Algorithms for Walking, Running, Swimming, Flying, and Manipulation

© Russ Tedrake, 2022

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**Note:** These are working notes used for a course being taught
at MIT. They will be updated throughout the Spring 2022 semester. Lecture videos are available on YouTube.

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So far in the notes, we have concerned ourselves with only known,
deterministic systems. In this chapter we will begin to consider uncertainty.
This uncertainy can come in many forms... we may not know the governing
equations (e.g. the coefficient of friction in the joints), our robot may be
walking on unknown terrain,
subject to unknown disturbances, or even be picking up unknown objects.
There are a number of mathematical frameworks for considering this
uncertainty; for our purposes this chapter will generalizing our thinking to
equations of the form: $$\dot\bx = {\bf f}(\bx, \bu, \bw, t) \qquad \text{or}
\qquad \bx[n+1] = {\bf f}(\bx[n], \bu[n], \bw[n], n),$$ where $\bw$ is a new
*random* input signal to the equations capturing all of this potential
variability. Although it is certainly possible to work in continuous time, and
treat $\bw(t)$ as a continuous-time random signal (c.f. Wiener process), it is
notationally simpler to work with $\bw[n]$ as a discrete-time random signal.
For this reason, we'll devote our attention in this chapter to the
discrete-time systems.

In order to simulate equations of this form, or to design controllers against them, we need to define the random process that generates $\bw[n]$. It is typical to assume the values $\bw[n]$ are independent and identically distributed (i.i.d.), meaning that $\bw[i]$ and $\bw[j]$ are uncorrelated when $i \neq j$. As a result, we typically define our distribution via a probability density $p_{\bf w}(\bw[n])$. This is not as limiting as it may sound... if we wish to treat temporally-correlated noise (e.g. "colored noise") the format of our equations is rich enough to support this by adding additional state variables; we'll give an example below of a "whitening filter" for modeling wind gusts. The other source of randomness that we will now consider in the equations is randomness in the initial conditions; we will similarly define a probabilty density $p_\bx(\bx[0]).$

This modeling framework is rich enough for us to convey the key ideas; but
it is not quite sufficient for all of the systems I am interested in. In
*dimension of the state vector* may change in different realizations of
the problem! Consider, for instance, the case of a robot manipulating random
numbers of dishes in a sink. I do not know many control formulations that
handle this type of randomness well, and I consider this a top priority to
think more about! (We'll begin to address it in the output feedback chapter.)

Roughly speaking, I will refer to "stochastic control" as the discipline of
synthesizing controllers that govern the probabilistic evolution of the
equations. "Stochastic optimal control" defines a cost function (now a random
variable), and tries to find controllers that optimize some metric such as the
expected cost. When we use the terms "robust control", we are typically
referring to a class of techniques that try to guarantee a worst-case
performance or a worst-case bound on the effect of randomness on the input on
the randomness on the output. Interestingly, for many robust control
formulations we do not attempt to know the precise probability distribution of
$\bx[0]$ and $\bw[n]$, but instead only define the *sets*
of possible values that they can take. This modeling is powerful, but can
lead to conservative controllers and pessimistic estimates of performance.

My goal of presenting a relatively consumable survey of a few of the main ideas is perhaps more important in this chapter than any other. It's been said that "robust control is encrypted" (as in you need to know the secret code to get in). The culture in the robust control community has been to leverage high-powered mathematics, sometimes at the cost of offering more simple explanations. This is unfortunate, I think, because robotics and machine learning would benefit from a richer connection to these tools, and are perhaps destined to reinvent many of them.

The classic reference for robust control is

We already had quick preview into stochastic optimal control in one of the cases where it is particularly easy: finite Markov Decision Processes (MDPs).

We've already seen one nice example of robustness analysis for linear systems when we wrote a small optimization to find a common Lyapunov function for uncertain linear systems. That example studied the dynamics $\bx[n+1] = \bA \bx[n]$ where the coefficients of $\bA$ were unknown but bounded.

We also saw that
essentially the same technique can be used to certify stability against
disturbances, e.g.: $$\bx[n+1] = \bA\bx[n] + \bw[n], \qquad \bw[n] \in
\mathcal{W},$$ where $\mathcal{W}$ describes some bounded set of
possible uncertainties. In order to be compatible with convex
optimization, we often choose to describe $\mathcal{W}$ as either an
ellipsoid or as a convex polytope*invariance* of a region (if I start inside some region, then I'll never leave).

In some sense, the common-Lyapunov analysis above is probably the wrong analysis for linear systems (perhaps other systems as well). It might be unreasonable to assume that disturbances are bounded. Moreover, we know that the response to an input (including the disturbance input) is linear, so we expect the magnitude of the deviation in $\bx$ compared to the undisturbed case to be proportional to the magnitude of the disturbance, $\bw$. A more natural bound for a linear system's response is to bound the magnitude of the response (from zero initial conditions) relative to the magnitude of the disturbance.

