Underactuated Robotics

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

Russ Tedrake

© Russ Tedrake, 2020
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Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Spring 2020 semester. Lecture videos are available on YouTube.

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Algorithms for Limit Cycles

The discussion of walking and running robots in Chapter 4 motivated the notion of limit cycle stability. Linear systems are not capable of producing stable limit cycle behavior, so this rich topic is unique to nonlinear systems design and analysis. Furthermore, the tools that are required to design, stabilize, and verify limit cycles will have applicability beyond simple periodic motions.

The first natural question we must ask is, given a system $\dot{\bx} = f(\bx)$, or a control system $\dot{x} = f(\bx,\bu)$, how do we go about finding periodic solutions which may be passively stable, open-loop stable, or stabilizable via closed-loop feedback? It turns out that the trajectory optimization tools that we developed already are very well suited to this task.

Trajectory optimization

I introduced the trajectory optimization tools as means for optimizing a control trajectory starting from a particular known initial condition. But the fundamental idea of optimizing over individual trajectories of the system is useful more broadly. Even for a passive system, we can formulate the search for a periodic solution as an optimization over trajectories that satisfy the dynamic constraints and periodicity constraints, $\bx[0] = \bx[N]$: \begin{align*} \find_{\bx[\cdot]} \quad \subjto \quad & \bx[n+1] = f(\bx[n]), \quad \forall n\in[0, N-1] \\ & \bx[0] = \bx[N]. \end{align*} Certainly we can add control inputs back into the formulation, too, but let's start with this simple case. Take a moment to think about the feasible solutions to this problem formulation. Certainly a fixed point $\bx[n] = \bx^*$ will satisfy the constraints; if we don't want these solutions to come out of the solver we might need to exclude them with constraints or add an objective that guides the solver towards the desired solutions. The other possible solutions are trajectories that are periodic in exactly $N$ steps. That's pretty restrictive.

We can do better if we use the continuous-time formulations. For instance, in our introduction of direct collocation, we wrote \begin{align*} \min_{\bx[\cdot],\bu[\cdot]} \quad & \ell_f(\bx[N]) + \sum_{n_0}^{N-1} h_n \ell(\bx[n],\bu[n]) \\ \subjto \quad & \dot\bx(t_{c,n}) = f(\bx(t_{c,n}), \bu(t_{c,n})), & \forall n \in [0,N-1] \\ & \bx[0] = \bx_0 \\ & + \text{additional constraints}. \end{align*} But we can also add $h_n$ as decision variables in the optimization (reminder: I recommend setting a lower-bound $h_n \ge h_{min} > 0$). This allows our $N$-step trajectory optimization to scale and shrink time in order to satisfy the periodicity constraint. The result is simple and powerful.

Finding the limit cycle of the Van der Pol oscillator

Recall the dynamics of the Van der Pol oscillator given by $$\ddot{q} + \mu (q^2 - 1) \dot{q} + q = 0, \quad \mu>0,$$ which exhibited a stable limit cycle.

Formulate the direct collocation optimization: \begin{align*} \find_{\bx[\cdot],h} \quad \subjto \quad & q[0] = 0, \quad \dot{q}[0] > 0, \\ & \bx[N] = \bx[0], \text{(periodicity constraint)}\\ & \text{collocation dynamic constraints} \\ & 0.01 \le h \le 0.5 \end{align*}

Try it yourself:

examples/limit_cycles.ipynb

As always, make sure that you take a look at the code. Poke around. Try changing some things.

One of the things that you should notice in the code is that I provide an initial guess for the solver. In most of the examples so far I've been able to avoid doing that--the solver takes small random numbers as a default initial guess and solves from there. But for this problem, I found that it was getting stuck in a local minima. Adding the initial guess that the solution moves around a circle in state space was enough.

example: simplest flapping robot

Lyapunov analysis

Recall the important distinction between stability of a trajectory in time and stability of a limit cycle was that the limit cycle does not converge in phase -- trajectories near the cycle converge to the cycle, but trajectories on the cycle may not converge with each other. This is type of stability, also known as orbital stability can be written as stability to the manifold described by cycle $\bx^*(t)$, \[ \min_\tau || x(t) - x^*(\tau) || \rightarrow 0.\] In the case of limit cycles, this manifold is a periodic solution with $\bx^*(t+t_{period}) = \bx^*(t)$. Depending on exactly how that convergence happens, we can define orbital stability in the sense of Lyapunov, asymptotic orbital stability, exponential orbital stability, or even finite-time orbital stability.

In order to prove that a system is orbitally stable (locally, over a region, or globally), or to analyze the region of attraction of a limit cycle, we can use a Lyapunov function. In particular, we would like to consider Lyapunov functions which have the form cartooned below; they vanish to zero everywhere along the cycle, and are strictly positive everywhere away from the cycle.

Cartoon of a Lyapunov function which vanishes on a limit cycle, and is strictly positive everywhere else. (a small segment has been removed solely for the purposes of visualization).

