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<!DOCTYPE html>
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<head>
<title>Ch. 4 - Simple Models
of Walking and Running</title>
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of Walking and Running" content="text/html; charset=utf-8;" />
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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>
<p style="font-size: 14px; text-align: right;">
© Russ Tedrake, 2022<br/>
Last modified <span id="last_modified"></span>.</br>
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<p><b>Note:</b> These are working notes used for <a
href="http://underactuated.csail.mit.edu/Spring2022/">a course being taught
at MIT</a>. They will be updated throughout the Spring 2022 semester. <a
href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture videos are available on YouTube</a>.</p>
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<chapter style="counter-reset: chapter 3"><h1>Simple Models
of Walking and Running</h1>
<p>Practical legged locomotion is one of the fundamental problems in
robotics; we've seen amazing progress over the last few years, but there are
still some major unsolved problems. Much of the recent progress is due to
improvements in hardware -- a legged robot must carry all of its sensors,
actuators and power and traditionally this meant underpowered motors that act
as far-from-ideal force/torque sources -- Boston Dynamics and other
companies have made incredible progress here. The control systems
implemented on these systems, though, are still more heuristic than you might
think. They also require dramatically higher bandwidth and lower latency
that the human motor control system and still perform worse in challenging
environments.<p>
<p>The control of walking robots is definitely underactuated: assuming we
cannot pull on the ground (and barring any aerodynamic effects!), then no
matter how powerful my actuators are, there is nothing that we can do to
accelerate the center of mass of the robot towards the ground faster than
gravity. But beyond the underactuated systems we have studied so far, the
control of walking robots faces an additional complexity: <b>controlling
through contact</b>. The fact that we can get non-zero contact forces if and
only if two bodies are in contact introduces a fundamentally
discrete/combinatorial element into the problem†<sidenote>†I've
had some interesting arguments with folks on this point, because it's
possible to write contact models that smooth the discontinuity, and/or to
model systems that have neglible collision events. But replacing a
discontinuity in the vector field with a stiff but smooth transition does not
remove the need to decide whether or not your robot should make contact to
accomplish a task.</sidenote>.</p>
<p>Contact is essential for many aspects of robotics (manipulation is my
other favorite example); it's sad to see so many robots going through life
avoiding contact at any cost. Controlling contact means that your robot is
capable of performing physical work on the environment; isn't that the point
of robotics, afterall?</p>
<p>I like to start our discussion of contact with walking robots for a few
important reasons. As you'll see in this chapter, there are a number of
elegant "simple models" of walking that capture much of the essence of the
problem with minimal complexity. But there is another, more subtle, reason: I
want to continue to use the language of stability. At this point we've seen
both finite-horizon and infinite-horizon problem formulations -- when we can
say something about the behavior of a system in the limit as time goes to
infinity, it's almost always cleaner than a finite-time analysis (because time
drops away). Walking provides a very clean way to use the language of
stability in studying behavior of systems that are sometimes in contact but
sometimes out of contact in the limiting case; we just need to extend our
notions from stability of a fixed point to the stability of a (periodic)
cycle.
</p>
<section><h1>Limit Cycles</h1>
<p> A limit cycle is an periodic orbit that is a <a
href="https://en.wikipedia.org/wiki/Limit_set">limit set</a> of the
dynamical system; they can be stable (the limit set of neighboring
trajectories as time goes to infinity) or unstable (the limit set of
neighboring trajectories as time goes to negative infinity). One of the
simplest models of limit cycle behavior is the Van der Pol oscillator. Let's
examine that first...</p>
<example><h1>Van der Pol Oscillator</h1>
<p>The Van der Pol oscillator is described by a second-order differential
equation in one variable: $$\ddot{q} + \mu (q^2 - 1)\dot{q} + q = 0, \quad
\mu>0$$ One can think of this system as almost a simple spring-mass-damper
system, except that it has nonlinear damping. In particular, the velocity
term dissipates energy when $|q|>1$, and adds energy when $|q|<1$.
Therefore, it is not terribly surprising to see that the system settles
into a stable oscillation from almost any initial conditions (the
exception is the state $q=0,\dot{q}=0$). This can be seen nicely in the
phase portrait below.</p>
<figure>
<table width="100%"><tr><td width="50%">
<img width="100%" src="figures/vanderPol_phase.svg"/>
</td><td width="50%">
<img width="100%" src="figures/vanderPol_time.svg"/>
</td></tr></table>
<figcaption>System trajectories of the Van der Pol oscillator with $\mu
=.2$. (Left) phase portrait. (Right) time domain.</figcaption>
</figure>
<p>However, if we examine system trajectories in the time domain (see the
right panel of the figure above), then we can see that our traditional
notion for stability of a trajectory, $\lim_{t\rightarrow \infty} |\bx(t)
