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	<meta charset="utf-8" /><meta name="hf:doc:metadata" content="{"title":"Control Systems and Their Limitations","local":"control-systems-and-their-limitations","sections":[{"title":"Differential Kinematics: A Smarter Approach","local":"differential-kinematics-a-smarter-approach","sections":[{"title":"The Key Insight","local":"the-key-insight","sections":[],"depth":3},{"title":"Differential IK Solution","local":"differential-ik-solution","sections":[],"depth":3}],"depth":2},{"title":"Adding Feedback Control","local":"adding-feedback-control","sections":[{"title":"Feedback Control Solution","local":"feedback-control-solution","sections":[],"depth":3}],"depth":2},{"title":"Why Classical Approaches Struggle","local":"why-classical-approaches-struggle","sections":[{"title":"1. Integration Challenges","local":"1-integration-challenges","sections":[],"depth":3},{"title":"2. Limited Scalability","local":"2-limited-scalability","sections":[],"depth":3},{"title":"3. Modeling Limitations","local":"3-modeling-limitations","sections":[],"depth":3},{"title":"4. Ignoring Data Trends","local":"4-ignoring-data-trends","sections":[],"depth":3}],"depth":2},{"title":"The Learning Alternative","local":"the-learning-alternative","sections":[],"depth":2},{"title":"Key Takeaways","local":"key-takeaways","sections":[],"depth":2},{"title":"References","local":"references","sections":[],"depth":2}],"depth":1}">
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	<link rel="modulepreload" href="/docs/robotics-course/pr_27/en/_app/immutable/chunks/stores.0a12db33.js"><!-- HEAD_svelte-u9bgzb_START --><meta name="hf:doc:metadata" content="{"title":"Control Systems and Their Limitations","local":"control-systems-and-their-limitations","sections":[{"title":"Differential Kinematics: A Smarter Approach","local":"differential-kinematics-a-smarter-approach","sections":[{"title":"The Key Insight","local":"the-key-insight","sections":[],"depth":3},{"title":"Differential IK Solution","local":"differential-ik-solution","sections":[],"depth":3}],"depth":2},{"title":"Adding Feedback Control","local":"adding-feedback-control","sections":[{"title":"Feedback Control Solution","local":"feedback-control-solution","sections":[],"depth":3}],"depth":2},{"title":"Why Classical Approaches Struggle","local":"why-classical-approaches-struggle","sections":[{"title":"1. Integration Challenges","local":"1-integration-challenges","sections":[],"depth":3},{"title":"2. Limited Scalability","local":"2-limited-scalability","sections":[],"depth":3},{"title":"3. Modeling Limitations","local":"3-modeling-limitations","sections":[],"depth":3},{"title":"4. Ignoring Data Trends","local":"4-ignoring-data-trends","sections":[],"depth":3}],"depth":2},{"title":"The Learning Alternative","local":"the-learning-alternative","sections":[],"depth":2},{"title":"Key Takeaways","local":"key-takeaways","sections":[],"depth":2},{"title":"References","local":"references","sections":[],"depth":2}],"depth":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="control-systems-and-their-limitations" 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="#control-systems-and-their-limitations"><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>Control Systems and Their Limitations</span></h1> <p data-svelte-h="svelte-1byyrzt">In this section, we step from kinematics to control. We first show how to reason with velocities (differential inverse kinematics, diff-IK), then close the loop with feedback, and finally summarize where classical pipelines struggle in practice.</p> <h2 class="relative group"><a id="differential-kinematics-a-smarter-approach" 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="#differential-kinematics-a-smarter-approach"><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>Differential Kinematics: A Smarter Approach</span></h2> <p data-svelte-h="svelte-2cj2a1">Instead of solving for joint positions directly, we can work with <strong>velocities</strong>:</p> <h3 class="relative group"><a id="the-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="#the-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>The Key Insight</span></h3> <p>If we know the relationship between joint velocities and end-effector velocities, we can control motion more smoothly:
	<!-- 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><mover accent="true"><mi>p</mi><mo>˙</mo></mover><mo>=</mo><mi>J</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mover accent="true"><mi>q</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{p} = J(q) \dot{q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></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:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span></span><!-- HTML_TAG_END --></p> <p>Where <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J(q)</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" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span><!-- HTML_TAG_END --> is the <strong data-svelte-h="svelte-1u2ru9w">Jacobian matrix</strong> - the relationship between joint and task space velocities.</p> <h3 class="relative group"><a id="differential-ik-solution" 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="#differential-ik-solution"><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>Differential IK Solution</span></h3> <p>Given a desired end-effector velocity $\dot{p}^*$, find joint velocities:
	<!-- 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><mover accent="true"><mi>q</mi><mo>˙</mo></mover><mo>=</mo><mi>J</mi><mo stretchy="false">(</mo><mi>q</mi><msup><mo stretchy="false">)</mo><mo>+</mo></msup><msup><mover accent="true"><mi>p</mi><mo>˙</mo></mover><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\dot{q} = J(q)^+ \dot{p}^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></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:1.0713em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><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 accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></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></span></span></span><!-- HTML_TAG_END --></p> <p>Where <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>q</mi><msup><mo stretchy="false">)</mo><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">J(q)^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0213em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><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 --> is the <strong data-svelte-h="svelte-1kr4y23">pseudo-inverse</strong> of the Jacobian.</p> <h2 class="relative group"><a id="adding-feedback-control" 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="#adding-feedback-control"><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>Adding Feedback Control</span></h2> <p data-svelte-h="svelte-puurbl">Open-loop tracking is brittle under modeling errors and disturbances. We close the loop by feeding back the tracking error.</p> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-planar-manipulator-floor-box.png" alt="Moving Obstacle" style="width: 100%;"> <p data-svelte-h="svelte-1vjqd7t"><em>Dealing with moving obstacles requires feedback control.</em></p> <p data-svelte-h="svelte-fjtl60">Real environments are <strong>dynamic and uncertain</strong>. We need feedback to handle:</p> <ul data-svelte-h="svelte-1cm1me4"><li><strong>Modeling errors</strong> - Our equations aren’t perfect</li> <li><strong>Disturbances</strong> - Unexpected forces or obstacles</li> <li><strong>Sensor noise</strong> - Measurements have uncertainty</li></ul> <h3 class="relative group"><a id="feedback-control-solution" 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="#feedback-control-solution"><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>Feedback Control Solution</span></h3> <p>Combine desired motion with error correction:
	<!-- 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><mover accent="true"><mi>q</mi><mo>˙</mo></mover><mo>=</mo><mi>J</mi><mo stretchy="false">(</mo><mi>q</mi><msup><mo stretchy="false">)</mo><mo>+</mo></msup><mo stretchy="false">(</mo><msup><mover accent="true"><mi>p</mi><mo>˙</mo></mover><mo>∗</mo></msup><mo>+</mo><msub><mi>k</mi><mi>p</mi></msub><mi mathvariant="normal">Δ</mi><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dot{q} = J(q)^+ (\dot{p}^* + k_p \Delta p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></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:1.0713em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><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="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></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="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:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span><!-- HTML_TAG_END --></p> <p>Where <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mo>=</mo><msup><mi>p</mi><mo>∗</mo></msup><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta p = p^* - p(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</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 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: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></span></span><!-- HTML_TAG_END --> is the position error.</p> <blockquote class="tip"><p>Start with small <!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">k_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><!-- HTML_TAG_END --> and increase gradually while monitoring oscillations. Use a watchdog (safety stop) and saturate commands to keep the system within safe limits.</p></blockquote> <h2 class="relative group"><a id="why-classical-approaches-struggle" 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="#why-classical-approaches-struggle"><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>Why Classical Approaches Struggle</span></h2> <p data-svelte-h="svelte-1772k61">With differential reasoning and feedback, many tracking tasks are solvable—on paper. In practice, the system still breaks under real-world complexity for the reasons below.</p> <img src="https://huggingface.co/robotics-course/images/resolve/main/ch2/ch2-classical-limitations.png" alt="Classical Limitations" style="width: 90%;"> <p data-svelte-h="svelte-ht1j1p"><em>Four key limitations of dynamics-based robotics approaches.</em></p> <h3 class="relative group"><a id="1-integration-challenges" 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="#1-integration-challenges"><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>1. Integration Challenges</span></h3> <p data-svelte-h="svelte-u0hem1">Classical pipelines are built from <strong>separate modules</strong>:</p> <ul data-svelte-h="svelte-9l3kjg"><li>Sensing → State Estimation → Planning → Control → Actuation</li></ul> <p data-svelte-h="svelte-9lhc3"><strong>Problems:</strong></p> <ul data-svelte-h="svelte-8b4yp8"><li>Errors compound through the pipeline</li> <li>Brittle when any component fails</li> <li>Hard to adapt to new tasks or robots</li></ul> <h3 class="relative group"><a id="2-limited-scalability" 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="#2-limited-scalability"><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>2. Limited Scalability</span></h3> <p data-svelte-h="svelte-1wt1ew">Traditional methods struggle with:</p> <ul data-svelte-h="svelte-1rdfawf"><li><strong>High-dimensional sensor data</strong> (cameras, LIDAR)</li> <li><strong>Multi-task scenarios</strong> (each task needs custom planning)</li> <li><strong>Multi-modal integration</strong> (vision + touch + proprioception)</li></ul> <h3 class="relative group"><a id="3-modeling-limitations" 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="#3-modeling-limitations"><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>3. Modeling