this human friendly uh uh goal and and look at uh the planning and control and look at the human robot interaction so let me start with the something that we we basically developed uh in the early '90s uh and uh I mentioned and probably uh you you saw all of you you saw this uh uh couple in our lab Romeo and Juliet the two platforms that we use to develop and explore the area of human uh friendly robotics so here is the video if we can have the projector please up so the idea is really not to think about the problem as well moving the platform stopping manipulating moving again stopping manipulating it is about how you can combine Mobility and manipulation in a way where you are controlling both so if you're holding an object and you have to move because of an obstacle you should be able to decouple the two we call this the task and this is the posture and we are able to control the whole robot to achieve a task and also to achieve the posture in a decoupled way so with this you can do a lot of things you can clean the carpet open the fridge thank you make a contact control the contact force oh this is my shirt I think it's the only shirt in the world that was ironed by a robot I I should add it was iron one time this is one of the most complex task you can imagine I mean you can imagine internal forces control macromania actuation uh cooperation between the two all of this well there involves a lot of models and things we will be discussing in advanced robotics well this was a a lished and demonstrated uh at the opening of this building uh the gates building and in fact Bill gate when uh uh he came to the inauguration he uh he was quite amused with those robots uh which you can in fact also uh used to interact with a human uh so uh my students used to dance with those robots uh here is another example of uh how you can interact with them by guiding the motion so the human is is is giving the intelligent task of the guidance and the robot is following and this is uh this is uh basically uh oh and Alan is dancing okay so by 97 uh when we accomplished this uh if I don't know if you remember I mentioned by 97 998 that there was uh the Honda robot that brought the first walking stably machine uh and it was remarkable achievement it dealt with The Locomotion aspect but it didn't really address the problem of manipulation and since then we had this challenge to bring together manipulation and mobility in the way we did it for Rome and Juliet on a more complex robot that involves many degrees of freedom branching structure we're not talking about contact with one end of factor there are many different points you are controlling you have multi contct you have many constraints you have a lot of things taking place and you really need to deal with all of these simultaneously and uh in a coherent fashion you cannot pull in One Direction without really accounting for what is happening in other directions so the idea now of dealing with this problem through inverse kinematic is crazy because you have a lot of degrees of freedom you do not want to commit your final configuration before moving because there are constraints and Joint limits and many that happens so the approach that we discussed in class earlier about task oriented control makes a lot of sense you pull toward the goal and the configuration emerges you do not decide it ahead of time and this approach results into a very simple uh way of controlling the robot directly in the task space so in the next two minutes I'll show you how we we do this uh here but I mean you can realize there are a lot of details let's go to the arm that's what we studied earlier you have a a force you have the gradient of the force how do we apply it torque equal transpose F okay very simple that that should work we said we cannot just do it like this we need to account for the mass properties we need to establish the model between acceleration and forces and by having a good estimate of this model we can go and correct this and now we decouple the system and we align the directions to follow the initial properties now the question is if we go to this problem how do you deal with all these points how do you combine how do you control this while you're controlling this if you move one one arm everything else is going to move how do you decouple the motion of the right arm from the left arm while balancing and doing all these things that is a good idea gyroscopes will give you a good uh sensing by integrating you can find the orientation of the body you can use that information but how can we deal with with this problem of uh finding the dynamic equations of a a system like this and controlling [Applause] yeah we want to use energy but first we need to get a model of the system for this task and that task and the head and the legs and so so we need to extend this operational space control or model not only to find these Mass properties but I I want to find all the Mass properties and the coupling between them it's not obvious but it's very simple so when you run into a problem like this just sit down relax and and just move back move back from the problem don't go too close to the problem move back what do you see you go to a higher dimensional space if you go to a higher dimensional space everything appears like one point so if we take this problem X1 X2 X3 and put them in a higher dimensional space in that higher dimensional space they will be a point a task involving X1 X2 X3 all of these and we are back to the beginning because now with this you can find a Jacobian when once you have a description you find a Jacobian you find your mass Matrix MX or Lambda what I call and on the diagonal actually you have the Mass properties at each of the point and of diagonal you have the coupling between them and now you're back to this beautiful you have the same model in a higher dimensional space and now you can use this model to control the the system it's very easy we we use the energy that is we we use a force of task to control it with J transpose F but now suppose I'm controlling the legs and the hands but I have all this motion that is possible Right This is the null space motion this is what we will discuss with redundancy next quarter and in there you have to guarantee that your control is consistent do not interfere with the first control and to do that you filter it with n the null space of the Jacobian and the result is that you guarantee that there will be zero acceleration coming from the control of the posture and that means when we move in the posture space these points will be fixed and that is very very important and once you you have this decoupling then you can control the system in a very effective way so the control of the whole body is Task filled to control the task posture field to control the posture and the two are decoupled so you can generate motions without any trajectory the question is how can we find those energies how we can we find those criterias well if we are working with the uh horses maybe we should look at horses but if we're working with human we should look humanlike robots we should look at human how human do it and this is really a good question I mean we already started to be inspired by the human the way human move Servo their hands the way they move their body and and we need to see how all of this can be captured through simple mathematical models that we can use to reproduce the motion so the starting point would be like [Applause] just motion capture capture the motion and then well I'm not going to replay the motion directly because if I record the motion and replay it then we cannot generalize what we what we what what we are after is not a specific motion we are after understanding so if you have a system you want to understand what do you do you shake it enough to find all the behaviors and to shake and then you identify it with the model of that system so you need a model of the system I I mean it is really interesting we were working with robots and now we are working with human model but human model are essentially articulated body systems the mass Matrix you computed is the same you can use the same mass Matrix the actuation is different so for the mass Matrix you can now take the rigid body part the skeletal model and now animate did and this is what you see in fact drop it in the gravity field a pendulum the idea is how can we go from the human to the robot right and with less degrees of freedom so we want to see the characteristics of the motion to see the characteristic of the motion you need to see the actuation also you cannot do it without the actuation so we modeled the muscles and all of this was done with the with people uh who are working in biomechanics the group of uh Scott Delp uh Who provided us with all the data about muscles and uh and um uh the skeletal system and skeletal model and once you have all of these you can start doing the study and the analysis and you start to to look at it and try to find what is going on so so to analyze this where do we start so we can we can say okay let's start at something very simple I'm going to give the human a task which is like to hold some object and the question is where the pure is going to be imagine you are pushing your it was very cold in the morning your car didn't start you're going to push your car how do you push the car you push it this way or you push it like this the question the answer is obvious actually the child when at Birth didn't know that so little by little you learn you learn and you discover something amazing about about the human body and then you start using it and what what is amazing is in any uh mechanical structure like this there is a mechanical advantage and when you discover the mechanical advantage you start to using it so you do not just use the rigid body mechanical advantage you are also using the way you're actuated and your muscles the child when he's moving first will will pull the muscles and it hurts and you have this feedback and little by little you start to to adjust the motion and move your body correctly so human little by little are learning discovering a way to minimize the effort produced by the muscle and this is is our speculation we said there is muscular effort minimization but it's not always like that it is uh there are also a lot of other constraints like it's much easier to bring the spoon to the to the mouth than the the I mean easier to bring the head the mouse to the spoon than the mo the spoon to The Mouse and the reason is the reason is uh uh mom says this is not polite you cannot do this you have to do it this way so there are a lot of social constraints there are a lot of other constraints physical constraints on on the system that you need to integrate but essentially you are minimizing the effort and if you say I'm minimizing the effort it's it's it's very important because now you bring the physiology in the model and you start analyzing the physiology and you look at it so a a robot produces a force by applying a torque J transpose F all of you now know this right a human produces the torque using another Jacobian L transpose this is the muscle attachment jacoban and these are the muscles now if you think about it you are really tuned to use this in a way to minimize your muscular effort but every muscle has a different capacity bigger muscles smaller muscles so you need to account for the capacity of the muscle so here is uh the our hypothesis we think human are adjusting their posture to minimize muscular effort so there must be an energy okay okay now pay attention we want to find this energy so the muscle is M extension of the muscle the tension is M if I'm minimizing the the muscle uh effort the energy is going to be in M square right but there is the capacity and what we speculated about is that this energy is weighted for different muscle with their capacity so here is the energy are you already sitting okay are you ready want to see it okay now if you look inside of this this is what you see this captures the following captures your task captures the mechanical advantage of the skeletal system captures the muscle attachment and the capacity of your muscles if a muscle is weak you compensate for it if a muscle is missing you compensate in for it etc etc so in fact it turned out that our student went to the motion capture lab and we did the analysis and it's amazing actually if you if you I mean I see you sitting comfortably why don't you just relax your back and and move forward without touching anything with your back let's try it we're going to to to to to see if it works also with you so Let's uh drink a cup of coffee don't drink it completely just go very close to drinking the coffee and comfortably and like you you you you relax and let's go okay now look around everyone is doing this not like this not like this it is like this angle and this angle correspond in fact to this angle correspond to the minimum of this energy so it is not uh a coincidence in fact if you do this analysis and you increase the weight this angles goes up and you can see it here as you move and increase the weight from 0 to 15 pound for instance this is the speculated angle that you will find you can check it on the it makes so much sense you are discovering how to use this machine and you are able to use it for the specific task in its optimal way and it's not only Jacobian that is it's not only the skeletal model it's also the muscle attachment and the way the muscle capacity is distributed so with this now we can go and simulate and create so we are doing I I'm not sure if you you understand this let me you remember I showed you uh I showed you stand bot I think I showed you this right we we saw stand bot you saw this before right okay quickly so we move to the goal we move to the goal we move to the goal and you you are able to just track that motion right and you remember can fall boom The Joint limits all right so now let's go back to this so this is what we're doing we are controlling one point and the body is adjusting through a criteria in the null space and now you can move it to robot not as a trajectory you're copying but as a criteria that you're applying to the robot well this is the environment we developed veled that also contain uh um neuromuscular models that can be used to analyze and uh look at human this is a recent work on taii analysis and uh the master here from beIN is performing a motion that we can record and then we can analyze the motion and you can you can start to see that well all these model can produce amazing things so that that essentially you can uh analyze skill of a human you can look at the behavior of the human uh synthesize it and in fact if you record that motion and if you play it back on the on the physical model of the robot it will fall over because you need control you have errors and unless you control it correctly it's going to fall so what we what we are going to show here is half of the body is following the taii motion and the other half is controlled and in fact you can see that you can achieve uh those desired have while you're controlling the robot with other behaviors so all of these are part of uh the the development that is now uh being uh uh implemented on uh human robotics like Asimo and a key aspect of of the implementation and the task oriented control is not only just motion and force control but how do you deal with constraints how how do you deal with the fact that if you have a joint limit and you're moving and you hit a joint limit what is going to happen so this structure of control that I talked about produces a very useful way to apply constraints uh essentially I mentioned about constraints in term of attractive forces repulsive forces now if you have a constraints it has to take the highest priority in the structure of task and posture control we have priority this task will not interfere with this so if we know a constraint and we know the Jacobian of that constraint then we can know the null space of that constraint then we can take this whole thing put it in the null space of that constraint and then control the constraint and now the robot is moving and you can see two different postures because you have a hip limits here you are stuck you cannot reach because of joint limits here the body is going to move away to avoid self Collision so the trajectory is going through the body and the body will move away automatically by these repulsive forces obstacles so so it is very very important to be able to create those interactive behavior that allow you to avoid Collision but at the same time you need to think about the global path not only about the low local behavior and well we have end degrees of freedom the problem is exponential in the number of of degrees of freedom you have really to to deal with a way of connecting the two and to do that we developed a concept of we call elastic planning which essentially connect all the plan but allows you to deform it in real time and by deforming this we are able to to uh change a feasible plan and adapted to the real environment and this is this is a an amazing thing because you are able now to be uh to change a trajectory that requires hours of of replanning if you didn't use this for instance you will have to replan the whole trajectory with the new uh obstacles and uh constraints at the same time you are reconstructing because when you are touching teleoperating uh interacting with surfaces you need to reconstruct the surface to fit and control your robot and that means you need to model contact correctly you need to deal with the contact models also in the control and this is a very very interesting result that comes directly from control you remember when we talked about uh Collision essentially we are interested in the collision with the multibody system but how can you resolve it well people usually remove the joints resolve the whole multibody free multibody collision and then they put the joint eliminate constraints but in here we are going to be able to reduce the problem to this because we can use the mass metrix in this direction we can use the effective inertia directly the Collision doesn't care about what humanoid robot you have you just need to know what Mass properties you have at the contact so with this we have a very effective algorithm that allow us to simulate and resolve Collision uh in real time and with the real time resolution then you can go and and use it in many different things I believe I showed you this at the beginning of the class but now you understand what what it means we are able to find the properties in Impact forces and we are able also to deal with this problem of contact and collision the problem is very difficult because when you are looking at a humanoid robot you just push it and it will tip over it is missing these six actuators it's not connected to the ground and you spend your time balancing and if you have any reaction force it will tip it over so the question is how can we move this body while maintaining dealing with those constraints how can you do it well just say it say I need to treat this as constraints what are your constraints these normals take the Jacobian take the null space and take this and put it in now you know how to do it right put it in the contact space and then you are able to to bring the two together and now this motion will be consistent with the constraints and you can control the forces at the constraints if you remember in in uh the beginning of the class we talk about Omega and Omega bar this will become your Omega bar and this is your Omega in the multi contct space so that was uh really uh uh a very important result to allow us to to implement Behavior with multi contct and motions and uh uh now uh Distributing the effort between different uh uh surfaces that you are touching uh moving um balancing dealing with Dynamic skills etc etc so then you can start to build Behavior over this that is now the the robot is looking watching it is if there is an obstacle it will move down if there are constraints uh you are able to deal with the constraints as it moves it is automatically generating the right motions to avoid uh hurting those constraints and as we are building we are moving up and up in the structure so what we we really have seen in the class is really the lower level in this system where we are looking at the execution of uh uh the motion controllers the force controller but on top of this you can build behavior that make use of those Primitives and you go up to levels where that now you require more abstraction of the sensory information your perception and as you are building a solid foundation you are going to be able to move Higher and Higher by integrating also skills and learning from the human hopefully we will be able to uh get Asimo to graduate all right well I guess uh I'm going to stop here and um tonight we will see the rest of the group yesterday we had a a session very nice session we will talk about the final and um those uh who I won't see tonight well I wish you good luck for the final and for those I will see we will talk about it more in the evening so 7:00 okay stop here |
Lecture_Collection_Introduction_to_Robotics | Lecture_13_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started so today's video is about juggling this is from uh Dan kich and it was presented at isrr International Symposium of robotic research in '93 video okay at the University of Michigan robotics laboratory we're interested in tasks involving dynamically de ous interaction between robots and their environments computers currently play chess better than all but a few of the best human experts but no machine has yet been built that can manipulate the physical pieces with anywhere near the skill and reliability of the youngest human chest novice our three degree of Freedom direct drive robot is endowed with a juggling algorithm that transforms the positions and velocities of a falling ball into desired joint positions and velocities which the robot is forced to track by use of a nonlinear inverse Dynamics controller smooth position and velocity estimates are produced by a linear Observer which in turn receives input from a real-time stereo Vision system the one juggle task requires the machine to bat a single ball into a stable periodic trajectory passing through a user specified Apex adding a second ball with an independently specified Apex Point defines a two juggle task the juggling algorithm shown here employs an urgency measure to switch the machine's interest between the reference commands corresponding to the two independent one juggles it's worth emphasizing that there is no planning in the conventional sense taking place in this system rather the robot's impact decisions are induced by its continuous motions in the effort to track a carefully distorted version of the positions and velocities of the two balls machine juggling skills in themselves seem unlikely to play a direct role in the social and economic impact of advanced robotics however we are convinced that the problems of controlling context focusing visual attention and coordinating in real time the constituent behaviors of such skills provides an invaluable laboratory for understanding what is hard about dynamical dexterity without a phase regulation control term the balls quickly wander in phase and eventually fall simultaneously in contrast with phase regulation again enabled nearly simultaneously falling balls are successfully sep separated in this experiment we failed to prevent a spatial Collision we hope in the future to better understand the nature of these and other dynamical obstacles in order to control around them more effectively of course there will always be situations from which the machine cannot recover okay so who's interested in juggling well those who are interested in juggling could try it uh next quarter in experimental robotics um in fact a lot of the projects in experimental robotics involves uh Dynamic skills throwing a ball into a basket uh playing ping pong or whatever so juggling is quite challenging actually well juggling requires control and uh here we are so so this is a little bit of a concept that we are going to see over uh the discussions on control and the concept is instead of really thinking about the robot as a a programmable machine where you need to find all the joint motions corresponding to your task so you want to move to some location and you want to uh be able to reach that location with some orientation of your affector well uh basically what you have to do is you have to solve this inverse kinematic problem to find the joint angles that would allow you to be in that configuration I'm not sure if a human do that human usually uh are really poor at computation so finding the inverse kinematics finding all the joint angles that that will uh put you in that final configuration it's really difficult so what do you think human do feedback of what look so you you you you sort of like think I mean it's try to reach for something try to reach for the like the chair in front of you how do you do it so you're looking at your hand you you look at the chair and you you have this visual feedback servoing so it's sort of like your your hand is attracted by a force pulling you toward that goal position you describe and this is the concept you see here it's sort of like potential energy where the minimum of this potential energy is located at the goal position and that is going to create create a force pulling your hand toward the goal your hand is going to just move toward this goal without a prior imagining or knowing where your final configuration is going to be the final configuration is going to emerge from your motion we will come this to this later but this kind of idea is really uh what we call task oriented or operational space control the idea of uh really doing the control not through this inverse kinematic and programming the robot well there is another method and most robots today are controlled through inverse kinematics that is we control The Joint motions so you first decide where you're going to position your hand so you need to find this configuration which means you all what you know is the position and orientation of the hand you don't know yet yet this so you need to do the inverse kinematic you saw the inverse kinematic for 6° of Freedom uh how about inverse kinematic for this I'm not sure but anyway you might maybe use a mannequin and you you just position it and you decide well this is a good configuration and you start from here and now you sery your joint angles to move to that final configuration well it doesn't really work very well with humanoid Robotics and in fact uh a lot of humanoid robotics today are suffering from this problem the fact that we are still controlling uh robots using inverse kinematics however for industrial robots well with few number of degrees of freedom or if you have a repeatable task you're repeating the same motion over and over basically you recorded the motion and what what is left is how to track that motion So today we're going to discuss uh uh the basics of control and we're going to really start slowly with just something like we saw on the video just natural systems like you're dropping a ball or you are looking at a pendulum moving and you are trying to understand just the relationship between the potential energy appli to the system and uh the kinetic energy resulting from its motion and then we will analyze how this Behavior would allow us to create something like PD control proportional derivative control well in nature we don't have too much of I integral action but we will we will be able also to add integral action if the error is large and then we will apply this to controlling robots in joint space so we can control this robot to follow a trajectory that is given in joint space and then we will discuss how we can uh apply control techniques directly to the task in the way we we do it human uh that is by directly applying a force not through the inverse kinematic or at the joints but directly to controlling the ector motion velocity and acceleration so at the end we will see that motion control is not really the only thing we need to do when we have a robot you really need to interact with the environment and in order to interact with the environment you need to control the contact forces so if you are sliding over a surface you are moving and at the same time time you are applying a force of contact and that is going to be uh a critical uh uh technique in order to interact with the world effect objects assemble move and cooperate with different robots so a manipulator like this one can be controlled directly through its joint motion by simply imagining that you have some sort of like Springs at the joints and if you put just Springs then this mechanism is going to oscillate if you disturb it so what do we need to add I'm I'm just talking about passive mechanism so we we we put a spring and now it's going to hold itself at some configuration and if you disturb it it will oscillate so what do we do to make it more like asymptotically stable put some damper so if you place a a spring and a damper at each of the joint you will basically go to that resting point of all the springs and that will allow you to be at that configuration right so to control the robot in joint space just imagine that the resting point of the spring is changing so little by little you're here and then you're moving it there and there and there and then you can control the joint motions so this is typically the approach that we will have in joint space control except the fact that we are not really dealing with the coupling with the inertial forces created and we will see uh little bit more about this we will see why this could work I mean it's not not obvious that it's going to work in with passive with passive devices if you put just spring and damper the system is passive and is going to somehow rest at some configuration there will be deflection due to even at a steady state like if you let it rest what what other forces will disturb the position so the Springs will go to their rest position but they will deviate little bit because yeah the gravity so the gravity will create little disturbance you need to compensate for the gravity to account for the gravity you need also during motion to account for this acceleration you are generating and that that are scaled by the inertias and the masses so they produce coupling as well as C Fugal coris forces so the equation of Dynamics is here and now we need to account for that but uh simply the concept of the control is just a spring damper system and the behavior is going to be very close to a mass spring damper except the fact you have coupling so if we start over and we want to control the robot directly in with respect to the task so we want to move this in effector to some location what can you do still using some passive Springs and den I'm going to give you one big spring and a damper and you just need to place it somewhere [Music] mhm yes perhaps specify a GPS position maybe it could go there I don't know that yeah the GPS position is good it will give us where the robot is and uh we know where the robot is going so we know the error between the two but I'm asking what is the concept in term of moving I mean implementing a controller that will work with the task instead of working with the joints so I want I don't want to use this if I place all the Springs at the joints I need to know the joint displacement I need inverse kinematics if I want to control the hand yeah so exactly I mean just pull it right just put the spring there anyway I gave you only one spring so you I have to just place it some all right so okay what is going to happen here is you are going to pull the end effector to that location to the resting position and it will do this and everything will fall you don't know where it's going to be but it will fall and we will see that the concept is as simply as this well in six Dimension X Y so the spring is like six dimensional spring okay all right so this is basically the concept of task oriented control I mean you can think about the spring uh as passive spring or some attractive potential energy that you are creating at the end of factor with a gradient that is pushing you towards the goal and that gradient is actually here coming from the spring the spring has a potential energy when it's Disturbed and when you go to rest you reach the minimum of that energy and it entially you're applying the gradient of the potential energy okay so in joint space this is what is happening as I said we have an inverse kinematic problem we have a task that is described in term of x y z the orientation of the ector Alpha Beta gamma whatever representation you have you need to compute the desired joint motions and then you have those desired Motion One desired motion two Etc and you look where you are so you measure from encoders your q1 Q2 qn you form a small error between where you want to go and where you are and then you are reducing this error by control independent controllers most of the time sent to each of the joints so you have Servo controller at each of the joint taking the joint from some value of theta to another value of theta the problem is you have this inverse kinematic all the time in the system another approach that came about as early as 69 70 70 one and there is a paper by Dan Whitney in 72 describing resolved motion rate control so rate means we looking at the derivative at the velocity and the idea is to instead of doing the inverse kinematic using the forward kinematic and taking its inverse the idea is to find a small displacement Delta Theta or Delta q that correspond to your desired displacement Delta X so what do you think we have at our uh in our menu as models that could help there so we would like to find a relationship between Delta X small displacement so I'm at X I would like to move little bit Delta x what would be the Delta Q so what model we should use and you don't say it someone else jaob yes the Jacobian actually the inverse of the Jacobian so so here is the Jacobian it relates precisely Delta X to Delta Theta right if you take the inverse you have to make sure you're not at a singularity otherwise you have to do special treatment of the configuration if you're outside of the singularity you will and if you you have six degrees of freedom regular case rectangular Matrix then you can take the inverse otherwise you have to resolve to generaliz inverses or pseudo inverses um so you compute a Delta Theta so for a small Delta X now you have your Delta Theta knowing your Theta knowing you where you are the next configuration you want to go to is what theta plus Delta Theta so so you start from your current position X you compute the error that is the X desired minus your current position and then you compute your Delta Theta and you add it to Theta so you you keep controlling the robot to this theta plus which is where you were plus the small displacement and this is a vector so here is the model now the Jacobian inverse is inside your Servo control Loop and you need to compute the forward kinematics which is easier to compute especially for you now right very easy forward kinematics so forward kinematics include the Jacobian you just have to inverse the invert the Jacobian and for a small robot you can get it almost in analytical form so basically you compute a deltaq and you distribute it to all the joints and you have controllers for each each of the joint to move and form this error in Delta qn and you move so now you're continuously moving well this has a lot of problems in term of uh uh the conditioning of the jobian the fact that uh the Jacobian has this uh strange thing about its metric because the space where you are measuring Delta X involve linear motion and angular motion so linear motion is measured in displacement in ctim or meters or inches but it has also rotational Motion measur in degrees or radians and it all included in the Jacobian so the metric of the Jacobian is not homogeneous and that create problems also you have the singular ities you have the redundancy you have all of that in addition to the fact that you have Dynamics so this works usually uh most of the time but it works best if you use it to find the trajectory you want to execute in industrial robots often you want a repeatable trajectory and this doesn't repeat you will drift so if you do this in simulation you will be able able to find a trajectory so resolve the inverse kinematic this way and then come up with a trajectory that you can execute okay well let's see how we're going to control the robot anyway we get joint angles We are following directly the trajectory whatever we do we need to control the robot we need to create a a a motor torque that is proportional somehow to the error so we drive the joints to move toward the goal so how does it work by the way how many of you have had some control classes okay that's what I thought so we we'll start assuming you know nothing forget everything now right okay so what is the simple simp system we can consider well I think a mass spring system would be the simplest you can imagine right so you have a mass resting on a surface with zero friction nothing it's sliding and you have a spring you pull it what's going to happen that you can imagine I'm sure everyone what's going to happen from rest you pull it little bit and let it go it will ass so we are really interested in understanding the this oscillation and how the oscillation is affected by and by what is affected so this problem could be resolved and looked at through this same equation we use to find the Dynamics we look at this Mass and we find its kinetic energy and we look at the system it has some potential energy where is the potential energy it is I'm sorry I cannot hear I'm sorry sorry the spring the spring yeah I'm sorry I didn't hear you the spring right so when you when when you're at TR the potential energy is equal to Zer I mean if you let it rest alone you're not intervening the potential energy is equal to zero and the kinetic energy is equal to zero the velocity is zero the kinetic energy is 12 MV s so if we disturb it and hold it what happens to the kinetic energy still zero potential energy going to be positive it will increase now if we let go the potential energy start to decrease and it that energy is transferred to the kinetic energy and then it keep decreasing keep decreasing we come to the minimum of the potential energy we will have the maximum kinetic energy and now the kinetic en the velocity start to reduce the kinetic energy reduce and we start building potential energy and essentially this oscillation is a transfer between K and V so this is K and we can write this equation so if you write this equation using K that we saw here you take the derivative with respect to x dot it gives you what MX dot the derivative of K with respect to X is zero so you get the time derivative of that quantity MX dot equal F and the potential energy of this spring is 1 12 K x² so that gives you the gradient when you take the derivative with the respect to X you get minus KX okay so the lrange equation is Newton equation in this this case Mass acceleration equal force and the force is minus KX it's a conservative Force so you you are transferring energy between minus K KX and uh the kinetic energy which is building velocity and acceleration so ah for some reason it's written twice I don't know what happened why it's written twice but now I move this minus KX to the left hand side of the equation and we have mass acceleration plus KX equal to Z so there is no external forces this KX is the gradient of a potential energy V and this is the acceleration of your mass now let's take a look at the response of this system so I not sure if you can see it you see this red potential energy over there this is the potential energy of the spring and let's imagine this Green Dot that is this point Mass we are going to drop so if we drop this point mass is going to fall and it will oscillate right there is no friction it will keep oscillating forever so this is time and we are looking at the frequency of Crossing this axis that is we're going from one side of X we're going to the negative side and there is a frequency of Crossing so so my question to you is in relation to these two parameter K and M what is the effect of M on the frequency so if your mass is heavy heavier and heavier what is going to happen to this frequency and if your K is smaller and smaller what is going to happen to this frequency so if K is very big what would happen if K is zero what is going to happen K is zero nothing would happen right K is larger the oscillation frequency so frequency increases with K and decreases with the mass there is this quantity that we call the natural frequency of the system and this is the square root of k / by m anyone knows why I mean where this Omega is coming from do you see it from this top equation somewhere yeah so so basically yeah say a multip s Omega and T and then very good you integrate the equation and and analyze it its response if you don't trust this result let's see how we can resolve integrate this equation so if we divide by m we get the acceleration plus k / MX now if you integrate this equation the you get the square root of this coefficient of x that will appear and and and we usually rewrite this equation on the top we rewrite it as Omega Square X so the K divided by m we is really the square of your uh natural frequency and if you write this equation and do the integration of this equation you get a senoidal response where Omega appears as the frequency of your sinal motion so in fact X that comes from the integration of this equation is some constant cosine Omega t+ 5 so what is five and C depends of what I heard initial conditions right so from the initial conditions of position and velocity you can determine C and fi and this is your response and you can see that this Omega is strictly the square root of k / by m well if you understand this we need just one more step and then you understand PD control it's very simple PD control actually is imitating the natural system to recreate a spring this K will become your stiffness K is the stiffness and M is the mass K will become your proportional gain and in few seconds we will few minutes we will see another K that involves the damping that also come in into the equation but not of conservative system of dissipative system system that dissipate energy because of friction and and then we will have the complete equation so if we are looking at only conservative system without any damping this is the response okay so in fact if we just add little bit of friction underneath the mass as it's moving there will be some dissipation of energy and this dissipation would be a force opposing opposing what opposing the motion opposing the velocity so it's sort of minus some coefficient time x dot and that friction if we add it to the system we have to add it on the right hand of the equation so the lrange equation is capturing the natural system on the left side on the right side we are putting a natural force which is friction but we cannot put it in this left side of the equation because this force is not conservative it is not a potential energy force it cannot be integrated in V so it appears on the right hand of the equation an external friction Force applied by the environment on the object and if we assume that this this force is simply proportional to the velocity it could be nonlinear friction can be nonlinear you can have column friction you can have stion all kind of if you add this Force to the previous equation that we had it appears here this is a second order equation general form of the equation my system is not anymore conservative because if you isolate now you're going to lose energy and little by little you lose energy and then you stop so the mass acceleration plus BX do plus KX is the general form of a linear system of the second order and if we take the system and analyze it so what we do we divide by the mass do you see Omega Square now so we have omega square and uh it's thank you very much finish so can continue now what we are going to do with B divided by m well when we integrate those equations this is going to appear in some form so what we would like to do is to to see how B is affecting this damping so for instance if you put a large B very very large B and you start falling you're falling you're falling well if B is very large are you going to cross you will just asymetrically reach that goal position without Crossing so you have sort of an overdamped system if B is very small you're going to oscillate and eventually you will lose the energy and you will converge towards the minimum of the energy so here is an isolatory Dam system with higher values of B divided by m we have an overdam system and as we move from here to here there is a special value at which we just go and reach the x-axis and this value is called the critically damped system and remember this we're going to use it a lot because we try to imitate this behavior when we control any of the systems we will try to make it critically damped system so we need to know for which value B divided by m reaches this value this state and the value of B divided by m is simply 2 Omega n so when B / m is equal to Omega n Omega is square Ro TK of k / M well then we have a critically damped system and this comes just simply from the integration of the equation this condition so now you can compute b b the critically dumped B is equal to what come on 2 Omega n * m all right so if you know your mass if you know your k then you can compute your B to be uh gain for your control system here B is the natural uh damping of the environment so the system is passive so let's take this two Omega n and and try to make it explicit in that first equation so I'm going to take this equation and I'm going to rewrite it as a function of Omega and as a function of this critically Dam system so to do that we take B / M which is the value the value that we have right now and compare it to the value that will give me critically damped Behavior okay so B ided by m is compared to this critically dumped B ided by m so this is a sort of ratio right it's uh damping ratio and well I I need to replace B ided by m by something that make two Omega and appear so I need to divide by this and multiply by this you agree so on the left I have this ratio and we call this the natural damping ratio and we use this symbol to represent it what do you call this symbol symbol S so we use S S represent the natural damping ratio it's B / m / 2 Omega n so for which value the natural dumping ratio gives me critically DED system okay for one when Zeta n is equal to one It means that b is simply twice the square root of k m and for this value I will be able to have a critically dump system okay so far so good not too confused so we introduced two Notions we introduce the natural frequency square root of K divided by m and we introduced the natural damping ratio this B ided by 2 square root of KM and now we can analyze our system and write it in this form so the acceleration it was M time acceleration we divide it by m and we can rewrite the equation in this way we can rewrite it acceleration plus 2 Zeta Omega velocity plus Omega squ velocity of X is equal to zero now the time response of the system requ us to integrate this equation and if we integrate this equation we will have this response because of the damping your senoidal the amplitude of the senoidal is reduced As you move there is a decrease and this decrease is exponential and this decrease depends on Zeta and Omega you have the senoidal motion which is function of your natural frequency Omega but it's also function of your damping ratio you can see when Zeta is equal to one this will become zero the cosine of zero and if it's greater then there is no cosine because you will have only the exponential so here is the response so you have this exponential and you have the frequency that is Now function of Omega n Square < TK of 1 minus Theta n uh Zeta square that is the period is 2 piun Omega n s < TK 1us Theta n² so it's not omega n anymore Omega n was the natural frequency but this thing that appears there is sort of a natural frequency that is affected by the damping so we call it Omega the damped natural frequency so it was Omega the natural frequency and now it is damped all right I think this is the last definition you need to remember and with this we can do almost everything except the nonlinearities we have to deal with little later but Omega when you have damping is really Omega and the natural frequency that appears in your spring scaled by square root of 1 minus Zeta which comes Zeta Square which comes from your damping B divided by the mass and the the the spring or the gain of your system okay all right so these are characteristics of a second order system and what we need to do is to uh just Inspire our controller by this and then we will be able to re recreate that behavior simply by selecting selecting what by the way so if you start with the mass and now you want to create a system like this what do you need to select H you need to select the spring which is K the stiffness of the spring and you need to select B so by selecting B and K you can create this second order behavior on AAS so if you have one joint with some inertia to create a behavior like this a closed loop behavior of second order with some natural frequency some damping ratio and some damp natural frequency you should be able just to select B and K and find your system okay so really the control of a system is going to be almost the same we are going to pick Omega we are going to pick Zeta which determine K and B and then you will be able to control the closed Lo yes what happens when uh V is a function of the configuration of when V I'm sorry when B when B is a function of the configuration does that like how do you handle that right right well b b could be uh actually the most general form of B is B is function of x and x dot and uh even higher order and then what you get is you do not get a linear system you get a nonlinear system so there will be additional disturbances on the system and you uh need to do things to do two things either to modrow your friction and then try to compensate for that and that's what we're going to do for centrifugal forces that's what what we're going to do for the fact that the mass is configuration dependent but uh once you model it you can integrate uh the model in your control and you can compensate for those uh nonlinearities then at the end after compensation you come to this form a linearized form so what we're going to do actually later is to go and compensate for the gravity compensate for centrifugal corus forces compensate for uh nonlinearities like friction and then reach a level where we have simply a decoupled system with M or n masses that we we can control using this now compensating for friction is very dangerous it's not easy you cannot just go and uh like well do some estimate and compensate for the friction uh you can compensate for the friction and uh well well we can try compos it for the friction if you want I'll show you how dangerous it is okay here is a Rob what is the name of this robot the Puma okay let's move it it's moving well it has some friction what this is this friction it has you see I'm moving it and uh well it has friction because it has natural friction but if I remove this friction okay now I remove the friction look what's going to happen I'm going to apply a small Force are you all attached we're going to [Music] move all right we saved it so so you can see uh if you compensate with the for the friction uh you you start to have you you will have quickly uh oscillations and you will have um instabilities so let's say let's say the robot is is controlled now with those Springs you have 400 400 uh these are the value of the springs and you have uh some value for the bees basically in here 40 this is Joint three so if we change this to be like uh 40 you see I'm I'm pulling now it is com it is little bit moving you can see joint three if I pull on joint two it is stiffer joint three is responding I mean if we can if we can reduce this gain to make it oh four is good no that's too much let's make it four now it's easier to move but you you see the damping it's still like over damped when you move it it is not responding so let's make this very small now there is little bit of motion if we make this zero so now if we start putting negative damping to compensate for the natural friction we're going to go unstable so this is uh small amount okay how about minus 9 uhhuh and let's put little bit higher again okay and obviously you're going to go unstable the the real time is not real that's why it is little bit weird but it is so so don't try this no negative damping Works damping positive dumping is really good okay all right so so remember this uh uh Omega is it bigger or smaller than Omega n so you get your Omega n from square root of K ided by m and now your Omega after damping is smaller good okay okay all right don't look now at notes please we're not hiding every you you have the answers but think about this what is your damped natural frequency we just saw the expression so don't look at your nose okay try to compute it so how do you compute your damp natural frequency you need to compute the endm natural frequency the natural frequency is what 8 / 2 square root is 2 your your uh damping ratio is what B divided by square root of kilometers km so km is 16 square root is 4 2 is 8 oh come on I'm not going to do it okay so this is your Omega you're right this is your Zeta is it's easy to remember B divided by two Square kilomet squar okay remember that and your Omega is 1.6 so you reduce it from 2 to 1.6 all right so we have another video segment next time but we will skip it now so let me all right we have a little bit more time so I'm going to to go over one degree of Freedom that we are going to control exactly as we did with a passive system so one degree of Freedom robot we're going to assume that the robot has just an inertia or maybe a mass so it's sliding maybe mass is better if we take a prismatic joint so a prismatic joint is going to involve just the of the moving link and we're going to move it with a force and we are going to create this Force as a spring and now we want to move to some location so that would be the resting position of the spring and then we can recreate exactly the same behavior on the robot so here is the the one link robot it's uh just simply a mass probably this is the simplest robot you can imagine it is a mass moving under this force of the motor so the motor is going to apply a torque translated into a force F and you want to move it from its current position to X design okay simple problem so we have mass acceleration equal force so now your force is your motor force and you your task is to create a force that will let you move this Prismatic joint from its current position to the goal position desired so the same thing as a spring what we're going to do is we are going to create this spring or potential energy whose minimum is at X desired so the potential energy that you're creating is positive everywhere except at XD is equal to zero so it's zero at your desired position and it means that you could have something like a quadratic potential energy with some gain KP which produces a gradient that is equal to what what would be the gradient of this potential energy F should be [Music] what so you take partial derivative with respect to X and that would be K time x - x d are so your system is simp simply Mass acceleration plus KP xus X desired okay so here we have the zero of the spring moving changing with your desired position that the only difference that's it and that doesn't change anything that just changes the zero so instead of talking about stiffness we are going to talk about the gain that you using for your error in position so we call it position game and immediately we can go to the equation and analyze what happens if I apply this controller and why whether this controller is going to be stable or not we can go go and do this analysis on your on the equation of flag Ranch on that system system and we analyze what happens when we apply a force that is the gradient of this potential energy so our potential energy is is this we took the gradient of the potential energy and we applied now going from this page to this page it's interesting because here I was looking at just one degree of freedom but if we go here we can show that whatever the number of degrees of Freedom if we apply a controller like this essentially so this could be 6 degrees of freedom 20 degrees of freedom whatever the number of degree in the system if your potential energy is in this form and if you are applying a force that is the gradient of that potential energy then what you can do is you see you see this equation here you're applying this Force there well you can move it to the left side and now you have your potential energy in this equation so what is happening here is that your lrange equation showing you that you have a system on the left hand side that involves only kinetic energy and potential energy equal to zero zero external forces so what do you expect in term of the stability of the system so if you have a mechanical system under potential energy and and kinetic energy with no external forces no external forces because all the forces are conservative are gradient so what we can say about this system huh it is stable so simply by selecting your control your motor controls to be the gradient of a potential energy you guarantee that your system is going to be stable so this is a very important result because now we we know that if we use this form of control sort of proportional to the error which is uh the derivative of a potential energy then we are going to be stable now stable is not sufficient because it can be aolly stable actually this system we know the response of this system it's going to oscillate so this force is not sufficient what should we do we should add some damping we should add some dissipative forces so in here I'm going to change this zero with some damping very good we're almost there so this force is not in the potential energy it cannot move to the left it is going to be there and I'm going to put an external dissipative Force I'm going to put a damping Force but I need to know in which condition what are the conditions on this Force what conditions are required in order to make this system asymptotically stable what does it mean asymptotically stable by the way you understand what what what it means so this is the behavior of isolation that are damped critically damped or over damped your system will converge toward the goal and reaches that goal this what we are trying to achieve and this thetive force that we saw before was doing something to the system do you remember someone here said something about that force that force was doing something to the most or so what is the condition on FS that is the question resist has to oppose the motion right so if you are moving in some direction you should oppose that motion so what would be the simplest way to do that FS should be if your motion is measured with a velocity x dot well you could always have a a force that's just constant opposite to the velocity right so if you are in higher Dimension basically this could be a force that is opposing your velocity and what you need to do to make sure f is what in order to oppose the velocity that the dot product between the two Vector is negative so if your force that product with the velocity is negative for any nonzero velocity you are ASM totically stable very simple so come on pick a one force that that satisfy this condition the simplest one you just said it so you can pick minus K VX dot with KV positive that would satisfy this condition so if we apply there FS equal to this linear damping then essentially we take this control the conservative part of the force and add to it the damping part okay these two pieces represent the PD control proportional derivative so if you want to move to a goal position all what you need is a term that captures this error this goal position could be far away even so it's like a step response you're you're stepping X to this goal position and you're you have damping that is trying to reduce x dot to zero because here you could have X do minus X do not desired but because you are not tracking a trajectory you are just going to a goal position you want to stop at the goal position so it's like X do minus Z okay well this is the PD control now how we Design This control how we pick KP how we pick KV depends on the similar similar characteristics we studied just earlier with passive system and dissipative system natural system so the KP is going to be picked so that KP divided by m the mass of the system gives you the Omega that you wish so when you are controlling your robot do you wish to have a small Omega or a large Omega what Omega does to the response Omega small Omega so if you want to move from here to here with small Omega you takes long long time so usually you want much faster response if you want much faster response I mean if you want to move slowly not only the time response but also the the the stiffness of your system because your stiffness is depending on KP your disturbance rejection is depending on KP We Will We Will analyze KP and see why we want higher KPS we will analyze KV and see its limitations and we will see how we pick those KPS and KVs for given performance of Omega and Zeta on Wednesday okay |
Lecture_Collection_Introduction_to_Robotics | Lecture_2_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started so as always uh the lecture starts with a video segment and uh today's video segment comes from 1991 and uh from uh the group uh at British Columbia and it deals with byet walking so there should be some sound [Music] okay so this is passive walk no Motors [Music] counting on Bal [Music] V well maybe we need some Motors right okay So today we're going to start covering kinematics and kinematics as I mentioned last time kinematics is very very important the models that describe the robot position uh the robot uh frames and links and Joints so we're going to go over the basics in uh descript describing a task the models that we can use to uh determine the position and orientation uh of the end of factor then obviously when we determine the location of a link we need to be able to transform that description to the next link or to describe the position and orientation of the end factor in a previous link so we need really to uh handle Transformations then we need to discuss how we represent uh the position and orientation there are many different ways uh through which we can describe a position or an orientation and we will discuss few uh different representations and uh I'm going also to describe a little bit what is a manipulator what is a robot arm and then uh what are these joints what are the degrees of freedom of a manipulator how we count represent uh the position of a manipulator so a manipulator is defined by a set of links connected through joints the first one the first of those link is fixed we call it the base and the last one is actually the this gripper the the whole purpose of the manipulator is really to move this gripper and place it in space uh to do manipulation obviously uh later we we will see that it is possible to use the body the links themselves to do manipulation we call it whole body manipulation but for now we are really interested in locating this end of factor uh at the same time locating any other links uh that is moving so we will see that there are two types of joints that we are going to consider there will be other possible type types of joints but we can see that any set of joints could be reduced to those two type of joints the revolute joints and the Prismatic joints a revolute joint allows you to rotate about a fixed axis and a prismatic joint allows you to translate about a fixed axis and this motion is done along or about one axis so it is one degree of Freedom motion so as I said we have links and those links are in number of n we're going to call n is the number of moving link plus one base link the fixed link and we have joints of two types revolute joints and Prismatic joints so the idea is we are going to work with one degree of freedom and this is interesting because knowing that we have only one degree of Freedom joints then we will be able to connect these to the generalized coordinates as we will see and as I said if we have a joint like a spherical joint would you know what how many degrees of freedom a spherical joint would would have two three three yeah three so what we would do is we will then use three revolute joints with zero links length and then we will introduce uh these joints and links to represent a pris uh a spherical joint okay okay now we have this manipulator in this configuration and the question is how can we represent the configuration of the manipulator what would be a good way to represent the configuration because we need to know where the manipulator is in space with respect to a fixed frame so well I mean there are many different ways we can go to each link and try to to fix that link so we can take uh maybe a given link and say we're going to locate this link with several vectors that lock it there so if we use like three vectors at three different points the link is defined and that would give us the configuration of that link now we are going to use in that case uh three vectors each Vector has three parameters in 3D so we have nine parameters to describe each link and we have n links moving links so we will need N9 NS a lot of parameters and this is this would be one of the representations so the description of the position using a set of configuration parameters can involve a large number of parameters and each of them is fine so any set of parameters that describe fully the configuration is called a set of configuration parameters so in this case here we have nine parameters per link now we are really interested in a particular set of parameters configuration parameters that has minimal number of parameters involved we don't need all these parameters in that three vectors three points because the points are fixed so so there is a Constance between these points and that is going to show us that these three vectors are not independent in fact we will see that we will be able to describe the configuration of the link with much much less parameters so that brings us to generalized coordinates so a jaliz coordinate is essentially a set of configuration parameters that brings parameters that are completely independent and working with those coordinates is very interesting because you can use them to find the Dynamics later as we will see we can use them we can count them and the number of geniz coordinates gives you directly the number of degrees of freedom of your robot so let's uh analyze how many parameters you would need to describe the configuration of a manipulator with a set of geniz coordinates so we have this manipulator connected through uh joints and in order to count how many degrees of freedom we have what I'm going to do I'm going to remove the joints so let's remove these joints now you have n rigid body in space right if we take one of those regid B this one of the links how many parameters you need to describe the position and orientation six three for the positions and three for the orientation so with six parameters we have a description of one rigid body now we have n moving rigid body in total we need 6 n now let's think about those constraints introduced by the joints if we put back the joints we are going to introduce constraints now a joint has one degree of Freedom right we said one degree of Freedom so how many constraints the placement of a joint is going to introduce each joint will introduce how many constraints five five exactly because it is going to allow only one degree of Freedom so if we think about the number of constraints we will see that we will have five constraints per joint and that leads to five n constraints yes re joint whether it's a um position parameter or an orientation parameter that's well let's let's look at it so if we have uh a joint it's going to introduce constraints on the rotations and uh the position so if we place a revolute joint it's not going to allow any displacement and it's not going to allow rotations about orthogonal axis to the axis of rotation so that's five P strengths three positions constraints and two rotation constraints in the case of Prismatic a prismatic joint is not going to allow any rotation it's just translating and translating about one axis so the two other axis are eliminated doesn't matter it is still five constraints so now let's do the count so we said we have six n parameters before placing the joints and now we place the joints and we have five n constraints so the question to you how many degrees of [Music] freedom it is going to be it is going it is going to be the difference right so 6n minus 5 n the answer is n Okay so indeed we have just n degrees of freedom which is really nice if we have one degree of Freedom joints in a manipulator in a robot we can we can be sure that we are going to have nend degrees of freedom the number of joints now I'm talking about a manipulator with fixed base if I take a humanoid robot and I do the same things at a given configuration if we lock one of the feet of the humanoid robot this is correct but a humanoid robot can move so in that case the base is moving and the base has six degrees of freedom so it will be n + 6 unfortunately those last six degrees of freedom of the base are not not actuated there there are no Motors and that what makes the control of a human age robot very hard however in the case of a manipulator if the basis fix we have n degrees of freedom for n jointed robot okay clear okay so let's go to the end of factor now the end of factor is this last RIT body in the system system so it has all the freedom be before to position and Orient the on end effect so we can think about a point at the end of factor that we are locating and we can think also about the orientation of the end of factor how we Orient this ector so there is a sort of frame attached to the ector rotating with it and that allow us to describe the position and orientation of the end Factor so if we have just one end of factor in the robot essentially we have just one rigid body at most we need depending on the freedom because some some robots can only move in the plane some robots can move with restricted rotations or orientations or positions so at most we're going to have six degrees of freedom for the end of factor but again we can represent that freedom of the end factor that is the configuration of the Endor with many different parameters so we can talk also about configuration parameters for the end of factor and we can see that some of those parameters can be dependent and when they are independent they form a set of generalized coordinates and then we can have a description of the end of factor using a set of generalized coordinates task coordinates or what we call operational coordinates so if they are just configuration parameters without the condition of Independence then we can talk about these M parameters describing the position and orientation of the end of factor so this is the definition that is we have a set of parameters describing the position and orientation with respect to a fixed frame let's see give me an example any example of set of parameters that describe the position and orientation of a rigid body and a factor so to describe the position and orientation of a rigid body this is arid bu here is arid bu I would like to describe the position and orientation with respect to this Frame so well for the orientation sometimes tilt uh ya and roll okay you you you use three angles so if we use uh like three alert angles or fixed angles we can uh find this position orientation for the ector and for the position XY X Y and Z so we can take a vector and locate one point fixed point so with one vector and three angles we can describe the position and orientation of the end Factor yes isn't that perfect six parameters oh I was talking about how a joint introduces five constraints on a rigid body and uh instead of so a rigid body so imagine the base imagine the next rigid body it's was free completely to move and now I put a joint and now it has only one degree Freedom left yeah so so here we have a very nice example of of uh um a representation that is minimal representation so anyone can give me a representation that is not minimal three points select three three vectors to three different points that will be nine parameters so that would be a set of configuration parameters well we will see also that have you heard about alert parameters how many of them you have four so you Quan or well the reason we will we will see later that when we use three angles we have a problem we have a problem tracking this rotation continuously there are configuration where the representation becomes singular and we need a different set of parameters really to keep track of the orientation and we introduce ler parameters have you heard about Direction cosines no vaguely okay you heard about rotation Matrix we're going to see this in a few minutes but you heard about rotation Matrix right so you have a frame you have a frame and you you're looking at the relationship between the two well the rotation Matrix if you take this rotation Matrix between two frames so the description of this frame with respect to a fixed frame essentially if you take this Matrix which is three columns of three vectors so it's nine parameters well if you take that Matrix these three vectors uh these three vectors form the so-called Direction cosines of the this frame with respect to this Frame and we can use directly that Matrix that description so we will have nine parameters only for the orientation plus the position three parameters so we will end up with 12 parameters describing the end of factor and we will see why we will go that way or why we go to other parameters and we will examine those singularities of the representation so a particular set of coordin that we are interested in is this set of independent coordinates we call them operational coordinates or task coordinates essentially there we are looking at the operational point where the robot is acting where we Define the task so for instance the task could be if I'm going to grasp the task is somewhere in the middle between the the two jaws as I move this is the point I'm controlling but if I have a tool the task will be here and the operational point would move depending on where I'm going to do the the interaction so this is the first definition is operational point so you have three degrees of freedom and then you add three degrees of freedom like the the angles the three angles to form a set of independent parameters and that gives you a set of uh generalized coordinates or operational coordinates so this number uh so before you remember it was M if it was not independent when it is independent we call it m0 to to to point that this is independent set of parameters and that gives us again the number of degrees of freedom of the end Factor so an Endor of a robot with six degrees of freedom moving in the three-dimensional space the end Factor itself can be positioned anywhere and oriented anywhere so it has six degrees of freedom and that is the most number of degrees of freedom it can have have now if we if we go to the plane if we have a robot J is moving just in the plane how many degrees of freedom you expect to see for the end of factor my plane a robot it's moving only in the plane cannot go out of this plane for for the position how many we need in the plane two X and Y and for the orientation one only one so three so if we have a planer robot then we will be talking about m0 equal 3 and you have different robots with different characteristics that ends up to gives you m0 that is equal to four or five or six but at most you have six for one and a factor so now we defined operational coordinates or task coordinates we defined joint coordinates and here is an example if we take a a planer robot so just uh three revolute joints Theta 1 Theta 2 theta 3 and this robot is moving in the plane so we have we have U sort of representation of the joint so for uh here you have uh I don't know 80° U 45° and uh 50° representing this configuration of the of the manipulator so one way to think about it is to go and represent this whole manipulator as a point in a three-dimensional space and that space is Theta 1 Theta 2 theta 3 and that would be the joint space where this point Theta which is the vector Theta 1 Theta 2 theta 3 represents the configuration of this manipulator so we call this the joint space or the configuration space and this space plays a very important role in motion planning we we talk about configuration space and we talk about planning motions in configuration space so planning the motion of theta and we talk also about all these obstacles that we have in this space in the physical space that we map to that space that becomes configuration space obstacles or C obstacles that represent the uh how obstacles for in the real world are mapped to that abstract configuration space and then we can do the planning around those obstacles now for the end of factor we as I said we locate the end of factor with a vector XY but that doesn't Define completely the pos position and orientation of the end Factor we need also to define the orientation or some angle so we need Alpha and then x y and Al represent fully the position and orientation of the ector and that defines the three coordinates operational coordinates for the IND Defector and obviously we have with that a space the operational space which is now the combination of X Y for this example and the orientation Alpha of the end factor and that is a point so the robot is reduced uced to a point Theta in configuration space and its end factor is reduced to a point XY Alpha in the operational space and that represents the manipulator and the Endor now these two spaces so the first space is fully describing the configuration of the robot but imagine that we add one more joint on this robot the ector is fixed but the robot configuration can vary because of the redundancy we introduce by adding one more joint so redundancy in this example here you can see we have four joints and the end Factor still has three degrees of freedom so for the same configuration you have different possible configurations of the structure of the links and that means we have redundancy so we talk about redundancy and we call the robot redundant if the number of degrees of freedom of the robot n is greater than the number of degrees of freedom of the end Factor m0 so if we have this situation m0 here is equal to 3 and N equal to 4 then the robot is said to be redundant and redundancy is very important in order to to reach and have accessibility you cannot just work with uh the motion of the robot with for instance three degrees of freedom in the plane you're going to hit obstacles so when you have redundancy that helps you to move around obstacles and position the robot in different configurations and we measure the degree of redundancy by the difference of N and m0 okay so essentially we're not going to really discuss redundancy which is very important but we're going to uh to focus on non redundant robot F first in fact uh in Spring we will cover extensively redundancy we will talk about the use of redundancy to control control the robot make make uh use of the mechanical advantage the dynamic reduction that is introduced by redundancy and also the motion planning uh in collision avoidance using redundancy and how we combine and control redundancy in a way that would allow us to achieve a task while redundancy is is maintained to achieve different criterias and uh different goals so redundancy we will not not come back to it uh in int robotics uh but it is a a very important notion that you you need to at least know about in term of the def its definition what we're going to do we're going to go to the basics now and we're going to start by building the models that would lead to the forward kinematics so I'm going to start with the simple definition of a vector that is def finding a point in space and we are going to go from there to building the models for representing an object in space which requires position and orientation and then we connected these objects but before that we will talk little bit about representations of those different parameters I mentioned earlier so we will talk about iller parameters we will talk about lert angles and Direction cosines and uh probably this week we will cover all of that we will cover Transformations how we move between frames and we will be ready next week to build the forward kinematics so a point in space a point in space p how can we Define a point in space and what are the things that really fix uh that point or Define it with respect to some uh reference so what is really important is to think about a point is the fact that the definition of a point everyone of you probably think vectors right we can use a vector to define the point but what really is going to determine the vector is another point a reference point the the origin that uh you're using to define the point if you change the origin you will change the vector so we will talk about the description of a point with respect to another point to some origin and this point is going to be represented by a vector p and this Vector P will describe this point with respect to this origin if we change origin we will change the vector okay so if we have two points we Define again the origin and then we will have two different vectors representing this point and these are those vectors built by taking the connection between the origin to the point very simple now I'm insisting in this because now we're going to introduce this origin and then we will be able to describe the vector component so the vector is independent of this X Y and Z which frame we are taking now we're going to put a coordinate frame and then we can express this Vector in the that frame if I change the orientation of the frame the vector is the same the coordinates the components of the vector will change and we're going to be interested in those Transformations between different descriptions that involve different frames we will work with oron normal frames X Y and Z and we will go from one frame to another so in this case X Y and Z and we did small rotation and so about the same origin and we end it up with a different frame and we need to know how we can take a description of the coordinate of p in the the frame X Y and Z and then go to a description in frame X Prime y Prime and Z Prime and this is going to happen using Transformations now this transformation is just rot in the the the uh the frame about the same point but we might have a prismatic joint and then we are going to translate the origin so there will be a translation so we need not only to deal with rotation in the frames but also translations of the frame well a frame really is related to something Beyond a point the fact that if if we we just are we're working with points we we just don't need really to worry about the whole orientation what is really happening is that you have different points on a rigid body and when we rotate we are kiing the distance but the orientation is changing so a frame is really related to the description of a rigid body so if we take this rigid body take a fixed point and attach to it a frame I'm calling this Frame B so the coordinates are going to be described with respect to those axis XB so this is the frame B we denote XB YB and ZB and the frame itself is B and the question is how we describe the frame B with respect to a fixed frame a so as you know we need to find the relationship between the origins so there is an origin of frame a An Origin of frame B we need this Vector between the two and that is defined by a vector and this Vector is going to locate the origin of frame B with respect to frame a and we have its description in frame a so this is PA the orientation of the frame is these vectors XB YB and ZB these vectors can have descriptions in different frames now we are describing them in frame a so XB in a and this is the notation we denote XB and here we put a to say these are the coordinate of XP in frame a so these vectors are going to describe the rotations of B with respect to frame a If It Moves so essentially if you think about those vectors and the relationship with the frame a we really going to find the rotation Matrix which is the the first uh model that we need really to use in order to describe the rotations of one rided body with respect to another rid body so because we are concerned with the just the rotations I'm going to slide this Frame and make PA equal to zero so we will just move to the origin and just think about the rotation and in this case we will focus only on the rotation Matrix and the rotations of that frame with respect to frame a so we're now concerned with the rotation of frame B with respect to frame a this rotation is described by a matrix it's called the rotation Matrix and it has nine components we are calling them R11 2 1 31 these three columns okay I know some of you know very well the rotation Matrix how many of you knows like perfectly the rotation Matrix I remember how many of you remember okay all right this is very important so pay attention those of you are seeing this for the first time but I I I think I'm sure everyone has seen it some in some form or another so what I'm going to do I'm going to say State a a description of what is the relationship between XB this Vector XP and XA XB Define find in frame B is obtained by this rotation Matrix and the vector the resulting Vector is XB in frame a so the description of XB in B is now transformed into a description of XB in a using the rotation Matrix what is XB in b 1 0 0 uh the X vector in its own frame has unit uh one unit along the X Direction so it's 1 0 0 so what about y it is the same thing 0 1 0 right about z0 Z1 you you these are the unit Vector x y z in their own frame right and now using the rotation Matrix we have the description we have their component in the a frame so the rotation Matrix is basically just this right I'm just using that definition so if you multiply the First Column is XB in a the second column so what is the rotation Matrix the rotation Matrix between b and a is simply the component of x b in a component of Y in a and component of Z in a okay so always remember this definition this is very very important because uh we are going to find the rotation Matrix uh through many different ways but sometimes you are looking at a problem and you are looking at the frames look at the component of that frame in the other frame and you you will make sense of your result and this is all always the definition the rotation Matrix essentially The Columns of the rotation Matrix are the component of the uh uh axis x y z of the new frame in the reference frame okay all right so this is the definition of the rotation Matrix from B to a it is XB in a YB in a and ZB in a so how do you obtain XB in a how do you obtain the component of XB in frame a you just do the dot product so essentially if you do the dot product of x b with X Y and Z you will obtain XB and a right you agree good so which means the rotation Matrix is is essentially the dot products of XB with a x y and z YB with X Y and Z ZB with X Y and Z right okay good so look focus on this and look look look look look look look at this row do you see anything any anything like special so here you have x x x XYZ here you have anything constant x a x a XA so it's the dot product of x a with XB YB and ZB which is you see this which is X A and B written as a row which means it's transpose this is very interesting because if we start looking at these properties we see that this rotation Matrix is either the component of b in a or the transpose of the rows of a in b so b in a a in B inverse relation going from B to a going from A to B so going from A to B we are able to see that it is just the transpose which means that b to a the rotation from B to a is equal to the transpose of a to B A to B is the inverse of P to a so we have this property if I'm trying to compute the inverse of the rotation Matrix B to a which means it's a to B it is simply B to a transpose so the transpose of this Matrix is simply I mean the inverse of this Matrix is simply its transpose and that's a very important property and this property comes naturally because the rotation Matrix formed with these unit vectors that are orthogonal so the Matrix is called utto normal and this uton normal Matrix is always going to have this property its inverse is its transpose all right example so can you compute the rot I think you have it don't look at your notes now for this examp so could you give me the First Column of the rotation Matrix from B to a what is the First Column going to be so it is the component of XB on a and xB B on a you see XB and x a are aligned so so anyone could say what are the component so one z0 what about y y has a component only along the z-axis right Y is only along the z-axis and zero above others what about Z Z has a component along the minus y AIS and its full component minus one z z very simple example but it illustrates what we have done essentially what we are doing we are doing either dot product between the two vectors or looking at the component of that Vector in frame a at the same time if you think about this this is XB in a YB in a and ZB in a this is our definition and in the other direction if you look it is X a transpose in B so if you take x a and express it in B it's going to be 1 Z 0 transpose so these rows represent x a in b x y a in b and Za in B all right so now we know the rotation Matrix let's let's build finally this representation for a rigid body so we know how to represent the frame the rotation of the frame we need the translation of the origin and by combining X the description of the rotation that is those vectors and by locating the origin of the frame B with respect to a we are going to Define fully the frame B with respect to frame a so the frame B is essentially defined by this rotation Matrix B2 a and by the location of the origin with respect to a Okay so there is uh different ways of thinking about the use of those rotation Matrix matrices and in fact uh we can think about it as we have done so far which is to say we have a vector P we have a frame a and a frame B and essentially we are expressing the component of frame B in frame a in frame B and we are looking at the relationship between the two so this is what we call mapping that is we are changing the description of a vector from one frame to another frame but the vector remain the same there is another way of thinking about it which is so this is uh the way we are looking at it you have a vector p and this Vector p is going to be described by it's dot product with a to give you its component in a and this is the same thing I'm just removing the vector outside and putting it as a matrix so it's a transpose Y and Z transpose you see this writing it's the same writing right I'm here I'm I'm doing Dot product and just if you move P outside it means you are doing that multiplication with the different uh rows so now if you have this relation you can say p can be expressed in any frame I mean this operation could be represented in any frame which is let's select B the frame B or C or D so but you have to be consistent you have to take the same description and if you do that essentially you are obtaining that mapping that is you are obtaining p in B rotated to give you p in a so this is a changing the description of the same Vector P from one frame to another frame in the translation we are going to start by considering the same problem but while maintaining the same orientation of the frame so I'm going to slide this Frame and change the location of the origin of the frame so frame B and frame a has the same orientation and we're going just to move uh along the direction of the vector P so if we have a vector if we have a point in space it is located with respect to origin B and described by the vector P the same point in space is described with respect to another origin a with a different Vector PA a so it is the same point but described with with respect to two two different points two different Origins attached to two different frames and you end up with two different vectors so this was our initial description in frame B and now we have a new description and this transformation is resulting in two different vectors you have to realize contrary to the case when we were doing just rotations now when we do a translation we are going to change the description by changing the vectors involved in the description so so this operation that involve a translation of a vector P defining this origin of the frame B it's changing the description from origin B to origin a and the relationship between the two that is the relationship between the Green vector and the red Vector is essentially this relation that is giving us the vector with respect to a origin a as the sum of the vector with respect to origin B and the translation of the origin so this fact that we have now two different vectors is going to uh appear later in the homogeneous transformation and make the transformation uh little bit different from rotation matrices because we are introducing a translation and we are introducing this nonhomogeneous relation in the model so when we come to uh General transformation so now I'm saying I'm going to have a description of this this same point P but with respect to an arbitrary frame B that is rotated with respect to a then we need to account for this rotation and that means that in the description here it's not simply the sum but I have to do the sum with respect to descriptions in the same frame that is I cannot add this Vector directly with this Vector I need to rotate this Vector to frame a and that means we have this relation this General relation that is we take the description of P we rotated to frame a we have its description of a and we have the origin description of a and now we can add them together and the result is this Vector we we haven't changed anything we still are talking about this Vector plus this Vector equal this Vector but what we have to make sure is that this description is rotated correctly to frame a all right well this is the general transform and in fact using this applying this between links we should be able to uh compute and propagate go from this link to the next to the next to the next but this description is not uh not simple to carry when you have multiple links because you you don't H it's not like a rotation metrix where with a rotation Matrix if you know the first rotation with the frame with respect between two frames and you know that the the two are rotated with respect to a different frame all what you need to do is to multiply the rotation matrices in here uh you have to carry sums and you have to carry those relations so a better way to handle this transformation is to try to put it in a homogeneous form how can you you do that this is sum of two vectors in three-dimensional space you cannot have it in homogeneous form but if you go to four-dimensional space then you can put it in a homogeneous form do you want to see how H yeah okay so I'm just rewriting the same thing and I'm going to just add one more Row for for nothing so this row is saying 1 is equal to one right if you multiply this vector by this Matrix you obtain the the first relation right and the second part is 1 = to 0 multiplied by the vector p + 1 so 1 equal to 1 but now we have captured the homogeneous property that is this Vector is transformed into this Vector this description is transformed to this description using the rotation Matrix and the translation do you see that and this is what we call the homogeneous transformation it's a 4x4 Matrix you have four component that are doing nothing except help them math to make this transformation homogeneous in that when we go from A to B to C to D then essentially we are going just to multiply matrices but we have to handle 4x4 matrices instead of 3x3 matrices okay so this is this is a very important component now by the way sometimes 0000 01 uh the definition uh is uh on the top so sometimes you define one p instead of but it is ex exactly the same uh computation so here is an example of the homogeneous transformation now we have two vectors I'm taking the frame you remember the frame we the example we saw earlier the example where we had a rotation Matrix with rot rotating about the x- axis so I'm just translating now the origin of B with this Vector 031 so the homogeneous transformation so it is the same rotation Matrix as before and 031 is the vector describing the origin of frame B in frame a and your homogeneous transformation is Here and Now using this homogeneous transformation you can compute the new position of the point P so the point p in frame B is described by 011 this is the the point I'm looking at and to find this Vector you all what you need is to take this transformation and multiply this vector by this transformation but you have to add one so we take the 0 one one we add one and the multiplication leads to 0 2 2 and one you drop the one the answer is 0 22 so if you look over there this was happening in this plane so essentially here we have one and this is two and one and two so it's just two two okay clear okay now that you understood everything I'm going to confuse you uh normal I I mean once you understood mapping now we have to to to to completely change the the intuition now instead of mapping we're going to to so a rotation Matrix I said allows you to uh describe the same Vector in two different frames but now I'm going to take the rotation Matrix and use it to rotate a vector so the vector was here I'm going to rotate it by uh the rotation Matrix so I said the mapping is changing descriptions okay an operator is moving those points in space well that could be useful so you have a vector you have one frame you have a description another frame you have a description now what I'm going to do I'm going to do a rotation and this rotation will rotate the vector so in fact a rotation Matrix which could describe the relationship between the component of a the same Vector in two different frames in this way can be also used as an operator that would operate on a vector P1 to produce a vector P2 and this is very useful later when we compute representations you will find very useful to compute transformation between different frame so all or here is operating on P1 to produce P2 okay so P2 is the rotation applied to the vector P1 and of special interest are those rotations that take place about some specific axis like the x axis or the y axis or the z- axis with some rotation so then you can talk about RX with some angle Theta and that will will be very useful in some of the operations so in general we can talk about a rotation about a k Vector not the XYZ but any arbitrary vector and it rotates a vector P1 into Vector P2 so here is an example this is a rotation about the x-axis so the xaxis is 1 0 0 and the rotation about the x-axis is of theta it is cosine Theta minus s you know this familiar Matrix rotation about the x-axis so if you take P1 which is has component on Y and Z so it's 0 to 1 0 to 1 if you take this much you end up with 01 2 which is over there 1 2 you see P1 and P2 so this is the vector 0 2 1 2 and 1 and now 0 1 2 is 0 1 and two okay translations the same thing we can now instead of describing a point with respect to uh two different frames we are going to translate that point using an operator so in the mapping we take took the vector with respect to B and produce a vector with respect to a in the operator so in the operator this is what is happening we are changing the point we're going from P2 to P1 so this point P1 moved so we have two different points and two different vectors so when you are translating you are thinking about it like in the rotation the rotation we had Vector we have P1 we rotated to P2 now we have here a vector P1 with this translation is essentially is producing is producing this point so I have to remove this you can see it better so we are moving P1 to P2 with the translation uh Q so P2 is P1 + Q so this is an an operator of translation and you can have this operator Q along the x axis y AIS Z Axis or any arbitrary Vector so a translation through the operator Q would result into a different Vector P2 and now you have to describe it you can describe it in frame a or any other frame but through always maintaining the same relation I mean same description for all the vectors when you apply the when you apply the components for describing the frame you have to make sure that Q is is described in the same frame as P1 and the result will be in the same frame so now we can think about this operator as the this operator 11 one that is there is no rotation at all but there is only translation QX qy qz and this is defined by the Q Vector so the translation is done by QX qy and qz and now you can combine the two you can combine the rotation and translation so you could have an operator operating by translating and rotating so the general operator that you can imagine is this operator rotating about some vector k with some angle Theta and translating Q so the point P which was in P1 now is rotated and translated and the result is P2 and this is without any definition of the frame then you define the frame in which you want to express all these vectors but you have to make sure that you are using the same frame for all the operations so this is the most general form of transformation between two different uh points it is the same homogeneous transformation it is the interpretation uh of that transformation whether it is a changing the description or operating and changing the points okay are you enough confused good so two things we we we we saw the transformation it is the same homogeneous transformation that is has two components rotation translation rotation Matrix translation of the origin of those frames and that can give us mapping changing the description and I just discussed the other uh interpretation that you could have which is to change the vector itself and that means change the points describing uh uh those uh uh rotations so when you rotate a a vector you are going from one point to another point or when you're translating you are going from one point to another and you can apply both of them and now we're going to look at the inverse now in the case of rotations the inverse is very simple what is the inverse of the of R rotation Matrix from B to a transpose now for the homogeneous transformation this is not the case we we cannot just say it is transpose that is this Matrix is not auton normal because the presence of this translation so the inverse is not exactly the trans trans transpose of this Matrix but it is almost because if you look at the inverse of this Matrix it involves the transpose of because if you take it by block you you will see that you have the inverse here this is the same the only thing you have here is the description of the inverse of this Vector what is this actually I mean if we were thinking about this inverse it is going from A to B so this is the origin of a in frame B so basically you are just writing here the origin of frame a in B that's it okay so now we know the forward we know the inverse of the homogeneous transformation and it is going to be essential the homogeneous transformation uh we're going to use it to describe the uh kinematic chain of the manipulator and we will see that the this this uh transformation it could be described by those four parameter the DH parameters so we will be able to describe each relation between successive links just using four parameters the DH parameters so I will not reveal the movie segment for next time but unless you have any questions we can stop here any questions no okay we will stop here and we'll see you on Wednesday for |
