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<!DOCTYPE html>
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<head>
<title>Ch. 6 - Model Systems
with Stochasticity</title>
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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, 2023<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="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 style="counter-reset: chapter 5"><h1>Model Systems
with Stochasticity</h1>
<!-- TODO: Adopt more of the 6.437 notation; it's excellent and compatible:
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<p>My goals for this chapter are to build intuition for the beautiful and rich
behavior of nonlinear dynamical system that are subjected to random
(noise/disturbance) inputs. So far we have focused primarily on systems
described by \[ \dot{\bx}(t) = f(\bx(t),\bu(t)) \quad \text{or} \quad \bx[n+1]
= f(\bx[n],\bu[n]). \] In this chapter, I would like to broaden the scope to
think about \[ \dot{\bx}(t) = f(\bx(t),\bu(t),\bw(t)) \quad \text{or} \quad
\bx[n+1] = f(\bx[n],\bu[n],\bw[n]), \] where this additional input $\bw$ is
the (vector) output of some random process. In other words, we can begin
thinking about stochastic systems by simply understanding the dynamics of our
existing ODEs subjected to an additional random input. </p>
<p> This form is extremely general as written. $\bw(t)$ can represent
time-varying random disturbances (e.g. gusts of wind), or even constant model
errors/uncertainty. One thing that we are not adding, yet, is measurement
uncertainty. There is a great deal of work on observability and state
estimation that study the question of how you can infer the true state of the
system given noise sensor readings. For this chapter we are assuming perfect
measurements of the full state, and are focused instead on the way that
"process noise" shapes the long-term dynamics of the system. </p>
<p> I will also stick primarily to discrete time dynamics for this chapter,
simply because it is easier to think about the output of a discrete-time
random process, $\bw[n]$, than a $\bw(t)$. But you should know that all of the
ideas work in continuous time, too. Also, most of our examples will take the
form of <i>additive noise</i>: \[ \bx[n+1] = f(\bx[n],\bu[n]) + \bw[n], \]
which is a particular useful and common specialization of our general form.
And this form doesn't give up much -- the disturbance on step $k$ can pass
through the nonlinear function $f$ on step $k+1$ giving rich results -- but is
often much easier to work with.</p>
<p>When implementing stochastic systems in Drake, we aim to be <a
href="https://drake.mit.edu/doxygen_cxx/group__stochastic__systems.html">rigorous
about randomness</a>, to facilitate advanced algorithms that are written
against stochastic systems and to enable, e.g. random simulation roll outs
but still enable deterministic replay (simulation is deterministic given the
random seed).</p>
<section><h1>The Master Equation</h1>
<p>Let's start by looking at some simple examples.</p>
<example><h1>A Bistable System + Noise</h1>
<p>Let's consider (a time-reversed version of) one of my favorite
one-dimensional systems: \[ \dot{x} = x - x^3. \] <todo>add plot</todo>
This deterministic system has stable fixed points at $x^* = \{-1,1\}$ and
an unstable fixed point at $x^* = 0$. </p> <p> A reasonable discrete-time
approximation of these dynamics, now with additive noise, is \[ x[n+1] =
x[n] + h (x[n] - x[n]^3 + w[n]). \] When $w[n]=0$, this system has the
same fixed points and stability properties as the continuous time system.
