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<meta charset="utf-8" /><meta name="hf:doc:metadata" content="{&quot;title&quot;:&quot;Understanding Robot Kinematics&quot;,&quot;local&quot;:&quot;understanding-robot-kinematics&quot;,&quot;sections&quot;:[{&quot;title&quot;:&quot;From Complex to Simple: The SO-100 Example&quot;,&quot;local&quot;:&quot;from-complex-to-simple-the-so-100-example&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;The Simplified Robot&quot;,&quot;local&quot;:&quot;the-simplified-robot&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Forward Kinematics: From Joints to Position&quot;,&quot;local&quot;:&quot;forward-kinematics-from-joints-to-position&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Inverse Kinematics: From Position to Joints&quot;,&quot;local&quot;:&quot;inverse-kinematics-from-position-to-joints&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;The Challenge of Constraints&quot;,&quot;local&quot;:&quot;the-challenge-of-constraints&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Key Insight&quot;,&quot;local&quot;:&quot;key-insight&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;References&quot;,&quot;local&quot;:&quot;references&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2}],&quot;depth&quot;:1}">
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 <link rel="modulepreload" href="/docs/robotics-course/pr_27/en/_app/immutable/chunks/MermaidChart.svelte_svelte_type_style_lang.4d36332b.js"><!-- HEAD_svelte-u9bgzb_START --><meta name="hf:doc:metadata" content="{&quot;title&quot;:&quot;Understanding Robot Kinematics&quot;,&quot;local&quot;:&quot;understanding-robot-kinematics&quot;,&quot;sections&quot;:[{&quot;title&quot;:&quot;From Complex to Simple: The SO-100 Example&quot;,&quot;local&quot;:&quot;from-complex-to-simple-the-so-100-example&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;The Simplified Robot&quot;,&quot;local&quot;:&quot;the-simplified-robot&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Forward Kinematics: From Joints to Position&quot;,&quot;local&quot;:&quot;forward-kinematics-from-joints-to-position&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Inverse Kinematics: From Position to Joints&quot;,&quot;local&quot;:&quot;inverse-kinematics-from-position-to-joints&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;The Challenge of Constraints&quot;,&quot;local&quot;:&quot;the-challenge-of-constraints&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Key Insight&quot;,&quot;local&quot;:&quot;key-insight&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;References&quot;,&quot;local&quot;:&quot;references&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2}],&quot;depth&quot;:1}"><!-- HEAD_svelte-u9bgzb_END --> <p></p> <div class="items-center shrink-0 min-w-[100px] max-sm:min-w-[50px] justify-end ml-auto flex" style="float: right; margin-left: 10px; display: inline-flex; position: relative; z-index: 10;"><div class="inline-flex rounded-md max-sm:rounded-sm"><button class="inline-flex items-center gap-1 h-7 max-sm:h-7 px-2 max-sm:px-1.5 text-sm font-medium text-gray-800 border border-r-0 rounded-l-md max-sm:rounded-l-sm border-gray-200 bg-white hover:shadow-inner dark:border-gray-850 dark:bg-gray-950 dark:text-gray-200 dark:hover:bg-gray-800" aria-live="polite"><span class="inline-flex items-center justify-center rounded-md p-0.5 max-sm:p-0 hover:text-gray-800 dark:hover:text-gray-200"><svg class="sm:size-3.5 size-3" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" fill="currentColor" focusable="false" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 32 32"><path d="M28,10V28H10V10H28m0-2H10a2,2,0,0,0-2,2V28a2,2,0,0,0,2,2H28a2,2,0,0,0,2-2V10a2,2,0,0,0-2-2Z" transform="translate(0)"></path><path d="M4,18H2V4A2,2,0,0,1,4,2H18V4H4Z" transform="translate(0)"></path><rect fill="none" width="32" height="32"></rect></svg></span> <span>Copy page</span></button> <button class="inline-flex items-center justify-center w-6 max-sm:w-5 h-7 max-sm:h-7 disabled:pointer-events-none text-sm text-gray-500 hover:text-gray-700 dark:hover:text-white rounded-r-md max-sm:rounded-r-sm border border-l transition border-gray-200 bg-white hover:shadow-inner dark:border-gray-850 dark:bg-gray-950 dark:text-gray-200 dark:hover:bg-gray-800" aria-haspopup="menu" aria-expanded="false" aria-label="Open copy menu"><svg class="transition-transform text-gray-400 overflow-visible sm:size-3.5 size-3 rotate-0" width="1em" height="1em" viewBox="0 0 12 7" fill="none" xmlns="http://www.w3.org/2000/svg"><path d="M1 1L6 6L11 1" stroke="currentColor"></path></svg></button></div> </div> <h1 class="relative group"><a id="understanding-robot-kinematics" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#understanding-robot-kinematics"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Understanding Robot Kinematics</span></h1> <p data-svelte-h="svelte-st78c5">In this section, we’ll build intuition for robot kinematics through a concrete, worked example. Kinematics describes the mathematical relationship between joint angles and end-effector positions - given the joint configuration, where does the robot’s hand end up? We’ll explore this fundamental concept by walking through a simplified but representative case.