- \bx^*(t)| = 0$, is not satisfied here for any $\bx^*(t)$. Although the
phase portrait clearly reveals that all trajectories converge to the same
orbit, the time domain plot reveals that these trajectories do not
necessarily synchronize in time.</p>
</example> <!-- end of VdP example -->
<p>This example shows that stability of a trajectory (in time) is not the
definition we want. Instead we will define the stability of an orbit,
$\bx^*(t)$, using \[ \min_\tau || \bx(t) - \bx^*(\tau) ||.\] Here the
quantity of interest is the distance between our trajectory and the
<i>closest point</i> on the orbit. Depending on exactly how this distance
evolves with time, we can define <i>orbital stability</i>: for every small
$\epsilon>0$, we can find a $\delta>0$ such that $\min_\tau||\bx(t_0) -
\bx^*(\tau)|| < \delta$ implies $\forall t, \min_\tau ||\bx(t) -
\bx^*(\tau)|| < \epsilon$). The definitions for asymptotic orbital
stability, exponential orbital stability, finite-time orbital stability
follow naturally, as does the definition of an unstable orbit. In the case
of limit cycles, this $\bx^*(t)$ is a periodic solution with
$\bx^*(t+t_{period}) = \bx^*(t)$. </p>
<p>As we begin to extend our notions of stability, allow me to make a few
quick connections to the discussion of stability in the <a
href="lyapunov.html">chapter on Lyapunov analysis</a>. It's interesting to
know that if we can find an <a href="lyapunov.html#lasalle">invariant
set</a> in state space which does not contain any fixed points, then this
set must contain a closed orbit (this is the Poincaré-Bendixson
Theorem) <elib>Strogatz94</elib>. It's also interesting to note that
gradient potential fields (and Lyapunov functions with $\dot{V}(\bx) \prec
0$) cannot have a closed orbit <elib>Strogatz94</elib>, and consequently we
will also have to slightly modify the Lyapunov analysis we have introduced
so far to handle limit cycles. We'll discuss this in some details in an <a
href="limit_cycles.html#lyapunov">upcoming chapter</a>.</p>
<subsection><h1>Poincaré Maps</h1>
<p>The first tool one should know for analyzing the stability of a limit
cycle, and one that will serve us quite well for the examples in this
chapter, is the method of Poincaré. Let's consider a dynamical
system, $\dot{\bx} = {\bf f}(\bx),$ with $\bx \in \Re^n.$ Define an $n-1$
dimensional <em>surface of section</em>, $S$. We will also require that
$S$ is transverse to the flow (i.e., all trajectories starting on $S$ flow
through $S$, not parallel to it). The Poincaré map (or return map)
is a mapping from $S$ to itself: $$\bx_p[n+1] = {\bf P}(\bx_p[n]),$$
where $\bx_p[n]$ is the state of the system at the $n$th crossing of the
surface of section. Note that we will use the notation $\bx_p$ to
distinguish the state of the discrete-time system from the continuous-time
system; they are related by $\bx_p[n] = \bx(t_c[n])$, where $t_c[n]$ is
the time of the $n$th crossing of $S$.</p>
<p>For our purposes in this chapter, it will be useful for us to allow the
Poincaré maps to also include fixed points of the original
continuous-time system, and to define the return time to be zero. These
points do not satisfy the transversality condition, and require some care
numerically, but they do not compromise our analysis.</p>
<example><h1>Return map for the Van der Pol Oscillator</h1>
<p>Since the state space of this system is two-dimensional, a return map
for the system must have dimension one. For instance, we can take $S$
to be the line segment where $q = 0$, $\dot{q} \ge 0$. It is easy to
see that all trajectories that start on S are transverse to it.</p>
<p>One dimensional maps, like one dimensional flows, are amenable to
graphical analysis.
</p>
<figure> <table><tr><td> <img width="100%"
src="figures/vanderPol_section.svg" /> </td><td> <img width="100%"
src="figures/vanderPol_retmap.svg" /> </td></tr></table>
<figcaption>(Left) Phase plot with the surface of section, $S$, drawn
with a black dashed line. (Right) The resulting Poincaré
first-return map (blue), and the line of slope one (red).</figcaption>
</figure>
</example>
<p>If we can demonstrate that ${\bf P}(\bx_p)$ exists for all $\bx_p \in
S$, (all trajectories that leave $S$ eventually return to $S$), then
something remarkable happens: we can reduce the stability analysis for a
limit cycle in the original coordinates into the stability analysis of a
fixed point on the discrete map. If $\bx^*_p$ is a stable (or unstable)
fixed point of the map ${\bf P}$, then there exists a unique periodic
orbit $\bx^*(t)$ which passes through $\bx^*_p$ that is a (stable, or
unstable) limit cycle. If we are able to upper-bound the time it takes
for trajectories leaving $S$ to return, then it is also possible to infer
asympotitic orbital or even exponential orbital stability.</p>
<p>In practice it is often difficult or impossible to find ${\bf P}$
analytically, but it can be sampled quite reasonably numerically. Once a
fixed point of ${\bf P}$ is obtained (by finding a solution to $\bx^*_p =
{\bf P}(\bx^*_p)$), we can infer local limit cycle stability with an
eigenvalue analysis. If this eigen-analysis is performed in the original
coordinates (as opposed to the $n-1$ dimensional coordinates of $S$),
then there will always be a at least one zero eigenvalue - corresponding
to perturbations along the limit cycle which do not change the state of
first return. The limit cycle is considered locally exponentially stable
if all remaining eigenvalues, $\lambda_i$, have magnitude less than one,
$|\lambda_i|<1$.</p>
<p>It is sometimes possible to infer more global stability properties of
the return map by examining ${\bf P}$. We will study computational
methods in the <a href="limit_cycles.html">later chapter</a>.