But let's examine the system when $w[n]$ is instead the result of a
<i>zero-mean Gaussian white noise process</i>, defined by: \begin{gather*}
\forall n, E\left[ w[n] \right] = 0,\\ E\left[ w[i]w[j] \right] =
\begin{cases} \sigma^2, & \text{ if } i=j,\\ 0, & \text{ otherwise.}
\end{cases} \end{gather*} Here $\sigma$ is the standard deviation of the
Gaussian. </p> <p>When you simulate this system for small values of
$\sigma$, you will see trajectories move roughly towards one of the two
fixed points (for the deterministic system), but every step is modified by
the noise. In fact, even if the trajectory were to arrive exactly at what
was once a fixed point, it is almost surely going to move again on the
very next step. In fact, if we plot many runs of the simulation from
different initial conditions all on the same plot, we will see something
like the figure below.</p> <figure> <img width="80%"
src="figures/cubic_bistable_particles_time.svg"/>
<figcaption>Simulation of the bistable system with noise ($h=0.1, \sigma = 0.3$)
from many initial conditions.</figcaption> </figure>
<p>During any individual simulation, the state jumps around randomly for
all time, and could even transition from the vicinity of one fixed point
to the other fixed point. Another way to visualize this output is to
animate a histogram of the particles over time.</p>
<figure>
<iframe style="border:0;height:300px;width:400px;" src="data/cubic_bistable.html?height=240px" pdf="no"></iframe>
<figcaption>Histogram of the states of the bistable system with noise
($\sigma = 1$) after simulating from random initial conditions until
$t=20$.</figcaption> </figure>
<p>You can run this demo for yourself:</p>
<script>document.write(notebook_link('stochastic'))</script>
</example>
<p>Let's take a moment to appreciate the implications of what we've just
observed. Every time that we've analyzed a system to date, we've asked
questions like "given x[0], what is the long-term behavior of the system,
$\lim_{n\rightarrow\infty} x[n]$?", but now $x[n]$ is a <i>random
variable</i>. The trajectories of this system do not converge, and the
system does not exhibit any form of stability that we've introduced so far.
</p>
<p>All is not lost. If you watch the animation closely, you might notice
the <i>distribution</i> of this random variable is actually very well
behaved. This is the key idea for this chapter.</p>
<p>Let us use $p_n(x)$ to denote the <a
href="https://en.wikipedia.org/wiki/Probability_density_function">
probability density function</a> over the random variable $x$ at time $n$.
Note that this density function must satisfy \[ \int_{-\infty}^\infty p_n(x)
dx = 1.\] It is actually possible to write the <i>dynamics of the
probability density</i> with the simple relation \[ p_{n+1}(x) =
\int_{-\infty}^\infty p(x|x') p_n(x') dx', \] where $p(x|x')$ encodes the
stochastic dynamics as a conditional distribution of the next state (here
$x$) as a function of the current state (here $x'$). Dynamical systems that
can be encoded in this way are known as <i>continuous-state Markov
Processes</i>, and the governing equation above is often referred to as the
"<a href="https://en.wikipedia.org/wiki/Master_equation">master
equation</a>" for the stochastic process. In fact this update is even
linear(!) ; a fact that can enable closed-form solutions to some impressive
long-term statistics, like mean time to failure or first passage
times<elib>Byl08f</elib>. Unfortunately, it is often difficult to perform
the integral in closed form, so we must often resort to discretization or
numerical analysis. </p>
<example id="logistic"><h1>The Logistic Map</h1>
<p>In fact, one does not actually need stochastic dynamics in order for
the dynamics of the distribution to be the meaningful object of study;
random initial conditions can be enough. One of the best examples comes
from perhaps the simplest and most famous example of a chaotic system: the
logistic map. This example is described beautifully in
<elib>Lasota13</elib>.</p>
<p>Consider the following difference equation: $$x[n+1] = 4
x[n](1-x[n]),$$ which we will study over the (invariant) interval $x \in
[0, 1]$. </p>
<figure> <img width="80%" src="figures/logistic_map.svg"/>
<figcaption>The discrete-time dynamics of the "logistic map" (plotted
alongside the line $x[n+1] = x[n]$).</figcaption>
</figure>
<p>It takes only a moment of tracing your finger along the plot using the
"<a href="simple_legs.html#staircase">staircase method</a>" to see what
makes this system so interesting -- rollouts from a single initial
condition end up bouncing all over the interval $(0,1)$, and neighboring
initial conditions will end up taking arbitrarily different trajectories
(this is the hallmark of a "chaotic" system).</p>
<figure> <img width="80%" src="figures/logistic_map_rollouts.svg"/>
