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    <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>
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      &copy; Russ Tedrake, 2020<br/>
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<p><b>Note:</b> These are working notes used for <a
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at MIT</a>. They will be updated throughout the Spring 2020 semester.  <a
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<chapter class="appendix" style="counter-reset: chapter 3"><h1>An Optimization
  Playbook</h1>

  <p>Coming soon... a collection of tips and formulations that can make
  seemingly non-smooth constraints smooth, seemingly non-convex constraints
  convex, etc.</p>

  <todo>PQ trick from LQR</todo>
  <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>

  <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>
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