Lecture_Collection_Introduction_to_Robotics | Lecture_7_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started so today's segment come on interesting this uh this is really an interesting development uh in 99 um a robot playing volleyball and um this was developed by by a company toiba which which is remarkable so robots in games and competition beach ball volley playing robot Japan a lot of [Music] people future robots are going to work in your houses and Hospital hospitals with humans tashiba has developed a beach bar volley playing robot as a demonstration of such a human friendly robot technology we consider that it is essential to interact with robot using everyday words such as let's play volleyball for the everyday word commands to work the robot needs to measure the target's relative position with respect to the robot position to know the mechanics and procedures of the tasks and to have a good database for the environment and the target with an integrated sensor feedback technology we have demonstrated a robot which can play beach ball volley with an opponent pick up a ball from the floor and shake hands with a human by only one respective everyday word command the robot uses two headmount CCD cameras to measure the target ball position in every 160th seconds the cameras always track the B the operator commands the robot to pick up the red ball then the robot starts to search for the red ball on the floor after finding the ball it moves moves above the red ball then it checks the arrangement of all the balls to make a plan to pick up the ball when the robot finds that the arrangement is too difficult to pick up the red ball it pushes the green ball aside and checks the ball Arrangement again then it picks up the red ball we use visual server control at the rate of 160 seconds to achieve this picking up motion this is the robot view after pushing aside the green ball the robot checks the arrangement again and approaches the red bone by using the visual Servo control also to show the human friendliness of the robot it can shake hands with a human it has a tactile sensor in its right hand to start handshaking it also has a six degree of Freedom Force torque sensor at its wrist to rily respond to the young lady see [Music] you okay so I'm not sure who was shaking the hands the human shaking the hands of the robot or the robot shaking the hands of the human um I wonder if you know why this whole environment is white color yeah I mean the vision is working picking the color uh and using Color you can go fast so so what is the challenge here you to have any color environment you would possibly mistake a blue back or a blue ball no I meant what is the challenging problem in this demonstration what was the most difficult part in this demonstration perception perception depth and uh and tracking the ball tracking tracking the ball tracking the ball is very important because you have to predict where you're going to hit by the way students from this class last year worked on a project in experimental robotics to duplicate this with the Poma and it was it was a lot of fun so you have the camera and you're you're tracking the ball you need to predict where the ball is going to be and then you you place because your arm is not going to to be positioned correctly with the right orientation unless you do that prediction and this is how human work human do a lot of prediction I mean the feedback is really slow but the prediction is amazing and that's how you can do this um there there are also uh other aspect of the problem they were talking about the grasping which is a very simplistic problem in here because they are using really twoand gripper for a large object so the the grasping part is but it was remarkable in 199 they managed to do it so that was uh quite interesting uh 90 798 Honda came with Asimo and that was uh another big uh uh uh achievement in robotics getting a robot to walk Upstairs Downstairs we will get to talk about this later but it is uh it was the time when uh suddenly just before the turn of the century we started to see all these results coming uh from the maturing of technologies that were bringing Vision uh good mechatronic integration uh to to make robots much faster to to do tasks like uh walking uh stably and also uh performing Dynamic skills like hitting the ball back okay the Jacobian again well today we're going to be able to look at a mechanism like this one and see the Jacobian I promise to you that you will take a look at the robot and you will see the Jacobian we're going to do it and like in few minutes you will see like all the columns coming in place and everything so this is the explicit form of of the Jacobian last week we saw a messy complicated way of looking at the Jacobian or extracting the Jacobian by propagating velocities from the base to the end effect Factor Computing the total linear and angular velocities as a function of the joint velocities recursively and then by doing this computation you have you have the expression of the Jacobian some where embedded in that uh final answer of your linear velocity and angular velocity so if you separate the The Joint velocities you will find what are the element of your Jacobian metrix but you don't have any idea about exactly what is the structure of this Jacobian matrix uh what would make it really uh singular how how uh uh the mechanism joint uh axis uh affect uh angular velocity and linear velocity Etc so now what we're going to do is we're going to go to each of the joint axis Prismatic or revolute and analyze its impact on both the linear velocity ities and angular velocities and by doing this analysis we are going to be able to immediately see how these joint axis are going to map to the jacoban in which columns and in which way and then basically we can do the reverse we look at the Joint axis and we will see the jacovan so this is what we're going to do all right so let's start with a joint pick a joint I don't know this one let's pick this one so I'm drawing the vector Omega I representing the angular velocity generated by Q do I so the direction of q. I is along this vector and its magnitude is proportional to Theta dot okay so this is basically Theta do Z which we are calling Omega I well to be to span all the degrees of freedom we need another joint that is Prismatic and I think we have one over there Prismatic joint so let's take oh well we'll come later so we we just are writing what is Omega I and here is the Prismatic joint this Prismatic joint is along joint axis J so along ZJ and the velocity I'm measuring is this Capital VJ which is proportional to what it is proportional to D do J D is the variable Prismatic variable along the the axis ZJ so the Prismatic velocity for joint I we I VI is z q do I in the case of Prismatic Q do is d dot and in the case of revolute it is Theta dot you remember the Epsilon I okay now the question that we we should answer is the following if we go to the end of factor so let's go to the end of vector and say I'm interested in Computing the impact of the linear velocity on the linear velocity of that revolute joint and that Prismatic joint so I'm going to put Prismatic joint and revolute joint and we are forming this table so on this table the question is what is the contribution of a prismatic joint moving at velocity VJ on the linear velocity at the end of factor you see the end of factor point is right here so at this point what if I have everything is still if I just have VJ what would be the end of Factor linear velocity so we need to answer that question then we will place it here I'm sure everyone now is looking at the notes don't look at the notes press it here and then we're going to find the impact on angular velocity so think about it this joint is translating with this velocity what would be the linear velocity do to VJ not all at the same time anyone just add it would be VJ what is the impact on the linear uh the angular velocities n so the VJ is transformed directly to the end factor and we will find that the Prismatic joint will have a contribution on the linear velocity identical to the Prismatic joint velocity along the same direction of that velocity on the angular velocity a linear velocity will not change the angular motion of the end effector so there will be no effect and I'm not going to hit return because I'm afraid that I will we will see revolute so let's think about revolute I don't remember the order so now it's little bit more complicated but remember the Apple that is we are going to look at the effect of this Omega I at the end effect Factor what is the linear velocity do uniquely to Omega I so do you remember the Apple so what information we need we need to locate the end effect Factor Point distance from from the axis so we really need to find to introduce a vector that connect joint I to the end of factor we will call it p i n connecting the origin of frame I to the origin of frame n so let's make sure the none is okay perfect so now on the revolute we we we do this connection and now we need to compute the linear velocities due to Omega I so that is going to be so you're rotating and you have the vector how do we compute that linear velocity cross product cross product but that's not enough you have two possibilities right Omega cross P Omega cross P everyone agrees it is Omega cross so Omega I cross p i n gives you the contribution of revolute joint I on the end Factor linear velocity okay what about angular velocity we said the angular velocity is not affected by the Prismatic is it affected by the revolute what is the contribution Omega I itself so if we have a revolute joint the total angular velocity is going to be the sum of all the Omega I and everyone is going to contribute with the same angular velocity now the total angular velocity you you walk through you take your Epsilon I and following the type Prismatic or evolute you can compute the total angular velocity so if we had a prismatic a robot with Prismatic joints only what would be the total linear velocity we call it V the total linear velocity is going to be equal to the sum from I = 1 to n of with Epsilon I need something because I'm going to use something like Q dots Epsilon so Epsilon I was equal to one for prismatic and equal to zero for revolution so sum I from 1 to n of Epsilon I so all Prismatic joints come on I mean if you if you just have Prismatic joints we are talking only about V1 to VN right you just add them together but can you write it with the Epsilon is it Epsilon I or Epsilon I bar Epsilon I times V okay right if we had revolute joints only the linear velocity would be okay someone else I think from this side someone was trying to answer anyone you we trying to answer the V okay I I have only revolute joints and I have those joints starting from 1 to n what is the total velocity due to those revolute joints would be would be the sum of all of these from 1 to n but I'm just looking for the Epsilon sum from IAL 1 to n of very good Epsilon I bar multiplied by this now if we have mixed joints then the linear velocity will be Epsilon ivi or Epsilon I bar Omega I I now you understand this yes zero when it's revolute and one when it's pris right so this Epsilon is equal to one for prismatic joint and equal to zero for revolute joint and Epsilon bar is the complement of Epsilon which means that this would be equal to one if the joint is Prismatic this would be zero or this would be zero and then that would be one either one or the other okay now what about the total the total angular velocity it's going to be if I have Prismatic joints if we have Prismatic joint you use Epsilon I multiplied by by zero he said he's like sitting like this have to yeah zero yes Epsilon I by zero so only revolute joints will contribute and that means me omega is equal to the sum from I 1 to n of Epsilon I bars all these Omega I so so Omega is just this expression okay all right so here is Omega and V so here so over there you have V the total linear velocity and Omega the total angular velocities now I don't know if you see the Jacobian yet but we're not far we're very close how do we go so now we have an expression for V an expression for Omega and the question is how can we extract the Jacobian so what is the Jacobian doing the Jacobian matrix is defined as the Matrix that connects Q dots to V and Omega so you have two part in this Matrix JV and J Omega JV is the Matrix that connect V to Q dots right and and J Omega connect Omega to Q dots the question is how we go from here to finding JV and J Omega could you could you help what do we need to do so right now I'm expressing v as a function of VI and Omega I omega as a function of Omega eyes what do we need to do yes okay so sorry yeah zq and then bi is with z q dot as well and if we do that then we will have an expression here and expression here that is linear on Q dots and this is a sum from Q do1 to q.n basically you are just moving through The Columns of the Jacobian matrix so we substitute in those expression VI with ziq DOT Omega I with Z IQ Dot and if we do this what's going to happen I'm going here to substitute this with zq DOI and there with zq Let's Do It come on oh magic now what do we have here we have q. I the same but depending on the type of the joint if the joint is Prismatic I'm using Z and if the joint is revolute I'm using Z cross p i n and here if the joint is revolute I have simply C Let's uh start with the simplest form so the simplest is this one and my question is what is the First Column of the Jacobian matrix J Omega so this is will be the J Omega this is a sum right so J Omega the first column of J Omega of a robot whose first joint is revolute okay I hear Z1 I hear zero oh Z1 Z1 Z1 everyone so the First Column of a robot that is first joint is revolute will be Z1 for J Omega what would be the First Column of JV for that robot same robot we have to go up here any volunteer to1 cross P1 n excellent so if the first joint is revolute would be one One n that Vector connecting joint one to Joint n my robot is completely Prismatic could you tell me what is JV so the First Column would be Z1 second column Z2 isn't that beautiful Z3 Z4 that's your Jacobian what about J Omega for a robot that is completely revolute Z could it be any simpler I think I think now you you will start to see the Jacob so if we take this expression and really expand this is what is happening V is now a linear combination of the joint velocities going from the first to the end and now if you take the coefficient of those Q dots you can write your JV and JV is saying that if the joint is Prismatic the column is just the Z the Z axis and if the joint is revolute it's the cross product between that z-axis with the vector connecting the joint to the end of factor okay and if we expand the second expression we are just looking at revolute joints so a prismatic robot will have zero columns and a revolute robot will have all its columns so it would be all these as just Z axis so it is you're just writing zero or Z okay all right well basically this is this is it now you combine the two JV and J Omega and you get your Jacobian so this recursive Comin computation we were doing as we propagated and we computed the velocities and when to each frame and make sure that the frame uh the expression is done in the same frame and transform it back to the zero frame all of this now is written here explicitly so you have this explicit form that is actually not expressed in any frame yet we said the Jacobian this Jacobian J Omega is function of Z1 Z2 ZN we didn't say in which frame if you decide now about the frame we have to decide where we express these vectors right so when we express vectors we have to express the Jacobian in a given frame zero all the vectors have to be in frame zero all the columns over there you are doing cross product computation well the cross product computation should be done in the same frame it could be frame 72 but then you have to transform it to the same frame where you're expressing your Matrix so you can do it in uh frame two for instance and then take it back to frame zero so this is a very general form expressing the Jacobian in a vector form and then the question is where you're going to need your Jacobian computation and you do that computation by expressing all the vectors in that frame so this jacoban which is JV and J Omega can be also simplified further if you already computed your forward kinematics because JV if you remember JV is related to the linear velocities so if I'm using spherical coordinates I cannot just difference iate spherical coordinates and get linear velocity but if I'm using cartisian coordinates then basically I can directly differentiate that vector and obtain JV so if you have X Y and Z computed already where can we find X Y and Z in the forward kinematics where where can we find it in which column in the fourth column and you get X Y and Z of the point that is expressed in some frame and you can take the differentiation of that and your Jacobian matrix if this Vector was expressed in frame zero the position XP now XP is X Y and Z of the end Factor point then you are taking the differentiation with respect to q1 Q2 to qn and that is also your your Jacobian associated with linear motion and this jacoban should be identical to the one computed from Z1 cross p1n P1 n P1 p2n to the last one all right so again there is no Comm commitment yet where XP is expressed so to go to a frame what is important is to make sure that everything in relation to all these vectors describing the Jacobian is expressed in the same frame so this is a vector representation of the Jacobian and if we go from the homogeneous transformation if we computed the homogeneous transformation then we should be able to do this computation immediately so once you have your homogeneous transformation you should be able to extract your Z1 Z2 ZN and XP and just do the simple differentiation and find your Jacobian matrix now most of the time we express the Jacobian in frame zero so in frame zero this would be simply expressing all these vectors in frame Z well it could be frame n it could be maybe sometimes you need it in frame n for some task and you want to transform it to frame zero so we need to understand also the relationship between the Jacobian in different frames but let's uh first look at these these vector and for these vectors the information about Z in frame zero so let's say I'm interested in z7 in frame zero so where can I find this information precisely so Z is the Z Vector of frame I so Z in frame I is equal to what what 0 0 1 so any Z in its own frame is going to be 0 01 how could how can we find 0 01 and transform it to frame Z what are we going to use I'm sorry I cannot hear you very well you have to speak louder transformation so we take the transformation from frame I to frame zero which transformation you want uh basically the rotation you rotate you you take the rotation Matrix the 3X3 which is in the homogeneous transformation so you take the rotation from frame I to frame zero and you can compute your Z in frame zero so so Z in frame I is always constant right so we call it Z I mean this is the Z Vector 00 01 and as you said we take it from frame I to frame zero which means that essentially we need to mult premultiply 01 by the rotation Matrix from frame I to frame Z what does it mean think now about this 3x3 rotation Matrix and you are multiplying the rotation matrix by 0 0 1 what are you doing there what is the the result if you multiply the 3X3 matrix by 0 01 you are picking the last column so basically this information is found in the rotation Matrix in the third column of the rotation Matrix 1 to0 2 to 0 n to0 4 the revolute joints okay so so now we have this form and we are going to try to apply it to a relatively complicated robot the Stanford Shaman arm and um on Wednesday uh I will bring it uh in the lab so you can see it and we can look at some of the properties related to the singularities and those axis and you can see the Prismatic joint but let's rely on the picture so you still have to imagine the robot and what I want you to do is to to see now the jacovan can you see it no all right maybe if I give you a drawing of it it will it will be better but before that let's go and summarize what we have seen we we said to compute V and Omega we compute this Jacobian from its two component JV the block JV and the block J Omega V is JV connecting Q do to V and Omega J Omega connecting Q do to Omega and the First Column I mean the first plock JV is either the Z of that joint or the cross product of this Z with the vector i n okay the Omega and the J Omega is simply the Z vectors corresponding to revolute joints all right so you you see this we because now I'm going to ask you to to give me to give me uh the Jacobian for this robot and this is the schematic of the robot and you can see here the Jacobian is empty you need to fill it and I hope you're not looking at your notes or you don't have the answer so all right let's see if anyone sees what should goes here now you have to match it with the with the joint of the robot and their types okay do you understand those types here joint one is revolute or Prismatic revolute second and third and then the rest is revolute okay here we go here Z1 okay here Z2 who said Z3 I heard Z3 no you said zero okay zero zero four five and six okay what about this one first one it's a revolute joint the first one is zero revolute joint it is Z1 cross P1 or zero because they are coincident but P1 2 N in this case n is that green frame it is on frame three so really two to three what about this one so 1 2 3 2 2 3 yes three why not six it could be seven if we attach the frame at any point like often for the end Factor we attach the frame additional frame but for this competition we said we will do the competition at the wrist frame which is the frame in this case it turned out to be the frame number three identical the origin of three is identical to four five and six the same the same frame so this one would be Z2 cross 2 three and here Z3 Z so you have Z1 Z2 Z there should be a Z3 it pushes up it goes up the Z3 goes up and you get zero so when you have a prismatic joint you you're getting a zero otherwise you have the revolute joint so let's check so you get C3 so the next column is going to be zero because now we are measuring the distance from 3 to four it's the same point yes can you explain all these Z's what they represent are they the z-axis for each frame okay this is very important if you if we don't know the Z AIS uh come on my computer is what just one second let's go back to the beginning so the z-axis is the joint axis itself along which we are having this rotation and this is the in we when we go to to the frame themselves like when we go here this is z0 and if you remember there was Z1 uh attached to frame one and G Z2 we are seeing here Z6 so the these are the Z axis along the joint axis and those are the the the Z's that we are talking about and we are measuring the rotation about the z-axis and we are measuring the Prismatic motion or the linear motion along the Prismatic joint Z so when we were Computing Prismatic joint three the motion along along this direction is coming from along that joint okay everyone sees this this is important if you don't see the z-axis yes you have a question is always 0 01 right in its own frame but in frame zero it's it's whatever it is so Z3 in frame three is 0 01 and this is what we call the Z and what we saw is that we when we go and express this in frame zero we need to transform this Vector to frame zero as well as all the vectors should be expressed in frame zero okay now why do we have zero here so this should be Z what Z4 cross p 4 to three I mean basically we're going to that same point which is zero so here Z and here zero here we said we have Z1 Z2 0 Z4 5 and six okay all right can you really feel what is happening on this Jacobian compared to your mechanism do you understand it although we are in the same frame the origin at the same point rotations are going to add so there is columns for Omega for the linear velocity there is no effect of joint Q do4 5 and six on the linear velocity but angular velocity it is there the rotations are described directly by the z-axis of the joints and when you multiply by the Q dot to compute V and Omega you are multiplying here by q do1 q.2 q do3 4 5 and six okay yes just double check you could have also said Z1 cross p16 yeah I mean if if we uh uh if we called this is the end of factor frame in frame six yeah it will be six and if it was uh another frame seven uh at the end of factor it could be seven uh I I just write it here knowing the fact that three is the same point as all the following uh Origins okay now we're going to do this computation now we're going to find this Matrix and so maybe that's what it was doing just like thank you how can I turn it off I think my computer was like suffering trying to update something all right and it must do it every time I have a lecture I I think the camera just like looks and find me and and look at the black and white and then just oh he's here so let's do it now uh all right so it's quite simple but please pay attention now I'm going to show you how we can start from the beginning right after you did your FR frame assignment and compute both forward kinematics and your Jacobian in one shot so so let's let's uh let's walk through it so we know how to do this right you did the homework and you know how to do it and this is D2 a constant right over there so I'm going to use D2 D3 and we are going to use those variables and we are going to express our transformation as a function of theta 1 Theta 2 D3 3 four 5 and six which are the qes so what we are going to do from here we are going to compute the homogeneous Transformations every rows correspond to a homogeneous transformation that is computed from this these Expressions going from frame I minus one to frame I this is I and now I'm going to write it down so when we computed all of these we do this multiplication and we find the forward transformation from frame n to frame zero okay now the way you do this computation the steps you're going to take are very important in order to obtain your Jacobian in a given frame at the same time as you are doing your forward kinema so this is the transformation from frame 1 to zero 2 to 1 3 to2 these are coming directly from the formula you're not doing any multiplication you're getting 1 to Zero from the Row first row second row third row fourth fifth and sixth okay so I'm I'm just applying the formula to all the DH parameters and if we had another frame attached to the end of factor there will be 7 to six and that one will be constant it will not be depending on any joint so here is what we're going to do if we are going to express our Jacobian and this is what we do most of the time in frame zero because we would like to find our velocities in frame zero so what are we going to need we need we are going to need all the vectors in frame zero so in particular we need the zi vectors in frame zero which means we need the rotation the third column of the rotation Matrix Vector that take us from frame I to frame zero so for the first Z Z1 in frame zero is where Z1 in frame zero is right here it is 0 01 so when I do this computation I will compute two to Z by multiplying 2 to 1 by 1 to0 I get 2 to0 and that gives me Z 2 in frame zero what is Z2 in frame Zero from here minus sin 1 cosine 1 and zero the third column and this is Z3 in frame zero so these are very important in Computing the Jacobian but also are going to be needed so when I'm doing the mulp ication I always form from I to zero I will not compute 3 to one and then multiply it by 3 to Z I will be wasting too much time I will just go 1 Z two 0 3 0 4 0 5 0 okay and this way you are getting directly all the vectors in frame zero so you can do this for all of them and this is the last one this is what Z6 in frame zero okay so where is your Jacobian basically you already have it it is already here in the homogeneous transformation the J Omega J Omega 6 column fifth column fourth column third second First Column actually the third is zero but this Z is needed in the top of JV so it will go up yes what about the vectors okay we're getting to the P vors okay so the J omegas are already there all the columns and if the robot is Prismatic the same thing we have Jacobian JV yes when you were saying to save time not multiply the T10 and tc1 inal one so do we just figure out the columns of T10 and T20 directly from just inspection well actually the third column yeah but actually you are going to need you are going to need the final transformation to compute uh the orientation of the robot so you need this column and this column for the description of the orientation for the forward kinematics so you need to do the multiplication but what I'm saying is do multiply starting from you you start from you have one zero but then you have two to one then three to two do not do not walk by multiplying from this side 3 to two by 2 to 1 then multiply it by 1 to0 you multiply 2 to0 I mean U you obtain 2 to0 from 2 1 multip by 1 Z and now you use this Matrix multiply by the next one you go from 3 to Z and this way you have these intermediate competition that are going to be needed in addition to your result you you still need the final the final Matrix because this is your homogeneous trans formation okay any questions about I mean this is just to make sure that you have these intermediate vectors expressed in frame zero yes you go back two slides two slides I guess I me it was when you were doing the formulas to compute is it from IUS one to I or to like you have one on the slide yeah I said it is a mistake okay sorry and uh I don't know if I should change it now or later but every year I have this it was it was a mistake from long time ago so let's see if we can change it no so ah yeah I need to go here and I think I can change it all right okay now it's correct thank you all right so now uh what I was saying is as you compute your homogeneous transformation make sure that you have those intermediate vectors that you will need for the jobian now okay are you still with me your attention all right look at the last column look at the last column in this transformation and compare it to the previous one to the previous one it is identical right do you know why we have the same form so actually this is the same as we have here when we reach frame three in fact we reach the end Factor Point okay so the question is how are we going to compute now as we go through this how are we going to compute JV well if we go and compute JV we have two ways of doing this either by Computing the vector P from one to n and express it in frame zero or the frame where the Z is expressed or as I said what what this represent X Y and Z the cartisian coordinates so you have already the cartisian coordinate in your homogeneous transformation so actually you don't need to compute the piece you just differentiate this and express your JV as a function of the partial derivative of X Y and Z with respect to q1 you get the First Column Q2 second column Etc so just to check if you are following what I'm saying could you give me we are on the first first Vector of the Jacobian JV and I would like to find the third element of the first column third element of the First Column of the of the Jacobian JV so what are you differentiating okay it gets you zero and the first element will be differentiating the first one and second one okay what is the third column of JV let's differentiate C1 S2 s S1 S2 and C2 okay can you check whether this your answer is correct independently yes go ahead it's just a z it is the it is because this is a prismatic joint joint 3 it should be identical to the Z3 which is here so when you differentiate with respect to D3 you obtain this column do you see that because the both computations are should give you the same answer so the third column we know the third column is simply Z3 three but if you still differentiate you get Z3 okay now you trust it all right so so basically this computation of the Jacobian is already in your homogeneous transformation I mean you are Computing at the same time you are Computing your forward kinematic X Y and Z and all what you need to do is to extract your Z vectors in the appropriate frames and fill your jacoban and for the JV part you are going to use this differentiation and you have your Jacobian so the Jacobian matrix for the shim Stanford shiman arm once you computed your homogeneous transformation from frame six to frame zero you have your for kinematics in here you have X Y and Z in here you have the rotation so you have Direction cosin you have all the description here you are you have your Direction cosin X Y and Z Direction cosin First Column second column third column and now you can take the intermediate steps where you computed the z i to fill your Jacobian matrix and the Jacobian matrix is here yes get two columns the first two columns we differentiated we differentiated this expression with respect to q1 this is X this is the first element 1 one the in the First Column second element is differentiation of this with respect to one and this with respect to one so what we are doing we are taking X Y and Z and differentiating with respect to q1 that will give you the First Column this column correspond to the contribution of joint one so you differentiate with respect to q1 this column is the contribution of joint two you differentiate with respect to Q2 the third column is with respect to Q3 or because this is a prismatic joint this could be obtained directly from Z3 in frame zero yes um so this XP is it the fourth column of [Music] t06 right this is the fourth column of t06 in here so if you if you read this and you manage to read this you will obtain the same thing XP is copied from X Y and Z in the omous transformation if we are selecting if we are selecting cartisian coordinates okay q1 there is no there's no like D1 in the first expression so yeah fortunately otherwise it would be a problem because we have C1 so what is C1 maybe maybe the confusion is that you don't see the dependency on q1 what is C1 and S1 what does it mean could you uh could you explain what is C1 and S1 anyone can help me finding what what1 what is C1 it's cosine of q1 or cos it is cosine q1 for joint one which is revolute which means cine Theta 1 the variable Theta 1 so q1 is in the cosine 1 and C1 means cine the angle q1 so when we differentiate C1 with respect to q1 what do we get if we just differentiate differentiate this first element uh first uh element of the fourth column you differentiate with respect to q1 are you there with me are you there with me could you differentiate the first element with respect to q1 now theine Justus so you differentiate with respect to q1 and you have cosine 1 that would beus sign one and and2 very good okay do you understand that everyone understand what what what this probably this is the maybe not clear S1 when we say S1 C1 S2 C2 we are referring to the variable Q2 okay all right so this is the The Matrix all right so so you you see after all the Jacobian is not that complicated the Jacobian is simply is simply these columns associated with your Za AIS for the lower part related to rotational motion for revolute joints for prismatic joints it is the same axis and for the other joints I mean for the linear motion all what you are doing is a simple differentiation of your X Y and Z and that gives you the answer for the jobian metrix now what is interesting about the Jacobian metrix is the fact that you cannot just go and use the metrix without really making sure that uh you are in the right configurations to perform a task because for for many of uh the interesting configurations for some robots with rist that uh has problems like the singularity you are going to not be able to move about some axis you are locally going to lose the ability of rotational motion uh for uh uh about some of the axis depending on that configuration so there is the notion of kinematic Singularity that you can directly examine from this Matrix and those singularities correspond to a singularity into into the Matrix itself so if we take a two degree of Freedom robot if we can have the camera on my arm please so if you take a two degree of Freedom robot and you are moving now in space in this plane I can move anywhere right instantaneously I can produce any displacement but if I move and stretch and stretch to the end of the workspace do you notice anything now I cannot move along this axis so the axis could be like this or like this but I cannot move instantaneously along this axis there is no displacement Delta Theta 1 Delta Theta 2 that produces a component in this direction and this is a singularity what happen in this configuration is the fact that now the two columns of your Jacobian matrix became dependent so when those columns become dependent you are going to lose rank the Matrix is not full rank this Matrix is 6x6 and we like it when it is full rank but if two columns become dependent immediately you lose rank and you lose the freedom to move so on Wednesday I will bring uh the Stanford Shaman arm here and we will be able to examine another type of Singularity related to the wrist point and the fact that joint four and Joint sex when they are aligned you lose the ability to rotate about this axis and that makes it really hard to perform motions uh in those locations not only in those locations but also around those locations the problem of Singularity is that when you come close to the singularity The Matrix become becomes ill conditioned the numerically it's ill conditioned if you want to compute the inverse it is going to give you uh large numbers and that is if you want to do a small displacement in Delta X you will need large displacement in Delta Q to produce those motions so we will examine this and we will talk about the kinematic singularities on Wednesday see you then |
Lecture_Collection_Introduction_to_Robotics | Lecture_12_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development all right so today we're going to the Alps an Innovative space rover with extended climbing abilities [Music] Switzerland autonomous mobile robots have become a key technology for unmanned planetary missions to cope with the rough terrain encountered on most of the planets of Interest new locomotion concepts for Rovers and micro Rovers have to be developed and investigated in this video sequence we present an Innovative off-road Rover able to passively overcome unstructured obstacles of up to two times its wheel diameter using a rhombus configuration this Rover has one wheel mounted on a fork in the front one wheel in the rear and two Bogies on each side here we can see the trajectory of the front wheel mounted on the fork an instantaneous Center of rotation situated under the wheel axis is helpful to get on obstacles to ensure good adaptability of the bogey it's necessary to set the pivot as low as possible whilst simultaneously maintaining maximum ground clearance this architecture provides a non- hyperstatic configuration allowing the bogey to adapt passively to the terrain profile Motion in structured environments for climbing stairs it's necessary to have a good correlation between bogey size and step [Music] Dimensions we can see that the Rover is able to climb regular stairs effortlessly [Music] Motion in an unstructured environment we also made some outdoor tests on Rocky terrain as can be seen the Rover demonstrates excellent stability on rough terrain it advances despite a lateral or frontal inclination of 40° and is able to overcome obstacles like rocks even with a single bogey in a Next Step the robot will be equipped with adequate sensors for fully autonomous [Music] operation so it's quite interesting because when we think about uh Wheels the robots we think about mobile platforms working in indoor environments and it's really difficult to imagine these machines going around and being able to uh like go over obstacles so we usually try to use what do we use in uneven terrain I mean what would be the solution the other solution the alternative solution mhm lagged so we always think likeed Locomotion would be the solution and uh in in fact legged Locomotion is very I mean adapted to those problems because now you can move each leg and go over obstacles but in here we can see a a way that you you in fact putting the that uh compliance uh inside the structure and moving uh the structure as you are adapting to the environment so this is really a quite nice solution but there was also design that that combines uh leged Locomotion together with wheeled Locomotion so you have hybrid solution where uh you are uh using the wheels and pushing with the legs so uh there are several ways of even going further Beyond just uh modifying the the chassis itself going to uh adding some proportion by the legs so uh in fact this project uh was pursued further and uh I'm not sure if we will have more videos on on this one okay so let's go back to Dynamics and uh I think today we will uh finish uh that portion of the the lecture I think uh I emphasize Dynamics I also emphasize the fact that Dynamics is very very uh closely related to control and we really need to understand those equations of motion in order to be able to control well uh the the robot so let's go back a little bit to what we saw on Monday about the lrange equations we saw that we can describe the dynamic behavior of the robot that is its motion uh it is it's as a function of the torqus applied to the robot through this legrangian equation that involves the kinetic energy and the potential energy so K is the kinetic energy and U is the potential energy and we saw that because our potential energy is only function of the configuration we can separate this equation and find uh the structure Rel ated to the inertial forces and the forces applied to uh the robot that is the gravity and the Torx which means that we can uh write the equation put the inertial forces on one side and then analyze uh the Dynamics that is the inertial forces Dynamics and then see the effect of those forces applied to uh the system that is the external torqus through the motors at each joint we have a motor and the motor is applying a torque at the joint or a force at a prismatic joint and also the effect of the gravity that are coming so if we analyze this equation we saw that we can write it in this form uh in the form of a mass Matrix multiplied by the acceleration at all the joints plus some additional forces that are fun function of the velocity and function of the fact that the mass Matrix is configuration dependent and those forces could be obtained uh if we express our kinetic energy in its quadratic form uh expression that is 1 12 q. transpose M function of the configuration Q dot this is a scalar and if we do the derivation of the scalar we find in fact that we have this equation that is this part will brings those element the mass Matrix the acceleration and this part which represent the centrifugal coris forces and we can see from this expression that when Q do is equal to zero this will disappear or if the mass Matrix was constant this will also disappear so we saw the proof of this I think that you remember so in this form the equation now can be written in term of the mass Matrix the derivative of the mass Matrix and the velocities and the gravity forces equal to the applied torque because the mass Matrix is this quadratic form now we can if we are able to extract the kinetic energy from our analysis of the different motions of the links we should be able to find the mass Matrix directly from the kinetic energy then we put it there and because V is solely function of the mass Matrix and the velocities we should be able to obtain V and that will give us the full equation simply by Computing the kinetic energy okay so that's what we're going to do we're going to do this analysis and find how we can find the mass Matrix and once we found the mass Matrix we will find V and then G is a piece of cake okay so how are we going to proceed so here is an example of a an arm and and we are looking at one of the joint one of the links link I so this is a rigid body and this rigid body has a mass distribution has some inertia and has some Mass so what we're going to do we're going to look at this kinetic energy associated with that specific link and the idea is if we are able to describe and find this kinetic energy we will be able then to go over all the different lengths and do the sum and find the total kinetic energy and because we have this relation we can put the two together and we can then identify the total kinetic energy from the individual kinetic energies so for link I the total kinetic energy is going to be the sum of the different kinetic energies so link I will be here and when we walk through all the links we are going to find different kinetic energies that we can sum and find the total kinetic energy but then we will say the kinetic energy of the system should be expressed as a quadratic form onized velocities and and the question is how can we then do this identification and extract M so you see the algorithm just doing this part and then identifying M from that sum all right so I'm sure half of you remember the kinetic energy and the other half does not remember so how we Define the kinetic energy those of you who remember so you have a rigid body at rest and we move it to some velocity V what is the kinetic energy yes yeah but what does that represent is it work you need to to rest to uh yeah uh I I think I understood what you mean but could you repeat it clearly so everyone can so it's um a work needed to to move the SP from R zero to from rest to its current uh State yes that's the definition so it's the work done by the external forces to bring the system to its state from rest and that means we are going to take a look at this point mass and its final velocity V and that work will come to be2 MV s now this is for a point Mass but we are working with lengths and links are rigid objects right and they have rotational motion and which means each of the particle is moving at different velocity Etc and there is a a quantity that allow us to evaluate that inertial forces generated by those particles and this is the inertia so the inertia of the rigid object is going to intervene now if we look at rotational motion so the kinetic energy associated with arid body who inertia is i c c here represent the inertia computed with respect to the center of mass as we have everything is represented at the center of mass will take this rigid object from uh rest the zero kinetic energy when we reach Omega there will be some kinetic energy and this kinetic energy is going to be you gu half transpose I half Omega transpose in the ini in so if this was just uh um one degree of Freedom this would be basically a scalar ey representing the inertia about that axis Omega square but we we are looking here at a spatial motion so I is a matrix and uh the Omega is a vector so we need to write it in this form okay so if we take this rigid body and put to it together V and Omega that is it is undergoing both angular motion or uh rotational Motion Plus linear motion then the two will combine and the total kinetic energy will be the sum of these two kinetic energies okay we're clear about this is very important once you understood the kinetic energy you have all the Dynamics the rest is just uh just uh math basically so you understand this for one regid bud good then we're going to be ready we just tried this all right now pay attention from here we're going to find the dynamic equations of multibody articulated system directly by summing these K kis if you understand this you will see immediately that m is going to emerge this Mass Matrix for all the structure so the kinetic energy of one link is this sum combining the kinetic energy associated with linear motion and the kinetic energy associated with angular motion and let's do the sum so the total kinetic energy is the sum and I'm going to write it just uh to remind you we have selected for this structure we selected a set of generalized coordinates q and that means we have a set of generalized velocities Q dots so these are the minimal number of parameters or configuration parameters needed to represent this configuration once Q is defined the configuration is locked right so now we can say because we know these generalized coordinates and generalized velocities we can say the kinetic energy is also this expression of half q. trans ose mq dot where m is this positive definite symmetric positive definite Matrix that appears in this quadratic form so we have these two different ways the first one is here we are saying we can compute individually the kinetic energies without worrying about the joints the connection the constraints right we are just going and looking at every link and we are evaluating its kinetic energy and saying the total kinetic energy is going to to be the sum without even thinking about what type of joints we have now we are saying if if we write the expression as a function of the geniz velocities we have this expression and the two are equal it's the same kinetic energy are you still following here good and this is the key now we are going to identify this expression with the sum obtained by the individual links and somehow we are going to like work little bit here to come up with this Mass Matrix okay so now help me what is needed in here in order to to extract M and find its expression so I heard Jacobian that's correct we need to translate the velocities to um joint space so in the left hand side we have half and here we have half right in the left hand side we have q do transpose Q dot around the m in here we have v c the linear velocity at the center of mass of each of the lengths the and the angular velocity at each of those lengths so what you are saying is we need somehow to express these VC and Omega eyes as a function of Q dots and if we do that then we can say we can put it in an expression similar to this and extract m is that a good way you all agree should we do it why not you have it in your hands but let's do it okay so to do this we are going to use the Jacobian and what we need to do is somehow to come up with a Jacobian this is not the same same Jacobian we talked about before that is if we look at what we developed before we developed a Jacobian at this point do you remember at the last link we had a Jacobian that allow us to compute the linear and angular velocities at this point and these were called what those jacobians we had two jacobians anyone remembers before the term JV which is the Jacobian associated with linear motion and J Omega so easy J Omega J Omega the jobian associated with angular motion right okay so but this was defined here at the end of factor and this was the velocity corresponding to the velocity generated by all the cues all the Q dots right but now I'm talking about the linear and angular velocities at this point so it's not going to be this VC I is dependent on which velocities q do1 q22 q do I IUS one or I we're not sure well you see we are at VCI which is the center of mass of the length so q.i is just before so it is dependent on up to q. I okay all right so I will Define it later but essentially we need to come up with a Jacobian that will capture all these Q's because I need Q dot at the end I don't want just to go from Q do1 to q.i I need to be able to write this Matrix so that I can multiply it by Q dot so we will Define jvi in a bit but we need a matrix like this you agree what about Omega well Omega is going to be in the same form we will have another Matrix we call it J Omega I this is different from JV and J Omega when we say JV and J Omega we mean the Jacobian associated with the end Factor when we put the I it is really related to the velocity at that specific one okay all right now let's go and plug this in this equation so now you substitute VCI with jvi Q Dot and the transpose of that so the what is the transpose VCI I transpose would be q q. transpose JV CI transpose so when you do this you're going to have the transpose of Q do here the jvi and the jvi coming from the VCI and the same thing for the Omega all right we're almost there you're going to see the mass Matrix emerging someone can help with the next step we're almost there actually some of you already see what is M but let's let's uh what do we need to do factor out the KE so we we notice this I we have an ie here we have these eyes but this these Q's are independent of the sum so we can take them outside right and that's it do you see the Jacobian in the mass Matrix do you see the mass Matrix M it is quite amazing your mass Matrix is simply those Jacobian transpose jacobians scaled by your Mass properties that's it so all what this Mass Matrix is is just to take the Jacobian associated with those specific points of the center of mass and see their impact on the velocity because they are capturing the effect of the velocity and you are scaling them by Mi or ICI so if your robot was one degree of Freedom so I will be one basically this would be M1 and JV jv1 transpose jv1 and that will be the inertia of that first link now if if you have multiple links what you can see is you can see that the Jacobian matrix of all of these links are going to contribute so you can think about this as the sum of M M1 M2 M3 MN and you can see the impact of each of the links on the total mass Matrix so we we will take an example little later but you can see how each of those links is going to affect the mass Matrix and as we propagate and move from one link to the next we are capturing the inertial properties coming from the lower joints and moving down to the end so that is your mass Matrix and now the rest is really competition just getting uh this uh V VOR from the partial derivatives of the mass Matrix that's it by the way uh I still haven't defined what is jvi it is I said this is the Jacobian associated with that Center of mass and I think we need to Define it more carefully so in order to be able to capture jvi at the center of mass we and and express it as a function of all the cues what we're going to do is we're going to take this Vector locating the center of mass and taking its partial derivatives up to the point Qi so this is column I and every column after that will be zero so by definition this Matrix is the Jacobian matrix computed with this Vector PCI and up to this point and then we are adding those zero columns okay so what about J Omega I what would be J Omega I without looking at your notes J Omega used to be what Epsilon I Bar Z 1 Epsilon 2 bar Z2 so we will do the same thing up to Z and we add zeros columns okay so now you know this definition of jvi j Omega I and now you can compute this okay you already have the DH parameters for the Stanford Shaman arm let's compute the mass Matrix you have 10 minutes no well it will take more than 10 minutes I'm sure so I will take an example of two degrees of freedom in fact before even going to the example let's uh let's do something uh something uh of just the analysis of this Mass Matrix we talked about so I'm not not sure if you you really uh see what is in here but when you think about this Mass Matrix as I said it is uh sort of like symmetric positive definite and it has a lot of properties and you can connect those properties to the structure of the robot so first of all let's see what M11 repes present M11 so I have a robot I'm going to use my arm to illustrate it if you can have so this is the first joint rotation about this axis and this is the second joint okay so it's in the plane so could you tell me what M11 represent so let's take this manipulator and lock it it is one link the whole thing we lock it right and I'm going to rotate about this axis so I apply a torque there is an acceleration basically inertia time acceleration equal the torque right in that case just one degree of Freedom like a pendulum the inertia of the pendulum so so that inertia of the pendulum is captured where in this Matrix can you see it no not yet it is M11 M11 is representing this inertia now this inertia is function of what of the configuration so if I change this configuration like this I'm going to change the value of M11 if I move like this it is lighter heavier okay okay M22 the same question it represents what oh come on we did M11 so M22 is lo so if we if we lock all the joints after M22 uh I mean after joint two M22 would represent what so it is really the inertia perceived at joint two so all these diagonal element are representing the inertia perceived the effective inertia perceived at each of the joints okay let's go to mnn hey wake up come on mnn is this last link okay so this is representing the inertia perceived about this axis right right it's function of what so this is link n and uh it is not really my hand it is just a constant link it's a rigid body it cannot move so mnn is representing that perceived inertia and it's function of which variables last joint coordinate I'm sorry just the last joint coordinate it is the last joint coordinates okay everyone agrees with him okay those who agree with him please show your hands one two you are in minority democracy take over so it's not correct actually in this case uh it's really the law of of physics that are going to play and the law of physics says that if I'm moving one link about some axis so it is the case of this link everything is locked and the inertia is could you tell me if it's function of anything except the weight and the inertia Mass Distribution on the link so it's constant so mnn should be constant it's not function of the qn but thank you for for just making uh the point uh to make sure that we we emphasize this uh mnn is not function of the lost Link uh L joint it is constant now the question to you the previous joint the previous joint here well let's take this one so you can see that previous joint the inertia is going to depend on the joint so so MN is constant the previous joint will depend on the next one right and as we move we see M22 is function of what M22 is going to be function of all the joint that are following so the robot if we think about M11 is M11 function of joint one no M11 will not be dependent on the configuration of joint one wherever m q1 is the inertia depends only on how we're displaying the structure so the following joints everyone sees this good you're saying it's just constant no M11 is function of what I'm not saying it's it's independent of the first joint because as I move this around if I fix all the other joints and I move about this I'm not changing the inertia about this axis but it's function of all the following joints so M11 is function of what Q2 to qn M22 is function of Q3 to qn and the last one is constant okay there are a lot of interesting properties about this metrix what is the relationship between M12 and M21 identical it's symmetric if if you have a robot one degree of Freedom robot mnn and you model the mass Matrix of that robot and then you hook it to another structure well when you find the total mass properties of that structure the mnn is exactly the same that you computed for that robot so if I take if I take this structure a robot with joint 2 to Joint n with its Mass properties I find the mass Matrix of that robot and now I attach this robot to an additional joint look what happens in the new Mass Matrix you will find the same block the same Matrix is completely inserted here and what you are adding now you are adding the mass Matrix the Mass properties perceived by this joint but immediately you are creating coupling you are creating this coupling between the first joint and each of the joints of the structure you are adding so M12 M12 is representing the coupling between the acceleration of joint two on joint one M21 is the opposite joint one on joint two so think about this if you multiply this matrix by Q double dot Q dble do one it will be M11 qou do1 plus M12 qou do two all of this is going equal to torque one so the first equation is the Dynamics of the first joint and that dynamics of the first joint if you try to imagine what that equation will be it will be M11 Q do1 Plus all these coupling accelerations equal torque one so when you have just locked joints you have M11 q do1 no accelerations here but as soon as you relax the lock and leave it away as you start moving this is going to produce coupling forces you see that okay what does it mean that m is positive definite means that no matter what Q dot you use the answer has to be greater than zero unless Q dot is identically zero very good so that is physically you cannot talk about an object with zero Mass right so the object has to have a mass and a mass is always positive right if you have a particle with some Mass it is always positive now when we go to articulated body systems it's the same property but it is in a matrix form and this m has to be positive all the time whatever so if we think about the kinetic energy it's 1 12 Q do transpose mq dot well the kinetic energy is always positive right or zero if we are at rest if we are at rest the Q dots are zero so the quantity k 1 12 Q do transpose mq dot is going to be positive and zero only if Q do equal to Zer okay okay to discuss the this V Vector I'm going to simplify the problem and we're going to analyze it with two degrees of freedom I think that will make it little easier but not completely easy because what we need to do is now little bit of computation using lrange equations going to those vectors we computed M doq do minus this big Vector uh all of these competition you do them once forever and then you know the structure and then that's it but I I want you to understand where those equations are coming from so we're going to analyze it on this okay okay so this is a two degree of Freedom manipulator and now I'm writing these equations for two degrees of freedom so you can see here what is the first equation could someone re uh read the first equation for me you have two equations here right this is one vector equation that you could write in two equations so could someone read the first equation yeah M11 q11 2 q.2 + V1 + G1 so the first equation equal torque one represent the Dynamics of the first link the second equation represent the Dynamics of the second link if we lock the first the second link M11 qou do1 plus V1 + G1 equal torque 1 if if the second one is moving there is coupling coming from q.2 on the first one and the opposite is on joint two so what I'm going to do is to compute the V okay we're going to compute V1 and V2 and we are going to do this by going to by going to the equation of V this is uh scary equation but don't worry it's all there if you remember when we did the the computation of the kinetic the derivative of the kinetic energy we came up with v equal m doq do minus this Vector you remember that right everyone remembers so basically V is is simply M do Q do minus half the first element is q. transpose M q1 and mq2 what does it mean mq1 well mq1 is means that this is the Matrix the mass Matrix all the element of that Matrix are taken as derivative with respect to q1 and the second one with respect to Q2 so I'm rewriting this equation here here in more explicit way I'm saying it is m dot so what is m dot it is the derivative of the element of the Matrix right you agree with this and what is mq1 mq1 is the partial derivative of M11 with respect to q1 M12 with respect to q1 basically the Matrix with respect to q1 so I'm writing M11 so it is partial derivative of 1 one with respect to one okay and here with respect to two Okay uh do you see this notation basically the notation is saying the element one one is taking here partial derivative with respect to two or to one depending on the variable the last element okay so we just re wrote V now for m the time derivative of M11 we will write it in this way we write the time derivative of of M11 is the the partial derivative with respect to one q do1 and the partial derivative with respect to two q.2 you agree these are the the only variables that are involved in in the Q so I'm I'm just expressing expanding little bit this equation all right okay so this is just rewriting the equation and now we are going to this is little bit more about what is 11 one it is partial derivative of M11 with respect to q1 partial derivative of 22 with respect to Q2 Etc so let's rewrite this and if we go further and develop this computation and rewrite it so this is how it comes if we write the V Vector we can develop it as the sum of these two terms so I mean from the top there you can see that the vector is involving velocities product of velocities Q do1 multiplied by by Q do the vector Q do so it will result into product of velocity Q do1 Square Q do2 square and Q do1 q.2 right and we are grouping all of these in this form so this is the answer I I'm just doing it for saving time you don't have to do it so this is the answer what is V is as a function of those partial derivatives and the velocities okay do you accept that I mean you can do it but basically you can in fact you you might be surprised why I'm writing plus m121 minus m121 writing there 1 one + 1 1 minus 1 one it is really interesting form but what it turned out is that under this form there is some pattern that is taking place and if we look at this expression this expression has a pattern that is repeated all the way and when we go to end degrees of freedom we find this pattern over again so it's sort of m i j k with permutation first element plus minus and there is a permutation involving those three elements so this expression is going to help us find finding those matrices that are are going to scale the velocities the product of velocities that is if we think about two degrees of freedom we have square of the first velocity square of the second velocity and product of the two but if you go to like six degrees of freedom you have the square of all the velocities and many product of velocity q1 q.2 q do1 q do3 to q6 and you have all of these so we can always put them in term of Matrix multiplied by the velocities square of the veloc ities and Matrix multiply by the product of velocities and always involving those elements anyone knows what this represents well sir Christal discovered uh this pattern and we call them the Christal symbols the b i j K's that we can form from combined finding the permutation of uh the partial derivatives of M J so you start with an element m j and you take its partial derivative and then you form these uh symbols a b j k is 1/2 the element Mi i i j taken with respect to K and then ik K taken with respect to J minus J K taking with respect to I well so this is the first element and using these symbols we can simplify the writing of what we saw here these two matrices and write them in this form so you have a matrix multiplied by the square of the velocities and another Matrix multiplied a column Matrix in this case multiplied by the product of velocities when when we go further this will generalize and this Matrix is function of Q we call it C this is the centrifugal force Matrix because this Matrix is multiplied by the square of the velocities and B is the Coriolis Matrix and this Matrix is multiplied by the product of velocity and this generalize in this way it's for n degrees of freedom so the centrifugal force Matrix is this Matrix that when multiplied by the square of the velocities gives you the centrifugal forces and the coriolus first Matrix when multiplied by the product of velocities give you gives you the vector B Q do q. we symbolically we we put it this way this will give you the Sun the uh coris forces so the C Matrix centrifugal force Matrix has to be of Dimension what how many square of velocities we have we have q dot how many Q dots we have n so how many how many squares we have n square of velocity so this Matrix will be an N byn Matrix okay this Matrix B how many Q do Q do how many product of velocities we have beside the square well it turned out we have n -1 * n / by two M basically a column so for 6 degrees of freedom this is what 5 * 6 ided 2 which is 15 so if you go from 1 two to 16 and then two three to the end you have a long Vector of product of velocities how many rows we have here the dimension of V is always going to be six or n so we will have always n rows so you have more columns how many columns this is the 15 columns for 6 degrees of freedom so this is a wide Matrix multiplied by this long Vector to produce your C your coriolus forces the coriolus forces are B multip by Q doq do and C also has n rows but it is square Matrix okay so we can compute V simply by finding the B's and these B's are simply function of the partial derivatives of the element of M once we computed M we just do this differentiation and do the computation very simple well very simple if you're not doing it by hand but if you're doing it by hand for n degrees of freedom it's complicated but but let's let's take an example in in two minutes and you see it's not that difficult for few degrees of freedom and you get the sense of it okay you get the idea here I mean the main idea is to remember M gives you V and V is obtained by two matrices c and b and these matrices are invol Ving element that are the partial derivative of of M and that's why if m is constant everything here is zero so if the mass Matrix was constant there is no centrifugal cor forces okay one more thing left is the gravity I mentioned the gravity uh and we need to deal with the gravity so how do we compute the gravity so you you have you have each of the link somewhere and you have the center of mass right and and as you move up and down you have different potential energy right higher better so the height is very important and you can compute the height so and then compute the potential energy of your specific link and then you add them together so the the potential energy for uh to compute hi you have a vector we we already found this Vector locating the center of mass we have the height we have the gravity pointing South so we take the minus gravity Vector we multiply it by the PCI with that product we compute H and that gives you mi now what is the gravity forces well it is just the gradient of that you just take the partial derivative with respect to q and you find it and what is the partial derivative gives you usually gives you the columns of the Jacobian matrix so essentially your gravity is simply this minus multiplied by the JV I times this Vector M1 M2 to MN actually a very simple way to think about it is let's let's look at it this way you have this manipulator you have all these links and you have the center of masses if you are standing like this it's almost like you are at each of the link you have a force pulling you down and you are trying to compute the torque corresponding to that Force so what is the force pulling down this is the mass of that partic joint multiplied by the gravity right everywhere like this right you have weights so what is the torque corresponding to this so let's start with the last one the F what is the torque torque equal J transpose F so J transpose in this case is VN F which is m n g right just add them all together and how and now you have your gravity okay so now we know the gravity we know how to compute V let's take an example okay this is uh do you see this robot uh it's a little bit uh I'm not sure can you see it so this is a two degree of Freedom robot and the first joint is revolute what happened to your voice come on the second joint is oh yeah that's better all right so we have a revolute prismatic joints to simplify to really uh so what we are doing is we are selecting we are selecting uh this point and representing the end Factor at this point so this D2 is measured from here to here okay I mean we are not putting it in here we are putting it at the center of mass and uh we are looking at a first link that has a mass of M1 a second link of M2 and inertia tensor of i1 at the center of mass and ic2 we are locating the the vector pc1 by this distance from uh the the origin and the origin located here at the axis so X1 y1 and you have Z1 so could we find the dynamic equations of this robot how to proceed how are we going to do it what is the mass Matrix yes uh the first one would be the uh the first well we starting from the from the end you want to say the uh mass of the SE second link times the distance from the so I mean we found the mass Matrix is the sum from I = to 1 to n of Mi i j v I transpose JV I plus Omega so we we can just take that expression and just write it okay write it out you can you can start from the end from but basically you need to to compute this okay do you agree with this I mean this is what we we we we established so let's just write the equation so what do we need to do in this equation now what is missing we need to compute these jvi and J Omega I right how do we compute jvi we need PCI the vector locating the center of mass do you see that Vector well it is there you have to be careful how you you write it so the PC one is do you agree L1 cine1 and L1 sin1 you agree if anyone doesn't agree please make sure well sometimes there are mistakes so there is no minuses everyone agrees okay okay now once you have the P Vector you do the partial derivatives and you compute your Jacobian all right the first Jacobian jv1 we compute up to one so the first column and add a zero column the second one we compute up to column two right basically all of them and now we want to make find the expression M1 jv1 transpose jv1 so you get these two expressions so the first element there is this 2x two Matrix and the second element is that so the mass Matrix is equal to the sum of those four elements the first element is this we have three zeros and just this so Mass one the contribution of mass one to the total mass Matrix appears here what is this if everything was zero this is telling you that the center of mass of mass one multiplied by its distance to the axis gives you some inertia and this is the contribution of the center of mass one to the mass Matrix right you understand that and that Mak sense Mass to contribution is to Joint one and its contribution is appearing by the distance square of that Center of Mass to the axis make sense but Mass two has another contribution on the second joint okay so you see how this is this is added you start with one element Mass one and you see its contribution Mass one will never appear anymore Mass two is going to appear here and here and in different robots it will appear also in here in the coupling but as it happened because of the Jacobian it it's not going to appear anymore the inertia with the Z1 and Z2 we are going to have the jacobians here are the two jacobians and when we do the multiplication we see the contribution of the inertia of Link one appears only on joint one the contribution of the inertia on joint two appears only on joint two because joint two is Prismatic so the total mass Matrix is here so the mass Matrix as it happened for this robot is decoupled there is no the uh of diagonal terms you can see that M2 is appearing here M2 is constant as we said M2 is representing the inertial properties viewed along axis two that is what if you lock joint one you are moving a mass you see and that's all what you see right okay let's lock joint two and look at joint one when you rotate about joint one you're going to see the inertia of joint one link one and Link two you're going to see the distance of the center of mass by the scale by the mass of one and two right okay this is yes why is it that like the term ml squared looks like an inertia it is the inertia of the center of of mass uh when when you are looking at the linear motion and then you have the angular motion bringing all the rest of the contribution of the inertial properties you have both of them and and the the mass M1 which represent the mass of the link at the center of mass brings in a in for a revolute joint brings a contribution to the inertial uh forces by the square it has to be I mean to be uh homogeneous in units you you you you basically need distance Square so you have M1 L2 plus all the rest of the contribution of the inertia of the link because it is it is a rigid body and we computed its inertias already the next question is what about how this is varying as we move this M2 as we extend the location of M2 this is function of the distance you see that so it's varying okay we don't have much time time but I think we can do the centrifugal corol forces so we need to compute B jks all right so I said it is simple and I didn't say why it is really simple because there are a lot of things that just disappear what about these M jks what what things you you you would remember from what we said we said something about this like element mnn is in dependent of any variable M22 is function of M22 is function of the configuration so if you take the partial derivative of M22 with respect to Joint one you're going to have zero joint two zero it's function of three four and and the rest so many of those elements like if I take IJ with respect to a k larger than I then it's going to be zero and that leads to properties that all the bi II are zero all the b i j i are zero for I greater than J and in our robot the only variable is just M11 the only element that is changing so with M11 we have only to consider the the it's um it's derivative with respect to two that means that we have only m112 that is non zero and m112 is simply the element was M2 D D2 squar so it will be 2 m 2 D2 and when we write the matrices that appears like this so it appears in the B Matrix and it appears in the C Matrix so there there are indeed centrifugal forces uh that will appear and the Matrix V appears like this could you tell me where the centrifugal forces are appearing which joint is going to have to see the effect of centri forces joint two because you can see here you have theta. 2 multiplied by this and this goes to V2 you see that which joint will have coriolis forces which joint louder joint can you read this is a vector you multiply this by the top it goes to V1 so V1 has a centrifugal joint one has a coris force and Joint two has a centrifugal force and it's very easy if you rotate this you see that M2 will will will start going away and that is joint two centrifugal and the first one is the product of velocities corus okay so the V Vector is like this the gravity Vector is already there because we already know jv1 and jv2 just you computed be careful on your gravity Vector it is pointing down along the minus y direction and this is your vector so when we are in this configuration the center of mass one and two at their distances will produce a force projected with the cosine and the on the joint two we have only M2 that is affecting the gravity and finally here is your model wow on time okay now we can use it next time see you on Monday for |
Lecture_Collection_Introduction_to_Robotics | Lecture_8_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started so today's uh video SE m is okay it's about parallel parking I know a lot of you don't know how to do the parallel parking so now you get into the car and you press the button and the car will park itself yes please over here okay automatic parallel parking and returning to traffic Maneuvers [Music] France at inria Ron Alp we have developed and tested automatic parallel parking and pulling out Maneuvers on an experimental electric car the car can be driven manually or move autonomously with automatic steering and velocity control it is equipped with various sensors including sonas to monitor its surroundings when he wishes to park the car the driver starts the automatic parking mode and leaves the vehicle the planning and decisional architecture allows the sequence of autonomous motions to be controlled in real time in classic traffic situations within a parking area the car moves autonomously in the lane its realtime motion sequence control ensures safe oper operation in a dynamic environments the car avoids collisions with other cars and obstacles it continues the motion when the lane is free when it detects a free parking space the car stops at a starting position in front of the space the system computes appropriate steering and velocity controls for a nominal trajectory leading to the park location a backward parallel parking maneuver into the space is then performed nice par once the motion is carried out the autonomous system decides whether a suitable Park location has been attained if necessary additional iterative motions are performed within the parking space when the lateral displacement of the car is sufficient the car adjusts its location to midway between the front and rear vehicles to ensure a proper [Music] distance for automatic pulling out the required steering and velocity controls are planned and executed the car moves to an appropriate spot at the rear of the parking space then control commands are planned for the subsequent motion the car turns and moves forward to leave the parking [Music] space it stops in the traffic lane and is then ready for further manual driving our future work aims at integrating the automatic Maneuvers we have developed into a novel Urban transport system based on a fleet of dual mode electric cars a preliminary manual mode version of this system with 50 electric cars is currently under evaluation near Paris the future version will include all the automatic Maneuvers we are currently developing Okay so a lot of progress was made in the in the last years in navigation of the car so now you just get to your place where you want to go and drop the car and the car will go alone and parked itself and when you are ready to leave you just with your cell phone you just call the car and the car will come and and pick you up what do you think about that cool well we'll see um there is uh two announcement the first one is regarding the midterm which is on Wednesday as you know next Wednesday it will be in class and uh um preparing for the midterm we're going to have two review sessions you have them on your schedule Monday and Tuesday so it is uh the same review session we will go over the midterm of a past year and it's very very important for you to attend now our group is very large and that's why we are going to split into two groups so uh there is uh uh a sheet of paper you sign your name and pick which day you would like to come Monday or Tuesday and uh uh uh I I think it will be done just at the time when you are very hungry so don't worry about it we will have pizza for you and also we will have uh we'll have some demonstrations of some robots in the lab so the review sessions will be upstairs in the robotics lab in the open area and uh you just uh you just uh should be there at 7:00 we will we will start uh with the demos and the pizza and then we will move to the review okay now uh probably some of you would are wondering about what is included in the midterm and uh what is included in the midterm is all the lectures up to today that is uh we are going to cover everything up to the Jacobian that we we will complete hopefully today now over here you have the Stanford shiman arm we saw the Stanford shiman arm in picture many times and this is actually the one of the early uh uh arms that was built in the early '70s so you can see here the first joint The revolute Joint about this axis then there is another joint which is here and that allow us to rotate up and down and uh I'm not going to be able to actuate it because it needs uh to be connected to the controller so we release the brakes there is a brake right now that is blocking it in the horizontal position and I imagine you understand why otherwise it will slip if we because we have the third joint which is which is Prismatic joint so the Prismatic joint is really uh an interesting feature about this arm because you can now take this end factor and reach once you decided the direction you can quickly reach in that direction and uh Advance the the arm well the this has a limitation that is you can move up to here and you have another stop about here I mentioned about the wrist and uh the wrist is here and it has 3 degrees of freedom so if we look at this wrist you can rotated about this axis you see this motion this is Joint four and Joint five okay and Joint six which is this axis so something very interesting happens when joint five goes to zero and this is what Singularity and it's really difficult to see exactly what that Singularity is so I'm going to try to illustrate it the idea is when you go to Joint five equal Z this is the zero of joint 5 what happens is this axis of rotation number four and the axis of rotation number six do you see them one like this and one like this these two axis become colinear right so for this position and orientation when the these two axis are cinear you are losing locally a degree of freedom and this degree of Freedom that you are losing is the rotation about I don't know if you can tell me which axis I'm going to take a a pencil and try to to find this axis about which we cannot rotate now so here here here here we can rotate about this right we can still rotate about this what about this one you're you're locked and if you move little bit out you will be able to rotate but you see there is a large rotation of joint five when we are at Singularity what happens also is the fact that you have a freedom here it's internal right and at this Singularity for all these configurations of combination of joint four and six you still have the singularity because joint five now is set to zero so you cannot rotate about this axis okay you see it all right well you will have a chance at the end of the class to come closer and check it out but uh let's uh go to all right we're there so the definition of the kinematic Singularity is here that locally the in effector loses the ability to rotate or to translate about a given Axis or a given Direction and this direction is called the singular Direction and for that location we have a singularity so the study of the singularity really depends on on those Za axis we saw here Z4 and Z6 aligned and you remember in the in the Jacobian this z-axis appears in the columns so what really happens is that the Jacobian if we look at the Jacobian with its columns the rank of the Jacobian is going to be reduced at the singularity that is there are a number of those columns that become dependent and you have a singularity so how do you study the rank of a matrix or you know that the Matrix is angular you cannot invert it what do you we use the determinant if this Matrix is square you take the determinant and when the determinant is equal to zero you have you have those values for the singularity the way we do it we we go and say let's compute the determinant as a function of the joint angles and what we will obtain is a set of expressions multiplying each other and those will be the determinant and those expressions are function of Q so when we are looking for the singularity all what we need to do is to take each of those expressions and set it to zero and that will give us the singularity associated with that particular expression so one thing to notice is as we look at the Jacobian the Jacobian expressions of are more complicated in some frames than others so it is very important to rewrite your jacoban in a frame that is that gives you the simplest form and knowing that the Jacobian Express in frame J would give you a determinant that is identical to the jacoban in frame I we can do this transformation and not worry about the rotation Matrix so the determinant of the Jacobian is independent of the frame uh do you know why anyone can tell me why yes it because the determinant is preserved by conjugate by U conjugation by matrix yeah by the rotation Matrix because to go from a Jacobian in frame J to a Jacobian in frame I all what we need is to premultiply by a rotation Matrix like this right so if you are in the 6x6 spatial case linear velocity and angular velocity the Jacobian INF frame a can be transformed into a jacoban in frame B by premultiplying the jacoban with a matrix that has two blocks of the same rotation Matrix that transforms the descriptions from A to B and that will give you the Matrix in B now if you take the determinant of these the determinant of the Jacobian in B is equal to the product of the determinant of the two matrices and since the these matrices are uton normal their determinant is one and that means we have this relation okay okay good so so remember this and remember that you can use it to simplify the expressions and immediately uh analyze your determinant now once we said we are going to set the the determinant of the jacovan to zero we immediately see the singular configurations because in the expression of the determinant as I said the determinant will come up in this form so it is product of expression S1 1 S2 S3 Etc and each of the those expressions when it vanishes it makes the determinant equal to zero so if you have multiple singularities they are here and you just set each of those expression to zero and you get your Singularity okay let's take the simplest example possible which is a two degree of freedom and see if we have a singularity for this Jacobian associated with this mechanism do you see a singularity here which is where when Theta 2 goes to zero or to Pi or k Pi you align the two points and you're going to have a singularity let's see it in the mathematics so I'm I'm going to take X and Y X L1 cosine Theta 1 + L2 cosine Theta 1+ theta2 you remember all of that and now if we differentiate we have this Chu right so as I said if we determine our expression for the determinant it comes to be L1 L2 sine of 2 and this vanishes when the sign of two is equal to zero which correspond to all the configuration when Q2 is equal an integer time Pi so as I said also so you see the Jacobian here you see its expression well it will become much simpler if we write it in frame one to take it to frame one we premultiply this Jacobian by the rotation Matrix 0 to one we are doing the multiplic the rotation in the plane so it is a 2X two Matrix don't be confused all right so we're just doing it in the plane and at Singularity this is the singularity when when we move the last link to be aligned with the first one well in that configuration when the sign vanishes this is the Jacobian yes the notation you say cosine 2 versus sign 2 sign 2 that's the difference so here we say cine Theta 1 + Theta 2 and we denote it C12 or S12 would be the sign for the sum okay Ju Just to simplify to to have a more compact representation so so this is this is what is happening so what do you see here you see directly how the these two columns are dependent and the rank of the Matrix is equal to one one so if we think about the displacement Delta X and Delta y associated with this configuration so if we make a small displacement Delta Theta 1 Delta Theta 2 what would be the displacement Delta X and Delta y well we know Delta X is equal the Jacobian Delta Theta so you multiply and this is what you obtain Delta X is going to be zero the first row is zero and the second row I'm multiplying with the second row Delta Theta 1 with the first element and Delta Theta 2 with the second element you see this you understand this okay what does it mean for any Delta Theta 1 or Delta Theta 2 Delta X is going to be equal to zero and the reason for that is the fact that when we are in this configuration and when we make a small displacement Delta Theta one so if we rotate about this axis this point will move on the tangent of this circle right if you rotate about this axis with a Delta Theta 2 you rotate about this you go on this tangent that is we whether we move Delta Theta 1 or Delta Theta 2 both of them are contributing to the displacement with a small displacement along this tangent so there is no displacement that will take place along this direction and that's why Delta X is equal to zero do you see that yes how did you getan in the last SL how do we get this Jacobian here in in the last slide well differentiate well I I thought we we we we're done how do we obtain the Jovan from X and Y for this is going to be just the linear jacoban JV so we differentiate Delta X the First Column will be what partial derivative of x with respect to cosine to Theta 1 so it will be minus okay you understand this everyone okay so when we rotated with this Matrix I'm multiplying and I get this Jacobian at zero and we are here and we're saying that Theta 1 displacement Theta 2 will be always in the this direction that is will be orthagonal to this direction so there is no displacement Delta X and I'm looking here in frame one in frame one so this is y and this is this is y and this is X okay so let me show you this on a bigger robot what is the name of this robot pum the Puma very good so uh right now the Puma is floating so if I apply a force to it it's going to rotate it's floating so I'm pulling like this I'm pulling like this if we pull here what is this configuration if you try to move the Puma in this direction you apply a force it's not moving if you try to move in this direction it it moves because there is a small angle but if this angle is straight then you apply a force and there is no motion and this is the singularity when you extend the arm you cannot move along this direction so if you apply any force in this direction immediately you have a motion if you apply whatever Force you apply here there is no displacement along this direction okay you you understand the singularity good so let's close this window all right so now you know the singularity that configuration that we are calling singular configuration but the really the problem is not the singularity itself it is the region around the singularity because when you come close to that Singularity you start to have a metrix that is ill condition and if you try to uh make a displacement along this direction so you you were in this configuration and if you you are very close to it you can move but it's very heavy it's going to take a lot of force to pull it and make this goes up the elbow it goes up so what I'm going to do I'm going to take a small displacement Delta Q yes the last slide why is the Jacobian different from why is it l yeah why is it L2 why is the top right L2 S2 instead of L2 s can anyone answer this question frame one instead of frame zero so this Matrix is expressed in frame one no as in as in that one is frame zero right no the multipli the the multiplication by Z to one actually this Matrix uh this Matrix is multiply by this give you in frame zero so this is j0 this one here is it I think you're confused because of this this is in frame zero and this Matrix in frame one there is a program running that's why everything thing become slow okay so this is in frame zero okay so in frame zero because this is the rotation from 1 to zero and this is the Jacobian in frame one and when you make S of 2 equal to Z you get 0 0 L2 and L1 + L2 okay all right so let's go to the next slide so rather than Computing the Jacobian exactly I'm going to make a small approximation of the Jacobian and I'm going to compute Delta q that correspond to a Delta X where we are in the that singular configuration so if Theta 2 is equal to zero uh I mean very close to zero Theta 2 is Epsilon you can write this inverse of the Jacobian you so what do you do if if Theta 2 is very small to approximate your SS and cosiness S of theta 2 when Theta 2 is very small is almost equal to to zero and the cosine one so if we do that and compute the inverse in frame one this is the Matrix you will obtain and from this Matrix you can find the displacement to make a small displacement Delta X you can compute the displacement required for Delta Q so that means if we want to make a displacement Delta X1 small displacement Delta X1 and small displacement Delta y1 this would be the expression for Delta q1 and this would be the expression for Delta Q2 okay so this is not exact this is an approximation where we are uh working around the configuration Theta 2 equal to Z but it is very useful to look at it because as Theta 2 gets smaller and smaller you are dividing by zero you can see that so to achieve any Delta X you are going to have a large displacement Delta q1 or Delta Q2 do you see that in fact if we look at it graphically so here is the configuration very close graphically if we look at displacement of theta 2 that comes closer and closer to zero the displacement Delta q1 increases and the same thing if we come from the other side of zero we have so we have this this continuity that makes it that we have large value of delta q1 that will be needed or large values of Delta Q2 that will be needed and when you get to the singularity there is no amount of value you need infinite value okay so this means that we have really to deal with the singularity not only at zero but also around the region where the singularity is taking place now in your homework in our discussions we always place the problem in the spatial case but often the robots you are dealing with become uh R planer and you have this large Matrix and you are going to study the Jacobian of a matrix that is actually uh nons squar this is the case for if we take XY and we take uh the J Omega 00 01 001 we end up with a large description with all these rows of zero so we talk about the reduced Matrix that you can analyze in the uh case Cor responding to a linear motion for a two degrees of freedom or a planer motion so if you have the case of two degrees of freedom your analysis is done on this Matrix on the 2x two Matrix on the JV part and you don't have a Z part so it is just a 2X two Matrix okay in the case if you have so this is what what you will be finding and this would be coming from this 2x two Matrix if you have a 3 degrees of freedom if you have a three degrees of freedom then you are going to have a matrix like this x y and the Z and then you have the J Omega 0 01 so the reduced Matrix will be in the plane would be x y and the last row basically you are dropping those three rows so this is your metrix and you study this determinant and you will be able to find the singularities associated with this Three Degree Freedom manipulator so yes three R uh because in in the case of a planer robot that is that has three degrees of freedom The Matrix that you're going to analyze is going to be XY and the rotation about the Z axis so you can do it actually with this Matrix but uh to do it to analyze the the the um uh to analyze the determinant of this Matrix you do not know how to do it how do you take the determinant of this Matrix so to analyze the singular configuration you can do it from here by forming the Matrix J transpose J and reducing the Matrix to a 3X3 Matrix which correspond directly to eliminating the three rows so the Matrix on the top captures the characteristics of uh rank of this Matrix yes but will it be the same determinant yeah it will be exactly well uh when basically if the Matrix is not Square you have to go to a singular value DEC composition and the singular value you you basically look at the singular value by forming the square of the Matrix so the Matrix here that you will be form will be the square of that Matrix but because we have zero rows we can just reduce it to those three independent so we know exactly that we have x y and the rotation okay this is just a simple trick to help you do the analysis without going to singular value de composition all right okay so here is the stem for Shaman arm and uh we just analyzed I mean we we just uh saw the singularity of the shiman arm by looking at joint five when it goes to zero and in fact uh if we do this computation we multiply the matrices we do the differentiation you remember we did this last time and we obtain this metrix so let's let's see what happens when Theta 5 goes to zero in this Matrix so could you tell me what is happening what we're interested in is to see if there is immediately dependency between columns whether when at that configuration columns becomes dependent so this is column 1 2 3 4 5 and the last column six four and and six becomes this yes okay let me write them for you that's what happened so the column four and the column six are identical and which means that now you lost the rank the rank of this Matrix is five so you have a singularity so basically the Jacobian captures the fact that you see in the physical robot the fact that when these two columns these two axes are aligned essentially what is happening is you are getting two columns in the jauan that become dependent okay everyone sees that good all right so now we're going to answer some concerns about this last frame you remember many times I repeated and said well the robot and the factor might be operating here if I'm grasping or if I have a tool I'm concerned with this point here at the end of the tool or in the middle of the tool depending on the task so what we're going to do we're going to use the intersection of the three axis as the point origin of all the frames and we develop the Jacobian for that and the Jacobian we saw was here this in this configuration at this point now I say it will be very easy to go from here to Here There is a constant transformation this is both for computing X so X this x is X RIS X dou and this x here is the X the factor x e so when we go from here to Here There is a constant Vector so XE is equal to xw plus that Vector so now we're going to see how we transform the description of the Jacobian from a Jacobian associated with this point to a Jacobian associated with that point yes I think your mic fell off I'm sorry I think your microphone fell off yeah you're right I can see it thanks okay so so essentially we have we have two frames one frame associated with the frame we selected for the last link in this case was this Frame or the wrist point and the the second frame is this additional frame we are attaching to the end of factor so now think about it you have a vector connecting the origins this is the vector p n e connecting the last link frame to this location of the end of vector now the velocities linear velocities and angular velocities at that point are transformed to linear velocities and angular velocities at the end point and the question is what is the difference between the two so in term of if we go back to the robot and if we think about we have some velocity linear velocity here moving this point and this at the same time is rotating with Omega so the question is what is the Omega here and what what is the V here could someone tell me what would be the linear velocity at this point given the linear velocities that we know at this point and the angle of velocities of that point so given we know VN and Omega n what is v and Omega e they're the same okay so Omega e is equal to Omega n everyone agrees all right that's correct and V is equal to VN okay who who disagrees with that statement how many I disagree I I disagree to all right so some of you still think it is the same velocity V equal to VN you remember the Apple there is an additional velocity coming from the fact that this there is an angular velocity here that is going to affect this angular velocity even if there was Zero velocity at this point V I mean VN there will be some linear velocity to due to the rotation so the relationship between the two is like this V is equal to VN plus Omega n cross PN e all right and the other relation is is fine this is correct so I wonder why I I switch and put minus anyone knows yes they mean the same thing if you cross something corre cross them in the opposite direction corre but um um I mean so I I just replace that relation and put it in this form uh why I'm after this form so basically V Omega n is now linear in VN and Omega n we put it in the other side and the reason is I would like to connect the two and do you see how can we connect the two what is the relationship between the vector VN Omega n as a vector and V and Omega e as a vector what is the there is some Matrix and this Matrix involves what identity matri is because this is identical and and PN e cross what is PN e cross how do we make PN e cross as a matrix exactly so we use the cross product operator so actually this writing allow us to find this Matrix immediately connect VN Omega n to V Omega e what is on the diagonal because V is equal to VN minus P NE e so it is the first uh block diagonal block is identity and the second Omega e is going to be identity times Omega n so there will be zero identity and what is on this upper block on the right it is minus the cross product operator so so basically we have the identity on the diagonal we have zero and we have minus p and E hat okay all right so now we have a linear relation between the two and it involves just this Vector connecting the wrist to that point and we are taking the cross product operator of that Vector now when we do this computation you have to be very careful to make sure that you know your p and E Vector is described in frame n most of the time it's it's just like a straight line along the z-axis uh it is just a small length along that z-axis so it will be 0 07 but if your VN Omega n is described in frame zero you cannot do this multiplication so you need your operator to be described in frame zero so we have to pay attention to this and to make sure that this computation is carried correctly so that we obtain a Jacobian in the same frame the Jacobian in frame n is now related to the Jacobian I mean the Jacobian at the end of factor is related to the Jacobian n by this Matrix and we have to make sure that the Jacobian here is expressed in the same frame to produce the same frame Jacobian at the end of factor yes involved between the last joint to the ector well this is capturing both JV and J Omega there is a rotation now the question is uh I'm talking this is General expression of the relationship when we have a Jacobian in frame zero we have to to make sure that this operator is in frame zero n is the rotation in the last joint right Omega n is the total it Omega VN and Omega n are the end uh are the linear velocities and angular velocities produced by all the joints expressed at the end Factor frame uh not in the factor frame at the last link frame frame n and we are now Computing the velocities at the next frame the N Factor frame and that's why we can substitute VN and Omega n with JN q.n and this would be the Jacobian eq. e and this is true for any Q so this is the relationship so what you see here is that the Jacobian add theist point in this case is transformed to the Jacobian at the end effect Factor very simply by taking the cross product operator associated with the origins very simple okay so this is the relationship if we we are in frame zero I said this operator should be in frame zero now uh there you are going to make the mistake so I'm going to tell you not to do it explain why and then you have the relation correctly for this operator in frame zero so suppose you have your operator in frame n how do you transform it to frame zero so I take the vector 0 07 in frame n and I take that operator and then I want to transform it to frame zero so you so what do you do you premultiply or you you take the operator and REM multiply by the rotation Matrix to frame zero how many agrees with him no one come on at least you're alone so how you do it then you don't agree with him so how do you do it you're not really alone but no one is really convinced yet so have to find so how do we how we do that the the problem is his statement is wrong you cannot just take an operator and transform it this way by just a rotation Matrix in fact this is a similarity transform you have to premultiply and post multiply and the reason is think about it from the beginning let's go back to this blue equation and think about it so we are saying that let's go back to the vector representation so P what we are transforming is the vector p in frame n multiplied by the vector Omega in frame n and transforming them this is this is correct this is Vector relationship so now what we are going to do we are going to take this vector and now I'm going to express this Matrix in the frame and that leads to this relation and the transformation is so if you if you did your operation in frame n you transform it using this similarity transform you are because you you are multiplying you assuming you are multiplying by the Omega in the appropriate frame which is frame n so you have to go and post multiply it by n to zero now the best way to do it perhaps is just to go to p p in frame n transform it to frame zero and then just do the operation I mean the cross product operation in frame zero and then you will have it directly so if you start with p n transform it to frame zero then you can get the directly the operation otherwise if you have the operator in frame n you need to do this okay don't do that mistake all right so here are the two important relations we learned the first one is a Jacobian in frame I is transformed to a Jacobian in frame uh J uh and the relationship involves only premultiplication by a rotation Matrix in the case of a Jacobian in frame n that we would like in frame zero we have to to do this diagonal transformation from frame and to zero but for the operator you have to put the similarity transform okay good well there is uh here an example of uh a Jacobian computed at the rest point a Jacobian computed at the end Vector point and we are Computing this Vector W from E to Zer I'm going to skip it because we really don't have time to to to do it but once you you I think you have it in your handout just go through it and you see that you can compute J from this J Omega simply by computing in this operation associated with the vector connecting W to e and what we did is we computed p in frame zero and found the operator in frame zero directly okay so please take a look at this and uh so I'm going to to move because we we are going now to discuss um uh something important about the Jacobian and this is the fact we we we already started to introduce it little bit but the fact that the Jacobian is very useful not only to analyze velocities but also to to control the robot most industrial robots actually are controlled just with the Jacobian because this Jacobian is always connecting your desired position and orientation of the end effector you have some Delta X here and if you know how to control the joint motion all what you need is to say okay I'm at this configuration I would like to make a small displacement Delta x what would be the small displacement Delta q and that can be obtained from the Jacobian because the Jacobian is giving you this relation and if you are outside of the singularities what can you do can you compute Delta Theta that correspond to your Delta X well this is this is the relation that can be used directly to compute displacement Delta Theta to follow what you would like to achieve with Delta X and you start from Theta you know the Theta you know your X you can compute it from the forward kinematics you want to go to x desired then you can compute aulta X by taking X Des are minus X hopefully this is small so you can use the the jacoban and then you can compute the corresponding Delta Theta to this Delta X using this relation right and then you know Theta so you control the robot to move from Theta to theta plus okay you see this algorithm very simple you measure where you are you know your Theta you know you know your X and you want to make a small displacement Delta X all what you need to do is to compute this Delta Theta from your jobian so basically you have a sort of controller that is using using the jauan to compute this error between your desire position and measure position computed from the forward kinematics as a function of your cu's and that will give you a Delta q that will distribute to each of the joint and ask the joints to track okay so this is uh the resolve motion rate control proposed by Dan Whitney back in the 70s this simple algorithm if you don't really care about Dynamics if you don't care about Force control uh contact the robot with the environment this is fine that will and if you are outside of singularities this will take you through but obviously it's not going to work very well because you're going to have problems with accelerations the uh if you're tracking a motion you are basically doing this and you will have a lot of errors and but still if you have just a small displacement and slow motions this is going to work okay the last Point uh in our discussion deals with static forces and again the jobian is going to give us the answer and that's why uh now now with this Jacobian we saw that we are able to understand the velocities produced at the end of factor giving the joint velocities but we are going also to see that the Jacobian is part of this relationship between forces exerted by the end factor and torqus produced at the joints the same Matrix so how can we start with this okay let's go upstairs so now we're in the robotics lab uh James was uh the TA I took that photo when he was ta but it was a couple of years ago back uh so how do you open this door come on we're doing it with the so so you take your hand and you start after turning the knob you push little bit and it's going to rotate you see how it's going to work so let's analyze it so you're pushing here right at some location and you have the axis of rotation right so there will be a linear velocity you're pushing with some velocity so at that point there is some linear velocity corresponding to that velocity there is some angular velocity about the axis you see that all right now we can also think about the forces I'm pushing with so the linear velocity is related to angular velocity by this relation we saw this the forces if I apply a force here there will be a torque actually I could have this door actuated with a motor producing a torque and that correspond to a force here and there is a complete relationship between the two and do you know what is the relationship between the two torque and force the cross product but in this in this way the torque is equal to P cross F so the torque now is on the other side and the f is on this side okay so V and F you see V and F they are crossing in the two two equations actually the first equation represent sort of a Jacobian relating velocities V and Omega or Theta Dot and the second relation is representing sort of transpose of that Jacobin and we're going to see that so let's go and analyze this so we have omega we have linear velocity and this is the relationship we agree here we can write it in this form we can say V is related to Omega by minus P hat right okay torque so if we want to produce a force there you can apply a torque and that would be equal to P cross F which can be now represented by P hat F okay let's take uh that and put it really similar to this so it's minus P hat so it is identical to that you see this transpose I'm writing uh because the operator is asymmetric you get minus the operator equal the transpose okay so I'm just writing minus P hat T so basically this is the Jacobian and this is the transpose of the Jacobian and if we want to really to check it out just put a frame and compute this operator it will be minus p y PX these are the coordinate of this Vector PX py and if you put it there it is this so basically there is a duality between this is for one degree of Freedom a door is one degree of Freedom so there is a duality between velocities and forces we can compute the torque corresponding to the forces acting at the point by looking at this Matrix minus py PX X or we can compute the velocities through this so this is the Jacobian and this is the Jacobian transpose and the relationship between forces involves the same Matrix in transpose okay so remember these two relations they are fundamental relations in the kinematics of robots but they are going to play a very important role as we go to Dynamics and control velocity and forces okay well there are many different ways of actually establishing this relation so you remember when we wanted to establish the relationship between velocities at the joints and velocities at the ector we said we can do propagation of velocities and compute the propagation from in fact the base up to the end of factor and find the relationship and extract the Jacobian well you can do the same here you can say my robot is in contact with the environment and it is applying to the environment a force F and a torque or moment n and you can start from the end and go down and propagate velocity uh I mean propagate forces and find the relationship between forces at the ector and torqus apply to the joint and this propagation is going to lead to the Jacobian transpose after elimination of all the internal forces because when you are pushing with the robot the structure of the robot is taking part of the forces and the other part is going to the joints and the joints are going to support it with motor torqus so if we do this elimination we have to be careful that we project at each step those forces on the axis of the joints to make sure that we are looking at the active part of the force so we're going to just briefly look at this and then I will show you another way of doing the analysis using the virtual work principle so how do we do this so if you have a problem with the application of forces and you are reaching a static equilibrium and you are saying I'm going to try to establish the relationship between those forces and those torqus it is at static equilibrium because you're applying these Tor and these are pushing the environment and the environment is pushing with equal and opposing forces and moments so how do we do it well we just break it down we break it down into pieces and we take each of the rigid bodies and we say each of the rigid body is going to have to be at static equilibrium so we start from the top and we say first of all this is the forces apply to the environment which means the lust link is receiving the reaction forces of the environment minus F and minus n and it's on the other side of the link at the Joint it's receiving some forces F3 and N3 on link three and you do the same for the other ones okay then then what do you say well if each of these is at static equilibrium how do you express that what are the conditions some of the forces equal to zero and some of the moment computed at with respect to any point on the rigid body are equal to zero that's and you have static equilibrium so this is the last link and now if we take one of them and express what you just said so static equilibrium when some of the force equal to Z and some of the moment with respect to any point equal to zero then then you can find a relationship and those relationship are going to be in this form forces plus equal to zero moments plus these moments plus the moment associated with the force f i + 1 is equal to zero if I'm Computing at that point and that is the origin of frame I and from this you have have recursive relations so you can compute back propagation fi as a function of I + one n i as a function of this you back propagate and you extract the Jacobian but before that what you need to do is to project on the axis and make sure that you are getting the forces as you propagate so at the Joint you are if the joint is Prismatic you are going to project the forces because if I'm doing something here this will the only thing that will come is a force that will make it translate so that would be a force for a joint revolute joint the forces are supported by something what counts is the moment so you project in there the moment and the torque will be so this is projected by dot product with the Z AIS and you get this algorithm and now you prop propagate from the forces you're applying back to the uh ground and from there you can find your Jacobian transpose well again you you in this case uh the Jacobian is embedded inside your uh numerical calculations that is you are propagating from the end to find your Jacobian so the Jacobian can be computed by velocity propagation you can compute the Jacobian transpose by back propagation of forces or we saw and this is uh the the the simplest way is to compute explicitly the Jacobian by looking at the structure and once you obtain the Jacobian that way the explicit Jacobian you can in fact immediately go and state what is the Jacobian associated with the force we can redo the analysis completely in a different way this analysis was relying on taking the mechanism and breaking it into pieces and eliminating the internal forces to find the Jacobian well we know something about the mechanism and about the forces about active forces about internal forces that would allow us to make a statement more general statement that will not require us to go and do this Force elimination and this is the virtual work principle are you familiar who's familiar with the virtual Works principle 1 2 3 4 5 7 okay seven all right so it's very simple and it's very important in fact in Dynamics later the idea is if you take a mechanism you can just drop all the internal forces and just worry about active forces because internal forces are workless they are not going to move anything they are supported by the structure so I'm going to avoid them and I'm going to make a more general statement about static equilibrium which is so we're going to think about the the work you know work work is force times displacement right so we we can think about the work uh done in small displacement and if there are displacement there are forces acting along those displacement and what we're going to say is that at static equilibrium so this is the statement can you read it okay what does it mean that is if we take all these active forces that are acting along the the the axis of displacement so all external forces are active if you push on the robot is going to move if you apply a torque at the Joint it's going to move internal forces are not going to produce any motion they are taken by the structure so now we need to deal with two things the forces in that Vector F F and N the the forces applied by the environment and the torqus and what I'm going to say I'm going to write this relation I'm going to say the virtual work done by all these forces has to be equal to zero if I'm in static equilibrium otherwise there will be a motion so what is the work done by the torqux well the work done by the Torx is torque transpose Delta Q the dot product between Delta q and the Torx and what is the the work done by the external forces the dot product between between F and what is the vector name Delta huh Delta X because f is acting along X Direction X Y and Z okay that's it so the whole thing now is captured by this equation we're saying that if we are really at static equilibrium these torqus are supporting the forces okay just extract the relationship from here how do you do it well you have Delta X you have Delta Q do you know any relationship between Delta X and Delta Q Jacobian very good so let's use the Jacobian and that gives you this relation for any Delta Q which by transpose gives you your answer all right so basically this whole thing comes to be very simple if you just look at principle that abstract the problem I mean you can go and look at internal forces then you compute them then you eliminate them and then you find the relationship between the active force torqus and the external forces of f or you just state so we have this Duality and you have here uh an example showing you how you can compute the Torx to support one newton it's very simple you you take the Jacobian transpose and do the multiplication and the answer is going well we we comput it for L1 equal 1 L2 equal 1 Theta 1 equal 0 and Theta 2 equal 60° and you need this amount of torque to support that one newton Force now if I'm going to apply one ton I mean 1,000 Newton so let's see we do the same thing we put uh one k of forces along the z-axis I mean the y axis in this case here along this direction and I'm going to compute it at this configuration L1 L2 Theta 1 = 90° Theta 2 is equal to Z do you know do can you imagine this configuration this is this configuration oh you cannot see it it is the configuration when the whole arm becomes vertical why it didn't come anyway when when join joint one and Joint two are aligned and the answer is zero what happened you're putting the arm here and you're pushing down you can take any Force because of the singularity so at Singularity we cannot move but we can sustain a lot of forces or produce a lot of forces with a small amount of torque when you come closer to a singularity to produce a large Force you just make small small torque and you have huge amount of forces and when you at the singularity you can sustain and that's why when you are lifting something heavy you go like this you don't lift it like this way you just go like this you use your Singularity of the all right so um the homework is due now on Friday I I hope everyone get the message and the homework that is distributed today will be du on the week after uh the midterm I will see you on Monday |