<figcaption>Two simulations of the logistic map from different initial
conditions. Remember, there is no stochasticity here -- the dynamics
are entirely deterministic!</figcaption>
</figure>
<p>Here's what's completely fascinating -- even though the dynamics of any
one initial condition for this system are extremely complex, if we study
the dynamics of a distribution of states through the system, they are
surprisingly simple and well-behaved. This system is one of the rare
cases when we can write the master equation in closed
form<elib>Lasota13</elib>: $$p_{n+1}(x) = \frac{1}{4\sqrt{1-x}} \left[
p_n\left(\frac{1}{2}-\frac{1}{2}\sqrt{1-x}\right) + p_n\left(\frac{1}{2} +
\frac{1}{2}\sqrt{1-x}\right) \right].$$ Moreover, this master equation
has a steady-state solution: $$p_*(x) = \frac{1}{\pi\sqrt{x(1-x)}}.$$ </p>
<figure> <img width="116%" style="margin-left:-8%" src="figures/logistic_map_master.svg"/>
<figcaption>Plotting the (closed-form) evolution of the master equation
for the logistic map, initialized with $p_0(x) = 1$ for $x\in[0,1]$,
reveals surprising simplicity, and rapid convergence to a stationary
distribution (dashed orange line).</figcaption>
</figure>
<p>For this system (and many chaotic systems), the dynamics of a single initial condition are complicated, but the dynamics a <i>distribution</i> of initial conditions are beautiful and simple!</p>
<p>Note: when the dynamical system under study has deterministic dynamics
(but a distribution of initial conditions), the linear map given by the
master equation is known as the <i>Perron-Frobenius operator</i>, and it
gives rise to the Liouville equation that we will study later in the
text.</p>
</example>
<p>The slightly more general form of the master equation, which works for
multivariate distributions with state-domain ${\cal X}$, and systems with
control inputs $\bu$, is \[ p_{n+1}(\bx) = \int_{\cal X} p(\bx|\bx',\bu)
p_n(\bx') d\bx'. \] This is a (continuous-state) Markov Decision
Process.</p>
<p>Continuous-time formulations are also possible -- these lead to the
so-called <a
href="https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation">
Fokker-Planck equation</a>.</p>
<todo>mention Chapman-Kolmogorov?</todo>
<todo>mention Ito Calculus?</todo>
</section>
<section><h1>Stationary Distributions</h1>
<p>In the example above, the histogram is our numerical approximation of
the probability density. The logistic map example had the remarkable
property that, although the individual trajectories of the system <i>do
not</i> converge, the probability distribution actually <i>does</i>
converge to what's known as a <i>stationary distribution</i> -- a fixed
point of the master equation. Instead of thinking about the dynamics of
the trajectories, we need to start thinking about the dynamics of the
distribution.</p>
<p>The most important example of this analysis is for systems with linear
dynamics and additive Gaussian noise; for this case we have closed-form
solutions (in arbitrary dimensions). But to keep things simple, let's make
sure we understand the scalar case first.</p>
<example id="linear_gaussian"><h1>A One-dimensional Linear Gaussian
System</h1>
<p>Let's consider the one-dimensional linear system with additive noise:
\[ x[n+1] = a x[n] + w[n]. \] When $w[n]=0$, the system is stable (to the
origin) for $-1 < a < 1$. Let's make sure that we can understand the
dynamics of the master equation for the case when $w[n]$ is Gaussian white
noise with standard deviation $\sigma$. </p>
<p>First, recall that the probability density function of a Gaussian with
mean $\mu$ is given by \[ p_w(w) = \frac{1}{\sqrt{2 \pi \sigma_w^2}}
e^{-\frac{(w-\mu)^2}{2\sigma_w^2}}. \] When conditioned on $x[n]$, the
distribution given by the dynamics subjected to mean-zero Gaussian white
noise is simply another Gaussian, with the mean given by $ax[n]$: \[
p(x[n+1]|x[n]) = \frac{1}{\sqrt{2 \pi \sigma_w^2}}
e^{-\frac{(x[n+1]-ax[n])^2}{2\sigma_w^2}}, \] yielding the master equation
\[ p_{n+1}(x) = \frac{1}{\sqrt{2 \pi \sigma_w^2}} \int_{-\infty}^\infty
e^{-\frac{(x-ax')^2}{2\sigma_w^2}} p_n(x') dx'. \] </p>
<p>Now here's the magic. Let's push a distribution, $p_n(x)$, which is
zero-mean, with standard deviation $\sigma_n$, through the master
equation: \begin{align*} p_{n+1}(x) =& \frac{1}{\sqrt{2 \pi
\sigma_w^2}}\frac{1}{\sqrt{2 \pi \sigma_n^2}} \int_{-\infty}^\infty
e^{-\frac{(x-ax')^2}{2\sigma_w^2}} e^{-\frac{(x')^2}{2\sigma_n^2}} dx',\\
=& \frac{1}{\sqrt{2 \pi (\sigma_w^2+a^2 \sigma_n^2)}}
e^{-\frac{x^2}{2(\sigma_w^2 + a^2 \sigma_n^2)}}. \end{align*} The result
is another mean-zero Gaussian with $\sigma_{n+1}^2 = \sigma_w^2 + a^2
\sigma_n^2$. This result generalizes to the multi-variate case, too, and
might be familiar to you e.g. from the process update of the <a
href="https://en.wikipedia.org/wiki/Kalman_filter">Kalman filter</a>.