<elib>Koditschek91</elib> describes some less computational stability
properties known for <em>unimodal</em> maps which are useful for the
simple systems we study in this chapter.</p>
<todo>the paragraph below treats x_p as living in the reduced space, whereas the text above kept it in the original space.</todo>
<p>A particularly graphical method for understanding the dynamics of a
one-dimensional iterated map is the "staircase method": Sketch the
Poincaré map -- a plot of $x_p[n]$ vs $x_p[n+1]$ -- and also the
line of slope one. In order to "iterate" the map, take an initial
condition on the plot as a point $\left( x_p[k], x_p[k] \right)$, and move
vertically on the plot to the point $\left( x_p[k], x_p[k+1] = f(x_p[k])
\right)$. Now move horizontally to the next point $\left( x_p[k+1],
x_p[k+1] \right)$ and repeat.</p>
<figure id="staircase">
<img width="70%" src="figures/vanderpol_staircase.svg"/>
<figcaption>The "staircase" on the return map of the Van der Pol
Oscillator with $\mu = 0.2$, starting from two initial conditions
($\dot{q}_p[0] = .5$ and $\dot{q}_p[0] = 3.5$).</figcaption>
</figure>
<p>We can quickly recognize the fixed points as the points where the
return map crosses this line. We can infer local exponential stability if
the absolute value of the slope of return map at the fixed point is less
than one ($|\lambda| < 1$). The figure above illustrates the stability of
the Van der Pol limit cycle; it also illustrates the (one-sided)
instability of the fixed point at the origin. Understanding the regions
of attraction of fixed points on an iterated map (or any discrete-time
system) graphically requires a little more care, though, than the
continuous-time flows we examined before. The discrete maps can jump a
finite distance on each step, and will in fact oscillate back and forth
from either side of the fixed point when $\lambda < 0.$</p>
</subsection> <!-- end of Poincare -->
</section> <!-- end of Limit Cycles -->
<section><h1>Simple Models of Walking</h1>
<p>Much of fundamental work in the dynamics of legged robots was done originally in the context of studying <i>passive-dynamic walkers</i> (often just "passive walker"). We've already seen my favorite example of a passive walker, in the first chapter, but here it is again:</p>
<figure>
<p>
<video width="80%" controls title="If your browser cannot play this video, then download it using the link below.">
<source src="http://ruina.tam.cornell.edu/research/topics/locomotion_and_robotics/3d_passive_dynamic/from_angle.mpg" type="video/mp4"/>
<source src="figures/passive_angle.ogg" type="video/ogg"/>
</video>
</p>
<figcaption>A 3D passive dynamic walker by Steve Collins and Andy Ruina<elib>Collins01</elib>.</figcaption>
</figure>
<p>This amazing robot has no motors and no controllers... it walks down a
tiny ramp and is powered entirely by gravity. We can build up our
understanding of this robot in a series of steps. As we will see, what is
remarkable is that even though the system is passive, we believe that the
periodic gait you see in the video is actually a stable limit cycle!</p>
<p>Well before the first passive walkers, one of the earliest models of
walking was proposed by McMahon<elib>McMahon80</elib>, who argued that
humans use a mostly ballistic (passive) gait. He observed that muscle
activity (recorded via <a
href="https://en.wikipedia.org/wiki/Electromyography">EMG</a>) in the
"stance leg" is relatively high, but muscle activity in the "swing leg" is
very low, except for at the very beginning and end of swing. He proposed a
"ballistic walker" model -- a three-link pendulum in the <a
href="https://en.wikipedia.org/wiki/Sagittal_plane">sagittal plane</a>, with
one link from the ankle to the hip representing a straight "stance leg", the
second link from the hip back down to the "swing leg" knee, and the final
link from the knee down to the swing foot -- and argued that this model
could largely reproduce the kinematics of gait. This model of "walking by
vaulting" is considered overly simplistic today, as we now understand much
better the role of compliance in the stance leg during walking. His model
was also restricted to the continuous portion of gait, not the actual
dynamics of taking a step. But it was the start. Nearly a decade later,
McGeer<elib>McGeer90</elib> followed up with a series of similarly inspired
walking machines, which he coined "passive-dynamic walkers".</p>
<subsection id="rimless_wheel"><h1>The Rimless Wheel</h1>
<figure>
<img width="40%" src="figures/rimlessWheel.svg"/>
<figcaption>The rimless wheel. The orientation of the stance leg,
$\theta$, is measured clockwise from the vertical axis. </figcaption>
</figure>
<p>Perhaps the simplest possible model of a legged robot, introduced as
the conceptual framework for passive-dynamic walking by
McGeer<elib>McGeer90</elib>, is the rimless wheel. This system has rigid
legs and only a point mass at the hip as illustrated in the figure above.
To further simplify the analysis, we make the following modeling
assumptions:</p> <ul> <li>
Collisions with ground are inelastic and impulsive (only angular momentum
is conserved around the point of collision).</li> <li> The stance foot
acts as a pin joint and does not slip.</li> <li> The transfer of support
at the time of contact is instantaneous (no double support phase).</li>
<li> $0 \le \gamma < \frac{\pi}{2}$, $0 < \alpha < \frac{\pi}{2}$, $l >
0$.</li> </ul>
<p>Note that the coordinate system used here is slightly different than
for the simple pendulum ($\theta=0$ is at the top, and the sign of
$\theta$ has changed).</p> <todo> update the coordinates (here and in
drake)</todo>
<p> The most comprehensive analysis of the rimless wheel was done by
<elib>Coleman98a</elib>.</p>
<subsubsection><h1>Stance Dynamics</h1>
<p>The dynamics of the system when one leg is on the ground are given by
$$\ddot\theta = \frac{g}{l}\sin(\theta).$$ If we assume that the system
is started in a configuration directly after a transfer of support
($\theta(0^+) = \gamma-\alpha$), then forward walking occurs when the
system has an initial velocity, $\dot\theta(0^+) > \omega_1,$ where
$$\omega_1 = \sqrt{ 2 \frac{g}{l} \left[ 1 - \cos \left (\gamma-\alpha
\right) \right]}.$$ $\omega_1$ is the threshold at which the system has
enough kinetic energy to vault the mass over the stance leg and take a
step. This threshold is zero for $\gamma = \alpha$ and does not exist
for $\gamma > \alpha$ (because when $\gamma > \alpha$ the mass is always
ahead of the stance foot, and the standing fixed point disappears). The
next foot touches down when $\theta(t) = \gamma+\alpha$, at which point
the conversion of potential energy into kinetic energy yields the
velocity $$\dot\theta(t^-) = \sqrt{\dot\theta^2(0^+) + 4\frac{g}{l}
\sin\alpha \sin\gamma }.$$ $t^-$ denotes the time immediately before the
collision.</p>
</subsubsection>
<subsubsection><h1>Foot Collision</h1>
<figure> <img width="50%" src="figures/rimlessWheel_collision.svg"/>
<figcaption>Angular momentum is conserved around the point of
impact</figcaption> </figure>
<p> The angular momentum around the point of collision at time $t$ just
before the next foot collides with the ground is $$L(t^-) =
-ml^2\dot\theta(t^-) \cos(2\alpha).$$ The angular momentum at the same
point immediately after the collision is $$L(t^+) =
-ml^2\dot\theta(t^+).$$ Assuming angular momentum is conserved, this
collision causes an instantaneous loss of velocity: $$\dot\theta(t^+) =
\dot\theta(t^-) \cos(2\alpha).$$</p>
</subsubsection>
<subsubsection><h1>Forward simulation</h1>
<p>Given the stance dynamics, the collision detection logic ($\theta =
\gamma \pm \alpha$), and the collision update, we can numerically
simulate this simple model. Doing so reveals something remarkable...