Typically, this is done with the a scalar "$L_2$ gain", $\gamma$, defined as: \begin{align*}\argmin_\gamma \quad \subjto& \quad \sup_{\bw(\cdot) \in \int \|\bw(t)\|^2 dt\le \infty} \frac{\int_0^T \| \bx(t) \|^2 dt}{\int_0^T \| \bw(t) \|^2dt} \le \gamma^2, \qquad \text{or} \\ \argmin_\gamma \quad \subjto& \sup_{\bw[\cdot] \in \sum_n \|\bw[n]\|^2 \le \infty} \frac{\sum_0^N \|\bx[n]\|^2}{\sum_0^N \| \bw[n] \|^2} \le \gamma^2.\end{align*} The name "$L_2$ gain" comes from the use of the $\ell_2$ norm on the signals $\bw(t)$ and $\bx(t)$, which is assumed only to be finite.

More often, these gains are written not in terms of $\bx[n]$ directly, but in terms of some "performance output", $\bz[n]$. For instance, if would would like to bound the cost of a quadratic regulator objective as a function of the magnitude of the disturbance, we can minimize $$ \min_\gamma \quad \subjto \quad \sup_{\bw[n]} \frac{\sum_0^N \|\bz[n]\|^2}{\sum_0^N \| \bw[n] \|^2} \le \gamma^2, \qquad \bz[n] = \begin{bmatrix}\sqrt{\bQ} \bx[n] \\ \sqrt{\bR} \bu[n]\end{bmatrix}.$$ This is a simple but important idea, and understanding it is the key to understanding the language around robust control. In particular the $H_2$ norm of a system (from input $\bw$ to output $\bz$) is the energy of the impulse response; when $\bz$ is chosen to represent the quadratic regulator cost as above, it corresponds to the expected LQR cost. The $H_\infty$ norm of a system (from $\bw$ to $\bz$) is the largest singular value of the transfer function; it corresponds to the $L_2$ gain.

Coming soon...

Coming soon... See, for instance,

The standard criticism of $H_2$ optimal control is that minimizing the expected value does not allow any guarantees on performance. The standard criticism of $H_\infty$ optimal control is that it concerns itself with the worst case, and may therefore be conservative, especially because distributions over possible disturbances chosen a priori may be unnecessarily conservative. One might hope that we could get some of this performance back if we are able to update our models of uncertainty online, adapting to the statistics of the disturbances we actually receive. This is one of the goals of adaptive control.

One of the fundamental problems in online adaptive control is the trade-off between exploration and exploitation. Some inputs might drive the system to build more accurate models of the dynamics / uncertainty quickly, which could lead to better performance. But how can we formalize this trade-off?

There has been some nice progress on this challenge in machine
learning in the setting of (contextual) multi-armed
bandit problems. For our purposes, you can think of bandits as a
limiting case of an optimal control problem where there are no dynamics
(the effects of one control action do not effect the results of the next
control action). In this simpler setting, the online optimization
community has developed exploration-exploitation strategies based on the
notion of minimizing *regret* -- typically the accumulated
difference in the performance achieved by my online algorithm vs the
performance that would have been achieved if I had been privy to the data
from my experiments before I started. This has led to methods that make
use of concepts like upper-confidence bound (UCB) and more recently
bounds using a least-squares squares confidence bound

In the last few years, we've see these results translated into the setting of linear optimal control...

Coming soon...

L2-gain with dissipation inequalities. Finite-time verification with sums of squares.

Occupation Measures

SOS-based design- "Essentials of Robust Control", Prentice Hall , 1997. ,
- "Robust Control Design using H-infinity Methods", Springer-Verlag , 2000. ,
- "Convex Optimization", Cambridge University Press , 2004. ,
- "Dissipation inequalities in systems theory: An introduction and recent results", Invited Lectures of the International Congress on Industrial and Applied Mathematics 2007 , pp. 23-42, 2009. ,
- "Linear {Matrix} {Inequalities} in {Control}", Online Draft , pp. 293, 2015. ,
- "Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games", IEEE Transactions on Automatic Control, vol. 18, no. 2, pp. 124--131, apr, 1973. ,
- "A Tutorial on Geometric Programming", Optimization and Engineering, vol. 8, no. 1, pp. 67-127, 2007. ,
- "Risk-sensitive optimal control", Wiley New York , vol. 20, 1990. ,
- "Beyond {UCB}: Optimal and Efficient Contextual Bandits with Regression Oracles", Proceedings of the 37th International Conference on Machine Learning , vol. 119, pp. 3199--3210, 13--18 Jul, 2020. ,

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