Lecture_Collection_Introduction_to_Robotics | Lecture_6_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development alright let's get started so today a video segment is video segment something happened okay we have a little problem I guess we can stop and start over if you would like yeah okay okay my computer just crashed okay okay let's start over all right take two so today video segment is about poly pod have you heard about poly pod now so these are little small robots that we connect to build a shape so that we call them reconfigurable robots and they are modular and there is a lot of interest in this area and one of the pioneer in this area was Mark him mark him was a PhD student in the robotics lab in the early 90s and he is now professor teaching robotics and building very advanced poly pod systems so I'm going to show you the concept that he proposed in 94 and the some of the realization he made at that time lollipop we could have the light of this poly pound is a reconfigurable modular robot it's made up of two types of modules called segments and nodes segments are two degree freedom modules with two motors force and position sensing and a microcomputer on board nodes are readied cube shaped housings for batteries segments may be mounted parallel to each other or they may be mounted perpendicular to each other modules may also attach on any face of a node simple locomotion gates are statically stable gates that move along straight line the rest of this video will quickly present a series of locomotion gates through simulation and implementation on poly pot each segment runs semi autonomously by controlling each degree of freedom with the sequence of behaviors all of the following motions use two simple behavioral modes called ends mode and Springs mode with the springs mode a degree of freedom acts as a well dam spring using force sensors with ends mode the degree freedom moves at a constant speed until it reaches a joint limit many of the gates shown here are extendable to an arbitrary number of modules by adding to the length of the robot this next set of locomotion gates combine simple modes to achieve more complex locomotion in this case segments are added perpendicularly interspersed between the original segments this gate may be used as a platform to carry objects though more modules used larger and heavier the object may be the next two gates will show the robot turning in this skate as the robot turns the two feet of the robot in contact with the ground rotate with respect to each other and so they must slide on the ground this would create problems if the robot were trying to navigate using dead reckoning or if it were walking on carpet here segments are again interspersed perpendicularly between segments since the segments are placed on the ground I went and picked up the other no sliding will occur the following two gates are called exotic gates are not necessarily efficient or useful they are still interesting this gate is called the moonwalk manipulation of large objects in locomotion can be considered equivalent here we are doing both the sly simulation is not a locomotion mode but it shows one possible dynamic reconfiguration of poly pod cool well the early realization were really difficult and very simple but today there are a lot of interesting devices that can carry all these simulations and I hope you will have the chance to see them later okay so today we are going to start on instantaneous kinematics and that is going to introduce the model I discussed many times before I refer to this kinematics model called the Jacobian matrix that is going to be a very important part of all what we are going to do later in term of not only the motion but also the dynamics of the motion so you remember our first task was to try to understand how we localize the in the factory so we know now this frame at the end of factor and we can describe the position and orientation of that end effector with respect to a fixed frame so if we start moving we are going to have small displacement that we can monitor in time and if we go to a small displacement from the current configuration so we are at a given configuration we have theta 1 to theta n and we know the x position but if we make a small displacement of theta so that would be a sort of Delta Theta that we are introducing to each of those joint joint angles we are going to have a small displacement Delta X it's not only Delta X in the position but also in the orientation and the question is what is this Delta X given that we know Delta Theta and we know the theta where we are so this question finding the relationship between Delta Theta and Delta X is answered by a linear relationship that connects the two Delta Theta and Delta X are connected by by I cannot really see with derivatives yeah it is derivatives but what is this model that is going to connect the two you guessed it now so Delta X is going to be related to Delta Theta by a matrix yeah digitally so Delta X Delta Theta or the derivative if we divide by time theta dot and X dot will be related to each other through this Jacobian matrix now X dot again involves two things remember our representation of X involves the position and orientation so there is a part that discusses the linear velocity and there is another part that represents the angular velocity in this vector so what we need to do is to find this relationship and establish this Jacobian matrix that connect those displacement so to study the Jacobian we are going to start by looking at differential motion and we are going to discuss what is how do we compute a linear velocity at a rigid object how angular velocity is compute computed as we propagate and how we can compute the impact of angular velocity on the linear and angular velocity of the end effector so this will take place through this propagation of velocities from one joint to the next and that is going to provide us with a sort of recursive relation that will allow us to find the velocities and in the factor we are going to examine another way of doing this analysis rather than propagating velocities we're going to examine the structure of the kinematics of a robot and its impact on the end of factor velocities and that would lead us to something very interesting we call the explicit form of the Jacobian matrix that is we are going to analyze the kinematics and we will see that in each of the column of these matrices we are going to have an association with the joint with a specific joint so if we take the first column this first column corresponds to the first joint and its impact on the velocity at the end effector velocity linear and angular so this explicit form is going to be very important in establishing the model that Connect displacement or velocities at the joint and at the in the factor and this model is going to be very important also in establishing the relationship between forces forces are acting on the joints depending on the type of the joints if we have a prismatic joint we have a force if we have a revolute joint we will have a torque now if we apply a set of talks to the arm there will be some resulting forces at the end effector it turns out that the relationship between torques and forces resulting in the end effector come from the exactly the same model from the same Jacobian there is a dual relationship between velocities and Static forces that we will use and this will lead us to it establishing the relationship between torques and forces so first you are going to analyze those displacement and what we need is a description of our general coordinates so we picked the joint angles as generalized coordinates but sometimes we have joint displacement if we have a prismatic joint so what we will use we will use a variable we call Q to represent to capture the joint angle whether it is a prismatic or a revolute joint and this is done by saying that by introducing Qi as theta I or D I epsilon is equal to 0 or 1 it's a binary number so if we have a revolute joint epsilon is 0 if we have a prismatic joint it's 1 and epsilon bar is the complement so the Qi is going to be either theta I or D I following the type of the joint so with the joint coordinate vector q1 q2 q3 now we can go to the representation find the relationship between the two that is between X and Q and then compute those differentiation so we can differentiate as you suggested to compute this relationship and this differentiation is going to involve multi variables so X Y Z whatever representation we have may be alpha beta gamma and then Q q1 q2 q3 so this is sort of a vector differentiation we have X 1 that correspond to the first function f 1 this could be just the coordinate X X 2 is y etc and we have all these functions so the jacobian could be computed simply by this differentiation partial differentiation we can compute Delta x1 as the partial derivative of F with respect to q1 and f1 is function of all the Q's so we take the partial derivative with respect to all the Q's and that provides Delta x1 Delta XM in the same way we take the last variable last function differentiate and obtain the relationship now here we have a set of equations how many equations we have can you count them M so we have M equations depending on how many variables M number of joints how many joints we have n so it is basically M equations that is expressed in function of all these variables in variables so do you have another way to write this equation so that is like little bit more compact because I'm not going to write this every time and so we can put it in a matrix form where Delta X 1 mm is a vector Delta X and Delta Q 1 2 Delta Q n is a vector Delta Q and the relation between the two is this matrix so do you see the first column of this matrix how we transform a set of equations into a matrix vector form so how do we do that so do you see first of all what would be the first column come on what is the first column where is it so all of the coefficients of q1 so the first column is the partial derivative of F 1 with respect to q1 2 partial derivative of F M with respect to q1 so basically this is the matrix and this is precisely your Jacobian matrix which is a matrix with M by n it's M by n and it's connecting the Delta qn and of them to your Delta X all right so here is the Jacobian but doing this computation is not simple you have different kind of representations there are different ways of expressing your position and orientation and if you go and analyze this Jacobian you are going to find yourself analyzing both the kinematics the representation that you used and it's very difficult to make sense of what is happening so if we take an example let's take an example where is the example oh yeah by the way this writing I'm writing here a connection between Delta Q and Delta X it is exactly the same matrix that connect Q dot to X dot because we are doing this differentiation if you take it with respect to the time then Q dot is connected to X dot through the same matrix so the element of this Jacobian is the partial derivative of the function I with respect to J and here is the example so it's very simple example two degrees of freedom with in the plane lengths l1 and l2 and our representation is going to be just X&Y so what is X in this case okay let's do the D H parameters and do the frame assignment and the propagation or maybe you can just give me X directly by looking at the figure and this is finished oh good good to know I didn't ask for this okay so what is X you have theta 1 and theta 2 so X will be on this direction so it will be the cosines so it will be L 1 come on cosine theta 1 plus L 2 cosine theta 1 plus theta 2 right because we are taking wither and the signs will give you the Y so here is L 1 cosine 1 plus L 2 cosine 1 plus 2 and Y so in this case to differentiate it's very simple the differentiation gives you this and now you have your matrix and you could see that the first row is minus y the second row is just X so the Jacobian in this case is quite simple and this Jacobian gives you this relation for small displacement Delta Theta you can compute the corresponding displacement Delta X and for small values for velocities in joint space you have velocities at the inter factor actually this matrix has been widely used to control the robot because you can say I'm here I would like to move the end of factor with a little displacement you can take that displacement and generate a trajectory and compute small displacement Delta Theta so you want to find the Delta Theta that correspond to your input Delta X you want to displace the in the factor by little bit you compute the corresponding Delta Theta so how do you do that how do you extract Delta Theta from this equation we take the inverse and using the inverse of this matrix we can compute a delta theta that correspond to Delta X and then we can drive the robot this way and many industrial robots are driven using the inverse of the Jacobian so let's take another robot we examined and see how this extends how the Jacobian becomes a little bit more complicated so this is a six degree of freedom that involves one prismatic joint and you remember the schematic of this robot theta1 theta2 theta3 theta3 actually is d3 and the lost rotations and we have this table of the h-parameters providing us with the description of links with respect to previous ones and using this we computed XYZ for the position these invisible three first component this color is really difficult but this is XY and Z and what is this vector so the direction cosines this is the x axis of the last frame that is the axis attached to the endo factor expressed in the base frame and the second set of ops and this is the y axis and this is the z axis okay so let's differentiate and the first one is very simple you differentiate the first one with respect to cosine 2 Q 1 you get minus y you get sine 1 sine 2 D 3 plus cosine 1 D 2 and and you differentiate with respect to Q 1 the last component that is the Z component you get 0 so now I want you to give me the second column so how do we find the second column of this Jacobian matrix we differentiate X Y Z with respect to Q 2 right so what this should be someone said to sign one I know I need any volunteers to to do the second count so the first variable is X which is it has is it has a function of q2 in sign to write so the derivative of sign to do you remember those derivative what is the derivative of sine to cosine to so it will be the versa because sine 1 cosine 2 D 3 and that's it the second component sine 1 D cosine 2 D 3 and the last one - sign to D 3 so that's correct okay so what about to the third column we derive with respect to D 3 or Q 3 this would be there is only one here so okay logger ah-ha I haven't sent it and go side excellent and with respect to Q 4 plus X Y or Z depend on P R for know so it should be zero zero zero okay well here is the part of the Jacobian related to X dot y dot and Z dot that was easy yes well they are not I mean XY and Z is independent you remember because x y and z were was chosen at the wrist point so when we move the end of factor the point is still fixed so you get zero columns four four five and six okay all right well what seems very simple for x y&z becomes a little bit more challenging for the orientation so let's take r1 r2 r3 you remember r1 r2 r3 are these three vectors representing the direction cosines and if we write the derivative it's going to be our dot one r dot to our dot three will be related to the Q dots by the partial derivative of R 1 with respect to q1 force column two etc to six so let's do that okay five minutes well I doubt mmm well I'm sure I mean you might be able to write a program to do it in fact and still it will be quite complicated to find all these columns but more than that what do we have here is we have a matrix that corresponds to this description and which means that we are computing spending this time to compute a Jacobian for the position and the orientation are represented with Direction cosines and this matrix will have dimensions of we found the first one three by six and the second one nine by six so it will be a twelve by six matrix so if you look at the rank of that matrix it's not square matrix its rank is at most six but you cannot really analyze this matrix and make sense of what is happening you remember there might be configurations that can bring singularities if we are using Euler angles or some minimal representation and that is going to be reflected in the Jacobian so our Jacobian is really really not giving us the properties of the mechanism in terms of the linear velocity and angular velocity rather it is mixing everything up mixing the representation properties with the properties of the mechanism itself so when we have an XP and XR we have we as we saw different representations for XP or for XR we have Cartesians freako cylindrical coordinate for the orientation might we might have a low angles fixed angles Direction cosines alert parameters and if we compute the Jacobian this way that is if we compute the Jacobian for the position from differentiation and from the orientation this resulting Jacobian is going to be depending on the representation and it will have dimensions that accommodate the represent so this is not something that you want to do you are trying to find the Jacobian of your robot you want to find the properties associated with the robot in term of its linear velocities and angular velocities so this is really what we are after we are trying to find how the inter factor moves when we put velocities at the joints when we make small displacement at the joint what is the linear velocity what is the angular velocity so the linear velocity and angular velocity will be related to Q dot and there is a matrix that provides that relationship it is the Jacobian matrix that connect this six by one vector linear velocity is V X V Y VZ angular velocity is Omega X Omega Y Omega Z and the q dot has the number of degrees of freedom and this Jacobian will be sort of like the basic Jacobian that is describing the kinematics independently of your representation what is interesting is this Jacobian J 0 will play a very important role in the kinematics but also in describing the velocities for your representation any representation that you will have can be connected to this Jacobian because any representation of velocities can be connected to linear and angular velocities these angular velocities are instantaneous so there is there are representation of the orientation if we take three three allure angles alpha beta gamma if you take the derivative of those angles alpha dot beta dot gamma dot they are related to Omega they are not equal to Omega the derivative of allure angles are not angular velocities but they are related to the angular velocities by a representation by a various simple model a 3x3 matrix and using those relationships we will be able to describe the Jacobian for any representation and connected to this Jacobian so if we have vertical coordinates the derivative of the spherical coordinate are connected to the linear velocity by a simple matrix that is only function of the representation of spherical coordinates if we have Eller angles we connect them to omega simply by a three by three matrix that only involve alpha beta and gamma so through this relation we can see that we can connect x dot for the position and orientation to V Omega and because V Omega is connected to Q dot then we can have a relationship between the Jacobian associated with this representation and the Jacobian associated with the kinematics so here are some examples if we take other angles so if we take alpha beta gamma and take the derivative we can see that these derivatives of alpha beta gamma are related to Omega by a matrix which is in this case the sine of alpha cosine of beta so it is only function of L or angles if we take Cartesian coordinates the matrix that connect Cartesian coordinates to linear velocity is simply the identity matrix because the derivative of Cartesian coordinates are the linear velocities so using this step of computing the linear and angular velocities we will be able to generalize and find the Jacobian associated with any representation so the Jacobian for a representation ax X is XP and XR will be a Jacobian we call JX for that specific X and this Jacobian will be related to the basic Jacobian by a matrix e where e is going to connect the descriptions of linear and angular representation to V and Omega through this relation so here is an example if we take X dot as related to V through this matrix and X that are related to Omega through this matrix because V and Omega are related to Q dot by J V and J Omega then we will have this relation which leads to this relation that is the part associated with the position representation is simply the Jacobian V associated with linear velocity pre multiplied by ep that is the wishes function only of the representation or ER for the rotations which means if we combine them together we obtain this relation that is the Jacobian associated with other presentation is related to the basic Jacobian by an e matrix that has a diagonal form where EP is the matrix associated with the linear motion and ER is the matrix associated with your representation of the rotations so if we select Cartesian coordinates EP will be what identity matrix so if we have always Cartesian coordinates we have only to worry about ER okay you get this point so now the focus is on J 0 we have to find J sir linear and instantaneous angular positive but first let's examine some ease for the position representation XY and Z we said is the identity matrix are about cylindrical coordinates cylindrical coordinates you get Rho theta and Z and here is the relationship between XY and Z and those coordinates all what you need to do is to relate the differential relationship between Delta X Delta Y Delta Z and those coordinates and you find the matrix and that leads to this matrix for cylindrical coordinates what is nice is yet you go and store this in your library and you have access to all this you don't have to recompute them so for cylindrical coordinates you have this matrix for spherical coordinates you have this matrix so these are things just are there if you want to change your representation you just change your e P now for the other angles we saw that we have another relation that is the 3 by 3 matrix anything special about this matrix you notice anything bad above this matrix no it's fine yeah awesome it's time bed here very good you remember what happened when we had sine beta equal to zero before we had the singularity of the representation it appears again in the velocities so when you get to that location you have a problem because every time beta is equal to some integer times pi you have a problem and that is the problem with a lower angle representation or any minimal representation you get a singularity in your representation that is going to lead to a singularity in your Jacobian associated with that representation okay so we have established this relation that we should not worry about the representation we will find e for each of the representation there is no problem or what we need is to find and establish this relation between V Omega and our Q dots so from now on I you remove the zero that jaco basic Jacobian is the Jacobian when we say Jacobian it is J and this Jacobian is relating the Q dots to your linear and angular velocities and our task is to find V and Omega as a function of all these Q dots okay so let's do that I'm almost certain all of you understand what is the vector of linear velocity but probably there is some confusion when we start propagating and moving and putting multiple rigid bodies and multiple frames so I'm going to go a little bit back and describe how we can compute linear velocities as we propagate our vectors and we go from one link to the next so here is a point and this point is moving with respect to something if it is moving with respect to the origin of frame a we talk about the velocity we talk about the velocity P of that point with respect to a a being the frame now you have to distinguish between this magnitude of the velocity and its direction as a vector and its components and where we are expressing this vector so this vector could be expressed in frame a right we can have its component in frame a but also we can express it in frame B and we can have its component in frame B but still this vector is the vector representing the velocity of frame of point P moving with respect to frame a now if you want we can put it in C so don't be confused about the vector and where we are expressing this vector and that's why I'm putting P /a specifically to show that a measuring the magnitude of the velocity the vector of velocity with respect to frame a now let's have the following situation I have this point that is moving with respect to frame a so I have a vector representing the velocity of point P with respect to frame a this vak now if frame a is moving with respect to another frame and this is the case of link 3 moving with respect to link 2 or whatever there is a velocity of the origin and if this velocity is the velocity of the origin represented representing the velocity with respect to frame B the question is what is the velocity of B with respect to frame B what would be that velocity obviously it's going to be the sum of the velocity in with respect to frame a and the velocity of with it is of the origin with respect to frame B so you just add these two vectors and that is the velocity of the point B with respect to frame B so here we discussed the motion of the frame frame a with respect to B and this motion is uniquely produced by a translation with a velocity V a of the origin of that frame with respect to B now it becomes little more complicated when you have a rotation so suppose you are translating and at the same time you are rotating so we need to introduce the effect of rotation and when we rotate our frame different point will move at different velocities right so if you're rotating an object depending on the axis of rotation this point is moving faster than this point if I'm rotating about this axis actually on this object there will be point that will have zero velocity do you know which points will have zero velocity the points that lies on the axis of rotation so if we take an apple and we rotate it about this egg there will be some points that will not be rotating but actually the points on the outside will be rotating more and what we are concerned with is if I have a vector of instantaneous velocity what are the linear velocities at different point located by a vector P and that is the question we need to answer first do you know the answer not yet three four minutes you will know dance okay so we have an axis of rotation we have all these fixed points so we're just doing a pure rotation so these points are fixed and let's have a schematic simpler to see so obviously the points closer to the axes are going to move at slower velocities let's pick a point P and we've concerned with the velocity at this point the angular velocity measured about this axis is Omega and Omega itself is representing the vector so it's vector and magnitude now what is what is the magnitude of VP and its value given that we know its location P and we know Omega well first of all we need sort of like locate we need to locate P with respect to some frame and we need to make sure that this frame is not moving so we put our frame on the axis so the origin of the frame is fixed with respect to this rotation and we locate it with a vector P okay now we have everything we need to find VP so let's see the magnitude of VP is going to increase with the magnitude of Omega that may sets we're going to increase with the distance from the center which is the vector P multiplied by the sine of the angle Phi between the two axes right the further away from the center the largest the VP is going to be and we can notice also that VP is orthogonal to both Omega axis of rotation so the velocities orthogonal to that and also orthogonal to the location the vector locating that point because we are taking the point from the fixed point so that means with all of these that means what VP is equal to when you have like three vectors that are orthogonal and you have the sine then you will have cross cross what product between the two which one is first so VP is equal to cross product of what by what well you have two choices I said well you will know in four minutes and it is almost four minutes so here is the answer where is the answer here so it's Omega cross P it's not P cross Omega by the way we will see P cross cross F to produce the torque there is a dual relation so here we have velocity you have Omega you have the point now you can imagine on the same figure that instead of Omega implying a torque and the result will be a force and there is a similar relationship between the two and in that case the relationship will be involving the distance first times the force and to produce the talk okay we will see that later probably on Monday though now we have this relation linear velocity is the angular velocity cross the distance it is represented as a vector marble here what we need to do in order to go to the matrices we need to introduce a matrix representation that is instead of writing that a vector representation I need to write this in a matrix form so how can we so B is a vector now and C is a vector and a is the vector transformed into this matrix operator that does for you the a cross B so we need to find this a hat the operator that is equivalent to a cross are you familiar with this the cross-product operator so this is essentially a skew matrix a skew symmetric matrix whose diagonals are 0 and which is formed directly from the vector a ax minus y z-component and it is non-symmetric matrix and you start with a vector a if you take this vector and put it in this matrix then you get the matrix that will operate on the vector to produce the cross product of that vector and that will be your resulting linear velocity in the case of Omega cross P okay so we represent a hat as this operator alright so in our case VP is Omega hat cross the vector P and Omega is this matrix so this is something you might need to remember and just write down and put it somewhere in your notes okay let's use it so now that we are going to combine linear and angular motion the new velocity there in that frame is going to involve this Omega cross P and this will be added to that vectors that is you have the velocity of the point coming from the linear motion the velocity of the origin and the velocity due to this rotation and this is omega cross the vector locating the point now be careful about where you express your vectors because if we say we are going to compute those quantities to express them in frame a the result has to be expressed of each term should be expressed in frame a so if we have the vector P expressed in frame B we have to transform it to frame a and then do the multiplication and we have to Express is in frame a so that the whole result is finally in frame a so you have to make sure that each term or each vector is expressed in the same frame it doesn't matter which frame it is but it has to be in the same frame and if you need a then everything has to be transformed to a Wow were you in trust good all right we're ready now so we're going to take those concepts and now apply them to our mechanism and I think you we're going to skip the movie segment we will see it on Monday so the way we're going to proceed is by taking those velocities from one frame and taking the those velocities and propagating them as we move from one link to the other until reaching the velocity at V and Omega so this propagation is going to involve two velocities the linear velocity V and the angular velocity Omega alright and when we get there we will have the Jacobian that will be in the total velocity that is when co-teach V and Omega you will have implicitly the expression of the Jacobian multiplied by theta which you can extract so let's start with the linear velocity so here let's take a vector V I and Omega I that is describing the velocities of this origin of frame I and the rotation of the frame with respect to Omega I that is representing the instantaneous velocity and let's go to the next frame where we have a new velocity linear velocity I plus one and the new is Santini's angular velocity Omega plus one so what is the resulting linear velocity VI plus one as related to the velocity VI so first of all don't look at your notes and let's see if you have the intuition about it so what would be VI plus 1 as a function of VI is it smaller larger so VI plus 1 is going to be equal to come on is it related to VI by any chance so there is a translation everything is moving with VI so the VI plus one is going to har VI in there without doubt but there are two terms that will appear and the first one is Omega I cross P I plus one you didn't tell me why but maybe now I'm showing you what is the term tell me why yes movement in this in this frame is going to add the linear the last you know so the linear velocity is computed at the origin of that frame right so the Omega involved will not be that Omega I plus one the large Omega I plus one this instantaneous rotation here if this is a revolute joint this will be rotating so this rotation will not change the velocity here what will change the velocity here is the rotation Omega I of this point by located by this vector so this would be Omega I which is here cross this vector locating this point so this is the first term now anyone can explain the second term yes prismatic joints so the second term only appears if that joint the z-axis was not a revolute joint but was a prismatic joint that is translating along the z axis so along Z I plus 1 and the magnitude will be the D dot I plus 1 D is a variable in this case for the prismatic joint so this is the D dot I plus 1 Zi plus 1 is the local velocity in a frame I plus 1 of that point and the Omega I cross P I is the contribution of the rotation of the frame and plus all the translation that were happening before I had a question it looks like the prismatic is on the Zi and it is always on the CI plus 1 so this will appear only if this was not revolute it was prismatic and it is translating along the Zi plus 1 they solve it the two-headed arrow over fly yeah I'm not representing D dots here okay okay now what about angular velocities we are concerned with what is Omega I plus 1 as a function of Omega I so if the joint I plus 1 was prismatic so there is no rotation between the two what would be Omega I plus 1 which will be identical to my guy if it is revolute then there will be the Omega I plus one that will be added and Omega I plus one is simply the dot of theta I plus one along Z I plus one so to propagate from one frame to the next what we need to do is to take this relate these two relations and go from the base the fixed base where V zero is zero it's attached to the ground and Omega zero is zero there is no motion and propagate to the end and when we reach the end we are going to have the V associated with the end effector the Omega associated with the end effector now in those relations what do you see you see that you are using d dot and theta log and the kinematics obviously the z axis so that means we are going to be able to compute the total velocities V and Omega in the factor as a function of the theta dot and the D dots all the Q dots yes these two frames rotating independently like is there a revolute joint in an eye and a revolute joint in I plus 1 there is a revolute ja there could be a gyro Vaalu joint in frame Omega in I plus 1 or a prismatic joint and depending on the nature of the joint you get either D dot or you get the Omega I okay so when you do this computation you are going to have an expression of the total linear velocities total angular velocities as a function of the D dots or the theta dot so for joint 1v1 and Omega 1 can be expressed in frame 1 and this is going to be sort of using this relation to compute the Omega I plus 1 from the initial frame this would be the expression for I plus 1 and this will takes us because we are expressing all these in the frame I plus 1 we are going to find the total expression of the velocities in the frame n as we propagate and once you reach the final velocities in frame n you can transform them back to the base frame and that will give you the total velocities at the end effector in frame 0 yes when you get me theta I theta the previous slide yes which was very well either Omega I plus 1 equals to theta I plus 1 where you get that theta I this one ok so Omega I plus 1 is the angular velocity associated with the motion of joint I plus 1 if the joint is revolute and if the joint is revolute the velocity vector is about the vector Z I plus 1 so this is a unit vector and the magnitude of that velocity is proportional to theta dot I plus 1 so this is actually joint I plus 1 if you take the derivative you remember we aligned the Omega the rotations of each revolute joint along the z axis so that is where it comes from yes that's why named di plus 1 enters defined along the z I always say that the dot product right what do you mean it's not that product that Zi plus one that's not adding anything is it at this point this is uh no this is a portion it is not that product it is it is preparation now theta that is a scaler and the Zi is a vector so no this is not a dot product this is just a scalar multiplication all right other questions yes expressed in terms of the four nets fry plus one if the forward kinematics you're gonna get the universe okay so the algorithm that we are using here for the propagation is taking the velocities and propagating to the end so we compute that in frame N and at the end if we need them in frame zero we do the transformation back to frame 0 this is the the way this algorithm is done you can do the competition backward and you compute everything in frame zero or you actually the most efficient place to do this competition you know we're not at frame 0 or at frame and it is in the middle because the those transformations becomes more and more complicated if you go from the base to the end in the middle the transformations are simpler so if you transform just to the middle as you propagate you will get the most efficient form but in this algorithm we are showing that if we use our I I plus 1 we will end up with velocities Omega N and VN that will give us the total linear and angular velocity in frame N and if we need them in frame and that is fine otherwise we transform them here now where is the Jacobian in all of this anyone can see the Jacobian I just showed you this recursive propagation we forward propagation computing the velocities but where is the Jacobian now you cannot see it well it is there I have to find it like you have to go inside so you write out the relationships you do the propagation what do we need to extract from those expressions to find the Jacobian you need to get the theta dot I plus 1 and the D dot I plus 1 out and everything else is going to give you these columns of the Jacobian so you cannot really see it but it is there now velocity propagation is really nice numerically but it doesn't give you any idea about the structure about the contribution of the joint about your kinematics and it is not really the best way to analyze your mechanism what you're going to do we're going to in fact analyze and work with this explicit form of the Jacobian that would allow us to really look at the mechanism and see immediately the Jacobian matrix and its columns and its structure so next time when we analyze the explicit Jacobian you will be able immediately to look at the mechanism so now you see the Stanford Shimon are you look at Stanford channel arm and you see this is the first column this is the second column this is the third column this is how how the Jacobian is going to be before we leave don't don't don't we still have few more minutes just there is an example this example is useful if you if you went really to understand little better the velocity propagation and this example is done over a very simple three degree of freedom mechanism we know the answer of the Jacobian you know the Jacobian for this is you write theta one you can write x y and z which is l1 cosine theta 1 + l2 cosine 1 and 2 and l3 cosine 1 2 3 + sine and you do the differentiation and you find your Jacobian at the same time if you do it through velocity propagation using these relations you go to the first propagation the base is at 0 you compute the linear velocity at p2 you compute it you get this expression you compute the velocity of 2 so just take a look at this example and once you completed the propagation you find that your Omega vector is the sum of these velocities which means essentially your total Omega is coming from these three rotations one two and three contributes to the total rotations of your end effector and the linear velocities are going to be this would be the Jacobian so if we extract theta dot one theta dot 2 out this is your Jacobian for the position and this is your Omega so this is the Jacobian for the Omega now this all this whole computation turned out to be well through but this numerical propagation but if you do it through the analysis what you're going to find is for J Omega you see here what do we have 0 0 1 0 0 1 0 0 1 what is this 0 0 1 this is the Z vector all the omegas are rotating about the z vector which means it's an essentially the Jacobian associated with a angular motion is simply the zi vectors associated with the joint angles what is this well I mean this could be directly computed from the partial derivatives or it could be computed from this cross product of those joint angle rotations with the point locating that point so we will see that structure next time and then you can see much better the properties of the Jacobian as relates to the kinematics of the robot I will see you on Monday |
Lecture_Collection_Introduction_to_Robotics | Lecture_3_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's uh get started so today movie segment is about a special actuator uh probably you saw this on the first lecture but we will see some more details so this is u a flexible actuat that comes from Toshiba and this was developed in the early '90s it was presented at 1991 video proceedings this new actuator is made of fi reinforced Rubber and is driven pneumatically or hydraulically it has three degrees of freedom pitch Y and stretch which are adequate for robot mechanisms such as fingers arms or legs the actuator has three internal Chambers and the pressure in each can be controlled independently through flexible tubes the rubber is circularly reinforced with fiber to resist deformation in the radial Direction the actuator can be flexed in every direction by controlling the pressure in each chamber I've developed actuators ranging in size from 1 mm to 20 mm in diameter this is the 4 mm actuator the design is easily miniaturized because of its simple structure this is a modified version the rubber is reinforced spirally with fiber so that rotational movement is possible we can apply these flexible microactuators to at robot manipulators by connecting them serially we get an arm with many degrees of freedom and snake likee movements this is a prototype consisting of two actuators and a mini gripper it has seven degrees of freedom including the gripper it can accomplish delicate tasks which could be handled only with great difficulty by conventional robots constructing miniature robot manipulators is easy because the actuators also act as the robot structure on the other hand combining the actuators in parallel results in a multi-fingered robot hand they form a dextrous hand with a delicate touch this prototype consists of four actuators each 12 mm in diameter and it has 12° of freedom is able to handle fragile and complicated work with ease because the actuator is deform to suit the shape of the workpiece itself a bolt is easily tightened with only rough settings of the position and orientation of the hand because the actuators have such good compli miniature robots with a soft touch and no conventional links can be created using these actuators we foresee the use of flexible microactuators so what do you think what would be the advantages of using pneumatic in this way yes probably safer for a lot more objects safer you said oh safer so sa safety is a very very important uh uh aspect of the design of a robot especially if the robot is going to interact with humans and you you really don't want this robot to just go crazy and hit and uh make a large impact So Soft actuation using pneumatic is is very good because it uh it basically it's compliant right now another implication of the fact that you are using pneumatic is is the structure of the robot is going to be lighter because if you think about operating these fingers with motors or if you like think about an arm with motors you need to carry the motors you need to put gears you need a lot of structure to to to handle it so definitely this is lighter safer more flexible compliant all of that any disadvantage you can see yes hard to control yeah basically well I mean um depends what you want to achieve but uh uh you canot expect to achieve all the task what kind of task you cannot control with this type of actuation precise motion precise motion or fast Dynamics if you want to change directions and uh because [Applause] what is the problem with the with the actuators yes frequency so yeah I mean basically the the response of the system is slow because you are using uh air pressure and you cannot push the air pressure to a point where you can really get fast Dynamics uh well later on in the quarter we will see uh a concept that combines this idea of using pneumatic which is light to carry and uh would result into a nice uh light structure combine it with uh other type of actuation to bring hybrid actuation in a way that combines both the advantages of the light structure and the fast dynamics that we need to achieve uh all the different tasks that would require the robot to respond quickly yes is there any good wa and you close the loop on the movement like you don't actually know where you're the last sequence you would see uh the robot was like turning yeah okay well okay uh anyone would like to answer this question I'm sure you you have an idea but but uh I mean uh this is really not fundamental to the the robot design it is more on the fact that we have no external feedback or no no touch or we are not using the information about the touch to realize that we already left uh the that context but uh this is actually uh something that you can add on top of the design uh to like with what kind of sensor you would use a pressure sensor uh touch sensor yeah yeah you mean like you you want to know if you are touching or not I mean pressure would be I guess um like for rotary joint you got an encoder you know exactly like how far you are um but with something like that like curvature is going to change if I I want it's touching and everything like yeah you could put something that exactly tells you the position of the end so we we can put a sensor at the end uh that uh is look localized at the tip of the finger and then we can feel whether the sensor is is on or off that is just touch but if you want more information about the pressure you need a sensor that tactile sensor that would measure but then then the problem I mean now now we we come into a much harder problem which is the fact that if you're holding something let's let's imagine with your two fingers you're holding something and uh there is always slip so you need to measure uh the slip so that you you can apply larger pressure and to do that you need the sort of uh Dynamic tactile sensing so there has been a lot of work actually in the group of Mark katuski uh uh a lot of research on tactile sensing Dynamic tactile sensing and uh also the idea of using uh uh pressure is a very good idea because in fact by measuring the pressure and the the control pressure If there is a difference between what you are expecting and what you see you will be able to deduce some information about the contact there was a comment here I say you can put uh stretch sensors on the surface and use those to find the position MH now if you instrument the environment uh obviously you will be able to get a lot of information about the environment but that is costly so you want more to put the sensors on the robot but there is another type of sensor that will give you more information about the environment that uh especially uh about whether you are in I mean close or not or I mean like to to localize and see where things are to see where things are to see to see what what do we Vision put couple of cameras and you would see where where you are with respect to the world and uh so you have a mechanism you have the controllers but really you need to close the loop but to close the loop you need perception and perception could use sensors in the environment sensor uh external that are monitoring the environment or instrumentation on the robot itself anyway this uh cute design actually was pursued for a couple of years they built even a a a big robot that is walking with those legs and I'm not sure if you will see it but uh uh then uh this project just uh I mean didn't go any further uh it is like many of the designs that make use of uh uh air pressure only you end up with really limitations a lot of limitations on the use of um on on the ability of the robot to perform tasks and in fact there is a lot of work today in this area uh that is uh with artificial muscles to create uh faster uh muscles that use air pressure and uh there are uh many different solution that will push this a little further but still you have limitations and as I said we will discuss a little bit more about those issues of design especially in the context of safety because safety really really is uh is becoming a very important aspect in robot design because uh we have been working mostly with robot uh with industrial robots so industrial robots are working alone or working with with parts and objects you don't really worry too much if there is an accident just between uh well uh the robot and that environment but if you are going to work with humans you really have to make sure that there is no danger to the human and that is really a challenging problem so we will come back to this later any other comment about this okay all right so let's uh go back to uh the lecture uh so last time we we saw this tool we call homogeneous transform and the homogeneous transform uh really uh has several interpretations or can fulfill uh several functions uh and the first one of them is uh the fact that a transformation like this allow us to describe the frame so frame B is described with respect to frame a given this transformation so if I know the homogeneous transformation between b and a that is this 4x4 T Matrix that describes a uh B in with a relation to a then I have a description of this Frame B and this description contains the rotation of the axis of frame B with respect to a and the location of the origin of frame B with respect to frame a now we saw also that there is another uh role this homogeneous transformation can play and the second role or third role whatever uh do you have an idea what what can we use this transformation for what can we what can this transformation help us do yes operations operations that is uh we can interpret the transformation as an operator that is acting on a vector and changing this Vector rotating the vector or rotating and translating that Vector so this is a second interpretation the transformation as an operator or one more so if you have a vector in space describe with respect to some frame B and you want it descript ion in a different frame a can you use this transformation right so this is uh the mapping what we call mapping that is we take a description of a vector p in a frame well B and we map it to a description in frame a so the vector P this green Vector over there is now now the red Vector that is this one describing this point in frame a and you can see here we have two different vectors if there was no translation then basically it will be the same Vector with two different set of components and as you said we have also the description of uh the homogeneous transformation as an operator that is we take a vector P1 and change it to a vector P2 so the vector P1 is now P2 after this translation so these different roles uh of the homogeneous transformation use the same mathematics but the application the interpretation is going to be different and uh we have to pay attention to the way we apply by that definition so the next question that we have to address is how we now use so if you remember when I talked about homogeneous transformation I said by building this Matrix 4x4 Matrix in a higher dimensional space we are able to have a homogeneous relation between vectors so this property is going to help us propagate and go from one frame to another and have descriptions that uh are related by the individual Transformations between frames so here is an example you have uh this camera monitoring the environment and uh here in fact you have uh a robot a mobile manipulator this is Romeo that is that is uh did I introduce Romeo and Juliet to you no not yet okay well maybe we'll have a chance little later so Romeo is essentially a mobile platform homic mobile platform with with an arm and it allows you to move in the environment and manipulate the environment but because of the platform uh this is done uh everywhere in the world not like when you have just one arm fixed on a table where you bring material to be proc here you are able to explore the human environment and uh in fact uh the robot is moving and its location is always difficult so the question is how can we for instance locate this robot so you need to find uh actually the transformation between the base frame of the robot with respect to the camera this is little bit difficult uh unless you are able to well to find elements and different things so suppose the camera is monitoring the end of factor so you have the end effect Factor here and you can see it so if you have this relation that is in a given frame you are able to see and identify the end factor which is let's say this Frame then through the other path that is going from here to the base through the those transformations to the end of factor you have another path you know this and you know this then you can compute this and we need to be able to propagate and resolve this transform equation so this comes everywhere yeah well Romeo is capable of ironing I will show you maybe later if we have a chance little time in the lecture uh so again where you're ironing where is the base and you have this Loop in the environment if you are observing the environment and you need to be able to go through the Transformations from the base to the uh end effect Factor you have always the forward kinematics through all the Transformations between links but uh obviously with the ground you have slippage uh and you cannot determine exactly the relationship between a fixed camera and the location of the base so first of all how we we combine Transformations let's consider two frames and consider that now we're going to introduce a third frame uh C so we have a transformation from C to B we have a transformation from B to a and obviously I'm interested in the transformation from C to a The Total Transformation I would like to combine these two Transformations so if we have a vector vector described in C that is we have a point P described in uh frame C the question is what is the description of this Vector in frame a well the result is obvious I mean probably you already get it we are going to multiply these two Transformations and to prove it let's first compute what is the transformation to frame b p in B is simply going to be obtained by the homogeneous transformation from C to B right now if we write the same thing for this point in B we can go to a through the homogeneous transformation B to a now if we substitute this with the expression ex that uses C we will obtain this relation that is the vector in C is transformed into the description in frame a using this form which means that essentially we are going from description in C to a that is from C to a and that means essentially the transformation corresponding to these two frames successive frames is going to be the product of the matrices the homogeneous transformation C B and B to a and you can see this has a nice form you you just eliminate the B and you're going from C to a with this notation so that is really the advantage of com I mean homogeneous transformation is that that you have a matrix and then when you have multiple frames all what you need to do is to multiply these matrices so now if we multiply C to b b to a this is what we will obtain we will see here that the structure of this transformation is going to maintain the structure with respect to the rotations so we have the same properties of rotation it is C to a a through CB and B to a and this Vector which is the origin the vector locating the origin of C with the respect to a well it is essentially computed by locating the origin of C with respect to B rotating it to the right frame and then adding the offset due to the origin of B with respect to a so I mean the the logic is very simple it's Vector computation but again when you do the multiplication it is automatically taking into account and you get those results so now that we have this relation that would allow us to combine two rotations we can go and do the transform equation the transform equation is this uh um equation that is going to let us extract some information given that we know some other information about relationship between frames that is the robot base is fixed at a given instant the end Factor position could be computed with respect to the space now if we know where the end effect factor and if we are able to have that total relation then we can compute where the base is and in this relation there is some very intuitive property about this fact that if you start from a and go all around walk all around and come back to a you should obtain a transformation a homogeneous transformation that is equal to Identity so we have an identity transformation when we walk from a and go all around and come back to a and using this we can say essentially from a to d d to C C to b b to a if we multiply them be careful with the order you're going to eliminate all of these and you get a to a which is identity so now I don't know what is missing let's say we are missing D2 see then you multiply by the inverse of this from one side and the inverses of the others and eliminate and compute the transformation that you need that is here we have four Transformations one of them is unknown then you can extract it and find it in the case of computing A to B simply you are going to find that you go from A to B by a to d d to C and C to B okay so just and don't be confused I mean sometimes the Transformations are not written in the right direction so you will use the inverse but you have always you have to walk in the same direction and then you can say identity a to d and then you can compute any element that is unknown okay well we're done with Transformations any questions about Transformations good well I think you have a homework about that so hopefully you will find out if you have questions all right well now we are then at uh the last point in special descriptions and this is uh the representations so what do we mean by High representations the questions that we're concerned with is this question we have this end factor and the end factor is really the purpose of the whole manipulation problem that is we are concerned with how to position this IND factor in space and how to move it to some location so we need to say well our end of factor is at some location X Y and Z and and it has some orientation now it's not enough to say well I have a homogeneous because this description we we know we know the position and orientation it is embedded in the homogeneous transformation we know the relationship from the Endor to the base frame through T and you're going to find that relation when we finish the forward kinematics when we uh uh complete the forward kinematics you're going to be able to say my end Factor transformation to with respect to the base is given by T and this T will come from the description of the manipulator so the link length the angles uh of tilting and and characteristics of the of the links and also it will be function of what what other parameters will be involved in t so here is the manipulator it's moving at this configuration how can I compute this position and orientation I'm going to use this length and the Angles and also I'm going to use come on I'm tired here left this the joint The Joint angles The Joint angles I need the joint angles so what is variable what is changing as I'm I'm moving so T is going to be function of coefficient that are constant depending on this manipulator and the joint angles so T will vary with joint angles right so the configuration of the robot will affect where this ector is going to be so we will be able to find the T as a function of the joint angles and it will be a matrix 4x4 and this Matrix will contain for each configuration the description of the homogeneous transformation okay so at a given configuration we measure the encoders we read the encoders and then can we can compute T and say my homogeneous trans formation for the end factor is this now the question is where where is the position of the end factor and what orientation it the in Factor has the question is important because I'm not only concerned about positioning the end of factor I'm going later to move it following a trajectory I need to think about the velocity so I need some description of coordinate that would give me like coordinate X then I can compute x dot eventually acceleration so I need a set of coordinates okay so what we need to do then is to extract from T the position description and the orientation description and say I have a description something like I don't know I have the position and I have the orientation I have a set of parameters X XP for the position and XR for the orientation right because I cannot take T and take the derivative of T and just uh say well I'm going to build a trajectory directly with t you understand all right so where can we find XP what is the information where is the information about XP in t the first three element of the last column that's what you said correct because this is going to give you basically this Vector this Vector is in the transformation on the last column of T so you can compute this point and that would be a description X Y and Z of the in Factor position where is the orientation at least where I can find information about the orientation int yeah so it will be the first three columns corresponding to this R Matrix the rotation Matrix okay now how should I represent X because you said X Y and Z but there are other possible representations so for the position I could say I take that position and it is represented by X Y and Z so this is the cartisian representation but I can select these angles Theta and the projection of that on the plane XY and come up with a cylindrical representation or I take that other angle and come up with a spherical representation so there are several way ways of representing that position and most of the time we will use cartisian coordinates but in cases like if you if you have a uh dexterous manipulation with a a tool and you want to move in this direction it might be more advantageous to use cylindrical coordinates because your task is aligned directly with that vector and you can just extend the vector and move and the angles are fixed so you control them and just you move along that Vector so other representations are also very useful in different tasks for different tasks okay well still in here we're talking about X Y and Z row Theta Z or Theta fi always three parameters three independent parameters so not much of fun simple the fun starts when we think about rotations and that is where most of the difficulties with uh the kinematics uh of robot systems lies in fact we're going to start to see some aspect of those problems but those problems will carry on all the time as we we start to consider uh instantaneous rotations instantaneous accelerations and their relations to representations the space of rotation is really different from uh the space of uh linear of space of linear motion that is the space where we describe a position and we're going to little by little see this problem so the first complete representation that contains all the information about rotation is our rotation Matrix so in this rotation Matrix everything is there about the orientation I have an object held in end Factor basically there is a frame X Y and Z that is going to rotate with this object and if I take X Y and Z frame and take its component with respect to reference frame that will be my total rotation so from T the homogeneous transformation I can extract this rotation Matrix and in wish I see that R1 this is a vector this is these are the component R1 R2 R3 are these three vectors so R1 is what in this Frame if I take the X Co axis R1 represents what what is the definition of the basic definition of the rotation Matrix we say that the First Column is the component of X on the base frame so R1 is really the component of X R2 is the component of Y and R3 component of Z okay now I can build a representation with this I can just say my XR this x that is going to represent the rotations and the orientation of the end of vector is simply the concatenation of R1 R2 R3 I take the three vectors and put them one on top of the other R1 R2 R3 it's a long Vector but this Vector actually contains everything you need about the orientation now obviously uh we're overdoing it so do we need nine parameters to represent the orientation obviously there is how many degrees of freedom we need how many parameters the minimal number of parameters would be three I mean we have three degrees of freedom in orientation but here we have nine so how come how come we have nine well each one of those vectors is unit Vector so the third component is already predetermined that okay so we have sort of three relations of normalization associated with R1 R2 R3 so this is three constraints good we need more 9 - 3 was still six I need three more yes those vectors is also perpendicular uhhuh okay so R1 R2 one constraints R2 R3 one constraints and R3 R1 3 six good let's check yeah you're right so six and you have nine basically you have yet yeah actually 3° of real now this is a redundant represent a redundant in the sense that the parameters we are using are not independent and that creates a problem that is you cannot say so what is the problem with that I don't know if you realize the problem think about motion I'm going to I'm going to take this and rotate it like this I mean I'm going to just put some water in the cup so you have a motion so you have some initial XR you have some final XR and you want to go from one to the next how can you create a trajectory you see the problem so okay there are multiple Solutions or well let's think about another uh the the simpler problem which is I'm going to move from A to B A is defined by x a XY z a and here XYZ B how can I move from this to this one so I X is equal uh 10 and XB x a is equal 10 and XB is equal 22 so slope and TR I can I can interpolate and every every point is valid okay now let's interpolate XR one I mean the first configuration and the final configuration if you do the interpolation you will violate those constraints if you just do linear uh interpolation between the two you cannot take that they are not independent so you have to monitor what is happening so it is it's very difficult to just work with this D directly as as X Y and Z what is happening with the rotation is that you are moving over a sphere s SO3 in that space whereas XYZ are moving in the free space and you have to deal with this these constraints okay so what other representations we have oh come on yeah you know them so if we don't use the vectors associated with the frames what can we say about the orientation there are these angles right what angles o angles other angles exactly other angles are very very useful uh now there are many different ways in fact of talking about an angular representation the three angle representations I mean you can count maybe 24 of them we distinguish between other angles the fixed angles that we use in aviation and uh each of them has like sort of 12 different representation and those representation come from the way we obtain these angles so you have a frame and you're going to try to find the frame B and the relationship between the two uh using three angles so what do you do you have many options you can go to this axis and rotate about this axis first and then once you rotate about this axis with some angle then you can go and rotate about this one and then after you can come back to the same axis and do the rotation or you go to a third axis and so you have uh and maybe you can start from the Y AIS first and so that's why you have a lot of different ways of doing it all right but you end up always with three angles but you you need to know which axis you're rotating about and whether after the first rotation you rotated about the first axis there is no problem you get some intermediate configuration but next you can rotate about the new axis that resulting from the first rotation so this is like relative rotation with respect to the frame created after the first rotation or you can maintain your rotations three rotations about fixed axis so we distinguish between fixed rotations fixed angle rotations and that give us 12 and relative uh rotations give us another 12 erer angles are the relative ones and the fixed ones are uh the what we call fixed angle rotations okay so let's start from the beginning I'm going to go to this configuration by first putting the two frames together okay all right let's rotate I'm going to rotate about the x axis with a some angle this is the first rotation the new frame that result so the idea is I'm going to rotate from a identical and then I'm going to get a first frame and then I will do another rotation and a third rotation to reach the final B the B which is in here okay so if the next rotation is done about the relative next axis y so from that b Prime I'm going to do another rotation about B Prime then this is sort of L angle representation and I have 12 set depending on the selection of axis if we do this next rotation on the blue axis the fixed one here I will obtain an intermediate one but when I continue this will give me the fixed angle rotations and you have 12 set now we will see that in fact every representation here has an equivalent representation here so in total we do not have 24 different representations we have only 12 that could be represented obtained from relative Axis or fixed angle axis okay so in total you have only 12 all right let's take an example I'm going to take the rotation Z YX which means we will start with a rotation about the z-axis then y then X and I'm talking about ER angles which means relative rotations so here is the first rotation you're rotating about the z-axis and the XP XB Prime and YB Prime are in the same plane as x a and ya a and you have an angle Alpha between those axes so the next rotation will take place about what about y so and this is the new y the new y that resulted from the first rotation and we will call it beta angle and now this rotation is in the plane XB Prime and ZB Prime right and the final rotation will be about X and it will be of an angle gamma and it will be in the plane y b Prime ZB Prime and that will take us to B now I need to compute this transformation and I need to compute what is the rotation from frame B Prime to frame a the transformation from frame bble Prime to B Prime Prime and the final transformation from B to B Double Prime and if I have those Transformations If I multiply them out I will find the total transformation good so that is what we want we want to compute this this this multiply them out you get B to a well maybe before so what was the last transformation was X rotation about the X with an angle of gamma so I'm going to write rotation about x with an angle of gamma rotation the one before was rotation about y with the angle beta and rotation about Z with an angle Alpha you have your Transformations so you take this so this is what this is one0 cosine the angle gamma minus cosine sin cosine and you you have your transformation you compute it and you find your rotation so you get this transformation as a function of alpha beta and gamma and now for a given orientation you know the position in rotation space you say it's Alpha Beta gamma now if if we do it with fixed angles essentially we are going to do a rotation about X about y about Z but those angles are uh done I mean these rotations are done about the fixed axis in fact this is what we call the roll pitch and ya and these are used in uh aviations because you're you're doing small angle rotations in in general and these are very intuitive to perceive what rotation you have made and the computation of these comes to be very simple because if you think about the rotation x with the angle gamma that first rotation it if you have a vector v this Vector is transformed by the rotation about x with this angle you you have an operator changing the vector v to the vector RX gamma V and with the next transformation you take the result and transform it with rotation about uh Y and the last one is going to apply the lust transformation and when you put all these Transformations you see that the total transformation is given by the product of X gamma r y beta and RZ Alpha so these operators that we saw before are very useful and in fact when we look at those Transformations essentially we are going to obtain the transformation directly from rotations about directly those axis so those operators are very simple because they are done about the X Y and Z and every one of them is like the operator about Z is a rotation here 0 01 about the z- axis with cosine minus s cosine the operator about Y is rotation about the Y AIS with this angle and this is the rotation about X so if we do this multiplication you will obtain a matrix that is only function of Alpha Beta gamma these I'm not showing the the terms because they are a little big but basically I'm showing just those element and you can now do the product and find that Matrix and now your rotation Matrix is expressed as a function of Alpha Beta gamma now let's go back to the beginning how do we compute the end of factor position and orientation how do we measure it we said we measure the encoders we use the forward kinematics to compute the homogeneous transformation and we know the rotation Matrix numerically now what we are saying is this Matrix is equivalent to this Matrix computed with those three rotations so how can I find those rotations what is Alpha what is beta what is gamma this is really the problem the problem is I need to be able to compute these values and uh to compute these values I need to find the inverse that is given the value of the Matrix I need to compute those values oh this is another example here so here we have zyx the Matrix looks like this if you have z y z you get this other Matrix so these the element that are very simple so you know this element cosine beta you have the numerical value and you identify cosine beta and you can invert beta and find what is beta for that uh for that specific configuration okay so if we take zyz representation what is the angle here L angles Z I'm sorry Z YX so what is alpha beta and gamma don't check your have to remove this can you think about it from here can you see it what is the which which one is non zero I mean I I'm I'm sure most of you saw the answer already but uh can you see it really so this is a rotation we went to the red frame from the blue initially with a rotation about which axis the x-axis of an angle of Plus 90 de yeah good you see it everyone no confusion good okay as I said if we take fixed angles and we take other angles I just I'm writing them together now the fixed angles this is the relation we obtain the ler angles this is the relation and now if you look at the two you can see that the rotation about ER rotation and fixed rotations are identical that is the XYZ with those rotation gamma beta Alpha is equal to the Z Prime y Prime X Prime with Alpha Beta gamma so for each rotation you have a corresponding rotation uh in the 12 sets of fixed and uh ear angles so gamma Al and beta are the same for or they different no the same in the same equation have to be the same all right so now we come to the difficult problem the problem I mentioned earlier which is that how you obtain this Alpha Beta gamma how you obtain your representation from your measurement so you are given the rotation Matrix numerically from your forward kinematics so you know R from B to a and the question is what is Alpha Beta gamma well I mean essentially I need to identify each element from B to a so I know R11 R12 r13 I need to identify them to the rotation Matrix so now you have the full Matrix and uh you have 5 minutes so how do we do that I can help you if you need some all right obviously the the the these terms are little bit too complicated this one and this one so let's see I have Alpha Beta here Alpha Beta here and beta here so I can I could use this right so how can we compute um beta if we Square this and square this cosine square and sin Square add to one and I will get cosine beta square right so so I can get cosine beta from the square root of this element and this element right now we're going to use a function that we call uh the inverse of the tangent two that takes two argument it takes the S and cosine to compute the tangent and that gives you uh all the angles within uh the the their area so where can we find the sign of beta well we have the sign of beta here in this element element 31 it is minus so it has to be minus so if I take this function and I place this here and here I can obtain beta now given that you obtain beta how can you compute Alpha well you go back to those element one one and one 2 one and you know beta so you can compute the cosine and sign the problem that you're going to have is this problem which is that if your cosine beta becomes equal to zero if cosine beta becomes equal to zero you're going to have a problem and this problem is that you cannot divide anymore by zero so you have some end determination and this leads to a singularity we call it Singularity of the representation we will see later kinematic singularities real singularities the robot is moving and when it reaches this configuration the robot can canot move immediately in this direction it's locked in this direction like this this is a kinematic Singularity and we will see this when we take the derivative of uh the forward kinematics but in here this representation is fictitious it's only you who selected this representation that you selected a mathematical model that fails at some configurations so in this case when cosine beta becomes zero you can only determine the sum of Alpha and Gamma or Alpha minus gamma and the reason is when beta is zero essentially the Z axis are aligned and you have a rotation in the same plane you do a rotation of alpha then zero and Gamma so you are not able to distinguish between the Alpha and the gamma I think I have an example so this is what happened if cosine beta is zero with the sign the S of beta positive or negative the rotation is done in the same plane and all what you have is the alpha minus gamma at that location or the alpha plus gamma if the beta is negative so what does it mean and why it is a problem I mean obviously you see it mathematically but why this is a problem okay so you are going to use your parameters Alpha and beta and you're trying to identify Alpha and beta As you move because you are measuring and you and you reach this configuration you have an and determination and at that moment you are not able to compute the velocity ass iated with Alpha as we will see that the Jacobian when we take the derivatives we we we have a singularity in that location and you are not able to track your motion so you cannot really determine the properties of the motion at that point and this is a problem if you want to move and produce smooth motion usually if you are doing a very tiny motion you might be able to select Alpha and beta so that they will never go to zero within that small motion but if you are performing large motions in space you are going to run into the singularities now we can change representation we can use another one but every single representation that uses three parameters is going to run into a singularity all three parameter representation will have a singularity some where and that is the problem so three angle representation has are efficient because they are minimal but they have a problem with the representation in term of the singularity of the representation direction cosiness are perfect in term of the representation no singularities whatsoever however you have redundancy and you have to handle all these constraints as you plan your motion there is another derivative representation that uh we can talk about a little bit before talking about uh the best solution and this is a representation that you can think about uh just by going back to the problem of uh moving from one frame to another frame so you have frame a you have frame B and you can show that there is always a vector k about which you can rotate with an angle Theta to go from a to that specific B so there is a rotation about a vector that will take you there so we can now think about a representation that uses K and Theta and this is the equivalent angle axis representation so how can you build a representation you can say I'll take Theta in radian I take the vector k I need to identify K obviously but I can find KX KY and KZ and then scale it with Theta in radians and that will give me three numbers representing my position and now I can oh what is nice you can interpolate and you can move and you can go between configurations and now the rotation Matrix associated with this is like this it involves both the K and the angle Theta through cosine s and also 1 minus cosine for this uh uh new variable and you are given this measuring this and you need to identify the element KX KY KZ which you can do through those two equations so you can compute your K by taking the difference and dividing by the S of theta and this is coming from identifying with the Matrix so do you see any problem well you have a singularity you can divide by zero when sign is zero so you have again the same problem so I said nine is too much three doesn't is is not going to work whatever we do with three is not going to work so what should we do four good let's try four well this is what what actually the ear parameters do they now take the same concept I take the vector I know I can rotate with a vector but instead of taking the angle and scaling it I'm going to track correctly everything we will talk about this in advanced robotics uh more in more details about it but just like to give you the intuition essentially we are going to take four parameters the following parameters I'm going to take W this unit Vector it's unit vector and I'm going to scale it by half this s of half of the angle Theta and then I'm going to add another parameter which is just the cosine of half that angle so Epsilon 1 Epsilon 2 Epsilon 3 are just Omega W the vector W scaled by S of half of the angle and the last parameter Epsilon 4 is just cosine half of the angle now when you there are a lot of reasons a lot of interpretations for this but what you can see here is a lot of interesting properties that uh comes with that way of selecting the representation uh W is a unit Vector sin square and cosine Square have very nice properties so now Epsilon is a unit Vector in four-dimensional space that is you have this normality condition if you add the square of epsilons you get one and that has nice properties uh that uh in The Operators and it carries to the derivatives and everything so let me just tell you that uh in again with this representation it's this represent represent solve the problem but there are really uh uh many details related to The Way We select the representation the way we track and compute the epsilons and when any of the parameters we are using to do the computation goes to zero something happens so you have to be very careful about it and if we analyze and try to identify you can see the relationship between the representation Matrix and your measure measurement and you can uh uh find that if you add the square of the diagonal it gives you 3 - 4 this quantity and this quantity between parenthesis is what is 1 minus the missing parameter Square so you can then compute this parameter and then you can compute everything else very simply but again you are dividing by this parameter what happens if this parameters goes to zero so if this parameter go goes to zero actually some other parameters will not go to zero and because the property I I mentioned that this is a unit Vector in a of the hyper sphere in four-dimensional space we have a property that shows that for all rotations at any time there is always one of the parameter that is equal or equal means all of them are equal to one have to build one otherwise it's not possible or otherwise there is always one parameter that is large enough and then you can do the same competition using that parameter that is large enough so the algorithm that we use is always to think about what is the largest parameter and resolve with respect to that larger parameter so as you rotate you are tracking the larger parameter and you are resolving if if it is Epsilon 1 you do you use this formula if it's Epsilon 2 you use this formula etc etc so you can you can always resolve it depending on which parameter is large but here you eliminate the singularity no more singularities so well we don't have time for the quiz what are the L parameter for this example it's a rotation about the x- axis with an angle 60 s 30 so it will be x axis is 1 0 0 S 30 is do you remember 1 12 so like this and the cosine of 30 that's your parameters and what what is the corresponding Direction cosine representation [Applause] the First Column is going to be one 0 0 and then you get the sign and cosine of the angle all right wow we're done I can't believe it on time okay so have a nice break and uh I will see you next Wednesday see e e e e e e e e e for |
Lecture_Collection_Introduction_to_Robotics | Lecture_9_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's uh get started so today uh it's uh really uh a great opportunity for all of us uh to have a guest lecturer one of the leaders in uh robotics Vision uh Gregory Hager from Johns Hopkins who will be uh giving uh uh this uh guest uh lecture and um on uh Monday I wanted to mention that uh uh on Wednesday we have uh the me term in class uh tonight and tomorrow we have uh the review sessions so I think everyone uh has signed uh on for those sessions and uh next Wednesday uh the lecture will be given by uh a former uh PhD student from Stanford University uh Cassi corov who uh will be giving the lecture on trajectories and uh inverse kinematics so welcome GRE thank you so it is a pleasure to be here today and thank you AMA for inviting me um so Asama told me he'd like me to spend a lecture talking about Vision uh and um as you might guess that's a little bit of a challenge uh last count there was there were a few over a thousand papers in computer Vision in peer reviewed conferences and journals last year so summarizing all those in one lecture is is a bit more than I can manage to do but what I thought I would do is try to focus it specifically on an area that I've been interested in for really quite a long time namely what is the perception and sensing you need to really build a system that has both manipulation and Mobility uh capabilities and so really this whole lecture has been designed to give you a taste of what I think the main components are and also to give you a sense of what the current State ofthe art is and again it's obviously with the number of papers produced every year defining the state-of-the-art is difficult but at least give you a sense of how to evaluate the work that's out there and how you might be able to use it uh in a um a robotic environment and so really I I want to think of it as answering just a few questions or looking at how perception could answer a few questions so the simplest question you might imagine trying to answer is uh where am I relative to the things around me I you know you turn a robot on it has to figure out where it is and in particular be able to move without running into things be able to perform potentially some useful tasks that involves uh Mobility the next step up once you've decided where things are is you'd actually like to be able to identify where you are and what the things are in the environment clearly the first step toward being able to do something useful in the environment is understanding the things around you and what you might be able to do with them the third question is once I know what the things are how do I interact with them so there's a big difference between being able to walk around and not bump into things and being able to actually safely Reach Out And Touch something and be able to manipulate it in some interesting way and then really the last question which I'm not going to talk about today is how do I actually think about solving new problems that uh in some sense were unforeseen by the original designer of the system so it's one thing to build a Materials Handling robot where you've programmed it to deal with the five objects that you can imagine coming down the conveyor line it's another thing to put a robot down in the middle of a kitchen and say Here's a table clear the table including China dinner wear uh glasses boxes things that it potentially has never seen before and needs to be able to manipulate safely that I think is is a problem I I won't touch on today but at least I'll I'll give some suggestions as to where the the problems lie so I should say um again I'm going to really Breeze through a lot of material quickly but at the same time this is a class obviously if you're interested in something if you have a question uh if I'm mumbling and you can't understand me just stop and uh we'll go back and talk in more depth about whatever I just covered so with that um the topics I've chosen today uh really in many ways from bottom if you will to top from lowlevel capabilities to higher level our first computational stereo a way of getting the geometry of the environment around you feature detection and matching a way of starting to identify objects and identify where you are and motion tracking and visual feedback how do you actually use information from Vision to manipulate the world and again I think the the applications of those particular modules areir obvious in robotic uh mobility and manipulation so uh again let me just Dive Right In um I don't actually how many people here have taken or are taking the computer vision course that is being taught okay so a few people for you this will be a review uh you can I guess hopefully bone up for the midterm whenever that's going to happen or something um and hopefully I don't say anything that um disagrees with anything that Jana has taught you but um so what is computational Stereo well computational stereo quite simply is a phenomena that you're all very familiar with it's the fact that if you have two light sensing uh devices eyes cameras uh and you view the same physical point in space and there's a physical separation between those viewpoints you can now solve a triangulation problem I can determine how far something is from the viewing sensors by finding this point in both images and then simply solving a geometric triangulation problem simple uh and in fact uh you know there's a lot of stereo going on in this room uh pretty much everybody uh has stereo although oddly about 10% of the population is Stereo blind for one reason or another so it turns out that you know in this room there are probably three or four of you who actually don't do stereo but you compensate in other ways um even so uh having stereo particularly in a robotic system would be a huge step forward sorry for the colors there didn't realize it had transpose them um so when you're solving a stereo problem in computer vision there are really Three core problems the first problem is one of calibration in order to solve the triangulation problem I need to know where the sensors are in space relative to each other a matching problem so remember in Stereo what I'm presented with is a pair of images now those images can vary in many different ways hopefully they contain common content what I need to do is to find the common content even though there is variation between the images and then finally reconstruction once I've actually performed the matching now I can reconstruct uh the three-dimensional uh space that I'm surrounded by and so I'll I'll talk just briefly about uh all three of those so first um calibration so again why do we calibrate well we calibrate for actually any number of reasons um most important is that we have to characterize the sensing devices so in particular if you think about an image uh you're getting information in pixels so if you say that there's some point in an image it's got a pixel location it's just a set of numbers at a particular location what you're interested in ultimately is Computing distance to Something in the world well pixels are one unit distances are a different unit clearly you need to be able to convert between those two so typically in a camera there are four important numbers that we use to convert between them to scale factors that convert from pixels to millimeters and two point two uh numbers that characterize the center of projection in the image so the good news for you is that uh there are a number of good toolkits out there that let you do this calibration uh they let you characterize What's called the intrinsic or internal parameters of the camera in addition we need to know the relationship of the two cameras to each other that's often called the extrinsic or external calibration that's also something that you can get very good toolkits to solve uh there's a toolkit for mat lab in fact that um solves it quite well so calibration really is just getting the geometry of the system so we're set up and we're ready to go now for um the current purpose is let's assume that we happen to have a very special geometry so our special geometry is going to be a pair of cameras that are parallel to each other and the image planes are co- planer and in fact the scan lines are perfectly aligned with each other so if I look at a point in the left image if I wanted to find the same corresponding point in the right image of some physical point in the world it's going to be on the same row and in fact that's going to be true for all the rows of the camera so it's a it's a really convenient way to um to think about cameras for the moment so for a camera system like that solving the the stereo problem really is uh from a geometric sense quite simple so what did I say I've got a point on One camera line I've got a point in another camera the other camera the same line what I can do is effectively solve triangulation by Computing the difference in the coordinates between those two points um again not to go into great detail but I can write down the equations of perspective projection which I've done here for two cameras for what I call now the x coordinate so in fact um on this last slide I don't I forgot to point it out but I'm going to use a coordinate system in fact through this talk which is X going to the right y going down in the image and I guess uh most of you should be able to figure out which direction Z goes once I've told you those two things right where's Z go out the camera so Z is heading straight out of the camera lens so that's the coordinate system we're going to be dealing with so the things that we can find fairly easily in some sense are the X and y's of the point the unknown is the Z so whenever I say depth you can think of I'm trying to compute the Z's so I can write down perspective projection for a left camera and a right camera which I've done here they are offset by some baseline which I've called B I've also got the Y projection but it turns out to be the same for both cameras because I've scan line aligned them so I've got three numbers XL XR and Y I've got three unknowns x y and z a little algebra allows us to solve for the uh depth Z as a function of disparity which again is the difference between the two coordinates the Baseline of the camera and this internal scaling factor which which um allows us to go from pixels to millimeters just a couple things to notice about this depth is inversely proportional to disparity so the larger the disparity the smaller the depth makes sense I get closer I my eyes have to keep going like this more and more and more to be able to see something uh it's proportional to the Basel line if I could pull my eyes out of my head and spread them apart I could get better accuracy and it's also proportional to the resolution of the Imaging system so if you put all that together um you can start to actually think about designing stereo systems from very small to very large that operate at different distances and with different accuracies the other thing to point out here is that um depth being inversely proportional to disparity means that close to the cam we actually get very good depth resolution as we get further and further away our ability to resolve depth by disparity goes down drastically you see out here you know at a distance of 10 meters one disparity level is already tens maybe even hundreds of centimeters of distance so stereo is actually good in here it's very hard out there and in fact anybody happen to know the human stereo system what its optimal operating point is anybody had a class where they talk about that it's right about here it's about 18 inches from your nose right in this point you have great stereo Acuity you get in here and your eyes start to hurt you get out here past about an arm length and it just turns off you actually don't use stereo at long distances you're really just using it in this workspace and of course it makes sense we're trying to manipulate right well I made a a strong assumption about the cameras I said that they had this very special geometry and back in the good old days when I was your age um these are those are the good days by the way just so you're all aware of that fact you're in the good days right now enjoy them back in the good old days we actually used to try to build camera systems that um had this geometry um because if you start to change that geometry you no longer get this nice scan line property so in fact if I rotate the cameras inward if I start to look at the the relationship between corresponding points I start to get these Rays coming out so if I pick a point here the Line is now some slanted line well it turns out luckily one of the things that's really been nailed down in the last decade or two is that that doesn't matter we can always take a stereo pair that looks like this and we can resample the images so it's a stereo pair that looks like that and in fact by doing this calibration process we get it so the good news is I can almost always think about the cameras being these very special scan Line to Line cameras and so everything I'm going to say from now on is going to um pretty much rely on the fact that I've done this so-called rectification process so again a very nice abstraction I'll just point out again I'm not going to talk a lot about it but um the relationship that I just described so how do I if I have a point in one camera how do I know the line to look for it on the second camera well it's a function of the relationship the ation and the translation between the two cameras and it turns out this is again something that really has been nailed down in the last uh couple of decades I can estimate this from images so in fact I could literally take a pair of cameras put them in this room do some work and I could figure out their geometric relationship without having any special apparatus whatsoever what it also means is instead of doing stereo I could literally just take a video camera and walk like this process the video images and effectively do stereo from a single camera using the video and that's because I can estimate this Matrix e and that relationship up there uh which if we worked out what e was it turns out to contain the rotation and translation between the two camera systems and once I know that I can do my rectification and I can do stereo so actually stereo is very a special case in some sense of taking a video and processing it to get motion and uh structure at the the same time so again that's a whole lecture we won't go in there but suffice to say that from a geometric point of view we can actually deal with cameras now in a very general way with relatively little operary assumptions about how they start okay so geometry calibration is done I now want to reconstruct but to do reconstruction I need to do matching I need to look at a pair of images and say hey there's a point here and that same point is over there and I want to solve the triangulation problem so there are two major approaches feature based and region based matching so feature-based obviously depends on features I could run an edge detector in this room I could find some nice edges you know the chairs the edge of people the floor and then I could try to match these features between images and now for every feature I'm going to get depth so if you do that you end up with these kind of little stick figure type uh cartoons here so here's a set of bookshelves and this is a result of running a feature-based stereo algorithm on that so you can see on the one hand it's actually giving you kind of the right representation right the the major structures here are the shelves and it's finding the shelves but you notice that you don't get any other structure you're just getting those features that you happen to pull out of the image so the other approach is to say forget about features let me try to find a match for for every pixel in the image a so-called dense depth map every pixel's got to have some matching pixel or at least up to some occlusion relationships so let's just look for them and try to find them and so this is so-called region matching method I'm going to actually pick a pixel plus a support region try to find the matching region in in another image and this has been a cottage industry for a very long time and uh so people pick their favorite ways of matching their favorite algorithm to apply to those matches and so on and so forth so again a huge literature in doing that um there are a few Al a few match metrics which have come to be used fairly W widely probably the most common one is something called the sum of absolute differences it's uh right up there um or more generally I've got something called zero mean sad or some of absolute differences and that's probably the most widely used algorithm not the least of which because it's very easy to implement in hardware and it's very fast to operate so you take a region you take the difference take a region take a region take their difference take their absolute value sum those and assume if they match that that difference is going to be small if they don't match it's going to be big all the other metrics I have up here are really different variations on that theme of take this region take that region compare them and try to minimize or maximize some measure so if I have a match measure now I can look at correspondence again I get to use the fact that things are scan Line to Line so when I pick a point in the left image or pixel in the left image I know I'm just going to look along a single line in the right image not the whole image so simple algorithm uh for every Row for every column for every disparity so for every distance that I'm going to consider essentially uh I'm now going to compute my match metric in some window window I'll record it uh if it happens to be better than the best match I found for this pixel sorry if it's better than the best match I found for this pixel record it if not uh I just go around and try the next disparity so you work this out what's how much Computing are you doing well it's every pixel rows and columns every disparity so um for example for my eyes if I'm trying to compute on kind of canonical images over a good working range you know 3T to a foot maybe 100 disparities 120 disparities so rows columns 120 disparities size of the window which might be let's say 11 by 11 pixels it's a pretty common number so there's a lot of computing in this algorithm a lot a lot of computing and in fact up until maybe 10 years ago even just running a stereo algorithm was a a a feat just point out that it turns out that the way I just described that algorithm although intuitive is actually not the way that most people would implement it there's a slightly better way to do it this is literally mat lab code so um for anybody in computer vision this is the mat lab to compute stereo on an image um and you can see the main thing is I'm actually looping first over the disparities and then I'm looping over effectively rows and columns doing them all at once the reason for doing this without going into details is you actually save a huge number of operations in the image so if you ever do Implement stereo and I know there are some people in this room who are interested in it don't ever do this it's the wrong thing to do do it this way it's it's the right way to do it and if you do this you can actually get pretty good performance out of the system Just One Last twist to this so one of the things that's going to happen if you ever do this is it's going to you're going to be happy because it's going to work reasonably well probably pretty quickly and then you're going to be unhappy because you're going to see it's going to work well in a few places and it's not going to work in other places uh someone in computer vision can you give me an example of a place where this algorithm is not going to work or unlike is unlikely to work a situation very simple situation rotated I'm sorry rot I can't hear you well if the but I'm assuming rectification so uh yeah the boundaries are going to be a problem right because I'm going to get occlusion relationships will change the pixel so that's good occlusions are bad how about this I look at a white sheet of paper what am I going to be able to match between the two images almost nothing right so it's put a stereo algorithm in this room it's going to do great on the chairs except at the boundaries uh but there's this nice white wall back there and there's nothing to match so it's not going to work everywhere it's only going to work in a few places how can tell when it's working you know it's one thing to have an algorithm that does something reasonable it's another to know when it's actually working uh and the answer turns out to be a simple check is I can match from left to right and I can match from right to left now if the system's working right they should both give the same answer right it's I can match from here to here or here to here it shouldn't make any difference well when the right when you have good structure to the image that will be the case but if you don't have good structure to the image it turns out you usually are just getting random answers and so I said something like a 100 disparities right so the odds that you pick the same number twice out of 100 is really pretty small and so it turns out that almost always the disparities differ and you can detect that fact it's called the left right check uh on a multicore processor it's great because you have one core doing left to right one core doing right to left at the end they just meet up and you check their answers here's some disparity m uh these are actually taken from an article by Cork and Banks some years ago just comparing different metrics uh and here I just happen to have two one is so-called SSD sum of square differences the other is zero mean normalized crosscorrelation um couple just quick things to point out so like on the top images you can see that the two metrics that they chose did about the same you can also see that you're only getting about maybe 50% data density so again they've done the left right check half the data is bad so kind of another thing to keep in mind 50% good data not 100% not 20% either 50% good data Maybe second row why is there such a difference here well there's a brightness difference I don't know if you can see but the Right image is darker than the left well if you're just matching pixels that brightness difference shows up as uh just a difference it's trying to account for in the right column they've taken so-called zero meaning they've subtracted the mean of the neighborhood of the pixels gets rid of brightness differences pulls them into uh better alignment in the brightness range and so they get much better density and this is just an artificial scene I think the most interesting thing there is it's an artificial scene you've got perfect photometric data and it still doesn't do perfectly because of these big areas where there's no texture and it doesn't know what to do so it's producing random answers the left right check throws it out so any there are no guarantees that's correct it could make a mistake but the odds that it makes a mistake it turns out are quite small it actually is it I should say it's a very reliable measure because usually it's going to pick the wrong pixels occasionally by chance it'll say yes but those will be isolated pixels typically usually if you're making a mistake you don't make a mistake on a pixel you make a mistake on an area and what you're trying to do is to really kind of throw that area away but a good point it will it's not perfect but actually in the world of vision it's one of the nicer things that you get for almost free I'll just say that um these days uh a lot of what you see out there is real-time stereo so starting really about 10 years ago people realized if they're smart about how they implement the algorithms they can start to get close to real time now you can buy systems that run pretty much in software and produce 30 frames a second stereo data now it's kind of a just a mention that you know I said that the data is not perfect well the data is not perfect and as a result if you could imagine what you'd like to do is take stereo and build a geometric model right I'd like to do manipulation I want to take this pen I want to have a 3D model and then I want to generate or manipulate 3D well you saw those images stereotypically doesn't produce data that's good enough to give you high Precision completely dense 3D models it's an interesting research problem and people are working on it but it's a great modality if you just want a kind of rough 3D description of the world and you can run it in real time and you now get in kind of real time course 3D and so I think that's why right now realtime stereo is really getting a lot of interest because it gives you this course wide field 3D data 30 frames a second that let you just do interesting things I just thought I would throw in one example of what you can do so this is actually something we did 10 years ago I guess you're a mobile robot and you want to detect obstacles and you're running on a floor so what's an obstacle well it's positive obstacle something that I'm going to run into or it could be a negative obstacle you know I'm coming to the edge of the stairs and I don't want to fall down the stairs so I'm going to use stereo to do this and so the the main observation is that since I assumed I'm more or less operating on a floor I've got a ground plane and it turns out planer structures in the world are effectively planer structures in disparity space so it's very easy to detect this big plane and to remove it and once you've removed that big plane anything that's left has got to be an obstacle positive or negative doesn't matter what it is so you run stereo you remove the ground plane if there's something left that's something you want to worry about avoiding so here's a a little video showing this so there's a real-time stereo system so first we put down something that's an obstacle it shows up here's something which if you were just doing something like background subtraction you might say that that newspaper is an obstacle it's different than the floor right so you would drive around it because it it could be something you don't want run into but here since we've got this ground plane removal going we can see that this is very clearly an obstacle that is very clearly something that's just attached to the floor and disappears and you can go by it and this is something real cheap easy simple to implement and I think that really is the great value of stereo and Robotics right now today is that most stereo algorithms can give you this core sense of what's around you and where you're going and you can use it for Downstream computations in the world of stereo and research there are a lot of other issues that people are trying to deal with how do you you increase the density increase the Precision deal with photometric issues like shiny objects deal with uh these differences between the images just due to brightness um lack of texture how do I deal with the fact that in some places I don't have lack of texture how do I infer some sort of depth there and also geometric ambiguities like occlusion boundaries how do I deal with the fact that occasionally there'll be parts of the image that the left camera sees and the right camera doesn't see so there's ongoing research there most of these methods I'll just say try to solve stereo not as this local region matching problem I mentioned but as a global optimization problem and so there's a lot of work in different uh Global optimization algorithms and you know there is hope that ultimately stereo will get to the point that it really can do um the sort of thing I mentioned I want to pick this up and I really want to get a 3D model and use that for manipulation probably not there today I should just say that there's one simple way to get a huge performance boost out of your stereo algorithm something that people often do and if anybody can guess uh one way to just take a stereo algorithm and make it work a whole lot better think about like how I could change this sheet of paper to be something that I could actually do matching on you want to add texture how could you add texture light there you go just put a little light projector on the top of your stereo system and you'll be amazed at uh how well it works suddenly this thing which you couldn't match before becomes the world's best place to do stereo because you get to choose the texture and you get to um match it so um that's the other thing that people have looked a lot into you'll see in the literature structured light stereo is another way to get better performance out of stereo okay so that's stereo again the message here is by using two cameras you can get at this point data density and accuracy that still exceeds pretty much anything else you could imagine in the laser range finding World um it's not as reliable as laser range finding and that's probably the thing that is still the the main topic of research any quick questions on that before I shift gears okay so I'm a robot I'm running around I've got realtime stereo I don't run into things any more if somebody walks in front of me I Scurry away as quickly as I can so that I don't hurt them but I have no clue where I am in the world or I have no clue what's in the world around me so if you said you know go over to the printer and get my print out you know where is the printer where am I where's your office who are you so how can we solve those problems and in fact this I think is an area of computer vision that I would say in the last decade has undergone a true Revolution ution 10 years ago if I would have talked about object recognition in this lecture we really had no clue there are kind of some interesting things going on in the field but we had no clue and today there are people who had claimed that at least certain classes of object recognition problems are solved problems we actually know very well how to build Solutions and there are actually commercial Solutions out there so what is the problem with object recognition well it's a chicken and egg problem if I want to recognize an object there are many unknown so I look at an image I don't know the identity of the object I don't know where it is and most importantly perhaps I don't know what's what's being presented to me so I don't happen to have a oh here we go so you know if I say recognize find my cell phone in the image you don't know if you're going to see the front of the cell phone the back of the cell phone the top of the cell phone you don't know if you're going to see half of the cell phone hidden behind behind something else you don't know what the lighting on the cell phone is going to be huge unknowns in the appearance let alone then finding it in the image and segmenting it and then actually doing the identification so there's a sense in which if I could segment the object if I could say here's where the object is in the image then solving recognition and pose would be fairly easy or if I told you the pose of the object solving segmentation and recognition would be easy to do or if I tell you what the object is finding it and figuring out its pose is easy to do doing all of them at the same time is hard so for a long time people tried to use geometry so maybe the right thing to do is to have a 3D model of my cell phone and to use my stereo Vision to recognize it well we just said stereo is not real reliable right probably not good enough to recognize objects so another set of people will said well how about if we recognize it from appearance so let's just take pictures you know the way I'm going to recognize my cell phone is I've just got 30 pictures of my cell phone and you find it in the image so um you can do that and in fact you can see here this is some work of Shri ner about uh 10 years ago um they're doing pretty well on a database of about 100 objects but you notice some other things um not in the database um black background the objects actually only have in this case one degree of Freedom it's rotation in the plane so yes they've got 100 objects but the number of pictures that or the number of images they can see is fairly small no occlusion no real change in lighting so this is interesting you know in some sense it generated a lot of excitement because it's probably the first system that could ever do a 100 objects but it did it in this very limited circumstance and so the question was well how do we how do we bridge that Gap how do we get from hundreds to thousands and do it in the real world and really the answer you can almost think of it as combining both the geometry and the uh appearance-based approach so the observation is that views were a very strong constraint giving you all these different views and recognizing from views worked for 100 objects but it's just hard to predict a complete view it's hard to predict what part of the cell phone I'm going to see it's hard to get enough images of it to be representative it still doesn't solve the problem of occlusion if I don't see the whole thing so views seem to be in the right direction just not quite