the wheel starts rolling, but then one of two things happens: it either
runs out of energy and stands still, or it quickly falls into a stable
periodic solution. Convergence to this stable periodic solution appears
to happen from a huge range of initial conditions. Try it yourself.</p>
<example><h1>Numerical simulation of the rimless wheel</h1>
<script>document.write(notebook_link('simple_legs'))</script>
<p>I've setup the notebook to make it easy for you to try a handful of
interesting initial conditions. And even made an interactive widget
for you to watch the simulation phase portrait as you change those
initial conditions. Give it a try!</p>
<p><a
href="https://github.com/RobotLocomotion/drake/blob/e08fc2bb0d289deddb5d76d0235f3a75ef3795d3/examples/rimless_wheel/rimless_wheel.h"
target="_blank">The rimless wheel model</a> actually uses a number of
the more nuanced features of <drake></drake> in order to simulate the
hybrid system accurately (as do many of the examples in this chapter).
It actually registers the collision events of the system -- the
simulator uses zero-crossing detection to ensure that the time/state
of collision is obtained accurately, and then applies the reset
map.</p>
<figure>
<img width="80%" src="figures/rimless_wheel_phase.svg"/>
<figcaption>Phase portrait of the rimless wheel
($m=1$, $l=1$, $g=9.8$, $\alpha=\pi/8$, $\gamma=0.08$).</figcaption>
</figure>
<p>One of the fantastic things about the rimless wheel is that the
dynamics are exactly the undamped simple pendulum that we've thought
so much about. So you will recognize the phase portraits of the
system -- they are centered around the unstable fixed point of the
simple pendulum.</p>
</example>
</subsubsection> <!-- end of forward simulation -->
<subsubsection><h1>Poincaré Map</h1>
<p>We can now derive the angular velocity at the beginning of each
stance phase as a function of the angular velocity of the previous
stance phase. First, we will handle the case where $\gamma \le \alpha$
and $\dot\theta_n^+ > \omega_1$. The "step-to-step return map",
factoring losses from a single collision, the resulting map is:
$$\dot\theta^{+}_{n+1} = \cos(2\alpha) \sqrt{ ({\dot\theta_n}^{+})^{2} +
4\frac{g}{l}\sin\alpha \sin\gamma}.$$ where the $\dot{\theta}^{+}$
indicates the velocity just <em>after</em> the energy loss at impact has
occurred.</p>
<p>Using the same analysis for the remaining cases, we can complete the
return map. The threshold for taking a step in the opposite direction
is $$\omega_2 = - \sqrt{2 \frac{g}{l} [1 - \cos(\alpha + \gamma)]}.$$
For $\omega_2 < \dot\theta_n^{+} < \omega_1,$ we have
$$\dot\theta_{n+1}^{+} = -\dot\theta_n^{+} \cos(2\alpha).$$ Finally, for
$\dot\theta_n^{+} < \omega_2$, we have $$\dot\theta_{n+1}^{+} = -
\cos(2\alpha)\sqrt{(\dot\theta_n^{+})^2 - 4 \frac{g}{l} \sin\alpha
\sin\gamma}.$$ Altogether, we have (for $\gamma \le \alpha$)
$$\dot\theta_{n+1}^{+} = \begin{cases} \cos(2\alpha)
\sqrt{(\dot\theta_n^{+})^2 + 4 \frac{g}{l} \sin\alpha \sin\gamma}, &
\text{ } \omega_1 < \dot\theta_n^{+} \\ -\dot\theta_n^{+} \cos(2\alpha),
& \text{ } \omega_2 < \dot\theta_n^{+} < \omega_1 \\ -\cos(2\alpha)
\sqrt{(\dot\theta_n^{+})^2 - 4\frac{g}{l} \sin\alpha \sin\gamma}, &
\dot\theta_n^{+} < w_2 \end{cases}.$$ </p>
<p>Notice that the return map is undefined for $\dot\theta_n = \{
\omega_1, \omega_2 \}$, because from these configurations, the wheel
will end up in the (unstable) equilibrium point where $\theta = 0$ and
$\dot\theta = 0$, and will therefore never return to the map.</p>
<p>This return map blends smoothly into the case where $\gamma >
\alpha$. In this regime, $$\dot\theta_{n+1}^{+} = \begin{cases}
\cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 + 4 \frac{g}{l} \sin\alpha
\sin\gamma}, & \text{ } 0 \le \dot\theta_n^{+} \\ -\dot\theta_n^{+}
\cos(2\alpha), & \text{ } \omega_2 < \dot\theta_n^{+} < 0 \\
-\cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 - 4\frac{g}{l} \sin\alpha
\sin\gamma}, & \dot\theta_n^{+} \le w_2 \end{cases}.$$ Notice that the
formerly undefined points at $\{\omega_1,\omega_2\}$ are now
well-defined transitions with $\omega_1 = 0$, because it is
kinematically impossible to have the wheel statically balancing on a
single leg.</p>
</subsubsection> <!-- poincare map -->
<subsubsection><h1>Fixed Points and Stability</h1>
<p>For a fixed point, we require that $\dot\theta_{n+1}^{+} =
\dot\theta_n^{+} = \omega^*$. Our system has two possible fixed points,
depending on the parameters: $$ \omega_{stand}^* = 0,~~~ \omega_{roll}^*
= \cot(2 \alpha) \sqrt{4 \frac{g}{l} \sin\alpha\sin\gamma}.$$ The limit
cycle plotted above illustrates a state-space trajectory in the vicinity
of the rolling fixed point. $\omega_{stand}^*$ is a fixed point whenever
$\gamma < \alpha$. $\omega_{roll}^*$ is a fixed point whenever
$\omega_{roll}^* > \omega_1$. It is interesting to view these
bifurcations in terms of $\gamma$. For small $\gamma$, $\omega_{stand}$
is the only fixed point, because energy lost from collisions with the
ground is not compensated for by gravity. As we increase $\gamma$, we
obtain a stable rolling solution, where the collisions with the ground
exactly balance the conversion of gravitational potential to kinetic