</p> <p data-svelte-h="svelte-1wao2sz">Let’s examine how traditional robotics approaches robot control using a specific example you can follow step by step.</p> <h2 class="relative group"><a id="from-complex-to-simple-the-so-100-example" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#from-complex-to-simple-the-so-100-example"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>From Complex to Simple: The SO-100 Example</span></h2> <p data-svelte-h="svelte-12wxhrh">We’ll start with a familiar robot platform and systematically simplify it to isolate the core kinematic principles without unnecessary complexity.</p> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-so100-to-planar-manipulator.png" alt="SO-100 to Planar Manipulator" style="width: 70%;"> <p data-svelte-h="svelte-rvs88y">The SO-100 arm simplified to a 2D planar manipulator by constraining some joints.</p> <p data-svelte-h="svelte-lon8ap">The SO-100 is a 6-degree-of-freedom (6-DOF) robot arm. To understand the principles, let’s simplify it to a <strong>2-DOF planar manipulator</strong> by constraining some joints.</p> <h2 class="relative group"><a id="the-simplified-robot" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#the-simplified-robot"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>The Simplified Robot</span></h2> <p data-svelte-h="svelte-in1eji">To keep the math clear, our simplified robot has:</p> <ul data-svelte-h="svelte-vrkuzb"><li><strong>Two joints</strong> with angles θ₁ and θ₂</li> <li><strong>Two links</strong> of equal length <em>l</em></li> <li><strong>Configuration</strong> q = [θ₁, θ₂] ∈ [-π, +π]²</li></ul> <h2 class="relative group"><a id="forward-kinematics-from-joints-to-position" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#forward-kinematics-from-joints-to-position"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Forward Kinematics: From Joints to Position</span></h2> <p data-svelte-h="svelte-1g43rhv"><strong>Question:</strong> Given joint angles θ₁ and θ₂, where is the end-effector?</p> <p><strong data-svelte-h="svelte-megsjq">Answer:</strong> We can calculate the end-effector position mathematically:
<!-- HTML_TAG_START --><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>l</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>l</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>l</mi><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>l</mi><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">p(q) = \begin{pmatrix} l \cos(\theta_1) + l \cos(\theta_1 + \theta_2) \\ l \sin(\theta_1) + l \sin(\theta_1 + \theta_2) \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span><!-- HTML_TAG_END --></p> <p data-svelte-h="svelte-5iiezm">This is called <strong>Forward Kinematics (FK)</strong> - mapping from joint space to task space.</p> <blockquote class="tip"><p data-svelte-h="svelte-18lhek9"><strong>Understanding the Math:</strong> This equation comes from basic trigonometry!</p> <ul><li>First link: endpoint at <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>l</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>l</mi><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(l \cos(\theta_1), l \sin(\theta_1))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span><!-- HTML_TAG_END --></li> <li>Second link: starts from first link’s end, rotated by <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\theta_1 + \theta_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><!-- HTML_TAG_END --></li> <li data-svelte-h="svelte-1k9lyd7">Final position: sum of both link contributions</li></ul> <p data-svelte-h="svelte-kbodls"><strong>Why it matters:</strong> For a simple robot, FK is relatively easy - given joint angles, we can always compute where the robot’s hand is. More complex robots - for instance, dexterous hands, are more challenging to model.</p></blockquote> <h2 class="relative group"><a id="inverse-kinematics-from-position-to-joints" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#inverse-kinematics-from-position-to-joints"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Inverse Kinematics: From Position to Joints</span></h2> <p data-svelte-h="svelte-1o0owgi">Now let’s try to invert the mapping: given a desired hand position, what joint configuration achieves it?</p> <p data-svelte-h="svelte-165jagt"><strong>Question:</strong> Given a desired end-effector position p*, what joint angles should we use?</p> <p data-svelte-h="svelte-1faxicm"><strong>Answer:</strong> This is <strong>Inverse Kinematics (IK)</strong> - typically more intricate to solve, as it deals with the Jacobian matrix of the forward kinematics function!</p> <p>We need to solve: <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>p</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">p(q) = p^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span><!-- HTML_TAG_END --></p> <p>In general, this becomes an optimization problem:<!-- HTML_TAG_START --><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mrow><mi>q</mi><mo>∈</mo><mi mathvariant="script">Q</mi></mrow></munder><mi mathvariant="normal">∥</mi><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>−</mo><msup><mi>p</mi><mo>∗</mo></msup><msubsup><mi mathvariant="normal">∥</mi><mn>2</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\min_{q \in \mathcal{Q}} \|p(q) - p^*\|_2^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.6304em;vertical-align:-0.8804em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.3557em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span><span class="mrel mtight">∈</span><span class="mord mathcal mtight">Q</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∥</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span><!