there so um Cordelia Schmidt did a thesis in 1997 with Roger Moore where she tried a slightly different approach she said well what if instead of storing complete views we store think of it as interesting aspects of an object you know if you were going to store a face what are the interesting aspects well they're things like the eyes and nose and the mouth you know the cheeks are probably not that interesting there's not much information there um but the eyes and the nose and the mouth they tell you a lot or you know my cell phone you've got all sorts of little textures so what if we just store if you think of it this way thumbnails of an object so my cell phone is not a bunch of images it's a few thousand thumbnails and now suppose that I can make that feature detection process very repeatable so if I show you my cell phone in lots of different ways you get the same features back every time now suddenly things start to look interesting because what the signature of an object is not the image but it's these thumbnails and I don't have to have all the thumbnails you know if I get half the thumbnails maybe I'll be just fine if the thumbnails don't care about rotation in the plane that's good if they don't care about scale even better so really it starts to become a doable approach and in fact this is really what has revolutionized this area and in particular there is a set of features called sift scale invariant feature transform which I know you've learned about in computer vision if you've had it uh developed by David low which really have become pretty much the industry standard at this point in fact you can download this from his website and build your own object recognition system if you want let me just um talk a little bit about a few details of the approach so I said features is is where we want to go well there are two things that we need to get good features one is we need good detection repeatability I need a way of saying there are features on this wall at this orientation features on this wall at this orientation I should find the same features so detection has to be invariant to image Transformations and I have to represent these features somehow and probably just using a little thumbnail is not the best way to go right because a thumbnail if I take my cell phone and I rotate a little bit out of plane or I rotate even in plane the image changes a lot and I'd like to not have to represent every possible appearance of my cell phone under every possible orientation so um we need to represent them in a coordinate and variant way and we need lots of them and when I say lots I don't mean 10 I mean a thousand we want lots of them so sift features solve this problem they do it in the following way they do a set of filtering operations to find features that are invariant to the detection is invariant to rotation and scale it does a localization process to find a very precise location for it it assigns an orientation to the feature so now if I redetect it I can always assign the same orientation to that feature and cancel for rotations in the image builds a keypoint descriptor out of this information and a typical image yields about 2,000 stable features there's some evidence that suggests to recognize an object like my cell phone you only need three to six of them so from a few thousand features you only need one or 2% and you're there so again just uh briefly the steps set of filtering operations what they're trying to do is to find features that have both a local structure that's a maximum of an objective function and a size or a scale that's a maximum of an objective function so they're really doing a three-dimensional search for a maximum once they have that they say aha this area is a feature key Point descriptors what they do is they compute the so-called gradient structure of the image so this allows them to sign assign a direction to features and so again by getting by having an assigned orientation they can get rid of uh rotation in the image so if you do that and you run it on an image you get uh confusing figures like this what they've done is they've drawn a little arrow in this image for every detected feature so you can think of a long arrows being a big feature so a large scale feature and small arrows being fine detailed features and the direction of the arrow is this orientation assignment that they give and so you can see in this house picture uh it's not even that high res picture it's 233 by 189 and they've got 832 original key points filtered down to 536 when they did a little bit of throwing out what they thought were bad features so lots of features lots of information that's being memorized but discreetly now not the whole image just discreetly if you take those features and you try to match features it turns out also they're very discriminative so if I look at the difference in match value between two features that do match and two features that don't match it's about a factor of two typically and so there's enough signal there you can actually get matches pretty reliably now I mentioned geometry so right now A an object would just be a suitcase of features so if I going to memorize my cell phone I just said I gave you some thumbnails so most systems actually build that into a view so I don't just say my cell phone is just a bag of features but it's a bag of features with some spatial relationship among them and so now if I match a feature up here it tells me something about what to expect about features down there or more generally if I see a bunch of feature matches I can now try to compute an object pose that's consistent with all of them and so in fact that's how the feature matching Works uses something called the Huff transform which you can think of as a voting technique just generally voting is a good thing I'll just say as an aside this whole thing that I've been talking about all we're trying to do is to set up voting you know so and we're really trying to be in an election where we're not going down to the August convention to make a decision of who's winning the primaries this is an election that we want to win on the first try so these features are very good features they do very discriminative matching of objects you add a little bit of geometry and suddenly from a few feature matches you're getting pose and identity of an object from a very small amount of information so here are a couple of results from David Lowe's original paper so um you can see there's just a couple objects here a little toy train and a froggy uh and there's the scene and you know if you just are given that Center image I think even a person has to look a little bit before you find the froggy and the toy train and down the right are the detected froggy and toy train including uh a froggy that's been almost completely hidden behind that dark black object you just see his front flipper and his back flipper but you don't see anything else and the systems actually detected in this case two instances I think well not actually it's one one box so we got one instance it even got got it realized that even though it was occluded it's one object out there so I mean these again I think it's fair to say are fairly remarkable results considering where the field was at that time since then there's been a cottage industry of how can we make this better faster higher stronger so um this happens to be worked by pon pon and rothganger where they try to extend it by using better geometric models and slightly richer features so they have 51 objects uh and they can have any number of objects in a scene and this is the sort of uh recognition plots that you're starting to see now so you know we're not talking about getting it right 60% of the time or 70% of the time they're getting 90 plus% recognition rates on these objects now of course I've been talking about object recognition just point out that you can think of object recognition as there's an object in front of me and I want to know its identity and its pose or you can think of the object as the world and I want to know my pose inside this big object called the world and so for example if I'm outside I might see a few interesting landmarks and recognize or remember those landmarks using features and now when I drive around the world I'll go and I'll look for those same featur teachers again and use them to decide where I am and so in fact this is um again work out of UBC where they're literally using that same method to model the world recognize you know I'm in this room I see a bunch of features I go out in the hallway I see a bunch of features I go in the AI lab I see a bunch of features store all those features and now as I'm driving around in the world I look for things that I recognize if I see it then I know where I am relative to where I was before moreover remember I said that for stereo to get Geometry we didn't have to actually calibrate our stereo system we could have one camera and it could just walk around we could compute this so-called epipolar geometry automatically and then we could do stereo so in fact what they've done in this map is they've from One camera as they're driving around they're not Computing just the identity but the geometry of all these features around them so they can build a real 3D map just to like You' build with a Laser Rangefinder but now just by matching features and images and in fact we edited a joint issue of uh jcv and JRR about I guess six eight months ago and probably half the papers in that special issue ended up being how can you use this technique to map the world in different variations and flavors so again it's it's a technique which really in many ways is there you can download it from the web practically and put it on your mobile robot and make it run and in fact this is my favorite um result so um this is the work of Ryan Eustace who uh was a postto at Hopkins uh with L witcom who does a lot of underwater Robotics and so this is the Titanic this is actually a Boward Expedition where they flew over the Titanic with a camera and um the goal was obviously to get a nice set of images of the Titanic but the problem is that underwater it's really difficult to do very precise localization and odometry and so what Ryan did is he took these techniques and he built effectively a mapping system very high Precision mapping system that was able to take these images use the images to localize the underwater robot and then put the images together into a mosaic and so this is a mosaic of the Titanic as I flew over it uh you can see the numbers up there they actually ran for about 3.1 kilometers they have what is it uh how many images there it is over 3,000 images 3,500 images matched successfully computed the motion of the robot successfully filtered all this together and we're able to produce this Mosaic so really an impressive impressive Rive system do have they actually have um so from this actually do get the 3D geometry I me in this Mosaic they they've basically projected it down but they are Computing uh up to the set of features that they're able to use the the 3D geometry and actually the little red versus brown up there the red I believe is the original odometry that they thought they had on the robot and the brown is actually the corrected odometry um that they computed or vice versa I don't know which is which now I can't remember if they did the two separate pieces or they did just one piece but really impressive work very nice also it mentions here doing a cman filter that he built a special purpose common filter that operated on the space of um reconstructed images so um that is kind of the next piece of this puzzle oh just I guess I had one other three in here so a lot of people interested in 3D now so this is Peter Allen um also putting together images here he just showing the range data but you can imagine if you have range and appearance now you can actually do interesting things using both 3D and appearance information I know there's some work going on here at Stanford in that also so that's kind of chapter two so now a set of techniques that not only let me avoid running into things in the world but a set of techniques that let me say well where am I and where are some things that I'm interested in so you can now actually imagine phrasing the problem I want to pick up the cell phone and you could actually have a system that recognizes the cell phone and is able to say hey there's the thing that I want to pick up so I'll just finish up with what I thought was the last piece of this puzzle namely how do I pick it up well I'm not going to tell you exactly how to pick it up there's a lot of interesting and hard problems in figuring out how to put my fingers on this object to actually pick it up but at least let's talk a little bit about the hand eye coordination it takes for me to actually reach over and grab this thing or even better if I do that I don't want to drop this um if I do that um how do I actually catch it again which luckily I did otherwise it'd be a very sad lecture so um I'm going to talk about this in two pieces so one piece is going to be uh visual tracking so we're now really moving to the domain where I want to think about moving objects in the world and having precise information about how they're moving how they're changing so visual tracking is an area that attacks that problem and I know you saw a video maybe a week ago of a humanoid robot that was playing um bad mitt or pingpong or volleyball volleyball I think it was and so I I think uh Professor KB had already explained that you know they're doing some simple visual tracking of this big colored thing coming at them and they're using that to to do the feedback well so big colored things are nice uh unfortunately my cell phone is not daylow orange so it's hard to just use color is the only thing that you can deal with but tracking has been you know a problem of interest for a long time tracking people tracking faces tracking Expressions um all sorts of different tracking so what I think is interesting is first to say well what do you mean by tracking to begin with it's kind of cool to write a paper that says tracking of X but no one's ever defined what tracking is so I have a a very simple definition of visual tracking which simply says um I'm going to start out with a Target you know my face is going to be the the canonical Target here so at Time Zero for some reason you've decided that's the thing you want to track and the game in town is to know something about where it is at time T and the something about where it is is something is what you in principle get to pick so there going to be a configuration space for this object you know I could the simplest thing is your big uh orange ball it's just round so it's got no orientation it just has a position in the image so its configuration is just where the heck is the the orange Ball but you can imagine you know my cell phone has an orientation so presumably orientation might be part of the configuration or if I start to rotate out of plane you get those out of plane rotations in fact if it's a rigid object how many degrees of freedom must it have they better know the answer to this six yes believe me there's no trick questions and he knows what he's doing so if he told you it's six it really is six there's no no question there um six so sure if this is a rigid object in principle there must be six degrees of freedom that describe it though of course you know if it's my arm then it's got more degrees of freedom I wonder how many more it has anyway um okay so there's going to be a configuration space for this object and ultimately that's what we care about is is that configuration space the problem is that the image we get depends on this configuration space in some way and so here I'm going to imagine for the moment that I can predict an image if I knew its configuration if I knew the original image and so you can imagine this is like the forward kinematics of your robot you know I give you a kinematic structure and I give you some joint values and now you can say aha here's a new kinematic configuration for my object so the problem then is uh I'm going to think of a tracking problem so I know the initial configuration I know the configuration at time T I know the original image now what I'd like to do is to compute uh the change in parameters or even better just the parameters themselves I don't know what d stands for so don't ask me what d stands for um I want to compute the new configuration at time t+1 from the image at time t + one and everything else I've seen or another way to think of this is look I said I believe I can predict the appearance of an object from0 to T I can also think of the other way around I can take the image at time T and if I knew the configuration I could predict what it would have looked like when we started and now I can try to find the configuration that best explains the starting image and so this really is effectively a stabilization problem I'm going to try to pick a set of parameters that always make what I'm predicting look as close to the original template as possible so in this case I'm going to take the face and un rotate it and try to make it look like the original face and so my stabilization point now is an image and so this gives rise to a very natural sort of notion of tracking where I actually use my little model that I described my prediction model to take the current image apply the current parameters produce what's hopefully something looks like the image of the object to start with so if I started with my cell phone like this and later on it looks like that I'm going to take that image I'm going to resample it to hopefully it looks like that again if it doesn't look like that there's going to be some difference I'm going to take that difference run it through something hopefully that something will tell me a change in parameters I'll accumulate this change in parameters and suddenly I've updated my configuration to be right now what's interesting about this perhaps let's just um skip over this for the moment so we can for a planer object we can use a very simple configuration which turns out to be a so-called apine model so how do I solve that stabilization problem well again I said I'm going to start out with this predictive model which is kind of like your kinematics and if I want to go from a kinematic description talking about positions in space to velocities in space what do I use Jacobian imagine that hey we've got you know kinematics in the rigid body World we've also got kinematics in image space let's take a Jacobian if we take a Jacobian we're now relating what changes in configuration space to changes in appearance just like the Jacobian robotics relates changes in configuration space to changes in cartisian position so there you go so I'm going to take a Jacobian it's going to be a very big Jacobian so the number of pixels in the image might be 10,000 so my I'm going to have 10,000 rows and however many configurations so 10,000 by 6 so it's a very big Jacobian but it's Jacobian nonetheless and we know how to take those well now I've got a way of relating change in parameter to change in the image suppose I measure an error in the image which is kind of locally like a change in the image so an error in the in the alignment well suppose I effectively invert that Jacobian now again I have to use a pseudo inverse because I got this big tall Matrix instead of a square Matrix well so I take this incremental error that I've seen in my alignment go backwards through the Jacobian Al lo and behold it gives me a change in the configuration and so I close the loop by literally doing an inverse Jacobian in fact it's the same thing you could use to control your robot to a position in cartisian space through the Jacobian really no different so it really is a set point control problem um again I won't go into into details right now this is a huge big time VAR in Jacobian it turns out that you can show this is work that we did in and um uh um name slips my mind CMU also did work showing that you can make this essentially a Time invariance system which is just a way of implementing things very fast what does a Jacobian look like well the cool thing about images is you can look at jacobians because they are images so this is actually what The Columns of my Jacobian look like so this is the Jacobian if you look at the image for a change in X Direction Motion in X it kind of makes sense you see it's basically getting all the changes in the image along the the rows or yeah along the rows Y is getting a change along the columns rotation is kind of getting this velocity field in this direction so on and so forth so that's what a Jacobian looks like if you never saw a Jacobian before um it turns out that I what I showed you is for planer objects you can do this for 3D so my nose sticks out a lot if I were to just kind of view my face as a photograph and I go like this it doesn't quite work right so I can deal with 3D by just adding some terms to this Jacobian and in fact you'll notice you know what can I say I've got a big nose and so that's what comes out in the Jacobian of my face is my nose tells you which direction my face is pointed um again we can deal with illumination and this is actually probably a little more interesting I can also deal with a clusion while I'm tracking because if I start to track my cell phone and I go like this well lo and behold there's some pixels that don't fit the model so what I do is I add a little so-called reting Loop that just detects the fact that some things are now out of sync ignore that part of the image so you put it all together and you get something something that looks not like that so just so you know what you're seeing remember I said this is a stabilization problem so if I'm tracking something right I should be stabilizing the image I should be subtracting all the changes out so that little picture in the middle is going to be my stabilized face I'm going to start by tracking my face and this is actually the big linear system I'm solving my Jacobian which actually includes motion and includes some illumination components too which I didn't talk about so I'm just showing you uh the Jobe and you can kind of see a little frame flashing this is a badly made video was back when I was young and uninitiated and so now you can see I've I'm running the tracking this is just using planer tracking so as I tip my head back and forth and move around it's doing just fine uh scale is just fine because I'm Computing all the configuration parameters that have to do with distance um I'm not accounting for facial expression so I can still make goofy faces and they come through just fine uh now I'm saying to an unseen partner turn on the lights and so I think some lights flash on and off yep there we go so it's just showing you can actually model those changes in illumination that we talked about in Stereo 2 um through some magic that only I know and I'm not telling anyone at least no one in this room so actually it's not hard it turns out that for uh an object like your face if you just take about a half a dozen pictures under different eliminations and use that as a a linear basis for illumination that'll work just fine for elimination model notice here that I'm turning side to side though and it clearly doesn't know anything about 3D so uh you can actually make my nose grow like Pinocchio by just doing the right thing so I just let this run a little long okay so now what I did is I put in the 3D model and so the interesting thing now as you see my nose is stock still so I actually know enough about the 3D geometry of the face and 3D configurations that I'm canceling all the configuration all the changes due to configuration out of the image as a side effect I happen to know where I'm looking too so if you look at the backside I'm telling you at any point in time what direction the face is looking and uh here I'm just kind of pushing it eventually as you start to get occlusions it starts to break down obviously cuz I having modeled occlusions and I wish I could fast forward this but it's before the days of fast forwarding here my face is falling apart uh he wasn't supposed to be there it happens it's uh faculty at yeld saying hey what are you doing uh and this is just showing the what happens if you don't deal with occlusion and vision can see that I'm kind of knocking this thing out and it comes back and then eventually it goes Kaboom and now we're doing that occlusion detection so I'm saying hey what things match and what things don't match and uh there you go cup sees a cup says that's not a face so you can take these ideas and you can then push them around in lots of different ways um this is actually using 3D models uh here we're actually tracking groups of individuals and regrouping them dynamically as things go on um here's probably the most extreme case so this is actually tracking two Da Vinci tools during a surgery uh where we learned the appearance of the tools actually as we started so there's um 18 degrees of freedom in this system so it's actually tracking in an 18 degree of Freedom configuration space uh during the surgery okay very last thing I have 10 minutes I'm racing for home now so I uh I can track stuff cool so what uh it's fun but um the thing I said I wanted to do eventually was to finally manipulate something I want to use all this visual information and I want to pick up the stupid cell phone and call my friends and say the vision lecture is finally over in robotics we can go out and do something else but the question is how do I want to do that so I I've got cameras they're producing all sorts of cool information I've got a robot that I want to make drive around where do I drive it to or how do I drive it so what should I put in that box any suggestions you can assume I've got I've got two cameras so I've got stereo I've got pretty much anything you've seen what's anybody think of a I don't care which way you want to think of it what could you put in that box what information would you use and what would you put in the box it's going to be on the midterm well look the simplest thing you could imagine right is I've got if I said I've got two cameras I can actually with those two cameras measure a point in space I can actually calibrate those cameras to the robot and so I could just say hey go to this point in space end of story good thing bad thing good or bad thing anybody think why it could be good or bad yeah it's bad because if you run into anything on the way then you can't really accommodate for right so okay you're not monitoring but I could monitor in real time right so that would get rid of at least that problem what if my robot's not a real stiff robot what if my kinematics aren't great like turns out the D Vinci kinematics aren't that great so I could reach out to a point in space but maybe my arm goes here or there instead right so the cameras just tell me to go somewhere but it's not not really closing the loop so what if I do one better what if I compute the position of My Finger track it let's say and I compute the position of the phone in 3D space now I can actually close the loop I can say I want to make this distance zero and we could write down a controller that would actually do that pick your favorite controller I know Osama has some ideas of what they should be but pick your favorite so and that will work in fact that'll work pretty darn well it turns out but suppose that my cameras are miscalibrated and in fact suppose that I say well what I want you to do is to go along a line defined by the edge of the cell phone I want you to be here for some reason turns out you can show that if you do it in position space and reconstructed space and your cameras aren't perfectly calibrated you can actually get errors you can in fact you can get arbitrary errors if you want to it's not real likely but it can happen so there's one other possibility which is I'm looking at this thing and I'm looking at this thing what if I close the loop in the image space what if I just write my controller on the image measurements themselves well it turns out if you do that and what this is called it's called an encoding so if you can encode the task you want to do like touch this point of the cell phone to my finger and do it in the image space not in the reconstructed space well you've defined an error that doesn't mention calibration right it just says make these two things coincident in the images if you can close that Loop stably think of jacobians again for example um you can actually drive the system to a particular point and you've never said anything about calibration in your error function which means that even if the cameras are miscalibrated you go there in fact there's pretty good evidence that's what you do you don't sit there and try to figure out the kinematics of your arm and the the position in space and then kind of close your eyes and say go there you're watching right and you're actually using visual space and we know this because I can put funny glasses on your eyes and after a while you still get pretty good at getting your fingers together so um again I'm running out out of time I won't go into great detail but the interesting question is really when can you do this en coding when can I write things down in the image domain and the answer again depends a little bit on what you mean by cameras but suffice to say you can do a set of interesting tasks just by doing things in the image space and closing the loop in the image space and the interesting fact is that a lot of sort of tasks that you might imagine like putting a screwdriver in a screw or or putting a disc in a disc drive you can write it all in the image space and you don't need to calibrate the cameras or you don't need well calibrated cameras and I'll have to say so this is how why did I ever get into this because I was sitting in this stupid Lab at Yale and I started to do this tracking and just for the heck of it I built this robot controller to do Vision so you see I'm tracking and I'm controlling here and um like the usual cynical young faculty member I never expected this thing to work the first time I hadn't calibrated the cameras I just gu what the calibration was you know I just threw the code together I turned this thing on and it worked and I mean it worked within a half a millimeter it wasn't like it just worked it was right and then I started to think about it and I realized of course it worked I didn't need to calibrate the cameras and so then we actually spent the next few years figuring out why it was that I could get this kind of accuracy out of a system where I literally put the cameras down on the table looked at it and said I think they're about a foot apart and ran the system system and this is you know the moral equivalent of that so it's out there doing some stuff and you know I'm doing the moral equivalent of pulling your eye out of your head and moving it over here and saying okay see if you can still do whatever you were doing and uh you know just to prove you can do useful things with it we had to actually do something with the floppy so there you go you can also see how long ago this was by the the form factor the Macintosh that I'm putting the floppy disc into and anyway all right so I'm about out of out of time but I hope what I've convinced you of is that at least a lot of these basic capabilities we've got you know we've got stereo real time gross geometry we can recognize objects we can recognize places we can build Maps out of it we can track things and we even know how to close Loops in a way that are are robust so we don't have to worry about having finely tuned Vision systems to make it work so why aren't we running around with robots you know playing baseball with with each other well you know I've given you kind of the the simple version of the world obviously if I give you complex geometry objects you haven't seen before it's not clear we really know how to pick up a random object though we're hopefully getting close a lot of the world is deformable it's not rigid what's the configuration space how do I talk about tracking it or manipulating it and a lot of things are somewhere in between you know rigid objects on a tray which yes I could turn it like this but it really doesn't accomplish the uh the purpose in mind so understanding those physical relationships uh in the real world there's a lot of complexity to the environment it's not my cell phone sitting on an uncluttered desk it's well my desk would be I'd be happy if it were that uncluttered uh and I'm telling you to go and find something on it manipulate it so complexity is still a huge issue and it's not just complexity in terms of what's out there it's complexity and what what's going on people walking back and forth and up and down things changing things moving so imagine trying to build a map when people are moving through the corridors all the time oh it's good I can't there's some movie I can't prepare and uh in fact again I know um this is something of Interest here human computer interaction you know I can track people so now in principle I can reach out and touch people what's the safe way to do it when do I do it how do I do it what am I trying to accomplish by doing so how do you actually take these techniques but add a layer which is really social interaction and safe uh social interaction to the top of it and I don't know if you've noticed but I think these are not just uh you know there's a a research aspect to it but there's also a a market aspect to it at what point does it become interesting to do it you know what's the first killer app for actually picking things up and moving it around it's cool to do it but can you actually make money at it and then the last thing and I at this beginning I said this you know the real question is when are you going to be able to build a system where you don't pre-program everything you know it's one thing to program it to pick up my cell phone it's another to program it to pick up stuff and then at some point have it learn about cell phones and say go figure out how to pick up this cell phone and do it safely and by the way don't scratch the front because it's made out of glass so again there's a lot of work going on but I think this is really the place where I have to stop and say I have no idea how we're going to solve those problems I know how to solve the problems I've talked about so far but I think this is where really things are um really open-ended and there are a lot of crosscutting challenges of just building complex systems and putting them together so they work so uh I'll just close in by saying you know the interesting thing is all this that I've talked about is getting more and more real um this chart I'll just tell you is uh dating myself but I built my first visual tracking system in my last year of grad school cuz I wanted to get out and I needed to get something done and it ran on something called the microvax 2 and it ran at I think 10 Hertz on a machine that cost 20,000 bucks so cost me about $2,000 a cycle to get visual tracking to work and so I still have that algorithm today in fact and I just kept running it as I got new machines so those numbers are literally dollarss per Hertz dollars per cycle of vision that I could get out of the system and so it went from $2,000 till when I I finally got tired of doing it about seven years ago when it was down to 20 cents of Hertz so you know literally for pocket change I could have visual tracking up and running so you know all the vectors are pointed in the right way in terms of Technology knowledge I think we've learned a lot in the last decade I mean it's cool to live now and see all of this stuff that's actually happening I think the real challenge is putting putting it together so if you look at an interesting set of objects and an interesting set of tasks like be my workshop assistant which is something I proposed about seven years ago at ikra that you could actually build something that would literally go out and say aha I recognize that screwdriver and he said he wanted the big screwdriver so I'll pick that up and I'll put it on the screw or I'll hand it to him or whatever and oh I've never seen this thing before but I can at least figure out enough to pick it up and hand that over and say what is this and when he says it's a appliers um I'll know what it is so I think the pieces are there is the interesting message um but nobody has put it together yet and so maybe uh one of you will be one of the people to do so so um I think I'm out of time and I think I've covered everything I said I would cover so if there are any questions I'll take questions including after uh class thanks very much thank you so much |
Lecture_Collection_Introduction_to_Robotics | Lecture_4_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development all right let's get started so today video segment is about a small device called the hummingbird the hummingbird was developed at IBM wedson Research Center and in the early 990s and it was published in the proceedings of ikra 1992 the hummingbird mini positioner is a compact device designed for the ultra high-speed positioning of low mass payloads developed for the contact and non-c contct probing of planer objects it can probe at over 50 cycles per second with accelerations exceeding 50 gs the five bar linkage of the mini positioner covers a 13 mm Square workspace to reach larger regions the entire mini positioner can be moved the two main links are driven by high performance moving coil actuators and the link positions are sensed by low mass Optical encoders preloaded bearing pairs of the joints provide High stiffness and zero backlash for accurate XY positioning and a miniature linear servo motor provides 1 mm of z-axis travel the hummingbird system can generate peak accelerations of 50 to 100 G's in all three axes to avoid shaking the structure holding the mini positioner the mechanism is designed to be reactionless during XY motion the dynamically balanced linkage assembly generates no net xy reaction forces during motion because the net center of gravity of the moving Parts remains fixed for all linkage orientations the net z-axis reaction torque generated by the two link actuators is actively cancelled by a third rotary actuator which oscillates only very slightly because of its high rotary inertia one of many applications for high-speed probing is the electrical testing of high density circuit boards when driven by a custom multitrans Servo controller the hummingbird can provide such testing at unprecedented rates at 40 tests per second as shown here the camera and the human eye are unable to follow the probe motion the use of a strobe light however reveals the three large XY moves and the small Z moves in this repetitive pattern here a pattern of smaller moves on a 225 Micron grid is being demonstrated at only 10 tests per second two quadrants of the grid are probed systematically while the other quadrants are probed only partially but more randomly here that same pattern is being probed at 20 tests per second and finally the complete 372 move pattern is performed at a full 50 tests per second details of the motion are shown in the high-speed video being replayed 33 times slower than Real Time Each probing cycle includes one XY and two Z moves and takes only 20 milliseconds to complete yielding a total of 150 distinct moves per second at this speed the XY probe placement accuracy is approximately 5 microns to provide a more familiar size reference the probe is shown here above some ordinary table salt the hummingbird was programmed to contact each three selected grains at 49 different locations at a rate of 50 probes per second the inset shows one of the salt grains after probing and the characters formed by the tip the letters are about 140 microns tall with a typical dot spacing of 20 microns unb the hummingbird mini positioner resulted from the interdisciplinary team effort of these and many other contributors so unbelievable H well obviously when you want to move very fast you have to make everything very light in order to to achieve that if you have moving structure it's going to be impossible to do it and now integrating all that structure in a way where you can get all the stiffness void vibration is not a simple problem okay well uh today's uh uh lecture uh really now is going to take what we have learned uh about the frame assignment the descriptions and take these uh to a manipulator what we're going to do is we're going to take this manipulator and start by looking at a link and try to see how we Define this link and from that description we're going to introduce uh this uh promised description I mentioned earlier the DH parameters or the denavit harenberg notation that would allow us to describe the link and its connections to the next or previous link this is going to allow us to then uh precisely Define the frames that are going to move with the links but that will also allow us to connect the base through the structure to the ector so that will give us the forward kinematics the forward kinematics is this relationship between the lust frame and the base frame you remember we talked about this end effector placed uh at some location in space connected through those links to the base so if we have frame here if we have a frame at the end of factor these two rigid objects in between you have all these links and all of them are moving and the question is how are we going to define those frames how we're going to attach frames to the different links obviously you can go to each of the link and say okay I'm going to go to the center of mass of the link and put a frame and you you you still have freedom in in assigning that frame but that would be fine then you will have to find the relationship between frame on the base link one with your your selected frame and the next frame and the next frame so if you think about it let's say I'm going to put frames at the center of mass this is just one example what is going to happen in term of the relationship between two successive frames how many parameters we're going to need to describe the these two two frames any idea well you have frame another frame you have a homogeneous transformation how many parameters six six all right now this link We Know It Moves just with one degree of freedom and there are restrictions right so if we just go and place frames arbitrary at the center of mass or any other point point we're not going to take really advantage of the fact that there is some set of constraints associated with this mechanism so the purpose of what we're going to do is really to take advantage of those constraints and come up with a minimal description that allow us to uh somehow emphasize this variable this joint variable that is rotating and have it explicitly in the description so what we're going to do we will start with the link description and then we take a look at the link connection and from there we will identify those parameters and we will identify the variable some of the parameters are fixed the length of the link is fixed the relationship between axes so if we think about axis I and axis I minus one there is some fixed relationships between them right as we move this axis there are some parameters that are constant what kind of parameters are constant there what what do you see as as constant between these two axes H the dist distance so basically these axis are maintaining a distance right and in general these axes are not parallel so there is a tilt between them and that tilt is going to be maintained and uh there are uh some offsets that will be introduced there is an angle that is taking place but we cannot see it with just the axis we need to assign the frame and we will start to see that relationship so the link description I'm going to take two axis and arbitrary axis so that we will not just take parallel the parallel axis case so axis IUS one and axis I are connected somehow through this link so if we take a link at the extremities of the link we have the joint axis okay so what are the things that are constant so you say distance how we going to define the distance between two axes come on faster perpendicular so perpendicular perpendicular to a plane perpendicular to an axis but we have two axis so it is a common common perpendicular right something that is perpendicular to both that would measure that distance so if we take this common perpendicular to both axes then we have a sort of I mean this is going to be unique right except if the the axis are parallel then you have infinite way of placing that common perpendicular okay you agree with that selection that makes sense we take the common perpendicular and that will give us this distance so we call it a so now you have to pay attention to the notation because we we're going to describe link IUS one with the parameter a i i minus one which is the common perpendicular to those two axis so IUS one is the common perpendicular between axis IUS one and I all right what else we need to introduce so if I slide axis I along this common perpendicular and I come to the intersection there will be an angle a Twist angle this angle you see it we slide it up to the intersection and there will be an angle we call it the link twist and it is the parameter Alpha I minus one which measures this angle and what we will do is we measure the angle along the vector a i minus one in the right hand sense and you're going to learn how to use your everyone knows how to to measure the angle in the right hand sense just make sure use your right hand not it happens okay so we have two parameters in fact U we're going to see that in total we need four parameters one of them is variable The Joint angle or the joint displacement if it's a prismatic joint and now we identify two alpha and a are constant all the time so once you design your robot these Alphas Alpha z alpha Al 1 to alpha and are going to be con the same for a now if we look at most mechanisms we're going to see that the axes are not always apart most of the time they are parallel and sometimes they are intersecting so if we take the Puma you have this first joint axis and then you have the second one and they are intersecting if you take the rist you have three intersecting axes so when we have intersecting axis the question is what is the common normal so you have this axis intersecting here where is the common normal so we take the plane formed by IUS one and I and take a perpendicular to that plane and that will be a vector perpendicular to both axes which direction so we have this angle but how we Define it because I can take a vector in the plane or out of the plane and that changes the direction of the angle so we have sort of a free variable here to decide in which direction we're going to select Alpha typic what we do is you have the base and you're moving toward the end of factor so you are putting this a the the the vector a you are pointing the a towards the end of vector so it it is very uh intuitive to to create those vectors and once you have a defined then you will be able to to say well a is in that direction and now I take the angle in the right hand side or if it is in this direction you will take it in the other direction okay that's for Alpha now what we're going to do next is to connect those links so we defined the link through these two axes the distance between them the common normal and the twist but if we move further we're going to have another link now that other link will have another common normal right and this common normal will be between axis I and axis I + one so there that common normal will intersect with axis I right it intersect at some location with axis I so we know this point where we have this intersection now what we need to do is to introduce these two other parameters that defines those uh connections and obviously this is perpendicular to the axis I so I don't know if you see this Vector how can you define this Vector with respect to this line a that is I I I use this color it's can you see it you you see this this Vector you see it okay and you see this Vector what are the variables that we need to introduce to Define it so you need angle between the two yeah there is an angle between the two that's correct and this angle can be found if we slide this Vector to the plane we will find it this is going to be so when the link the that following link is rotating we will see this axis rotating with it and that angle increases and decreases and there is one more parameter which is the distance this this offset so what we will do is we will project this Vector on that intersection point and then we can measure the distance Di and now you can see the angle so di is defined by as the link offset and this di is going to be constant for revolute joints but for prismatic joints it's really the direction along which the joint is going to uh affect the motion of the following link so di will be variable if the joint is Prismatic so for a revolute joint Theta is the variable so if like in this figure the this is a revolute joint as in here this Theta will be the variable so Theta I is called The Joint angle and it's variable for revolute joints basically now we have everything that we need for each of the links and if we identify Alpha a d and Theta we will be able to go from one frame to the next as we attach frames here frames there and propagate so here is a a short uh movie segment if we could put the light little bit up please lower the light uh so we have this manipulator and it was designed to show you different properties about different angles and uh you see the end Factor moving carrying an object so we see axis of joint two joint three 4 five six so you have all these axis of rotation uh and now the last joint so let's go and go back to the the beginning so we have uh one axis here we have another axis there and you see this distance this common normal do you see it this is the common normal so as we move this this is fixed but basically the and this is the angle between the axis so what we're going to do is along this first axis we're going to attach a frame we will see that in more details and along the X Direction along the common normal we put the x-axis so now here is a case of common normal between two parallel axis there are many many possibilities but we are not going to make the commitment of the assignment until we place the next joint you see there is this point and we do not want to introduce this road so we will move the common normal there so these are the rules that we will see in our frame assignment now that we decided this that will decide the rest and little by little we build that structure that would allow us to do the frame assignment so now we assigning the frames and the frames are assigned along the z-axis most of the time at the intersection between Axis or along the Comm common normals so this is the frame assigned here for this joint and when we rotate the next joint you have this angle so you have another frame when you rotate about this so all these frames are assigned with the same origin this is the three intersecting axis you see that now there is always an additional frame that goes for the object object and often we assign it depending on the task or the need so you end up with a structure like this basically these frames and each of them is rotating just with one variable and from there you can go and then build your u connection from the base to the end of factor and as you end effector is moving it's covering the space so in the uh just coming here you can start to see the workspace of the robot this is what we call the workspace of the robot the the space where the end effector can be positioned given the joint limits given the structure of the robot there are areas inside of this that will not be reachable because there are joint limits on the different joints and you cannot uh be able to access all these points so the workspace is this volume of the space where we can position uh the end factor and uh you will uh uh we will discuss this later uh more more more precisely about the definitions of the workspace but this is basically what is going to happen in term of how you define the workspace where are these configuration that are not reachable because of the joint limits and uh because of the uh the different length you have on your uh uh structure so this is this is an example of the workspace that we need to study in order to uh position the base of the robot so that the end of factor of the robot can reach uh in different areas okay let's go back to the link so now that we discussed those intermediate links and their connections we need to to to be little bit concerned about how we Define uh the beginning uh of the structure that is the frame as attached to the base and how we deal also with the lust links and there is a lot of freedom there this Freedom comes from the fact that those frames can be can be assigned and moved as long as you are able to find a frame that is fixed with respect to that rigid Val so we saw that for axis I AIS I + one we are taking the common normals we are taking uh those twist angles and if we think about it this Ai and Alpha I are depending on I and I + one so when we say AI the definition of AI and Alpha I is going to depend on axis I and axis I + one which means that from 1 to n having those axis will determine A1 to a n minus one and Alpha 1 to alpha n minus one then now we have to somehow decide about what is a z and a n what is alpha0 and alpha n and that comes from the way we Define the frame attached to uh the last link and the first link so there are many different conventions and those conventions can can vary following your task and requirement but essentially what we try to do is to to carry in the forward kinematics the maximum number of zero parameters because when you put an alpha angle of 30 or 40 or whatever you need to compute the cosine and S and that introduces more constants so what you need to do is to to try to uh Set uh select your your frame in a way that makes a0 a n equal to Z that will simplify the forward kinematics alpha0 and alpha n to be equal to zero and and we can we can do it here so this is Axis one and we have axis 2 and three Etc so for axis Z axis Z is is essentially connected to the base it's fixed with respect to the base but it has a freedom of being defined so what you can do is you actually you can move this axis and make it parallel to axis one and you can even make it coincident with axis one and now you can by make putting them uh uh uh along the same axis you are setting a z and Alpha 0 to zero so that simplifies the number of parameters you're carrying in your forward kinematics obviously you might uh need a different frame but that different frame can be uh computed with respect to this Frame that you are introducing uh by a constant transformation and you can do that separately you don't have to to carry it in the forward kinematics so for the Ender Factor the problem of the end effector is that you your end effector is doing many different things you are carry carrying a a tool and now you you need to compute the forward kinematics to this point you your tool can change you are carrying just this and your task is to control this point or you're you're you're carrying a glass and your description is really related to this object so ultimately you will need a an imposed frame by by the task itself but in the forward kinematics the frame n can be simply the most of the time can be obtained simply by going to the intersection Point associated with the rist and there you get the simplest form of that description well this is not all the time uh uh the case I mean sometimes we we we we uh give you specify well I need this Frame and I want you to find that transformation to that frame so you you can do it that way but in general this Frame that is associated with the tool we uh with origin o n + one that is frame n+ one can be arbitrary placed with respect to that L rigid body and it depends on the tool you're carrying or the object you're handling so for the last link that is for frame n and axis n what we're going to do is we're going to remove this frame in the same way and make it coincident with with axis n which means that we will have a n and alpha n equal to zero and that simplifies again the forward K kinematics so this is uh the summary that we are really moving all these frames and we are putting the frames for Alpha and a the first one and the last one to zero we still have to decide also about the Theta and the D because Theta I and Di I depend on IUS one and I and that means essentially that now we Define Theta 2 to Theta n minus one and D2 to d n minus1 so we still have D Theta 1 D1 Theta n and DN and those will be fixed once we decide those other axes so the convention again is so you remember Theta I and Di one of the two is constant Theta I could be the variable the angle Theta I Theta n will be the variable if it is a a revolute joint so in that case di is constant DN basically so what you want to do is to set the constant parameter Theta I or di I to be zero the variable has to be variable so that means if the variable is D1 then we make Theta 1 uh if the variable is D1 we make Theta 1 equal to zero and if Theta is the variable we make D1 equal to zero so here is an example for the first link what we will do is we will we we selected the axis we reduced those uh uh parameters Alpha and a but now what we will do we will move the point that will become the origin of the frame we will move it to the same point of the intersection reducing this D1 so there is no more D1 and D1 becomes zero so this is by moving the Axis or actually orienting that axis so to make Theta 1 equal to zero for the last link we are going to do the same thing we are going to reduce Alpha and a and make the selection of the point of that axis to reduce either DN or Theta n by selecting the direction of that lust frame because the lust frame associated with that virtual axis n + one is not yet defined so when you define it you define that axis so the result is the DN or Theta n becomes equal to zero so with this we with this convention and those four parameters now we can uh essentially uh I mean basically we defined the DH parameters because the denate heart parameters are in fact those four parameters that we just saw that is Alpha I AI Di and Theta I you have four parameters defining each of the links and each of Mo three of those parameters are going to be constant one of them is variable so in the case of Prismatic joint Theta I will be the variable in the case of revolute joint di will be the variable the first parameter D1 will be set to if we have a revolute joint D1 will be set to what zero if it is a prismatic joint Theta 1 will be set to zero the same thing for DN by the type of the joint so as I said three fixed link parameters and one joint variable and this variable is either Theta I or di Theta I for the revolute case and Di for the Prismatic case so as we said the first two of those parameters Alpha and a are describing the link because we have the link and we have the two axes and the distance AI the alpha I I describe that link what about the D and Theta what what do they do describe joint basically describe how a link is connected Ed to the next one so you have one link and D would give you this translation between them if this is a prismatic joint you have one link you have the other link and you are describing the translation through D if it is revolute link it is going to be this angle so Di and th I describe the Link's connection that is we go from one link to the next and how we connect them it is really through this di or thet one of them is variable okay now you know the DH parameters good so actually uh our task uh and probably the homework will involve Computing finding these DH parameters so you take a link and you go and find Alpha i a i d I Theta I and you go through all the links and once you have them basically you need to use them to compute your transformation so we need somehow to use those parameters uh in our definition of the frames that we are going to attach so that when we go from one frame to the next we are able to describe the relationship using those parameters so the frame attachment is a very I mean it it will become for you very simple once you did a few uh few examples but it's very important because the way you attach your frames uh you will uh simplify or make it more complex and uh we are going to very carefully uh look at this problem and make sure that as we attach the frame we are going to use these parameters in the homogeneous trans formation describing the relationship between two successive frames and then we will be able to to have that transformation and then we propagate Transformations and we can compute the end effector with respect to the base frame by multiplying out all these homogeneous Transformations so how do we proceed with the frame attachment any help what should we do where should we start okay take a look frames what is what what what is a frame what is what is the most critical thing about the frame origin someone said origin so we we really need to decide a rule about how we we select the origin of those frames and we also need to decide something about the axis right a frame is the origin and the axis so you have X Y and Z and I have to make some decision also about the the the axis so any any good idea what what selection you would do for your x axis or Y axis or z-axis what convention you're going to to you need to do something systematic so if you noticed what we did we selected IUS one and I and we said these are the axis and we are looking at the distance between them and the offset between them so this axis has a very important role you want your frame to be aligned with those axis right so you want the these joint axis to to to be uh the axis that Define your one of the vectors basically X Y or Z so let's let's let's quickly Pick One X Y or Z which one okay now all together Z yes so so what we will do is it's very simple you have I mean these you you you built your mechanism you designed it now you have joint axis just along this axis I'm going to pick the z-axis all right so so we we already we already made made a big decision that if we pick the z-axis now the rest I mean once you pick the z-axis and you you if you pick the xaxis their intersection is the origin or if you you think about where the origin has to be then you have little bit of freedom but basically you're almost done so so the z-axis along the joint uh angle is very interesting why because later when when you are going to rotate for instance about a joint immediately you measure the angle Theta about that axis and that is your joint angle and that's going to be very simple uh and if the axis are well selected you will end up with just one transformation so we are going to take the origins as you suggested and this origin has to again make use of the information we we we just displayed with those parameters what is what is unique about those parameters well we said we are using the common normal this point of intersection with the common normal is very important so now we're going to take the origin of our frames at the intersection with the common normal this is the this is the frame that will be assigned along I minus one and this intersection point is very important where is the next important point I know you cannot point because you don't have any pointer but yeah just try to describe it to me yeah somewhere along the x- axis oh no no no I I said this is the first origin of this Frame now there is another frame here for the next joint where where where the origin is going to be and here okay at the intersection of the common normal with this point wow right on okay if you understand this yeah well it is but it's very important I mean sometimes you you you some if you if you select it down there you you run into a lot of other problems you're not using the DI parameter properly and everything becomes so okay we we take the intersection of the common normal with each of the axis and pick that point as the origin simple you remember that okay then we take the z-axis to be along the joint axis that's also simple so Z - one will be defined along axis IUS one and the other one obviously is here okay good so what is left to define the frame x-axis so what do we do with the x-axis so we know the origin and we're going to say the x-axis has to be somehow orthogonal to Z and you are proposing to take this orthogonal to be along a minus one because this axis a minus one is orthogonal to Z IUS one because it is it is the axis orthogonal to the Joint angle so we take and we take it in this Direction I think our problem is solved now because we assigned we we we pick the origin we pick the Z we pick the X the Y is defined in the right hand rule so in this case where is y Yus one it is inside the plane right x y z right you you know this rule like this okay yeah you have to use your hand otherwise you if you end up with the like um the opposite frame all your transformation is going to be well wrong okay good and XI is along there now we need to verify that we can go from zi minus one to Z using the these four parameters do you think this is possible so we have two frames we assign two frames and we should be able to describe this frame with respect to this that is the homogeneous transformation uniquely using those parameters and and we we will do it just shortly but there is no problem we should be able to do it okay uh this simple uh uh well illustration of the frame assignment is much more complicated because there are all these special cases where you have intersecting axis you have parallel axis and you have uh that lust frame that is still not defined so you you you have to to see where to Define it and you have freedom that is added and you have to make your selection and that's why the convention of always reducing getting zero parameters so when you're doing the assignment you're going to run into a lot of special cases hopefully we will span them in your homework midterm and final so you will be mastering that problem all right so always remember this right hand rule it's very important and try to practice with it if you you're not familiar with it okay so the summary for the frame attachment is the following is we we pick the normals okay so you have the axis you pick the normals take the intersection and pick the origins along the origins you place your axis and then you define the x-axis along the common normals and that's it so you have these four steps finding the normals the common normals finding the origin at their intersection taking the z-axis along the joint axis from the point of the origins you selected and then placing the x-axis okay now here is the x-axis which which are placed from the intersection Point toward the next link so let's see the case of intersecting joint axis which happens uh very often in the beginning and the end of the mechanism in the beginning usually joint one and two are intersecting and Joint four five and six are intersecting I mean in in six degree of freedom manipulators because we use wrist with the three intersecting joints uh the mechanics of that is uh is well understood and and it's uh quite easy to build and also the uh there are a lot of advantages in term of the work space and the freedom uh except they have problems with respect to singularities as as we will see later in when we studied the Jacobian but uh essentially we're going to run into this case so we know this point and what we said is that well we we picked the intersection we pick the origin and now this is the origin for frame I and we place the z-axis but the question is where do you select your X Direction and that defines your so if you place your XI in this direction your angle twist angle will be measured about that so it will be in this sense in this direction and if you place it in the opposite direction so this is One Direction if you place it there you get a different definition of your Alpha ey and that is fine because whatever you do you have this Freedom when once you place XI you defined your Alpha and you are going to carry that transformation through the the propagation and and this will be captured in your in your uh homogeneous transformation and because you are going to find the next transformation between XI to the next one everything will work out so that freedom will be accounted for you can come up with a description that use the minus sign or the plus sign and that will work out with the next joint the next joint will account for it and everything will work out you had a question okay so the this direction and the sign of alpha depends on your speaking of X so here is uh an example of the first link um I'm taking a revolute link what we would like to do for a revolute link if the first joint is revolute like in the case of the Puma what you would like to do is to almost say that the fixed frame and the moving frame are identical when Theta is equal to zero so so you are really setting a alpha to be zero and D1 equal to Z for the revolute joint and the only variable is Theta and the Z of theta is when the two frames are identical and that gives you the the simplest form so as you rotate about this joint you are measuring Theta from zero to the value of theta okay for a prismatic joint what you are saying is I'm going to take the two frames to be identical this is this is imposed this is your selection of the base frame and you are placing the frame so that when these two are identical D1 the Prismatic joint so you have a translation up and down measured by this variable and when D1 is equal to zero when D1 is equal to zero you you have the two frame coincident so this is the frame Z and the frame R are identical when D1 equal to Z your variable is equal to zero all right for the last link if we have a revolute link we are going to select we we are going to select the frame so that the DN equal to zero which depends on the following frame that is we are saying DN equal to Z and that frame just measure the angle Theta n and when Theta n equal to Z basically we have xn and x n minus one and xn aligned Theta measures this angle between the x- axis basically as you rotate and for the Prismatic joint we do the same thing to set Theta n to zero that is we have when DN equal to Z xn comes down to be aligned with xn minus one so these are the conventions that you are going to try to enforce in your frame assignment and using this you will end up with the simplest form of the U forward kinematics but then as I said again the tool frame that you will add you will add one more frame that will account for your task and what is nice is once you have all this relation between the base frame and end frame all the the other transformation are constant that is the next transformation will only involve constant parameters so it is a very simple transformation okay let's see the total summary now so what we said is we need to introduce for each joint we need to introduce these four parameters AI Alpha I Theta I and uh Di and what AI is doing AI is measuring the distance between frame Z and I'm sorry between axis Z and axis Z + 1 along the X I Axis Alpha I measures the angle between axis Z and access Z + one about the X I Axis in the right hand sense di measures the distance between the x axis x i minus one and x i along the Z axis and Theta I measures the angle between the x-axis about Z now this summary is very very useful make a copy of this and keep it next to you you're going to be confused about the eyes you are using and about the definition of these make sure that you have a copy of those definitions not far away and you will see that this is very useful so in these definitions we have two distances and two angles A and D are distances a measure distances between the z-axis D measure distances between the x-axis along the opposite axis okay Alpha and Theta measure angles between the Z axis and the x-axis about the other a now what is important is to notice the fact that we're looking at z i + 1 and here we're looking at x i - 1 x i for the DI so be careful with which ey we're talking about okay you're confused enough to make a copy of it right yeah keep a copy I think it is it is really useful all right so let's uh take an example very simple example I'm going to take a planer robot and uh this planer robot is just set of three revolute joints so we are talking about Theta 1 Theta 2 and theta3 so where are the joint axis someone made a sign I think I understood but everyone do you see the joint axis so the J joint axis are coming perpendicular to the plane okay so how do we pick the origins in order to pick the origin so you see three parallel axes so how do we pick the origins what do you need to do you need the common normals right so you have uh between two parallel access many possible common normals but because this is in the plane we are going to use the common normals in that plane which is the plane of this screen so where are the common normals from this axis to the next one to the next one do you see it basically this is the first Common normal the second common normal and if there was a frame there that will be long so you you know the the common normals directly from here so if you have the common normal the common normal are intersecting at this point so the common normal is intersecting here intersecting there there and this will become the origin of the of uh the those uh frames that we are going to assign so for frame X I mean frame one the common normal is intersecting here the Z1 will be out of the page X1 is along the common normal and y1 complete the direct frame so basically you have this as the first frame right any question about this one very simple okay do you agree with this second one X2 is along the common normal and Z2 is coming out of the plane and Y to complete the frame and we are placing the last frame we have the Z3 and we are placing the origin so X3 is taking along the direction perpendicular to the Z axis and along the direction to L3 and that measures the angle theta 3 so between these frames the only variables that you're going to see is Theta 1 Theta 2 theta 3 and now you need to introduce the first frame so for the first frame we said we are going to simplify we are not going to select a z0 oh I mean in an arbitrary direction we select z0 along Z1 so this way we will select the x0 to be coincident with X1 when Theta 1 is equal to Zer and Theta 1 is measured from here so x0 will be along this direction it's too simple all right so with this Frame assignment what we're going to have is the following we defined for each of the joint those parameters and now we have to uh identify those parameters and make sure for one 2 3 4 six whatever number of degrees of freedom step by step we are writing down these parameters so we for form a table and this table is like this you have Alpha I minus one a i minus one that describe the links D and Theta I that describe the connection of this link with the next one and we say joint I one so for joint I alpha0 is equal to Zer there is no angle between Z z0 and Z1 no distance no no no distance between the two it's zero no offset zero and the only variable is Theta 1 so we're going to go through this one by one and because of the fact that in this case the only variables that are going to be introduced are due to L1 L2 so L1 measure what which variable is measured by L1 a the a and that means A1 is going to be L1 and A2 is going to be L2 and basically now we have the description and the connection of each of the joints so we build this table now you have to notice something very important is that Alpha we set the value of alpha zero a we set the value these are constant D in this case all the it's revolute joints so we set the the values but for Theta I didn't measure I didn't go and measure this is uh uh 32° and put Theta 1 replace it with 32° because this is going to move so I'm in this table I'm setting the variable if the second joint was a prismatic joint the D will not appear as a zero but it will appear as a variable so Theta or D will be a variable depending on the type of the joint usually we add one more column the column where we we say configuration shown in the configuration shown you set the value of the variable so Theta 1 in the in the configuration shown is equal 32° you put 32° in that column if needed and sometimes we ask you to measure that variable so but now with those four parameters and this table we should be able to describe the forward kinematics that is we should be able to describe the position and Orient ation of the Endor yes how do you get the position of the Endor of L3 is yeah right now I'm going to frame three now the transformation from frame three to the that point that blue point involves another transformation and sometimes we assign a frame four and we put four here and then you you can find that transformation but for now we are just looking at the risk point this is the wrist point and the variable is already in the wrist point the only thing is a translation that is constant okay we're ready we're going now to find the transformation between two frames and once we found one transformation using these parameters we will generalize it and we will we will multiply out all the Transformations and find the forward kinematics okay I see everyone ready let's go so so here is a frame I and I'm going to compute the transformation from IUS one to frame I and we have these four parameters and you're going to help me to do it we need four Transformations four parameters I'm going to use I mean I can do it directly but I'm going to use four Transformations I will do one at a time so I think about di as an operator and now if I use this di as an operator I basically slide frame I to some other frame that just use this transformation from here to here so this would be a frame here then I rotate this Frame by Theta then slide it by a and then tilt it by Alpha and I have four transformation four simple Transformations that will give me The Total Transformation do you agree very easy so let's let's do it so First Transformation I'm going to call this transformation that takes me from uh I down to p zp and then we introduce zq rotating to to this and then we go here to ZR and then we tilt ZR to z i minus one so basically the way we're going to do it we will move from this Frame to this Frame to this Frame Q to this Frame and slide it up and reach this one and the transformation between each of them involves only one operator so what is the operator between Z and zp you remember those operators that uh uh I think we called Q I think Q of the of the vector so this Q is along the z-axis with a distance di the second one is a rotation about the z- axis with an angle Theta the next one is a translation about the x- axis the next one is a rotation about the x-axis and that will lead us to so if I find the transformation from I to p p to q q to r r to IUS one then the total transformation between my transformation and this is the most important thing you have once you have this expressed in term of the a D's alphas and thetas then you have a general transformation that you can use in all your frames and then you can build your uh forward kinematics so this one as I said is a translation about the z-axis with di this one is a rotation about the z-axis with Theta I this one is a a translation about along the x-axis with a minus one and this one is a rotation about the x-axis with Alpha IUS just multiply them out you get your transformation and we're done and this transformation is simply this so you have the answer you don't have to do it well I have the answer look I to i - one is given as a function of a i - one alpha i - one Di and Theta that's it so we have this homogeneous transformation between this first frame and that final frame and now we can apply it once we have these parameters that is once we form this table we apply this transformation we know those parameters we have the homogeneous transformation correct once you have one of those transformation you you can go and multiply transformation between frames you start from frame n and you go all the way to frame zero and you have your transformation homogeneous transformation from n to zero which is Now function of the parameters that are constant of the links the A's and alphas and D or thetas and the variables D or thetas and with that you have this information about the position of the ector which is contained in this transformation The Lost Vector remember the Lost Vector of this transformation contain X Y and Z and the rotation Matrix in this transformation contains the orientation of the frame with respect to the first frame so you have your forward kinematics great do we have a homework today okay so you're going to to we have we have something about uh FR gr good so you're going to have fun this weekend all right see you next uh next Monday |
Lecture_Collection_Introduction_to_Robotics | Lecture_5_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development ok let's get started so today we're going to the zoo and this video segment which comes from 93 this is from Nagoya University it's quite interesting the moving style of Gibbons shown in this video is called brachiation the brachiation robot is a dynamically mobile robot modeled on the Gibbon it moves from branch to branch and swinging its body like them [Applause] the brachiation robot which we have developed has two arms and no body the total length is one meter and the total weight is 4.8 kilograms the arms and grippers are actuated with DC motors through harmonic drive gears this is the movement without actuation at first the robot doesn't know how to move at all now the robot is going to learn how to locomote to the next bar motion planning of such a robot is a difficult problem because of its non Helana me our robot is able to generate desirable motions by itself using our new heuristic method the algorithm is based on trial and error of animals and human beings to obtain better motions after the motion learning process the robot can locomote from branch to branch forward and backward the posture of the robot is measured by gyroscopes built into the arms and the joint angles are met by encoders the robot is calibrating its gripper position and closes it when it approaches the target bar this motion is so-called the under hand motion the robot can perform another motion called the overhand motion this motion is naturally more difficult than the underhand motion because the robot has to stop the turning of the arm against the gravity force and the movement is liable to become unstable however our robot also succeeded in performing continuous overhand locomotion it needs more torque than the underhand motion but this is more efficient motion with respect to time and energy consumption when the robot fails to catch the target bar it can recover by swinging with arms to obtain energy by a method based on parametric excitation by continuously performing two motions namely the motion to control the swing amplitude and the motion to approach the target the robot can catch the target bar from any initial State indeed well this project continued and probably you will see some more about it alright let's get back to this we're over there so today I'm going to cover a few examples I know this is might be boring topic but we really need to understand how we can do frame assignment because this is the only way you can generate the forward kinematics and what we're going to see is that once we have the D H parameters essentially we will have the forward kinematics which means we know the position and orientation and little later in fact on Wednesday we will see how this can help us finding the Jacobian and later on we will see how can we find the dynamics so these are very important parameters once you define these parameters you define your robot kinematics you need to add the masses and few additional things inertia and then we we have all the models okay so you remember last time I discussed the attachment frames and we emphasized the fact that we have those axes of the joints that are going to play an important role because those axes will provide us with first of all the information about the distance between axis which is the common normals so we identify this you have one here one here one here and when they are intersecting basically that distance is zero and that point is very important then we continue with finding all the origins which are defined do you remember how do we define the origins anyone hey how do we define the origins for for the first so we take the intersection of the common normal with the previous access and that give us this origin this origin etc we said the this joint axis will be aligned with the z axis so we are selecting our z axis along those axes so we have Z 1 Z 2 Z 3 Z 4 D 5 ok and we said once we have those z axis we have the origin now we take X to be along the common normal or perpendicular to the plane containing the successive z axis okay and that is in this case we are going to have those axis and we also defined the different parameters the for D H parameters two distances and two angles a it's called what the length of the links links links length and this is the distance between the Zi axis so ai is the distance between Zi and Zi plus one measured along baek's I axis okay alpha I is the angle between these two z axis the same axis Z I and I plus one measure about X I in the right hand direction and the two other parameters are one of them is variable either the D which measured distance between x axis so it's measuring the distance from the previous X to the current X I along the Zi axis and this could be a prismatic displacement or a constant offset and the last one is this is the most common thing that we're going to see it's the joint angle because those most of those joints are revolute joints and we're going to measure this angle about the zi axis between X I minus 1 the previous X and the next X so I'm going to take an example and we're going to work out this example and this is a typical example of the things that you have in your homework in midterm in finals these are simple examples that we we try to design so simple enough so you can solve them and also interesting enough with the different difficulties of assignment and and everything in them so you see this red frame we are introducing we are giving you a frame we are calling it X 5 Z 5 this is a frame attached to the end effector this is our task and the the goal is to find this transformation from the base to that frame so we need to find the D h-parameters up to this frame and this frame is given to you here it can be not given we can ask you to assign it but we are giving you the last frame so also I should notice this is the typical way we describe the kinematics of schematic kinematics of a robot and I'm not sure if all of you are familiar with it but what you see here is that you have an axis this is the joint axis this representation describes our revolute joint the output of which defines the next axis so this is coming at the output of joint one and the striding joint here that slides like this is a prismatic joint so we represent it this way and the output is coming to define the link that leads to the next joint so if I can get my cursor back so this is the output and here you have the next revolute joint when we represent it this way we mean that the axis is perpendicular to the plane when we represent it here we say the axis is in the plane okay and the output is defining what is this joint revolute joint and the output of the revolute joint is connected to the Indo factory yeah so this prismatic joint actually is sliding along this axis you see these two points these are the sliders so it moves so it moves to the left and to the right so we have only here one joint that has an axis coming out of the plane and this one is the revolute joint you're asking me if we have the case of a prismatic joint that is coming out of the plane well that might happen and you will see the difficulty there is a way of representing that but when we do it we we usually use a sort of 3d representation to show you how it's coming out of the plane so one thing an important thing to remember is this is not mechanism describing one configuration you should try to imagine how this mechanism is moving so you have to somehow capture the fact that if we rotate about the first axis the end of fracture will move out of the plane you see that it rotates about the first axis if it slides so just move each axis and see the in the fracture motion can you see that it slides now if you rotate about the third joint what is going to happen we move in the plane if we were in that configuration and when we move about the last joint we go out of the plane about that axis you see that so you need really to imagine that motion because if we go to analyze the workspace of the robot you need to fill that workspace and you need to find a way to imagine the three-dimensional motion of this mechanism and find the volume that is spanned by the motion of the endo factor now what we need to do is to do the frame assignment and we are going to start by assigning origins and z-axis and in this case we have a lot of things that are already there so you see the first joint axis the second one what particular entity about those axis we have so joint 1 and joint 2 our second intersective so the intersection point is going to be an important point in this case joint 3 and join two are the intersecting so they are sort of what the axis are exactly I mean they are sort of parallel basically you view the other in two parallel planes and the next one the last joint is intersecting with the revolute joint number three so we call this mechanism revolute prismatic revolute revolute and that's our P or R okay any questions here before we start good all right so let's start and I'm going to start by putting the z axis and the origins so the z1 is a long axis joint 1 z 2 along Z 2 Z 3 is coming out of the plane Z 4 is along that rotation and there is basically no question all of these are directly assigned very simple and we are introducing already a distance and this is coming from in fact it will be clearly defined when we we put the origins so origin 1 is there but could you explain why we are selecting origin 1 at that point at the intersection of axis 1 and axis - yes is it minimize number of variables that you've got well they're all 0 I didn't I was not asking about origin 0 if for origin 0 you're right but for origin 1 this is imposed because you have the intersection of z2 and z1 at that point this point becomes the origin of frame 1 so oh one is imposed but we put all 0 the origin of frame 0 at that location because we want to minimize is the number of parameters if we put zero lower or higher we will have a distance that we need to account for it in the parameters here so for origin three I mean the next origin Oh two or two comes to be here if we and how did we assign it at this location it is the common normal intersecting with axis Z 2 we get o - what about Oh three mmm-hmm what is going to be so 3 will be somewhere here here yeah alright okay so far so good now Oh for so how do we define Oh for is along the common normal between uh-huh what did you say between so when we define three we said it is the intersection of three with four right before four it is four and five so because we define five we have the common normal between the two how many common normals you have infinite so we have a choice we can put all four in different location we're going to select all four there and that will contribute to as we will see to minimizing the number of parameters okay now let's introduce those parameters so you are given this problem but there are not enough I mean so what is what is a 0 a 0 is the distance between which axis 0 1 and they are coincident so it's 0 ok now what about a 1 so it is again distance between the z axis 1 & 2 0 what about 2 & 3 that will give us a 2 so there is some distance so there are distance that are not written on the figure and we are going to introduce them so you introduce any time those distances so now we are going to need a parameter that describe the distance from o to 203 and that distance we just put an L you call it L whatever L L 2 we are going to need that distance which is actually depending on the location of v and we are going to need this distance 205 to access v the distance between the x axis I'm sorry well the this you you will see this will come to be associated with the with the following frames that's why we are putting that indices you're not going to use DL you're not going to have need for an l-3 okay so when we start building the the table so we're going to to need to assign the x-axis I mean so far I'm introducing those parameters without really assigning the axes and deciding what are the signs of those parameters that what we need to do is to put the x-axis so what is the first one X 1 what is X 1 going to be to so it will be X 1 is perpendicular to so it's perpendicular to the plane containing Z 1 and Z 2 so it could be in or out of this plane we are selecting x1 out if we select X 1 in that is fine but that will change the sign of the the angle between the X 1 and the following X so X 2 X 2 is how do we define X 2 it is along the common normal and it points from o to 2 O 3 and that is X 2 you see X 2 okay a lot of fun now X 3 where is X 2 Z 3 Z 4 intersecting situation it could be the same direction as x3 or in the opposite direction we are putting x3 in that direction up you agree if you put it down the angle will change when you measure it and that account for it and since then its relation to x4 will be accounted for everything is fine and x4 because we selected over there x4 will go from oh four to the axis z5 so it will be down okay so it is along the common normal from four to five all right so now I'm going to show you the table and you are going to see all the parameters and I think you have them in your hand so I wanted to see if you can compute them without checking your your table can you do that without looking so don't look and let's try to find to find a few of these parameters so if we were to find the parameter alpha 2 so this is alpha 2 so what is alpha 2 alpha 2 is an angle between axis 2 and 3 right measured about x2 so you go from 2 to 3 you see access to you see access 3 and you're measuring about x2 so how many how what is the didn't the angle 90 plus or minus minus because you're going this way okay alright so I'm going to show you the table and then let's check more little more just to make sure you so what is it - alpha 2 is minus 90 so that is correct in the configuration shown in this configuration could you give me what is the value of theta 1 what is Theta one zero that's correct why the theta one is measuring measuring an angle between the x-axis between X from X 0 to X 1 about the z axis and the two are coincident in this configuration what is the variable in joint 2 what is the variable in joint 2 D 2 so if I think about a variable Q Q 1 Q 2 so Q 1 is Theta 1 Q 2 is d 2 the variable so what is the value of Q 2 in the configuration shown it's about this so I know how many inches that is the configuration shop okay so you just measure it and that would be the configuration what is that what about theta 3 which is the value of theta 3 in the configuration shown theta 3 is measuring from X 2 to X 3 about Z 1 X 2 is you see X 2 to X 3 you see x2 it's up and x3 is up they are quality okay correct so let's do something more yeah D for D for as you label origin or okay D for is the distance between X 3 and X 4 where is X 3 and X 4 0 so it's 0 ah so this [Applause] how come this is not corrected where is it let us fix it thank you yes there was that question yeah I had I had another question in the reader you said that a convention would be to pick your intersecting axis X so that the Alpha was always positive is that or did I miss read that oh well the III don't think you you need to I mean you cannot maintain that so I mean there are so so many different frames you cannot keep your L you pick it for the first one and then it becomes negative in the next one so I don't think this is possible you cannot you can never maintain the sign of the alcohol from X I - one - I would always be so you propagate from yeah so I mean as long as you you can you have a choice and you can select it positive then you do it and that is fine but if you put it negative it is fine as well I mean it's it's overall if you start with the - the way you assign your X in or out will change the sign but any selection you make in or out will result in the same answer because the angles will account for that there was a question here okay all right okay well I'm going to take a more more realistic example and this is a real robot we will see this robot here later this is the Stanford the Scheinman arm Vic Simon designed and built this robot this is a six degree of freedom robot it's among the few robots in the world that has a prismatic joint so the third joint on this robot is translating so when you pull here you are moving along this axis the axis of translation so this is a prismatic joint and it has this wrist that has 3 degrees of freedom three intersecting axes actually as you walk in and out you can see a yellow arm in the museum exposition on the first floor and you can see this arm over there so what we're going to do we're going to find the forward kinematics for this zone and this is going to require us to go through the D H parameters first but then we are going to compute these transformation associated with the D H parameters to find all the transformation for this robot then we multiply them out and find x y&z at the end of factor and find this orientation of the endo factor giving the joint angles and the prismatic joint location all right we're ready yes all right so first how can we do this well we need a schematic we need to see these joints axes and we need so here is a schematic that we're going to use it is sort of 3d so in this case the first joint so always we are using the same concept you have a rotation the output is going to the next joint a rotation so let's let me explain so the first rotation the first rotation of the arm this rotation about this axis here and then this is connected to the second one so I'm showing it this way so this here is in the plane containing this axis and this is orthogonal to it so when we rotate about this axis this whole structure rotate ok and when we rotate about this axis the following structure will rotate about this axis now this whole structure translates in and out along this joint the output of this joint is attached to the three joints revolute joints that are intersecting at this point and that form the wrists structure so this is in the wrist you have the last joint joint six joint five joint four and joint three is prismatic joint one and two are orthogonal and joint one rotate the whole structure joint to rotate the structure following about this axis so you have these two first intersecting X so do you see this mechanism now a little bit now the last three joints are going to affect the orientation so when we think about this this is when we move the structure and the intersecting point is here so this this point of is the intersection this structure rotate about this axis it rotates about this axis this is joint five so joint four is like this joint five is along this axis and joint six is along here so if you put your hand here and rotate this you rotate about the z axis if you pull it out it will rotate about this axis and if you extend it completely and rotate your joint six this joint six and joint four will become aligned that is this joint and this joint become aligned if you rotate this up and make this joint and this joint parallel and this is a special configuration a bad configuration we call it kinematic singularity because when we move this joint up and the two are aligned we cannot rotate any more about this axis we are locked it's called wrist lock so when joint five is equal to zero okay so we're going to do the forward kinematics and place the first axis you agree z1 z2 going to be a long second John z3 I wonder why I put it there [Music] c4 c5 and c6 okay any question about the z-axis so now we have our z axis these are the axis of the joint along the axis of the joint we place this axis and obviously we place Z 0 to be aligned with Z 1 in that configuration and that places the origin z 1 z 0 at the same origin z 2 y the origin is there with Z 3 ok for 500 ready so this point is going to be the origin of three four five and six all of them okay common normals perpendicular to 1 2 X 1 you see it 1 2 Z 1 Z 2 X 1 2 & 3 X 2 now 3 4 5 & 6 so X 2 is the common normal in the plane two and three now three and four with yes we can put it upward we can put it upward it doesn't matter it doesn't matter so if you if you put it upward or down downward all what that changes is the Alpha angle and the theta angle and the following angles and everything will be accounted for so there is no problem so x3 will be along z 6 X 4 and X 5 is perpendicular to 5 & 6 so it will be selected this way you can select it the other direction and 6 is selected along the same direction of 5 so we are saying this is the position theta 6 equal to 0 okay obviously x0 is going to be selected here if we select it in the opposite direction it will introduces a theta that is different from zero here the theta one is equal to zero so we introduce distances now we need a distance between the x1 and x2 axis this is d2 we need a distance between the origin of z2 and z3 and what else we need I think that is what that's it we just need these two parameters okay and here we have our definitions to assign and find the values so this time I'm going to introduce these 1x1 years you don't have any information left about the distance between for example the wrist joint and manipulate so the wrist joint and like the end of the oh that was so we are we going to introduce the last frame here yeah so there is no frame g7 we did not introduce it and this frame g7 could be introduced later in any way and without any problem the point all 403 I mean that intersection point is part of the link the last link so it is just the first point on that link and anything you use select later would be fine because you are not introducing any variable it's a constant transformation this is what I said last time because the end effector is here we can think about this point but we might think about the end of the tool held by the in the factor so that frame is always changing so most of the time what we do is we compute the transformation to the respond and then we do an additional transformation for the task okay so we're going to do the first one and we need alpha 0 a 0 D 1 and theta 1 so so we can start like and just we fill it or quickly so what is alpha 0 Y alpha 0 is the angle between z axis which one's 0 to 1 and they are coincident so 0 and what about a zero because the intersecting and what about D 1 D 1 is what distance between x axis 0 anything left theta one so what is theta one huh okay good theta 1 and it is equal to in this configuration is there because X 0 and X 1 are coincident so as we rotate we are going to rotate X 1 away from X 0 and that will give us the volume okay number 2 we let's start from the other side what is Theta 2 it's variable its theta 2 what about D 2 D 2 measures the distance between between what between what X's X 1 and X 2 X 1 and X 2 you see the two so it's D - and what about I shouldn't put D - I should put the l2 next time make it t to equal L - yeah I always didn't I mean I never liked the use of D - later on I call it to Row 2 so we are really not confused with with D the constant but anyway we call it D - I'm not going to confuse you with that D 2 could be a variable here it's a constant what about a - I mean a 1y a measure distance between a1 will measure 1/2 to 1 and 2 are intersecting okay so the variable was alpha why it's minus 90 what is this alpha 1 alpha 1 is measuring the angle between 1 & 2 Z 1 and Z 2 you go from Z 1 to Z 2 about X 1 so right this way minus 90 so if you put X 1 and ay that remember whose question was that if we put the X we will just get a different angle but now on the following one you are going to account for it so you get positive for the first but you pay for it later okay okay 3 is what is Theta 2 43 I mean what is Theta 3 for the third joint theta 3 X 2 2 X 3 so X 3 is down X 2 is down measure about Z 3 Z 3 is this way yeah anyone cattle what is Theta three for joint three zero everyone sees it zero is facing downward from the transformations it looks like you should be facing the same way x1 space okay let's look at it okay oh you say x2 okay so x2 is facing down because it is perpendicular to the plane containing z2 and z3 right so it could be up or down fine well if you put it up x2 if you put it up the transformation between X 1 and X 2 will change and then from 2 to 3 will change but right now with this selection I mean once you have the selection you are accounting for it so now x2 and x3 are parallel the angle between them is 0 so theta 3 will be 0 what about D 3 D 3 is the distance between the x axis right x axis X 3 to X and what is this distance between x3 - x4 along the z3 this is the displacement this is the you see as you move translate this prismatic joint you are going to slide d3 d3 will increase so it's the variable so it is the e3 the variable and so zero d3 a is the zero a is between the z axis this is 0 and alpha is 90 degrees so alpha 3 is measuring Z 3 and Z 4 and this is 90 degree about X 4 okay now once you reach this location 3 actually you have the transformation to frame 3 which is at the intersection point of there is and basically you have your position of the inter fracture I mean the the position at the intersection at the wrist point and this position is not going to change anymore because all the other frames are going to have the same origin so this distance between the origin here and here is going to remain the same later what's going to change with the next joints is the orientation of this frame how these joint move will affect the orientation of the end-effector so z4 will have only one variable theta 4 and z 5 will have a theta an alpha 4 that is equal to minus 90 degrees the Alpha is the angle between z5 and z6 and the last one 90 degrees with theta 5 theta 6 okay yeah I was too - you have a D - there shouldn't it be like L I didn't hear but why is it for I equals 2 you have equal you have e 2 and the DI not variable D 2 is not variable well I talked about it earlier I have if if you want I can change it online but so call it L 2 please I'm sorry it's so every year I have to change it and forget to do it maybe I can do it can I do it can you do it like wow I can do it huh all right all right that is better that's better all right now we have the table whoo well you're going to do a few of these and there will be always one more column configuration shown and you have to remember that you're looking at this configuration and you start to move it a little bit and look what is the value of the variable okay so now that we have oh I have to change the next one all right later okay so each of these each of these I'll do it now because I'm sure I will forget that - let's do this and save it I'm sorry oh yeah thank you okay so now every row that you have here 4 1 2 3 4 5 6 is going to give you a transformation you're going to use this information and you're going to have a transformation so if we go here we're going to find the transformation from this is not 1 this is I my god what is happening to this guy someone didn't find all the mistakes all right so this is going from i to i minus 1 this is the transformation that you are going to compute as a function of your alpha a d and theta so you have these four parameters and you are going to use them to compute these transformations ok and as long as you you are doing the assignment following this convention you can apply this formula so we're going to start here is the table and we do the first transformation you have five minutes I'm kidding so let's do the first transformation and then the first transformation is very simple the only thing on the first row that is variable is theta one and you can find theta one cosine theta one minus sine theta one sine theta 1 and cosine theta one and then you have a 1 here so this is what you obtain the transformation from frame 1 to frame 0 is simply a rotation about the z-axis ok and the origin is 0 the same audition alright now the next transformation is going to be from frame 2 to 1 not to 0 you have to be careful so it will involve only one variable and that transformation involves cosine 2 and sine 2 and because the origin is different you see D 2 appearing which is now should be l2 oh my god have to this is going to this is going to be really tough if we keep going with this I cannot even change it all right all right what about 3 2 1 3 2 2 is so the transformation 3 2 2 involve only this variable d3 there are no variables in the rotation there are 90 degrees which gives you cosine and sine of zero and then basically you get constant and the only transformation involved there is d3 only variable okay so we continue for two three five two four and six two five okay have you written everything down we have it okay now how can we compute the the whole kinematics so we're going to multiply that is we're going to say the total transformation from frame n to frame 0 is we start from N and minus 1 2 1 1 0 and we reach frenzy multiply them out so here is 1 2 0 here is 2 2 0 how did you obtain 2 2 0 now to do this multiplication you can start from here you can multiply n n minus 1 by this and you get one matrix you multiply this and there are many ways of do this doing this multiplication but I'm going to just emphasize the fact that we are going to do it in a specific way because later the way we will do it we will find intermediate computation that will help us find the Jacobian so the Jacobian actually we will see that the Jacobian relies on this vector that is the Z vector of the rotation in frame zero so we need the Z vector in frame zero in order to find the Z vector in frame zero we will start doing this multiplication by taking 1 2 0 2 2 1 by 1 2 0 that give us 2 2 0 and now I have the vector Z 2 in frame 0 this is Z 2 in frame 0 and this is Z 3 in frame 0 you see it this vector so this vector is going to play a very important role later so we do the multiplication always this way we take 1 - 0 - 2 0 3 to 0 by just multiplying one matrix by the previous one so we start from the left and do the multiplication you caption good ok well when I reach 3 0 look what we have here this represent the rotation of frame 3 with respect to frame 0 this is X 3 in frame 0 y 3 in frame 0 Z 3 in frame 0 and this is what the origin of frame 3 in frame 0 ok now notice this is going to remain the same as we continue because we selected the same origin so the origin will not change so four to zero you see that lost column is the same so this last column here because sine 1 D 3 s 2 minus s 1 D 2 well if you go here it is the same it is the same it is the same but what is growing is what is this what is this what this represent transformation 6 in 0 what is this this is the end effector lost frame frame 6 I have x y and z 6 z 6 in frame 0 is so complicated you have the sine cosine of 1 cosine of 2 cosine of 4 sine of 5 all of these are the component to compute the component of that last vector Z 6 in frame 0 this is its X component Y component and Z component you see it now this these three columns are the rotation matrix from frame six to frame 0 now in this transformation I have everything about this lost frame we have the x axis component in frame 0 the y axis component in frame 0 and the z axis component I'm not showing you these two because there is no room and the origin of frame six with respect to the origin of frame 0 so you have everything so now you can form a set of parameters to describe your position and orientation you can say I would select X Y Z to represent my in defector position and then for the orientation what are you going to select pick something so what options we have do you remember we we had this in the first lecture I think or second lecture we have angles like three angles quaternion Euler angles or Direction cosines what are the direction cosines these just pick these you have one two three four five six seven eight nine numbers that will give you your direction cosines representing essentially all the parameters in the rotation matrix involve with the orientation so so this is what we're going to do going to put the position X P the coordinate of the position I will use Cartesian coordinate XY and Z and that will be these three and now I'm going to use the three columns R 1 R 2 R 3 to represent the rest so this is big this is big and this is big so what I'm showing you is the smallest part so this is what this is the Z 6 axis in frame zero this is the y 6 axis in frame 0 and this is X 6 axis in frame 0 all right so the forward kinematics if we don't care about the orientation is very simple but if we care about the orientation and we do care about the orientation it is not very easy to capture and now go ahead and find me the three angles associated with these right how do we do that well we go and say this matrix rotation matrix is identified to three rotations and we compute the inverse of the cosine alpha you remember beta and then we make sure it's not zero no singularity and then we extract alpha and and gamma and and all of that requires you to take those values numerical values once you plugged it because what does it mean c1 cosine theta one right so you plot you measure theta one from your encoder you get the value of that and you have now cosine theta one and sine theta two so the orientation part is quite interesting and that is going to be all the way interesting as we go to the Jacobian in fact this part the Z vector is very very important because as we will see later the Jacobian is going to be formed by just the z vector Z 1 in frame 0 Z 2 in frame 0 Z 4 to the end that gives you a half of the Jacobian matrix and the other half of the Jacobian matrix turns out to be just simple differentiation of these 3 x y and z coordinate so once we understood the forward kinematics in terms of the position and orientation and then once we understood how when we move the different joints when we produce velocities of the joints we affect the in defector velocity with linear and angular velocities then we are going to be able to create this transformation between joint velocities to end effector velocities that uses the knowledge of the axis of rotation of different joints this vector is important because about this vector that we have the velocity of the joint the z axis you remember we selected rotation about the z axis so the theta one is measured about the z-axis and that is going to play a very important role okay any questions all right well I'll see you on Wednesday and we will start working with the Jacobian you |