energy. As we increase $\gamma$ further to $\gamma > \alpha$, it becomes
impossible for the center of mass of the wheel to be inside the support
polygon, making standing an unstable configuration.</p>
<figure> <img width="100%" src="figures/rw_retmap.svg"/>
<figcaption>Limit cycle trajectory of the rimless wheel
($m=1,l=1,g=9.8,\alpha=\pi/8,\gamma=0.15$). All hatched regions
converge to the rolling fixed point, $\omega_{roll}^*$; the white
regions converge to zero velocity ($\omega_{stand}^*$).</figcaption>
</figure>
</subsubsection>
<subsubsection><h1>Stability of standing still</h1>
<p>I opened this chapter with the idea that the natural notion of
stability for a walking system is periodic stability, and I'll stand by
it. But we can also examine the stability of a fixed-point (of the
original coordinates; no Poincaré this time) for a system that has
contact mechanics. For a legged robot, a fixed point means standing still. This can actually be a little subtle in our hybrid models: in the rimless wheel, standing still is the limiting case where we have infinitely frequent collisions. [Details coming soon.]</p>
<todo>call out to the fact that I left the origin OUT of the poincare map on the van der pol.</todo>
</subsubsection>
</subsection> <!-- end rimless wheel -->
<subsection id="compass_gait"><h1>The Compass Gait</h1>
<p>The rimless wheel models only the dynamics of the stance leg, and
simply assumes that there will always be a swing leg in position at the
time of collision. To remove this assumption, we take away all but two of
the spokes, and place a pin joint at the hip. To model the dynamics of
swing, we add point masses to each of the legs. I've derived the
equations of motion for the system assuming that we have a torque actuator
at the hip - resulting in swing dynamics equivalent to an Acrobot
(although in a different coordinate frame) - but let's examine the passive
dynamics of the system first ($\bu = 0$).</p>
<figure>
<img width="45%" src="figures/compass_gait.svg"/>
<figcaption>The compass gait</figcaption>
</figure>
<p>In addition to the modeling assumptions used for the rimless wheel, we
also assume that the swing leg retracts in order to clear the ground
without disturbing the position of the mass of that leg. This model, known
as the compass gait, is well studied in the literature using numerical
methods <elib>Goswami96a+Goswami99+Spong03</elib>, but relatively little
is known about it analytically.</p>
<p>The state of this robot can be described by 4 variables:
$\theta_{st},\theta_{sw},\dot\theta_{st}$, and $\dot\theta_{sw}$. The
abbreviation $st$ is shorthand for the stance leg and $sw$ for the swing
leg. Using $\bq = [ \theta_{st}, \theta_{sw} ]^T$ and $\bu = \tau$, we can
write the dynamics as $$\bM(\bq)\ddot\bq + \bC(\bq,\dot\bq)\dot\bq =
\btau_g(q) + \bB\bu,$$ with \begin{gather*} \bM = \begin{bmatrix}
(m_h+m)l^2 + ma^2 & -mlb\cos(\theta_{st}-\theta_{sw}) \\
-mlb\cos(\theta_{st}-\theta_{sw}) & mb^2 \end{bmatrix}\\ \bC =
\begin{bmatrix} 0 & -mlb\sin(\theta_{st}-\theta_{sw})\dot\theta_{sw} \\
mlb\sin(\theta_{st}-\theta_{sw})\dot\theta_{st} & 0 \end{bmatrix} \\
\btau_g(q) = \begin{bmatrix} (m_hl + ma + ml)g\sin(\theta_{st}) \\
-mbg\sin(\theta_{sw}) \end{bmatrix},\\ \bB = \begin{bmatrix} -1 \\ 1
\end{bmatrix} \end{gather*} and $l=a+b$.</p>
<!-- My derivation is here: https://www.wolframcloud.com/objects/02914093-8fcd-4bd2-abb3-e76ca196b393 -->
<p>The foot collision is an instantaneous change of velocity caused by a
impulsive force at the foot that brings the foot to rest. The update to
the velocities can be calculated using the derivation for impulsive
collisions <a href="multibody.html#impulsive_collision">derived in the
appendix</a>. To use it, we proceed with the following steps:</p>
<ul> <li>Add a "floating base" to the system by adding a free (x,y) joint
at the pre-impact stance toe, $\bq_{fb} =
[x,y,\theta_{st},\theta_{sw}]^T.$</li> <li>Calculate the mass matrix for
this new system, call it $\bM_{fb}$.</li> <li>Write the position of the
(pre-impact) swing toe as a function $\bphi(\bq_{fb})$. Define the
Jacobian of this function: $\bJ = \frac{\partial \bphi}{\partial
\bq_{fb}}.$</li> <li>The post-impact velocity that ensures that $\dot\bphi
= 0$ after the collision is given by $$\dot\bq^+ = \left[ \bI -
\bM_{fb}^{-1}\bJ^T[\bJ\bM_{fb}^{-1}\bJ^T]^{-1}\bJ\right] \dot\bq^-,$$
noting that $\dot{x}^- = \dot{y}^- = 0$, and that you need only read out
the last two elements of $\dot{\bq}^+$. The velocity of the post-impact
stance foot will be zero by definition, and the new velocity of the
pre-impact stance foot can be completely determined from the minimal
coordinates.</li> <li>Switch the stance and swing leg positions and
velocities.</li> </ul>
<!-- My derivation is here:
https://www.wolframcloud.com/objects/60e3d6c1-156f-4604-bf70-1e374f471407
-->
<example><h1>Numerical simulation of the compass gait</h1>
<p>You can simulate the passive compass gait using:</p>
<script>document.write(notebook_link('simple_legs'))</script>
<p>Try playing with the initial conditions. You'll find this system is
<i>much</i> more sensitive to initial conditions than the rimless wheel.
