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
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In this chapter we will finally start considering systems of the form: \begin{gather*} \bx[n+1] = {\bf f}(\bx[n], \bu[n], \bw[n]) \\ \by[n] = {\bf g}(\bx[n], \bu[n], \bv[n]),\end{gather*} where most of these symbols have been described before, but we have now added $\by[n]$ as the output of the system, and $\bv[n]$ which is representing "measurement noise" and is typically the output of a random process. In other words, we'll finally start addressing the fact that we have to make decisions based on sensor measurements -- most of our discussions until now have tacitly assumed that we have access to the true state of the system for use in our feedback controllers (and that's already been a hard problem).

In some cases, we will see that the assumption of "full-state feedback" is not so bad -- we do have good tools for state estimation from raw sensor data. But even our best state estimation algorithms do add some dynamics to the system in order to filter out noisy measurements; if the time constants of these filters is near the time constant of our dynamics, then it becomes important that we include the dynamics of the estimator in our analysis of the closed-loop system.

In other cases, it's entirely too optimistic to design a controller assuming that we will have an estimate of the full state of the system. Some state variables might be completely unobservable, others might require specific "information-gathering" actions on the part of the controller.

For me, the problem of robot manipulation is one important application domain where more flexible approaches to output feedback become critically important. Imagine you are trying to design a controller for a robot that needs to button the buttons on your shirt. Our current tools would require us to first estimate the state of the shirt (how many degrees of freedom does my shirt have?); but certainly the full state of my shirt should not be required to button a single button. Or if you want to program a robot to make a salad -- what's the state of the salad? Do I really need to know the positions and velocities of every piece of lettuce in order to be successful? These questions are (finally) getting a lot of attention in the research community these days, under the umbrella of "learning state representations". But what does it mean to be a good state representation? There are a number of simple lessons from output feedback in control that can shed light on this fundamental question.

To some extent, this idea of calling out "output feedback" as an
advanced topic is a relatively new problem. Before state-space and
optimization-based approaches to control ushered in "modern control", we
had "classical control". Classical control focused predominantly (though
not exclusively) on linear time-invariant (LTI) systems, and made very
heavy use of frequency-domain analysis (e.g. via the Fourier
Transform/Laplace Transform). There are many excellent books on the
subject;

What's important for us to acknowledge here is that in classical control, basically everything was built around the idea of output feedback. The fundamental concept is the transfer function of a system, which is a input-to-output map (in frequency domain) that can completely characterize an LTI system. Core concepts like pole placement and loop shaping were fundamentally addressing the challenge of output feedback that we are discussing here. Sometimes I feel that, despite all of the things we've gained with modern, optimization-based control, I worry that we've lost something in terms of considering rich characterizations of closed-loop performance (rise time, dwell time, overshoot, ...) and perhaps even in practical robustness of our systems to unmodeled errors.

Just like some of our oldest approaches to control were fundamentally solving an output feedback problem, some of our newest approaches to control are doing it, too. Deep learning has revolutionized computer vision, and "deep imitation learning" and "deep reinforcement learning" have been a recent source of many impressive demonstrations of control systems that can operate directly from pixels (e.g. consuming the output of a deep perception system), without explicitly representing nor estimating the full state of the system. Unfortunately, the success or failure of these methods are not yet well understood, and they often require a great deal of artisanal tuning and an embarrassing (sometimes prohibitive) amount of computation.

The synthesis of ideas between machine learning (both theoretical and applied) and control theory is one of the most exciting and productive frontiers for research today. I am highly optimistic that we will be able to uncover the underlying principles and help transition this budding field into a technology. I hope that summarizing some of the key lessons from control here can help.

One of the extremely important almost unstated lessons from dynamic
programming with additive costs and the Bellman equation is that the
optimal policy can *always* be represented as a function $\bu^*
= \pi^*(\bx).$ So far in these notes, we've assumed that the controller has
direct access to the true state, $\bx$. In this chapter, we are finally
removing that assumption. Now the controller only has direct access to the
potentially noisy observations $\by$.

So the natural first question to ask might be, what happens if we write
our policies now as a function, $\bu = \pi(\by)?$ This is known as
"static" output feedback, in contrast to "dynamic" output feedback where
the controller is not a static function, but is itself another input-output
dynamical system. Unfortunately, in the general case it is *not* the
case that optimal policies can be perfectly represented with static output
feedback. But one can still try to solve an optimal control problem where
we restrict our search to static policies; our goal will be to find the
best controller in this class to minimize the cost.

We've already seen an example of a very simple linear control problem where the set of stabilizing feedback gains formed a disconnected set -- which is suggestive that it could be a difficult problem for optimization. For some other problems in control, we've been able to find a convex reparametrization.

Unfortunately,

Just because this problem is NP-hard doesn't mean we can't find good controllers in practice. Some of the recent results from reinforcement learning have reminded us of this. We should not expect an efficient globally optimal algorithm that works for every problem instance; but we should absolutely keep working on the problem. Perhaps the class of problems that our robots will actually encounter in the real world is a easier than this general case (the standard examples of bad cases in linear systems, e.g. with interleaved poles and zeros, do feel a bit contrived and unlikely to occur in practice).

Searching for the best controller within a parametric class of policies is generally referred to as policy search. If we do policy search on a class of static output feedback policies, how well does it perform? Of course, the answer depends on the particular governing equations (for instance, $\by = \bx$ is a perfectly reasonable output, and in this case the policy can be optimal).

Bilinear alternations with SOS, Policy search with SGD, ...

Here is a simple extension of the LQR with least-squares derivation... (it's a work in progress!)

Given the state space equations: \begin{gather*} \bx[n+1] = \bA\bx[n] + \bB\bu[n] + \bw[n],\end{gather*} Consider parametrizing an output feedback policy of the form $$\bu[n] = \bK_0[n] \bx_0 + \sum_{i=1}^{n-1}\bK_i[n]\bw[n-i],$$ then the closed-loop state is convex in the control parameters, $\bK$: \begin{align*}\bx[n] =& \left( {\bf A}^n + \sum_{i=0}^{n-1}{\bf A}^{n-i-1}{\bf B}{\bf K}_0[i] \right) \bx_0 + \sum_{j=0}^{n-1} \sum_{i=0}^{n-1}{\bf A}^{n-i-1}{\bf B}{\bf K}_{j}[i] \bw[i-j],\end{align*} and therefore objectives that are convex in $\bx$ and $\bu$ (like LQR) are also convex in $\bK$. Moreover, we can calculate $\bw[n]$ by the time that it is needed given our observations of $\bx[n+1], \bx[n],$ and knowledge of $\bu[n].$

We can extend this to output disturbance-based feedback: \begin{gather*}
\bx[n+1] = \bA\bx[n] + \bB\bu[n] + \bw[n],\\ \by[n] = \bC\bx[n] + \bv[n],
\end{gather*} by parametrizing an output feedback policy of the form
$$\bu[n] = \bK_0[n] \by[0] + \sum_{i=1}^{n-1}\bK_i[n]{\bf e}[n-i],$$ where ${\bf e}[n] = ...$

- "Linear Systems Theory", Princeton Press , 2009. ,
- "Feedback systems: an introduction for scientists and engineers", Princeton university press , 2010. ,
- "{NP}-hardness of some linear control design problems", SIAM journal on control and optimization, vol. 35, no. 6, pp. 2118--2127, 1997. ,
- "Robust Output Feedback Control with Guaranteed Constraint Satisfaction", In the Proceedings of 23rd ACM International Conference on Hybrid Systems: Computation and Control , pp. 12, April, 2020. [ link ] ,

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