Lecture_Collection_Introduction_to_Robotics | Lecture_10_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development good afternoon my name is crir karov um I am going to be teaching the lecture today I'm also the co-author of the notes for the course so if you have any complaints directed to me if you have any Praises directed to AMA um I did my PhD here at Stanford about 16 years ago um so I was in your shoes and I've been kind of doing few lectures as well as some of the classes uh completely since um I'm not working in the robotics area right now but I'm staying pretty current in that the the topic of the lecture today is trajectory generation and um it's quite relevant to the video that you saw because um in addition to the control functions the sensor functions uh the underground the the underlying um mathematics is really planning for a path for um object among other objects and that's basically trajectory planning um so what we're going to be talking about today is really the basic mathematics in and that can be used um at higher level planning including the one uh with the navigation vehicle so we're going to define the project first so we have a manipulator arm um and it's starting um we want to remve the manipulator arm from some initial position um Don out it with the frame T sub a um to some goal position which will be the desired position t subc um and the manipulator has is based in a stationary frame um which is s in this case so we want to plan a motion for the manipulator arm from tsub a to tspc um in addition to make things more interesting uh we might have to go through some intermediate points uh like for example this a b and we have that because if we have an environment with obstacles and the manipulator is moving in that environment you want to make sure that you're avoiding the obstacles in which case you introducing intermediate points for the manipulator to move for so U this is the basic problem and uh some terminology we have path points the initial the final point and the Via points that we'll be going through um and then what we want to plan is the trajectory the trajectory in the simplest case is a Time history of the position I velocity and acceleration for each of the degrees of freedom for the purposes of this lecture and basically this class will be uh planning each of the degrees of freedom independently and then assume that the motion is realizable as a whole okay because once you put them all together it starts getting very complicated so for each of the degrees of freedom will be planning the trajectory what kind of constraints uh do we expect to see so there can be spatial constraints obviously obstacles in the environment that we don't want to collide with uh time constraints if the motion has to be done in a particular uh time frame for especially if we have a industrial operation that we are trying to achieve and everything is going on a conveyor and you have to do it in a particular time um and smoothness you want the manipulator to have a smooth motion because that uses much less energy and it's easier to control so these are the type of constraints that will be into the problem um okay so that's the setup for pretty much everything that we'll be discussing today initial Point goal Point intermediate points constraints and we're going to be charting the time history uh from a mathematical point of view it's very simple problem right we're planning path we can look for the solution in several spaces two main spaces really um there is the joint space for the manipulator that's the native space for the manipulator right so we want to go in that case um it is easy to go through this intermediate points because we will know exactly what the joint configuration is going to be for the robot in this intermediate points um so at those points in order to get the actual uh joint uh characteristic will be solving the inverse kinematics at all the path points and then we'll plan for a motion in that space so let's say we have the coordinates of each the points that we want to pass through we solve the inverse kinematics for all these Target points we get the corresponding joint coordinates and then we plan in joint space the trajectory okay so that's pretty simple and it's much less calculations uh there is no problem with the singular ities because the singularities occur in joint space that's where the manipulator cannot move in a certain way because the physical structure is precluding it from doing that okay so in joint space that's immediately obvious less calculations we are only doing the inverse kinematics at this target points a problem we cannot follow a straight line right that's the simplest problem let's say we calculate the joint coordinates for the immediate in the initial point and the target Point forget about intermediate so we have that we convert it in joint space we plan a path in joint space but we have no idea whether that path in actual cesan space where the robot is moving is a straight line or what it is it is what it is okay so we cannot force a particular trajectory very easily okay if that's not uh if that is not a a problem for us if we are okay if it is not exactly straight line but it approximates it that's fine so we have less calculations but if we want to follow a particular trajectory doing it in joint space is very difficult that's much easier to do in cartisan space right we can actually track a shape because we're putting the points that we want to go through in the act cartisan space where the manipulator moves if we give it a straight line to move on it will move on straight line right so um and how do we do that basically we plan in a cartisian space using the coordinates and then to find the orientation of the robot we can use any of the mechanisms that you've learned so far equivalent aess Oiler angles you know all these mechanisms to compute the corresponding angle angles for the joints you can use those formulas right so we can track a shape here it is much more expensive at runtime because what happens um we have a initial point we have a goal point we tra we plan a trajectory how do we make sure that the robot actually goes along that trajectory basically at real time we have to sample points along that trajectory often enough to force that kind of trajectory and then compute for that all the joint coordinates and make sure that that we feed that to the actual robot to go through that right so it's much more computationally intensive to force that particular trajectory um okay and the other major problem is that we have a discontinuity problems here uh because if we are planning in cartisian space we have our nice straight line that we are following in cartisan space we convert to joint space it might turn out that it is impossible to do that in joint space and we'll see some examples right now of this kind of problems so both spaces have pluses and minuses in reality you usually use some sort of a hybrid approach um to limit the computation but also to make sure that you're not colliding with obstacles along the way any questions if you have just ask before I forget the answer so um so let's look at the planning difficulties we have a robot we have a starting point a we have a Target point B it's a relatively simple um two link robot uh this is the reachable space right of the manipulator when you stretch both links you're Travers ing the outside um Circle when you collapse one into the other you're traversing the inside so the um gray shaded area is the reachable space for this um robot so we have two points initial a goal point B they're both reachable for this manipulator right they are in the rable space now if we plan a straight line in cartisan space you see that it goes through a space where we cannot reach this point C is not reachable so the intermediate point is not reachable we wouldn't know that until we actually start doing this computations along the path and suddenly find out that we are running into a singularity okay so that's one type of problem let's say they are all reachable okay we have um starting point a goal point B everything along this um along this path is reachable okay the singularity is right there in the middle well as we approach that singular position your joint velocities go to infinity and obviously you won't be able to follow this straight path it will cause deviation again we wouldn't be able to uh know that in advance if we plan in cion space until we actually reach that point so here is another example in which both points are reachable everything along the path is reachable without infinite you know velocities however the two solutions are actually different are reachable in two different joint space areas so we cannot go from from point A to point B in a continuous motion along that path because point a is reachable from the left configuration if you want point B is reachable from the right configuration and they are not intersecting in the middle okay so um this is um the type of problems that we can encounter when we are planning in cartisan space only so so far we kind of set up the problem and see what kind of difficulties um that can be now let's put some formulas down on how do we actually plan um we'll assume one generic variable U and uh not me U so it's X U Can Be XYZ if you're doing the cartisan uh coordinates it can be um Alpha Beta gamma if you're using Direction cosiness uh it can be thetas Theta 1 Theta 2 Theta 6 if you have a uh 6° of Freedom manipulator with uh a joint U angles so um we'll use that generic variable to denote any of those and throughout the entire um uh lecture here we'll be using that U as the common variable just think about it that when you do the actual planning you you will substitute U for X for y for Z and then plan for all of those independently so um okay we want to go from one point to another Point any space one variable what's the simplest way straight line right you I go from here to there along a straight line that's my simplest uh uh trajectory um the problem with the state line is that we have discontinuous velocities at the path points right because a straight line only gives you basically two parameters and you're not in control of uh how fast you go or acceleration you know there isn't such so um here is an example you want to go from point A to point D via point B's and C's so a is the initial Point D is the target point and B and C are intermediate points right so the simplest trajectory is we go from A to B B to C C Tod along straight lines and as we said we'll see it in the formulas in more detail but basically if we plan a straight line from A to B you know we can't guarantee that the straight line from B to C will have the same velocity at point B as it was the ending velocity of the previous SE M okay so it's going to be a discontinuous jerky motion if you're going if you're starting and stopping intermediate one and then you're starting from the intermediate and in stopping in the next intermediate that's fine but usually those intermediate points are introduced there so that we don't collide with obstacles or we can uh achieve certain tasks in the middle they're not you know necessarily to stop at them and we probably don't want to stop at them because we're wasting uh energy so we want to kind of go continuously from a to d avoiding those obstacles on the side going as close as possible to B and C that's usually the goal so what do we do we do straight lines with Blends in those intermediate points so we start you know usually again we we have time to accelerate on the robot it doesn't just start from scratch so we have have a small blend then we maintain a straight velocity for a while then we get a curve around B to maintain the continuous velocity then a straight line then a curve a straight line and then we decelerate and stop gracefully at the end okay so you can think if you want about this vehicle that we saw if it's planning a path um same kind of way um so that's the next the next uh level of planning and then of course another way to do it is instead of using straight lines we can actually do a cubic polinomial so the obviously PowerPoint here is it almost looks like a straight line with uh Blends but think of it as a cubic polinomial right so you're actually having a higher degree of Freedom curve between each of the points along the path okay that will be slightly more complicated from a formula point point of view so everybody following it's pretty simple but um and then finally if you have a lot of constraints that you want to satisfy along the way uh you might want to use a higher order pols like quintics or septics or whatever uh because uh in this case in the cubic polinomial case uh we we have a cubic polinomial as we'll see in a moment has four parameters so you limit it on how many constraints you can satisfy say you're starting from certain position you're ending at certain position you're starting with velocity zero you're ending with some velocity that's about it if you want to control acceleration deceleration things like that you need higher degree polom because you need more coefficients to satisfy that motion and we'll see that more in detail um but of course the higher degree polinomial you use the formulas get more complicated usually we try to get away with the simplest thing we can okay so far and that again is the planning for each of the degrees of freedom each of the positions each of the angles for each of them you can do that kind of planning independently so let's look at the actual formulas here is a single cubic polom a general um equation for that would be uh of course that's in uh as a function of time would be a0 plus A1 t plus a t A2 T2 plus A3 t to the 3r so as we said we have four parameters here which can um we can use those parameters to satisfy certain conditions for the motion typically what we will have is we'll have as initial conditions some starting point and some ending point right those things will be given you know where do we start the motion and what is the position for each of the intermediate points and the goal point where we want to go so this is two conditions right so at time z u of0 is u0 which basically immediately gives us the value for a Sub 0 um and then at time T sub F which is the duration of this particular interval uh we have some value U subf and that will give us one equation uh for the remaining three unknowns A1 A2 and A3 okay and then we can have more conditions for example for the velocity this is a graph of the velocity of that function now the velocity uh has only A1 A2 and A3 that's just the the derivative with respect to time and again as initial conditions let's say that we want to start at velocity zero so start at rest and finish at rest we will probably not be finishing at rest at the intermediate points but this is the simplest case so U do at Z is Zer U do at TF is zero so that immediately gives us a value for A1 which will be zero if you doted zero Z and then another equation for A2 and A3 so now we have two equations for A2 and A3 one that will come from U do of T subf and one that will come from U of T subf right so two linear equations two unknowns that's the beauty about working with polinomial is that everything is linear right so we can find um the solution um and [Music] as far as the um acceler uh the acceleration is concerned we are toast basically it's fixed we can't control that whatever it is that it's it will come from the formulas right so that's why I saying that if you want to control the acceleration then you need a higher degree polinomial to give you more parameters to work with so basically here is the solution I'm not going to spend the time to derive it right now but it's pretty simple two linear equations with two unknowns A2 and A3 these are the values of A2 and A3 A1 we set is zero because the velocity at the beginning is zero and a0 is u0 because that's the position at Time Zero um so as you can see the trajectory depends on the initial position goal position and the time that we want to traversee that segment okay pretty simple so far now if we are using intermediate points then what we want to do is that at the intermediate points in the middle we don't want to stop so we don't want the velocity there to be zero right so uh for continuous motion with no stops we need velocities at their intermediate points so at Time Zero the velocity will be some value U sub Z Dot and a Time T will be some other value let's assume for a second that those are given we'll see later how to deal with that so those will be added to the initial conditions we'll have the position in the beginning the position at the end the velocity at the beginning the velocity at the end four conditions for parameters a0 A1 A2 A3 again linear equation for yes why do you say the acceleration is a constant because if you differentiate the original equation twice you still get I'm sorry it's not a constant obvious I show a line right so but it is fixed in terms of the value of the acceleration is fixed because we don't it's a function of time but the parameter we cannot control we cannot uh it will be a fixed line so to say right in your prev 63 should be 683 * T I think I probably had it at a particular oh you're right yeah uh no no no no no no hold on a sec U dot no that's the third derivative right this is the this is yeah the third derivative is 6 A3 here um this is the acceleration u u double dot right of the it's a straight line what I meant is that the values A2 and A3 have already been computed using the conditions that we had before right because we had four conditions four unknowns we are Computing it so we really don't have a control over that so at every time it will will be fixed we don't have extra conditions that allow us to control the acceleration so there is no control of the acceleration it's fixed in a sense it comes out whatever it is going to be based on the other computation if we want to control the acceleration so have certain variables that we can introduce there then we need a higher degree polinomial to you know to use for conditions for that right so I I'm sorry if I misspoke I didn't mean to to be fixed in terms of a value I also showed you the the curve is obviously not fixed like it's line but the numbers that Define that line are fixed we cannot we cannot compute them uh based on goal configurations like I cannot say I want the acceleration at Point T Sub 0 to be certain value it will be basically 2 * a sub 2 which will be the determined from before so I have no control over that why is acceleration going down why is why is the slope negative ah this particular curve is you know uh based on the conditions that were put in that particular curve it it it doesn't really it doesn't have to be that way right it it depends on what the numbers actually come with that's for that particular curve because we were starting at that conditions okay okay so let's see where we were here so we have different initial conditions we have certain values and now obviously the formulas are going to be different um a Sub 0 is still U subz because we start at uh from t0 we have this U Sub 0 is the initial point a sub one now is going to be U Subzero dot because that's the condition and then for A2 and A3 we again have two equations with two unknowns and they will be function this time not only of the positions and the time but also of the target velocities okay so if we know the target positions U Sub 0 U subf the target velocities U subz do us subf Dot and the target duration for that segment we can compute the trajectory given those conditions okay now how to find those well um obviously if we know the um actual velocity and um and angular velocity for the robot that we want to have for the actual coordinator uh then we can use the inverse of the Jacobian to find the U dots in the cartisan space right so if we say okay we want the each of the links of the manipulator to have certain velocity because when you have an industrial robot they are all targeted for certain velocities and certain angular velocities that are the characteristic of the mechanical structure you put those in the conditions you can calculate some uh Target velocities another way is um you can have the system choose some reasonable velocities using heuristics for example if we have several um if we have several links that we want to Traverse in the trajectory in addition of having continuous velocity um maybe we want to have some averages on each side of the motion so that you can have a blended continuous motion with the least amount of energy dissipated so you can put this kind of heuristics whatever is important for your type of motion that you're planning for and have the system automatically compute the velocities um and then finally you can actually introduce additional constraints so for example we have U1 is the velocity for the first um U1 dot is the velocity for the first segment uh U2 dot is the velocity for the the second segment so we want to make sure that at the via point where the two segments meet we have continuous velocity so then U do one U1 dot at T subf will be U2 dot at zero right because it's the same velocity and we probably want the same acceleration as well so then here is the second derivative comes in you know as an additional condition and add that to the system system okay obviously if we do that then we have to take some other constraints out because we might have too many constraints if we have four for each of the four for each of the segments then we don't have enough constraints to satisfy we don't have enough parameters to satisfy all the constraints um so these are the type of these are the type of reasoning the type of reasoning that you can use to compute the those velocities typically you will not be given the velocities you'll just want to go from one point to that point to that point and some time frame and by the time might not be even given as well you just might want to use the time by heuristic like you know how fast we want to do as fast as we can do it without actually um going over the the speed limit for that particular motion for the um link okay so this is so far this is cubic splines um interpolation right um any questions so we'll move to linear now uh so linear templation straight line right we have starting time t0 goal time T subf position at the beginning U sub Z position at the end us subf and that's it there is there is no more there is no more conditions that we can satisfy in this case because we have two parameters a Sub 0 a sub one and we have two conditions and that uniquely defines our motion right that was the problem is that if we take an acceleration here it will just be a sub one whatever is the value that comes in from those two conditions and we cannot control it so we have discontinuous velocity now we're going to introduce this parabolic Blends that we were talking about so we have the T Sub 0 is the starting time the t subf is the ending time um and then we have this blend and the blend the first blend occurs at time T subb and the next blend is at time T subf minus t sub B so we'll be assuming that the length of each of the Blends are the same for Simplicity I you know why make our life more difficult so in this case the the equation for the parabolic blend itself is um U subt is 12 a t² so we have one parameter which is a um and we want to introduce some conditions for this parameter so that will give us one more condition that we can satisfy which in our case is velocity you know we want to make sure that the velocity is continuous throughout the motion um so the velocity here is simply a * T and if we put a condition for a a constant acceleration uh for example then that will give us that value a um or um in that case basically we'll have U subt is 12 uble do t² where uble do is the acceleration okay and that acceleration we'll see in a moment how we can determine for continuous motion so following so far so another um so if we want to have continuous velocity um then basically we can calculate the time for the blend from a condition for a continuous velocity so we want this function at u subf u sub Z and uble Dot will give us the value for the blend for that particular region that achieves continuous velocity around okay um if you want to see the actual condition the actual equations um there in the book they're a little bit more complicated but it's basically a second degree of Freedom polinomial where T here is the duration of the entire motion from T Sub 0 to to to t subf so we basically have here the equation for the motion of using straight line with parabolic Blends in a continuous fashion from time T Sub 0 to time T subf everything here on the right side is given and it's function of things that are given T Sub 0 t subf u Sub 0 U subf and we will see u double dot the acceleration is not given right away but we'll see how to compute it very easily yes that Green Point TB to the left is that TB or is actually t0 plus TB it's T subf minus t subb it makes it I'm sorry the left that's t0 that's T subb we we want to make sure that oh uh you're right it should be huh interesting we want to make sure that the time of the blend is the same right which is T subb so that should really be uh T Sub 0 plus T subb because you want that to be the location of the blend okay let me move over that I can't resolve it right now but I mean it makes sense the the aidea is that the Blends have the same amount of time right okay I guess people are assuming that t Sub 0 is zero and that's why it's t subb but it doesn't have to be so from that point of view that's right and I think that these formulas these formulas are computed with that in mind let me just check okay we'll check on that and get back to you um okay any other questions I'm glad you guys are paying attention that's good um so now if we have several segments things kind of get a little bit more hairy um let's say we have the positions of the different points that we want to go through we have um the slope of the different linear Blends uh of the different linear portions which will basically give us the velocity um then we have the time directions which are in this particular case um I to J is a segment so a typical segment is from uh T subi to T subj and the duration of the entire segment will be t d i j and then um the duration of the next segment is T sub DJ K Etc then we have the duration of the actual Blends will be denoted as t subk for each of the blend um we have the slopes and then the duration of the straight line segments will be denoted with t sub JK which is the straight line between position J and position K okay so these are the parameters that we are introducing here um so then we have t sub I is the first blend then T sub i j is the straight line segment this subj is the next blend etc for all of them okay the slopes we already denoted with u dot i j j k k l l m so what is given here we we'll come back to the picture uh actually let's go back and look at that um what is given is the positions U sub I U sub j u sub k u sub L those are the points that we want to go through right the initial the final point and the intermediate point points that's one of the things that is given then uh the next thing is the desired time durations for the entire trajectory from one point to the other so this whole thing is the only thing that is given we're not going to be we'll calculate all the others we just want to have this blend like linear section with the blend for the entire portion for the next portion Etc so those T desired i j t desired JK etc those are the things that they're given the ti the ti JS Etc will compute those okay yeah you don't actually go through the points your desired positions hold on to that that's we'll address that very good point yeah so desired time duration and then the other thing um that is given or can be computed is the magnitude of the acceleration so usually this is certain limit for the particular um joint that we using say you know it can't go faster than that or with with faster acceleration than uh than that and that would be your number so those here this is not denoted here obviously because it's not a graphic term but the magnitude of acceleration is given as well so using those we want to compute the blend times so how long each of the Blends are the straight segment types times the velocities the signed accelerations because here we just had the magnitude of the acceleration but we don't know whether we are accelerating or decelerating in and that will depend on the motion uh and that's basically it so those formulas are given in the book the notes uh but we'll also look at them here briefly um one note is that the system usually calculates or uses default values for the acceleration based on the particular robot based on the mechanical structure you know how fast you want to drive it you know what's the workspace Etc um and also the system can calculate desired time durations based on default velocities all right pretty simple so here is the formulas uh they will be different clearly for the first segment and the last segment than the intermediate ones uh because if you remember the picture that's the one thing about okay if you remember the picture we starting here with a full blend in the beginning and we are ending with a full blend so that we can sort of accelerate and decelerate smoothly and then in the middle we have kind of half Blends for each of the segments so the formulas will be slightly different uh for the different ones but it's all computed based on the conditions uh that we had so for the first segment we given we given um U sub one U sub 2 and then we given the magnitude of the acceleration or the system has computed that so then we can compute the actual acceleration based on the sign of the difference between the positions whether it's an accelerating or decelerating so once we compute that and then we know the the duration of the entire uh blend uh part not blend part but the entire segment uh then we can compute T sub one using that uh then we can compute the velocity for that segment uh U do one two and then we can compute the time for the linear part of the BL of the um of that part um then moving on to the inside segments uh um again we can find the velocities just simple you know position over time um then we can find the signed acceleration the same way as for the first one we can find the time for the linear blend uh by using the velocities that we found and the acceleration and then finally we can find the time for the um straight line segments by just subtracting from the whole time for the segment uh the times for the Blends and as you see here the Blends here are half half on each side um and then when we get to the end similarly to the first one we find the actual acceleration the signed acceleration we find the time for the last blend we find the velocity and finally we find the time for the last straight line segment okay this is I mean it kind of looks hairy but it's very simple formulas very simple derivation basically you know second degree equations for the uh times up there and SAR linear equations for the velocities and the accelerations so using those sets of formulas we can go from the beginning to the end and compute the trajectory and I don't know exact what kind of homeworks you guys are getting but you might actually have to do that for a for a project so that you can understand how it works okay so so far fine yeah yes can you have a question maybe to what he was asking yes I I'm I'm coming to it right now you mean about not going through the point right okay here we go so um what you actually see here is you're not going through the actual point right you're going around them now remember that we introduced this via points the main reason really to to have this via points is when we are planning a motion for a robot with obstacles we want to introduce this kind of intermediate points to make sure that we go around the obstacles so it's really not that important that we go through the exact points unless you know we want to force it and we'll see right now that we can force it but in princi principle the Via points is just to make sure that we avoid certain spaces in the workspace okay so it's not that important but if we do want to go through them then we have you know several mechanisms uh we can introduce Pudo via points so here is the original via point that we want to go to we can introduce on both sides you know on a small distance to Sero VI points and do the planning for that and then the straight line will go through the vi point if we plan it the right way okay if they're close enough uh we can also Double U okay the other thing is we can use a sufficiently high acceleration to actually force it through a particular Point um or if we want to stop there we can simply repeat the vi point and then we'll have you know uh we'll make sure that we go that through that in particular Point um that will obviously affect the formulas we want to make sure that we don't have division by zeros Etc but there are mechanisms um the bottom line is that these via points are really there so that we have a general motion around the space that is avoiding obstacles okay now these were the two mechanisms we can use um cubic pols or uh straight lines with um parabolic Blends um as we said if we want to satisfy more conditions then we can use higher order polinomial for example um let's say we given two positions two velocities and two accelerations for that particular segment right now we have six conditions basically right so you know we can use a quintic we have fifth five fifth degree polinomial six parameters plug in those conditions there six equations six unknowns you know because some of them are relatively simple you Subzero us subf we'll be taking parameters down so it's not going to be that complicated if we use linear equation for that okay um the formulas are actually in the book if you're really curious to see what they look like um now another thing think is we can use different functions we're using polinomial because they result in a linear equation so it's relatively simple to solve uh if you want to uh if you feel particularly challenged that that particular day you know you can use exponential functions trigonometric functions whatever you want you know to plan the the trajectory uh in the space yeah what are you talking the not uh oh uh yeah the the cig book The the recommended book uh is that might not be in the notes here right yeah it's in Craig's book sorry they're not in the notes but they are available and I think that 718 is actually refers to the to Craig's book I don't think you should worry too much I don't think you'll be getting that on test or anything like that but you know um okay so so far um well let's stop for a second and and see um so far we basically looked at different mechanisms to plan paths uh given conditions uh for the trajectory okay uh and it it's been very theoretical right these are just you know math formulas of what you can do which is the underpinning that will be good to know um now what do you do when you're actually doing the planning for for the robot so uh runtime path generation uh so basically we need to feed something to the control system to tell the different um joint uh uh you know positions velocities etc for the different joints of the robot right so Theta here stands for you know Theta 1 Theta 2 theta 3 Theta 6 depending on you know whether it's a six degrees of freedom revolute joints or free Prismatic free revolutes whichever you know it's a generic generic terms and those are the actual values that we feed to the to the robot um so what do we do um the path generator computes the path at some update time at some update rate um so we saw the beginning Point intermediate points we can compute all those values using the formulas that we saw if we are planning in joint space directly right um let's say we're using cubic spines we can change the set of the coefficients at the end of each segment and feed that to the control system right so we start with certain uh set of coefficients we fit it to control the mo the robot is moving when we approach the other one say okay at time this you know feed those numbers at time that feed those numbers Etc and we get that um cubic spline um if we're using linear with a parabolic blend then we have to check at each update whether we are in the linear portion or in the blend portion because we have different formulas for the different and depending on where we are we Feit those kind of values right and we're saying here updates basically you know at certain frequency you know that you're updating the control system you compute the points and you feed them right you following so far okay so at the cubic spines we have those particular update points at the linear with parabolic we have to figure out which part of the formulas we are it's not a big deal we have the formulas it's simple computation um the problem of course is that we're not following a particular trajectory we're just moving kind of continuously in the space if we're doing the planning in cartisan space uh then we calculate the cartisian position and orientation at each update Point using this same formulas and then we have to calculate the joint space coordinates using either inverse Jacobian and derivatives or uh the find the equivalent frame representation and then use the inverse kinematics functions to do theta Theta Dot and Theta Dev Dot and you should know how to do that now from you know the kinematics that you've done so far right and the inverse kinematics so this is how we can compute all that um on top of that you have to remember that this is typically what we saw so far is just for one parameter uh so we have to be careful to make sure that the motion is continuous if we're planning for all three of the parameters that the times are the same so things get more complicated when you're trying to build a full system system but the underlying the underlying technology is what it is here okay um so that's that's so much about the uh trajectory planning and different um you know parameters and how we do that computation um now if you don't have any questions we we can talk a little bit about obstacles um so if we have and there is a whole course on um you know motion planning um at least there was when I was at school Jean CLA latam was teaching it I I assume that he's still doing that um it's a very very cool course that's that's one of the I did my thesis in that area so it it's a lot of fun um so in in that case uh basically there is several considerations that we need to keep in mind um if if we let's say we have a six degrees of freedom Puma arm right um so the question then is do you plan uh do you PL you uh plan a path for the whole manipulator when you're doing that are you dealing with the global Motion in the space or the local motion where just the end effector is so typically you do some sort of a combination between a global and a local motion planning you're using Global motion planning when you're moving from a relatively empty space and you know you know you're not going to be hearing you know hitting obstacles and then when you get at the place where it's more cluttered with obstacles then you switch to a local more precise motion planning for the end Defector only um so that's one type of um that's one type of uh planning um another one is a configuration space approach and I'll show you a few slides on that in fact let's switch to that let's see what do I have um okay let's switch to that um okay so let's say we have a we have an environment with obstacles here okay and we have a point robot a small circular Point rot that we want to move around this environment okay um and that robot actually has certain uh this circle has a radius okay it's not just a point but it's it has a substance um so what we can do is we can plan in configuration space uh basically we take the obstacles and we grow them with the size of the uh Point robo if you want okay um in which case if you look at the Red Dot there uh the idea is that if we plan a path for the Red Dot that doesn't collide with this grown obstacles then we know for sure that the circle uh the circle is not going to collide with the smaller Target obstacles okay and so then we just have a planning for path for one point in that space which can be done many different ways um geometrically these gr obstacles are called uh c space configuration space obstacles um and this is a a c-space planning approach and what we can do is we can put the GD around it and then plan the path of the point around this grd so that it doesn't collide with the grown obstacles um and then when we get back to the circular robot it's not going to collide with the obstacles itself okay if we have a um several degrees of freedom robot or we start adding orientation uh then this configuration space obstacles can be not only planner which is here but then you get a free dimensional obstacle because you're thinking about the orientation that you're approaching with or you can get many dimensional obstacles if you have many degrees of freedom so in its most generic form that becomes plan planning for a n dimensional um path in a m dimensional space okay then you can start talking about um uh you know kind of high uh High math there let's go back okay so we have um c-space planning for a point drop it uh so you can put a graph representation of the free space build a quad tree and know which part of the free space you are and then path a plan uh plan a path and you can use things like the artificial potential method to tells you like as you go closer to an obstacle you can have a force that is re you know repulsing you from the obstacle and as you have the goal you have a force that is attracting you to the goal and based on that you you can build an artificial potential field which is by the way what um Osama did way back on his thisis was very revolutionary work at that time and and then use those and you get in all kinds of interesting things of getting into local Minima Global Minima Maximas Etc um so it's a fun thing um and then to add on top of that uh you can have multiple robots moving in the same environment um this is very much similar to to the video that we saw in the beginning like if you have a planning path for autonomous vehicle if it's the only vehicle that is moving on the streets then it will be the previous approach if you have several Vehicles then you have certainly multiple robots so you have to use those other robots as a uh detractors and then there will be repulsing force from them uh you can have moving obstacles you can have uh moving robots you know things kind of get interesting right uh but that's not going to be covered here if you are interested in that you know check out the uh motion planning um area course I think um I'm about done unless you guys have any questions I think the TA had some announcements to make first do you have any questions on the lecture is there any way that you can post on the website the uh well yeah I think that's that shouldn't be a problem not to talk to but most of the this all this material should be in the notes so but we can we can certainly do okay let me give you the doesn't want to go so stuck all right uh couple announcements first of all home the next homework is due on Monday at 5:00 p.m. so can either turn it in to class on Monday or just put it in the Box um and the next thing is we're going to pass out the exams and solutions right now we're going to put them on this desk and I think they're ordered in some way um um Solutions will be right here Solutions are right there I'm going to put trying to divide into three so uh a through H through H can be right here J through P yeah J through P so as a result for a boring lecture you get to get your exams if you stay until the end to the end do remember what the average was 80 uh |
Lecture_Collection_Introduction_to_Robotics | Lecture_14_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development okay let's get started so today video segment is about about uh tactile sensing now I wonder what is difficult about building tactile sensors anyone has an idea so what is the problem with building a tactile sensor oh you used to see the video first okay so yeah do you need motion to be able I mean do you need a perturbation to be able to see what you're touching sometimes um well yeah sometimes you I mean uh human tactile sensing is amazing so you have uh the static information so if you grab something now the whole surface is in contact and you can determine the shape right so so what does it mean in term of uh like designing a tactile sensor just if you think about the static case must be soft and valuable Mal yeah you need some softness in the skin you are putting then you need to take this whole information uh what kind of resolution you need if you are touching you feel the edge you need a lot of pixels right so how can you take this information and first of all how you determine that information what kind of um procedure you yes well I mean there's there's an element of pressure like how hard you're the average how hard you're touching on all these okay so you can imagine maybe a sort of uh resistive or cap captive sensor that will deflect little bit and uh give you that uh information how many of those you would need you need sort of an array right so how large like let's say this is the end Factor I'm trying to see if you get that problem you're going to have a lot of information here and you need to take it back and you have a lot of wires you have a metrix and you're going to have a lot of basically information to to transmit so the design of tactile sensors bring this problem of how we can put enough uh sensors and how we can extract this information and take it back so these guys came up with a an interesting idea here it is uh the light please a novel tactile sensor using optical phenomenon was developed in the tactile sensor shown here light is injected at the edge of an optical wave guide made of transparent material and covered by an elastic rubber cover there is clearance between the cover and the wave guide the injected light maintains total internal reflection at the surface of the wave guide and is enclosed within it when an object makes contact with it the rubber cover depresses and touches the wave guide scattered light arises at the point of contact due to the change of the reflection condition such tactile information can be converted into a visual image using this principle a prototype finger shaped tactile sensor with a hemispherical surface was developed a CCD camera is installed inside the wave guide to detect scattered light arising at the contact location on the sensor surface the image from the CCD camera is sent to the computer and the location of the scattered light is determined by the image processing software using this information the object's point of contact on the sensor surface can be calculated to improve the size and the operation addal speed of the sensor A miniaturized version was developed the hemispherical wave guide with cover the light source infrared LEDs a position sensitive detector for converting the location of the optical input into an electric signal and the amplifier circuit were integrated in the sensor [Music] body the scattered light arising at the point of contact is transmitted to the detector through a bundle of optical fibers by by processing the detector's electric signal by computer it is possible to determine the contact location on the sensor surface in 1.5 [Music] milliseconds through further miniaturization a fingertip diameter of 20 mm has been achieved in the latest version of the tactile sensor it is currently planned to install this tactile sensor in a robotic hand with the aim of improving its Dex [Music] terity okay a cool idea right uh because now you're taking this information and transmit and taking it into a visual uh image and transmitting the image and in fact this was done long time ago um I I believe um the emperor of Japan was visiting that laboratory and I he saw this and he was quite impressed um before starting the lecture just wanted to remind you that uh we are going to have two review sessions on Tuesday and Wednesday uh next week and uh uh we will uh again sign up for two groups I hope we will have a balance between those who are coming on Tuesday and Wednesday uh we will uh do the signing up next uh Monday so those who are not here today be sure to come on Monday to sign up all right okay last lecture we discussed uh the control structure we were talking still about one degree of freedom and we are going to pursue that discussion with one degree of Freedom so we are looking at uh model the dynamic model of a mass moving at some acceler option X double dot and controlled by a force F so the control of this robot is done through this proportional derivative controller involving minus KP x minus X desired and minus KV x dot so the KP is your position gain and the KV is your velocity gain now if we take this blue controller and move it to the left the closed loop behavior is going to be written as this second order equation and in this equation we can see that we have uh sort of mass spring damper system whose wrist position is at the desired XD position so KV is your velocity gain and KP is the position gain now if we rewrite this equation by dividing it by m we are going to be able to see what Clos Loop frequency we have and what damping ratio we have and every time at the lecture time this finishes so so what is your Clos Lo frequency KP is equal 10 and uh the mass is equal one what is the Clos Lo frequency square root of 10 and what is the damping ratio little bit more complicated but we can rewrite the same equation in this form 2 Zeta Omega and Omega Square where Omega is your Clos Loop frequency and where Zeta is this coefficient KV divided by 2 square < TK of km and Omega is simply the closed loop frequency square root of KP ided by m so you remember this but now the difference with before before we had natural frequency so we were talking about natural frequency and natural damping ratio now this is your gain and you are closing the loop so this is your control gain it's the closed loop damping ratio and the closed loop frequency okay so the only difference is instead of a natural system with spring and damper now we are artificially creating uh frequency through this closed loop or we're creating this damping ratio through KV so basically this is what you are going to try to do you're going to take your robot you're going to find those gains KP and KV and try to control the robot with those gains so again thinking about KP and KV KV is affecting Zeta right and KP is also affecting your Omega now when you're going to control your robot what what is the objective what are you going to try to do let's think about it you're trying to go somewhere right or you're trying to track a trajectory so what do you want to achieve with those I mean here is your behavior what do you what would be good to achieve here yes high frequency and critical D so we want to have a critically damped system most of the time so we will uh reach those goal positions as quickly as possible without isolation so KV would be selected to achieve that value and for that critically Dam system what is the value of Zeta anyone remembers it was only two days ago Zeta is equal to for critically Dam system Zeta is equal to Unity One when Theta is equal to 1 that is when KV is equal to 2 square < TK of KPM you have critically Dum system so basically if you know your KP if you already selected your KP and if you want critically Dam system then immediately you can compute KV from M and KP right for that value for Zeta so basically you are trying to set SATA what about Omega so now we need to set KP in order to compute Zeta and how do we set Omega someone M okay no idea so you have your your Puma you go and you want to control let's say joint three we can do it if you want where my glasses and get it out here is the simulator oh that doesn't have an F Factor let's take this one so here are your gains and right now if we ask the robot to so the robot is floating and if we ask the robot to go to its zero position it's going to just move and it's moving with a KP equal 400 and KV equal 40 these are the gain we set for the robot but in fact uh this is controlled also with Dynamics so we will get to this little later but if we want to to see the control without Dynamics we take this probably non dnamic joint control so this one so let's float it a little bit actually I can exert a little Force outside and see if it can move it's really solid well you can not move it too much so let's reduce the gain here so the springness KP is 40 so see now if I apply a force there is a deflection right and when I'm going to release it's going to go there aate little bit tiny bit not too much in fact this has a lot of friction natural friction if we remove the friction and do the same thing it will probably ulate more m not enough okay wow still there is friction no so let's put little bit minus how much - two this is minus 20 I think it will go unstable wow so so we we we see that your gain cannot be negative it will oh can you stop okay we need some friction otherwise it will not stop ah so so in fact uh you can see there is a lot of coupling I move just one joint and everything else is moving let's make this game bigger this is Joint one so if I pull on joint two and release look at joint three what's is happening so there is an inertial coupling coming from joint two on joint three just by moving joint two you are affecting joint three you can see again joint two release and Joint three is moving so in order to avoid that uh disturbance coming from the dynamic what what should we do with KP make it smaller or bigger yeah you're not sure should we try it so let's uh make it bigger how big 400 okay 400 now you realize with 400 this is not damped enough because we need to compute uh uh this to make it little bigger so let's make it 20 okay so now what do you expect the disturbance will increase or will be reduced when I'm going to release more disturbance or less hey less who agrees with less okay and who disagree with L everyone else okay so this is less yeah it is less actually you you're you're moving little faster and you are you are still isolating and the isolation is because we don't have enough damping here so if we increase the the damping it will oscillate less and if we increase the gain do you see what is happening now it's going very quickly to its position so in fact the the coupling this is the degree you look at the 90° between joint link two and Link three it is maintained almost in fact if I increase joint two as well it will be hard to move it so what is happening now with the with the response do you see the response when we went to 1,600 faster or slower slower faster so so the dynamic response of the close loop is faster with higher gain well then should we increase it like keep increasing I don't know we can try a limit at some point so what is the limit so let's make it 3,000 now joint three is locked it's not moving anymore should we make it more okay so what's going to happen it's not moving anymore it's now the problem if this was a real robot with 30,000 work why your motor is going to saturate at some level you have big motors yeah the saturation of the motors is one thing but suppose you you have really big motors it's not the limitation you have some sort of um air drift well we we'll discuss it a little later but essentially what is going to happen is that you remember uh inside inside the structure you have Motors you have Transmissions you have gears and all of these are going to move and they have flexibility in the structure this flexibility makes it that you start to excite those mode of the flexible system and as you start moving the motors start to vibrate and if you have flexibility in the structure the structure start to vibrate and when you hit those frequency of vibration this is system will just go unstable so our KP this KP that we went oh we closed it just one second let's go back to so this KP we have here this KP cannot go too high we want it as high as possible to increase what what what it does when KP is high disturbance rejection because errors uh coming uh Dynamic coupling coming from other links will be rejected it's stiffer however KP cannot go too high because KP is deciding the closed loop frequency and this closed frequency can go as high as those unmodeled flexibilities actually we cannot even come close to them we have to stay away from them so Omega cannot be too high which means KP can has a limit but we want to achieve the highest KP so what is the relationship between KP KV and those performance so from those two equations we can write KP is M Omega square and K V is M to Zeta Omega right just rewriting these two equations and Computing KV and KP so when we are controlling a system we are going to set what we're going to set really the Dynamics of the system which means we we need to set Zeta and Omega so we set Zeta and Omega and we can compute our KP and KV most of the time Zeta is equal to one so KV is M2 Omega and so all what is left is to set Omega so for 400 Omega is equal to what in the case of the fora in the simulation we had 400 KP so Omega is equal to square root divided by we we well m is equal to one let's say in that case it's 20 it's 20 multiply what is the frequency the real frequency Omega divided by 2 pi right so what is your frequency about let's say divide by 6 20 divided by 6 so it's very low 34 Hertz in fact if you're lucky you can go well to 10 Hertz I mean this would be great so when we go to 1,600 this is this is uh this is really nice 40 divided by six well in practice you start with very low gains and you start tuning your gains up up up up and suddenly you're going to hit that noise start to vibrate so go down but we will see uh some ways of doing this uh in more uh precise way but again what you are seeing here is KP and KV now if we think about two different links one link that is heavy and one link that is light m equal 1 and m equal 100 your gain KP is going to be for the same frequency is going to be much much bigger for the bigger link so that gain is scaled by the mass and because it is scaled by the mass we can think about the problem of setting the gains for the unit Mass system you remember we said if I'm moving joint two the inertia of joint two is changing big small so we need to be able to somehow account for the fact so I set my frequency I set Omega and set Zeta and now I computed KP and KV but M doubled so I need to update my gains right if I want to move with the same Clos Loop frequency I need somehow to update my gains and that becomes nonlinear control so we talk about the unit Mass gains so let's just imagine that your system this Mass was unit Mass your gains will be simply Omega square and 2 Zeta Omega which is for one this would be 2 Omega very simple just set Omega and you get your KP and KV okay but we know the system is not going to be i unit Mass so for the this Mass system what are the gains gains from this KP Prime and KV Prime what would be KP for m a system with mass m using KP Prime KP will be M time KP Prime and KV just linear so you take M and you scale your gains okay well what is the big deal about this why I'm talking about it well the big deal is that m is going to change so even for one uh changing Mass you can make this nonlinear and scale and track a constant frequency and constant damping ratio but for a system with many degrees of freedom we have a mass Matrix and we are going to use the same concept we are going to say I look at the unit system and then I scale the unit Mass system with the mass Matrix and everything will work exactly in the same way and I will be compensating for the variation of the mass this is the nonlinear Dynamic decoupling that we're going to introduce and it is based on the idea that I design the unit Mass system and then I will scale the unit Mass system with the mass Matrix well in this case it is just a a scalar simple Mass so this is what we call the control partitioning if I have a system with a mass m I basically de compose it in the mass and the unit Mass system so the blue is the unit Mass system and M is the the scaling of the unit Mass system so I can now design a controller for the unit Mass system with KP Prime and KV Prime and then the KP and KV for the original system will be just scaled by that mass so here is my controller F I'm going to write it as M time F Prime where F Prime is this quantity a PD controller designed for unit Mass so we always denote these as primes of KV or KP so when we say Prime we are talking about the unit ma Mass system the controller of the unit Mass system F Prime and F is M * that F Prime that will make more sense when we go to the multi- degree of Freedom controller because M becomes the mass Matrix okay so essentially we have our initial system that is now controlled as a unit Mass system scaled by the mass itself self and the behavior of the whole system is like this well the dynamic Behavior the dynamic response and the damping ratio are like this but we have to be careful about other characteristics like disturbance rejection stiffness they are not and we will see that in a second the dynamic behavior of the closed loop is is like this so you design your controller for the unit mass and basically if you scale with that mass then you have the behavior of the unit Mass okay so in this case what is Omega for this system it is simply the square root of KP Prime and uh okay now we are going to introduce one more element we talked about it on Monday and this is just a tiny nonlinearity let's add some friction so we started with the system without any nonlinearity and now I'm just adding a little bit of of friction nonlinear friction like uh some stion on that joint so the equation changed completely that is it's not L nonlinear anymore we cannot uh just treat it as a linear system and we have to deal with uh a controller that is going to be nonlinear so how can we deal with this come on ideas so you have your your joint and it has a gear with like some some friction that is uh or even uh U it has some gravity or whatever yes well if you've got a certain type of friction you can uh like if it's velocity you can put that into the motion equation and and change your B value your k d k v you mean yeah the KV so if if it is linear yeah I I think uh you can in fact in integrate it directly in the KV but if it is not nonlinear like just the gravity so what do we do if we if we have the gravity what do we do with the gravity we model it I know the model because I know the mass the center of mass all of these things so if I can model it I can somehow like anticipate what the gravity is going to be and try to comp compensate for it very good so we can compensate for the gravity well if we have a nonlinear term what we will do is we put that compensation in the controller so now the controller it has the linear part which was F Prime alpha fpre alpha fime actually is mass F Prime and now we are going to add another term beta which will attempt to compensate for B you do not know B exactly you know sort of a model with some estimate of B you don't know X exactly you don't know x dot exactly you have estimate of these what we call the X hat x. hat and B hat now B has a structure if it's the gravity it's going to be I don't know ml cosine that angle and you can estimate your mass estimate your length estimate the the the position and come up with an estimate of B which which would be B hat so in that case you can say Alpha is simply the mass an estimate of the mass minus plus one G probably you will find it and your B hat is going to be an estimate of of B given the state your estimate of the state and you probably have 10 epsilons little bit more of error so we're assuming that we are going to have some errors but by compensating for those nonlinearities estimating the gravity and taking it out later estimating centrifugal coriolis forces and trying to taking them out we should be able to bring the closed loop system closer to a system that is a unit Mass system because with this compensation if everything was perfect we compensated perfectly B then basically beta will take out B for each configuration each velocity beta is exactly compensating for B it takes it out and the system is linearized right well this will never happen in reality but we will be very close so this is what we can write we can say this is our system and this is the controller you understand this controller this controller is a nonlinear controller but it is attempting to render in the closed loop your system to become the coupled linear system so here is the result if B and B hat were identical if B hat was compensating perfectly for B and if the estimate of the mass Matrix later this Mass was identical to M then your system will behave this way so what you design for f Prime will be part of the closed loop of the whole system we're talking about one degree of freedom but if we are later we will see 20 degree of Freedom it would be the same okay well here is how we can write this system so our our system was F with the output x x dot the state basically what we are doing is we are looking at the model of the system and we are using x and x dot to estimate B the nonlinearities in the system and compensate for them so f is going to have a component which is b b hat in addition our input control which is f Prime is going to be scaled by an estimate of M the mass of the system so that there is a virtual system here that would look like a unit Mass system with an input fime and this same output and this big box the red box is like a system that is linear with unit mass and that is the purpose of this design later this will be centrifugal Corola gravity forces and this would be what right the mass Matrix so so in fact with many degrees of freedom we will be able to do the same thing where this becomes the mass Matrix and here we will have V and G you remember V centrifugal coris and G gravity and you can add the the friction as well okay so essentially we are designing a non linear controller to compensate for centrifugal coriolis gravity and to decouple the system to decouple the masses the inertial forces and to achieve a unit Mass system Behavior okay so let's see our design for f Prime F Prime is in this structure in the decoupled control structure and if you have a dard position XD what would be F Prime just a goal position so a goal position we have X desired F Prime will be minus minus something who remembers I'm sure you remember F Prime isus K * x- K Prime * x - x you meant minus KP Prime x - XD so minus KV Prime X do minus KP Prime x - XD and the Clos Loop Behavior would be very nice so we linearize the system all right well most of the time you're not just going to a goal position most of the time you are tracking a trajectory and on this trajectory you might have like you might have uh different accelerations at different point you have different velocities and whereas in this controller we are just reaching to the goal position KP Prime is trying to reduce the error and KV Prime is trying to put just damping to bring the velocity to zero at the end point but if you are tracking a trajectory you have all of these desired things you have desired position function of time desired velocity and desired acceleration so we need to design a controller that is more suited for this so what F Prime would be so see now we forget about the system because we we know we can decouple it make it linear let's think about the unit Mass system how you would design a unit Mass System Controller and then you put it in that structure so what is the objective if you have all these desired things what should F Prime be okay so you see on the top here is f Prime I have some desired acceleration I have my acceleration unit Mass acceleration equal F Prime and I know my desired acceleration it's X double do Desir so if this was is really a perfect system and you are trying to track this acceleration desired what F Prime should be I think the question is so simple that you cannot believe it that come on this is very simple too simple so my system is X double dot and I know the desired Accel cation it's double do desired what should F Prime be come onus a Conant xou dotus x d dot yeah I I think you you you you went too far that is that is correct but I'm just saying if if the system was able to respond directly to F Prime with no errors nothing and my system is X double dot and have the desired acceleration it's double do desired what I would do with f Prime just make frime equal to x. Des right right okay okay you you you see what what what we're talking about you have your acceleration desired so just put xou do equal x do desart and everything should just you apply this force and the system should follow X double. desart right well it won't it will drift because there is really no feedback you you have your acceleration and you are saying it's double that desired this is my acceleration desired and as soon as you start the system will start accumulating errors and it will drift so what should we do we should do the PD part and that's why now we are going to add proportional control to the error the position error X as you said minus k p Prime x - x Des what about the error in velocity because now I have X do desired what would be the term that I should use to follow x. desired so that would be minus KV could you finish it minus KV x do- x exactly I form the error x - x do and I will so here is the controller so this time if I have the full trajectory I will form errors on the position on the velocity and I would feed forward the acceleration so essentially you are telling the system follow this desired acceleration it's not going there will be errors and I'm tightening these errors so the Clos Loop behavior of this is going to be controlling the error in acceleration in velocity and in position if I have the full trajectory in time and that will basically if I call x minus X Des are the error then I'm I'm really controlling the error as a second order linear system all right okay so now we have to make sure that we can do this with the whole robot and we have to make sure that uh this controller could work with those gains that we are trying to achieve and we start analyzing the system so let's imagine that I designed the system the compensation with the B hat I'm sorry they uh they are not appearing as hats but this is B hat and M hat and I get everything over there but then now we are talking about the real system so when we were running the simulation already we we saw that a small external force will disturb the system so there are a lot of uh forces coming from the errors in Dynamics errors in the gravity estimates uh nonlinear forces coming from the gears and the friction that will affect this behavior and as we start introducing disturbances in the system we are going to see that these gains that we set are going to play a very important rool in disturbance rejection so let's add a little bit of disturbance here so if we add some disturbance I'm going to take a very simple type of disturbance like a bounded disturbance that we are adding from some uh like type of error in the gravity imagine you have this link and you have little disturbance coming from the gravity so what is the effect of this disturbance on the closed loop now so here is our controller with f Prime scaling it by the mass estimate and B estimate we are getting this closed loop so this Clos Loop now is going to be equal not to zero with the disturbance Force it's going to be equal to this disturbance Force divided by m right uh by the way in some textbook uh this is not divided by m it's directly applied as if it was applied to the uh decoupled unit Mass system so because the disturbance is at the in out input of the system you have to remember that we are dividing by m so this is divided by m okay so let's see what is going to happen to your errors I'm not sure how many of you know what we mean by stady state error what is a stady state error here what do we mean by stady state error yes Wasing equilibrium velocity how what it's what its position are is going to be so steady state is like when the acceleration stops the velocity stops and you get you reach that position but you are not reaching it exactly there is a small error so it's like a spring Mass system if you apply little bit of perturbation with an external Force it's not going to go to its rest position it will be very close but not there and that would be the steady state error so this is what happens when we have the Velocity and acceleration Earth equal to zero and that means the last term the error term KP Prime is equal to F disturbance divided by m and that means your error is going to be the disturbance Force divided by m KP Prime again in some textbook there is no M appearing there and this is very important KP Prime is the gain the position gain for the unit Mass system it renders your frequency constant over all the motion when we vary m KP prime your frequency is square root of KP Prime however your stiffness your position gain is M * KP Prime and if m is changing your stiffness is varying so you do not have constant stiffness over the workspace as you vary M so your error your steady state error will vary depending on where you are and how big your m is and you better get a large KP Prime so to guarantee that you have a minimum uh disturbance rejection so I'm not sure if you you you are seeing it KP is your stiffness is your closed loop stiffness KP Prime is not it's m KP Prime okay so this is very important to remember this because we will use it when we go to n degrees of freedom it is the same structure okay well here is the example I mentioned earlier if we think about just a disturbance for for this uh one degree of Freedom it's sort of a spring mass and you are applying the disturbance force and that means the steady state error x - x desired will be given by this equation which means you are going to rest not at XD but at XD plus this Force divided by the stiffness you have so this is your Delta X Okay so the disturbance is going to produce an error Delta X that is given by the disturbance divided by your stiffness and this is your M * KP Prime don't confuse KP Prime for the unit Mass system and the KP okay now how can you get rid of that error the steady state yeah I know I I I HD that slide now you know you don't know all right go ahead High yeah do you think he didn't see the next sline no you didn't see it all right okay why Why by adding I we can we can remove that there because over time the eye grows large if you you have a space ster M and it'll correct for that yeah basically you detect the error and as long as you have an error you are adding adding so so essentially the idea is now your fpre will be in will include an additional term that magnifies this error with some gain and keep magnifying and keep adding until you you overcome uh that disturbance now integral action is is very good to reduce errors but you have to be careful about the way you use it better to use it close to your goal position not all over especially uh if you uh you're moving fast and accelerating winding and unwinding that integral might create instabilities so the way you can analyze the disturbance is that now you have this equation this closed loop and and if you take the derivative of this equation basically you see that now the steady state is going to be equal to zero so if you build that integral then you will take the error to zero actually integral action is very nice when we go to force control we will talk about it later in motion control you have to be very careful about its use there is another element of this when we looked at the simulation of the Puma we are looking from outside but let's go a little bit inside and see what is happening so you have the motor you have a gear with some gear ratio that depends on the diameter of those two spheres r and little r so what is the gear ratio here come on we have a lot of mechanical engineers is it big or small so what is a typical uh gear ratio for for a robot what do you think the gear ratio for uh joint two when I move joint two here what is the gear ratio H that's that's not that's reasonable actually the gear ratios for the Puma vary between like 50 and 150 uh Mo some robots has 300 gear ratio and they have multiple stages uh all the joints here like uh one two and three have two stages and you get a lot of flexibilities a lot of vibrations that appears and all of that so it is uh it's actually very nice here because we have a very small gear ratio but I'm not sure if you know what is the gear ratio what is the gear ratio about how how big the gear ratio 20 40 no one knows okay the motor is going faster or the link is faster I'm sorry motor the motor so the gear ratio is reducing the speed the link is going slower so how slower one time two twice three he he he he said like two like two yeah it is two I was worried okay so the gear ratio is really r divided by the other R the smaller R okay so that's your gear ratio and um the speeds the link speed is smaller by that ratio than the speed of the motor right how about the torque well as it's written so we have a g ratio of two here the torque at the link is twice as big as the torque at the motor so you take a motor if the gear ratio is one basically you have direct drive and you need a big motor to produce a torque right but this big motor is heavy and with the robot putting a heavy motor here is a problem so what do we do we put higher and higher gear ratio reducing the size of the motor and we can achieve the torque because we are using a high gear ratio now just I wanted to show you this to to illustrate the another problem it's not about just the speed reduction and torque uh increase there is another effect that comes with the gear ratio when we start moving so we have this inera it's varying with this right big inertia small inertia as we move now the question is what this motor is going to do to this inertia the inertia of the link is here there is another inertia coming from this rotation of the motor and this inertia is going to be affected by the gear ratio and the effective inertia perceived at this joint is going to be bigger than the link inertia so what is the effect of the inertia of the rotor of that motor there on the inertia of the link so the inertia of the link alone is I the inertia of of the motor is IM what do you expect the real I the effective iil to be so it's going to be equal to I plus something and this something is again you oh him okay him to twice tce in motor twice the inertia of the motor so you you mean in general this is n times the inertia of the motor that's what you mean yeah so it is in the inertia of the mot which is similar to the torque increase the torque at the link is n times the torque at the motor you're saying the inertia is the same in the same proportion it is n times the torque I mean the inertia of the motor well you're right it is bigger you're right there is an increase by n but it is not linear by n anyone can give me a better estimate now you know but they are not looking that's why no one is able to tell me so inertia of the motor is reflected at the link by it's not a torque it's not going to be reflected by n because the motor is moving much faster the the acceleration at the joint is moving also slower by that gear ratio so the effective iner shell reflected by the motor is how many times I am n Square much better so you have a gear ratio of 100 it is big the coefficient that you are going to carry is really big and that makes the robot very dangerous actually because the effective enertia that you see at the joint is correlated with the impact force that you might produce if you have uh a sudden collision and if you are reflecting this small inertia by n square of the gear ratio you're going to produce a large impact Force so here is the analysis we can write the dynamic equation on the side of the motor or we can write it on the side of the link and when we write it on the side of the link we see that it's i l+ n sare or AA Square I and this is your effective inertia now your effective inertia is always by the square of the gear ratio of the motor you are using well the direct drive case is really ideal we use it with Scara type robots where you do not have to carry the gravity but with robots that are articulated and that need to move in space it's very difficult to build robots that have big motors and can carry uh the structure now variation of the effective inertia makes it that if you have variation of your I I is going to change I is constant it's the same motor so the question is if I is changing what is the effect of that well the effect of that is changing your time response because your KP if you remember is M * KP Prime so it affects directly your Omega so you might do an analysis and see if you are using constant gains that the best way to select your gain is to go to the geometric average and look at your minimum the minimum value of your ilil and the maximum value of your IL and take that uh mid-range in geometric sense and that gives you some uh estimate that you can you can use over all the range of motion okay well I think uh we are coming to a point where we can break so let me remind more remind you that next Tuesday and Wednesday we will have the review sessions and uh on Monday we will be signing up for those review sessions so please those of you who are not here please make sure you come and sign up see you on Monday |
Lecture_Collection_Introduction_to_Robotics | Lecture_11_Introduction_to_Robotics.txt | this presentation is delivered by the Stanford center for professional development all right let's get started so the video segment to today is quite interesting it was uh presented at the 2000 uh uh International Conference on Robotics and Automation and uh I'm sure you're going to like it a robotic reconnaissance and surveillance Team USA heterogenous multi-root system for surveillance and exploration tasks at the first tier of this team is the Scout Scouts are small mobile sensor platforms used in a cooperating group at the second tier is the Ranger Rangers are larger robots used to transport deploy and coordinate the Scouts Scouts are wholly original robots with cylindrical bodies 40 mm in diameter and 110 mm in length the Scout carries a sensor payload used to relay environmental information to other robots the most common Scout payload is a small video camera but other payloads such as microphones are also used video data is broadcast to other systems via an analog RF transmitter Scouts communicate with other robots using an RF data link one specialized Scout has a camera mounted in a custom pan tilt unit allowing the robot to view its surroundings independently of the orientation of its body the Scout has two modes of locomotion to allow it to navigate different kinds of terrain and obstacles the first mode uses its Wheels allowing it to drive over smooth surfaces here the Scout demonstrates its ability to climb a 20° slope the second mode of locomotion is the hop The Hop is accomplished by winching the Scout spring foot up around its body and then releasing it suddenly here the Scout jumps over an obstacle Scouts are deployed by Rangers the ranger is a modified commercial allterrain robot the ranger uses a launcher to deploy Scouts into the area in which they will operate a ranger can carry and shoot up to 10 Scouts from its launcher Rangers supervise the scouts while working with other Rangers rangers report to a human group leader the scouts are designed to withstand the impact of landing and of being shot into and through obstacles such as these simulated Windows the Scout small size its deployability through launching and its multiple Locomotion modes and sensor payloads give it the ability to explore difficult to reach areas and Report useful data combining the scouts with Rangers which provide the ability to travel longer distances and to have greater computational resources forms a useful reconnaissance and surveillance team okay what do you think I guess we need a robot to do the laun I mean to load these uh devices in so one more robot is still needed okay so after completing the forward kinematics uh after finishing the jacoban we're ready now for Dynamics are you ready all right so well Dynamics and then we will do the control and that's it you have the basics kinematics Dynamics and control well here is uh an example of robots that involves a lot of Dynamics just imagine like moving the hand little bit you can see all these coupling forces coming on the other hands on the body as you start moving you have all these articulated body dynamics that are going to appear and the Dynamics of this system is quite complicated in fact um if we go to this problem we we we find that we we need really to understand the the Dynamics of just one rigid body and then combines these different Dynamics together to understand the articulated uh uh articulated multibody system so to do that that is to find the Dynamics of an articulated multibody system there are several formulations in fact there are many many formulations we will examine two of them one is the Newton ER formulation have you heard about Newton yes so what does it say to you Newton ER what what does it tell you so Newton Law is you were saying so Mass acceleration equal to the force applied a rigid value right so that is if you apply a force to a particle it will accelerate along the same direction with an acceleration that is equal to the force divided by the mass okay so what about erer what does erer do with Dynamics here you know ER angle you know uh ER parameters huh so er was looking at Angles y angles measure what rotational motion so linear motion because Force acceleration of a mass a particle it's just going to be a linear Dynamics and ER is dealing with the other side of Dynamics rotational motion now if you have a particle then there is really not rotational motion to talk about so we go to the rigid body and we find that we need to address the problem of uh angular rotation angular motion and that is uh the formulation the com combination of Newton and other equations extended to the problem of multibody so we will examine articulated multibody Dynamics and we will uh find similarly to the way if you remember we found the jacoban by analyzing the static forces propagation you remember we we break all the the joints remove the joints and look at the stability of each of the rigid bodies we are going to do the same thing with Dynamics now then we will examine another formulation a formulation that captures the whole Dynamics linear and angular in one equation uh that is the lrange equation and this formulation is relying on the energy that is the kinetic energy of the system you know what is a kinetic energy most of you what is the kinetic energy associated with a particle moving at a velocity V Let's 12 MV squ very good and also the potential energy and that will lead us to a very interesting form that will give us the dynamic of articulated body system in an explicit form you remember how we did the explicit form for the Jacobian we can find the Jacobian as a sum of contribution of the different uh velocities at the different links well we're going to do the same we're going to find the Dynamics of the whole articulated body system uh as a sum of the contribution to the Mass properties inas and masses uh we we established something called the mass metrix associated with the Dynamics and we will see that from finding the energy just finding the kinetic energy of the system of or at least of each of those links uh adding them all together to find the total energy we will be able to obtain the dynamic equations so this form is really important and we will uh examine this form uh probably on Wednesday but let me just uh to start give you an idea about what is happening when we look at the Dynamics this is a robot from France it's called the ma man plor 23 it is a cable driven robot so all the motors are in the back and the mo the cables are driving the structure so if we go and analyze the inertia view from just one axis let's say the first axis of rotation so you have big inertia smaller inera right by putting the masses away from the axis you are increasing the inertia perceived about this axis so this inertia then depends on all the mass distribution the length the the the the load Etc that is associated with with the manipulator if we go here we have also changes in the inertia perceived but it's independent of the previous length so there is a structure to the way the inertia is uh affected by the motion of the structure and the configuration is going to change the value of that inertia that you are perceiving if you want I can show you the equations I don't know if you can see them but here we go so the first enera perceived from this joint is like half the page It's s cosine and all of these things and depending on but obviously I mean we can obtain these equations it's not a problem the problem is to understand what what the structure uh of this equation is and uh how we can find those properties and how we can understand them what when we analyze later when we later analyze the explicit form you will see that essentially we're going to be able to again like with the Jacobian are going to be able to see the Dynamics of a manipulator just by looking at the structure of the robot all right here is another robot this is this was uh analyzed in my thesis this is a robot uh that can carry 80 kg that was quite heavy robot it's a a robot that has six uh uh degrees of freedom and we are performing just some cidal motion on the different joints uh on joint uh I believe joint five or joint four we have uh basically no motion just letting the The Joint to be controlled to its zero position so what do we see we see that on the top the lower joint on the robot during the senoidal motion of the other joints there is little effect I mean you can see some errors but the errors are sort of filtered somehow on the lower joints four and five you see large errors and that is reflecting the fact that if you have a heavy joint if there are disturbances the inertia the this big inertia of the of the robot is going to play the role of a filter it's going to uh somehow reject the disturbances and we will see little effect on those joint because they are quite heavy and the inertia is just taking I mean think about a Truck moving fast and you hit it with I don't know a fly it's not going to to be affected but for the fly it is really terrible so so this is the fly and that up there is the truck so if you want here is the equation we're going to establish this equation the equation is a vector so gamma is the the torqux applied to the robot sometimes we call it toe or gamma what is uh G what would it be some Force affected dependent on the configuration the gravity yeah g like gravity very good and it's dependent on the configuration for instance if we take this joint and if we put a w you you're going to feel a torque right if I take this joint to here what is the torque due to the gravity zero very good so you get it that is so if we go up basically the gravity is going to act on the structure and not on the joint qou dot is the acceleration and M is Mass acceleration so m is okay qou dot is the vector of accelerations and M very good it's a matrix so Matrix post multiplied by a vector will give you a vector so mq dot is going to be the inertial forces generated at Z zero velocity by the motion by the acceleration of the joints so it's it's playing the role of a mass so if I had one degree of Freedom you will understand it Mass acceleration m is a mass but because this is a multibody system it's going to be a mass Matrix and the mass Matrix if we had a robot that is uh Prismatic so three Prismatic joints uh essentially along the diagonal will you will see the element representing the masses the total masses reflected here here and at the last joint and it will be diagonal matrix but for articulated body with revolute joints you're going to have off diagonal terms that represent the coupling that is the motion of one joint will accelerate the other joints what about V this is a vector that is function of Q and Q dot so oh I forget my I had uh something to well maybe later we'll show you um so what would be this thing that will depend on the velocities so let's use the microphone if I I do this I'm wondering why it's standing like this if if I I stop moving it will fall and you rotate so there is a force pulling here it's called what centrifugal force yes now if you have multiple multiple body that are moving in addition to centrifugal you will have coris forces that is the product of velocities that will be involved uh will cause both centrifugal and corus forces so Q as I I said is your generalized coordinates m is called the mass Matrix or the kinetic energy Matrix because this Mass Matrix is associated with the kinetic energy of the system if you have just a particle M what is the kinetic energy for a mass m a particle of mass m what is the kinetic energy associated with the with this Mass when it's moving at a velocity v h one 12 MV Square yeah that's correct no good well it turned out it is the same for a mass matrix multibody it's going to be2 how do you do MV Square for a mass uh associated with articulated bodies and the the mass is a matrix so you get a v transpose and M and V which makes it quadratic form and basically your kinetic energy is just 1/2 Q do transpose mq do these are the centrifugal coriolis forces and we will see actually that these forces disappear if the velocity was zero or if the mass Matrix was constant that is the V term solely depends on the velocity product of velocity and on the fact that all the element involve are derivative partial derivative coming from partial derivative of the mass Matrix so if you have a mass Matrix that is constant then the derivative are zero and V will be zero yes that's not a dash that's a dash no no Mass Matrix or kinetic energy Matrix and the gravity forces and Gamma is the generalized forces so gamma one will be acting along or about axis one so you have q1 Gamma 1 Q2 gamma 2 right so gamma could be a force if the joint is Prismatic now again in term of the formulations it's very simple we saw this figure before if you have a rigid body you can study that the the stability of the you can State the static equilibrium of the rigid body under the forces applied and you say the sum of the forces should be zero if the rigid body was at static equilibrium and the moment computed about any point going to be equal to zero as well now if this rigid body is moving so here the rigid body is a static equilibrium we we do that analysis but if the rigid body was moving then the rigid body the masses and the inertias are going to generate an additional Force we saw this Force here the first Force mq do plus v actually these are inertial forces they are created by by the fact that this is a rigid body with mass so if we can compute the forces and the moment associated with this rigid body yes where do we have our damper and our spring forces well they will come they will come uh little later we can create the those spring and D damping in the control to stab ize and control the robot or the robot might might have some damping uh at each of the joint because of friction but we will come to that later so if we if we uh consider those forces then we can restate the static equilibrium by saying as it is moving we should have a a static equilibrium uh that will equal will be equal not to zero but rather to F and to n to those moment and forces appli so then we can come up with uh relationship so first of all for the Newton equation you we saw the equation earlier the new Newton equation is describing the linear motion the ear equation is descri describing the angular motion so Mass acceleration equal force moment equal inertia time acceleration angular acceleration plus this term that creates the centrifugal coris forces and by stating that equilibrium and then doing the projection on each axis you remember in the static uh analysis we projected on the axis to see the forces acting at that axis so this essentially eliminate the inertial uh forces uh internal forces acting on the structure then we will be able to find uh the equations to do that we will project like we did in the static case and find those component that is the Torx applied at each of the joint axis so this is basically the Newton ill formulation now the lrange formulation doesn't uh go into the uh specific motion of each of the joint doesn't require any elimination it looks at the problem differently from a energy analysis point of view so essentially what we're going to do there we take the whole articulated body system and we take the kinetic energy of each of the links so the kinetic energy of a link is let's say it's k i the total kinetic energy of the system is the sum of all the Ki and we have also the potential energy now once we decided about our system of generalized coordinates in this case it will be Q's q1 to qn we can write the kinetic energy in this form earlier we we said the kinetic energy of an articulated body system would be just this Q do transpose m q do and this means means that K the scalar K is half the velocity Vector transpose multiplied by The Matrix post multiplied by the vector that gives you a scalar now if you take the scalar and combine it with the potential energy you will be able to immediately find those equations now once you found those equations actually you realize I know the equations actually directly from M but by Computing the kinetic energy you identify M which will go here so you know m qou dot from the potential energy you KN you you know the gravity what is left is V and we will see that once we know M we can find V so the whole equation uh of the Dynamics can be obtained simply by taking the potential energy taking its gradient and that gives you g the gravity vector and by taking M and Computing M how can we compute M well we can say the kinetic energy should be if if uh if we have a system of jiz coordinate it's a quadratic form on the jiz velocity so from the top we can say the kinetic energy could could be computed for each rid body and by doing the identity between the first expression and this expression we can identify M we will see that on wednes step so this is a slide we saw before that is the idea of breaking the structure and analyzing each of them and then eliminating the internal forces well this is exactly what we're going to do again now but by adding the inertial forces so in addition here we we are at static equilibrium the rigid body is not moving if it is moving there will be these forces and then we can say the forces fi are equal to uh the fi+ FI + 1 from this relation the sum of the forces should be equal to the linear acceleration and the sum of Ms should be equal to the angular uh inertial forces and this is the algorithm basically that we will find uh this algorithm would allow you to compute the Dynamics so the newon algorithm does the following it propagates velocities we know we did that for the Jacobian you propagate velocities you as you propagate velocities you compute your accelerations and from the accelerations you can compute the inertial forces as you're propagating and then it has a a back propagation that is the projection on the axis by taking these forces and starting from the end and going back you propagate your forces and when you reach the ground basically then you have all your talks so we're going to start with the basics and uh the reason for that I want you to understand uh a very important equation related to the rigid body and this is the erer equation we need to understand what is the inertia of arid body if we were working just with a single particle of mass m the problem would be very simple we don't need a equation we are working with rigid body when they move this rigid body has a mass distribution and we need to capture the mass distribution at some point and that result into this inertia tensor that we are going to use to describe the rotational motion of arid body now this might be scary but it's going to be really really simple if you just pay attention we will start from a particle we take a rigid body and look at it as a collection of particles and then we will look at the linear velocities of each of those particles as they move and we group them together and we will find the inertia associated with the rigid body okay sounds good all right let's try so as I said for a particle M if we apply a force there will be an acceleration that is equal to F ided by this Mass so the mass is resisting to accelerate so if the mass was infinite no motion right the lighter the faster with the smaller forces so this is the law I think everyone is familiar with there is a velocity actually with this acceleration that is not aligned with the acceleration depending on how the trajectory is going and and we can think about this same equation in the following way so I'm going to group the mass and the acceleration using the derivative of the Velocity so you can write the same equation in this form the partial derivative of MV is equal to the force what is MV by the way I'm sorry momentum exactly it is the momentum associated with this linear motion it's the linear momentum MV is the linear momentum of the particle so this linear momentum is playing a really interesting role can you see what what is this role here if we think about this linear momentum the rate of change of the linear momentum is equal to the apply Force nice so the rate of change of the linear momentum is equal to the force we're going to show that actually ER equation all what it does it says the angular momentum if we know the angular momentum if we take the rate of change of the angular momentum is equal to the moment applied and with this symmetry basically then we will be able to compute this tensor or this in inertia Matrix associated with the rigid value okay so the rate of change of the linear momentum is equal to the applied force and we call the linear momentum fi okay can you remember that fi is equal to MV C Dynamics is not not that complicated all right let's talk about the angular momentum how can I make an angular momentum with a particle so we have an so we have this particle rotating and we have an inertial frame and I'm going to compute the angular momentum well basically I need to to cross product this with a vector right to compute the moment of the force and then I can have the moment of the inertias and so if we take the moment of this force on the right with respect to the origin O then we can take M MV dot is like a force right it's an inertial Force MV dot you agree so so we're going to take the moment of this equation with respect to o so we need the vector that connect o to the particle to the position of the particle right okay let's take the moment you remember the moment is p cross F on the right or P cross MV dot okay this is a very important step if you understand this step everything that we will talk about later will become just adding all of them together so I had a linear motion I'm just looking at it from angular motion point of view and I'm just taking the moment of this equation with respect to a fixed Point all right P cross f is a moment we call it n what is this let's hide it it's too complicated all right okay so you said earlier MV is the linear momentum I'm going to take P cross MV and if I take P cross MV and I need to to get the rate of change of that quantity this is sort of the angular momentum and I'm Computing it ahead of time just to show you what it's going to be so if we do the computation it will be P cross MV dot V cross MV and V cross MV is equal to zero so that gives you because when you cross product a vector with itself it gives you zero so that gives you the rate of this quantity equal to the moment and that is the angular momentum so the linear momentum is MV and the angular momentum is the vector locating M cross MV okay we put it down so now you have to remember MV linear momentum P cross MV angular momentum all right okay well once we go to arig ready all what we need is to do the sum so these P will become p i m i and VI I and we are going to add add all these particles a lot of them many of them so instead of doing this with just uh um a sum we will do an integral and because we have objects in three-dimensional space we will triple integral in DX Y and Z and basically you get your inertia for the rigid body so let's call it first this quantity P cross MV we call it fi it's the angular momentum so so I'm going to now think about this equation in the context of a rigid body the angular momentum of a particle we know we want to find the angular momentum of all the particles and we assume that this rigid body is moving at some velocity and acceleration related to the instantaneous angular velocity and acceleration we we know okay so we need to locate each of the particles so we have pi and the linear momentum is I'm sorry the linear velocity VI of that particle is going to be we studi that in the angular rotations it will be Omega cross PI right so what is the angular momentum the total angular momentum of this rigid body is going to be the sum of all the pis locating these Mi masses moving at VI velocity so now we're going to take VI and bring them down here and we will have p i cross m i Omega cross Pi I so we will have a more complicated expression so we have omega cross Pi so this would be the total angular momentum of the rigid body right do you agree if I can count them all all the particles it's difficult to count but that's what's nice about the mathematical models that you can use you can assume well let's assume that I can count them all now in this equation do you see anything that is constant that is not changing that is independent of the regid value louder louder Omega all these particles are from moving at rotating about that axis with Omega so Omega is independent so so what are we looking for if Omega is independent and if I'm going to go to an integral I need to get this Omega out of the sum right we don't need it in the sum so how do you do that I'm sorry can use a triple product so yeah I mean you you you basically the Omega is on the left side you can if you if you put minus you can bring it to the other side Omega cross P or minus P cross Omega right you remember this and the other thing is the fact that mi mi repres presenting that mass for that particle if we assume that we have a a homogeneous object then we can represent it represent the mass by the small volume multiplied by the density right so using this and using this well I didn't do it yet still so we go we go from that sum to this integral with the same equation before tripping this but this is what we need to get this out of this equation it is independent of the equation so the FI is your total angular momentum so let's rewrite this in this following way I'm going to rewrite it minus P cross p and Omega out and that means and also substitute with the cross product operator so P cross is p hat and that leads to this form in here what you see is minus P hat multiplied by P hat R DV all of this is variable depending on the particle Omega is constant so f is equal to this integral time Omega and this integral is essentially your inertia the inertia of the rigid body what we call the inertia tensor so we can write it simply like this F is equal I Omega so maybe I went too fast you you see this relation where I is this integral so for linear motion f is equal MV for one particle and we have Newton equation which says the rate of change of f is equal to F okay it's another writing of ma equal f no one remembers this one the five Prime equal F people are afraid of momentum okay now with the angular momentum we're going to see the same thing the rate of change of the angular the angular momentum is equal to the applied moments so fi is equal I Omega and er equation is simply the rate of change of Y is equal to the applied moments couldn't be any simpler right then you will be missing something now five Prime is not as simple as as the linear motion when you take the derivative of f you get I Omega do this is like Mass acceleration but also because because of Omega we have I Omega in another product of velocities and that produces centrifugal and coriolus forces so we have the two equations for a rigid body so a rigid body has a linear motion if I throw this like straight at you it will it's going to rotate there there will be some air resistance and you will feel some some rotational motion but this combination of linear motion and angular motion is captured by this for one rigid body well I have to deal with multi rigid bodies attached by these joints and when you put a joint you are putting a constraint you're throwing the thing it's going to to be pulled and pushed and and that means we need to eliminate the internal forces in order to find the actual motion so a very important thing that we established with this relation is this I and now we need to really I mean this is the thing that I need you to remember you need to be able to compute I for each of the rigid body of your robot so this is something that is absolutely needed and in order to to do that you need to see a little bit more into the structure of the inertia Matrix or the inertia tensor so we said the inertia tensor I is basically this integral over the volume of all the vectors locating all the points on the rigid body and scaled by the density of masses on the body so if we take this quantity P hat p hat so this is the cross product operator it turned out that you can rewrite it in this form so P hat is what you remember P hat the cross product operator it's a 3X3 Matrix so if you multiply a 3X3 matrix by itself you're going to get another Matrix right now this Matrix could be written as P transpose p ppose p is is what a scalar so you are scaling ppose p is the square of the vector P so you are scaling the identity Matrix so on the diagonal of the identity Matrix you have the square of the component of the vector uh P minus minus p p transpose which is a 3X3 Matrix so using this relation we can rewrite the inertia tensor in this form so let's take this computation so P transpose p is x² + y s + z s and p transpose P time the identity gives you this quantity right nothing complicated the other one gives gives you this if I have X Y and Z I'm going to multiply p p transpose that will give me this Matrix and the result will be this Matrix so minus P hat p hat in that integral is this okay essentially this is locating the the position X Y and Z with this vector and now when we do the multiplication it appears like this all right so we're almost there we need to control compute the integral of this and put the density and that then we will have the inertia tensor so the inertia tensor is you see this equation here now I'm going to put it in that integral and I will find i iixx i y y i ZZ and iix Y Etc anything remarkable about this Matrix you can tell me properties y symmetric what else positive definite unless if you're at zero if you're at zero you're you're in a black hole okay so this ixx y y zz's and the the are like this just we I'm using the previous Matrix to recompute this element and this is your inertia tensor cool you understand now the inertia tensor all what it is this 3x3 matrix it's going to each of the points finding the distance and then you're walking and integrating all of these from each of the component to find what is the weighting the inertial properties uh about the axis XX about the x axis y axis z-axis and what are the coupling between different axes so the XX y y ZZ are called the moment of inertia and the other one are called the product of inertias all right here is an example if we take a if we take a a rigid body that is nice symmetric like cylinder or parallel pad or cube uh basically the property of symmetry when you you do the integration uh you're integrating between one side to the other and that leads to some nice properties because if you do this computation at the center of mass you are going to be able to find the uh uh inertias and you will find uh most of the time zero product of inertias and then if you need to do this competition at a different point all what you need to do is now to look at your vector so if you start from this Vector this point you will be able to reach all these different points then if if you start from a different point to reach this point you can go like this and plus this this one you already used to find the inertias about the center of mass all what you need is to add this Vector but because you're going to add the same Vector for all the points it turned out that you have a very nice property which is the property of parallel axis theorem that tells you the inertia about any point a is the inertia about the center of Mass plus the mass of the object multiplied by this quantity that is the quantity that let us compute PC this Vector in addition because all the masses are basically Can Be Imagined like concentrated at this point so this property is very nice because you can uh do simple computation at the center of mass and then uh just do your transformation to move to a different point so there is a an example of a cube where we do the competition at the center of mass you have this example and the answer is you get you get the I XX at the center of mass i y y at the center of mass and I ZZ at the center of mass all equal to ma a² ided by 6 for this CU now if you want to do this competition at one of the edges like at this point all what you need to do is to add this quantity that represents the distance from that point to the center of mass and scaled by the center of mass and you obtain this quantity so this is a very useful theorem if you are doing this computation and what you notice also is that then you will have product of inertias so the at the center of mass you have no product of inertia you have a diagonal matrix when you compute the inertias at a a different point of the center of mass you are going to have product of inertias all right so now we have the inertia of arid body then we can apply the equations of Newton ER and we do this propagation and we will uh compute all the inertial forces as as we propagate and then we project back to compute the forces and we find the dynamic equations so as we saw these are the two equations the translational motion and the rotational motion and what you need to do also is to compute accelerations you remember here we're saying you need to compute velocity find the accelerations and then you can find the inertial forces so we can go over this um now I'm not asking you to uh to really follow the details of this computation but essentially you you're going to start from the vol velocities uh the recursive relation of velocities as we propagate Omega I + 1al Omega I plus the joint velocity of that revolute joint and you have this relation now if you take the derivative of this relation you will find this derivative involving uh the acceleration of your uh revolute joint you will also need to compute the linear accelerations you start with the velocities you take the derivatives and you get your recursive relations okay so we we are forward propagating to to go to the last joint and you have a lot of derivative depending on the type if D is revolute Prismatic or not and we're not done because we have to find the velocities and acceleration at the center of mass not only at the Joint angles so you need to do a small addition the linear velocity at the center of mass you need to multiply by this Vector Omega I + 1 and that gives you the velocities acceleration at the center of mass and now you're ready so at the center of mass you can write the forces the linear and angular forces so this is the inertial forces acting at the center of mass and these are the linear forces acting at the center of mass and this is the inertia tensor computed at the center of mass now you take this moving accelerating rigid body and you say uh that all the forces applied to this rigid body should be equal to the inertial forces and the acceleration are related to the moment through the equation and you write these two equations and now you do the recursive relations with these expressions and you get your recursive relation so now you see it is fi as a function of I + one so you're back propagating and as you back propagate you have to make sure in order to find the torque to project on the axis so we did this we do this and now we have the recursive relation and then you project at each of the axis your n you comput it from here or F comput it from here and you have those relations oh my God I'm lost so you can see it's I mean it's very difficult it's a but it is wonderful algorithm in fact to compute uh precisely your your torqus as a function of the velocities acceleration inertias and masses and everything but you have no idea about what's going on right it's very difficult to see you need to okay so this is outward iterations from Z to five and inward iteration with don't forget this to eliminate your okay yeah well I mean it can for Asimo it can go to 25 but it works yeah what what about the gravity we didn't talk about the gravity here we forget the gravity oops so what do you do for the gravity how do we account for the gravity in this algorithm so okay you remember I said the algorithm you start from the base you assume the base at Z Z velocity zero acceleration and you move out and you come back now if we want to account for the gravity what should we do accelerates at G yeah just set uh very good so if you say I have a linear acceleration of equal to 1 G from the