It actually took some work to find the initial conditions that make it
walk stably.</p>
</example>
<p>Numerical integration of these equations reveals a stable limit cycle,
plotted below. Notice that when the left leg is in stance (top part of
the plot), the trajectory quite resembles the familiar pendulum dynamics
of the rimless wheel. But this time, when the leg impacts, it takes a
long arc around as the swing leg before returning to stance. The impacts
are also clearly visible as discontinuous changes (decreases) in velocity.
The dependence of this limit cycle on the system parameters has been
studied extensively in <elib>Goswami96a</elib>.</p>
<figure> <img width="90%" src="figures/compass_gait_limit_cycle.svg"/>
<figcaption>Passive limit cycle trajectory of the compass gait. ($m=5$kg,
$m_h=10$kg, $a=b=0.5$m, $\phi=0.0525$rad. $\bx(0)=[0,0,0.4,-2.0]^T$).
Drawn is the phase portait of the left leg angle, which is recovered from
$\theta_{st}$ and $\theta_{sw}$ in the simulation with some simple
book-keeping.</figcaption> </figure>
<p>The basin of attraction of the stable limit cycle is a narrow band of
states surrounding the steady state trajectory. Although the simplicity
of this model makes it attractive for conceptual explorations, this lack
of stability makes it difficult to implement on a physical device.</p>
</subsection>
<subsection id="kneed_walker"><h1>The Kneed Walker</h1>
<p>To achieve a more anthropomorphic gait, as well as to acquire better
foot clearance and ability to walk on rough terrain, we want to model a
walker that includes a knee<elib>Hsu07</elib>. The mathematical modeling
would be simpler if we used a single link for the stance leg, but we'll go
ahead and use two links for each leg (and each with a point mass) because
this robot can actually be built!</p>
<figure>
<img width="60%" src="figures/kneedwalker.svg"/>
<figcaption>The Kneed Walker</figcaption>
</figure>
<p>At the beginning of each step, the system is modeled as a three-link
pendulum, like the ballistic
walker<elib>McMahon80+Mochon80+Spong03</elib>. The stance leg is the one
in front, and it is the first link of a pendulum, with two point masses.
The swing leg has two links, with the joint between them unconstrained
until knee-strike. Given appropriate mass distributions and initial
conditions, the swing leg bends the knee and swings forward. When the
swing leg straightens out (the lower and upper length are aligned),
knee-strike occurs. The knee-strike is modeled as a discrete inelastic
collision, conserving angular momentum and changing velocities
instantaneously.</p>
<p>After this collision, the knee is locked and we switch to the compass
gait model with a different mass distribution. In other words, the system
becomes a two-link pendulum. Again, the heel-strike is modeled as an
inelastic collision. After the collision, the legs switch instantaneously.
After heel-strike then, we switch back to the ballistic walker's
three-link pendulum dynamics. This describes a full step cycle of the
kneed walker, which is shown above.</p>
<figure>
<img width="100%" src="figures/kwcycle1.svg"/>
<figcaption>Limit cycle trajectory for kneed walker</figcaption>
</figure>
<p>By switching between the dynamics of the continuous three-link and
two-link pendulums with the two discrete collision events, we characterize
a full point-feed kneed walker walking cycle. After finding appropriate
parameters for masses and link lengths, a stable gait is found. As with
the compass gait's limit cycle, there is a swing phase (top) and a stance
phase (bottom). In addition to the two heel-strikes, there are two more
instantaneous velocity changes from the knee-strikes as marked in the
figure. This limit cycle is traversed clockwise.</p>
<figure>
<img width="90%" src="figures/kwLimitCycle.svg"/>
<figcaption>Limit cycle (phase portrait) of the kneed walker</figcaption>
</figure>
<todo>reproduce this model in drake.</todo>
</subsection>
<subsection><h1>Curved feet</h1>
<p>The region of attraction of the kneed-walking limit cycle can be improved considerably by the addition of curved feet...</p>
<todo>also knee "de-bouncers" (suction cups). Simulation model?</todo>
<figure>
<iframe width="560" height="315" src="https://www.youtube.com/embed/WOPED7I5Lac" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<figcaption>Tad McGeer's kneed walker. <a href="https://www.youtube.com/watch?v=Uv7gDLWCZug">Here is</a> Matt Haberland's guide to launching it successfully. It's nontrivial!</figcaption>
</figure>
</subsection>
<subsection id="strandbeest"><h1>And beyond...</h1>
<p>The world has seen all sorts of great variations on the passive-dynamic
walker theme. Almost certainly the most impressive are the creations of
Dutch artist Theo Jansen -- he likes to say that he is creating "new forms
of life" which he calls the <a
href="https://www.strandbeest.com/">Strandbeest</a>. There are many
variations of these amazing machines -- including his <a href="https://youtu.be/LewVEF2B_pM" target="_blank">beach walkers which
are powered only by the wind</a> (I've actually been able to visit Theo's
workshop once; it is <i>very</i> windy there).</p>
<p>These results are very real. Robin Deits (an exceptionally talented
student who felt particularly inspired once on a long weekend) once
reproduced one of Theo's earliest creations in <drake></drake>.</p>
<figure>
<iframe width="560" height="315"
src="https://www.youtube.com/embed/H6fL-8ScUnU?rel=0" frameborder="0"
allow="autoplay; encrypted-media" allowfullscreen></iframe>
<figcaption>Robin Deits' <a href="https://github.com/RobotLocomotion/drake/tree/last_sha_with_original_matlab/drake/examples/Strandbeest">simulation of the Strandbeest</a>.</figcaption>
</figure>
</subsection>
</section> <!-- end of walking -->
<section><h1>Simple Models of Running</h1>
<p>Just like walking, our understanding of the dynamics and control of
running has evolved by a nice interplay between the field of biomechanics
and the field of robotics. But in running, I think it's fair to say that
the roboticists got off the starting blocks first. And I think it's fair to
say that it started with a series of hopping machines built by Marc Raibert
and the Leg Laboratory (which was first at CMU, but moved to MIT) in the