Limitations</span></h3> <p data-svelte-h="svelte-rzztsa">Real-world physics is complex:</p> <ul data-svelte-h="svelte-171gwna"><li><strong>Contact dynamics</strong> - Hard to model precisely</li> <li><strong>Deformable objects</strong> - Beyond rigid-body assumptions</li> <li><strong>Friction and compliance</strong> - Difficult to characterize</li></ul> <h3 class="relative group"><a id="4-ignoring-data-trends" 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="#4-ignoring-data-trends"><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>4. Ignoring Data Trends</span></h3> <p data-svelte-h="svelte-1rw0j38">Classical methods don’t leverage:</p> <ul data-svelte-h="svelte-17rjybc"><li><strong>Growing robotics datasets</strong> - Millions of demonstrations available</li> <li><strong>Cross-robot learning</strong> - Insights from other platforms</li> <li><strong>Community knowledge</strong> - Decentralized data collection</li></ul> <h2 class="relative group"><a id="the-learning-alternative" 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-learning-alternative"><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 Learning Alternative</span></h2> <p data-svelte-h="svelte-wr3smj">To address these limitations, we contrast a classical modular pipeline with an end-to-end learning policy.</p> <div class="flex space-x-2 items-center my-1.5 mr-8 h-7 !pl-0 -mx-3 md:mx-0"><div class="flex items-center border rounded-lg px-1.5 py-1 leading-none select-none text-smd border-gray-800 bg-black dark:bg-gray-700 text-white">classical </div><div class="flex items-center border rounded-lg px-1.5 py-1 leading-none select-none text-smd text-gray-500 cursor-pointer opacity-90 hover:text-gray-700 dark:hover:text-gray-200 hover:shadow-sm">learning </div></div> <div class="language-select"><p data-svelte-h="svelte-e4nl4x"><strong>Classical Robotics Approach:</strong></p> <div class="code-block relative "><div class="absolute top-2.5 right-4"><button class="inline-flex items-center relative text-sm focus:text-green-500 cursor-pointer focus:outline-none transition duration-200 ease-in-out opacity-0 mx-0.5 text-gray-600 " title="code excerpt" type="button"><svg class="" 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> <div class="absolute pointer-events-none transition-opacity bg-black text-white py-1 px-2 leading-tight rounded font-normal shadow left-1/2 top-full transform -translate-x-1/2 translate-y-2 opacity-0"><div class="absolute bottom-full left-1/2 transform -translate-x-1/2 w-0 h-0 border-black border-4 border-t-0" style="border-left-color: transparent; border-right-color: transparent; "></div> Copied</div></button></div> <pre class=" "><!-- HTML_TAG_START --><span class="hljs-variable">Perception</span> → <span class="hljs-variable">State</span> <span class="hljs-variable">Estimation</span> → <span class="hljs-variable">Planning</span> → <span class="hljs-built_in">Control</span> → <span class="hljs-variable">Actuation</span><!-- HTML_TAG_END --></pre></div> <p data-svelte-h="svelte-1qtanlh"><strong>Challenges:</strong></p> <ul data-svelte-h="svelte-1hswlr6"><li>Each module needs expert tuning</li> <li>Errors compound through pipeline</li> <li>Hard to adapt to new tasks/robots</li> <li>Requires precise world models</li></ul> </div> <p data-svelte-h="svelte-1dnv6az">This is the promise of <strong>robot learning</strong>!</p> <blockquote class="tip" data-svelte-h="svelte-3mscvj"><p><strong>The Best of Both Worlds:</strong> Modern robot learning often combines classical insights with learning. For example one can combine learning with safety constraints from control theory</p> <p>Pure learning vs pure classical is a false dichotomy - hybrid approaches have had their successes</p></blockquote> <hr> <h2 class="relative group"><a id="key-takeaways" 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-takeaways"><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 Takeaways</span></h2> <ul data-svelte-h="svelte-1ptu27b"><li>Classical robotics relies on explicit mathematical models and expert knowledge</li> <li>Forward kinematics is straightforward, but is viable only in quite simple scenarios. Inverse kinematics is more general, but it can be challenging to develop in practice</li> <li>Differential kinematics works with velocities rather than positions for better control</li> <li>Classical approaches struggle with integration, scalability, modeling accuracy, and data utilization</li> <li>Learning-based methods offer solutions to these fundamental limitations</li> <li>The future lies in hybrid approaches that combine classical insights with learning capabilities</li></ul> <blockquote class="tip" data-svelte-h="svelte-1vcn74v"><p>Up next, we’ll show how learning-based methods (reinforcement learning and imitation learning) absorb some of this complexity by optimizing directly from data.</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-facmj1"><li><p><strong>Feedback Systems: An Introduction for Scientists and Engineers</strong> (2008)<br>
	Karl Johan Åström and Richard M. Murray<br>
	A comprehensive introduction to feedback control systems, covering the principles that underlie closed-loop control in robotics.<br> <a href="http://www.cds.caltech.edu/~murray/amwiki/index.php/Main_Page" rel="nofollow">Book Website</a></p></li> <li><p><strong>Real-Time Obstacle Avoidance for Manipulators and Mobile Robots</strong> (1986)<br>
	Oussama Khatib<br>
	A seminal paper introducing the artificial potential field method for obstacle avoidance, demonstrating how feedback can be used for reactive control in dynamic environments.<br> <a href="https://doi.org/10.1177/027836498600500106" rel="nofollow">DOI:10.1177/027836498600500106</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/unit2/4.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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