beginning mean you will be including the gravity good a movie segment now all right so good we we uh we will skip this and to skip it I will go to here and we go to the lrange equations all right so what we saw the thing you have to remember and not forget get is how to compute your inertia we need still to compute the inertia because when we go and compute the the kinetic energy that is needed in lrange formulation you are going to compute the linear motion kinetic energy which is 12 MV squ and what about the angular motion the kinetic energy associated with angular motion Omega so it is 12 Omega transpose I Omega so we need this I anyway you cannot Escape so you have to know how to compute I all right okay lrange equations actually I mean we can skip that if you want an Innovative no no no uh what I meant is we we really don't have to to know all the details of the equations but I really want you to understand lran equations because they are going to be very useful for you when you when we get to control and uh when we are going to control the robot the robot has its own Dynamics and also we are going to apply to it external forces to control it so these external forces are going to affect the dynamic of the robot in some way and we need to understand the lrange equation not only to compute the Dynamics all what we need for the Dynamics is the kinetic energy and we know the answer from lrange equation or from Newton or uh from any formulation of the dynamic equations we will have the same structure Mass qou DOT plus V plus G equal torque and to compute the mass all what we need is to compute the kinetic energies but we really need to understand what this what is the structure of flr equation how uh uh under applied forces a mechanical system is going to move so this has a very important role not only for the Computing the Dynamics but really to understanding uh the control okay how many of you have seen this equation before okay five six all right so let's imagine this equation in the scaler case so this equation would be just simply uh an equation where the torque is uh the torque applied to to one revolute joint so this is a scalar equation what is L those of you who have seen it before so it is the lagrangian and it is simply the kinetic energy minus the potential energy and what is this Q so when we when we uh look at okay maybe I think I have I have K minus U so L is K minus U I don't know if I will put it in the so most of the time when we are talking about natural Gravity the U is only function of where what height you are at so it's function of Q so we can rewrite this equation in this form right I'm separating the kinetic energy from the potential energy because the potential energy is independent of the time and so so you can write it in this way and essentially what you have in here you have mq double dot and plus v if you have multip and here you have G the gravity so here you just have the gravity Vector so in fact if we move the gravity Vector to the other side let's let's not worry about the gravity you're in space or the gravity Vector is just the gradient of your potential energy so essentially it's saying that your inertial forces are equal to the torqux minus the acting gravity and if we think about it this part of the inertial force this is what is going to give you when once we do the derivation that that left part will give you Mass acceleration plus uh centrifugal corus forces equal to these torqus minus the gravity so let's look at this little bit as I said the kinetic energy if we have generalized velocities if we say Q is our generalized coordinates Q do is our generalized velocities we can write the kinetic energy as a quadratic form on the velocities so K is Q do transpose mq do and if we take the derivative so I'm I'm going now to to take this K from there and differentiate with respect to Q dot okay so partial derivative of K with respect to Q dot of this quantity and this is Q do transpose mq do so what do you think the answer is so let's do it in the scalar case so you are taking let's say uh I'm I'm taking m v² 1 12 MV square and I'm taking the partial derivative with respect to V what would be the answer MV well in the in the vector case it will be MV as well it will be mq dot so nice this is very nice so now take the second derivative of this I mean this the derivative with respect to time so if you take the derivative of mq do with respect to time you get mq dot plus M do Q do you see why m dot because m is function of Q right function of Q so you get M do Q dot so let's write this we computed this right and it is uh this first part is mq do plus M do Q do what about the other part you see K over there I need to find the partial derivative of this K with respect to the q's so in the kinetic energy what is dependent on the cues can you point that to me okay the kinetic energy is2 Q do transpose was M of q q dot so what is dependent on Q's M all right so I'm going to write it as a vector because this is what does this mean it's partial derivative with respect to q1 partial derivative with respect to Q2 partial derivative with respect to qn so it's like this so I'm just writing exactly that quantity partial derivative with respect to q1 to qn okay you agree so everyone who have never seen lrange equation before agrees also is this clear so this means Mass acceleration mq M do Q dot so if Q do was zero this disappears right if M was constant not configuration dependent this would be zero and and this would be zero right okay this messy thing is your centrifugal coris forces it's product of inertias if you take any element of this it's uh product I'm sorry product of velocities you have always Q do Q dot that is multiplying so this is what we call v v is this Vector minus half of this vector and this is V okay so this equation the partial derivative of the kinetic energy leads to this equation as I said because we we see the answer you need to compute M right but m is there in the kinetic energy and these things are function of M so we really have just to compute M from the kinetic energy and we know the the answer by the way do you know why this we already saw this example what is the answer is MX dot okay so this is I mean a way to think about it is to form a vector v which is M square root of m q Dot and then your equation will be V transpose V for the kinetic energy and then you can show that Etc all right if you if you want to to to see why it is equal to mq do all right so the equation of motion using lran equations are in this form Mass acceleration plus these two vectors leads to this V Vector that is function of Q and Q Dot and if Q dot is equal to Z V will be equal to zero so this is the structure of our Dynamic equation and what we know is inside the kinetic energy there is this m if I can compute the kinetic energy some other ways then I will find M and what we're going to do we're going to go to each of the rigid vales and compute kinetic energy for K1 K2 K3 to KNN add them all together and identify that expression to this expression and we can extract M and once we have M using this relation between V and M doq do and the vector we will be able to compute V so this is what we will be doing on Wednesday for |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 20_The_Early_Middle_Ages_2841000_Intellectuals_and_the_Court_of_Charlemagne.txt | PAUL FREEDMAN: I've put these names on the board because today we're talking about the so-called Carolingian Renaissance, the revival of learning, under Charlemagne and his successors. But the figures of this Renaissance, the intellectuals of Charlemagne's court, of the court of Louis the Pious, the court of Charles the Bald are not household names. And I just want to give you a sample-- and it is no more than a sample-- of these men. And I'm afraid they all were men. And most of them monks or other kinds of clerics. And what I want to do today is remind you for the first, to start, we said last time that Charlemagne's rule was based on a combination of traditional war leadership, similar to that Merovingians; an alliance with the Church, similar to the Merovingians, but much closer - Much closer in the sense of seeing the legitimation of rule, up to the point of emperor, as a product of what later would be called "sacred monarchy". In other words, that the kingship was not just rulership over the populace, or over warriors-- it was that-- but it was also a responsibility-- a ministry is the word that's used-- to see to the salvation of the public good. Not just the secular benefit of the public good, as in a modern state, but to the salvation of the people that God would hold the emperor, the king, responsible for this. And then the third element was to revive the Roman Empire. And not just as a political entity, but the Roman Empire as a state. As a form of rulership for the benefit of a diverse population, not just one nation, like the Franks or the Lombards or the Saxons, but of a kind of imperial confederation. The Carolingian Renaissance is a planned one, unlike the regular old Italian Renaissance, which is a spontaneous revival of learning in several different cities on the parts of intellectuals. The Italian Renaissance, like most revivals of learning, might have been patronized by secular rulers, but its impetus was not the state. In this case, however, we have a program of reviving Latin, reviving the classical literature and texts, teaching these subjects. And all of this not in aid of what we would call a purely educational program, but to help the state and to help its mission, which as we've just said, is both a Christian and a Roman one. The revival of letters was intended, then, to restore and deepen the piety of the population, the understanding of Christianity, the restoration of the Church as an intellectual force, the preservation of learning from both Roman secular and religious texts. So that Charlemagne's cultural program is the culmination of a longer period of trying to save something from the wreckage of classical civilization. And thus, as a Christian ruler and as a Roman emperor, Charlemagne surrounded himself with these and other intellectuals, monks, and scholars. In order to understand this mission, however, beyond just saying oh, Christianity or classical culture, we have to go back to some topics we touched on very lightly when we were talking about monasticism. And that have cropped up now and then. So I want to start out with the preservation of learning up to the point that Charlemagne really starts putting together this program around 780 AD. You'll remember, and you will have seen from Augustine's Confessions, among other things, that the ideal of Roman culture at the time of the Empire was cultivated leisure on the part of wealthy, well-educated lay people. That is to say, even after Constantine's conversion, the intellectuals of the Empire tended to be not clergy, but wealthy people who could afford the leisure, the time, and the expense of procuring books, and of discussing them. These were the people who represented the continuation of the literary and philosophical traditions of classical Greece and Rome. And this also had a practical benefit just as, rather in a different way, it does now. In order to enter Roman governmental service, which was the most successful career path of the time of Saint Augustine and had been for some centuries, you needed to have an education of a certain rather rarefied higher sort. And this is not unique to the Roman Empire, it was true of Imperial China, most obviously. In China, you had to pass a very tough exam, a very selective exam, in order to be a part of the highest level of the imperial service. And the exam was not a political science exam or a business decisions kind of exam. It wasn't like you have a certain kind of budget and you've got to allocate resources. It's not a problem set. It was poetry. It was ancient culture. And the same thing is true of the British Empire in the nineteenth century. In order to enter its highest levels, you needed to know Greek and Latin. Not because you were going to use them governing some colony in Southeast Asia, for example. But because that was what made you an educated person. So the centers of learning in the ancient world included Athens, the Platonic and Athenian academies; Pergamum in Asia Minor, the name of which city gave its name to parchment; Alexandria, which had an entity called the Museum, which was not what we now call a museum, but rather kind of library and research center; Constantinople, later. Alexandria is the most famous because it had this magnificent library as part of the museum, and the mystery of what happened to this library, allegedly burned by the Muslims in the eighth century on the grounds that you didn't need to know anything except what's in the Koran. And this is not true. The library had disappeared long before the eighth century and probably was the victim of the kinds of disorders that began in the third century, when we began the course, the kind of disruptions of local society, opportunities for plunder, neglect. And the Museum was actually closed by the emperor Caracalla in the third century. So the major problem of the Roman Empire in terms of the diminution of learning is not the burning of Alexandria or such dramatic events, it's a much more gradual process. And this gradual process involved what we have at various times, using Wickham's language, called "the radical simplification of material culture". Which means not only a lack of imported goods, more primitive accommodations, less trade and commerce, but also much less in the way of books and learning. This is also related to the disappearance of lay literacy. The people who were capable of reading and writing in Latin, as of, let's say, the sixth century, are overwhelmingly clerics. And not all that many of them, either. The last grand figures of Roman classical culture are Boethius and Cassiodorus. And we've mentioned them already, but they bear [clarification: bear repeating]-- Boethius, 480-524. And Cassiodorus, 490-585. Cassiodorus was a monk. And it is his accomplishment to join monasticism to the preservation of learning. And he had a very long life to do it as well. Reminding us that not everybody just kicked off of the age of thirty-five in the pre-industrial world. Boethius didn't have a particularly long life, because he was executed by the Ostrogothic king, Theodoric, suspected of treason and plotting with Byzantium. Boethius could still work in his personal library in Rome, where the light filtered in through alabaster windows-- the Beinecke Library idea-- Onto cupboards stacked with Greek and Roman papyrus books. But Boethius was among the last secular intellectuals in the west, and particularly one of the last to know Greek as well as Latin. He conceived a project of translating the great works of Greek philosophy into Latin, which would have included all of Plato and Aristotle. But he was cut off by his arrest, torture, and execution after just having done some introductory textbooks and just a smattering of Plato and Aristotle. After him and the Justinianic invasion of Italy, Italy was devastated. And even though it will remain a repository of manuscripts, its learning was preserved as a guttering flame by monks like Cassiodorus and, almost accidentally, by the monks who followed the rule of Saint Benedict. Saint Benedict does, as you'll recall, require reading. But it's usually during Lent and it's a kind of penance. The contribution of Cassiodorus is the organization of a more library-like kind of monastery. And the notion that classical culture is not just a collection of text about a discredited religion, but necessary for Christian interpretation of the Bible and Christian learning. At his monastery of Vivarium in southern Italy, Cassiodorus developed a notion of the liberal arts as an aid to religious truth. The liberal arts is not his invention, but the notion of the liberal arts culminating in a program that has a purpose in which classical culture is fused with Christian culture is his doing. He was not so much interested in the aesthetic pleasure of these classical texts as in their use for interpreting the Bible. The Bible, according to his, Augustine's, and virtually everybody's understanding of the time, is not a text that makes perfect sense in every respect literally. It is the book of life that God has set up for us, but it requires interpretation. It is not just a literal text, it is a figurative and prophetic one as well. And in order to get at what it really means, what its real messages are, you have to know things like mathematics, astronomy, geometry, even music, and certainly grammar, rhetoric, and logic. These are the seven liberal arts of Cassiodorus's curriculum. They're arranged in the three basic-- the trivium, as it came to be called. The three, not because of trivial, although that's where the word comes from, The trivium. I-U-M. T-R-I-V-I-U-M. Logic, grammar, rhetoric. Now, I'm in this sort of Rick Perry moment. The quadrivium, there are four of them. OK? Arithmetic, astronomy, geometry, and music. OK. So I got all of them. The quadrivium are the sciences. You don't think music is a science? In the late Roman medieval imagination, it was the science of intervals and of pleasing forms and of modes. The replication of this plan of Cassiodorus is the accomplishment of this period in which classical learning and its texts were endangered. The Benedictine monks of Italy-- question? Something I missed? STUDENT: I'm sorry. What was the four? PROFESSOR: The four-- the quadrivium. Quad, Q-U-A-D-R-I-V-I-U-M. Sorry, what were they? STUDENT: Yeah. PROFESSOR: Oh, you're putting me on the spot again? Geometry-- geometry, music, arithmetic, and astronomy. Right? This is the period in which the Irish are critical to the preservation of learning. There was a book that was popular a few years ago called How the Irish Saved Civilization. Like many popular books, this is a little bit reducing things to a kind of easy formula. But it is Irish as well as Italian and very much English monasticism, under the influence of both Italy and Ireland, that has a program of establishing monasteries, correcting and preserving Latin, which has become increasingly distant from what people actually speak. And of course, in Ireland, not part of the Roman Empire, they had no tradition of speaking Latin. The monks had to learn Latin as an artificial or completely foreign language. And in a way, that made their Latin better, because they didn't think that kind of on-the-way-to-French or on-the-way-to-Spanish language that was being spoken in general could be confused with Latin. In the critical period, 550, end of the Gothic wars in Italy, devastation in Italy, to 750, beginning of the Carolingian ascent, there are only 264 books that survive. That is 264 manuscripts. And all but twenty-six of them deal with religious subjects. Of this twenty-six of them, eight deal with law, eight with medicine, six with grammar. Now, one reason that very little survives is that these were mostly written on papyrus scrolls, which is not very durable. Though there is a change underway from about 400 AD to writing on parchment, which is very durable. And accordingly, along with this change to parchment is a change from the scroll to the codex. So the triumph of the codex is roughly 400 AD to 2010. We're going back to the scroll, actually. We even use the word for the process of going through a document on a computer or a Kindle. How big are these libraries? How many texts do they have? The largest library before Charlemagne's ascent would have been the two libraries associated with the-- in the north of England. And perhaps one hundred books, 120. But it's at this setting, the England of Bede, the early eighth century, that we start to have the monasteries that really conform to our image of monks diligently copying manuscripts. The invention of what's called the monastic scriptorium, a place for writing. So there are monks who not only read, but copy other books, maybe borrowed from other monasteries, too. This is a laborious process and one that we can follow, because we tend to know where our texts of classical learning come from, and by how slender a thread they arrive for us. The monks would copy these documents. They would be preserved in the library. But again, these monasteries close, are plundered. The wanderings of these manuscripts then increase the fragility of what exists and what we now. Yeah? STUDENT: How do monks in Northumbria get a hold of texts that's mainly from libraries and external... PROFESSOR: The question is how do these monks in far off Northeastern England get a hold of the classical texts? They're brought up with the missionaries. There are evidence of constant communication with Italy, which is probably the source of most of them. And they'd learned very quickly-- remember, I said that there is this magnificent codex of the Bible that's now in Florence, that was a gift of the English to the Pope in the eighth century. So there is traffic in these extremely precious objects. But it's very small scale and very fragile, as I've said, until Charlemagne. What Charlemagne does is not so much innovate or invent a program, as make it into something that's not just in artisanal enterprise dependent on a few monks in a few monasteries. And it's part of what we've said is a campaign to salvage classical learning. This is encapsulated in one of Charlemagne's governing instructions to his administrative cadres. He says, "we are concerned to restore with diligent zeal the workshops of knowledge, which, through the negligence of our ancestors, have been well-nigh deserted. We invite others by our own example as much as lies within our power to learn to practice the liberal arts." As Wickham points out, this is an unusual period of intellectuals participating in, and even to some extent, directing government. It's always necessary for governments to have some economists or foreign policy specialists. But intellectuals in the sense of promoting a program of liberal arts is unusual. Or intellectuals who are not just decorators, ornamentors, people who do nice, illuminated manuscripts, or beautiful fountains out of silver with chirping birds. The court of Charlemagne is a moment of intense intellectual effort. And Charlemagne and his successors were rulers who gave tremendous power, privileges, even wealth, to people who knew something about classical learning. The purpose of this is to standardize education in the Church and to make sure that the Church ran in an effective manner. And also to make sure that the government ran in an effective manner. By effective, I mean uniformity and discipline. So that you could go into a church in Barcelona, the southwestern corner of Charlemagne's realm, or in Aachen at the court of Charlemagne, or in Rome itself, and hear the same kind of liturgy or ceremony or ritual. That from one corner the Empire to the other, the church laws would be the same. That the discipline meted out to misbehaving priests would be the same. That preaching and missionary work would be coordinated and uniform. Even that spelling and the shape of the letters might be standardized. It is to Charlemagne and these intellectuals that we owe the very beautiful way that we write. As you can see, my minuscule print is the product of a monastic education. These little letters, right? These minuscule letters are Carolingian. This is how they write. In the Roman era, they tend to write in a different fashion. And in the Merovingian court, Visigothic courts, in a still different and quite difficult to read-- at least for us now-- fashion. The spelling of Latin is regularized at this time. The notion is that the people would receive basic education. And of course, there's some debate as to how wide a spectrum of the populace this involved. How much education ordinary people were to receive. But from the point of view of the intellectuals fashioning this, what was really important was that the clergy receive a good Latin education. The problem of clerical literacy is a persistent problem of the Middle Ages. Clergy, who may know some formula to recite, who don't actually know how to read, or who don't actually know what the Latin words mean. So this is a program not just to save texts, but to use them, to apply them. And also to apply them to a level below that of intellectuals sitting around and discussing the ancient world. At the highest level, monastic centers collected and copied Latin classical manuscripts. Collected, commented, collated. This is not an intellectual program that aims at entrepreneurship or intellectuality in the sense of innovation. Wickham talks about Amalarius of Metz, one of the people I didn't put on here. Amalarius, remember, is condemned for certain liturgical innovations. And he's astounded at being accused of innovating. He acknowledges that this is not his job, although he makes it worse by saying that he had found truth within his own spirit. Saying "I've found truth within my own spirit" in 840 AD is not a proper answer, because that makes it sound as if your spirit determines what truth is. Whereas truth is determined by a much longer, biblical, classical tradition. And that's what orthodoxy is. So he sort of had to accept his not incredibly severe punishment, but punishment nonetheless. The Carolingian Renaissance was organized by a group of people who are from all over the place. Theodulph of Orleans is probably a Visigoth. Paul the Deacon is probably a Lombard. Einhard is a Frank. The greatest of these figures, Alcuin, who was more specifically commissioned by Charlemagne to-- and Alcuin's dates are 730-804. Alcuin is from Northumbria. Alcuin is from the school established by Bede. These scholars, as you can see, have different areas of specialization. Their enterprises, or their duties, were to regularize liturgy, to revive rhetoric, moral theology, logical argumentation, poetry, and to argue against theological unorthodoxy. The two theological problems of the Carolingian Empire were iconoclasm-- debates about that, which we've already seen in the Byzantine Empire-- and a heresy called "Adoptionism". Adoptionism is the notion that Christ was a human who was essentially adopted by God to become his son. It's a heresy that, at the time of Charlemagne, was strong in Spain, or at least defined as a heresy and wiped out. We said that there are twenty-six non-religious manuscripts from between 550 and 750. For the ninth century, there are 290 classical manuscripts; 150 for the tenth century. So you can see that the ninth century is kind of a golden age of this. Although again, we're not talking about an incredible number of things. Not an incredible number, but they're very, very important. Many of you have taken Directed Studies-- first year, freshman liberal arts, great books program. Well of course, DS doesn't like the Middle Ages for a number of reasons. So it's not as if DS decided that Lupus of Ferrieres is up there with Plato and Thucydides, but the things that you do read in DS include Polybius's history of the Roman Empire, Livy's history of the Roman Republic, Tacitus. Polybius, Livy, and Tacitus are authors who depend basically on one manuscript. And that manuscript is a Carolingian manuscript. For Livy, for example, there's a lot of his incredibly long history that we don't have. The so-called lost decades of Livy. Livy's work is divided into books called decades. The fact that we have as much of Livy as we do is due to the Carolingian copying program. The fact that we have Tacitus at all is due basically to one manuscript. Cicero's Republic, rediscovered in the nineteenth century, is a palimpsest. And you'll have heard in section-- and in the Beinecke tour, a palimpsest is a manuscript that has been scraped of its old text, a new text put on. But the old text is usually more interesting to us than the new one and can be restored and read. Most of these are regarded as discoveries of the Renaissance. But where the Renaissance scholars found them was in northern monasteries, where they had been copied and kept since Carolingian times. Where then, did the Carolingians get their manuscripts? Where did they get the things to copy? Which is really a version of the very perceptive question that was asked before. Italy, for the most part especially Ravenna, Monte Casino, and Rome itself. So, for example, the German Monastery of Lorsch obtained from Italy an ancient manuscript of Virgil and copied it. They also have, or copied, Livy's fifth decade-- that is part of Livy's Roman history-- the unique manuscript of the fifth decade, found in the Renaissance at Lorsch. Copied at Lorsch in the ninth century from some papyrus Italian original, long since lost. In addition to Lorsch, the centers of learning include Aachen itself, the court of Charlemagne, Corbie,-- C-O-R-B-I-E-- Tours,-- T-O-U-R-S-- St. Gall,-- Saint-- S-T Gall, G-A-L-L. And these libraries, we can see their growth during this period. So, for example, the Monastery of Reichenau, on an island in Lake Constance. In 800, the year that Charlemagne was crowned as emperor, Reichenau had fifty books. In 846, it had 1,000. Well, all of this emphasis on education and classical learning is extremely important. And indeed, it merits those kinds of "How blank saved blank". "How the Carolingians saved classical civilization", in this case. But it does not mark a break with the piety and spirituality of the era. and. That's part of the point of the Einhard reading. Einhard is a critical figure in the court of Charlemagne, and of his son, Louis the Pious, about whom you'll be reading shortly, or have been reading. And in Einhard, we see an example of an intellectual of this period and of this place. Einhard is active in the court of Charlemagne. He would be one of Charlemagne's biographers, as we know, having read him. He's also a monastic entrepreneur. I did say that this society doesn't encourage intellectual entrepreneurship in the sense of coming up with new ideas, but it's a very entrepreneurial society in the sense of getting saints, getting relics, establishing monasteries, becoming popular. It's a very competitive business environment. And Einhard, as you've seen, was very good at this. He had several monasteries and churches under his jurisdiction. But his pet project, not that far from Frankfurt on the Main River, is Seligenstadt. And it's at Seligenstadt that eventually those relics of Saints Marcellinus and Peter come to rest. Note that the saints, Marcellinus and Peter, were acquired from Italy, just like the manuscripts. Because Italy is where there are an awful lot of saints from the Roman martyr period. There are even more saints, actually, in places like North Africa or Egypt, but they're tough to get at, because it's a long trip and they're Muslim. Nevertheless, we know that lots of relics were liberated-- to use a rather 1970s term-- from Muslim captivity. Liberated-- and, of course, it's not as if Einhard, in talking about what is called the "translation" of relics, says that he bought them, or that he came by them, quote, honestly. Translation meaning here, the movement, the moving of the relics from one place to another. Marcellinus and Peter were buried in Rome and in a church. And Einhard's agents negotiate with this dubious kind of character, who maybe can get them some relics and maybe not. And maybe it will require more money, or he's got to talk to a couple of guys. And it's this intrigue. And that's part of the story. The implication is that Marcellinus and Peter want to move. They let their tombs be found. Remember that in looking for St. Tiberius, it's too heavy to lift the altar up. They can't do it. On the other hand, with these saints, no problem. And once they come to Seligenstadt, they perform all these miracles, which shows they're happy there. Sure, they were stolen. And in fact, as Wickham remarks, and as you've read, Hilduin, a competitor of Einhard's, had stolen some of Saint Marcellinus. And Einhard got him to return it, pressured him to return it. And eventually, Hilduin was disgraced. And I can't remember if he's executed or merely exiled. But this is a tough business. So in 827, these relics get moved. Let me read you, since this is in a part of the Einhard book that we didn't assign. Let me just-- Yeah. It begins on page 69, if you're interested in following this. And they even have the itinerary of the relics. So "after a fast of three days, they"-- his emissaries-- "traveled by night to that place without any Roman citizens noticing them. Once in the Church of St. Tiburtius, they first tried to open the altar under which it was believed his holy body was located, but the strenuous nature of the job they had started foiled their plan, for the monument was constructed of extremely hard marble and easily resisted the bare hands of those trying to open it. Therefore, they abandoned the tomb of that martyr and descended to the tomb of the blessed Marcellinus and Peter. There, once they had called on our Lord, Jesus Christ, and adored the holy martyrs, they were able to lift the tombstone from its place covering the top of the sepulcher." Then they take it; they wrap them in a clean linen shroud. They take just St. Marcellinus, and they wrap him in a shroud. And then they put the tombstone back in place, quote, "so that no trace of the body's removal would remain." But then, just as they're about to make their escape, the deacon, one of this group, says, "Oh, my God, we can't leave without Saint Peter. These two were companions in life. We can't just take one of these bodies. We have to go back and take the other one." They go back, and they get the body. So this is an adventure story. But it's a sacred adventure and it involves the relics of people who still have some say in where they want to be. The very special dead, as Peter Brown refers to them. And what's very special about them is that they're not exactly what you would normally call "dead". Their relics are very much active. Now, we read in the series of miracles performed by these saints, along with another saint they picked up, Hermes, various medical cures. Notice that these miracles, unlike Gregory of Tours, are pretty exclusively in one category. Gregory of Tours has a lot of what might be called "scoffers punished" miracles. Somebody either tries to fake a miracle, and then is the victim of a real one. Or somebody tries to defy the bishop. Or somebody doubts the efficacy of the saint. In this case, most of these miracles, almost all of them, are people with paralysis, disease of the bowels, horrible sores, blindness. Where to start? How great are these? I know you could not put this down. And I hope you read it to your molecular biology roommates just to show how great this is. I think I have this in the wrong place. Anyway, remember the curing of the angry humpback? These guys are carrying the relics to Seligenstadt and they stop to rest in a field. And this angry peasant with a pitchfork wants to chase them away. He also has a humpback. And they show him the relics and say what they're doing. And he falls to his face in adoration. He is cured of his afflictions. And the people of the village insist that the emissaries stay with them, have a kind of party all night, bring other villagers in. This is an interesting moment. And I mention it because here, the piety of a person like Einhard is the same as the piety of the villagers. The Carolingian Renaissance is mostly about an elite program to try to preserve elite forms of knowledge. True, it is an effort to educate the populace, but that is probably the least successful aspect of it. The literacy rate in 875 was hardly any greater than it had been in 775, if you could do a statistical study, and it wasn't very high. But when it comes to the saints, and to the meaning of the supernatural in regular life, the most learned people have the same attitude as the villagers of this little story. OK. So we've reached the height of the Carolingian Empire in two days. And then, our next discussion will be about its precipitous and, in many respects, unfortunate decline. That's what we'll talk about on Wednesday. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 02_The_Crisis_of_the_Third_Century_and_the_Diocletianic_Reforms.txt | PAUL FREEDMAN: But the advantage of talking to you, even on an inconvenient day, is Diocletian. We don't use terms in the professional world of history like awesome very much. But the guy certainly deserves that. He rescued the Roman Empire, and we'll begin by saying what he rescued it from. As a preliminary, the sections are meeting this week. If you're wait-listed for a section, just go to it. We will have enough sections for however many people take this course, right? So there are currently four. The additional ones that were added since last week are Wednesday at 2:30 and Thursday at 7:00. And again, if you're wait-listed, just go. We'll figure this out. There will be enough sections for everybody. So I referred last time to the third century crisis. A crisis of the Roman Empire that preceded the accession of Diocletian, from 235 to 284. And we mentioned several interrelated weaknesses of the Empire that might be seen as long-term causes of this crisis. The size of the Empire. It's sheer, massive size. The problem the succession. That is, it was never quite clear, and we'll talk about this in more detail, how one emperor succeeded another. The urban-rural imbalance. This was an empire built on cities. And to some extent, although heavily debated among historians to what extent, but to some extent, the cities may be said to have drained off the energy, or been parasites to the productivity of the countryside. It was an empire that was, according to one point of view, more cosmopolitan than it had been, according to another point of view, more barbarian. In other words, this was an empire whose Roman population was less dominant, partly through its own success in co-opting other peoples. Some of the peoples that it co-opted we're not actually originally inhabitants of the Empire. Particularly, this is visible in the armies, which tended to be staffed by so-called barbarian tribes. So called by the Romans, who referred to them as barbarians. Another problem is the East-West imbalance, where the East tends to do better economically and demographically than the West, demographically meaning population. We live in a world where one of the great threats is over-population. It is therefore not self-evident, although true, that for most of the time, historically, most places have trouble reproducing their population. And indeed, we are starting to enter into a period of great demographic decline. The infant mortality, the low life expectancy, the death of women in childbirth, the prevalence of disease, and to some extent, military threats of invasion, or if not threats, the reality, made it hard for one generation to produce children to replace itself. And this is true up until the dawn of modernity. So these are fundamental problems of the Roman Empire. The real question is why do they explode in the third century? And this is a question, as I think I said last week, with any great empire. It's easy to point to the flaws in a complicated system. Often size is one of them. Often bureaucracy. Often overspending on the military. But some of these go on for a very long time, more or less success of successfully, despite the flaws. And the Roman Empire went on longer than most, as we said. The immediate problems that explode in the third century are invasions and succession. Invasions by, first of all, Persia. Persia is the old enemy of the Roman Empire. Indeed, as many of you know, the old enemy of the Greek city-states that precede the Roman Empire as far back as the first historian of the Western tradition, Herodotus-- first week of directed studies, for those of you who are nostalgic for that experience-- Persia is the enemy. Now we never study Persia. It's kind of like offstage all the time. And that's the great benefit of the Department of Near Eastern Languages and Cultures. If you want to learn about Persia really from within, instead of oh, my god, the Persians. And it's oh, my god, the Persians in 370 BC, [correction:470] and it's going to come to an end. It is going to come to an end in the seventh century. But that's because of Islam, just to anticipate. But in the third century, Persia becomes resurgent. Having been rather passive, it has this frontier with the Roman Empire in the East. More or less, buffer states are Mesopotamia, present-day Iraq, and Armenia. It's not exactly the same as the present-day state of Armenia, but more like eastern Turkey. The dynasty that controls Persia, and that is more aggressive than its predecessor are the Sassanids, and this appears in your reading. The Sassanid Dynasty, the rulers of Persia, just more aggressive and more adventurous. Beginning in 224, they start to probe that frontier along the Armenian and Mesopotamian border, and eventually cross it and start to wreak havoc in some of the eastern provinces of the Roman Empire. The climax of this is the Emperor Valerian was captured by the Persians in 260. Valerian actually one of the longer-reigning Emperor's of this chaotic period, 253 to 260. They kept him for a little while. Displayed him in chains, maybe flayed him. Anyway, he died in captivity. They did a kind of a job on him. I think maybe he was flayed after he died, just not to be too gruesome, but I think they displayed the skin. As I said, I'm not a Persian specialist. But the second invasion is across the Danube-Rhine frontier. The Danube and the Rhine form what the Romans thought of as a natural frontier with the barbarian states. That didn't mean that they didn't cross them. In fact, many of their great fortresses and establishments were on the eastern side of the Rhine. But they considered those as bulwarks against a dramatic invasion across the Rhine. The Rhine and the Danube almost meet-- the Rhine going from modern Netherlands down to Switzerland, and the Danube going also from Germany eventually Austria, Hungary, into the Black Sea. As we'll see, Charlemagne, in the early ninth century, tried to build a canal between the two of them. And there are actually traces of this immense and completely unsuccessful project. And there now is a canal between them. So this is sort of the frontier of the Roman Empire, and this is the line above which wine grapes are grown, [correction: wine grapes become scarce] no olive oil is pressed. And it's got to be protected, but not worth conquering. So we have pressure on the Rhine-Danube frontiers in the third century, and another Emperor, Decius, died fighting the Goths in 251, Decius. And Valerian dies in 260. So the Emperors are certainly out there as leaders, but that actually has to do with the fact that they're military guys. And that is part of the problem. The major problem besides the invasions was succession. It depends on how you count-- is someone a real Emperor, or is he merely a pretender?-- there are at least 30 Emperors between 235 and 285. Many of them ruling for months, most of them being killed. They're assassinated by the Praetorian Guard, that is, by their own troops. They're killed in battle against foreign enemies, like Decius and Valerian. They're killed in battle against other people claiming to be Emperors. But the most common thing is they're assassinated by their own troops. How had Roman Emperors succeeded each other? There were several ways. One is what you would expect, and that is dynastic. But that's not all that common. Dynastic, in other words, families rule. And the family is recognized as a ruling family, and therefore it goes from father to son, or if there's no son, father to person daughter marries, or father to nephew, something like that. But that was not actually so common. Sometimes the next Emperor was chosen by his predecessor. This is characteristic of the second century AD, the era of the so-called good Emperors. In theory, this is a great system. You have no family prejudice. You simply, as a good Emperor, pick someone who looks to you like he's going to be a good Emperor. So that is another possibility. Another possibility is some guy is powerful, and uses his troops to take over. And that's what we see in the third century. We see not only the militarization of the Empire, but the interference of the army in raising successors, in raising new Emperors. The army was able to make and unmake Emperors. Just as in some countries with unstable political structures, the military is able to make and unmake rulers. What is interesting about the third century is that they're able to do it far from Rome. Some of these armies are in North Africa. Some of them are on the frontier fighting the Persians. Rome is becoming less and less relevant as the dominant city of the Empire. And I mentioned that now, because we'll see on Wednesday the result of this is will be the establishment of another capital, a second capital in the East, in what would be called Constantinople, modern Istanbul. Rome is fortified for the first time. There is a wall that you can still see in many parts of Rome, built by the Emperor Aurelian in 271. And this is a significant thing, because until 271 for centuries, Rome had not been walled, because it was not threatened. And fortification in 271 starts to indicate things to come, or at least things that we know are to come. Namely, barbarian invasions, just as the marginalization of Rome begins at this time. Rome in the third century is ruled, if one can call it that, by a succession of generals. Not members of an upper class elite, but men who have come from the provinces. Men who are not particularly well-educated, who would have trouble recognizing a tag from Horace, there being no internet just to look these quotes up. They hold the traditional Roman elite in contempt. The Senate of Rome is the embodiment of that elite. The Roman Senate is a collection of extremely wealthy people, from good families, extensive property, and very, very fine education. The people leading in the third century now are generals raised by their troops. One of the reasons that the troops both raised up generals, and then killed them, was that they tended to get a kind of reward from the new Emperor-- a thing called a donative. A donative is money that you get when there's a new Emperor. A bad idea from the point of view of the Emperors, because it encourages double-dipping, or triple-dipping. OK, now we've gotten our money from this guy, let's kill him and get money from another guy. I don't think that's too cynical to say that that is some of what is going on. Now of all the crises of the third century, this is the one that leaves the most visible impression. There's absolutely no doubt that there are all of these different Emperors, and that the top of the government is unstable. The question is how much does that carry over into the lives of ordinary people? One measure of the effectiveness of a civilization is that is survives, and even people don't comment very much on political instability. In a way, we are testing that now. For many people, the fact that the government is polarized doesn't really matter, in terms of their everyday life. How long can that go on? That's partly because all sorts of institutions are functioning perfectly fine. Here we are. We are meeting. There's no problem of the supply of water suddenly, or scarce provisioning, or barbarians massed outside, and we have to sort of protect against during our class. It hasn't come to that. Yet. But if you look at the lives of ordinary people in the third century, they are not saying not another Emperor, I can't take it anymore. The philosopher Plotinus, for example, one of the great intellectual figures of this time. Now it's true, he has a very otherworldly way of thinking about things. He is a follower of Plato, he's the leading so-called neo-Platonist philosopher, flourishes during the third century, travels a bit. But we never hear of any difficulty, any kind of problems raised by these unstable conditions. The one that might have affected more people than the instability is inflation. Roman coinage was silver and bronze, and based on ratios of these metals to gold. The ultimate standard of value was gold, but bronze and silver had attributed values in relation to gold. The need to reward the army with those donatives, and the dislocations of the invasions led to tremendous government expenditures. And this was a society that did not have debt financing. There are no bonds. There are no ways of the government anticipating future revenues and borrowing money against them. That is actually an invention of the Middle Ages. This is an example of something that the Romans didn't have that Medieval cities in Italy pioneered, debt financing. I don't have to tell you what that is. And it's a great advantage, up to a point. Great advantage, because it gives you leverage, allows you to do things now. So how does a government deal with problems like this if it can't borrow money? It does what is called debasing the coinage. To debase, that is degrade, the coinage means that you just don't put in as much silver or bronze, let alone gold, and try to get people to accept it for its value anyway. So up until the 1970s, the American currency was based on gold, and dollar bills were called silver certificates. They said that you could hand this bill in for a dollar's worth of silver. It was, technically speaking, not currency. It was merely a representative of currency, which you could trade in. And before that, of course, the coins were metal. Dollars were both silver and paper, with the silver one being, in some sense, the more real. It is in the nature of modern government financing that, at some point, the government could just say forget about it. And even long before the 1970s, of course the government didn't have enough silver for everyone to trade them in, just as it doesn't have enough gold in Fort Knox to float the entire world's currency. It has a lot. And that's important that there be some real kind of value, or bullion, or bottom line kind of gold bars. But the fact that it's not enough is not intrinsically a problem. But here, so let's say the government receives, in the form of taxes and stuff, a thousand gold units. And it issues two thousand units using that gold. The coins say they're worth a unit, but they're actually, in terms of precious metal value, worth a half. So it pays its expenses that way. It pays the army. It gets people to accept this money. But when most people go to buy stuff, they are going to find that the stuff is 50% more expensive, because their coins are not actually very good. The governments tend to do this gradually, hoping the people don't notice in the first place, or that if the inflation is say, 10%, it's not so bad. But once having embarked on the debasement of the coinage, this tends to get out of control because of the famous Gresham's law-- an 18th century economist-- that bad money drives out good. If I have good sestertia, or Roman currency, with the full measure of silver that they say they have, one full unit, and I've got other coins that say one full unit, but only have half, I'm going to try to get rid of the bad ones and hoard the good ones. And only spend the good ones if I absolutely have to, or demand a premium on them. So therefore, there are all these crappy coins circulating like mad, and the good ones retreat into peoples' wealth-- they don't really have mattresses then-- but into their store boxes, or under their beds, creating in itself more and more inflation. So you have a fierce inflation. Prices go up. People don't know the value of things. They start bartering. And the Roman economy is very, very adversely affected, as hyperinflation tends to. And a final problem is the ruin of the local elite, partly because of this economic chaos, but partly from deeper causes. The importance of this is something that I alluded to last time, and that is that the Empire could not be held together, really, by the government, however big the government was. It required the cooperation of wealthy people with ties to their native city. It was these people who sponsored games, civic improvements, maintained the temples, and kept a kind of local order. The third century crisis undermines this elite. This elite is undermined by the militarization of society. They are, to some extent, ruined by taxation and increased taxation. They're just powerful enough to be well-off, but not powerful enough to evade taxation. But they also tend to be undermined by an Empire that's more cosmopolitan, where local elites don't matter as much. Where, for example, military people who move around a lot, are more important. And I emphasize this, not only because of these elites themselves and their role in holding the Empire together, but we're going to talk about this when we talk about Christianity. Because when the local elites are ruined, so is local religion. And local religion means polytheism. If I am of a grand family of, let's say the city of Sardis in Asia Minor, in modern Turkey, and I feel that my ancestors have always been involved in the worship of the goddess Cybele and I am a votary, or an officer, or like a member of the governing board of the club, or society that runs the cult of Cybele, I'm going to feel very loyal to that local deity. But if I come from North Africa, and I'm in Sardis because that's where my army is, I'm not going to don't care about some local. No more than you might care about a club that's important at Yale, if you went to the University of Illinois. No more important than you might feel about pizza in New Haven, if you came from somewhere else and didn't like pizza. It's just like this local cult, and you don't understand it, and you don't care. So the kind of things you will care about, we'll see, religiously. But they will tend to be religions that cross borders, like Christianity. Religions that are not identified with one place and one god, in the sense of local god, local temple, my people. So into this mess Diocletian-- because Diocletian defeats his predecessor, Carinus in battle in 284. Diocletian is a general. In 284, it would have seemed like more of the same. But Diocletian rules until 305 and he abdicates. He passes the power to someone else. He is, however, very typical of the military class of the Empire. He came from nowhere, socially. He was the son of an ex-slave from Dalmatia, modern coastal Croatia. You can still see his palace in the Croatian city of Split on the Adriatic. His retirement palace, actually. Under his severe guidance, the Empire was reformed. And the way it was reformed was that it was, in effect, militarized. Diocletian was not a great general, but he was a brilliant manager. And he was a brilliant bureaucratic organizer. I used the term bureaucracy, not to mean inefficient, useless administration, but administration. Administration officers of the state, who are capable of doing their jobs, or maybe not capable of doing their jobs. But who are nevertheless-- I'm not making a value judgment with the term bureaucracy-- I simply mean the proliferation of government and government offices. Diocletian is responsible for the militarization of society. That is, building society around the army in order to protect it. And he is responsible for a more efficient, and ultimately, burdensome form of taxation. The two are enlinked because, as we know, you have to pay to have a large and effective military. Diocletian did not set out to be a revolutionary. His aims were conservative. He wanted to save, preserve, restore, the Roman Empire of the pre-235 era. His methods were radical. He was willing to undertake radical measures. And the debate among historians, now somewhat muted. Many historians at one time felt that he had basically destroyed the Empire. By making it so bureaucratic, so militarized, so heavy-handed, in terms of government, it no longer was the Roman Empire. It was something else. Now the reason this is no longer exactly considered to be a big problem, or a big controversy, you'll see when we come to read Wickham. The Empire has an impress on society. There is what he calls the Burden of Empire, but it is, at the same time, not a totalitarian empire that controls everything. Society has an identity that's different from the government. So he has three goals. One, solving this problem of the imperial succession. Two, stabilizing the economy. Three, protecting the frontiers. Of these three, he's actually only really successful in the third, protecting the frontiers. He devises a system, that we're going to talk about in a moment, of succession, but it does not really outlast him very long. The economy does get fixed but, not exactly because of his policies. What he's really successful at, and what changes most dramatically, is what rulership means. That includes the figure of the Emperor, who becomes more sacred and more powerful, in terms of imagery, as well as administration. Changes in the administration, of taxes in particular, and then that goal of his to change and grow the military. The size of the army grew, probably doubled. Maybe, just as a ballpark figure, from 200,000 troops to 400,000 troops. This is a major, major increase that had to be paid for. And it had to be paid for by taxation, and from a population not particularly eager to volunteer to pay more. Ultimately, it looks as if Diocletian didn't so much increase taxation, as increase the efficiency of its collection. In order to increase the efficiency of his collection, he had to increase the bureaucracy charged with monitoring the taxes. And that means first making an inventory of taxable resources. There is no income tax in this society because it's an economy based more on land than on salaries. An income tax is easy because you can keep records of what people are earning. The government, to this day, finds it much easier to take a portion of your wages, because it knows from your company what you're being paid. If you're being paid in some other form, like you're a waiter, and a lot of your money is in tips, that's harder for the government. If your wealth is in property, it's hard to put a value on that. You own estates. You have people who are free tenants, who rent land from you, and pay you in money, or produce, or labor. You have some slaves. You have a water mill. How do you pay taxes on all of this stuff. The opportunities for evasion are greater. So the first bureaucratic tasks was just to have a lot of people could value things, who could come into a territory and say OK, this farm is worth so much, and it has these many people. They develop a system to evaluate productive units of things like land, and population, and to tax them according to a formula. So I have a lot of land, but it's not very good land. I don't have a lot of people making a living off it. I will be taxed at a lower rate than someone was maybe half that land, but twice the people, better soil, more clearance of forest, whatever the reason. It changes. So every 15 years, the government changes its estimation. A second means, once you have this taxation system in place, of bringing the Empire together and dominating it more, is to have the army really have first call on the resources of the state. We now start to have a state supply system for the army alone. This allows the state to avoid dealing with that debased currency. So for example, the state just goes in and takes wheat and gives it to the army, without taking money, buying wheat, giving that to the army. Diocletian binds the Empire together, also very effectively, by a postal system. Post in this case meaning a system of riders, horses, communications, that allows the Emperor to go faster than anyone else, to have news quicker than anyone else, to send orders quicker than anyone else. And also, more punishment for things like tax evasion. It's not just that people get killed for not paying their taxes, or imprisoned, or tortured, but that groups are responsible. Not only that if I don't pay my taxes, I get punished, but if you and I live in the same village, and you don't pay your taxes, I get punished. Or at the very least, taxed for the part that you evaded. But the most important change in government is the establishment of what's called the Tetrarchy. The Tetrarchy is the rule of four. Four rulers Diocletian divides the Empire, first in two-- East, West. A very significant move that will have consequences for the next 1,200 years. He then appoints a co-Emporer to rule in the West, while he rules in the East. And they each appoint a helper, number three and number four. The two Emperors are called Augusti, Emperors, and the two helpers are Caesars. So they're subordinate to their respective Augusti, and they're supposed to help them. Why this system? This is really to overcome the problems of size, communication, administration. It's a statement that the Empire is too big for one man to rule. But the chief Emperors are also exalted now. There's no longer a pretense that they're just first citizens. or princes. They are clothed in purple. They don't move a lot in public audiences. We're familiar with this kind of dichotomy between the political figure as distant authority, versus the political figure at least pretending to be just like you and me. We're in the era of the latter. The Diocletianic period ushers in a period when the Emperor is distant, glimpsed, product of ceremonies, wearing a lot of very funny-looking, but fancy clothes. He doesn't appear a lot in public. He's a god. You don't go up to him and shake hands, or say hi. You throw yourself at his feet, and don't look at him until he tells you to. So the tetrarchy, great idea, it really doesn't work. Because, first of all, the Emperors don't necessarily cooperate. The Caesars don't necessarily cooperate with the Emperors. And so in 285 Diocletian nominates a Caesar, and then makes him, in 286, a co-Augustus. And this man is named Maximian. In 293, Diocletian and Maximian appoint two Caesars. I think I'm not going to burden you with the personnel. I will hand out something on Wednesday that gives you some of this information. Diocletian and Maximian appoint Galerius in the East and Constantius in the Western-- you don't have to remember who these are-- and each of them marries a daughter of their respective Augusti. This looks like a great system. In 305, Diocletian and Maximian abdicate and then two people become Augusti. The two Caesars rise up to be Augusti and they appoint their own new Caesars. It breaks down beginning in 306. One of the sons of one of these Augusti is not appointed, and he's mad, and he revolts. And then the Augusti don't get along. Out of this chaos between 306 and 312 emerges one Emperor. And that is Constantine. And we'll be talking about Constantine on Wednesday. So the Tetrarchy fails. Diocletian's second big initiative was over this question of the economy, and ways of combating inflation. Diocletian issued a so-called edict on prices. The edict on prices attempted to set a fixed price for goods. And if you sold them for more than that, you were to be severely punished. This is the kind of classic example of the state trying to combat inflation by dictating prices. Most of you are not familiar with inflation, because we have lived in an era of very low interest and fairly stable prices. But if you think of those commodities whose inflation you are familiar with, like petroleum, it is very dislocating. It starts to create panics, and the panics then feed into the inflation. Just as, if people keep on getting gas because they think that it's going to go up in price, then there's a greater demand for gasoline, and it goes up further in price. Eventually, if the thing is really just speculative, it deflates again. And that's what's happened with products that we're familiar with in recent times. But there is also a kind of structural, longer-term inflation such as America experienced, for example, in the '70s. And in theory, if you have resource crises and things that are becoming scarcer, then you ought to have more and more experience with inflation. The government in the 1970s in the United States tried also to have an edict on prices. Under President Ford, there was a kind of administration of maximum prices. The problem with this is that it creates a temptation for black markets, creates a version of what we were just talking about with Gresham's law of coinage. If you say that tomatoes can only be sold for a dollar a pound, then those tomatoes that are being sold for a dollar a pound in a climate of severe inflation will be terrible, will be rotten. If you want to pay $2.00 a pound in secret, we have some nice tomatoes for you. And if you want to pay $5.00 a pound, we have some really nice, locally-grown tomatoes for you. All of which may be illegal, but the legal market is empty. And this is what happened in response to the edict of prices. This doesn't mean that the government cannot-- I mean it still remains debatable, obviously, the degree to which government can or cannot intervene in such things-- but certainly, in the case of the Roman Empire, this failed. What did bring back a measure of economic stability is the reform of the taxation. The fact that the state simply started getting in more resources, that less was being withheld by private people, and so the state could actually pay for its administrative and military costs. So Diocletian succeeded in abdicating peacefully, spent his retirement in Split in his palace, and lived to see the breakdown, or the partial breakdown of the Tetrarchy. And in certain respects, his policies clearly failed. The edict on prices had to be abandoned, the Tetrarchy did not work, and Diocletian failed in trying to suppress Christianity. We'll talk about this some more, but Diocletian, in the late part of his reign, a couple animals were split open to see what the future would be. Right? Isn't that what we all do if you want to figure out what's going to happen? You slaughter an animal and check out its liver. Maybe its kidneys and spleen, too. But the liver is really what you want to look at. And I don't exactly know what the animal was, what the organs were, and why they didn't splay out right, but the liver told Diocletian that the Christians were responsible for this, and that he'd better go after them. So there is this big persecution of Christians in the first part of the fourth century. And he certainly didn't succeed in that. Not only did the Christians not crumble, but of course Diocletian's most effective successor, six years after he abdicated, would convert to Christianity. But Diocletian is extremely important, and in many respects, extremely successful. He did more than prop up a tottering Empire. He did more than just transform a tottering Empire into a kind of tottering tyranny. He saved the Roman Empire. He saved the Roman Empire for 100 years. When you take a course like this that goes for 700, 800 years, you start to hurl centuries around and get confused among them. But any polity that exists for 100 years is fairly impressive. Or a polity that looks like it's about to collapse, and then is restored for 100 years. The Roman Empire, conventionally speaking, is thought to have collapsed in the West in the late fifth century. In the East however, arguably, Diocletian's reforms last for more on the order of 1,200 years. The Eastern Empire, the Byzantine Empire would fall in 1453. And to its last day, it was modeled on Diocletianic administrative and military forms. People at the time clearly thought that they'd been saved from disaster. If you look at fourth century artifacts, things like mosaics on the floors of dining rooms, people often put mottos there. And their mottos are things like "Joyful times everywhere." or "A world restored." The fourth century is interpretable as an era of increasing gloom, because we know that in the fifth century things are going to collapse. But people in the fourth century are not saying to each other, I'm so glad I'm alive in the fourth century, because I don't want to see what's going to happen in the fifth. They are just happy that the barbarians are back across the Danube and the Rhine, the Persians are more or less controlled along the frontier. Yes, taxes are high. Yes, the elite is some sort of riff-raff, and not as well educated, and they're military people. But basically, things are working, prosperity is restored to the people who had been prosperous before, the local elites have declined, but it's not so visible as it had been. But there are some changes. Changes that we can, with the proverbial benefit of hindsight, see. Changes in the center of gravity. The dominant places are now places that are military bases. They are great cities, with all the amenities of Rome. That is, stadiums, gateways. One of the best-preserved of these to this day is Trier in Germany. Trier, on the Moselle behind [correction: west of] the Rhine, has a wonderful collection of Roman ruins. A gateway, the Porta Nigra, a theater, a sports arena, a basilica, a law court turned into a church. The reason trier was great was because of the frontier. It was one of the most important cities in the Empire because of the military, because of its strategic importance. And there are other cities that are like. Milan, for example, becomes more important than Rome. Because Milan is further north, it's a good place to get to the Danube and to the Rhine quickly, whereas Rome is buried in the Mediterranean. So we're in a new world. We'll discuss more of the new world on Wednesday. And we will see that Constantine, in some cases, completely revolutionizes things. And in other cases, continues Diocletian's work. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 03_Constantine_and_the_Early_Church.txt | PAUL FREEDMAN: Today, we're going to talk about what is arguably the most important event of the first part of our course, the conversion of Constantine, the Roman emperor, to Christianity. Important not because of Constantine's own particular opinions. The fact that he embraced Christianity is, as we'll see, a little hard to explain on purely strategic grounds. But its importance is that it represents a permanent change. It represents the beginning of the Christianization of the Roman Empire, a very unexpected result. Because not only had Christianity been illegal in the prior history of the empire, that is to say, for over 250 years, but of course, the god-man Jesus had been put to death by the forces of the Roman Empire. And as we've discussed, Roman religion, with its emphasis on what we've called "civic polytheism" or the performance of ceremonies in public, ceremonies that have to do with local patriotism, emperor worship, the tradition of the Olympian gods, and, above all, polytheism, was very foreign to Christianity. The Christian religion thus seemed to be a kind of annoying epiphenomenon of Roman society when, in fact, with this event, Constantine's conversion, it becomes first a tolerated religion, then a favored religion, and very quickly, within the course of the fourth century AD, the official and almost the only religion of the Roman Empire. How can this be? We'll discuss both the specific events today, and their meaning, and how they play out. We recall then that what is called "paganism," a traditional religion of the Roman Empire, was polytheistic, was many gods, ceremonial, had a lot of local variation, and it was eclectic-- eclectic meaning that you could worship different gods in different places, different gods for different purposes, different gods for different times of your life. There was a certain emotional vacuum, or at least a perceived vacuum, in this religion because it seemed to deny individual longing and longing in general, that longing, that sense internally that there is more to life than there appears to be. So that many adherents of other religions, including, but not limited to, Christianity believed that some part of their body was immortal, or the soul was immortal, or that the immortal soul had to be healed by religion, and not that religion should simply be a pathway to good fortune or to easing the anxieties of the material world. So Christianity, we've said, is not so much otherworldly, focused on heaven. It is that. But even more important perhaps is its innerness, its inner worldliness, the sense that people have a interior soul that yearns for something eternal and more significant. And then Christianity was accompanied by other so-called "mystery religions," religions that also spoke to an immaterial, heroic, non-civic, non-urban type of piety, Mithraism, for example, the worship of the mother goddess Cybele. Christianity had certain advantages in terms of reaching a population, the promise of an afterlife, the commitment that it demanded of people, a religion that appealed both to the elite and to the common people, and a very strong local organization. But Christianity was alien to the Roman Empire. The Romans did not always persecute Christianity, as they did under Diocletian. But they found Christianity alien. They didn't like the fact that Christianity was intolerant. Every other religion of the Roman Empire, with one exception, accepted other gods. If you worshipped Isis, you had nothing against other people worshipping Jupiter. If you worshipped Cybele, you had nothing against other people worshipping Mithra. But Christianity, of course, ridiculed all of these gods. The only other religion that was like this was Judaism. But Judaism made some accommodations with the Roman Empire, recognized the authority of the Roman emperor, and did not defy the state in the way that Christianity at least appeared to. And Christianity was not a Roman religion in many of the ways that it rejected worldliness, rejected engagement in or enjoyment of the material world: the pleasures of the theater, the circus, the celebrations of civic paganism, emperor worship, law courts. Well, law courts may not be pleasurable. But this sort of civic involvement of the emperor and the Empire are rejected by Christianity. Christianity, when you see what Roman pagans write about it, is a kind of killjoy religion. It's a religion of people who seem to have their eyes focused on anything but the actual process of getting ahead in Roman society. All of this notwithstanding, it should be emphasized that Christianity was not persecuted constantly nor was the persecution very intense. We have Nero in the late 60s AD, the Decian persecution of the mid-third century, and of course, the great persecution under Diocletian. Christianity received just enough persecution, one might say, to fortify its spirit, to give it some backbone, but not enough to break it. Constantine emerged from the chaos following Diocletian's abdication. Diocletian, and as you'll recall, had created this four man rule, the Tetrarchy, in order to divide what was perceived as an excessively large empire with an excessively large administrative structure. The Tetrarchy was, at least we can say with hindsight, doomed to failure. These four emperors would not cooperate. They would tend to be rivals. Constantine was the son of one of the caesars, one of the subordinates. Remember there were two augusti, two caesars. His father was Constantius Chlorus who was appointed when the Tetrarchy began in the West in 293. So there was an Augustus of the West and a Caesar of the West. The Caesar of the West was Constantius Chlorus. The young Constantine was sent east to serve the eastern Augustus, who succeeded Diocletian, Galerius. Constantine was left out of the succession when Diocletian abdicated. Galerius appointed somebody else, and Constantine rebelled. Constantine, in 306, raised an army in faraway Britain, marched on Gaul, and eventually was grudgingly recognized by Galerius as caesar. At the same time, another disinherited son of an augustus, a man named Maxentius, rebelled in Rome. And I will not burden you with the whole working out of these intrigues, of the fightings of armies, of the quarrels of augusti and caesars. But basically, in 311, Galerius, who had been ill with cancer, died. And Galerius was succeeded by an emperor in the East named Licinius. And Licinius allowed Constantine to deal with the usurper Maxentius in the West. So we have Licinius in the east in 311, and then in the West, Constantine and Maxentius fighting it out. Galerius has died. Constantine defeated Maxentius at a battle not far from Rome, the Battle of the Milvian Bridge. The Battle of the Milvian Bridge in 312. And Constantine was now Augustus in the West, Licinius, Augustus in the East. The Battle of the Milvian Bridge is the context for whatever had happened that changed Constantine's mind about his religious orientation. Just before the Battle of the Milvian Bridge, something happened. There are two stories that purport to explain the event. One is that Constantine had a dream. And in this dream, an angel spoke to him and ordered him to paint a symbol combining the Greek letter chi and the Greek letter rho on his soldiers' shields. The rho, the R in Greek, and the chi written as an X. The two letters symbolize, or at least were taken later to symbolize, Christ, the first letter being a chi, the second a rho. The second version, which is later, that is, later in circulation as a story, but seems perhaps to have been attested by the emperor himself to his biographer, Eusebius. According to Eusebius, Constantine was marching with his army before the battle. And he, along with the army, saw a cross in the sky. And superimposed on the sun, against which background the cross appeared, were the words, "In this sign, you will conquer." Hard to say which version, if either, is what Constantine thought happened to him. The argument for the second one is partly Eusebius's description, partly the fact that angels in a dream are a standard kind of story. On the other hand, the Chi Rho symbol is not previously a sign of Christianity. So the very fact that there isn't a background to that, that this is something that we hear of now for the first time, might indicate that that's the true story. But more important than what actually happened is that there's no reason to doubt Constantine's sincerity. There is no reason to believe that this was a calculating, cynical, or politically astute move. This is not because Constantine wasn't devious. He was. But because it's hard to imagine any emperor thinking that Christianity was a good idea. Because Christianity was subversive of Roman values. And it was particularly subversive of the values of the Roman army, whose crucial aid Constantine depended on and of which Constantine had to be the leader not only in order to defeat Maxentius, but simply to survive in power. Christianity was pacifist. At this time, it took more literally than it would later the admonitions of Christ in the Gospels not to fight, not to hit back, not to engage oneself in the pursuit of worldly gain by means of violence. So it's hard to imagine anything more unlikely than an emperor becoming Christian and gaining the support of his followers. Now that doesn't mean that Constantine became some sort of monk, interpreted the Gospels literally, told his soldiers to put down their weapons. It's clear that Constantine regarded the Christian god much as other emperors had regarded, say, the Invincible Sun, or the genius of the divine emperor, or any other pagan deity that brought victory in war. Constantine, like all emperors, saw himself as a child a fortune, as someone who was favored by fortune, depended on fortune, and who needed to placate, to mollify, to please whatever god it was that controlled fortune. What's unusual is that he would deem the Christian god to be this sort of god, a leader of war, a giver of victory in battle, a companion to the emperor. None of this would seem, at first glance, to be likely in Christianity. The fact that not only does it work, but that it would work for centuries later is just part of the cataclysmic nature of this event, or if not cataclysmic, at least unexpected. Constantine was not ignorant. He's someone who had studied philosophy, who was quite literate, knew Greek pretty well, familiar with Latin literature. But nevertheless, he was obviously a man of affairs. He's not an intellectual, contemplative person, poring over philosophy books. He's a man of power, decisiveness, strategy, and not a little cruelty and brutality. And we can see that after his conversion experience-- and indeed, I should point out he did beat Maxentius--he accepted the Christian god, he went to battle with the usurper, and he defeated him. But even after his victory, he doesn't become, in every respect, a totally committed Christian at least in terms of the symbols of power of his office. His coins, for example, which are a very good mark of propaganda and self-regard, his coins kept the imagery of the earlier pagan deity associated with the emperor, the Invincible Sun. After a little while, you start to see the Invincible Sun on one side of the coin and the cross on the other. And only later in his reign do we have just the cross. Constantine's first substantive act as a Christian or as someone who favored the Christian church was the Edict of Tolerance. The Edict of Tolerance or Edict of Toleration issued at Milan in 313 was jointly the product of Constantine and Libanius, now the two last guys standing. The Augustus of the East, Libanius, and Constantine, the Augustus of the West. Libanius was a pagan. He did not share Constantine's bizarre enthusiasm, but all right. If he wanted to tolerate Christianity, this was fine. This was part of their-- I'm sorry. It's not Libanius. Libanius is a philosopher. Licinius. Licinius. Constantine and Licinius. Licinius was a pagan, but he was willing to go along with toleration. At this point, Christianity was legalized. But in the west, Constantine came to favor the Church and do more than merely accept it as legal. For example, he returned property confiscated in the Diocletianic persecutions. He exempted the Church from state taxation, an incredible gift, and allowed church officials, bishops and others, to use the imperial communications system, the so-called post system whereby they could get fresh horses to go from one place to another, greatly speeding up their journeys and making the journeys, in effect, chargeable to the state. Constantine left the pagan and ceremonial center of Rome alone, for the time being at least, and built two great basilicas on its outskirts. One, Saint Peter's. The St. Peter's that stands today is, of course, a product of the Renaissance and the Baroque. But the old church that was destroyed in the sixteenth century was that of Constantine. And he also built the Lateran Basilica. Both of these outside the walls of Rome. As we'll discuss, he also attempted to mediate in disputes involving the church. He never, however, completely marginalized the old religions. He emphasized the diversity of religious practice. He didn't require a single form of worship. But by the time he died in 337, the pace of conversions was such that perhaps as much as half of the Empire had embraced Christianity. And this brings us to a crucial question, of course, that we'll be discussing really throughout the semester, and that is what was the effect of Constantine's conversion on the Church? Or beyond the mere event of 312, what did it mean for the Church to go from persecuted minority to established majority? What explains Constantine's ability not only to change the course of the Roman religious practice and tendencies, but to do so permanently? For the Church, was this turnaround a providential sign or a kind of Trojan horse gift in which the Church would now be so tied to the official culture that it would never be able to shake off Rome, administration, and bureaucracy to get back to its original, charismatic, individual, powerful foundations? The era of Constantine establishes the problem of the Church in the world for the Middle Ages and, indeed, beyond. This problem is is the church a collection of special people who have their eyes fixed on heaven or, is it a kind of universal society that is hard to distinguish from just worldliness and engagement with the world of business, life, death, and other banalities? It is Saint Augustine who is going to deal with this most forcibly in terms of theory, but that's a century later or so, well, 75 years later. Externally, the Church adapted very quickly to success. We can see this in terms of the pace of conversion, as I said. Not only were 50% of the people, perhaps, Christian by 337 when Constantine died, but by 390, the time of the Emperor Theodosius and his death, 395, probably 90% of the population was at least nominally Christian. The reasons for this success. In other words, how could Constantine's particular gesture have such a decisive impact? Some of this has to do with Christianity's willingness to adopt to the customs of the Empire. Some of it may have to do with the weakness of the official religion of Rome and of the urban elites who were its chief support. Those who held out against Christianity were, on the one hand, people in the rural areas, so peasants, whose fundamental beliefs tended to be directed to agriculture, local deities, deities that controlled the weather, and water, and things like that. The army, for reasons I've just said, that is, Christianity is not, at first glance, congenial to people who fight for a living. And then the third group that held out were the intelligentsia, particularly of Roman and Athens, the people who had a substantial cultural investment in Greek and Roman philosophy, the intellectual side of the old elite. Well, Constantine fell out with Licinius. And after some small skirmishes, Constantine managed to defeat him at a place called Chrysopolis in 324. Licinius fled from the battlefield, Constantine's forces caught up with him, and Licinius was executed. This event, this Battle of Chrysopolis, important in itself-- P-O-L-I-S-- important in itself was even more important because it showed Constantine the importance of the small fortress city of Byzantium, not far away. Byzantium who is the ancestor of the city that Constantine would found there, Constantinople. And of course, modern Istanbul in its twenty-first century incarnation. Byzantium commanded a strategic point of access east-west and north-south. It was the point of access between the Black Sea and the Mediterranean. The Bosphorus is a narrow strait that separates Europe from Asia. Byzantium, Constantinople, Istanbul stands on the west bank, the European side, but it commands and controls the channel by which anyone would go from the Mediterranean to the Black Sea. And since the Black Sea is the gateway to central Asia, it, in effect, controls communications between two commercial, strategic, and military zones. It also controls the route from the Balkans, southeastern Europe, into Asia, into, specifically in this case, Asia Minor or the Asian part of what's now Turkey. Byzantium is, therefore, strategically located in terms of communication and, at the same time, located so that an army can get to two of the most dangerous frontiers of the Roman empire in a reasonable amount of time without having to commit itself to one or the other totally. It is not far from the Danube frontier, which was, as we said, one of the points at which the empire met the Barbarian tribes and which the empire had sort of decided on as its natural frontier. And Constantinople was also not that far from the eastern frontier of the Roman Empire, the frontier with Persia, which ran along what's now eastern Turkey, Armenia, western Iran, and Iraq. It was the city, also, within the richest part of the empire. As we said, one of the problems of the Empire in its later years, its later centuries was that the east was becoming richer, more urban, more commercial. The west was lagging behind, more rural, less successful in its commerce. Constantine wanted an eastern capital for both strategic and for economic reasons. For strategic reasons having to do with the movement of the armies and the protection of the frontier. For economic reasons having to do with taxation and administration. The city of Rome itself was somewhat isolated, strangely enough since, of course, the whole empire had grown up around Rome. But Rome was the historical origin of the Empire, but not, in the fourth century AD, its actual living capital. It would be too much to compare it, say, to the relationship between Portugal and Brazil. It's not quite that lopsided. But Brazil is a former colony of Portugal. They speak the language of Portugal. Yet on the world stage, Brazil is larger than Portugal, more important than Portugal, richer now than Portugal. So whatever preeminence Portugal or its capital, Lisbon, has within the world of culture, no Brazilian would take Lisbon to be the be all and end all of the Portuguese cultural world. So similarly, by this time, Rome has become less important even within the western empire. And this relocation of the capital to Constantinople, the relocation of the capital to the east is significant because it shows us the permanent result of the Tetrarchy. As we've said, Diocletian's experiment was a failure in the sense that the emperors and caesars would not cooperate. And such a scheme was never tried again. But the division of the Empire between east and west would be something that would eventually become permanent. Its first traces are with Diocletian, and that's one reason why we begin the course with him. It is also something that continues under Constantine without the addition of the caesars. Constantine ruled over the whole empire. He did not divide it himself, but he facilitated its conceptual, and eventually real, political division by creating a new Rome, a new capital in the former fortress of Byzantium, a town that he modestly named after himself. Constantinople, as this town was called, was planned to be a new Rome. Like Rome, it would have a forum; it would have civic spaces; it would have races and sporting events. It would have imperial palaces and gardens; it would have victory columns, triumphal arches, aqueducts, the whole panoply of classical civilization. It wouldn't have a whole lot of temples. Churches would be more important than temples, not that Constantine totally banned temples from Constantinople. But these were not the highlights of the city. It is an ideological statement like other planned, great, imperial sites. So we could compare it to, in the modern world, Saint Petersburg, created by the czars as a certain kind of statement, with a certain kind of plan, and a certain kind of look evoking western Europe in particular. Or Versailles, not a town at all, but rather a kind of palace city fit for the king of France. At this point, Constantine becomes considerably more devout and somewhat more intolerant. We start to see him interact with the Christian Church in its most intimate way, that is to say, doctrine. Constantine is appealed to by the Donatists, schismatics-- well, we're calling them schismatics-- or heretics, as they were decided to be, from North Africa. The Donatists taught that the priests who had given over the scriptures under persecution at the time of Diocletian were not legitimate priests. And we'll talk later about the implications of this. The implications, briefly, are that the Church cannot cover for priests, that the office is not greater than the man. If the man has committed a sin, such as what was called treason, the handing over of the scriptures to the persecutors, he no longer can baptize validly, he no longer can perform the sacraments with validity. Donatism, then, implies that the Church itself is really just as good as the character of its officials. The Donatists were strong in North Africa, and they appeal to Constantine against decisions that had been made against them within the Church. The fact that Christians are appealing to the emperor already, as early as 317, shows the acceptance of the emperor as a Christian arbiter. But it also shows a kind of, in retrospect, dangerous intermingling of what we would consider to be church and state. Similarly, Constantine would get involved in controversies over the relationship between God, the Father, and Christ, the Son. This, too, we'll go into in more detail, but this is the Arian heresy-- Arian with an "i," not with a "y"-- named after a priest named Arius who taught that while Christ is God, he is, in some sense, subordinate to God the Father. This is a controversy over the nature of the Trinity in which Christ is seen as coming from God, as emanating from God. And as I think I warned at the beginning of the course, if you don't like doctrinal and theological controversy, I'll try to spare you all its ins and outs, but you can't teach this course without it. Again, what we're talking about now is not the content of Arianism, who embraced it, why, but the fact that the Emperor gets involved in these controversies. On the one hand, this shows the quick adaptation of the Church to imperial rule. On the other hand, because Constantine was able to solve neither the Donatist nor the Arian division, at least not definitively, and at least not yet, it shows how difficult it was for an emperor who could conquer all of his secular rivals, who could control this vast realm from Gibraltar to the Tigris and Euphrates, but couldn't get a bunch of North African peasants to obey his orders about how to worship or Egyptian priests either. Constantine, we can see, is frustrated by this. You can see in the reading from Jones, his difficulties in dealing with this in the usual way. The usual way being the emperor is petitioned by people, he appoints some arbiters or judges, the judges make a decision, and then the emperor announces to these people that that's what it's going to be. The problem is that, of course, people like the Donatists were already used to martyrdom. Threatening them with imprisonment, threatening them with torture, denouncing them, trying to use the awesome, awe-inspiring power of the emperor against them was not going to be sufficient. Nevertheless, Constantine, far from abandoning Christianity in frustration, becomes more and more engaged in trying to, if not officially Christianize the Empire, at least legislate as a Christian emperor. By 330, he has come to see himself not merely as an emperor who has a kind of peculiar favor or a peculiar god that is following him, but as the implementer of the mission of the Church. So for example, he starts promulgating laws against married men having concubines-- ineffective-- or the seduction of wards by their guardians, or punishing rape by burning, all orientation towards sexual crimes that shows a more Christian horror of them than the more easygoing Roman attitude. Constantine favors the church, enacts legislation recommended by the church, favors the bishops, and even in the 320s, presides over the first ecumenical council of bishops of the Church called at Nicaea across the Bosporus from Constantinople, the Ecumenical Council of Nicaea called to deal with the Arian controversy. And here, we see Constantine as something different from an emperor merely the companion of Christ or the companion of God, but the emperor as, in some sense, head of the Church. Constantine appears at the council, he is deferred to by the bishops. Nevertheless, he is not himself a bishop. He is not himself, however imperial the Church may look, able to legislate by himself for the church. Because unlike many other religions-- and certainly when we come to Islam, you'll see the contrast-- the political leader of the Roman Empire is not the designated leader of the religious practice of the Church because he is not a priest. Now who is the designated leader of the church is not clear yet. Certainly, it's not yet the pope in the 320s. It is the collectivity of bishops, but in that case, then some bishops have more power than others. Nevertheless, this is the beginning of an era in which we have a blending, but not a total equivalency of secular rule, imperial rule, on the one hand, and spiritual or church rule on the other. And that's one of the things that, of course, characterizes our image of the Middle Ages, a period in which the church and the state were overlapping if not actually fused. Constantine in relation to Diocletian, to conclude. Differences and similarities. Obviously, their similarities are great. Both Diocletian and Constantine remade the Roman Empire as a much more tightly administered state, a more bureaucratically complicated state, and a more militarized state. Constantine continued Diocletian's military and administrative structure. Like Diocletian, in order to do this, he had to rely on very heavy taxation. If anything, his taxation had to be greater because he had exempted the Church and its clergy, and someone was going to have to make up the difference. But Diocletian had persecuted the Catholic church, whereas Constantine would favor it. And that is, of course, a crucial difference. On the other hand, even here there are some connections. Under Diocletian, the emperor was a god. The emperor was a distantly glimpsed figure. He was no longer, even in pretense, first citizen, guy just like you and me, hand-shaker, baby-kisser, anything like that. But this was also true with Constantine. Constantine, too, had a ceremonial, distant, and-- because of his association with the Church-- semi-sacred status. He couldn't be worshipped as a god, to be sure, but he was something a bit more than merely a follower of Christianity. Constantine ended the Tetrarchy, but he really set the seal on the division of the Empire east and west, as we've just said, by the establishment of Constantinople. And finally, Constantine was a little more successful economically. Diocletian did not have the means available to Constantine who had a certain amount from the old pagan temple treasures that he was able to confiscate. And also, by virtue of his victory over Licinius, he was able to rule pretty tightly over the Empire. The fourth century often is seen as a period of decline because we're focused-- we-- historians are focused on the collapse of the Roman Empire in the late fifth century. But obviously, people in 337, the year that Constantine died, did not know that in 476 the Western Empire would collapse. They did not know that in 410 Visigoths would invade and plunder the city of Rome, no more than we have the faintest idea of what's going to happen 75 or 100 years from now. From their point of view, the Empire had been restored. The Empire, which had been endangered in the third century by invasions, inflation, armed forces out of control, chaotic imperial succession, was now stable. It was clear who the emperor was. The barbarians had been pushed back behind the frontiers. Trade, culture, civilization seemingly flourished. And if we trust the impressions we have of contemporaries, both formal, written work and informal, things like the slogans that people put in their dining room mosaics, for example, good times had been restored. This seems to be the constant theme. And I emphasize this because, again, it's a lesson in how history cannot be seen from the front backwards. You can't use hindsight to tell what people should have felt. People in the fourth century at the time of Constantine were optimistic. No more so those people who had embraced Christianity as the coming thing, as the religion of not only truth, but of success. What is odd is, of course, that thus far, Christianity would have seemed to be unlikely. Christianity would have seemed to be alien from the Empire. And even if some emperor embraced it for weird reasons of its own, his own, it wouldn't have seemed to have been the most favorable context for the preservation of the Empire. And indeed, of course, the Empire would fail in the west within a century and a half or so of the embrace of Christianity. And it's no surprise, then, that the English historian of the Roman Empire, Edward Gibbon, whose Decline and Fall of the Roman Empire sort of sets the agenda for any course like this one. It's no accident that Gibbon blamed Christianity for the fall of the Empire. But indeed, in the fourth century, it seemed that Christianity was one of the forces that had saved the Empire. And not only that, as we will see as this course unfolds, much of what was preserved from the debacle of 476 and successive problems of the preservation of civilization would be preserved through the action of the Church. Thanks very much. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 09_The_Reign_of_Justinian.txt | PAUL FREEDMAN: We are starting to use primary sources for our understanding of the historical periods under discussion. Primary sources? Writings by people who lived at the time. And as with all such sources, the great advantage is vividness, immediacy-- the people lived through it. And the problem is distance from us and strangeness. Procopius and Gregory of Tours, who we'll be starting out with next week, are very different writers. Procopius much more conscious of style, a layman was somebody operating within the classical tradition. Gregory of Tours, certainly a person for whom style is not paramount. Or at least, it's not the classical notions of rhetoric, smoothness, and vividness that Procopius has. He is a bishop. He's very concerned with supernatural events and the Church. Or let's say, supernatural events controlled by the Church. Procopius, as you've seen, is not very concerned with Christianity, and the supernatural events that concerned him, such as Justinian walking around the palace with no head, are not Christian supernatural. They're from some other older supernatural tradition. But both Gregory of Tours and Procopius require an effort to figure out. Why not just read something by a writer, a historian living now who may be easier to figure out? And who is writing with you and me in mind? Because of the vividness and because of the trickiness of trying to reconstruct not only what happened, which is hard enough, but also what the mood of people was, and what the reaction was. We're talking about Justinian today. So an emperor whose rule occupies most of the sixth century, 527 to 565. So we're concentrating on the sixth century as part of this overall survival and crisis of the Eastern Roman Empire. His reign, or more precisely, the earlier part of his reign until about 540, is the height, apogee, maximum power of this empire which succeeds in shall we say, reconquering or conquering. Taking back or adding the parts of the Western Roman Empire, many parts of the Western Roman Empire that had been lost effectively to the barbarian invasions of the fifth century. If you still can refer to your map, or if your memory of geography is-- OK. The major areas of conquest of Justinian beyond the borders of the old eastern empire are first North Africa-- this is the coast of modern Libya, Tunisia, Algeria, and even Morocco, held by the Vandals and seized by Justinian; parts of Spain, coastal Spain, Mediterranean Spain, held by the Visigoths; and Italy, held by the Ostrogoths. This is the centerpiece of Justinian's reign. And for a time, it looked as if he had, in effect, recreated the empire of Constantine and Diocletian. But as we'll see, this is a triumph with a terrible price. The terrible price being that it weakened Byzantium. Now when we say of figures in the past, or even figures in the recent past, that their policies were a mistake because it turned out that the future enemy would be something other than what they were fighting, we can say that with the advantage of being able to see what was going happen. In other words, there are people who argue that the invasion of Iraq was a folly or that the expenditures on the aggressive foreign policy of the first years of the twenty-first century was foolish, because as it turns out, economic problems, domestic problems, the mortgage bubble, was really the problem that people should have been addressing. Or they should have been addressing the deficit. You can say that. In its own way, it's a fact. But it doesn't necessarily tell you what people at the time should have thought of. Thus, we know from last lecture that, first of all, Justinian should have concentrated on the Persians. The Persians on his eastern frontier who didn't interest him, who he just wanted to sort of pacify in order to go west and make his conquests. The Persians would turn out to be the biggest enemy of the Empire. And so, you know if you were plotting this out as a kind of international political strategy, you could say: "Forget about the Ostrogoths. Forget about the Vandals. Build up that frontier. Invade Persia, keep your army there." And indeed, with a little more hindsight, we can say, "Oh my gosh, in eighty years, the Muslims are going to take the eastern part of your empire." Well, obviously, there's no way he is going to be expected reasonably to know that. Except, if you're looking at a distance, from over 1,000 years, 1,500 years. Then we can say, sure, the eastern frontier turns out to be the point of vulnerability. So a classic kind of historical problem, or early Middle Ages midterm question is: "Justinian, overreacher or reasonable guy?" Or "The conquest of the west: folly or grandeur?" And it's both. It is a classic example of over-extension, over-extension of empires, meaning that empires weaken themselves at some point fatally by simply getting either too big or spending too much money. And the two are linked. If you get too big, you have to spend more money to defend yourself. Not really having the resources to keep what you have. The British Empire, to take a reasonably clear and neutral example, at some point is simply too large for the resources of a weakened Great Britain. And our colleague Paul Kennedy has explored, quite memorably, empires that simply could not maintain their commitments. The Spanish Empire, the British Empire, and, as it turned out after Kennedy wrote his book, the Russian Soviet Empire. This is a pattern in history that repeats itself. The question, however, is, under the circumstances, and assuming the existence of that empire, what are reasonable policies to preserve it or to extend it? We know about Justinian's wars of conquest and of defense-- he did have some wars against the Persians-- his wars of conquest and defense largely, although not exclusively, through Procopius. He is our best source in two works. One, the Secret History, and the other, much more extensive, a series of books called The Wars. And they're divided in Persian wars, African wars, Italian wars. In The Wars, you can see that once the Italian war starts to go badly, Procopius's opinion of Justinian and of the great general Belisarius tend to change from a kind of admiration and go, kill, get'em spirit to uneasiness, to blaming, to a kind of finger pointing. So we're dependent on Procopius. And when you first read The Wars, it seems very, very different from the Secret History. It seems like it's by Thucydides or some other sensible, objective Greek writer. And he indeed is writing in that tradition. Those of you who've read Thucydides will remember he describes, often, folly and very terrible events, but soberly, factually. And in a fashion of Olympian sorrow at the folly of policymakers and generals. And to some extent, Procopius has that tone, which seems to contrast very much with the vehemence of the Secret History, leading some people to assume that he was crazy when he wrote the Secret History. Or off balance, let's say. Or that The Wars represented the real Procopius, and this represented his "evil twin." The term, evil twin, doesn't appear in Gibbon, but it could. It could. What makes it more complicated is a third work of his called Buildings. Buildings is, as the name implies, a book about Justinian's building campaign, which includes, but is no means limited to, the church of Hagia Sofia in modern Istanbul, which is, continues to be, to this day, an extraordinary building of such immensity and such space in interior. A dome that seems unsupported by anything and that seems to cover half the earth when you're inside it. Both splendid and an extraordinary engineering feat. And then Justinian built churches. He built churches that stand in Ravenna with unbelievably beautiful mosaics, Ravenna in Italy. And these are important because Ravenna was outside the zone of territory controlled by the iconoclasts. And, therefore while the iconoclasts tended to take down or whitewash representations of anything divine, their reach did not extend as far as Ravenna. So in a way, the best examples of Byzantine mosaic art of the earliest period-- Not in a way, but absolutely are outside of the eastern Mediterranean, and in Italy. Buildings though, is not just an account of Justinian's architectural essays, but a panegyric, a praise of Justinian. Almost as slavishly adulatory as the Secret History is a condemnation. And as I suggested last time, these actually go together in a society where a tremendous power is concentrated in one person, or one court, or one setting, the reactions of people tend to be adulation which is, to some extent, forced out of them, or at least invited by the ruler. So again, to take an obvious analogy: Stalin, for his seventieth birthday was pleased that the greatest museum of Moscow, all the permanent exhibit was set aside and warehoused, and the whole museum was given over to gifts to Stalin on his seventieth birthday from a grateful people. He didn't have to order it. Somebody came up with the idea and he said, "Oh, don't go to any trouble." They had the thing, this adulatory. This is what later would be called the "cult of personality." And it's just one of hundreds of examples. Naming cities after him, lauding him as the "Great Gardener", "the Friend of Children," "the successor of Lenin", and so forth. The other side of that is a kind of hatred and diatribe, more or less secret. There were lots of jokes about Stalin. You could and people were sent to Siberia or executed for telling these jokes. But they were very good jokes, under the circumstances. This is some of the explanation for how you can get, at the same time, adulation and demonization. The interesting thing, of course, is it's in the same guy, Procopius. And although people at one time thought, "Oh well, he wrote The Buildings earlier and then became disillusioned." He did become disillusioned. Everybody became disillusioned, because after 540, things started to go wrong. There's a huge plague in 542 that kills off a third of the population, for starters. But it looks as if he's writing this stuff more or less at the same time. The Secret History is not finished. That's why it begins so oddly, not with Justinian, but with Belisarius and Belisarius' wife, being kicked around by his wife, and Theodora and you sort of don't know who these people are. And then suddenly we're at Justinian. Well, the order of this thing is not yet set. He probably did not finish it. He did, however, want it to be published after his death. It's called the Secret History or the Anecdota, sort of stories, by later writers. It survives in only one manuscript, as I think I remarked. Nevertheless, because it has a highly rhetorical style, it clearly was to be read by other people. It's not just a set of jottings for his own satisfaction. It is a work that he hoped would be widely published when he was safely dead. And Anecdota literally means, not "stories" as it would now, anecdotes, the false cognate, it means "not to be published". So in the Secret History, Justinian is a monster. Let's set that aside for a moment and talk about what Justinian actually did. Justinian was the power behind the throne of his uncle, Justin the First. So in a way, his rule goes back to the 510s. Justinian's character, as portrayed by Procopius in both The Wars and in the Secret History, is very smart, hard - working-- Procopius says he almost never slept-- devoted to details, capable of immersing himself in many different things: architecture, church ceremonies, theology, and law. He was of humble birth. His uncle, and his family were soldiers. They were from modern oh, Croatia, more or less, the Balkan peninsula, the former Yugoslavia-- Illyrians, as would have been the term used at the time. He grew up speaking some form of Latin, and he is, as I said last time, the last emperor whose native language was Latin as opposed to Greek. He dressed very simply, and he was approachable. He did not have that awe-inspiring splendor of Diocletian or Constantine, for example. He was seldom angry, but he was cold and seems to have had no trace of mercy or kindness. He reminds me of some professors of mine. He was intolerant; he was unforgiving; and he was merciless. He had a grandiose conception of the Empire. And he was willing to tax his subjects heavily and to endanger the security of the eastern frontier in order to expand his territory and his prestige. I think that is a fair judgment to make. He believed that his predecessors had, through neglect, lost what the ancient Romans had conquered. And he believed that you couldn't call it the Roman Empire if all it consisted of were possessions in the eastern Mediterranean. And, as we've said, he did indeed conquer, at great cost, North Africa, parts of Spain and Italy. He had a-- I think it's wrong to use the term totalitarian, but certainly a very strong conception of imperial rulership. He tried to impose doctrines on the Church in order to resolve the age - old Monophysite question. He was no more successful than Constantine or Theodosius by the way, but, for example, just to give you a sense of his methods, he kidnapped the Pope in Rome, tried to browbeat him, and exiled him to the Crimean Peninsula where he died. Theodora. One of the most interesting things about Justinian is that he gave so much power and respect to his consort, Theodora, who was of even more humble birth than he was. Now, I don't think we have to believe Procopius on the details of Theodora's youth. He certainly reserves his most hysterical diatribes for Theodora. I think it's fair to say that Procopius was not a great admirer of competent women. The historian Bury, J. B. Bury, one of the great historians of late Rome and Byzantium, who wrote about 100, 120 years ago, describes her youth as stormy. An adjective that I like, because it could be anything. Her stormy youth. Probably her father was a bear keeper. Somebody who kept bears for the entertainment of people at the circus. An animal trainer. She was the mother of a legitimate [correction: illegitimate] child. She may have had a background of amateur or quasi-professional, semi-pro prostitution. Notice that Procopius condemns her, first for being a prostitute, and then for suppressing prostitution once she became Empress. There's a logic to that. Procopius is not opposed to prostitution. One has the sense that he's, if not a connoisseur, at least a now-and-then partaker. But for prostitutes to be anything other than this firmly subordinated class, that is, for prostitutes to have some sort of voice or opinion, or for people to endeavor to help them, or respect them, is, in his mind, ridiculous and scandalous. Procopius is a conservative. He doesn't like the weakening of the senatorial classes. He represents the land - owning interests. He doesn't like too much imperial power. He's quite happy to respect the emperor, but is angry when the emperor seems to be taxing rich people. He doesn't like upstarts. Upstarts like Justinian. Who is he? A soldier's child. Upstarts like Theodora. Upstarts like Antonia, the wife of Belisarius. Justinian and Theodora ruled as a team. They had very different personalities. A very interesting team. Theodora loved sleep, luxury, was sympathetic to the Monophysites. Justinian was completely the opposite: an insomniac, somebody who dressed in extremely ordinary clothing and firmly anti-Monophysite. They, in fact, supported different factions in the circus. Here is a Giants-Jets marriage. The circus. The circus was a arena attached to the palace, where the emperor would make his appearances at sporting events. Although we've said Justinian was approachable, by that we mean that people in the government or in high positions could see him without too much ceremony. That doesn't mean he's approachable just to anybody. In an absolutist state, there are certain kinds of events at which the ruler has to show himself, or traditionally shows himself. So in the Soviet era, the May Day parades. There's a reviewing stand in Moscow at the tomb of Lenin. And foreign correspondents and intelligence people would try to see who was in and out of power by who appeared with the leader, who wasn't there, where they were standing. The Hippodrome, the horse racing arena in Constantinople, was a bit like this. The Emperor had his own box and the people could make sort of celebratory gestures to him, praise him, and if they were in a rebellious mood, criticize him as well. There were of circus factions, as they're called. That is, people who were cheering for one side or another, the most important of which in Constantinople are the Blues and the Greens. The Greens tended to be somewhat pro-Monophysite. And Theodora was a partisan of the Greens. The Blues, anti-Monophysite, Justinian was a partisan of theirs. In 532, the circus factions revolted. Partly, it's a tax revolt. Partly it's factions fighting. It doesn't do to try to probe what these factions represented too much. After a while, they're simply factions. They're simply people who like to fight. Or who like to root for one side or another. But they are rowdy, and even criminal. They have very outlandish costumes. They expend all their money and all their energy on sporting events and on rowdiness associated with them. This is not completely unfamiliar. The prefect of the city arrested seven people for rioting and condemned them to death. Two of them escaped when the rope broke. It always pays to maintain your-- I mean, this is a tip from a historian-- always pays to maintain your coercive equipment. Once these guys escaped, then they were heroes. And they were shielded from the crowd. They were put in a monastery where they had sanctuary. And conveniently enough, one was a Blue and one was a Green. So the Blues and the Greens united. They ran through the streets demanding pardon for the escapees. And when Justinian refused, a riot took place. The battle cry of these rioters was "Victory." Right? Nika, not to be confused with sporting equipment. Nika - victory. The crowds tried to overthrow Justinian and Theodora. And in the process, they burned down a lot of the city. Justinian is reported by Procopius as being ready to flee. But Theodora stiffened his resolve, basically telling him she preferred to die in the shroud of the imperial robes, rather than flee in disguise, and mobilized the generals, Belisarius and Narses. We've met Belisarius already. And they cracked down on the mob and killed maybe 40,000 of them. How many people attend a Yankee game? About 80,000? So 30,000, 40,000 people, and that ended the riots. Constantinople was partially burned. Justinian loved building. This was a great opportunity. He couldn't have asked for a better moment, in a sense. Of course, it required more taxes, but people now had seen the problems with resisting taxes. And so this is where we start the building of the new Hagia Sofia that we see today. Built in five years. Compare this to grand projects like you know an exit on the Connecticut Turnpike, which take fifteen years or so. The way you build something in five years is by an incredible number of workmen. And lavish expenditure of money. The patriarch's throne in Hagia Sofia was made of silver. It weighed 40,000 pounds. The columns are of porphyry, many of them. It uses a lot of glass in order to emit light. And the light comes from so far away that it forms these wonderful patterns, depending on the time of day. Justinian also rebuilt the Senate, the baths, the imperial palace, and in the Church of Saint Irene, the Church of the Apostles, et cetera, et cetera. He started his wars against Persia before the Nika revolt. And the war with Persia is one episode of a multi-century war. In this case, it's over influence in the Caucasus. But it's really about trying to protect Byzantium from Persian invasion. But as I said, Justinian's interest was not really in Persia. He was interested in peace with Persia and in securing enough of the frontier so that the Persians couldn't launch, at least not easily, a surprise attack. And in 531, the Eastern Roman Empire and Persia signed a perpetual peace. And Justinian then moved his troops to the west, the site of his real ambitions. The Vandal War in North Africa was a triumph. What we're seeing is one of those cases in which a policy seems to succeed miraculously easily. The Vandals fell, it seemed, without a fight. Here, the people who had been the terror of Rome 100 years earlier, who had sacked Rome in 455, who had seized the granary of Rome in 430, fell almost, it seemed, without a fight. True, the native Berber population who were subordinate to the Vandals, desert people, revolted. And they were able to raid the coast and to undermine the position of the Byzantine occupiers. The next stop was Italy in 535. 