early 1980's. At a time where many roboticsts were building relatively
clumsy walking robots that moved very slowly (when they managed to stay
up!), the Leg Laboratory produced a series of hopping machines that threw
themselves through the air with considerable kinetic energy and considerable
flair.</p>
<p>To this day, I'm still a bit surprised that impressive running robots
(assuming you accept hopping as the first form of running) appeared before
the impressive walking robots. I would have thought that running would be a
more difficult control and engineering task than walking. But these hopping
machines turned out to be an incredibly clever way to build something simple
and very dynamic.</p>
<p>Shortly after Raibert built his machines, Dan Koditschek and Martin
Buehler started analyzing them with simpler models
<elib>Koditschek91</elib>. Years later, in collaboration with biologist Bob
Full, they started to realize that the simple models used to describe the
dynamics of Raibert's hoppers could also be used to describe experimental
data obtained from running animals. In fact, they described an incredible
range of experimental results, for animals ranging in size from a cockroach
up to a horse<elib>Full99</elib> (I think the best overall summary of this
line of work is <elib>Holmes06</elib>). The most successful of the simple
models was the so-called "spring-loaded inverted pendulum" (or "SLIP", for
short).</p>
<subsection><h1>The Spring-Loaded Inverted Pendulum (SLIP)</h1>
<p>The model is a point mass, $m$, on top of a massless, springy leg with
rest length of $l_0$, and spring constant $k$. The configuration of the
system is given by the $x,z$ position of the center of mass, and the
length, $l$, and angle $\theta$ of the leg. The dynamics are modeled
piecewise - with one dynamics governing the flight phase, and another
governing the stance phase.</p>
<figure>
<img width="40%" src="data/slip.svg"/>
<figcaption>The Spring-Loaded Inverted Pendulum (SLIP) model</figcaption>
</figure>
<p>In SLIP, we actually use different state variables for each phase of
the dynamics. During the flight phase, we use the state variables: $\bx
= [x,z,\dot{x},\dot{z}]^T.$ The flight dynamics are simply $$\dot{\bx} =
\begin{bmatrix} \dot{x} \\ \dot{z} \\ 0 \\ - g \end{bmatrix}.$$ Since
the leg is massless, we assume that the leg can be controlled to any
position instantaneously, so it's reasonable to take the leg angle as the control input $\theta = u$.</p>
<p>During the "stance phase", we write the dynamics in polar coordinates,
with the foot anchored at the origin. Using the state variables: $\bx =
[r, \theta, \dot{r}, \dot\theta]^T,$ the location of the mass is given by
$$\bx_m = \begin{bmatrix} - r \sin\theta \\ r \cos\theta \end{bmatrix}.$$
The total energy is given by the kinetic energy (of the mass) and the
potential energy from gravity and from the leg spring: $$T = \frac{m}{2}
(\dot{r}^2 + r^2 \dot\theta^2 ), \quad U = mgr\cos\theta +
\frac{k}{2}(r_0 - r)^2.$$ Plugging these into Lagrange yields the stance
dynamics: \begin{gather*} m \ddot{r} - m r \dot\theta^2 + m g \cos\theta
- k (r_0 - r) = 0, \\ m r^2 \ddot{\theta} + 2mr\dot{r}\dot\theta - mgr
\sin\theta = 0.\end{gather*} We assume that the stance phase is entirely
ballistic; there are no control inputs during this phase.</p>
<p>Unlike the rimless wheel model, the idealization of a spring leg with
a massless toe means that no energy is lost during the collision with the
ground. The collision event causes a transition to a different state
space, and to a different dynamics model, but no instantaneous change in
velocity. The transition from flight to stance happens when $z -
l_0\cos\theta \le 0.$ The transition from stance to flight happens when
the stance leg reaches its rest length: $r \ge l_0$.
</p>
<example><h1>Simulation of the SLIP model</h1>
<p>You can simulate the spring-loaded inverted pendulum using:</p>
<script>document.write(notebook_link('simple_legs'))</script>
<p>Take a moment to read the code that defines the
<code>SpringLoadedInvertedPendulum</code> system. You can see that we
carefully register the hybrid guards (witness functions) and reset
maps, which tells the numerical integrator to use zero-crossing event
detection to isolate precisely the time of the event and to apply the
reset update at this event. It is quite elegant, and we've gone to
some length to make sure that the numerical integration routines in
<drake></drake> can support it.</p>
<p>We used the same witness functions and resets to simulate the
walking models, but those were implemented in C++. This is a purely
Python implementation, making it a bit easier to see those details.</p>
</example>
<subsubsection><h1>Analysis on the apex-to-apex map</h1>
<p>The rimless wheel had only two state variables ($\theta$, and
$\dot\theta$), and since the surface of section drops one variable, we
ended up with a one-dimensional Poincare map. We have a lot more state
variables here. Is there any hope for graphical analysis?</p>
<p>For this system, one can still make progress if we define the
Poincare section to be at the apex of the hop (when $\dot{z}=0$), and
analyze the dynamics from apex to apex. We will ignore the absolute
horizontal position, $x$, and the leg angle, $\theta$, as they are
irrelevant to the flight dynamics. This still leaves us with two state
variables at the apex: the horizontal velocity, $\dot{x},$ and the
hopping height, $z$.</p>
<p>Importantly, at the apex, these two variables are linked -- when you
know one, you know the other. This is because no energy is ever added
or removed from the system. If you know the (constant) total energy in
the system, and you know the height at the apex, then you know the
horizontal velocity (up to a sign): $$E_{apex}(z, \dot{x}) =
\frac{1}{2}m \dot{x}^2 + mgz = E_0.$$ To make our analysis less
dependent on our specific choice of parameters, we typically report
any analysis using a dimensionless total energy, $\tilde{E} =
\frac{E}{m g l_0},$ and a dimensionless spring constant, $\tilde{k} = \frac{k l_0}{mg}$ <elib>Geyer05</elib>.</p>
<p>Putting all of these together, we can numerically evaluate the
apex-to-apex return map, given a known total energy of the system.</p>
<example><h1>Numerical SLIP apex-to-apex map</h1>