</p>
<p>Taking it a step further, we can see that a stationary distribution for
this system is given by a mean-zero Gaussian with \[ \sigma_*^2 =
\frac{\sigma_w^2}{1-a^2}. \] Note that this distribution is well defined
when $-1 < a < 1$ (only when the system is stable).</p>
</example>
<p>The stationary distribution of the linear Gaussian system reveals the
fundamental and general balance between two terms in the governing equations
of any stochastic dynamical system: the stability of the deterministic
system is bringing trajectories together (smaller $a$ means faster
convergence of the deterministic system and results in a more narrow
distribution), but the noise in the system is forcing trajectories apart
(larger $\sigma$ means larger noise and results in a wider distribution).
</p>
<p>Given how rich the dynamics can be for deterministic nonlinear systems,
you can probably imagine that the possible long-term dynamics of the
probability are also extremely rich. If we simply flip the signs in the
cubic polynomial dynamics we examined above, we'll get our next
example:</p>
<example><h1>The Cubic Example + Noise</h1>
<p>Now let's consider the discrete-time approximation of \[ \dot{x} = -x +
x^3, \] again with additive noise: \[ x[n+1] = x[n] + h (-x[n] + x[n]^3 +
w[n]). \] With $w[n]=0$, the system has only a single stable fixed point
(at the origin), whose deterministic region of attraction is given by $x
\in (-1,1)$. If we again simulate the system from a set of random initial
conditions and plot the histogram, we will see something like the figure
below. </p>
<figure> <img width="80%" src="figures/cubic_histogram.svg"/>
<figcaption>Histogram of the states of the bistable system with noise
($h=0.1, \sigma = 1.5$) after simulating from random initial conditions
until $t=1$.</figcaption> </figure>
<figure>
<iframe style="border:0;height:300px;width:400px;" src="data/cubic_histogram.html?height=240px" pdf="no"></iframe>
<figcaption>Histogram of the states of the bistable system with noise
($\sigma = 1.5$).</figcaption> </figure>
<p>Be sure to watch the animation. Or better yet, run the simulation for
yourself by changing the sign of the derivative in the bistability example
and re-running:</p>
<p>You can run this demo for yourself:</p>
<script>document.write(notebook_link('stochastic'))</script>
<p>What is the stationary distribution for this system? In fact, there
isn't one. Although we initially collect probability density around the
stable fixed point, you should notice a slow leak -- on every step there
is some probability of transitioning past unstable fixed points and
getting driven by the unstable dynamics off towards infinity. If we run
the simulation long enough, there won't be any probability density left at
$x=0$.</p>
</example>
<example><h1>The Stochastic Van der Pol Oscillator</h1>
<p>One more example; this is a fun one. Let's think about one of the
simplest examples that we had for a system that demonstrates limit cycle
stability -- the Van der Pol oscillator -- but now we'll add Gaussian
white noise, $$\ddot{q} + (q^2 - 1)\dot{q} + q = w(t),$$ Here's the
question: if we start with a small set of initial conditions centered
around one point on the limit cycle, then what is the long-term behavior
of this distribution? </p>
<figure> <img width="80%" src="figures/vanderpol_particles.svg"/>
<figcaption>Randomly sampled initial conditions pictured with the stable
limit cycle of the Van der Pol oscillator.</figcaption> </figure>
<p>Since the long-term behavior of the deterministic system is periodic,
it would be very logical to think that the state distribution for this
stochastic system would fall into a stable periodic solution, too. But
think about it a little more, and then watch the animation (or run the
simulation yourself).</p>
<script>document.write(notebook_link('stochastic'))</script>