533 - 534, the conquest of Africa. But Italy would take twenty years, not one. And in the process, Italy itself would be devastated. And with that devastation, a lot of classical culture would be lost. What wasn't destroyed by the fifth century invasions-- and remember we said the Ostrogoths were pretty reasonable occupiers-- would be destroyed by the Romans themselves. I will not tax you with the ins and outs, and ups and downs of this campaign. Suffice it to say that the general, Belisarius, at first was able to triumph in Italy. The Ostrogothic resistance, however, proved to be much stronger than he expected. And Justinian recalled Belisarius. Almost all of Italy was reoccupied by the Ostrogoths and it was only the second general, Narses, who from 552 to 555 is able to take over Italy. 540 is the year that Ravenna falls to the Byzantines, and it seems to be the zenith of Justinian's reign. In that year, the Persians invaded. That perpetual peace had lasted nine years. And the Persian invasion was quite successful. It resulted in the sack of the largest city of the Empire after Constantinople and Alexandria, the city of Antioch in the Eastern Mediterranean. And this was followed then by a plague. The so-called Justinianic Plague, which seems to be related perhaps to the plague of Peracles' Athens or the Athens of the Peloponnesian War, and maybe to the Black Death of 1348, 1349. Hard to say. And in fact, research now being done on the DNA in mass graves from that plague will perhaps tell us what the disease really was. Although so far, apparently, it hasn't. So from 540 to 565, the death of Justinian, his policies are officially successful. 555, the fall of Italy. The plague eventually goes away. The Persians are pushed out of Antioch, at least. But the Empire in the later years of Justinian is clearly staggering under the weight of taxation, economic downturn, declining population, and over-extension. They had conquered Italy, but the Italy they had conquered was ruined. And this empire, stretching now from Sicily to the Persian frontier is clearly too big to hold onto. So this is some of what Procopius' anger is about. But he's bitter and disillusioned. He says, "But I grow dizzy when I write of such suffering. And pass on to future times it's memories." Here, he's speaking about the Persian invasion of Antioch. "For I cannot understand why it is the will of God to exalt the fortunes of a man, or place him and cast them down for no reason that we can see." Now if you contrast him with what you've read in Augustine, in The Confessions, you can see that Augustine has some reasons why this happens. Procopius resists the Christian explanation here. And this is led some observers to think, in general, that he's not really somehow a Christian. He is, but he's writing in a classical tradition. And he is also, remember, an "elitist" a conservative. I use the term elitist in a fairly neutral sense. It's hard to expect someone whose writings come down to us all this length of time to be, somehow, an ordinary person. Yes, he represents a class. But doesn't really like religious controversy. But doesn't really like all of the fussing about the natures or nature of Christ. But there are other things that are not in Procopius that are somewhat surprising. Justinian is best known for architectural monuments like Hagia Sofia; to historians, for what we are essentially talking about today, the Western conquest; and for his legal reforms, the Justinianic Law Code, which is the basis of all European law. European, that is, as opposed to Anglo-American. Anglo-American law is a separate tradition. European law is based ultimately on a reworking of Roman law precedents. And so I want to talk a little bit about his legal accomplishment, which Procopius, a man who would be familiar with law courts, with legal systems, doesn't tell us anything about in his works. Virtually nothing. Justinian essentially codified the Roman law. And this is important, not only because it's the basis of European law, but law is related to political and administrative order. However much we may hate bureaucracy, or denounce administration, that is how governments provide whatever it is they are providing for their citizens. And since the alternative to government is anarchy, and since there are examples before our eyes of anarchic societies, it won't do to underestimate the benefits of law, however cynical we may be about its implementation. Roman law at the time of Justinian was, as law tends to be, learned and unwieldy. If you wanted to know how to resolve a question, you could go through the thousands and thousands of what are called "responsae", or you could look at legislation. Just as in the Anglo-American tradition, and some of you will learn this very soon in law school, you can either look at statutes passed by legislatures or court cases-- precedents. The equivalent of a statute, Connecticut passes a law saying that you can't have a gun in your car. Whereas Texas has laws that say you could have a gun in your car under such and such circumstances, OK? So you have a whole set of statute law, which would be imperial statutes in the Roman Empire, imperial legislation. Or, if the statutes don't cover a particular situation, or you want something that has the particularity-- a tree on my property falls on my neighbor's-- did I mention this already? Yeah, that one-- on my neighbor's garage. Who's to blame? OK, you go and you say, well, this case came up in Cincinnati in 1949, and this is what the judge found. In the absence of computers, the search for this stuff is very hard. In Anglo-American law this is called "precedent." In Roman law, they're called "responsae." And interestingly enough, this term is also singular, plural. It applies to Jewish law. A response is a response. A judge, an expert, a law professor, in effect, is asked his opinion on something. And his response becomes preserved as a kind of precedent. These were voluminous and represented centuries of law. And even more, of course, the responsae conflicted. One judge says, "You have to pay because it was your tree." Another says, "It's an accident, he's responsible for his own remedies." What do you do if you have a conflict of judges? What would you do if you have two kinds of contradictory responses? You've got to decide who is more authoritative? Which one is better? So the work of Justinian's compilers was to sort out legislation, statutes, and the responses, and also to decide among contradictory ones. What is in this law? Well, what's in any law? We think of law as having mostly to do with criminals and stuff like that. But criminal law is actually very simple. It's like the Burgundian code. If you murder someone, this is what's going to happen to you. There may be different kinds of killing. If you murder them with intent and premeditation, that's worse than if you murder them in a fight and spontaneously. Manslaughter is different from murder. Manslaughter is where you didn't intend to kill the person, but you did. You punched him, and you didn't know they had a weak heart and he died. That's manslaughter. You punched them. You intended to hurt him, but you didn't intend to kill him, but he died. Vehicular manslaughter. What's the difference between negligence-- you should have seen something and you didn't-- versus criminal intent? You did it deliberately. But it's very simple, the criminal law. There aren't a whole lot of gray areas, and you can get through the criminal code pretty quickly. But what about contracts? What about property? This is endless. This is endless. So you know in law school, criminal law will be the cream, or the tip of the proverbial iceberg, or some little side issue. Most of your time is going to be spent on-- those of you who go for this option-- on property and contracts. And that's what the Justinian law is mostly: property and contracts, legal arrangements for buying and selling, inheriting, partnerships, guardianships, security, surety, obligations. This is a very advanced science in Roman law. As advanced as it is anywhere, at any time. This is very different from "You cut off one finger; you pay five solidi", which we were looking at last week. The work that ensued, the so-called Justinianic code, or the Corpus Iuris Civilis, the body of civil law, was drawn up in five years. Here, again, is an example of unbelievable rapidity, compared with the length of time it takes now, merely to reform the Connecticut tax code-- for that matter. It was undertaken by a commission. Four books were issued. The first is a collection of statutes, and it's called the Codex. Collected laws of the Senate and imperial laws of previous centuries. The largest book is the Digest, or in Latin, Digesta. The Digest is the weeded-out responsae, organized by subject. So this would be where you would go to try and figure out what happens if a river changes its course a little, and your land seems to be now taken over by your neighbor. Is the river the border, or is an artificial line the border? The third book is a kind of textbook, or a survey of the whole law and what it's supposed to mean called the Institutes. And the fourth is called the Novella, or new laws, because obviously, new laws would have to be made. The Codex, the Digest, the Institutes are in Latin, because Latin was the language of the Roman Empire. But the Novella are in Greek, because Greek was the language of the Empire now. "Now" being 534 when this work was finished. The Justinianic Code is more, however, than a rearrangement of old laws. It displayed a consistent philosophy of government where law is more than precedent, is an active force in society. The Emperor is seen as the servant of the law, the implementer of the law, but he's also the master of the law. He is an absolute power. He is the embodiment of the law. This is a well-run, immense, burdensome empire. Procopius gives us, unreliable though he may be as to Justinian being a demon, et cetera, Procopius gives us a vivid picture of a highly-governed, even efficiently-governed, but oppressively-governed and very ambitious society. Now for next week, and a little bit after, remember we have no class on Wednesday. We have class on Monday. We are going to read from Gregory of Tours about Clovis and the Franks. And it will seem more violent and more primitive than what we've been reading. But violence and primitiveness, unfortunately, are part of history and government at almost any time. And so, enjoy the intrigue. I'm not going to test you on the names. You'll see there are lots of great cat names of the Frankish barbarians. But pay attention to the figure of Clovis, and to the attitude of Gregory. Because, as with Procopius, we've got an interesting, if not completely reliable source. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 14_Mohammed_and_the_Arab_Conquests.txt | PAUL FREEDMAN: Now we're going to talk about Islam. And here we enter into a segment that, in a sense, is the most relevant, if relevance is a criterion, sort of is, to today's world. Because these controversies that play out today Shiism versus Sunni, for example, the nature of Islam, its appeal as a religion on a world scale are obviously established in the period that we're talking about. They were established slowly. One of the things that you'll have noticed from the assignment is that the author, Berkey, emphasizes very much that Islam is slow in formation, that it's not fully grown as this militant movement with a set of rules in 632, the year that Mohammed dies. And because of that, then, it's not to be understood as some militant, conversion-oriented, jihado-centric religion from the start. I said at one point when we were summarizing the end of the Roman Empire that there were three heirs to the Roman Empire. One was the Church-- ironic because, of course, the Church had grown up persecuted by the Empire. The other was the Byzantine Empire, the Eastern Roman Empire. Remember that it called itself simply "the Roman Empire." So that's the most clear claim being staked to succession to the Roman Empire. And that the third was Islam. And a couple of you said, well Islam, that's the most surprising of all, actually. And I didn't really elaborate at that time. And that's not the center of what we're going to be talking about. Because time moves on. We're in the seventh century, the collapse of the Roman Empire in the west is established two and a half, two centuries earlier. But Islam, although developed in Arabia, outside of the Roman empire, and although very strong in places like Persia or the western part of India, would in many respects take up the inheritance of the Roman and Byzantine Empires. The conquests on the Mediterranean, in the east, and in the south; its architectural and artistic style; its administrative structure; and not least, the translation and elaborations of Greek science, medicine, and other academic forms, including, for example, the translation of Aristotle and the influence of Aristotle would be signs of, evidence of the significance of Islam within-- it's pointless to debate whether it's the Western tradition or what the Western tradition means-- but within the classical legacy, within what it means to say that Rome as a polity ceased to exist in the west and yet the classical world has continued to influence society and ideas to this day. So in talking about the history of Islam, one is inevitably going to be emphasizing its revolutionary nature. And so I'm going to go against the reading in some sense. Or the reading is intended to go against the conventional way of presenting the early history of Islam. And that can be sometimes annoying. The writer is cautioning you against views you never had. Or the writer keeps on saying, "Well, we should not think that this"-- and I hadn't thought that. Just tell me what you think happened in early Islam. My apologies for that. Writers are always writing against other writers. Scholars are always writing against a prevailing interpretation. Berkey is a continuist, and that is the scholarly consensus now, arguing against this notion of Islam bursting forth like some kind of pent-up explosion in Arabia. But Arabia is off the map. Islam as a movement, and certainly the Arab conquests, are unpredictable events. They may be understandable later in terms of developments in the Near East, both religious and cultural, but up until the seventh century Arabia was on the periphery of the two great controlling empires to its north, namely Persia and Rome. And we are taking Byzantium as the heir of Rome in this sense. The problem with Arabia is that it is really dry. And before the discovery of oil, or more precisely, before the discovery that oil was useful, important, and valuable, it was a strikingly impoverished land in terms of natural resources. "A terrible land," as Isaiah the prophet says. Isaiah 21:1; "the burden of the desert of the sea. As whirlwinds in the south pass through it, so it cometh from the desert, from a terrible land." Now, Isaiah actually grew up in what most of us would consider to be pretty dry circumstances. The eastern Mediterranean, modern-day Israel, Lebanon, Syria are hot and dry compared to bounteous climes like New Haven. So the Eastern Mediterranean, however, is a land of fertile oases, river valleys like the famed Tigris/ Euphrates/ Fertile Crescent area, coastal cities with commerce, whereas Arabia is essentially a vast expanse of barely habitable terrain. It is isolated by the sea on three sides. It has very few natural harbors. There are no lakes, no forest, no grasslands, even, and no rivers that run year-round. The only intrinsically favorable part of it is the southwest corner, modern Yemen. This was in the ancient world, or at least to the Roman geographers, "Arabia Felix," Happy Arabia. Nice Arabia. And indeed, there were two kingdoms that emerged here around 1000 BC. Harbors, oases, and these two kingdoms controlled the spice and incense trade from India and from the Horn of Africa. These are two extremely valuable kinds of products for religious, gastronomic, and medicinal purposes. And most of the spices come from India. And most of the incense comes from modern Somalia, Ethiopia, places along the Red Sea. By the time of Mohammed, however, southern Arabia's best days had passed. And Mohammed is, of course, from central, and in the sense of its orientation, really, northern Arabia. The worst desert in Arabia is the south. So just when you get out of Yemen going northeast you come to an area that still bears the not really encouraging name, the Empty Quarter. This is really serious, no-oases desert. Further north, however, in places like Mecca or Medina, there is enough water for settlement, though not enough for agriculture. A key event in the history of Arabia is the domestication of the camel, which can be situated around 1000 BC. Arabia, outside of the Yemenite kingdoms, was mostly nomadic, though there were trading cities of which the most famous are Mecca and Medina. These cities were rather cosmopolitan. They had Arabs and other peoples. They had Christians, some of whom were Arabs, some of whom were not, and Jews, many of whom were also Arabs. They controlled overland trade, again, from further east, bearing exotic products like spices from India, organizing caravans. At times they would form kingdoms, at times there would be Arab kingdoms in the north, but generally these are feuding societies organized along tribal lines. Looked at from the Roman or Persian empire, Arabia was a little bit like the forests of Germany: a hostile terrain-- from the Roman point of view, Persian point of view-- inhabited by useful but presumably barbarian and presumably not very interesting people. This is the point of view of the Bible as well. They are primitive people bearing interesting products, inhabiting a land that's not worth having. Not worth invading, not worth owning, not worth dealing with. The first mention of Arabs seems to be 854 BC when, according to a Syrian inscription, a certain Gindibu, the Arab, contributed 1,000 camels to forces revolting against the Assyrians. The Arabs, henceforth, after 854 mentioned often in Babylonian and Persian texts. They're always frontiersmen or people inhabiting a land beyond a frontier. Questions so far? OK. Not all Arabs are nomads, but the Bedouin, Bedouin in this sense meaning Arab nomad, is sort of the Ur Arab, the original Arab, the defining archetype and the original colloquial meaning of the word Arab. In a way the Bedouins are a little like the Germanic tribes, an analogy I'll mention but I don't want to push too hard. They form extended kinship units, that is to say they know who their second cousins are and care about them. And these extended kinship units form a kind of tribal structure. This notion of tribe remains important to this day. You'll have seen, in accounts of post-Qaddafi Libya, for example, the cliche and probably accurate, is there is no tradition of government. Qaddafi didn't govern, he just tyrannized. And there are no civil institutions. It is divided by tribal loyalties. What does tribe mean in that? I actually don't know in terms of Libya. It is a way of saying the people do not have loyalty to the state, but to some extended kinship group, or what's sometimes called "fictive kinship group." I'm not really related to you, but we have either the same name or we're from the same place or we consider each other kin and therefore have a certain protective sense about each other. This is very useful if you don't have a government. We've already spoken about this in terms of our question of what held Merovingian society together. Your second cousin becomes important to you if there's no police force, if there's no way of making sure that someone is not going to kill you just for fun or because they got angry at you. Under the circumstances we live in, we don't care about our second cousins. We don't know who they are. We don't expect anything from them. In a society in which family is not just a sentimental attachment, but it actually is what is protecting your life, kinship is very important. Bedouins as well as Germans, then. The problem with kinship is that while it's a very solid attachment, it's also an irritant. And here we're not talking about arguments over whose Christmas is better or who gets the Venetian glass vase after Mom and Dad's death, we're talking about terrible arguments, feuds that are within a kinship group. So you have feuds within the kin groups or tribes, and feuds between tribes, accentuated in this case often by water. Water, a scarce resource, obviously, and one that people fight about a lot in terms of territorial feuds. So the Bedouins tend to have more feuds than the Germans. There's no Bedouin equivalent of wergild. Remember, the wergild is the price that you have to pay to make someone have peace with you, even if you've killed their relatives. It's the worth of a person. It's compensation. And then other people can be assessed on the basis of some tariff. So women may be 2/3 of a man, a pregnant woman may be one and a half times a man, and so forth. We saw this in the Burgundian Code. Key to the Burgundian Code is this notion of compensation, that money related to the nature of the loss. One finger, two fingers, right hand, left hand, is compensation. Bedouin don't have that idea of compensation, of tariffs, of wergild. In both the Bedouin and the German societies, the ruler has a limited amount of power. These are, I wouldn't want to say democratic societies in terms of some theory of representation, but they're not societies in which one person's will is obeyed unquestionably. They are consultative. They are more like gangs in that sense. There's a leader, but his control is conditional on the loyalty of his most powerful subordinates. And his most powerful subordinates are quite capable of overthrowing him. The Bedouin sheik is a little different from the Germanic king. And by Germanic king I mean not the kings that we've seen in the settled post-Roman, Merovingian empire, but the kings as described by Tacitus with possible greater or lesser accuracy. In the Germanic tradition, the king is a war leader. In the Bedouin tradition, the sheik is an arbiter, a settler of disputes. Both societies exalted custom, and both had an exacting standard of masculinity. The Germans practiced agriculture and herded animals. The Bedouins don't have agriculture, and they supplement their herding of animals by raids on wealthier society. These raids, called "razzia," are important because the Islamic conquests that we're going to be talking about on Wednesday may be said to begin as raids. They begin as raids, and then they discover that there's almost nobody there. That the Byzantine army and the Persian army are crippled by fighting against each other. So what begins by raids becomes conquests. So we come to Mohammed. At first glance it would seem that Mohammed is a religious leader whose career takes place in what a French scholar of religion called "the full light of history." 620s AD may not seem like the full light of history, but Mohammed as a historical figure at least seems to emerge more clearly than Abraham or Moses or Jesus. But, as you have read, Mohammed's biography is hopelessly entwined with legend. What we know about Mohammed is what later Islamic and Arab commentators wanted to have happened to Mohammed. There are several sources for the life of Mohammed, and for thus the early years of Islam. There are formal biographies, called sira, S-I-R-A. The problem with these is they were written long after Mohammed's death-- a hundred years, at least. There are collections of oral tradition, called hadith, H-A-D-I-T-H, to which, similarly, there are sayings, proverbs. These are also questionable, because although they were put together within fifty years of Mohammed's death, they're very heavily influenced by the first civil war of the 650s, which we will be talking about the day after tomorrow. And then there's the Koran, which is supposed to represent the words of Mohammed as composed by divine inspiration. The Koran itself is a text that scholars outside of the Islamic tradition have questioned in terms of when it was put together, how much by Mohammed, how soon after Mohammed. The problem with all these sources is not that they are unreliable in the sense of fabrication, but that they tend to shape events in light of what the writers already know happens and in light of what they think should happen. So that, as an example, we'll be talking about this in a moment, but you're all aware that in 622 Mohammed moves suddenly from Mecca to Medina. He maybe can be said to flee Mecca. This is called the Hegira, or Hijra in the Berkey book. H-E-G-I-R-A or H-I-J-R-A, depending on just how faithful you think you're being to the Arab original. The Hegira is a key event in Islamic history. It is the point from which Islamic dating is done. That is to say the Islamic calendar starts with the Hegira. So this year is the year of the Hegira such and such. I can't do 2011 minus 622 immediately, but that's the Islamic year. According to the traditional historical record, the Meccans tried to assassinate Mohammed, and he escaped, narrowly, this attempt. There's no real evidence of this degree of hostility on the part of people in Mecca. There's no evidence of an assassination attempt, or at least independent evidence. And the assassination attempt seems to be something that is important for the story, for the way that the story is presented later, to dramatize something that may not have been at the time as dramatic as it seemed. It may have been that the Meccans simply didn't listen to Mohammed and then he accepted an invitation to Medina. It may be that they were trying to sort of shut his movement down. But that they resorted to assassination does not seem to be very likely. OK, so having given you all of these fatiguing caveats about what we do and do not know, let's say Mohammed was born between 570 and 580. He was born into a reasonably prominent but not really very affluent family of Mecca. He may have been a merchant. It is usually assumed he was, and this is partly because the Koran has a lot of mercantile similes. In order to elucidate various points, comparison is made with trade, but there's no real evidence. We don't really know what he did for a living. We know that he married well. His first wife, Khadija, K-H-A-D-I-J-A, was from a wealthy family, a higher class family than Mohammed's own. And we also know that Mohammed got his start as a religious thinker, as a prophet, at the age of forty, an encouragement to those of us who are slow to get our careers off the ground. The discouraging part is that his career only lasts a relatively brief time. He dies ten years after the Hegira, but he does accomplish an awful lot. What was his religious experience? What was the revelation vouchsafed to him that he preached to the citizens of Mecca beginning around 615 to 620? It is certainly a message of monotheism against what was considered to be a prevailing paganism, or polytheism on the part of the merchants and tribesmen of Arabia. But as we've said, Arabia had lots of Jews and Christians as well. And it's a little tricky to tell how much Mohammed would have known about Judaism and Christianity. But it looks as if he did. And indeed, it looks as if his preaching begins as a kind of biblical monotheism for the Arabs. It is a message to the Arabs congruent with the message of Judaism and Christianity, the message of Judaism and Christianity being understood as a statement of the unity of God and a progressive interpretation of God's message by a series of prophets, a series of prophets beginning with Abraham, including Moses, Isaiah, Jesus, according to Islam, and Mohammed. Mohammed is then in the line of prophets. The degree to which this means that Islam takes on its own identity is hard to say. And the tendency of scholars outside of the Islamic tradition, that is people like Berkey, is to say it takes quite a while. Takes quite a while for there to be the confidence that Islam is a religion that is different from Judaism and Christianity, while it is clear from the start that the people who are embracing it are different, even though there are Arab and Jewish Christians, and we'll see what that means in a moment. Mohammed's first converts are his family circle and a group of key friends. And they're all important historically. His wife, his cousin Ali, who would marry his daughter Fatima, a merchant named Abu Bakr, a powerful member of one of the leading clans or family groups of Mecca, Uthman, sometimes spelled-- I think in the Berkey book spelled with a U, so we'll use the U. Uthman, right, Ali, I've mentioned, Abu Bakr. Uthman is a member of the Umayyad family. And Omar. These are sort of considered to be the inner circle of very, very early converts. As I said, according to tradition the ruling circles of Mecca became fearful as what had been a fringe movement started to gain more converts. The people who ruled Mecca feared that their control was slipping from their grasp, and they feared that this new movement was popular with a lower class. The leading clan of Mecca were the Quraish. These are the people who are most powerful in Mecca and begin as the enemies of Mohammed and are responsible for driving him out, if indeed he was driven out. In 622 the city of Medina, another merchant center, invited Mohammed to come as a kind of arbiter or ruler who was above factions and who could then settle their internecine disputes. This is not an uncommon pattern. You'll see it in late medieval and Renaissance Italy, in fact. The podesta in Italian cities is an outsider who is empowered with very extensive police powers to quell feuds. In Romeo and Juliet, for example, there's a podesta, but he's not able to solve the feud. But that's the kind of scene that we can imagine as the western equivalent of Medina. Muhammad is invited as a wise man, as an arbiter, as a reconciler to the city of Medina, an offer that he accepts. And it is in Medina that we really start to see emerge what can be called an Islamic identity. I tend to think a little bit earlier than Berkey, because here Islam starts to differentiate itself from Judaism and Christianity. And indeed, the period at Medina culminates with the expulsion of Jewish groups who refuse to accept Mohammed. What he does at Medina also is to preach that the religious loyalty is more important than loyalty to the tribal group. The religious loyalty therefore is not simply a accompaniment to your already existing identity; it is the most important aspect of your identity. It's at Medina that Mecca replaces Jerusalem as the point of orientation for prayer. It's at Medina that Mohammed stopped celebrating Yom Kippur and institutes a month-long fast during the daylight hours, Ramadan. At Medina, Friday becomes the Sabbath, not Saturday or Sunday. And the Jews are expelled from Medina. And by Jews I don't mean some kind of foreign community of non-Arabs who happen to be living in Medina, but rather Jews, some of whom came from elsewhere, many of whom were Arabs. And it's also here that Mohammed perfects this notion as against the Christians and Jews who have scorned him that he is the Seal of the Prophets, the last of the prophets. From Abraham to Jesus, now to Mohammad. This is the last prophecy. God's message has not been packaged into one deal, one file too large for one email attachment. And it's come in a bunch of them, and this is it. Here are all the cute cat pictures and all of the-- whole file, which you now can download in segments. But it's over. This is it. The Jews and Christians are wrong to think that either it ended with Elijah or that the be all and end all of all prophecy was Jesus. No. No. This is the truth. But keep in mind that Islam did, and at least is supposed to, respect Judaism and Christianity as not merely precursors, but as part of the same tradition. "People of the Book" is the term often used in Islam to describe Judaism and Christianity. They share the same, if not scripture exactly, but the same kind of historical religious orientation. The criticism that Islam can make against Christianity is that it tends to be polytheistic. And as you know, Islam expands great care to make sure that the human form does not appear in art. And in some forms of Islamic art that not even animals appear. Depiction of the human figure is proscribed. Great care is made in differentiating Mohammed from what Muslims see as the exaggerated stature of Jesus. Mohammed is a prophet. He is a messenger of Allah, not to be identified with Allah himself. There is no depiction of Mohammed. People do not have pictures of him in their houses. There are no statues to him. Mohammed's greatest challenge was to overcome these tribal loyalties in favor of the umma, U-M-M-A, the community of the faithful. He allowed property and marriage to be decided still by tribal tradition, but prohibited feuds and required that disputes be arbitrated in religious courts. This is important. Because Mohammed, like the rabbis of the Diaspora in Judaism, is both a religious leader and a judge. He is a community leader and a spiritual leader. And indeed, these two things are not really distinguished. I emphasize this because it's really different from Christianity. In Christianity, there's a church and there's a legal, secular state. Sometimes, and of course the Middle Ages to some extent defined by this, the Church will have what look like secular powers. We've talked about this with regard to the bishops and Gregory of Tours. At times the papacy, later, would claim all sorts of political powers. But conceptually in Christianity, because Christianity was an illegal religious brotherhood within the Roman Empire for over two centuries, church and state are different. In Islam, one can't really talk about church or clergy. There are religious leaders who have political authority, but their authority is what we would call that of a judge and that of a religious leader at the same time. The political order and the religious community are the same. That's why when we talk about the Arab conquest or the Islamic conquest, we're talking about something in which the new territories are taken over by a state that's not a theocracy in the sense of the church's ruling the state, but something in which the church and state are not to be distinguished. We'll talk about this more when we come to the conquests. Within two years of the Hegira, 622, so by 624 Mohammed was planning to take over Mecca, to re-enter in triumph the city that he had fled, if not under cover of darkness at least under murky circumstances. A victory in battle in 624 gave Mohammed the confidence to expel Jews and Christians from Medina and to take on this title of Seal of the Prophets. And by 627 Medina gained the upper hand, and in 630 Mecca fell to Mohammed and his forces. And all of the tribes of Mecca and of the surrounding areas submitted to Mohammed. They recognized him as a political as well as religious leader. Again, the two things not easily to be distinguished. And then Mohammed died. In 632 he died, and what is remarkable is that the momentum he established was able to survive his demise. Because most of the tribes probably thought that their loyalty was to him as a prophet and a person, and not to some sort of institution that would survive his death. And indeed, his death would usher in a period of incredibly rapid expansion. Within a few years of Mohammed's death, Damascus, the great city of the eastern Mediterranean, Byzantine Damascus, would fall to the Arabs, the first of many such conquests that we will be marching through on Wednesday. But this question of religious loyalty would be exacerbated by splits within Islam, which we'll also be describing. The tenets of Islam, just to close by way of our last remarks for the day Islam means "surrender." And to surrender oneself to the power of Allah is the beginning of wisdom, beginning of faith. Acknowledgement of Allah is acknowledgement a strict monotheism, acknowledgement of Mohammed as the messenger prophet of Allah, and as the last prophet. The so-called "five pillars of Islam" are duties incumbent on the believer. And these are the confession of faith that I just mentioned, daily prayers, five times a day, the giving of alms, the observance of Ramadan, and the performance of the pilgrimage to Mecca if that's possible. What's not there is interesting, too. What's not intrinsic to Islam is a strong sense of sin. The Arabs do not like the Confessions of Saint Augustine, are not interested in this particular form of spiritual investigation. The believer can pray directly to Allah. There is really no Islamic clergy in the sense of presiding over sacraments or channels of grace. The mosque is a gathering place. it is not a place that has some kind of powerful holy objects in it, in the sense that a church in this era would have relics or Eucharistic hosts and other very powerful, sacred things. Islam is a religion of conduct and law, not of mortification and purgation. It emphasizes upright behavior: no drinking, no gambling, certain dietary restrictions. Not the renunciation of the world-- it does not say, "Sell everything you have and give it to the poor." It says, "Give alms." It is a moderate religion, actually. It is a "do-able" religion. These obligations may be somewhat inconvenient at times, Ramadan for example, but there is nothing in here to the degree that the New Testament, for example, prescribes behavior that most people are not going to follow. Or that traditional Judaism, possible but certainly onerous. Lots of obligations. The other difference, more important, with Judaism is that Islam would be very quickly universal. In other words, it would encourage conversion, although not, as we will see, with great enthusiasm. It is completely erroneous to think that these armies that burst forth from Arabia after the death of Mohammed were intent of getting everybody to follow Mohammed. They were intent on conquest, all right. But we'll see that their goals were a little more complicated than that of orienting everybody towards this new faith, whatever we're going to call it. And we'll see more about that the day after tomorrow. Thanks. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 05_St_Augustines_Confessions.txt | PAUL FREEDMAN: Alright, so you may be asking yourself, "Why are we reading the Confessions?" I think I gave a preliminary answer before, but since it seems to be perhaps more appropriate for religious studies or philosophy, let me remind you why we're struggling through this. First, the impact of Christianity on the Roman Empire-- that is to say, the social and intellectual setting of the rise of Christianity in the late fourth, early fifth centuries. The second is to understand some of the Christian moral and doctrinal problems and their implications. Once again, we're not exactly interested in these for reasons of theology or morality, but we need to get into the minds of people at the time in order to understand what bothered them, what controversies they were involved in, and how those controversies indeed divided the Roman Empire and the successors to the Roman Empire. Some of those problems-- well, under that second heading of Christian moral and doctrinal problems, let me just mention three, which by no means exhausts them, but are three that we can sort of, if not identify with, I think see their importance. One, the problem of evil. Second, the relation between body and soul. And three, the Christian understanding of sin and redemption. Now, it turns out these are all aspects of the same problem, and they are dealt with in Augustine's works most thoroughly, more thoroughly than any other thinker of the ancient world. The third reason we're looking at this is the interaction between Christianity and classical culture and religion. Roman life and politics, Augustine's career and his giving up his career, what that means, other ideas within the Roman Empire, such as Manicheaism, Platonism. And then finally, this is a document of philosophical and psychological investigation. And while that is not our primary purpose here, you should not get out of a liberal arts college program without reading this and pondering it a little. This can be summarized in terms of the importance of the humanities, even, or of philosophical investigation, as opposed to mere investigation of natural phenomena, in words that Augustine uses in Book X, which we have not read. After Book IX, Book X is a turning. It discusses time and the meaning of time. Books XI, XII, and XIII are a commentary on Genesis. Worth reading, if you like, and interesting to think about how they do or do not mesh with the more confessional parts of the Confession. But in Book X, he says, "Men go out and gaze in astonishment at high mountains, the huge waves of the sea, the broad reaches of rivers, the ocean that encircles the world, or the stars in their courses, but they pay no attention to themselves." They are busy looking at external phenomena and not examining their own heart. And if the Confessions is anything, it is certainly an examination of the author's heart. But it's not an examination of his heart in a purely emotional sense, in the way we're familiar with in so-called confessional literature. I had a tough upbringing. This happened to me. That happened to me. I struggled with addiction. I beat my kids. Now I'm a great person. Whatever. This is an intellectual investigation as well as an emotional investigation. And indeed, Augustine doesn't see these as separate, or to the degree that he does, it's in a more complicated way than just saying intellectual versus non-intellectual. He is an intellectual, obviously. And he awakened to being an intellectual, an experience that many of you may have had. Remember, he reads this dialogue by Cicero, now lost, called the Hortensius. And this convinces him that the life of the mind is the most important thing to pursue. And I wouldn't say that we all have had this experience, but maybe you-- what was the point at which you discovered that you weren't like other people, that they lived more for immediate sensations, or pleasure, or what Augustine would call debauchery, and you wanted to read or think about stuff or do lab experiments? I think the essence of Yale, if I understand it correctly, is I don't have to choose between fun and the intellectual life. So it's not actually perhaps relevant to your lives now, but particularly if you went to a public high school, grew up in a non-intellectual environment. Those of you whose parents are professors and went to some school where everybody was reading Latin at the age of six, I'm not talking to you. But I'm talking to the vast majority who woke up one day and realized, either with pride or with dismay, "I'm different from other people." Ideas have meaning to me. I'm going to suffer in life for that, though there're going to be some rewards. And I leave you to discern what the rewards are and to mull over what the suffering has been or maybe will continue to be. I hope not, and I suspect you'll have an easier time of it. But this book is about a search for truth and a search that takes a number of wrong turns, at least from Augustine's opinion looking back on the situation when he wrote this in the 390s. It's a confession of sin. It's also, as I said before, a confession of praise for the God whose love directed him back to the right path. It is personal, but exemplary. It is about spiritual yearning, but it is also about intellectual yearning for truth. It's a book about the education of a young man and the adventures of this young man and what he learns from them. Now, in the first place, he is both an intellectual and a passionate person. He is someone who is unusually frank about his desires. But notice that he's not just opposed to desire. He is not someone who believes that desire, love are to be simply repressed or ignored. Love is a psychological need. And he has a very discerning and interesting passage in describing his teenage years and his lusts when he frequented the brothels of Carthage. In Book III, at the beginning, he says, "I was in love with the idea of love." So he was not only in love, but he was in love with the idea of being an emotional being, of love that is both sexual and spiritual, in which these two things are not well marked off from each other. He is also a believer in friendship. And it's funny, because in our own culture, I think friendship has changed. When I started teaching, people had trouble dealing with the affection that he has for his friends, like Alypius and Nebridius, or the mysterious unnamed friend who dies after being baptized. Augustine is always surrounded by friends. Even in the most intimate moments, when he's undergoing this conversion, there're all sorts of people right around him. And as I said before, this seemed to be-- the explanation was, well, he must've been homosexual, or he must have these desires, or maybe it's part of Roman culture of friendship. But in recent decades or years, where we have a culture of friendship, where your friends are extremely important-- admittedly, if you have 900 of them, it's a little bit, perhaps, weakened-- but I think we can understand some of this better than might have been the case a little while ago. This friend who dies after being baptized, here is an example of another form of seriousness. They go out and have fun together. The friend becomes ill. The friend is suddenly very serious. The friend gets baptized, because to be baptized, as with Constantine, means that you are committing yourself to a much more stringent and moral life than before. And then he dies. And this certainly disturbs Augustine. What is life all about? Any of you who've had the experience of contemporaries of yours who have died will understand this, I think. Augustine's also ambitious. He is a successful person. Even though he's from a modest family-- his father, a pagan or non-Christian, middle class official of North Africa, his mother, a Christian-- He's clearly marked for success because of his unusual gifts, his unusual gifts being intellectual, ability to write, ability to argue. He's marked out as extremely smart. And at that time, success for such a person, the course of success was through government service-- this is the era of post-Diocletian, post-Constantine-- and related particularly to a combination of rhetoric and law. This is not all that different from societies familiar to us. That is to say, the training in law gives one access to a number of different kinds of political offices. But rhetoric is perhaps a little stranger to us. Rhetoric in this context means the art of persuasion. So it's very closely related to law and legal pleading. It is the art of writing well, of writing elegantly, and it is very, very highly valued in the Roman Empire. His mother, Monica, is extremely pious. In fact, the first section, class rather that I taught-- no, I guess I was a section leader-- when I was a graduate student, the first section I had, my students were arguing with me about Augustine's patronizing attitude towards his mother. And I said, well, no, he's not patronizing. He's smarter than his mother. His mother's just an ordinary person. After all, Augustine becomes a saint, and his mother doesn't. Some guy from Santa Monica Catholic School said, who do you think Santa Monica is? This is Augustine's mother. So anyway, I've learned. Augustine's mother is a saint. She is a more steadfast kind of person than Augustine. She's not someone who stays up all night wrestling with the problem of evil. Nevertheless, she does not want him to be baptized. She wants him to be successful. Like most mothers, she wants her child to be a good person. But even more than that, she wants her child to be a success. And that means delaying baptism, because if he's going to be a success, he's going to have to be involved in the world of high government, and that may mean-- well, that definitely means involvement in sinfulness, involvement in the shedding of blood, involvement in legal wrangling and stuff like that. And so he is encouraged to lead a normal life, "normal" meaning, at least in his own retrospective view, sinful life. This is what Augustine is giving up in his conversion. He is giving up a career. He is giving up a social expectation of success or social definition of success. He is giving up the pleasures attendant on that career, which range from parties to honors to sexual conquests and the whole life of a well-established member of the Roman elite. What is bothering Augustine? What bothers him is, in part, the problem of evil, which we've alluded to already. Why does a good and omnipotent god allow evil to flourish? A related problem is that compared to the works of the Greek writers and philosophers, the Bible seems awfully crude to him, rhetorically, in terms of style, and conceptually, in terms of its ideas. The Old Testament god-- and we're probably at various levels of familiarity with the Old Testament-- but the Old Testament god is temperamental, I think it's fair to say. Here's a guy who decides to destroy-- a guy-- a deity who decides to destroy the world by flood, destroys the cities of the plain, kills one of the people bearing the tabernacle back to Jerusalem because he stumbles. What kind of god is this? This bothers Augustine. And his anthropomorphism bothers Augustine. In the Old Testament, in the Hebrew Bible, a god speaks with people. Adam hears him walking in the garden. How can that be? How can this human-seeming god be the real God? So these two anxieties put Augustine in the camp of the Manicheans. Remember, the Manicheans believe that the solution to the problem of evil is that God is not omnipotent. God is trying, but there's another evil god who is opposing Him. And that evil god is the god of the flesh and the god of the Old Testament, Jehovah, the creator god, the god of matter and flesh. We are souls imprisoned in flesh. Our true home is the spiritual, and we have to renounce everything that has to do with the flesh in order to go there. So Manicheanism would seem to be extremely ascetic. You should have absolutely nothing to do with the world. But as is always the case with the statement, "everything that's material is evil, but we are material," Manicheanism also offers or affords you an opportunity to be completely involved in the world, totally involved in the world, because there's nothing you can do about it. All you can do is say, the flesh is evil, I'm in the flesh, I'm just going to have to deal with it until I am liberated into the spirit. So Manicheanism is not necessarily world-renouncing, but they do identify the source of evil with the body. The body is wicked. The immaterial soul is good. This is not Christianity, as Augustine discovers or elaborates. Even though we may think of Christianity as exalting the soul over the body, nevertheless, it also exalts the body. Christian doctrine is that the bodies of human beings will be resurrected, not just the souls. There will be bodies after the Last Judgment in heaven and in hell. God created the world, and it was good. What then explains the presence of evil? Augustine at this stage turns to Platonism for an understanding of the nature of evil. Evil is not a thing in itself. It is rather the absence of good. Now, if anyone has ever said that to you, you will have found that unconvincing, at least in its first iteration. Because we're not just talking about the absence of good, as in, this bowl of chili is not very good. It's not very flavorful. It's OK, but it's lacking in something. But that's not evil. Evil is not like, not particularly good. Evil is much more vivid, gratuitous, cruel, all-encompassing. The Platonists don't deny that. What they mean by saying it's the privation of good is that it is nothingness. Evil is, in fact, the absence of being and meaning. The reason it produces such spectacular effects as war, oppression, crime, is that people turn away from the good, or they turn away from what is truly good to prefer lower goods. They turn away from the things of the spirit to the things of the flesh. They prefer their own lusts and desires, their own ambitions and greed, to the common good or to the immaterial and spiritual good. And it's this turning away from the Sun, this turning away from good, that seems to be a human problem. Human beings, generally speaking, don't understand what they're on Earth for, according to the Platonists. Many of you are familiar with the metaphor of the cave from The Republic. This is the classic depiction of this wrong preference. The people in the cave are chained facing the back wall of the cave, and they see images of what's passing in front of the cave reflected on the wall. As time goes on, they come to believe that those images are reality. They forget that they're chained. They forget that they can't see the real things. They forget the Sun. If you were to liberate them and turn them around and show them the light, first of all, they couldn't bear it, because they're used to the world of shadow. Secondly, they'd kill you, because you are destroying their assumptions and their world. They would at least persecute you. They're not interested in the truth. They're interested in getting by. So for the Platonists, evil is the result of this error in perception, assuming that it's a great thing to get rich. Or assuming that it's a great thing to beat people up because you're stronger than they are. Or it's a great thing to conquer and subjugate. Or all of these things, some of which are evil but are really the preference of things that you should not be pursuing or that you should be pursuing for reasons inspired by spiritual truth. How do you get rid of this? In the Platonists' imagination, by education. That's the whole point of Plato's dialogues. That's why they are dialogues, many of them, with a question and answer. They're very didactic. They're like being in a class. Socrates quizzes people, and then he shows the solution. And they say, "Oh, wow, Socrates, now I understand. Now I'm going to be a Platonist, and I'm going to build a perfect society." End of story. The important thing to understand about Platonism is that it is not dualist in the way that Manicheanism is or the way that we instinctively think about evil. Evil is not opposed to good. It's hierarchically inferior to good. The Platonists' universe is like a ladder with many rungs going up from mud, bugs, rocks to the immaterial One, and with many, many steps, as I said, many rungs or steps or levels in between. Human beings are in the middle. And also, human beings-- unlike animals, mud, slugs, but also unlike angels and demiurges and deities-- human beings can move up and down the ladder. That's what free will is. That's what being human is. You can read Hortensius, read the Confessions, fall in love with the liberal arts, and ascend to some very high realm of the spirit. Or you can choose the downward path to debauchery and mere pleasure. It's a question of how free you are, but we do have the opportunity to move up and down this ladder, unlike the animals and all other created forces that are fixed. Now, the question is what makes us move up and down this? Or what'll motivate us to move up? And here we come to some key differences between Platonism and Christianity with regard to evil. Platonism tends to ascribe evil to ignorance. Christianity tends to ascribe evil to sin. The difference between sin and ignorance is that sin is deliberate. You know you shouldn't do this, and you do it anyway. You're not overcome by desire. So to anticipate one of the paper topics-- but I don't think I'm going to be giving away the answer-- what is it about the pear-stealing incident that makes it so important? Anybody want to venture a preliminary response to this? STUDENT: Well, it's the fact that he doesn't need the pears. He just does it because he feels like sinning. PROFESSOR: So the pears-- he doesn't need the pears. It is just a desire. I mean he doesn't say to himself, I want to sin. I haven't sinned in two days. He doesn't need the pears. What do they do with the pears when they get them? STUDENT: They chuck them. PROFESSOR: Yeah, they throw them out. They throw them to the pigs. So they're not hungry. It's not like, I was overcome by desire, and that led me into some sinful behavior. They weren't overcome by desire at all. How would you describe their pre-pear-stealing state? At least, what would you guess was their pre-pear-stealing frame of mind? There's one word that'll describe it, but if you want to, use a few more. Some guys go and they steal some pears from an orchard. They fool around with them. They throw them to the pigs. STUDENT: Bored? PROFESSOR: Bored. Bored. Now, here's something we can identify with. They're bored. They need to amuse themselves. They cannot amuse themselves by saying, "I'm a good person," or, "I'm going to contemplate the One," or, "I'm going to do some homework." It doesn't end. Young people are thought to be easily bored, but the boredom of old people, it's a different kind of bored. But there it is. It's persistent. That's not the only reason people sin, but it is a gratuitous reason. And that's what's interesting about the pears. It's gratuitous. It's not from need. The Platonists don't have a good response to why this happens, because it's not a question of education. Now, Augustine does not invent Christian ideas of sin. If you said to Augustine, "Come on, why are you so worried about the pears?" He's not worried about the pears as such. It's just a little emblem or a little example of a different kind of problem-- that is to say, knowing how to behave doesn't change us. Feeling how to behave-- to put it in Freudian terms, it's not the ego that decides. It's the id. It's the instinct, not the intellect. And that's what his conversion means. His conversion is not: "Suddenly, I was convinced that Christianity was true." He already knows that Christianity is true, but he knows it intellectually. The conversion is a conversion to an emotional apprehension of it. So however intellectual he may seem to you, however formed in the tradition of Greco-Roman classicism he was, however much the Hortensius awakened him to the life of the mind, he is ultimately a theologian and philosopher of the irrational, of the supra-rational. And indeed, Christianity in its history has an oscillation between intellectualization and the rediscovery of sin and God's grace. If you think of movements like the Reformation of Martin Luther, John Calvin, et cetera, in the sixteenth century, it takes issue with the notion that we can do works that give us merit in the sight of God, and that the Church tells us we have accumulated merits, and therefore we'll go to heaven. The Reformation teaches that we are face to face with God and that our so-called good deeds don't amount to anything. We are all sinful. If God operated according to justice, we would all go to hell. It's faith and grace that save people, hence the Reformation. But it's also the Great Awakening of Britain and America in the eighteenth century, the development of Methodism, the Fundamentalist movement, all of these tend to reject attempts to approach God contractually, attempts to approach God in terms of a deal. So if human beings are sinful and if education is not going to get them out of sin, what will? Now, the Augustine of the Confessions is different than the Augustine of 20 years later when he wrote The City of God. And we're not studying The City of God, but this book, written in response to the sack of Rome in 410, develops some ideas that are found in the Confessions about the nature of sin and how we get out of it. The nature of sin is the pears. How we get out of it is at least in part the conversion. We got the pears sufficiently for the time being? The conversion is started-- well, it started long before the event. But what precipitates it as a drama is this conversation with Ponticianus in Book VIII, Part VIII, who has traveled and describes the monks of Egypt. Now, we'll be talking lots and lots about monks, but the monks of Egypt are the first example of Christian monks, men who flee the world into the desert and there live on weeds, saline water, locusts, other insects, basically nothing. And they have visions, and they are sought out by ordinary people. It's key to understand that to be a hermit in this society does not necessarily mean that you have nothing to do with people. People start to want to find you, because you must have special power. Back in Alexandria, their baby is sick. "Maybe you, oh hermit, living on locusts and out in the desert, have some spiritual power to help my baby." This is shamanism. It happens in all sorts of religions. You can't just be a shaman, a medicine man, a wise man, and hold down a regular old job. Or you can, but it helps. And that's the conceit of a lot of TV ideas, secret heroes. They're the real estate agents, but they're battling the forces of darkness. But generally, most of the time, you've got to be special, and you've got to look special. And you've got to be a reject. You can't have a spouse, kids, a mortgage, a garden, a swing set. You've got to be a seer. You've got to have your vision focused on the other world. Ponticianus tells Augustine about these men, and his response is not only to be impressed by them, but to be humiliated by them. First of all, here are these guys who are intoxicated with God, while I'm still thinking about my career. But-- and this is the ancient world speaking-- they are uneducated, these monks of Egypt. They didn't study The Republic, the Hortensius, the Timaeus, the rhetoric of Quintilian, the Satires of Juvenal. They don't know anything about this. They're uneducated people. Many of them are illiterate. And yet they are closer to God. They have an apprehension of the divine that causes them to renounce the world, whereas we-- Augustine says of him and his circle-- we "lie here groveling in this world of flesh and blood, while they storm the gates of heaven." And this is the moment of his conversion. Now, after his conversion, Augustine's plan was to lead a life of contemplation with his friends. They would retreat from the world, meaning they would give up their careers, but it would be a little bit like one of your friend's parents have a lot of money and have this wonderful cabin somewhere in the Rockies or the Sawtooth Mountains. And you're going to figure out some kind of way of-- you'll be on the Internet and everything, but you're going to have this kind of beautiful, contemplative life. But the beautiful part is that it's not going to be uncomfortable. It's not the desert of Egypt. It's remote-- you're not going to be bothered-- but there are beautiful mountains, trouts in the stream. It's idyllic. And you and your friends are going to talk about reality and the spirit and philosophy and-- I don't know how idyllic this sounds to you, but it's certainly an understandable idea of a way of life. It is the ancient idea of what's called "leisure with dignity." And indeed, that's what being a professor was supposed to be when I signed up for it. Otium cum dignitate, leisure with dignity. "Leisure" meaning not wasting time leisure, but not responding to clients, or not responding to urgent scheduling phone calls, deals. You've got to show up to your classes, but that's not really onerous. At least, that was the idea. And I won't go into the frustrations of being a professor or the dissatisfactions. But the classical idea is otium cum dignitate, "dignity" meaning not being naked in the desert, not having to eat locusts and figure out how to-- "OK, I had curried locusts last night. Tonight, I think I'll have locust casserole." No, no, no. Something nicer than that. But in fact, he did not follow through on this. He did not lead a life of cultivated classical dignity with his friends. He went back to North Africa. He became a bishop. His years were consumed by disputes over doctrine or with heretical-- as he deemed them-- tendencies, like Donatism, most notably. And he died defending his city of Hippo, Hippo Regius, in modern Tunisia, from the Vandals, one of those barbarian invaders who will occupy us next week. He then was very much involved in the world. To be a bishop in the Roman Empire was by no means an office of dignified leisure. It was right in there in the political trenches. It's a position of honor, to be sure, but his understanding of the Christian's duty in the world was that you cannot lead a life of perfection. You cannot lead a life of sin-free contemplation. We all are sinners. He becomes more and more the theologian, philosopher who combats perfectionism. Perfectionism is a doctrine that human beings can be made radically better-- perfect, even. There are debates throughout societies about the degree of human perfectibility. This is indeed supposedly and to some extent I think really at the heart of debates between what is called liberalism in the United States and conservatism. Liberals believe in human perfectibility. If you educate people, if you help them, if you encourage them, if you provide government subsidies, you will build a better society. The response upon the part of conservatives to that is, people are the way they are because that's the way they want to be, or they made wrong choices. But all the help from some public authority isn't going to help, isn't going to really make a difference. Are people perfectible? People who believe in education tend to believe that they are. On the other hand, very well-educated people have been bad. Hitler loved classical music. So did Stalin. Just because you are a connoisseur of art doesn't make you a good person. Augustine is a radical imperfectionist, more so in The City of God than in the Confessions, which is teetering on the brink. The pears is a kind of imperfectionist moment. He glimpses the power of sin. By the time of The City of God, by the time that the end of the Roman Empire is at least glimpsed as a possibility and the rise of the barbarians, Augustine has become someone who does not believe that human beings can, in any way, earn salvation. Human beings are irrevocably sinful. Once again, if God judged people according to their merits, they would all be damned. Since the Christian belief is that some people are saved, they are saved by a mysterious process called "grace." Grace, by its very meaning, is undeserved. You don't show up at the door of Heaven with a ticket of admission earned by your deeds on Earth. What opens the doors to you is a generous, arbitrary decision. Well, "generous" may mean, I had good intentions. I didn't kill anybody. But "arbitrary" may mean that we can't figure out who's going to Heaven and who's going to Hell. It may even mean that since God knew before we were born, God predestined us for Heaven or Hell. This is a harsh doctrine. It gets periodically rediscovered and then dropped. It's at the heart of the belief of the people who settled Massachusetts and Connecticut. It is the heart of Calvinism and of Puritanism, the belief in the elect. The question is, are these elect visible or invisible? The elect are people who are going to go to heaven. Are they visible? Can we say, this guy is so good, he's going to heaven? This woman is so loving, nurturing, self-effacing, whatever, she's going to go to heaven? That's the notion of a visible elect. An invisible elect is, "We don't know, we have no idea." And this is a crucial difference. Because if you believe in the visible elect, even if you say they're not guaranteed, but anybody outside of this circle is for sure going to hell, then you have Puritan New England. You have a small community of people pursuing perfection. Or you have the Amish. Or you have any small pious community that believes that outside of it is more or less given over to sin and more or less doomed. Inside of it, maybe it's not guaranteed, but your chances are much, much better. But if you believe that we don't have a visible elect, that we have no idea, then everybody ought to be in the Church. Everybody ought to have access to the sacraments that provide initiation into the Church. You ought to start converting pagans, even savage pagans. You ought to be out there roping in as many people into the church, including people who don't want to be. Because you just never know. Maybe their kids will be. Augustine is behind ideas of things like forced conversion. As long as they're baptized, there's a chance of them being saved. And baptized as infants, preferably, because baptism does not any longer mean, in Augustine's world, perfection. It means the beginning. It means entering the process. So the three things that he is teaching that are implicit in the Confessions and that he is important for in terms of his intellectual impact are his opposition to perfectionism, his exaltation of grace, and the notion of sin as indelible, not solvable. Where this becomes of key historical importance is in the Church. The Church is a body that can either be sectarian and small, as with the Amish or Puritan New England, or it can be huge and universal, as with the medieval Catholic Church. Augustine stands behind the medieval Catholic Church, which is a political body, a body of doctrine, a structure ruled by princes, and a structure that has a missionary impact on the rest of the world. Now, the papers. You have the paper topics. If you didn't get them, come up and see me after. You can choose any one of them, or you can choose something else. But if you choose something else, please talk to your teaching fellow or to me. And talk to us anyway about these papers. We'll give you plenty of opportunity to bounce ideas off us. Now, next week, we talk about the fall of the Roman Empire. But this is implicit in what we've talked about today, because the bottom line is the Roman Empire is going to fall in the West, and the Church is not going to. And so we'll look at how that works next week. Thanks. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 06_Transformation_of_the_Roman_Empire.txt | PAUL FREEDMAN: Today we're going to talk about the transformation of the Roman Empire. And I use the somewhat neutral and undramatic word "transformation." It can be "fall of the Roman Empire," "collapse of the Roman Empire..." It's clear that we're talking about the fall of the Western Empire. Next week we'll talk about the survival of the Eastern Empire. From 410 to 480, the Western Roman Empire disintegrated. It was dismembered by barbarian groups who were, except for the Huns, not really very barbarian. That is, they were not intent on mayhem and destruction. All they really wanted to do was to be part of the Empire, to share in its wealth and accomplishments, rather than to destroy it. Nevertheless, 476 is the conventional date for the end of the Western Empire, because in that year, a barbarian chieftain deposed a Roman emperor. Nothing very new about this for the fifth century. What was new is that this chieftain, whose name is spelled all sorts of different ways, but in Wickham, it's Odovacer. Sometimes he's known as Odacaer, Odovacar, Odovacer. We aren't even sure what so-called tribe he belonged to. A barbarian general deposed the child emperor Romulus Augustulus, who by an interesting coincidence, has the names of both the founder of the city of Rome and the founder of the Roman Empire. The "-us" on the end is little. It's a diminutive. So a man with this grandiose name, a child, deposed in 476. And instead of imposing another emperor, Odovacer simply wrote to Constantinople and said, "We're going to be loyal to you. We will recognize you as the sole emperor." Constantinople, however, was far away. And while of symbolic significance, this pledge of loyalty by Odovacer had no practical significance. For all intents and purposes, the Western Empire had, in 476, become a collection of barbarian kingdoms. A kingdom is smaller than an empire. We use the term empire to mean a multi-national, very large state ruled from one center, but consisting of many different kinds of pieces. Kings, and the term and title "king", is of German origin. Kings are very powerful, but over a more limited territory. So there was a king of Italy now. There would be a king of the Franks, or Francia, the former Roman Gaul. There would be a king of the Lombards later in northern Italy. A king of the Visigoths, first in southern France and Spain. And we'll go over who is where at the beginning of next class. For now, we're going to talk about this collapse and its consequences. And we're going to orient ourselves around three big questions. One-- why did the west fall apart? And as a corollary to that question, was this because of the external pressure of invasions or the internal problems of institutional decline. Did it fall of its own accord or was it pushed, in other words? Question number two. Or big question number two. Who were these barbarians? And how Romanised or how different from Rome were they? And that's what we're going to talk about more on Wednesday, next class. And three, does this transformation mark a gradual shift to another civilization, or is it the cataclysmic end of the prevailing form of civilization, ushering in a prolonged period of what used to be called The Dark Ages? The Dark Ages-- roughly the sixth to eleventh century. This is a term we don't like to use. It implies a value judgment that is not only not necessarily accurate, but also expresses a certain kind of point of view of what are good periods in history and what are bad periods in history. But I'd like to just probe this third question first. That is, how severe a catastrophe was this? So is it the end of civilization, a la Planet of the Apes or Blade Runner, or any of those apocalyptic images we have? Or is it merely a shift in power and the survival of Roman institutions such as the Church, while Roman political infrastructure-- the emperor, the consoles, the pratorian prefects, and so forth-- while that collapses? A medieval historian named Roger Collins in a book called The Early Middle Ages writes, "The fall of the Roman Empire in the west was not the disappearance of a civilization. It was merely the breakdown of a governmental apparatus that could no longer be sustained." The key word here is "merely." The destruction of the Roman political apparatus may simply mean that the Roman state ceased to function, but that everything else continued. But really, the question is, could everything else continue in the absence of a state and of a political order? The destruction of the political order also means, after all, the destruction of the military system. When we opened this class, we talked about a civilization built on such things as the rule of law and the maintenance of peace. These are no longer possible if there is no military governmental structure. As we'll say a little later, to some extent people didn't know that it was the end. Because for a while, things seemed to go on as before. People were speaking Latin, they were living in cities, the cities were much less populated, but nevertheless, they were still there; there were still rich people; there were still poor people. In retrospect, though, we can see that things really did change. How much they changed is the subject of a lot of historical controversy. The world of the late Roman historians is divided, roughly speaking, between catastrophists and continuists. As you may guess, the catastrophists think the fall of the Roman Empire--, whether we date it 476 or there's some reasons to date it, really, 550 for reasons we'll learn in next week. Between 450 and 550, a catastrophe happened. A civilization was wiped out. And really, if not literally a Dark Ages, a more primitive, more war-like, more illiterate, and more rural period was ushered in. The disappearance of ancient texts, things that the Romans knew from that lost Hortensius dialogue of Cicero that Augustine was so fond of, to many other kinds of works that had been known to the Roman world, right? I can't remember exactly how many plays Aeschylus wrote, but it's something on the order of 60, and we have three. So the disappearance of text. The end of literacy, except for a very small portion of the Christian clergy. A more primitive architecture. The end of grand civic projects like aqueducts, coliseums, theaters, baths. A more isolated society without these urban centers. A diminished population spread across the countryside, mostly engaged in subsistence. Hence, the, if not end of trade, the radical diminution of trade. The continuists, people like Collins whom I just quoted, see the political changes as dramatic all right, but as essentially surface phenomena based partly on archaeology and partly on a more sympathetic understanding of Christian practices. In other words, they don't think that the proliferation of churches, saints, cults, is necessarily a sign of a primitiveness. So based on both archaeology and an understanding of Christianity, these continuists point to the survival of trade, the role of bishops and other church officials, as replacing the Roman governors. The Roman political order may have collapsed in terms of staffing by lay people and military people, but the bishops were now the rulers of the city. The bishops would now do things like ensure the food supply, rally the local population against barbarian invasions, educate the populace. And the barbarian kings themselves try, with some success, to perpetuate the Roman order. They collect taxes, for example-- that may or may not be a good thing. They engage in some kind of public works, some kind of maintenance of order. The civilization of the sixth and seventh centuries in what comes to be considered Western Europe, rather than the Western Roman Empire, is not radically more barbarized or primitive than the late Roman Empire. Thus, the continuists. My own position, but I don't hold to it dogmatically, is that of a moderate catastrophist. I think something really happened; I think it's pretty radical; and it didn't happen all at once, however. 476 is not the year of collapse. It is a process. I'm fascinated by the degree to which people were and were not aware of the cataclysm, but I believe there is a cataclysm. Wickham, the author of this book that we're starting now The Inheritance of Rome, Chris Wickham, straddles the fence, as you've seen. His chapter that you were to read for today is entitled, "Crisis and Continuity: 400 to 550." I would never use a chapter title like that, because it's really frustrating. Which is it, dude? He's the leading medieval historian in the English - speaking world. He is Chichele Professor at All Souls, Oxford. And if that doesn't sound impressive, well, it takes a lot to impress you. He's a very great historian, but I don't like that chapter title. As I said, I would emphasize crisis or change cataclysm. Well, let's ask what happened, beginning with the gradual involvement of the barbarians in the military and their entrance into the empire. We're using the term "barbarians", which goes back to the Greek term applied to outsiders. People outside but threatening. The Greeks defined barbarians as uncivilized by reason of their speech, which sounded to them incoherent, and by reason of the fact that they're nomads. People who lead settled lives don't trust nomads. Nomads almost extinct in our world, once dominated many geographical regions and were frightening, because they moved to around to people who liked order and familiarity. They didn't live in cities, whether they were nomadic or not. Barbarians were illiterate. This is the Greek idea of barbarians. In the case of Rome, there is no single definition of barbarian society. We can say that Rome was overthrown by a war-like, but not very fierce, group of enemies. And I use enemies in a very mild sense. The Romans perceived them as enemies; the barbarians perceived Rome as simply a nicer place to live. But there is no Mongol horde kind of event here. They're not that frightening. The Romans had known them for centuries. Most of them were even Christians Heretical Christians, OK. They're Arians, A-R-I-A-N-S, I remind you, but they're not unfamiliar, again, even in their religion. They've been at the borders of the Roman empire forever. Like most empires, Rome was at the one hand, very aggressive, and on the other hand thought of itself is peace-loving. It maintained the Danube-Rhine frontier as a kind of natural frontier, every so often crossing those rivers to punish German tribes who were probing the frontiers of the empire. But generally speaking, the Romans were not interested in what they perceived, somewhat inaccurately, as endless forests inhabited by primitive people. The continuists argue, with some justice, that between 250 and 600 what changed was not that primitive warriors conquered a civilized state, in the way that say, the Mongols conquered China in the thirteenth century, but that the ancient world became the medieval world. That is, an urban culture became more rural. A Latin culture became amalgamated to a German one. Pagan society became Christian. Having said this, it's nevertheless true that the most dramatic event to the fifth century is that people who had been outside the empire were now in it. If we ask why the Western Empire collapsed, the simple, most immediate answer is it was taken over by German confederations, tribes. They came not so much as conquerors as military recruits, or as allies, or as refugees. So rather than as guys with knives in their teeth hacking and slashing and burning, they came as pathetic refugees, maybe doing some hacking, slashing, and burning; as military recruits; and as military allies. Again, not without a certain amount of H. S. B.: hacking and slashing and burning. But not a cataclysmic amount. They admired Rome. They wanted to continue its institutions. They regarded Rome as a rich and as civilized. The last thing they wanted was to still live in little huts in the forest. They were not the bringers of a revolution. They were not even that numerous, amounting to some tens of thousands. Nevertheless, they ended Roman government, accelerated the changes we've already described towards depopulation, decentralization, ruralization-- a less cultivated, less literate, less Mediterranean- centered society. So I want to begin the description of this process by the changes in the Roman army. We saw that Diocletian, around 300 AD, militarizes Roman government, pays for the, perhaps, doubling of the military presence of the Roman army by changing the taxation system. So the twin pillars of the empire in the fourth century are army and taxation, the latter requiring a civilian governmental apparatus. The army was a problem in terms of the recruiting of soldiers. This may have to do with the population; it may have to do with the unattractive nature of military life, but nevertheless there was already, in the fourth century, a tendency to get the more familiar barbarians into the army as Roman soldiers. Because they were available, they were near the frontiers-- this may seem odd. Why hire your potential enemy to be soldiers? But there's a lot of precedent. Very often, empires don't really want to supply their own manpower. And the people who are the best soldiers are also the people who may, in the future, be most threatening. I don't want to pursue this simile, but the Afghan Mujahidiee were trained by Americans, because at one time they were opposed to the Russian occupation of Afghanistan. As it happened, in retrospect, that had some bad consequences. But at the time, it seemed like a good idea. So in the 370s a group called the Visigoths asks to be admitted to the Roman Empire as an allied army. In other words, the whole group will be federated with the Romans. And federati is the term given for barbarian troops serving under the Roman Empire. Why were they on the move? These are not really nomadic people. They don't live in yurts or travel across Central Asia. They tend to be settled in villages. They have dairy cattle rather than have some kind of nomadic sheep, or something like. They're pretty settled. Nevertheless, in 378, they were on the move. And we don't know why. Some enemy pushing them across the Danube into what's now Romania? It may be the weakness of the Empire. They may have seen that the empire was not so strong and made a proposition, kind of like a takeover. You don't seem to be doing so well in your stock or your finances, so we're going to infuse some capital into you, i.e., our soldiers. They also may have been hungry. Certainly, once they crossed the frontier, the Romans were rather inept in feeding them, in supplying them, and the Visigoths rebelled. Thus far, nothing incredibly new. What really was new was that the emperor came with an army to suppress them. And rather to his surprise and everybody else's, the emperor Valens was defeated at the battle of Adrianopole. Defeated by the barbarians. Yeah. STUDENT: So, being involved in this federati, what did they get from the Roman Empire? Did they agree to fight for them and then they'd get land? PROFESSOR: They agreed to fight for them and they got a combination of land, or supposed to get land or territory, and some kind of maintenance in kind and or money. The question was about what the Visigoths, as federati, got out of this deal. Or were supposed to get. The defeat at Valens was not immediately cataclysmic, because, even though he was killed at this battle, even though it sent shock waves throughout the empire, in fact, it would not be this area that succumbed to the barbarians--, the East. Romania, or the Balkans would be part of the Eastern Empire. And indeed, both Adrianople the city, and Constantinople, the even greater city, would withstand Visigothic attempts to take them. In 382, the Visigoths were officially recognized, and they were allowed to settle in the Balkans as federati. And in fact, they were reasonably useful troops to the Roman Empire in the 380s and 390s. What this does show, however, is the barbarization of the army. And another aspect of that is that the army tended to be commanded now more and more by barbarian generals. These barbarian generals, at the top, bore the title magister militum-- master of the soldiers. So I'm using the term "general" as an anachronistic one, since that's what we're familiar with. These magistri were powerful leaders, charismatic leaders, of German or other tribal groups, who then ruled in the name of, or behind the throne of the emperor. They couldn't be emperors themselves, at least in these years, it was impossible to envisage a barbarian emperor. But they held more power than the emperors. Two of these generals, war leaders, magistri, Stilicho and Alaric. Stilicho was a Vandal Alaric was a Visigoth. Alaric wanted territory, food, treasure from Rome. The Visigoths were moving from the Balkans into Greece, eventually into Italy. Stilicho played a kind of game with Alaric, trying to keep him in check in the name of the Western emperor, but also negotiating with him. The emperors moved from Milan in the north to Ravenna, a little bit to the east. Ravenna, then, was in the marshes and impossible for a barbarian army to take. This is the last capital of the Western Roman Empire. Kind of romantic and mysterious, but strange as a place to end up. These are the Visigoths then, who are on the move in the 390s and the 400s. Eventually, Stilicho would be executed by the Roman emperor of the West and Alaric would invade and plunder Rome in 410. It was the Visigoths who engineered the so-called Sack of Rome that so shocked Augustine and his contemporaries. Where, you might be asking in all of this, was the Roman army? Alaric was wandering around the Balkans and Italy for two decades before he sacked Rome. The army, which had consumed so much of the resources of the Roman Empire, is curiously absent in the history of the fifth century. This is not the Eastern Front in World War II. This is something altogether different: the collapse of an empire that expended huge amounts of treasure on its army. Its army seems to be invisible and supports, to some extent, or that fact supports to some extent--, the argument that the Roman Empire collapsed of its own internal disorders, since we don't see it losing pitched battles to outside barbarians. Or maybe the army doesn't disappear, it becomes indistinguishable from the invaders. The army is the invaders. Creepier. Now within this, there are some real barbarians-- the Huns. The Huns are kind of nomadic. OK, they didn't actually cook their meat by holding it between their thigh and the horse hide, and the sweat and heat of the horse heated up the meat. This is a widespread myth of nomadic peoples. The Chinese say this about the Mongols, the Romans about Huns. But they were pretty mean. They were interested in the Roman Empire mostly for plunder. And they didn't care if that destroyed the economic base, because they weren't thinking in such terms. And indeed, they may have frightened the rather nice German tribes that stood between them and the Roman Empire. In the 450s the Huns were united under the leadership of Attila. And Attila certainly threatened the Eastern Empire first, but the Eastern emperor defeated the Huns, discontinued tribute to them, and in a pattern that we'll see repeated again and again, the Huns decided that Constantinople was too tough. That the Eastern Empire as a whole, access to which was more or less controlled by Constantinople, was too well guarded. And they turned to the west instead. Not as rich maybe, but much easier pickings. They show up in Gaul in 450. They were defeated by an army of Visigoths allied with Romans. They then went to Italy. They went into the heart of the Empire, sacked cities in the northeast of Italy, and there's no army. The emperor is holed up in Ravenna. basically shuts the door, gets under the bed, and waits for it to go away. The one power of Italy willing to try to deal with Attila is the Bishop of Rome, whom we haven't heard of yet, but we're going to be hearing about him a lot. And indeed, in the course that follows this, even more. The Bishop of Rome-- the pope. Pope Leo I, along with two senators from the Roman senate, goes up to northern Italy to remonstrate with Attila, to visit the leader of this barbarian tribe in 453 to try to get him to stop plundering Italy. Whether they were successful or not doesn't much matter, because Attila died shortly thereafter of a brain hemorrhage. And with his charismatic leadership, the Huns came to an end as a military force. That is, with the end of his leadership, the Huns no longer had as imposing a military force and quickly disintegrated. What's significant is that it's the pope who is taking over what we would think of as the Roman imperial responsibilities. And this will be a pattern, not only in the assertion of papal power, but in the way in which the Church starts to take over many of the roles abandoned by the empire. After this, the barbarian generals, in effect, take charge. The Huns are defeated, but the other groups now pour into the empire. The Vandals have taken over North Africa by this time, by 430, cutting off the grain supply to Rome. They are unusual among the barbarian groups in that they have a navy. They know how to use boats, and indeed, they plunder the city of Rome in 455 in a sack that might have been worse than that of 410. By 470, the Visigoths control southern Gaul, what's now southern France; a group called the Suevi are in Spain; the Vandals in North Africa; a group called the Ostrogoths in what's now Hungary; the Angles and the Saxons in Britain. All that effectively remained of the Western empire when Odovacer overthrew Romulus Augustulus was Italy. And in 476, that's it. A little coda, however. In 493, the Eastern emperor in Constantinople convinced the Ostrogoths to get out of Hungary, stop threatening the Eastern Empire, and take Italy from Odovacer. Once again, the Eastern Empire is capable of deflecting barbarians into the west, because they're too strong. So in 493, our friend Odovacer was overthrown by the Ostrogoths and their leader Theoderic. So what's the impact of all of this? On the ground, if you were looking around in 480s 490s, you would see a kind of accommodation. The Roman elite accommodated themselves to, compromised with, negotiated with, their new rulers. So, for example, a member of a very wealthy Roman family, a man named Sidonius Apollinaris in southern France, was a bishop and a great landowner. And we have a lot of letters of his that tell us about his negotiations with the Visigothic king Euric. He found the Vsigoths uncouth, hard to deal with, not knowledgeable of the Latin classics, but not very frightening, either. Not particularly formidable. So accommodation, improvisation. We have a saint's life that is a biography of a saint, a man named-- I'm sorry that I'm writing on the board so much today. Usually, as you know, I'm a little more in control. But these are great names. And some of them are good cats names or dog names, too. Severinus of Noricum. You know, "Stop scratching the furniture, Severinus." That kind of thing. Severinus of Noricum. A saint in what's now, more or less, Austria. His life tells us that he learned of the end of the Roman Empire this way: "At the time when the Roman Empire was still in existence, the soldiers of many towns were supported by public money to guard the frontier. When this arrangement ceased, the military formations were dissolved, and the frontier vanished. The garrison of Passau, which is still a town in modern Bavaria, the garrison of Passau, however, still held out. Some of the men had gone to Italy to fetch for their comrades their last payment." This resembles a corporation-- somebody, actually, was telling me yesterday they worked for Eastern Airlines, a company that went out of business in 1990. And so sudden was the collapse of Eastern, even though it had been predicted, that she was a flight attendant and had to get on another airline in order to get home. She lived in New York; she was in Florida; Eastern ceased to exist. So these soldiers are in the same position. They want to get their last paycheck. They were never heard from again. Nobody knew that they, in fact, were killed by barbarians on the way. "One day, when Saint Severinus was reading in his cell, he suddenly closed the book and began to sigh. The river, he said, was now red with human blood. At that moment the news arrived that the soldiers had been washed ashore by the current." Interestingly enough, he doesn't just stay in his cell and pray. He starts to organize this society. He is active, although some of it involves some miracles, in poor relief. He deals with the local barbarian king, the king of the Alamanni, remonstrates with him. He helps in diverting Odovacer into Italy. Again, like Pope Leo, we have a member of the church, and in this case somebody that you would think was a recluse, indeed had been living like a recluse, nevertheless taking over the responsibilities for a population abandoned by its civilian government. That is then one of the forms of accommodation. Another aspect of this era, however, is decline. The urban population declines. The society and economy experienced what Wickham euphemistically calls, "a radical material simplification." The term he uses, I believe, on page 95 and 105. "Radical material simplification" means that your standard of living plummets. Cruder ceramics. Instead of that nice, north African red slip ware, you've got mud that you baked at home. Fewer imports, no pepper. More homemade, crude building materials. Fewer luxury goods. The Vandal control of North Africa meant the end of the Roman wheat supply. The countryside of Rome had not grown enough wheat to feed the city since 200 BC. So for 600 years, minimum, Rome was dependent on other sources of supply. Southern Italy, Sicily, North Africa. The moment the Vandals cut the supply, the city could no longer support its massive population, could not feed everybody. When you multiply this phenomenon, it's not a surprise that the city's decline in population, and that the society becomes more rural, more agricultural, more subsistent And here's where I think Collins is naive to speak of merely a political decline. Without a government and military structure, trade could not take place on the scale it had before. And without that trade, cities could not survive. There is no denying a decline in culture, economy, and population. Let's just look at Roman population figures, based on things like pork supply figures, public-- well, I mean, nobody took a census in Rome. We don't really know exactly how many people lived there at any given time. But historians and archaeologists looking at things like food supply, public welfare payments, water delivery figures, for aqueducts, and the abandonment of houses and of building sites. Probably in 5 BC, the Roman population was 800,000. That would be a fairly conservative estimate. Maybe as much as a million, but definitely 800,000. 5 BC. Yeah? STUDENT: This is just the city of Rome? PROFESSOR: Just the city of Rome. Yes, just the city of Rome. At the time of Constantine, sort of where we begin the course, more or less, in the early fourth century, the population had declined probably to 600,000. After the sack of Rome in 419, probably 300,000 to 500,000. Obviously, these are very rough figures. But after the sack of Rome, more than half of the population that had existed in 5 BC is gone. With the end of grain shipments from North Africa, we don't really know immediately. We can estimate that by 590, there could not have been more than 150,000 people in Rome. This is after not only the Vandals, but after a catastrophic war in Italy launched by the Byzantine Emperor Justinian, who we'll be talking about next week. In 800, on Christmas Day, Charlemagne was crowned in Saint Peter's in Rome as Roman Emperor by the pope, an act whose implications we will be exploring towards the end of the class. On that day, Rome must've had maximum, maximum, most optimistic estimate, 30,000 people. This does not necessarily mean that they were primitive, but they were living in the Coliseum, for example. People built houses in there. They used the walls of the Coliseum as a fort. There is a certain Planet of the Apes quality, in fact. Rome, still to this day, is filled with picturesque ruins, even though it is a city of two and a half, three million people. As I said, people were not necessarily aware of this change. For example, lots of churches were built at this time, and some of them have mosaic pavements that have mottos about the grandeur of the Roman name, and the usual classical kind of mottos. But then again, people often aren't aware of what's happening to them. I mean, what if somebody in the future points to the fact that New Haven, in 1920, had far more people living in it than it does now? New Haven lost a third of its population between 1950 and 1980. What if some future historian is scandalized at the fact that in order to get into Yale a hundred years ago you had to know Greek and Latin. If you look at what those gentlemen C Students had to study, or were responsible for, in say, 1925, it's extraordinary. It's not very impressive in the sciences, but the decline of the humanities, if by decline we mean things like knowledge of classical literature, is stunning. Somebody may decide in a few hundred years that the Dark Ages began in about 1950. And that those pathetic people in, say, 2011, impressed with their little technological toys, nonetheless didn't know anything. Now I don't actually believe that. There are some people who do. There's a philosopher at Notre Dame named Alasdair MacIntyre who really believes that the Dark Ages began a long time ago, and we simply don't know. We simply refuse to recognize this. I was impressed by an obituary for a man named Patrick Leigh Fermor, who died at the age of 96 earlier this year. This is the last of the great British characters of the twentieth century. He not only was classically trained, wrote a lot about Greece, lived in Greece, he, in World World II, disguised himself as a Greek shepherd in Crete, engineered the capture of a German general, and the delivery of that general after three weeks of hiking through the mountains of Crete to a British destroyer. It's in a movie called Ill Met By Moonlight, if you ever want to check this out. Not a great movie, but-- Patrick Leigh Fermor also wrote two books out of a projected three about walking from Holland to Constantinople or, Baghdad actually, I think, in the 1930s. But the obituary describes a conversation he had with this German general, whom he is trying to get across Crete. And the general at one point, over some fire in the wilderness, quotes a line from Horace, the Roman poet, that then Patrick Leigh Fermor finishes is for him, and indeed, quotes the next two stanzas. Well, that world is over. That world is over. I don't pretend to be part of that world, either. And that's a world that would have existed in the time of Horace, or the years after Horace, who lives at the time of Augustus. This would have existed in 300 A D. It would have existed, at least, in a few monasteries in 800 AD. It would have flourished in the Britain of the eighteenth and nineteenth and early twentieth centuries. So again, I don't think that civilization came to an end. What came to an end was a civilization, a certain kind of society. It has some heirs, however, like all dead entities. There are four heirs to the Roman Empire. One is the Byzantine Empire, the Eastern Roman Empire, which calls itself the Roman Empire. It doesn't call itself the Eastern, doesn't call itself the Byzantine, it calls itself the Roman Empire, even though it does so in Greek. The second heir are the barbarian kings. We'll be talking about them on Wednesday. They are attempting to prop up the remnants of Roman culture, civilization, and material society. The third heir in some ways, is Islam, which we meet in the seventh century, the century of its invention. And the fourth heir is the Church. Even though the Church grew up in opposition to the Roman Empire, it will preserve Latin, cities, learning, classical civilization. OK. So barbarians on Wednesday. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 18_The_Early_Middle_Ages_2841000_The_Splendor_of_Byzantium.txt | PAUL FREEDMAN: We in the midst of the laughs, attempted to look at the seventh century last time as a turning point in the history of the period that we're dealing with, the post-Roman world, the early Middle Ages. Certainly among the major shifts was the rise of Islam and the consequent radical changes in the Mediterranean territories, particularly, of course, the areas conquered from the Eastern Roman Empire by Byzantium, namely Syria, Palestine, Egypt, and eventually North Africa. The seventh century, therefore, changed the shape of the Byzantine Empire, and so its orientation as well as culture. Here I have a kind of periodization. We've spoken about Justinian's expansion. Very shortly after his death, there begins what seems in retrospect, at least in part, to be a reaction to imperial overreach. Imperial overreach is a phenomenon seen throughout history, described most memorably, perhaps, by our own Paul Kennedy, the tendency for empires simply, in order to protect themselves or in order to fulfill their ambitions to get too large for their own ability to hold onto their possessions. This is an economic problem, a logistical problem, a resource problem, and even a cultural problem. In general, it's hard to say what provokes the crisis. That is, with the Roman Empire as with the Abbasids, we can say, "Oh, well it was too big". On the other hand, it was too big and did just fine for centuries. In the case of the Abbasid Empire, maybe not centuries, but 150 years, which as these things go is a pretty long time. Here, however, we're talking about something that is much more obviously related to some kind of overreach. Justinian formed his expanded empire, which is the first map in the handout. And merely a few years after his death, it started to unravel. You'll recall that we said that he had at great cost conquered Italy. After the easy conquest of North Africa, this looked like it would be easy as well. But in fact, while North Africa, occupied by the Vandals, fell within a year or two, Italy took twenty years to wrest from the Ostrogoths. And the peninsula was devastated and in radical economic decline when Byzantium took over. A mere three years after Justinian's death in 568, Italy was invaded by yet another barbarian tribe, the Lombards. The Lombards did not take over all of it. The Byzantine Empire-- Eastern Roman Empire-- managed to hold on to Sicily, much of southern Italy, the east coast, particularly the capital of the Byzantine province, Ravenna. But nevertheless, the Lombards occupied most of the peninsula. And as we'll discuss in a moment, other disasters piled up in this period that I have just called contraction. But it's not just a question of the empire getting smaller. It's really a radical crisis and an ongoing crisis with the appearance of many enemies and the radical shrinking of the borders of this empire. So again, look at the empire in 565. It's making a good attempt at Justinian's death to mimic the Roman Empire at the beginning of our course, at its height in the third century. It doesn't have northwestern Europe, but it has most of the Mediterranean. And to the degree that the Roman Empire was, as we with fatiguing repetition have said, a Mediterranean-centered empire, this empire of Justinian's does a good job of restoring that Mediterranean orientation. If you look at the second map during this period of what the author of this book that I took the map from, Haldon, H-A-L-D-O-N, calls "the process of devastation", you can see how much has been lost. The Empire at this point consists mostly of Anatolia, a little bit of Italy, a few islands. Even the Balkans is mostly occupied by Slavs and Bulgars. There then follows a period of reconstruction, of the stabilization of borders, of taking back some lands in the Balkans and in Anatolia from the Arabs in Anatolia and from various groups in the Balkans. This is also the period of the iconoclastic emperors. And then finally, an expansion of the Byzantine Empire. The golden age of the Byzantine Empire is this period after the settlement of Iconoclasm in 843, until the appearance of a new enemy, the Seljuk Turks, who won a devastating victory in 1071 and begin a process of what ultimately-- but, in this case, ultimately means 350 years-- what ultimately would be the final crisis for the Byzantine Empire, which would be extinguished by the Turks in 1453. So if you look at map number three, the Byzantine Empire around the year 1000-- Ignore Bulgaria for the time being; that is a separate kingdom, and we're going to talk about it soon. This is a compact empire compared to that of Justinian. It is not in control of the Mediterranean. It has two bases, really, the Balkans and Asia Minor. And it has a little bit of territory in Italy still. But this empire made logistic sense, was never easy to keep together. Nevertheless, it was stable, had enough money for its substantial military expenses, and even to create a kind of cultural efflorescence. So here we have a story not merely of survival, but of survival, adaptation, and expansion. And that's what I want to discuss with you today. The reason we're doing this is because this is part of early medieval history. We've talked about the legacy of the Roman Empire, a three-part kind of legacy in our formulation. On Wednesday, we'll start talking again about Europe, northern Europe and the degree to which Charlemagne in particular it is a self-conscious heir to the Roman Empire. He revised the title of Roman Emperor. In the case of the East, these people have never abandoned the title of Roman Emperor. And indeed, the key thing about the Byzantine crisis in contrast to the crisis of the West is that the Empire never falls. It is never destroyed. But it does undergo many of the same crises and consequences of the crises that the West did in the fifth century. I said last time that to some extent, while we don't like the term "Dark Ages", certainly after the collapse of the Western Roman Empire in the fifth century, Europe enters into a period of radical material simplification: That is to say, it is more rural. There's less commerce. It is politically more decentralized. It is militarized. And its culture is the preserve of a relatively small group of clergy. A lot of the same things happen in the Byzantine Empire in the seventh and eighth centuries. This is a kind of dark age for Byzantium. We don't have a lot of written sources, for example. There seems to be an almost complete desertion of the ancient cities, with the exception of Constantinople itself and one or two others, the same kind of ruralization of society. It is a militarized society, although as we'll see, with more central governmental role. But it is a society built around warfare. And it is a society in which culture is also somewhat restricted. The libraries don't get destroyed exactly, but they certainly don't get a whole lot of use, at least in terms of things that we can understand or follow. And the Byzantine Empire faced an awful lot of enemies. The history of Byzantium is to some extent the history of emperors you've never heard of, writers you've never heard of, a lot of really neat art, heresies that you may have never heard of before but you have to learn anyway, and invaders, some of whom you've heard of, some of whom not. But heresies and invaders are really the story. I didn't make this history, but it is very important. And actually, I think heresies are kind of neat, although who am I to say since I went into this for a living? So I must've liked them. The most dramatic enemy of Byzantium in this period is Caliphate, the Arab empire of Islam. And so in terms of the shocks and crises of this period of contraction after Justinian, the rise of Islam, and in particular the two dramatic sieges of Constantinople in 674, where a naval battle finally defeated the Arab forces, and the siege of 717. So 674, 717, two sieges of Constantinople by the Caliphate. So what was the problem of imperial overreach-- just to take this back to its origins? Justinian in particular, as we said, was focused on the conquest of the Western Roman Empire, which he saw, of course, not as a conquest but as a reconquest, a restoration of territories taken unjustly by barbarian rulers, which he was determined to take back. He already had one enemy, however, and that was Persia on the eastern front. This is nothing new. We started the course with the Rhine and Danube frontier as one front and the Persian frontier in the East as another. The only difference is now the Rhine and Danube frontier has been breached. Justinian pacified the Persians, paid them off, made a treaty with them in order to have a free hand to undertake the conquests in the West. In retrospect, with the historian's ability to quarterback after the game, the lack of attention to Persia was a mistake. The Persians in fact invaded despite this peace treaty in the 540s. The Italian campaign didn't go well. Things were patched up. Justinian died in peace. But his empire was quite fragile. In addition to everything else, there's a plague in 541-542 that's kills a very large proportion of the population. And as I said, after Justinian's death in 568, the Lombards invade Italy. And beginning just a few years later, in the 570s, we start to have another force that's a little more difficult to describe. Just in the manner of the convenient term "Lombards" we don't really know what "Lombards" means. We don't really know who these people are. I mean, we know a bit about them. But it's a pseudo-ethnic name. We don't really know how people identified themselves. But it becomes even more confusing with the groups that invaded the Balkans. The conventional way of describing these is that the leaders were a group called the "Avars" and their subordinates, their sort of cast of thousands, their slave armies, whatever you want to call them, were "Slavs". The Avars are a Turkic or Mongol people. The Slavs are Indo-European Slavs. This is to some extent how the Slavs get into the Balkans. How this worked, relations between the Slavs and the Avars, is not completely clear. But we certainly know that they took over much of the Balkans, including Greece. Greece in this case has to be considered part of the Balkans. So the southeastern corner of Europe, modern Greece, Bulgaria, European Turkey, Albania, Croatia, Serbia, Kosovo, Bosnia, Herzegovina, all of these countries are a heartland of the Byzantine Empire in its post-Justinianic form, but now invaded by Avars and Slavs beginning in 570. The emperor Maurice was murdered by his troops in 602, campaigning in the Balkans against the Avars and the Slavs. He was succeeded by a cruel but hapless general named Phocas, whose disastrous rule as emperor involved not only the uprising or invasion of Avars and Slavs in the Balkans, but a Persian invasion in the East. Phocas was overthrown in 610. The Persians, Avars, and Slavs allied against Constantinople. And in 626, the city was besieged from both sides. The Persians camped out across the Bosphorus in the end of Asiatic Turkey. And the Avars and the Slavs camped out outside the great Theodosian Walls of the city that were on its land side, its western side. The fact that the city managed to overcome this was attributed to various miracles of saints, and particularly the protection of the Virgin Mary. And it's very important in terms of the later history of iconoclasm to point out that they had icons and all sorts of wonder-working portrayals of powerful saints on the walls in order to protect the city. And it worked-- or at least certainly it looked like it worked. The invasions were foiled. How did the Byzantine Empire fight? How did manage to withstand these invasions? Very briefly, one factor is the strategic position of Constantinople, a city extremely difficult to take by force. Indeed, it would not be taken by a flat-out assault until 1204, when Western crusaders, an alliance of the Venetians and the Franks, as they're called-- basically various Western European groups-- took Constantinople. How they took Constantinople; why they didn't go to Jerusalem, which was the original plan; how they took a Christian city instead of the Muslim-held city of Jerusalem is a topic for the next semester. But that is the first time that the walls were breached by force. There are very successful sneak attacks. So for example, in our period, the emperor Justinian II had been deposed and had his nose and tongue slit. This form of mutilation was to prevent him from coming back again. It was considered more humane to mutilate someone than to kill them, but the mutilation was considered to be disabling. You can't have an emperor whose nose is basically gone. But they were wrong about him. In fact, he came back with a force of people from-- if you look at that third map-- Chersonesos, the Crimean peninsula that's sticking out into the Black Sea. They besieged the city, and he and some followers snuck in at a point where one of the aqueducts met the wall. There was a little sort of space to sneak in. And they managed to, by a surprise attack and by acclamation of the noseless Emperor Justinian, win the city for themselves. So this is an example of a successful siege, but not a flat-out open assault. Justinian had a reign of terror for another six or seven years after 705, the date of the siege. He was a very angry man. And he got his revenge on as many people as possible, including people who had nothing to do with his mutilation. And he was eventually killed. An awful lot of these emperors died violent deaths. They're they're tough guys, but it's a fairly tough time. So Persian/Avar/Slav siege of 626, the seeming victory of Heraclius over these forces, and then its virtual undoing by the Islamic conquests of much of the Byzantine Empire. Remember, the conquest of Syria in the 630s, the capitulation of Alexandria in 642? We looked at all of these from the Muslim side very recently. Meanwhile, while Byzantine was busy losing much of its empire in the East, the Slavs were permanently settling in the Balkans and in Greece. And they were now joined by another invading people, the Bulgars. The Bulgars-- that is, the people who live in Bulgaria now-- are a Slavic people. But the original Bulgars were a more Turkic people who formed a kind of what was called khanate-- that is, a state ruled by a military leader, a khan. But here again, who are the Bulgars ethnically? How many of them have Slavic followers? It's both very controversial-- that is, the modern Bulgarians hate the notion of being thought of as ancestrally Turks, so what I'm saying is heresy in some quarters. But these ethnic designations are not as meaningful, or at least as stable, as we often tend to think they are. The Bulgars would intermarry with the Slavs. And they would eventually convert to Orthodoxy. But they would be a problem for Byzantium from about 700 until just after 1000. So how was this empire rebuilt? It was rebuilt on the basis of control over the Balkans and Anatolia. It was also rebuilt on the basis of reorganization of the army. Obviously, the position of Constantinople and its walls is not enough to assure victory in battle once you get away from Constantinople. There's certain kinds of weapons that they had. There's this mysterious one that I mentioned before called "Greek fire", which is certainly part of the victory of 674, the naval battle against the Caliphs. It's a some sort of burning substance that explodes on impact and is useful to burn ships. But the army is reorganized. And this is one of the key aspects of this period of contraction and of reconstruction, of crisis and of recovery. And we'll come to that in a moment. During this era of crisis, it should not be thought that they didn't have time for religious controversies. In the seventh century, the big controversies continued to be those related to the natures of Christ. Even though the territories that were Monophysite had been taken by Islam, or at least the most Monophsite territories, there still are controversies trying to compromise this question of does Christ have one or two natures, how, if he has two natures, are these two related? Rather than saying he has two natures, the emperors proposed two possible solutions. Christ is both God and human, but he's got one energy, Mono-energism. Well maybe he's both God and man, but he's got one will, Monothelitism. The emperors in the seventh century tried to impose Monothelitism, or tried to impose Monotheletism and then prohibit discussion, or just prohibit discussion about it. The Papacy was adamant in upholding the two natures, one person. And this eventually won. But a huge amount of energy was expended in this controversy. And in the eighth and early ninth centuries, we then have Iconoclasm, which as we've spoken about, is the prohibition on the worship of icons. Icons, again, are a form of religious art that remains characteristic of the Eastern Church. And the difference between icons and other forms of religious art is that they're portable. You can carry them around. And they are non-narrative. They don't tell a story. So an icon doesn't have a depiction of the Annunciation or a depiction of events. They don't come in series, showing, for example, the life of a saint or the life of the Virgin. They are a single depiction. The iconoclasts were worried about this as a form of idolatry. We're not sure where this concern came from. There is no evidence that it comes directly as a reaction to Islamic or Jewish criticism, although of course Islam and Judaism patrol the frontier against idol worship or image worship much more severely than does Christianity. It does seem to be a reaction to the crisis. It does seem to be part of an effort to remake Byzantium into what I guess now would be called a leaner, more adroit, responsive, or that obnoxious business word, now, "robust", a robust response to the crisis imposed by Islam. Certainly it has something to do with that. But what it really shows us is the role of the emperor in religion. The role of the secular ruler in the West is not as extensive, even as we will see soon under Charlemagne. Remember Chlothar and his attempts to dictate religious doctrine to Gregory of Tours, who, scared though he is of Chlothar, basically laughs at him? That shows a ruler who does not really have control, at least over the doctrine of his church, even if he's got a lot of control over the wealth of the Church. The Western tradition would tend to separate out ruler of the state and ruler of doctrine. The East less so. This is partly just the way the Patriarch of Constantinople is situated. He's right next to the palace of the emperor, and so can be intimidated by the emperor. It is in the tradition of the Orthodox Church that there is less resistance to state authority on the part of the Church, more collaboration with state authority, and less controversy. This is the way it's normally taught, is that the emperor functions kind of like the Pope in the West. The emperor defines doctrine. But actually, however much deference is paid to the emperor, and however much the emperor tries to define doctrine, the real story, certainly for our period, is of the lack of success of the emperors. The emperors constantly come down on the wrong side or come up with compromises or doctrines that don't work. They propose what in a sense seems like a rational compromise over Monophysitism, but theology doesn't work that way. Just because you come up with a compromise doesn't mean everybody's going to accept it. In fact, often compromises mean that nobody accepts it. It's not like a negotiation over political trade-offs. This is a theological truth. So the emperors then create their own theological position. Iconoclasm is very much an imperial demand. And for a time, they succeed in imposing it. They succeed in imposing it for 120 years, off and on. Sometimes there will be an emperor who is a moderate iconoclast, but lets the icons come back. Sometimes there's an emperor's who's a moderate iconodule or iconophile. But the emperors, even in the East, even where the emperor is so heavily involved in religion, have limitations on their ability to define doctrine. Nevertheless, clearly, they are the leaders of a besieged people who sees itself both as Christians against pagans or Muslims, or at least infidels, and as a religious people and as a secular population. So with all of these crises, sieges, invaders, how did the Empire survive? We can see some of the plans of the emperors. One plan was simply to get out of Constantinople and rebuild a kind of Western empire based in the West. So Constans II-- you'll remember I said in 661-- moved his capital to Syracuse in Sicily. This was intended in part to guard the possessions in Italy and in part to hold onto North Africa. Constans was murdered in his bath in 668. The capitol was moved back to Constantinople, and within thirty years, all of North Africa had been lost to the Arabs. And you start to get the shape of the Empire to resemble that of the map that we looked at last, that of 1000, a Balkan/Anatolian empire. Constans is very important, though, even though his plans come to naught. Because it seems to be under his reign-- not exclusively, but very much forwarded by him-- that two things develop that are crucial for Byzantine success. One is the practice of deporting whole peoples and settling them on new lands. This extremely brutal practice-- you can imagine what it's like at any time. Of course, this is practiced a considerable amount, for example, in the former Soviet Union, where you just tell people to get up and you're going to move them 2,000 miles away. They're going to have better opportunities there or whatever. But in the meantime, some of them are going to starve. Many of them are going to fall to disease. This is a brutal kind of transporting of peoples, manipulation of peoples from one area to another. The peoples manipulated in this case were often Slavic prisoners or groups taken over as the Empire expanded, and often sent to Anatolia to resettle lands that had been deserted in the Arab invasions and now seized back from the Arabs. This policy, against what one might expect, actually seems to work, to the extent that it does increase the population of these key frontier regions in the East with militarized forces transported from the Balkans. Related to this is the organization of the army by locality instead of as a mobile and very large force. And this is related to the problems of paying the army. As far back as the beginning of the course, we said that a lot of imperial policy, just as it's true of the state today, is dictated by the need to pay for troops. The most expensive thing that most states, including the United States, does is maintain an army and use it. For a number of reasons, this is just a very, very expensive thing, and clearly a necessary thing, something that is not easy to dispense with. Diocletian's reforms indeed, the whole restructuring of the Empire in the late third century and early fourth century was oriented around increasing the size of the army-- basically, doubling it-- and figuring out a tax regime to pay for it. This no longer works in the post-Justinianic world, in the crucial era here and beyond. Because the state does not have enough money to pay these. It does not have a tax base. It cannot raise the money for a large army to operate in the Balkans and to oppose the Arabs without the tax revenues from its richest province, Egypt, and from other very wealthy and important provinces such as Syria. How, then is, it going to have an army at all? To some extent, it is going to do this by creating local armies paid for not so much money as by kind-- that is to say, grain, leather, weapons, and things manufactured locally, harvested locally, and that stay locally. This is the so-called theme system. And Wickham doesn't talk too much about it because historians are in the process of drawing back from this. This is one of those frustrating things about progress in history. A lot of progress in history is the dismantling of convenient kinds of formulae. So when I started teaching this, the themes were everything. I would have spent sixty percent of this lecture talking about the theme situation and how it saved the Byzantine Empire. The themes look like soldiers who are peasants. In other words, to some extent the deal that's offered to these troops is, we will give you land, and we will allow you to keep most of what you produce, rather than paying it as tribute to a landlord. But you've got to be willing to fight for it. You've got to owe military service. You've got to be ready for military service. And indeed, the land was often in places that were dangerous, places that were liable to invasion. This is to some extent true, but it's a little less of a kind of Homestead Act deal. If you look at the map number three, the year 1000-- see the things that are in italic capitals like Opsikion, Armeniakon, Anatolikon? These are themes. They are large agglomerations of provinces. They are what might be called "military provinces". And as such, they combine civilian and military rule. The Empire is now divided into military provinces. Nothing terribly new about that, even though the shape and the size is different. What's new is that, rather than the revenue being raised in cash from taxes paid all over the Empire, and then transported to Constantinople, and then disbursed to mercenaries or to standing armies, the money that's raised in the Opsikion tends to stay in the Opsikion. These provinces tend to be responsible for their soldiers and for paying their soldiers. The state is very heavily involved. This is not a militia in the sense of a bunch of trusty guys getting their muskets or whatever they used off the wall. This is not Paul Revere, or something like that. But it is an army that is closer to the population and in which the male population is largely involved in the military forces. Another word that I don't particularly care for in its modern use" "stakeholder". They're stakeholders in the sense that it makes a big difference to them. They're not just civilians in this case. This is, then, a militarization of society. It is a new way of paying for things in an economy that is less productive of revenues, at least of taxable revenues. And it is, above all, whatever its exact nature, an innovation that works. The basis for this expansion that takes place after 843, and indeed for the stabilization of the frontiers that proceeds it is a reorganization of the army, the theme system, a reorganization of revenues, and a kind of strategic plan to define the Byzantine Empire in a way that is ultimately feasible and defensible. From 717 to 843, the Empire recovers slowly. We choose 843 because that's the end of the iconoclast controversy. The Empire continued to have to fight on both fronts against the Arabs and the Bulgars. But it had a viable state, a viable military structure, and some capable emperors. After 843, we see a real rebirth and expansion. From 843 to 1000, the Empire grew by about one third. Even though that map, the third one, for the year 1000, doesn't seem very imposing, if you take a look at it, you'll see that its frontiers with Islam are much more secure than they have been before. The Empire extends eastward as far as Armenia, for a long time a frontier province, and it is on the borders of Syria. It has the great city of Antioch again. It has all of Anatolia. It has the islands of Cyprus and Crete, which had been occupied by Islamic forces for years. It has control over all of modern Greece and has held onto some possessions in Italy. This is an era of great splendor, ceremony, the restoration of education, and of learning. The Bulgars will be defeated definitively by 1019. Constantinople would be besieged by them twice. You can't have a century without a couple of sieges of Constantinople. The Abbasids are defeated, and the expansion of the Byzantine Empire, as you see, as far south as modern Lebanon and as far east as modern Armenia-- well, historic Armenia. This is a society that is still involved in religious controversy. This is-- in the ninth century with the Carolingians, there would be a controversy over the holy ghost in relation to the son and the father, the so-called "filioque controversy". But it is a world of great energy: artistic, cultural, and actually religious. Perhaps the most lasting and significant accomplishment of this era is the conversion of the Slavs and the conversion of much of Eastern Europe-- conversion to what would become the Orthodox as opposed to Catholic form of Christianity. The official split between the Orthodox world and the Catholic world won't occur until 1054. In 1054, the Patriarch of Constantinople excommunicated the Pope and the Pope excommunicated the Patriarch of Constantinople. I think they've taken it back, but obviously the churches are not joined. They have differences already, however, that are visible in the ninth and tenth centuries. Orthodoxy is less politically centralized. If, as we just said, it's very dependent on the ruler, if the ruler is different, then the Church has a sort of national identity. Thus, there is no Pope in the Orthodox Church. The Bulgarian Orthodox, Greek Orthodox, Russian Orthodox essentially form a confederation under their own patriarchs. They also have their own languages. The Bible could be translated into vernacular, and the liturgy was not in one language necessarily. So Slavonic, the ancestor of modern Slavic languages, would be instituted as the liturgy in many of these converted places. Priests are allowed to marry in the Orthodox Church. There were just, sort of, questions of style. For example, the Orthodox Church, even when it has admitted icons back, does not have statues, does not have three-dimensional statues, whereas the Catholic world, of course, does. Think of Notre Dame of Paris and its sculptural program or Chartres or the Michelangelo and Bernini sacred sculptures. This is partly because, while icons are OK because they're two-dimensional, statues were thought of as being too much like the idols denounced again and again by the Old Testament. Greek Orthodox priests have beards; Catholic priests don't. They're different styles of worship. In Orthodox churches, you stand; there aren't seats. Some of this is style. Some of this is theology. Some of this is just culture. But of course, this would be in the ninth century and remain a very strong difference. So you can see that those countries converted by the Catholics have a Latin liturgy and a Western or Latin or Roman orientation. Poland is Catholic. Russia, on the other hand, would be converted to Byzantine Christianity. The division is clear and unfortunately tragic in the former Yugoslavia. Serbian and Croatian are almost the same language. Serbian is written in Cyrillic, Croatian in Roman. The Croatians are Catholics; the Serbians are Orthodox. Their hatred for each other, and then of course the complication of having an Islamic people, the Bosnians, was the background to the tragic Yugoslav Civil War of the early 1990s. So these religious boundaries, which sort of correspond to ethnic boundaries-- even though those ethnic definitions are themselves invented, to some extent-- these religious boundaries continue to be very meaningful and to define culture in Europe. The conversion of this world is, then, one of the most important events of the renascent Byzantine Empire. The coronation of the king of Russia, the king of Kievan Russia, in 989 at Cherson on the Crimean Peninsula, presided over by emissaries of the Byzantine Emperor is one symbol of this. The conversion of the Bulgars. The conversion of many of the peoples of the Balkans. And indeed, there is a kind of aftermath of Byzantine civilization. The closest heir to the Eastern Roman Empire is the Russian Empire. The look of Orthodox worship, the look of the Russian churches, the icons, the gold, the imperial style is very closely related, self-consciously, to the model established by the Byzantine emperors. Indeed, in the sixteenth century, after the fall of Constantinople, Russian monks claimed that there had been three Romes. The Rome of the West, the original one, had fallen in the fifth century; the great Eastern Rome, that of Constantinople, had fallen in the fifteenth; and the third Rome was Moscow. I don't think that outside of the Russian orbit most people think of Moscow as the third Rome for a number of reasons, whatever its power and its own form of splendor. But the degree to which Russia in its history is the heir to the world that we have briefly described cannot be denied, even if to some extent it is a self-formed or self-conscious manifestation. Moving from Byzantium, on Wednesday, we will come back to friendly old Francia and discuss the rise and efflorescence of the Carolingians. Thanks. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 13_Monasticism.txt | PAUL FREEDMAN: We're going to talk about monasticism today. And monasticism in the popular imagination, and accurately, is linked to learning. We all have this image of monks quietly copying manuscripts, and those manuscripts being how the learning of the ancient world was transmitted. We're going to talk a little more about that when we come to near the end of the course on intellectuals in the court of Charlemagne. But monasticism and learning are linked in our mind, but they are not intrinsically linked. There is no logical reason why monks should copy manuscripts. They should pray. They should live in some kind of renunciation of the world. Generally speaking in the medieval West, they live in communities. Generally speaking, they are engaged in a kind of corporate rather than individual prayer. All of these things follow from the way monasticism was conceived. And the major text, though not the only one, but the most influential text about how monasticism is conceived, was the sixth century Rule of Saint Benedict. The Rule of Saint Benedict has some possible references to sacred reading as it calls it, or to some kind of program of knowledge. It assumes that the monks are literate, for example, a lot to assume at that time. But nowhere does Benedict say, "Please preserve the classical tradition by being scribes and writing and the Scriptorium." So how does this come about, is one of the problems that we will deal with. But what we're really interested is monasticism without the learning. And the reason we're interested in that is not only is this the prevailing spiritual movement of the early Middle Ages, but it has a tremendous influence on society outside the monastic walls. Because central to our discussion is a paradox. The paradox is that while the monks are trying to escape the world, the world is following them. The world is very interested in their prayers, because their prayers are thought to have a powerful real-world, this-world, effect. So as the monks become more distant from society, God hears their prayers with more and more sympathy. Therefore their prayers have a kind of power, a power to benefit others. This notion of power is like some kind of almost electrical utility. They're building up an incredible amount of electricity, if you want to call it that, or let's say spiritual energy to be more accurate. Way more than they need; way more than they can consume. They're like some little Persian Gulf state that is producing ten percent of the world's oil. There's no way they can use all of that. In this case then, how does the surplus get distributed? It gets distributed through the generosity of people outside the [correction: monastic] world worried about the condition of their souls. The notion that I, possessor of spiritual reserves and spiritual power, can pray for you, sinful knight, sinful king, sinful merchant, is called intercession. The notion that I can intercede for you-- and we've already seen this, haven't we? We've seen this with the saints in Gregory of Tours and in other texts. We've tried to emphasize how important the saints are, not just for our understanding of medieval religion, but for our understanding of medieval society. Remember that I tried to emphasize that one of the problems, once you're done with the Roman Empire, is how society is held together. In the Roman Empire, it's pretty clear. It's held together by institutions that, although not the same as our own, are translatable to our own: law, administrative structure, land holding, the whole panoply of what passes for civilized life. But in the early Middle Ages, after the collapse of the Roman Empire, we've seen the society does not have as much literacy, does not have very good records. The kings are thugs. The political order is very unstable. There's an awful lot of warfare. There's a lot of disorder. You can't just dial 9-1-1 and expect a response. So the question then becomes what holds that society together? And we mentioned some things, including the Church. And here we're looking at a particular instance of how that works. Because the monks, far from being kind of out there in the forest or desert or some remote region, or even if they are in the forest or desert and some remote region, are extremely important to how society functions. Because this is a society in which the spiritual, the military, the political, the economic, are not easily conceptually separated. This is the Middle ages, OK? You love the Middle Ages, otherwise we wouldn't be here together. But this is the part of the Middle Ages that is perhaps most medieval. What could be more medieval than monasticism? When I started teaching, which wasn't that long after the Middle Ages, but when I started teaching, monasticism was a real problem, because it was so alien. What are these people doing? In a way the situation is better, because monasticism is sort of chic, at least temporary weekend monasticism. People think of monasteries sort of like spas. You go there to get cleansed. In fact a lot of monasteries have taken on a lot of new business with retreats, and detoxification, and pilgrimage, and these kinds of concepts. But at the same time, while I think we understand the desire to renounce the world or to take time off from the world or to leave the personal digital devices at home and think about something for more than three seconds at a time, all of this is temporary. The whole point of detoxification is that you then go and re-toxify yourself or go back to normal life. What we have to understand here are people who have decided to embrace a world-renouncing way of life for good. Now monks therefore, are clergy. They are professional members of the Church. But they're not priests. It's key that you understand the difference. Priests interact with the laity. Layman, laity, are people who are ordinary people, believers but not clergy. The priests interact with the laity through mass, the performance of the sacraments, things like baptism, a little bit later than this period, confession, anointment of the sick. These are things in which the sacred is conveyed from the spiritual world to laypeople via priests. Priests are in that sense the intermediaries between the divine and the material. But because priests are involved in the world, there are certain aspects of their differentiation from the world, and there's a lot of debate in the Church at various times over whether that includes celibacy, not getting married. In the era that we're dealing with, there are married priests, or there are priests who are more or less married. And then there are priests who are celibate. But monks are not supposed to interact with the world. They are leading a life of contemplation and self-denial. True, they cannot focus only on their own salvation. Because that would mean ignoring the Christian's duty to others. One of the problems about being a contemplative in the Christian tradition is as soon as you say something like, "Boy, I am really contemplating great today." Or, "Wow, I am really seeing the mysteries of the universe." Or, "I can't believe I haven't had anything to eat or drink for three days and am feeling great." You are falling off. You're falling away. You're selfish. You're taking pride in your own accomplishments. So from the beginning is the notion that the monks have to abandon everything, including self-satisfaction. In fact even most importantly self-satisfaction. And this is where Benedict's notion of humility comes from, which is very strong in The Rule of Saint Benedict. Where does this desire to rid oneself of the world come from? It's very strong in Christianity. It's all over the New Testament. That is actually part of the Christian message. It's awkward, because most people don't follow that, including most people who are believers. They don't in fact give away everything they have to the poor and follow Christ. They don't renounce the pleasures of life, of the flesh, and so forth. But that is sort of what they are telling you to do. The first monk, the first guy who we know of to decide to run away to the desert and lead a life of contemplation is Saint Anthony of Egypt.-- Well Anthony, you know how to spell that.-- He lived from 251 to 356. I have trouble believing this. The sources are pretty good. I have trouble believing that anybody could live to be 105 in the Roman Empire. Or indeed, at any time before ten years ago or so. But there it is. At least we know that in 270, he heard the saying of Jesus in a Church: "Go sell all you have. Give to the poor and follow me." And he followed this literally. He established himself as a solitary hermit in Egypt. And Egypt is a great place for monasticism, because Egypt has a very narrow strip of incredibly fertile land on either side of the Nile. The Nile, which until the building of the Aswan Dam, flooded every year. And its silt, that it brought down from its sources, was so rich that it created this marvelous soil on which all sorts of things could grow. But once you got beyond that limit, you were in the desert. So you have absolute and total really Sahara-like desert, very close to fertile land, the best land of the Mediterranean. So that you could have a kind of interaction between-- I mean, it's not like you decide that you want to be a hermit and you live in Manhattan. And you drive and you drive and you drive, and you get to Long Island, and you're in the suburbs, and then you're in the ex-urbs. And then you're in sort of gas stations and strip malls. And then you're in the Hamptons. It's very, very hard-- I mean you can, actually, if you go north, pretty soon you'll get if not hermit country, at least a decent isolation. Of course the problem with New York is that it's really cold. It's great to be a hermit in September. But the Adirondack hermitages present problems in the winter. Egypt is hot, all right, but certainly no exposure to cold problem. Anyway, the first monks are in Egypt. And they are known as the Desert Fathers. Often, they live alone as hermits, but sometimes they live in communities. They have their own cells in these communities, but they can come together for prayer or for some sort of spiritual companionship. What's interesting is that right from the start, these hermits or first monks, appealed to the people who had no plans to become monks. They appealed to the people of the cities of Alexandria or Thebes, etc. of Egypt. The reason is, and the reason why people would become monks, is the establishment of the Church as the official church of the Roman Empire. There are no monks in the first years of Christianity, because just being a Christian means denying the world. The threat of death at the hands of the Roman authorities and the illegality of the religion means that you are already in a world-renouncing position. But once the church becomes established, once all sorts of people start joining it for motives that have nothing to do with spiritual reasons, or maybe ten percent spiritual, ninety percent it's time, my career, I want my kids to grow up in the sort of right faith. Then those of real spiritual bent, devotion, desire, have to present themselves as more than merely attending church as serious about Christianity in some sense. So monasticism has to be understood in its earliest years as a reaction against the compromises and comfort of official Christianity. And we see this in Augustine's Confessions. Where, you will recall, the emotional crisis that finally tipped Augustine over the edge into his conversion experience was hearing about the monks of Egypt from someone who had come to Milan and had been in Egypt, and described these men and actually women who were not well-educated were certainly not trained in rhetoric, law, and the classics to the extent of Augustine. Yet nevertheless, they had in Augustine's words, "stormed the gates of heaven." They had by their spiritual renunciation hence their spiritual power, become close to God in ways that he, Augustine, and his friends with all their knowledge, had not. And this contrast is what decides Augustine to embrace a way of life that although not monastic and much more active in the world, is a renunciation of the standard career, the standard definitions of success in the Roman Empire, and involvement in the world in things like marriage, property owning, etc. So in fact, if the monks are these not very well-educated people, if embracing their values means giving up Cicero and the classical tradition, it looks as if monasticism is an anti-intellectual movement. We've come back to this paradox of the monks as custodians of learning. It's not quite the fox as the protector of the hens, but it is not automatic that the monks would find themselves in the position of copying down Cicero. So the first forms of monasticism are those of the Egyptian desert. There's a kind of tension from the start between ascetic individualism and collective monasticism. Ascetic individualism means one person engaging in practices that dramatize the renunciation of the world. A classic example is not from Egypt but from Syria. These are the saints who sit on top of pillars for decades at a time. Individual, right? They're alone on this pillar, maybe it's this wide, or maybe this wide at the top, thirty feet high, fifty feet high. These saints collectively are known as Stylites. A "stylite" is a pillar. And the most famous of them, Saint Simeon of the Desert, or Saint Simeon Stylites, lived on top of this pillar for what is it, thirty-five years? Something like that. Thirty-five years up there, in the rain, in the sun. Imagine the sanitary arrangements. Imagine this. I mean this is world renunciation. But he was not alone. He was alone on top of the pillar. He was a hermit. But people came and visited him. There are pictures of ladders going up to him, people climbing these ladders, sometimes delivering a little message: "Would you pray for my child who is dying of"-- well I don't know what he's dying of. Asking for things in the world: "Please deliver me from a bankruptcy;" "Please deliver me from illness;" "Please protect me on this voyage." Why don't they ask God themselves? Save a trip to the desert out of Antioch? Save perhaps a donation to the Saint Simeon Stylites Foundation? Because God is going to listen to Simeon, right? Why? STUDENT: Because he's worthy to speak to him. PROFESSOR: And why is he worthy? What has he done to make himself worthier? You're quite right. He's renounced the world. STUDENT: And he knows Latin. PROFESSOR: Yeah. He's not unbelievably cultivated, probably Syriac in this case. Probably those notes are in Syriac. But his world renunciation has imbued him with power. This is not unique to Christianity, right? Shamanism, the idea of a person who is an outcast in society, or who has renounced the normal comforts of society. What's the problem with just getting married, having kids, having a job? Wondering what's for dinner? From the Christian point of view. That's what most practitioners are doing. STUDENT: You're focusing on the world and not on the-- PROFESSOR: You have to focus on the world. Supposing you wake up one morning and say, "I'm going to be a much more spiritual person." And then there's wailing from your kids, and your spouse is nagging you about fixing the dripping faucet. I mean this is stuff you've got to look forward to, most of you. I don't expect you to go into monasteries. But this is a distraction from what the New Testament tells you to do. The New Testament doesn't say, "Go ahead. Be happy. Amass property. Get a great job. Make a lot of contacts. Get ahead Have a bunch of children. Get them into good schools. Get them coaches for the SAT's." And so forth and so on. If you have done that, because you just got trapped-- well, one thing led to another and here you are. If insofar as you have spiritual anxieties and desires, then you're going to want to have a patron. You're going to want to have someone who can intervene with you, just like way back when you had somebody write a letter recommendation because they were on the board of some company, or they had some influence. Here your patron is a spiritual patron, and it is monks or hermits. So the role of the holy man in society is of somebody who has a heroic ability denied to most ordinary people. This is a spiritual superhero who, like comic book superheroes, isn't just a superhero for his own benefit. He doesn't just fly around because he likes the sensation of flying around. But who helps those who are weaker than him. And the intervention of somebody-- a very important Saint like Simeon Stylites, transcended the merely curative. Simeon was without fear of the emperor, for example. What could the emperor do to him that was worse than living at the top of a pillar, after all? The emperor Constantius, one of the sons of Constantine in the early fourth century, was going to punish the city of Antioch for defying the tax collectors. Tax had been collected in Antioch. There'd been a riot. Lots of people were killed. Normally, you'd expect the wrath of the emperor to come down on the city and punish it very severely. Simeon was able to intervene with Constantius to prevent this from happening. Constantius listened to him because he was a little scared of him. People are a little scared of those who are not only not playing the game, but who are playing a different game according to unique and very difficult to imitate rules. So there's a paradox here. Again, as the withdrawal from society becomes more dramatic, the imputed spiritual power becomes greater. If Simeon's renunciation were limited to something kind of small. Suppose he became a vegan. Would people believe in his power? And then this brings up the question of, then, what is the power of The Rule of Saint Benedict? If you read The Rule of Saint Benedict, there's nothing in there about living outdoors all the time at the top of a pillar. There's nothing about extreme asceticism. Yet Benedictine monasticism would prove more durable than pillar sitting. It would prove more durable than the desert saints even. Monasticism was brought to the West in several forms. But generally speaking, although there are lots of hermits, we hear more about the communities of monks. Monks who live in an establishment. And beginning in the late sixth, and particularly in the seventh century, a lot of monasteries were established in rural parts of Europe, rural or small town parts of Europe. Places like St. Albans in England, which is not far from London. Or Bede's monastery of Jarrow in Northumbria, more remote. Or Fulda in Germany. Reichenau, a monastery in southwestern Germany on a little island in Lake Constance. These monasteries owned property, and indeed many of them became very rich. Because one way of affiliating yourself with a monastery was to give to them, to donate land, money, serfs, coins, booty, whatever. And the reason people donated is because of the violence of their lives. The people who had the stuff to donate usually had gotten it by violence. Because it's a society that, as you've seen in the pages of Gregory of Tours particularly but not exclusively, it is a society organized around warfare. All of these guys in Gregory of Tours have blood on their hands. You cannot be successful without a certain amount of the infliction of pain on other people. And although the rich and wealthy in any society do not believe themselves to be rich and wealthy for vicious reasons, indeed the rich and wealthy generally speaking think they're great. In this society, the rich and wealthy think they're great all right, but they're also very anxious. It's not just a question of, what is the phrase that's often used by donors to universities, "give back to" Yale, for example, or give back to society. It's not just a question of giving back to society: "Yes, I made $40 billion, so I feel I need to give back something." It's a question of, "I'm going to hell. What can I do? I know I'm going to hell, because I killed 80 people in the course of just business deals. The closing of this deal required that 10 people be killed." Or 500, or 2,000. So there is a symbiosis, to be cynical about it. And as you know, I'm not really a very cynical person. I hope that has come through. I hope it's come through that I'm a really idealistic, even naive. But if you'll forgive a moment of cynicism, there is a symbiosis between the monks, who are amassing this huge quantity of spiritual energy, and the leaders of society, who are amassing this huge quantity of sins. It's a natural trade agreement. So there are paradoxical consequences of monastic wealth. As these places get richer and richer, they become too important to the kings, the leaders of society, just to be left to a bunch of weird, world-renouncing hermits. They start to be administered by people who themselves are from high families, of high lineage. In order to become a monk at Fulda or Reichenau, you can't just wander in and say, "I'm renouncing the world." You've got to be from a good family. You've got to come with an endowment. Generally speaking to get into Reichenau as a monk, the family's going to have to pay a huge amount of property, money, some form of wealth that endows that monk. So at some point maybe people will stop believing that Reichenau is such a great place. Maybe they'll stop believing in its spiritual energy. This would happen later. But not yet. Not yet. The preservation of the monasteries and their growth and success is due to the rule of Saint Benedict. The most important development of Western monasticism-- and there are all kinds of other monasticisms in the Christian world. But the monasticism that would characterize Western Europe is the rule of Saint Benedict, late sixth century, who devised a set of regulations for communal monasticism. So The Rule of Saint Benedict is a manual for the monastic life, how to set up and run a monastery. It's not the first. It's based on an older rule. We don't have to go into sort of who is responsible for inventing Benedictine monasticism, as it's called. But the monasteries of the West would be for the most part Benedictine until the twelfth century, when you start to have other orders. That is, there would be thousands of monasteries throughout Europe. And they would follow more or less the rule that you have read. Why is this rule so successful? For one thing, it is moderate. It is reproducible on a large scale. It is ascetic, all right. It does involve giving up a lot of things, but not to an extreme. Not to be compared to the life of a desert hermit or a desert pillar-sitter. Or the harsh monasticism of the Irish tradition. You don't have to go to some little island that's one square kilometer large and out in the middle of the windy Irish sea, and build some little beehive hut and live there. Ascetic monasticism of the extreme sort is all over the place. Because hostile environments are all over the place. Now it's not as if Benedictine monasteries are all located in cheerful, hilly, fertile countryside. But a lot of them are. A lot of them are located in productive land. But more important than that, they require a certain kind of asceticism. The asceticism of a Benedictine monastery is renunciation of self in favor of the community. In that sense, it's almost an opposite of the kind of monasticism that hermits or pillar saints practice. What's ascetic about a Benedictine monastery is not only the celibacy part, or the prayer part, or the isolation from worldly stimulus part, but the fact that you're not alone most of the time. You have to subordinate your will to communal rituals and life. You have to subordinate your will to the abbot. Remember how much Benedict emphasizes obedience. This is not just a good management tool for making it clear who's in charge. It is a form of self-abnegation, a form of renunciation. Benedict also enjoins manual labor. This is a penitential tool. But it's also something that has perhaps something to do with the economic success of these foundations. These are monks who are engaged in primarily two activities, labor and prayer. Prayer, we'll talk about in a moment And is quite understandable. But labor is more interesting and innovative. Because the ancient world despised labor. The whole idea of how you should live in pre-conversion Augustine's opinion, which reflects that of late Roman society, is what was called leisure with dignity. Leisure with dignity is what most professors aspire to. That is to say, leisure but not just to take naps and play with your dog or surf various dubious sites on the internet. Leisure, to read, to think, to engage in a kind of genteel contemplation. And this is the ideal of those Roman senators who were writing philosophical dialogues. But work, actual work, is degrading, horrible in the ancient world. Not to be engaged in by anybody who could call himself a gentleman. So by making people work, including people who come from the upper classes, this is a penitential labor indeed, particularly labor with your hands. Though other kinds of labor were envisaged. In Section 48, Benedict talks about labor and sacred reading. And this is one of those places where you can draw out from its meaning that copying manuscripts is a form of labor. Reading texts is a form of labor. So the transformation, or at least the addition of learning as part of the mission statement of monasteries, is implicit here, but certainly not drawn out by Benedict. Prayer. One of the things that is involved in this surrendering of your will to the greater communal good is the performance of prayers. Benedict emphasizes humility. The monk's life as a ladder of humility. Humility is encouraged by obedience, silence. In Benedictine monasteries, you weren't silent all the time. But there is a discouragement against mere chatter. Labor, all of these are penitential activities in which the individual will is suppressed. And if you think of experiences in which the individual will is suppressed so that the person focuses on the community, we are all familiar, in fact you are more familiar than I am probably, more immediately, with such activities. Any kind of training or boot camp-like thing, or a senior society, or an a capella group, makes you do stuff with the group that may be unpleasant, difficult, self-sacrificing, but that reinforces the esprit de corps. This is the heart of the military. Military beats you up in order to make you focus on the group. And many other organizations, including businesses, this idea of going out into the mountains and turning you loose with a tent and some rope and seeing which groups are able to survive, or whatever they do. This is part of this kind of training. It's not just to train you into depending on other people. It's to train you into focusing on the success of the organization or group as opposed to your own aggrandizement. The difference again is, you come back from the little upward bound experience. Or you come back from the retreat. Or you come back from the scavenger hunt. In this case, this is your life. But in addition to obedience, labor, silence, prayer is the most important renunciatory activity. Because these are not just prayers like, "OK, I'm going to go in for five minutes and recite some prayers." These are prayers that go on all day with some breaks. One of psalms seems to suggest that you should make prayers seven times a day. So they begin a little bit after midnight. Then they go to sleep for a little while. And then they get up a little before dawn and they pray. And then they go back to sleep for a little while. And then they have a kind of early morning session, and so forth and so on, seven times a day. And these are prayers they go through the psalms primarily. Different monasteries have different liturgies. A liturgy is a kind of ritual cycle. Some monasteries have more prayer and less contemplation, or less prayer and more work. But they all are engaged in the performance of group prayers, not individual but the community. And these go on and on and on. And they take on a kind of rhythm or a monotony, or a kind of visionary power that's such experiences can convey. And it is these prayers that are what are really storing up those spiritual reserves I was talking about as characteristic of monasteries. There's this tremendous power that all these repeated prayers have that cannot be duplicated outside the walls of the monastery. So this heavy round of prayer involves a significant sacrifice of comfort and of the self. This is a kind of sleep deprivation. Monks never really quite make up the sleep deficiencies. They sort of stagger into Matins, as the first hours are called, or Lauds, the pre-sunrise hour. But this is very impressive to the outside world. The outside world, the donors, love this. They love the buildup of these prayers. And they would like the prayers to take place in a nice place. Rather than having the monastery church being some kind of dank or fire trap wooden structure, they'll build beautiful churches for them and beautiful dormitories for them as well. And beautiful refectories for them. If you're a donor, you would rather that the stuff that you're donating for take place in nice surroundings. Donors like it if the lawns at Yale are well-clipped. And I think I've said this before. The closest thing to a monastery is a college, with some obvious differences. But communal living? The quadrangle is like the cloister focused in on itself. The emphasis on the group. Identification with the institution. The notion that your activities will benefit society. Not so much your prayers, but all that great research we're allegedly doing. OK. The parallel does not carry perfectly. But it is no accident that this university looks a little bit like a monastery. It is to evoke a tradition of contemplation, isolation. People speak about the Yale bubble, but it's supposed to be a bit of a bubble. It was constructed that way. That is the ideal. Now you may not have chosen to go to the places that really reproduce the monastic ideal. I went to the University of California at Santa Cruz. It was like up on a hill in a redwood forest. That was sort of like a monastery. Or Earlham College in Indiana, or Marlboro College in Vermont. I know you rejected these places, I know you just laughed at them just thinking about going to them. But there are people who've decided to go-- no? Devastated that you didn't get into Earlham? Well, OK. But this is an American ideal that is similar to the monastic ideal. It is a sort of renunciatory ideal. And it is an ideal of learning. So how do we get to learning? In the last few minutes I just want to trace these connections. The Benedictine Rule, as you will have noticed, does not encourage learning. And Benedict himself did not regard this as a primary duty of monks. He did expect the monks to read the Bible. He expected them to listen to readings from the Church Fathers. And very key, he says that the monks should take out a book from what he calls the bibliotheca. "Bibliotheca" is the Latin word for library. It's not clear when monasteries had libraries. But really the idea of collecting books and copying them comes from late Roman culture, from a desire to understand the Bible and from a transformation of that world of cultivated leisure, where the intellectuals like Augustine and his mentors, were the custodians of learning, to a world in which the clergy and particularly the monastic clergy, were the custodians of learning. Because the monks had three key elements: They had learning, that is they were literate. They had time, even though the prayers consumed a lot, they're not in the world. And they had wealth. It's not inevitable that time, learning, and wealth should lead to a cultural efflorescence. But they are certainly favorable conditions. So I leave you with that implication of the rule. We will develop it further in a few weeks. What we're going to be talking about beginning on Monday of next week is Islam. So we are moving into a different post-Roman reality. Thanks. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 01_Course_Introduction_Romes_Greatness_and_First_Crises.txt | PAUL FREEDMAN: So welcome to History 210, The Early Middle Ages. I'm Paul Freedman. And behind this innocuous title "The Early Middle Ages"-- I think we're going to have to jazz it up a little. I think we're going to put an exclamation mark on it, at least. But behind this innocuous title, you will see, I hope, if you stay for this course, a strange course. Strange, not because it covers the particular period 250 to 1000, but because it starts out very recognizable, and gets stranger and stranger, and seems to dissolve into a kind of a hard to grasp world. Hard to grasp, but fun. I will talk about both the strangeness and the fun aspects in more detail. There are several great themes in this span of centuries: the fall of the Roman Empire; its survival in the East, as the Byzantine Empire; the so-called barbarian invasions and kingdoms, set up on the ruins of the Roman Empire; the triumph of Christianity, which went from being an outlawed minority religion to the established faith of the Roman Empire; and then survived the extinction of the Roman Empire. We have two Teaching Fellows: Lauren Mancia, sitting at this end, and Agnieszka Rec standing in that corner. So far, there are sections scheduled for Wednesday at 4 o'clock and Friday at 10:30. We'll probably have two other sections. We'll see how large the class is next week. Let me know if those two section times-- well, probably, the sections to be added would be on Thursday: Thursday afternoon, and Thursday evening. Let me know if you have some special problem in terms of the scheduling of the sections. As I said, when some of you were already here, I'll have some pauses, so that if you are shopping and want to look at another course, it'll be, if not easy, at least possible for you to get up and leave. So I'll have several pauses during the presentation. But, I should say, I do want to give a full class discussion today, or presentation today. We only have so many opportunities to discuss things. And I'd like to set the scene for you. And I think that will also help you decide about taking this course. Now, this course is part of the Yale Open Courses Program. And, as you probably know, there are lots of-- well, a select number, but a substantial number, of courses that are offered free to the public via the Internet. And this is one of them for the fall. And I take this opportunity to greet our Internet students and Internet friends. So since it's part of this project, a Yale University broadcast team will be recording all the classes. And they'll be as unobtrusive as possible. The classroom experience will be essentially as it would be if they're not there. And it's their intention to videotape me, and not you, so neither your faces nor voices are supposed to appear. Your questions are unlikely to be heard. I will repeat the questions, so that people watching this on the Internet will have an idea. And I do encourage questions, both things that you haven't understood or things for elucidation. I have a slightly more formal lecture style than some people, perhaps. I try to have a reasonably structured lecture that doesn't wander off too much. Some of you have taken courses from me and know I have certain themes, or preoccupations, or diversions. But I'm going to try to be as coherent as possible, partly because we are filming. So I hope that you're enthusiastic about the fact that we are participating in this Yale Open Courses initiative. And, having said that, now you should just think it away. The broadcast team is not very conspicuous. And the objective is for us to interact in the classroom as we normally would. And this is part of the unique experience of teaching and learning at Yale, so don't hesitate to ask me if you have any questions or concerns. The syllabus, you all have copies of the syllabus, I believe. And, of course, you'll have seen it on the server. The books are at Yale Bookstore, and they are all there. I hope there will be enough copies; if not, we will get more. The first assignment is, conveniently in this sense, from the course pack. The course pack is at TYCO, the photocopy place on Elm Street. If you don't know where that is let us know. And the assignments for Monday and Wednesday, those first two assignments are from Peter Brown's The World of Late Antiquity and A. H. M. Jones' Constantine and the Conversion of Western Europe. Questions so far? Right. OK. So the requirements that you see on the syllabus are a short paper that's due October the 10th. A mid-term, that will be held in class October 17th. And a long paper, which is due December 5th. That long paper is a research paper. And we'll be glad to help you choose a topic, offer you suggestions, help you get started on that. It's 15 to 20 pages, and it counts for 40% of your grade. The mid-term counts 30% of the grade; the short paper 20%, and your section grade is 10%. Now, this course does not have onerous requirements. But I expect you to do the requirements that we have. There's no final exam. I urge you to blot out of your mind the temptation not to do the reading because there's no final exam, or the reading of the second part of the course. And if we think that this is a problem, judging on the basis of how the sections go, we reserve the possibility of giving you quizzes in the section in the section [correction: second] half of the course. Paper times, I'm going to be firm on this. I can't say absolutely no extensions on the paper, because I acknowledge the existence of overwhelming emergencies. But let me give you an example of an excuse that's not going to be accepted: "I have three other papers due that week." OK? Plan in advance. We are at your disposal. If you want to plan your final paper tomorrow, hey, this afternoon, talk to me. I'm eager to hear from you about that. In the sections, we don't want you to bring laptops. And the reason for that is, not that we think you're going to be Facebooking or answering your email, because we know that you never do that. The laptops, in our experience, interfere with the purpose of the section, which is partly to talk to each other. And rather than focusing on the screen, and then, in a sense, being a series, of archipelago of little islands, rather than a section, in the sense of give and take and interchange. If you think that that imposes some kind of hardship on you, I think you'll find that it is pleasant. And if there's some technical hardship, let me know. So logistical questions? Questions about the organization in the course, or any other aspect of this? Good, that means that I'm clearer than on some occasions. So, if anybody wants to leave now, this is one opportunity. But, since it looks like I have your attention riveted, let me introduce the course in terms of its actual content. We are beginning by looking at the crisis of the Roman Empire. And then we will be looking at its peculiar legacy. In the year 1000, where we stop, we will still be dealing with The Inheritance of the Roman Empire, the title of Chris Wickham's book, one of the books that we're going to be using a lot. The legacy is peculiar because, while the memory of the Roman Empire remains intact throughout the period, and beyond-- I mean, to this day, the head of the Catholic Church is in Rome. Until 1960, the transactions of the papacy were in Latin: the services of the Church were in Latin, the Catholic Church were in Latin. And Latin remains the official language of the Catholic church in its administrative head. So the most faithful preserver of Rome and its legacy, historically, is the Catholic Church. And this is a paradox because the Church begins its career, and, indeed, its first 250 years, as illegal in the Roman empire. And, indeed, there are periodic persecutions where people were punished, including killed, because they were Christian. The most faithful preserver of Rome, however, after the fifth century collapse of the Empire in the West, is the so-called Byzantine Empire-- the Byzantine Empire with its headquarters in Constantinople. Despite the fact that it would abandon Latin for Greek in the sixth century and turn into a very different kind of political and cultural entity, the Byzantine Empire went down in flames to the Turks in 1453 still as the Roman Empire. That was its official name to the end. Another heir to the Roman Empire, in a sense, is Islam, which begins in the seventh century, in the middle of our period. I don't have to emphasize to you the historical importance of Islam. But our task is to understand its origin and its astonishing expansion in terms of this era, 250 to 1000. To understand it in terms of its times, and thus how it arises and interacts with the Roman and Byzantine as well as, offstage, the Persian, Empires that it either destroys or weakens in the seventh and eighth centuries. Mohammed was from outside the former empire, from Arabia, and may be said to represent a very different kind of set of ideas. But the power of Islam would, for centuries, be concentrated in areas of the former Roman Empire: the Mediterranean, the Balkans, Egypt, Syria, North Africa. Of course, in the latter, it still is the overwhelming majority religion. And Islam then brings up a sort of question that I'm not going to deal with directly very much, but that will be at the back of our minds, and that is relevance. This is a pre-industrial course; it's in the very pre-industrial category in terms of requirements. It's far away; it's distant. That is part of its appeal, I think, is as I said, its strangeness. But the lessons from the material covered in this course are perhaps these-- worth thinking about. How, perhaps the most successful, multi-cultural empire of Western history, how it did that, how it endured for so long. The success of the Roman Empire and why it finally failed. And in that failure, how does a rich, literate, well-developed society come to be destroyed by a more primitive one? Primitive, at least, in the sense of material culture, economic complexity, urbanization, and literacy. Another important lesson is the power of religious ideas, not only intrinsically as part of people's lives and outlook, but socially and historically: how religion affects the political course of history. Having said this, I think I did mention, I was going to tell you what was fun about this course. This is what I think is fun, and I've already kind of alluded to this. We begin with a familiar world, in the sense that the Roman Empire, although obviously not technologically the same as the one we live in, is a very advanced society and a very complex one. Advanced? Well, go to Europe and look around, and see the engineering feats of the Romans. See the public life that the baths, stadia, temples, law courts, marketplaces, whose ruins still, in many instances, dwarf the towns that survived around them. See what an accomplishment that is. It is a huge empire, a bureaucratic empire, one with lots of literate people, a huge army, a huge civil service, a lot of commerce back and forth, all things that are familiar to us. But as it weakens and collapses, you get a kind of, if not post-apocalyptic, at least transformative experience. It gets stranger and stranger, more and more disorganized, harder to understand at first grasp. Basically we begin in the Shire and we end up in more dangerous territories. It's hard to describe the territory that we end up in, but that is what I think is intriguing about the course. You start out in a familiar world, and it just becomes something alien, but, I think, appealing. Appealing, but I do have one warning for you. Or one thing that I have seen students surprised at, and sometimes even annoyed at. And that is, we've got to talk about religion: both Christianity and Islam, and, to a more limited extent, Judaism, and also paganism, for that matter. But the one that tends to bother people actually is Christianity. So sometimes people will say, I thought I was taking a History course, and this turned into a Religious Studies course before my eyes. We're going to have to talk about some heresies. You're going to have to understand, actually, what people are fighting and killing each other over, when they talk about the nature of Christ, or the relationship among the three persons of the Trinity. This is unfamiliar, but, again, I think unfamiliarity is good for us. And unfamiliarity has a funny way of turning into familiar. When I started teaching-- well, dinosaurs weren't walking the earth, but they'd just departed, primitive birds and early mammals-- the assumption was that religion was safely-- religion as a political movement, not religion as a personal commitment, but religion as something that had an impact on politics was pretty safely gone, and that it was a feature of medieval history. Obviously the last years have shown us, in many ways, the power of religious ideas. The power of religious ideas, not solely personally, but collectively; not solely as sentiments, but as political movements. So we begin with the crisis of the Empire, the first crisis of the Empire in the third century AD. If we go back just before that-- this isn't officially part of the course, this is like the chef offers you this little amuse-bouche, this little snack to begin the meal. The period of the Good Emperors, the second century AD-- the term the Good Emperor is a term popularized by the great historian of The Decline and Fall of The Roman Empire, Edward Gibbon, who wrote in the 18th century. And it's one of those works that, whatever its myriad factual and interpretive inaccuracies, still sets the program for how we look at the decadence and collapse of the Roman Empire. Gibbon says, "If a man were called upon to fix the period in history of the world during which the condition of the human race was most happy and prosperous, he would, without hesitation, name that which elapsed from the accession of Nerva to the death of Marcus Aurelius," that is, 96 to 180 AD. He goes on, "Their united reigns are possibly the only period in which the happiness of a great people was the sole object of government." So it's this period of the so-called Good Emperors against which the subsequent decline traced in Gibbon's monumental work would take place. Now, as opposed to Gibbon, we're not so confident that the Roman Empire was so wonderful for everybody involved. We have a somewhat more egalitarian outlook than Gibbon. Gibbon never says, oh, well, what about the slaves? Or what about the peasant, or hey, the position of the women in society? But more than that, I think we have some more doubt and hesitation as to whether any state, particularly any powerful state, necessarily represents a standard of virtue or happiness. For the poor, the smooth functioning of the Roman government was less important than it was for the propertied classes. Because the Roman state, like most states, in so far as it practiced the rule of law, was set up to guarantee property, not rights. And you'll see Wickham, when we come to read him, he emphasizes, in the chapter on the burden of rule in the Roman Empire-- this is for September 19th-- the Roman Empire was not organized to reward ordinary people. To different degrees, and at varying times, it did rely on slave labor. This is easy to exaggerate. It's not a slave society. It's not an overwhelmingly slave-owning society, but certainly many of its enterprises involved slavery. Its laws were designed to protect the property of the wealthy, rather than to mete out equal justice. Rome was an imperial power, and, as I will say in a moment, it was an extraordinarily tolerant one. But it was tolerant as long as you conformed to their image of civilization. Like many great imperial powers, it assumed that there were certain areas of life that were optional. They were very tolerant with regard to religion, for example. But their definition of civilization was being like us. They were generous about that. They would make citizens of people from the Celtic lands of Britain to Egypt. But this meant conforming to a certain set of standards, beliefs, assumptions, and a way of life. Another thing that we now would dissent from Gibbon about is the efficiency of the Roman Empire. To Gibbon, in the 18th century, the Roman Empire appeared a marvel of efficiency. But really, how could it be efficient? The distances were so long, and travel was so slow. This is an empire that took weeks and weeks to traverse in the state of communications. And we know, from contemporary times that, even with great communications, indeed, with instantaneous communications, it's very hard to hold states together. And, in fact, one of the things that's happened in the last 50 years is the weakening of the state, strangely enough. Strangely, because what people thought was going to happen is what had been going on in the 20th century generally. States had become more and more powerful, more and more dictatorial, more and more tyrannical. And, indeed, George Orwell's 1984, written in the post-war era in the late 1940s, assumes that totalitarianism is what's going to be generalized. In fact, it turns out that the problem is not so much the state-- look at the great states of the mid-20th century: one no longer exists, the Soviet Union, and the other, the United States-- whatever our strengths are-- it doesn't seem to be the extraordinary power of the central government, which is, in fact, much reduced from what is was in, say, 1950, by any measurement. So we have a different idea of the state. We're more aware of the limitations of state power in past times. We're more aware of the discrepancy between the rhetoric of power-- and no polity equals the Roman Empire for its ceremonies of power, its architecture of power, its culture of power. But the emperor, in fact, however glorious, however much in the third century was worshiped as a god, his power was limited. His power was limited in terms of enforcement, if not in theory. And this will be important because, right from the start, we're going to assess the accomplishments of the Emperor Diocletian, at the end of third century, and the Emperor Constantine, beginning of the fourth century, who may be said to have saved the empire from collapse. The thing about the Roman Empire that is indisputable, and does not have a value judgment attached to it, is that it was enduring. The Roman Empire lasted for an incredibly long time; it was stable. In the year 410 the Visigoths sacked, plundered Rome. They entered the city of Rome, this so-called barbarian tribe, and they pillaged it. They pillaged it in a fairly orderly way, but, nevertheless, they pillaged it. This was the first time this had happened in the city of Rome in something on the order of 800 years. The Empire itself, by this time, was nearly 400 years old. The other indisputable accomplishment of the Roman Empire is that it controlled the Mediterranean. The Mediterranean was its center, referred to often as Mare Nostrum, as our sea. Our sea because they controlled all of the shoreline of the Mediterranean-- the only power that has ever done that. There have been great empires: the Ottoman Empire, the Byzantine Empire, the Caliphate, but none of them controlled more than about 40% of the Mediterranean at any one time. So even if we dissent from Gibbon's calm assurance that the best period of human history was the era of the Good Emperors in the second century AD, we shouldn't minimize the accomplishments of this empire and of this era. To live in security, with respect to both outside enemies and internal disorder, essentially peace and the rule of law, was unusual. And, unfortunately, it remains so. Rome was an immense empire. It stretched from England to the Sahara, from Spain to Armenia. It had a common language of administration, Latin. And two cultural languages, Greek and Latin, understood by the elite from one end of the empire to the other. The cities were not walled until the third century. Gibbon, in particular, emphasizes the tolerance of Rome, which appealed to his innate anti-clericalism. And, as many of you know, his explanation for the fall of the Empire was Christianity. It was Christianity that weakened the Empire, weakened its elite, turns its attention to foolish controversies over Trinitarian or Christological concerns, when they should have been concentrating on the barbarians. This is not accepted anymore, for reasons that will become clearer in these first few weeks. And its appeal though, the idea of toleration, is very important to Gibbon, writing in the 18th century, when Europe had just emerged from centuries of religious wars. Having seen the wars of the Reformation, well not literally, he hadn't lived through them, but being the inheritor of these religious wars, of the Thirty Years War in Germany in the 17th century, of religious controversy in Britain, Gibbon, like many members of the Enlightenment generation he was part of, thought that the world would be far better if religion remained either an exclusively private matter or just disappeared. So for him, the villain in history, and in particular, the villain in Roman history is religion. But, however tolerant, of course, Rome drew the line at Christianity, for reasons we shall discuss. But it is still, historically, quite unusual that the Empire should have permitted all of these other religions. Indeed, rather than regarding, say, the religions of Egypt as inferior, they simply brought those gods in. Monday, you might worship Zeus. And Tuesday, hey, if you wanted to see if Isis was going to help you with your impending business deal, why not go and see an Egyptian temple? So its eclectic. It's a little bit like the way Americans dine out. It makes no sense to most people, in most of the world, to say something like, "Oh, I don't want Japanese food. I had that for lunch." If you're in Japan you're expected to have Japanese food for lunch and for dinner, and the same with Italian food. The same is true of religion, most people don't just say, "Oh, I don't want to go to Presbyterian Church this weekend, I'm going to go worship at a Buddhist temple, just for a change." This is more in the nature of Roman religion. So this kind of tolerance is unusual. So tolerance, tolerance is a real virtue of the Empire, even if it's limited. Real virtue because it's unusual, historically. Peace-- the Roman Empire spent half of its state budget on the army. On the other hand, no empire this large could be held together by military means alone. It was held together by an elite that shared notions of civilization, that made certain sacrifices for the public good. Those games, the circuses, the competitions, the ceremonies were usually paid for by private people, not by the state, for example. It is an urban civilization, with an elite that is urban. The cities held their local gods. They had local administration. They built aqueducts, temples, law courts, all these edifices that I mentioned earlier. It was a cosmopolitan way of life. So it's diverse in the sense that there are many different peoples, but unified, in the sense that the elite shares a common language, and even the city planning is the same. If you went to London or you went to Timgad, in what's now the Algerian desert, you'd know your way around. You'd know where to find the marketplace. You'd know that it was laid out in a grid. You'd know where the law courts would be in relation to the temples. You could find your way, just as if you get off the interstate, you know that there will be a Wendy's or a Denny's or a Shell station. And it would be an exceptional place that didn't have them. If you couldn't find a Home Depot, even without benefit of technical aids, getting off an exit of the average interstate highway then you don't live in America, or you haven't been here very long. So the same thing, there's a sort of a mental picture of what a city looks like, from one end of the empire to the other. But it is an empire that is centered around the Mediterranean, and not just logistically or politically, but culturally. For the Romans, their empire included lands where olive trees didn't grow, and where wine grapes didn't flourish, but those places were not places they wanted to live. They wanted to control them, but they were beyond civilization. So a government official on the Danube, in what's now Hungary, writes home complaining that, quote, "The inhabitants lead the most miserable of lives for they cultivate no olives and they drink no wine, end of story. And you could imagine, there's a certain kind of East Coast discourse on the order of, "They have blueberry bagels out here, I can't live here." Or, you know, the nearest Starbucks is 30 miles away, and there's no substitute. This is an impressive empire then. So, its flaws-- it has an imbalance between the urban and the rural, not as great as historians once thought, but it is dominated by cities that depend on peasants, but that tend to drain the land of its vitality. It's also imbalanced East-West. Strangely enough, for an empire that was founded in the West, by the time we start, the East is more prosperous. It's more urban. It has more trade. It's better organized, more commercial. Another flaw of this empire is its size. It works for a long time, and then, when it doesn't work, this becomes a real problem. And then the army-- the army in the third century discovers that it can make and unmake emperors. So the immediate crisis of the third century, which lasts from 235 to the accession of Diocletian in 284, is that it has dozens and dozens of emperors. Most reign for less than three years. All but one die violently. So the two things are linked, the power of the army to proclaim an emperor, and the inability of that emperor to keep this power before the army or an army somewhere, in one province or another, rises up another emperor and kills the former one. So it's a durable empire, but an unwieldy, and in certain respects, exhausted one. So in the first part of the course, we're going to look at how this empire functioned. And we're going to look at its two great crises. The first of them, this third century crisis, which involves all these many different emperors, also has invasions from Persia, also has the first indications of barbarian invasions across the frontier of the Danube and the Rhine rivers, but it survives this first crisis. And that is the accomplishment of Diocletian, about whom you're going to be reading. The second crisis, that of the fifth century, is similar, in many respects, but more final in its results. The fifth century crisis witnesses the collapse of the Roman Empire in the West, the fall of the Roman Empire of Gibbon, that Gibbon made famous, and that continues to inspire a certain amount of fascination and fear today. So here are some questions that should be in the back of your mind, at least, while you do this reading for the first few weeks. When we're talking about the collapse or weakening of the Roman empire, did this happen because of foreign threats or internal weakness? The rhetorical topos, and it's often invoked with regard to empires in the modern world, is internal weakness. That's because the opponents don't seem very savage or very impressive. But it's not necessarily a given. As I said before, external enemies, even that don't appear to be that imposing, can, under certain circumstances, impress their will on what would seem to be a more powerful empire. To some extent that is, indeed, because of internal weakness. But it won't do to exaggerate that. But this is one of the problems. Another problem is continuity versus change. The East survives. The East even flourishes for a while. And even for a while seems to be on the verge of re-conquering the West from the barbarians. So how can we talk about the fall of the Roman Empire when, you know, only part of it falls? Another question is, how did the rise of Christianity affect the political and cultural fortunes of the Empire? As I said, Gibbon said it affected its fortunes by destroying it. But beyond that, I think, over simple explanation, how did Christianity change the Empire? Was this change a catastrophe or a transformation? And, how did Christianity triumph? It seems to be so alien to everything Roman. How does it become the official religion of the Empire? And, how does it become, indeed, identified with the Empire? All of these questions are currently very much debated by historians. I'm not going to have a definitive answer for you. I've certain opinions. I'll present the information basically in accord with that, but this is not something that has been scientifically proven or received universal acknowledgement. So we begin this course with the reign of the Emperor Diocletian, 284 to 305. And we do this because he solved a number of problems which threatened the survival of the Empire in the third century. These problems, as I said, instability of rule, Persian invasion, barbarian invasions, and then, one I didn't mention, which is inflation, tremendous economic dislocation. All of these are manifestations of the long-term flaws I just mentioned. The thing about long-term flaws-- I mean, you can point to long-term flaws in the Soviet Empire or long-term flaws in the British Empire. But why do they manifest themselves when they do? Or, to put it another way, why does the empire go on and flourish for a couple hundred years, or a few decades, and then collapse? Questions? Questions about this, or problems? You know how to reach me, paul.freedman@yale.edu. I put my office, 327 HGS, and office hours. Please come and see me. And I look forward to talking with you on Monday. Thanks. |
The_Early_Middle_Ages_2841000_with_Paul_Freedman | 04_The_Christian_Roman_Empire.txt | PAUL FREEDMAN: Good morning. Questions or problems? So today, this week, we begin to talk more seriously about religion. And I remind you, we're talking about Christianity in particular not because of its spiritual teachings, exactly, but we're interested in the Church as a power and as a power dealing with, allied to, but different from the Roman Empire. Everything changes with the conversion of Constantine. The Church becomes first tolerated, then official, and then finally the sole legal religion, except for Judaism, by order of the Emperor Theodosius the Great in 393. On the other hand, in order to understand the Church as a power or the politics of this period, we do have to understand the doctrine, because people fought over doctrine. And as you've seen in the Jones reading, they invoked the emperor's intervention. This is not a question as might be what you would expect today of a religion split about various problems of dogma solving those problems itself or splitting off into two factions. The questions over the relationship between the persons of the Trinity, Arianism, or the nature of priestly office, Donatism, provoked right from the year of Constantine's conversion, 312, divisive fights. And they're only the first. I urge you to have fun with heresies. Heresies are neat. Professor Carlos Eire is teaching a freshman seminar, I believe. Is it not called "Basic Heresies." What is it? STUDENT: "Essential Heresies." PROFESSOR: "Essential Heresies." Yes, even better. And I don't know of any quiz shows where you can demonstrate expertise on this, but there ought to be. Or there ought to be sites on the Internet where you can get a perfect score. We're only touching the surface, the proverbial tip of the iceberg, to use a cliche. There are lots of heresies. There's a lot of division in the church. Why not just say, "Hey, you believe that the Father is superior to the Son, and I believe that they're coequal. So what?" That's so 2011. Or actually, it's so 2000. Because we ought to be used to religious doctrines having military or other kinds of violent effects. So I'm not going to have to justify for you the fact that we're spending some time on religion, I hope. But if you do want me to, just remind me what the rationale is. If your roommates are bothering you because you're quizzing them about the relation between the persons of the Trinity, have them email me or set up an appointment, and I'll explain why it's important. I'd be glad to do this as a public service. Or my Facebook page, whatever. This is a situation created by Constantine. We've seen Constantine's fairly long reign transform the Roman Empire. We emphasized the creation of the city of Constantinople as a second capital and the favor shown towards the church, so that when Constantine died in 337, 40%, 50%, a large percentage of the Empire's population was Christian. When he died, the Roman Empire seemed to have been restored to its former glory. That chapter that begins the reading for today in Brown's World of Late Antiquity is entitled, "A World Restored." And he emphasizes this because, traditionally, historians saw the fourth century through the lens of the sack of Rome in 410 as a period of decline. In 410, as we will discuss next week, the city of Rome was pillaged and partly burned by the Visigoths, a-- I hate to use the word "tribe," but don't tell anybody-- tribe of barbarians who had entered the Empire across the Danube and the Balkans. This was a cataclysmic event, and it ushered in successive crises in the Western Empire that would lead to its collapse. This weakness of the Western Empire is partly military, partly political, and partly internal, a result of weaknesses, of instability, culture, and the economy. It's always hard to say why empires fall in some absolute sense, even though we can see in retrospect the signs of their decline. One of the signs Brown talks about in the reading for today fairly extensively, and that is the inequality among the rich and poor grows. On the other hand, this is a period of inequality, but also social mobility. There are a lot of paths open to barbarians, even. In this period, barbarians become important figures in the Roman army. And indeed, the army is a path of upward social mobility, a very important one. In general, as I've said, between 337 and 410, between the death of Constantine and the sack of Rome, people did not see themselves as living in a period of decline. They did see themselves as living in a period of real change, the most dramatic of which was the movement towards Christianity. And the most dramatic example of that movement is the reign of the Emperor Julian. I've given you a handout that shows the emperors of this period. Notice that some of them reign in the East, some in the West, and then some usually briefly reunify the empire. We're in a transitional period ushered in by Diocletian when the Empire is sometimes divided and sometimes not. With the death of Theodosius in 395, the third-to-last of those emperors on the list, he had been sole emperor for three years. With his death and the division of the Empire between his son Honorius, ruling from Ravenna-- the Western capital now, in what's now northern Italy-- and Arcadius, ruling from Constantinople, from that point on, we can really speak of two different emperors. And almost always, there were two emperors until the collapse of the Western Empire in the late fifth century. Questions at this point? So the reign of Julian the Apostate-- "Apostate" because he was born as a Christian and tried to restore the traditional Roman religion-- 360 to 363. His failure shows the weakness of what we can call paganism or traditional Roman religion. By this time, Christianity was too big to roll back. Julian tried. He tried to reestablish the temples. He tried to reestablish the vigor of the cults of the gods. He was an eclectic worshiper of many different gods, but he also was a philosopher. He is a wonderful, if sad, example of an intellectual in power. Very few of these emperors were what would be called intellectual in the sense-- many of them were cultivated, they liked the equivalent of classical music and art. But Marcus Aurelius is a philosopher, and Julian at least is a would-be philosopher. He had a beard. He studied in Athens. He was very unsuccessful, though. People did not follow his doctrines. He was a very good military leader. His troops, when he was on the Rhine frontier in what's now France and Germany, were extremely loyal to him. But as emperor, not only did the church oppose him, but many of his own followers did. He died in an unsuccessful campaign against the Persians of perhaps murdered-- unclear. But with his defeat, the end of civic polytheism, of that traditional urban, upper-class worship of the Olympian gods was in permanent eclipse. The decree of the Emperor Theodosius that I referred to already of 393 proclaiming Christianity the only legal religion, except for a small space allowed for Judaism, did not really mean that everybody in the Empire was a practicing Christian. The two groups [correction: most significant group] that would be outside of that definition are the people of the countryside, a very large number of people who would either continue their traditional polytheistic local worship of local gods or assimilate that worship to Christianity. We're going to be talking a lot about saints in a couple of weeks.. On some obvious level, which I don't want to exaggerate but which is nevertheless there, the saints-- and there are many of them-- they are holy figures. They're not gods, but they're more powerful than human beings. The saints are a kind of substitute for polytheism. We'll talk about why and how that works. The other class that tended to resist Christianity to the last was the very upper stratum of the Roman senatorial elite-- the people who were the most cultivated adherents of the traditional religion-- and philosophers, people who followed Plato, for example, known as the Platonists. And we're going to come to them, because they're very important for Augustine's Confessions. Constantine was succeeded by his sons. After two of the sons died, Constantius is ruled alone. Constantius assured that he would rule alone by massacring most of his family. Julian was one of only two that he allowed to survive, and he shouldn't have, because Julian overthrew him. This is brass-knuckle politics, I guess. The chief preoccupations of these rulers of the fourth century were the same as when we started out: the Persian threat on the eastern frontier, the barbarian threat on the Rhine-Danube frontier. But there are also other problems that the third-century emperors had not faced, and these are religious problems. They are, in particular, Arianism and Donatism. And we've seen a little bit about these from the reading about Constantine for the first assignment, but let's go over this. Arianism. Please remember that with a "y," this takes on a completely different meaning. Aryan is this discredited racial theory identified, but not invented by, Nazism. Arian with an "i" is a follower of Arius. Arius was a priest from Alexandria influenced by ideas from Plato about the absolute and unknowable nature of God. Any religion, but Christianity in particular, has a problem: If you make God too absolute and unapproachable, then why does he care about us? If you make Him too approachable, then how is he powerful enough to transcend the world? Platonism-- that is, the philosophy of followers of Plato-- emphasizes the absolute, emphasizes the inferiority of what we can understand with our eyes, with our other senses, and the superiority of the spiritual. The reason the spiritual is superior to the material in Platonism and in Buddhism and in many other philosophies is that the material is mortal, rots, dies, passes away. The problem for many people, not just people in the modern world, is that the material is right here, and the spiritual is hard to apprehend or prove. But most people who think about it most of the time in history agree that the spiritual is superior, because it is immortal. Therefore, those who emphasize the spiritual emphasize this kind of unknowability, the mystery, the not-easy-to-apprehend-ness of the supernatural. For Platonists and for Arians, God was placed infinitely high above us. Now, Christianity has as its great strength, in many respects-- just speaking of the power of ideas-- the fact that God becomes man. Christ is incarnated; that means he becomes flesh. This is not a common religious idea. Mohammed is not God. He is a messenger. In Judaism, there are prophets, but they are not made of the same substance as God. There are religions in which a holy man becomes supernatural-- a lot of religions of the East. But in Christianity, Christ is God. God is manifested in three forms-- the Trinity-- the Father, the Son, and the Holy Ghost. Arianism says that Christ is essentially different from God the Father in some fundamental way, that Christ was created by God as a way of interacting with the world. If you have studied Plato or were in DS and remember some of those key concepts, in Platonism, there's the One, and then there's the Demiurge. The Demiurge is this hard-working guy or hardworking force, whatever you want to call it, who interacts between the divine and the material. He's divine, all right, but he creates things. He's the one who is out there creating the spirit as well as matter, creating the world and the universe, indeed, as we know it, while the One is at rest. So similarly in Arianism, Christ is the one who's intervening on Earth, indeed, sacrificing Himself for humanity. God the Father remains in heaven. This is not what Christianity most of the time teaches. Christianity or the Orthodox or Catholic view is that Christ and God are of the same substance. The Arians therefore are subordinationists. They subordinate Christ to God. A convenient way of remembering them is their tagline: "There was a time when Christ was not." It's a supernatural, metaphysical time, but God made Christ. This goes against the creed enacted at Nicaea, which to this day is the standard confession of faith of Christian denominations. The Nicene Creed, as it is known, says that all persons of the Trinity are eternal, co-eternal. God did not create Christ. Now, Arianism appealed to people partly because they worried about the Trinity. The Trinity could be denounced by anti-Christians as actually three gods. How could God be both one and three? Also, people who worried about the physicality of Christ-- Christ becomes flesh, dies, suffers-- how can God be God if he dies and suffers? Arianism, you have to take seriously as something that is not just troubling the intellectuals. It is not just something for people with a lot of time and leisure on their hands. It's not for philosophy majors. It's not even for religious studies majors. It's not even for the Roman equivalent of college graduates. The barbarians-- and I deliberately avoid calling them the German barbarians-- but the first barbarians to convert to this religion were more or less-- and as time goes, on historians emphasize less-- but more or less Germanic tribes. The missionaries who converted them were Arians in the mid-fourth century, which is the high tide of Arianism in the Roman Empire. When the German tribes or the barbarian tribes entered the Empire beginning in the 370s, Arianism was all but over within the Empire. The Council of Nicaea in the 320s and the Council of Constantinople in 381 pretty much defined Arianism as a heresy, and by 381, it was on the run in the Empire. But ironically, the new rulers of the West would be mostly Arians well into the fifth and, in some cases, sixth centuries. Donatism, or the Donatists. The Donatists-- here we're not dealing with doctrine or metaphysical questions of the relationship of the persons of the Trinity. We're dealing with the nature of the priesthood, hence the nature of the church. It's a question of the office or the man. If it turns out that I do not have a Ph.D. from the University of California, Berkeley, or indeed that I am an impostor, that somehow I tricked the Yale computer, have been showing up here, teaching History 210, and am just some random guy-- like the guy who is obsessed with the New York City transit and has gotten various passes to get in to drive trains and maintenance yards and stuff like that-- if I turn out to be like that, you will nevertheless get credit for this course, I assure you. Because Yale, as an official body, is the one who grants your degree, gives you credits, and they were responsible for hiring me. So if I've imposed a fraud on them, you nevertheless receive credit. You don't have to do a background search of me. You don't have to find out about how many pets I have or my credit status or my mortgage, because you rely on Yale to do that. Think of this then in terms of the priesthood. If it turns out that the priest is not a nice guy-- and alas, that happens-- or a sinner-- or what the Donatists really cared about was he had buckled under threat of torture or the reality of torture during the persecution of Diocletian and had, say, burned the scripture or renounced Christianity-- if your priest is bad, is your baby baptized? That's what Donatism is about, and Donatism answers, no. No. It depends on the virtue of the officiate. You can't have a priest who was what they called a traditor. Traditor is an interesting Latin word, T-R-A-D-I-T-O-R, because it means both "tradition"-- or is the root of the word "tradition," but also of "traitor." It's the handing down or handing over of something, the giving up of something. So one can speak of the handing down of-- our family has cranberry sauce with slivered almonds, that's our tradition-- or handing over state secrets, military strategy, or the scripture to the authorities. The Donatists accused priests who had or they believed had capitulated to Roman persecutorial pressure in the last years of Diocletian. They branded them as traitors, and they said that they could not perform legitimate sacraments. Therefore you, in North Africa, where the Donatists were strong, you as an ordinary Christian had better do a background check on your priest, because your marriage is illegitimate, your baby is going to hell, your whole participation in the church is, as it were, short-circuited by this defect. Worse than that, supposing your priest is just fine-- in fact, supposing it's 330 AD and your priest wasn't even born during the reign of Diocletian, but the guy who consecrated him, the bishop who anointed him was a traditor, or maybe that bishop was not, but he was anointed-- it's like Christmas tree lights, at least the Christmas tree lights that I have, the analog ones. They're little bulbs, and if one of them goes, the whole string goes. If one is defective-- the proverbial bad apple rotting out the entire barrel. But the problem with that is that you can't organize a church on that basis, you can't organize a body spread from one corner of the world to another, unless you believe that the church will guarantee the sanctity of the individual minister. The office, the fact that the guy is a priest, has to be more important than the man, or else you can only have local communities. You can only trust, if you're a Donatist, somebody you grew up with and a community stable enough where you're confident of the ancestry. Donatism lends itself to what we call "sectarianism." A sect is a small, tightly cohesive group. It does not lend itself to a church that is universal, big, massive. And it also doubts that the church itself has the power to overcome the defects of its members. Now, Manicheans are not exactly a heresy, but they appear in the Confessions, and they're very important. So since we're doing doctrinal ideas, Manicheanism is a teaching about good and evil that can be applied to other religions besides Christianity. Manicheanism basically says that good and evil have a real existence. There is a war in the universe between a good god and an evil one. And this may be applied outside of Christianity or within Christianity. And within Christianity, the evil god is the devil, or according to the Manicheans of this period that Augustine for a while joined, the god of the Old Testament. Jehovah is the evil god, and the god of the New Testament, the Christian god, is the good one. Jehovah is the one who smites a lot of people. Jehovah is the creator god, because the Manicheans believed that matter is evil and is the source of evil. Spirit is good. The Christian god created spirit. Human beings are imprisoned in the body, and they have to figure out a way to liberate themselves from the dominion of the evil god. Vegetarianism, for a start, avoiding flesh. But salvation means casting off the flesh. How is this different from Christianity? And doesn't this sound to you like regular old Christianity, mistrust of the flesh? The devil is identified with sexual desire or physicality, generally. Manicheanism is very useful as an explanation of evil. And this may not be something that keeps you up at night, but it will at some point, intermittently. Where does evil come from? Why is there so much evil in the world? Why, if God is good, is there evil in the world? Sure, some of that may just be to test you-- you lose your job or your business fails, but you'll get a better one-- but that's not the same as horrible infant birth disabilities, or the death of people by starvation in the thousands, or choose your evil thing. Why does this happen? One explanation is that God didn't cause it. There's another god. What's the problem with that? Why is Manicheanism not a bigger-- I mean, in a way, it is. But you will travel far to meet anyone who says they're a Manichean. STUDENT: They think God's not omnipotent. PROFESSOR: Yeah, first of all, then God's not all powerful, and so then how is He God? And then instinctively, what's another problem, maybe, with it? I could say, so what? God is limited. You know, he's trying. Anybody else have a sense of the moral problem of saying, the devil made me do it? STUDENT: It takes away self-- PROFESSOR: It takes away individual responsibility. I didn't cheat those investors. The devil cheated them. I didn't mug that Ezra Stiles student. The devil did. Anyway, this is a debate. I'm not asking you to take sides, but notice in the Confessions the attitude of Augustine towards his Manichean experience. So the fourth century is a time of ideas circulating around as Christianity starts to define itself. This would merely be a chapter of the history of ideas, were it not that the Roman emperors had to intervene to settle many of these issues. So just forgive me if I am beating a dead horse, but heresy is important because of its impact in shaping the church through controversy and because it fell to the emperor to try to resolve heresy. And thus marks the beginning of a merger of church and state in a way that would become characteristic of the entire Middle Ages. The other thing to remark is that the emperor did not have a lot of success with this, almost ever. It's very frustrating for an emperor who can get someone killed immediately. All he's got to do is say, "You dropped that lark's tongue pie. Take him away and kill him." But he can't seem to do this with heretics, partly because heretics love persecution. It just toughens them. So you have the Prefect of Africa writing to Jones [correction: to Constantine as you've read in the jones book] and saying that this bishop in North Africa, Caecilian, has been attacked as a traditor. And Constantine does what emperors usually do, they appoint local judges as experts. They say, go and have a hearing and find out what's going on, and make a recommendation to me. He makes a recommendation telling the Donatists to go home and shut up. They don't. They appeal to the emperor. The emperor says, "Go home and shut up." And they don't. They defy the emperor himself. They go back to North Africa, and they continue preaching against traitors. Constantine, why does he intervene? Why didn't he just say, the hell with you-- pretty literally-- or, it's a matter of opinion, or, this doesn't get at the core the faith? STUDENT: He's afraid of upsetting God. PROFESSOR: Yeah. He's afraid of upsetting God. Remember, we said that Constantine was not much of a philosopher, not much of a contemplative. He had won a battle with the favor of the Christian God. Now the Christian God-- he was the companion of the Christian God. The Christian God has him by the hand. If the church starts dissolving into quarreling factions, God is going to be angry, and his favor will cease. We have to take the ruler's anxiety about religious unity seriously. So he changes his policy several times. Remember, we're talking about a guy who was able to defeat all his military rivals, sent Licinius running through Asia Minor, caught and executed him. A man of tremendous capabilities. Nevertheless, dealing with these peasants and poor townspeople and, from his point of view, riff-raff who were strong in North Africa, he couldn't get them to obey him. Sometimes he tried compromise, then persecution, then just saying, nobody can discuss this. And none of this worked. Not only did this not work, but now Arianism came to preoccupy him. And you can see in his letters, there's a kind of, "OMG, I didn't realize what I was signing up for," tone to it. At first he was anti-Arian, pretty totally. He saw it as a denial of Christianity. If Christianity by its very name is that Christ is God, then Arianism would seem to be antithetical to that fundamental thing. So after defeating Licinius, he learned more about Arianism, which was strong in the East, and his first reaction was maybe your reaction, or certainly the reaction of anybody who's more interested in power, politics, this world. He considered it over-subtle, philosophical, and ultimately trivial. But he couldn't get rid of it. He couldn't get rid of it, and now, instead of appointing judges, he summoned a council of the bishops of the Empire, what later would be called an ecumenical council-- "ecumenical" meaning worldwide-- a universal council, which met in the city of Nicaea, not right across the Bosphorus, but more or less-- there's an "a" in there sometimes-- more or less in the western part of Asia Minor. And it met in 325. At this first ecumenical council-- the last ecumenical council of the Catholic Church, by the way, was in the 1960s, the early 1960s under Pope John-- and this ecumenical council was summoned by the emperor, and the emperor was in the room and in certain respects presided, intervened, took sides. He was very deferential to the bishops. He is not a bishop, after all. He is not a priest. He does not have the sacramental orders, but nevertheless, he was, or attempted to be, the boss of this conference. And it came up with a formula, a definition of the creed, what every Christian must believe, saying that God and Christ are of the same essence, but different persons. God and Christ and the Holy Ghost, who's not really at issue yet-- there will be a heresy involving the Holy Ghost, but not until the end the course-- they are the same essence, but different persons. Same essence-- the Greek word is homoousios. Christ was begotten by the Father, but not made by Him. That's a tough one. That's a tough one. He was begotten by the Father, but not in time. There wasn't three minutes after creation. Does someone want to go out and see what that noise is? Or is it coming from outside, and we're hopeless? Thank you. Arius gave a kind of equivocal acceptance of this arrangement, but the Bishop of Alexandria refused to accept Arius back into communion. In other words, the Nicene Creed defined orthodoxy, but fighting raged for about 50 years over the relationship between Christ and God the Father. Constantine kept on changing his mind. Again, the poor guy only wanted unity. He wanted this to go away so that God would favor him. He died more or less an Arian. He came to believe that the Arian moderate position was the best. This is partly because of the absolute intransigence of the Bishop of Alexandria-- the Archbishop of Alexandria, Athanasius. Athanasius would then later be a saint, considered the upholder of the orthodox line. So Constantine was baptized just before he died. In this era, usually, baptism was supposed to mean that you embraced Christianity totally. It's Augustine, particularly, who's responsible for infant baptism. Infant baptism then means you enter the Christian community, but for sure you're going to sin. The understanding in the 330s was more, as a baptized Christian, I am no longer a sinner. So if you were in an office like Constantine, where you are shedding blood, giving judgments, leading troops, you would wait until your death-bed moment before getting baptized. So he would be baptized on his deathbed by an Arian priest. What's the danger of this, parenthetically? STUDENT: That you die unbaptized. PROFESSOR: Die unbaptized, right. The proverbial bus hitting you on the street or sudden death of another sort. And we're going to talk-- I'm sure this is going to be one of the things you will remember with pleasure-- but briefly about sin, why people sin, and the interaction of good and evil in human beings. Indeed, we're going to start very soon, because it's all over the Confessions. Let me give you a little bit of orientation towards the Confessions, which I hope you will read as much as possible. Why are we reading this very personal and autobiographical work? First of all, it is a classic. How many of you, may I ask, have already read it? A certain number. It is an autobiography, which makes it very unusual. It's an intimate autobiography. It's not just highlights of my career or my resume. He talks about anxieties, desires, doubts. It is, therefore, a portrait of the mentality of a thinking person in the ancient world. We get a better idea of the life of the spirit than from many other, most other, almost any other source. It shows the fluidity of the religious scene in the late Roman Empire. His father is a pagan. His mother is a Christian. He is first a Platonist, sort of, then a Manichean, then a Christian Platonist, and then a more fervent Christian. So it shows some of the religious and philosophical options of this era. In doing so, we do come across important and lasting ideas of the Western tradition about evil, sin, body, and soul. We also are looking at the impact of Christianity on the Roman Empire and the relationship between Christianity and classical culture. Some things to keep in mind as you're reading this. The book is uniquely self-revealing for its time. On the other hand, it doesn't tell us a lot of stuff we'd like to know. More time is spent going over a youthful prank involving stealing pears from an orchard than on Augustine's concubine. Concubine: a woman that you're not married to who's like a subordinate with whom you have a sexual relation. And indeed, he would have a child. Pears, Book II, towards the end, pay a lot of attention to that. The book has a lot of prayers, quotations from scripture. It's very Biblically infused. Don't be put off by that, because the book is a confession. "Confession" has two meanings in this era. One is its meaning now, the way we usually use it, confession of sin. But it's also a confession of praise. I confess God, meaning I acknowledge God's power, and I praise Him. Finally, I just want to go over Platonists, because he both admires the Platonists, feels that they saved him from the Manicheans, but ultimately comes to doubt them. This is described fairly well in Brown, beginning on page 73. But let me just take a few minutes to discuss this. Platonism or Neoplatonism, it's the same thing. Neoplatonism is a religious Platonism. It differs from Plato and his dialogues by being a little bit more mystical in the sense of focused on the One, trying to apprehend the One, and less interested in the city, the state, or the kinds of worldly problems that Plato dealt with. Platonism asserts the superiority of the spirit and the inferiority of matter. But unlike Manicheanism, which basically says that matter is evil and spirit is good, for the Platonists, everything is good. Everything is good, but there are inferior goods. So matter is not evil in itself. It's just inferior. Where evil comes from is the preference of the inferior to the superior, the preference of the material manifestation to the spiritual. So to take the most famous example-- because we still have this little funny vestigial notion of so-called Platonic love-- love of a sexual sort is not evil in Platonism. What is evil is assuming that that's all there is and not referring it to a higher spiritual, nonmaterial kind of love. Wealth is not bad, but the preference of wealth over spiritual things is bad. Here again, as we spoke at the beginning of the lecture, it's important to emphasize the superiority of the spiritual over the material, because the spiritual is eternal. Evil comes from not realizing the superiority of the material-- of the spiritual, excuse me-- or not acting on it, acting as if the material is all there is. In Platonism, evil is a falling away, or a turning of your vision, to evoke the cave metaphor. It is a misperception, and it's the result of poor education. If you educate people to the superiority of the spiritual, they will throw off their evil ways. The reason people misbehave, kill each other, oppress each other, is that they don't know any better, and if you teach them, they'll reform. Keep that in mind as we turn to Augustine and see what he thinks about it. He adores the Platonists because they're intellectuals. He admires them because they rescued him from the trap of Manichean dualism-- that is, good versus evil, just simple forces. Platonism is an explanation of evil, but for Augustine, it's not a sufficient explanation. And that will form part of the subject matter of our next talk. So I'll see you on Wednesday. Oh, and Wednesday, you'll get the paper topics for the first paper, which is due October 10. Thank you. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.