<p>Numerically integrating the equations of motion from apex-to-apex,
using zero-crossing event detection, can give us the apex-to-apex
map.</p>
<figure>
<img width="80%" src="data/slip_apex_map.svg"/>
<figcaption>Apex-to-apex return map of the SLIP model given
$\tilde{k} = 10.7$, $\tilde{E} = 1.61$, and a fixed touchdown angle
of $\theta = 30$ deg (the parameters used in
<elib>Geyer05</elib>). </figcaption>
</figure>
<p>Using our graphical staircase analysis (for discrete-time
systems), we can see that there are two fixed points of this map: one
"stable" fixed point around 0.87 meters, and one unstable fixed point
around 0.92 meters.</p>
<p>As always, you should run this code yourself to make sure you
understand:</p>
<script>document.write(notebook_link('simple_legs'))</script>
<p>I'd also recommend running the simulation using initial conditions
at various points on this plot. You should ask: why does the plot
cut off so abruptly just below the stable fixed point? Well the
apex-to-apex map becomes undefined if the apex is smaller that
$l_0\cos\theta$ (the height at touchdown). Furthemore, when $z$ is
too large (relative to $\tilde{E}$), then the system will not have
enough horizontal energy to transition through the stance phase and
continue forward motion: the center of mass will compress the spring
and then fall backwards in the direction from which it came.</p>
<todo>more videos. in particular a plot/video of the failure case of
falling backward would help a lot here.</todo>
</example>
<p>The system's conservation of total energy is quite convenient for
our return map analysis, but it should give you pause. The rimless
wheel actually <i>relied on the dissipation</i> due to collisions with
the ground to acquire its stability. Is there any chance for a system
that is energy conserving to be stable? The apex-to-apex return maps
visualized above reveal "stable" fixed-points, but you must understand
that the projection from $(z,\dot{x]})$ to just $z$ is using the
assumption that the energy is conserved. As a result, our analysis on
the one-dimensional Poincare map can only make a statement about
<i>partial stability</i> (i.s.L., asymptotic, or exponential); our
analysis does not say anything about stability against disturbances
that change the total energy of the system.</p>
<p>Despite this limitation, it is quite interesting that such a simple
system converges to particular hopping heights from various initial
conditions. The observation that these "piecewise-Hamiltonian" systems
(Hamiltonian because they conserve total energy) can exhibit this
partial stability generated quite a bit of interest from the
theoretical physics community. You can sense that enthusiasm when
reading <elib>Holmes06</elib>.</p>
<p><elib>Geyer05</elib> also gives a more thorough analysis of this
apex-to-apex map, including a closed-form approximation to the return
map dynamics based on a linear (small angle) approximation of the
stance dynamics.</p>
<todo>include Harmut's return map analysis here?</todo>
</subsubsection>
<subsubsection><h1>SLIP extensions</h1>
<p>The original SLIP model, perhaps inspired by the hopping robots
turned out to be quite effective in describing the vertical center of
mass dynamics of a wide range of animals <elib>Holmes06</elib>.
Remarkably, experiments showed that the dimensionless individual leg
stiffness was very close to 10 for a huge range of animals, ranging
from a $0.001~kg$ cockroach to a $135~kg$ horse
<elib>Koditschek04</elib>. But the SLIP model, and modest extensions
to it, has also been used to understand lateral stability in
cockroaches, resulting in some of my favorite experimental videos of
all time <elib>Jindrich02</elib>.
</p>
<todo>insert cockroach gunpowder video?</todo>
</subsubsection>
<subsubsection id="slip_control"><h1>SLIP Control</h1>
<todo>Hartmut's open-loop deadbeat, etc.</todo>
</subsubsection>
</subsection>
<subsection id="leglab"><h1>Hopping robots from the MIT Leg Laboratory</h1>
<figure>
<iframe width="560" height="315"
src="https://www.youtube.com/embed/Bd5iEke6UlE" frameborder="0"
allow="accelerometer; autoplay; encrypted-media; gyroscope;
picture-in-picture" allowfullscreen></iframe>
</figure>
<subsubsection><h1>The Planar Monopod Hopper</h1>
<p>A three-part control strategy: decoupling the control of hopping
height, body attitude, and forward velocity <elib>Raibert86a</elib>.</p>
</subsubsection>
<subsubsection><h1>Running on four legs as though they were
one</h1>
<p>bipeds, quadrupeds, and backflips...<elib>Raibert86</elib></p>
</subsubsection>
</subsection>
<subsection><h1>Towards human-like running</h1>
<todo>Add videos of 3D running from Katja (Schultz10).</todo>
</subsection>
</section> <!-- end of running -->
<section><h1>A simple model that can walk and run</h1>
</section> <!-- end of walk+run -->
<section><h1>Exercises</h1>
<exercise><h1>One-Dimensional Hopper</h1>
<p>In this exercise we implement a one-dimensional version of the controller proposed in <elib>Raibert84</elib>. The goal is to control the vertical motion of a hopper by generating a stable resonant oscillation that causes the system to hop off the ground at a given height. We accomplish this via a careful energy analysis of the hopping cycle: all the details are in <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/simple_legs/one_d_hopper/one_d_hopper.ipynb" target="_blank">this notebook</a>. Besides understanding in detail the dynamics of the hopping system, in <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/simple_legs/one_d_hopper/one_d_hopper.ipynb" target="_blank">the notebook</a>, you are asked to write two pieces of code:</p>
<ol type="a">
<li>A function that, given the state of the hopper, returns its total mechanical energy.</li>
<li>The hopping controller class. This is a <drake></drake> VectorSystem that, at each sampling time, reads the state of the hopper and returns the preload of the hopper spring necessary for the system to hop at the desired height. All the necessary information for the synthesis of this controller are given in <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/simple_legs/one_d_hopper/one_d_hopper.ipynb" target="_blank">the notebook</a>.</li>
</ol>
</exercise>
</section>
</chapter>
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