<todo>improve the visualization</todo>
<figure>
<iframe style="border:0;height:600px;width:420px;" src="data/vdp_particles.html?height=480px" pdf="no"></iframe>
<figcaption>Rollouts of the stochastic Van der Pol oscillator.</figcaption>
</figure>
<p>The explanation is simple: the periodic solution of the system is only
<i>orbitally stable</i>; there is no stability along the limit cycle. So
any disturbances that push the particles along the limit cycle will go
unchecked. Eventually, the distribution will "mix" along the entire
cycle. Perhaps surprisingly, this system that has a limit cycle stability
when $w=0$ eventually reaches a stationary distribution (fixed point) in
the master equation.</p>
<figure>
<img width="420px" src="data/vdp_particles.svg"/>
<figcaption>Sampled approximation of the stationary distribution of the stochastic Van der Pol oscillator.</figcaption>
</figure>
</example>
</section>
<section><h1>Extended Example: The Rimless Wheel on Rough
Terrain</h1>
<p>My favorite example of a meaningful source of randomness on a model
underactuated system is the <a href="simple_legs.html#rimless_wheel">rimless
wheel</a> rolling down stochastically "rough" terrain<elib>Byl08f</elib>.
Generating interesting/relevant probabilistic models of terrain in general
can be quite complex, but the rimless wheel makes it easy -- since the robot
only contacts that ground at the point foot, we can model almost arbitrary
rough terrain by simply taking the ramp angle, $\gamma$, to be a random
variable. If we restrict our analysis to rolling only in one direction (e.g.
only downhill), then we can even consider this ramp angle to be i.i.d.;
after each footstep we will independently draw a new ramp angle $\gamma[n]$
for the next step.</p>
<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>
<todo>Update this to have rough terrain.</todo>
</figure>
<p>In our original analysis of the rimless wheel, we derived the "post-collision" return map -- mapping the angular velocity from the beginning of the stance phase to the (post-collision) angular velocity at the next stance phase. But now that the relative location of the ground is changing on every step, we instead write the "apex-to-apex" return map, which maps the angular velocity from one apex (the moment that the leg is vertical) to the next, which is given by: $$\dot\theta[n+1] = \sqrt{\cos^2\alpha \left( \dot\theta^2[n] + \frac{2g}{l}\left(1-\cos(\alpha + \gamma[n])\right)\right) - \frac{2g}{l}(1 - \cos(\alpha - \gamma[n]))}.$$ </p>
<p>More coming soon. Read the paper <elib>Byl08f</elib> and/or watch the <a
href="https://youtu.be/TaaJgvjYBQc">video</a>.</p>
</section>
<section><h1>Noise models for real robots/systems.</h1>
<p>Sensor models. Beam model from probabilistic robotics. RGB-D
dropouts.</p>
<p>Perception subsystem. Output of a
perception system is not Gaussian noise, it's missed
detections/drop-outs...</p>
<p>Distributions over tasks/environments.</p>
</section>
<todo>Section on the liouville equation (ode describring the Perron-Frobenius along a path.</todo>
</chapter>
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<div id="references"><section><h1>References</h1>
<ol>
<li id=Byl08f>
<span class="author">Katie Byl and Russ Tedrake</span>,
<span class="title">"Metastable Walking Machines"</span>,
<span class="publisher">International Journal of Robotics Research</span>, vol. 28, no. 8, pp. 1040-1064, August 1, <span class="year">2009</span>.
[ <a href="http://ijr.sagepub.com/cgi/content/abstract/0278364909340446v1">link</a> ]
</li><br>
<li id=Lasota13>
<span class="author">Andrzej Lasota and Michael C Mackey</span>,
<span class="title">"Chaos, fractals, and noise: stochastic aspects of dynamics"</span>, Springer Science \& Business Media
, vol. 97, <span class="year">2013</span>.
</li><br>
</ol>
</section><p/>
</div>
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