-- HTML_TAG_END --></p> <blockquote class="warning" data-svelte-h="svelte-1smykon"><p><strong>Why IK is Hard:</strong></p> <ul><li><p><strong>Multiple solutions</strong> - Same end position can be reached with different joint angles</p></li> <li><p><strong>Nonlinear equations</strong> - Some functions make analytical solutions difficult</p></li> <li><p><strong>Constraints</strong> - Joint limits and obstacles further complicate the problem</p></li></ul> <p>This is why robotics engineers spend so much time on IK algorithms!</p></blockquote> <h2 class="relative group"><a id="the-challenge-of-constraints" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#the-challenge-of-constraints"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>The Challenge of Constraints</span></h2> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-planar-manipulator-free.png" alt="Free Motion" style="width: 100%; max-width: 200px;"> <p data-svelte-h="svelte-mfs8on"><em>Free to move</em></p> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-planar-manipulator-floor.png" alt="Floor Constraint" style="width: 100%; max-width: 200px;"> <p data-svelte-h="svelte-n24ipq"><em>Constrained by floor</em></p> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-planar-manipulator-floor-shelf.png" alt="Multiple Constraints" style="width: 100%; max-width: 200px;"> <p data-svelte-h="svelte-cwepal"><em>Multiple obstacles</em></p> <p data-svelte-h="svelte-1q6cvob">Real robots face <strong>constraints</strong>:</p> <ul data-svelte-h="svelte-15k5gmo"><li><strong>Physical limits</strong> - Can’t move through the floor</li> <li><strong>Obstacles</strong> - Must avoid collisions</li> <li><strong>Joint limits</strong> - Finite range of motion</li></ul> <p>These constraints make the feasible configuration space <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span><!-- HTML_TAG_END --> much more complex!</p> <h2 class="relative group"><a id="key-insight" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#key-insight"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Key Insight</span></h2> <p data-svelte-h="svelte-1jtbvqi">Even for this <strong>simple 2-DOF robot</strong>, solving IK with constraints is non-trivial. Real robots have:</p> <ul data-svelte-h="svelte-aj9t6f"><li><strong>6+ degrees of freedom</strong></li> <li><strong>Complex geometries</strong></li> <li><strong>Dynamic environments</strong></li> <li><strong>Uncertain models</strong></li></ul> <p data-svelte-h="svelte-x3qtr7">Traditional approaches require <strong>extensive mathematical modeling</strong> and <strong>expert knowledge</strong> for each specific case.</p> <blockquote class="tip" data-svelte-h="svelte-13eiwd4"><p>Mental model: FK is a direct calculator (q → p) and is usually easy; IK is a search (p → q) and becomes hard as soon as you add workspace limits, obstacles, or joint constraints. When IK gets brittle, we’ll switch to differential reasoning (velocities) in the next step.</p></blockquote> <h2 class="relative group"><a id="references" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#references"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>References</span></h2> <p data-svelte-h="svelte-12jfyok">For a full list of references, check out the <a href="https://huggingface.co/spaces/lerobot/robot-learning-tutorial" rel="nofollow">tutorial</a>.</p> <ul data-svelte-h="svelte-ldpzok"><li><p><strong>Modern Robotics: Mechanics, Planning, and Control</strong> (2017)<br>
Kevin M. Lynch and Frank C. Park<br>
Chapter 4 provides an in-depth treatment of forward kinematics with extensive examples, while Chapter 6 covers inverse kinematics with both analytical and numerical approaches.<br> <a href="http://hades.mech.northwestern.edu/index.php/Modern_Robotics" rel="nofollow">Book Website</a></p></li> <li><p><strong>Introduction to Robotics: Mechanics and Control</strong> (2005)<br>
John J. Craig<br>
A classic textbook with detailed coverage of robot kinematics, including the Denavit-Hartenberg notation and systematic approaches to solving forward and inverse kinematics problems.<br> <a href="https://www.pearson.com/en-us/subject-catalog/p/introduction-to-robotics-mechanics-and-control/P200000003519" rel="nofollow">Publisher Link</a></p></li></ul> <a class="!text-gray-400 !no-underline text-sm flex items-center not-prose mt-4" href="https://github.com/huggingface/robotics-course/blob/main/units/en/unit2/3.mdx" target="_blank"><svg class="mr-1" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" fill="currentColor" focusable="false" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 32 32"><path d="M31,16l-7,7l-1.41-1.41L28.17,16l-5.58-5.59L24,9l7,7z"></path><path d="M1,16l7-7l1.41,1.41L3.83,16l5.58,5.59L8,23l-7-7z"></path><path d="M12.419,25.484L17.639,6.552l1.932,0.518L14.351,26.002z"></path></svg> <span data-svelte-h="svelte-zjs2n5"><span class="underline">Update</span> on GitHub</span></a> <p></p>
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