Stéphane on Locomotion - Comments: Prototyping a walking pattern generatorhttps://scaron.info/robotics/prototyping-a-walking-pattern-generator.html/2022-05-12T00:00:00+02:00Posted by: Stéphane2022-05-12T00:00:00+02:002022-05-12T00:00:00+02:00Stéphanetag:scaron.info,2022-05-12:/robotics/prototyping-a-walking-pattern-generator.html//comment-reply-mpc-acceleration<p>There are two main ways to use model predictive control in legged locomotion: <a class="reference external" href="https://scaron.info/robotics/open-closed-loop-model-predictive-control.html">open loop and closed loop MPC</a>. Thank you for your question, which prompted this more detailed page 😀 Open loop MPC is the approach we follow in this tutorial. We can see the integrator in <tt class="docutils literal">run_com_mpc</tt>:</p>
<pre class="code python literal-block">
<span class="k">def</span> <span class="nf">run_com_mpc …</span></pre><p>There are two main ways to use model predictive control in legged locomotion: <a class="reference external" href="https://scaron.info/robotics/prototyping-a-walking-pattern-generator.html/robotics/open-closed-loop-model-predictive-control.html">open loop and closed loop MPC</a>. Thank you for your question, which prompted this more detailed page 😀 Open loop MPC is the approach we follow in this tutorial. We can see the integrator in <tt class="docutils literal">run_com_mpc</tt>:</p>
<pre class="code python literal-block">
<span class="k">def</span> <span class="nf">run_com_mpc</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span><span class="w">
</span> <span class="c1"># ...</span><span class="w">
</span> <span class="n">stance</span><span class="o">.</span><span class="n">com</span><span class="o">.</span><span class="n">integrate_constant_jerk</span><span class="p">(</span><span class="n">com_jerk</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span><span class="w">
</span> <span class="c1"># ...</span>
</pre>
<p>What this line does is, given the COM jerk <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>, compute the next state by forward integration:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>T</mi><mo>+</mo><mover accent="true"><mi>y</mi><mo>¨</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mi>T</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><mo>+</mo><mi>u</mi><msup><mi>T</mi><mn>3</mn></msup><mi mathvariant="normal">/</mi><mn>6</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mover accent="true"><mi>y</mi><mo>¨</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>T</mi><mo>+</mo><mi>u</mi><msup><mi>T</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>y</mi><mo>¨</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>y</mi><mo>¨</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>u</mi><mi>T</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\begin{align*}
y(k + 1) & = y(k) + \dot{y}(k) T + \ddot{y}(k) {T}^2 / {2} + u {T}^3 / {6} \\
\dot{y}(k + 1) & = \dot{y}(k) + \ddot{y}(k) T + u {T}^2 / {2} \\
\ddot{y}(k + 1) & = \ddot{y}(k) + u T
\end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5482em;vertical-align:-2.0241em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5241em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</span><span class="mclose">)</span></span></span><span style="top:-3.1359em;"><span class="pstrut" style="height:3em;"></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" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</span><span class="mclose">)</span></span></span><span style="top:-1.6359em;"><span class="pstrut" style="height:3em;"></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" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.0241em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5241em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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 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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</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 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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><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></span></span></span><span class="mord">/</span><span class="mord"><span class="mord">2</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 mathnormal">u</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><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">3</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord">6</span></span></span></span><span style="top:-3.1359em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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 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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</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">u</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><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></span></span></span><span class="mord">/</span><span class="mord"><span class="mord">2</span></span></span></span><span style="top:-1.6359em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">u</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.0241em;"><span></span></span></span></span></span></span></span></span></span></span></span><p>If we measured velocities and computed accelerations by finite difference as you suggest, we would implement closed loop rather than open loop MPC. Both approaches have pros and cons, you can check out the linked page for more details and pointers.</p>
Posted by: Seung Hyeon Bang2022-05-08T22:28:00-05:002022-05-08T22:28:00-05:00Seung Hyeon Bangtag:scaron.info,2022-05-08:/robotics/prototyping-a-walking-pattern-generator.html//comment-question-mpc-acceleration<p>Thank you for your great explanation on WPG but I have a few questions regarding the LMPC formulation you used in this post. The states are composed of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def …</annotation></semantics></math></span></span></p><p>Thank you for your great explanation on WPG but I have a few questions regarding the LMPC formulation you used in this post. The states are composed of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
y</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\dot{y}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\ddot{y}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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> while the control input is the COM jerk. In MPC control cycle, every state is measured first to provide initial states for each finite time horizon optimal control problem. My question is how you provide the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\ddot{y}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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> to the LMPC (QP) solver. Do you just measure the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\dot{y}(k)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\dot{y}(k+1)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</span><span class="mclose">)</span></span></span></span> and use</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>¨</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mi>T</mi></mfrac><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\ddot{y}(k+1) = \frac{\dot{y}(k+1) - \dot{y}(k)}{T},</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</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:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></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" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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">1</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 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;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0833em;"><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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
T</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> is the discrete time step of MPC?</p>