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<!DOCTYPE html>
<html>
<head>
<title>Ch. 24 - An Optimization
Playbook</title>
<meta name="Ch. 24 - An Optimization
Playbook" content="text/html; charset=utf-8;" />
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<header>
<h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a></h1>
<p data-type="subtitle">Algorithms for Walking, Running, Swimming, Flying, and Manipulation</p>
<p style="font-size: 18px;"><a href="http://people.csail.mit.edu/russt/">Russ Tedrake</a></p>
<p style="font-size: 14px; text-align: right;">
© Russ Tedrake, 2023<br/>
Last modified <span id="last_modified"></span>.</br>
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<a href="misc.html">How to cite these notes, use annotations, and give feedback.</a><br/>
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<p><b>Note:</b> These are working notes used for <a
href="https://underactuated.csail.mit.edu/Spring2023/">a course being taught
at MIT</a>. They will be updated throughout the Spring 2023 semester. <a
href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture videos are available on YouTube</a>.</p>
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<chapter class="appendix" style="counter-reset: chapter 3"><h1>An Optimization
Playbook</h1>
<p>This chapter is intended to provide a collection of tips and formulations
that can make seemingly non-smooth constraints smooth, seemingly non-convex
constraints convex, etc. (It's very much a work in progress...)</p>
<section data-type="sect1"><h1>Matrices</h1>
<todo>max eigenvalue. min eigenvalue. rank (nuclear norm, ...)</todo>
</section>
<section data-type="sect1"><h1>Ellipsoids</h1>
<p>Imagine an ellipsoid centered at the origin: $E = \{x | x^T S x \leq
1\}$ with $S=S^T\succ0$. The volume of this ellipse is proportional to
$\left(\det S^{-1} \right)^\frac{1}{2}$. We know that $\log\det(S)$ is a
concave function in the elements of $S$ <elib>Boyd04a</elib>.</p>
<p>maximizing volume.</p>
<p>minimizing volume.</p>
<p>volume of a semi-algebraic set. (via a contained ellipsoid)</p>
</section>
<section data-type="sect1"><h1>Polytopes</h1>
<todo>Sadra's linear encodings of polytopic containment</todo>
</section>
<section data-type="sect1"><h1>(Mixed-)Integer Programming</h1>
</section>
<section data-type="sect1"><h1>Bilinear Matrix Inequalities (BMIs)</h1>
<todo>PQ trick from LQR</todo>
<todo>Rank minimization</todo>
</section>
<section data-type="sect1"><h1>Geometry (SE(3), Penetration, and
Contact)</h1>
<todo>Hongkai's smooth non-penetration constraints for polygons</todo>
<todo>"Soft" absolute value: $|x|_epsilon \approx \sqrt{x^2 +
\epsilon}$.
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-2.,2.,101)
plt.plot(x, np.sqrt(np.power(x,2) + 0.01))
</todo>
</section>
<todo>Barrier function for friction cones in Grandia 2019, Feedback MPC for
Torque-Controlled Legged Robots.</todo>
<todo>Linearization of the rotation matrix in Real-time Model Predictive
Control for Versatile Dynamic Motions in Quadrupedal Robots</todo>
<todo>Briat15 chapter 1 has some good modeling tricks for LPV systems</todo>
</chapter>
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<div id="references"><section><h1>References</h1>
<ol>
<li id=Boyd04a>
<span class="author">Stephen Boyd and Lieven Vandenberghe</span>,
<span class="title">"Convex Optimization"</span>, Cambridge University Press
, <span class="year">2004</span>.
</li><br>
</ol>
</section><p/>
</div>
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