Transverse coordinates

How can we parameterize this class of functions? For arbitrary cycles this could be very difficult in the original coordinates. For simple cycles like in the cartoon, one could imagine using polar coordinates. More generally, we will define a new coordinate system relative to the orbit, with coordinates

  • $\tau$ - the phase along the orbit
  • $\bx_\perp(\tau)$ - the remaining coordinates, linearly independent from $\tau$.

Given a state $\bx$ in the original coordinates, we must define a smooth mapping $\bx \rightarrow (\tau, \bx_\perp)$ to this new coordinate system. For example, for a simple ring oscillator we might have:

A moving coordinate system along the limit cycle.
In general, for an $n$-dimensional state space, $\tau$ will always be a scalar, and $\bx_\perp$ will be an $(n-1)$-dimensional vector defining the remaining coordinates relative to $\bx^*(\tau)$. In fact, although we use the notation $\bx_\perp$ the coordinate system need not be strictly orthogonal to the orbit, but must simply be transversal (not parallel). Having defined the smooth mapping $\bx \rightarrow (\tau,\bx_\perp)$, we can always rewrite the dynamics in this new coordinate system: \begin{gather*} \dot{\tau} = f_1(\bx_\perp,\tau) \\ \dot\bx_\perp = f_2(\bx_\perp,\tau). \end{gather*}

The value of this construction for Lyapunov analysis was proposed in Hauser94a and has been extended nicely to control design in Shiriaev08 and for region of attraction estimation in Manchester10b. A particular numerical strategy for defining the transversal coordinates is given in Manchester10a.

A Lyapunov theorem for orbital stability

For a dynamical system $\dot\bx = f(\bx)$ with $\bx \in \Re^n$, $f$ continuous, a continuous periodic solution $\bx^*(\tau)$, and a smooth mapping $\bx \rightarrow (\tau,\bx_\perp)$ where $\bx_\perp$ vanishes on $\bx^*$, then for some $n-1$ dimensional ball ${\cal B}$ around the origin, if I can produce a $V(\bx_\perp,\tau)$ such that \begin{gather*} \forall \tau, V(0,\tau) = 0, \\ \forall \tau, \forall \bx_\perp \in {\cal B}, \bx_\perp \ne 0, V(\bx_\perp,\tau) > 0, \end{gather*} with \begin{gather*} \forall \tau, \dot{V}(0,\tau) = 0, \\ \forall \tau, \forall \bx_\perp \in {\cal B}, \bx_\perp \ne 0, \dot{V}(\bx_\perp,\tau) < 0, \end{gather*} then the solution $\bx^*(t)$ is locally orbitally asymptotically stable.

Simple ring oscillator

Perhaps the simplest oscillator is the first-order system which converges to the unit circle. In cartesian coordinates, the dynamics are \begin{align*} \dot{x}_1 =& x_2 -\alpha x_1 \left( 1 - \frac{1}{\sqrt{x_1^2+x_2^2}}\right) \\ \dot{x}_2 =& -x_1 -\alpha x_2 \left( 1 - \frac{1}{\sqrt{x_1^2+x_2^2}}\right), \end{align*} where $\alpha$ is a positive scalar gain.

Vector field of the ring oscillator

If we take the transverse coordinates to be the polar coordinates, shifted so that $x_\perp$ is zero on the unit circle, \begin{align*}\tau =& \text{atan2}(-x_2,x_1) \\ x_\perp =& \sqrt{x_1^2+x_2^2}-1, \end{align*} which is valid when $x_\perp>-1$, then the simple transverse dynamics are revealed: \begin{align*} \dot\tau =& 1 \\ \dot{x}_\perp =& -\alpha x_\perp. \end{align*} Taking a Lyapunov candidate $V(x_\perp,\tau) = x_\perp^2$, we can verify that $$\dot{V} = -2 \alpha x_\perp^2 \prec 0, \quad \forall x_\perp > -1.$$ This demonstrates that the limit cycle is locally asymptotically stable, and furthermore that the invariant open-set $V < 1$ is inside the region of attraction of that limit cycle. In fact, we know that all $x_\perp > -1$ are in the region of attraction that limit cycle, but this is not proven by the Lyapunov argument above.

Let's compare this approach with the approach that we used in the chapter on walking robots, where we used a Poincaré map analysis to investigate limit cycle stability. In transverse coordinates approach, there is an additional burden to construct the coordinate system along the entire trajectory, instead of only at a single surface of section. In fact, the transverse coordinates approach is sometimes referred to as a "moving Poincaré section". But the reward for this extra work is that we can check a condition that only involves the instantaneous dynamics, $f(\bx)$, as opposed to having to integrate the dynamics over an entire cycle to generate the discrete Poincaré map, $\bx_p[n+1] = P(\bx_p[n])$. While computing $P$ in closed-form happened to be possible for the simple rimless wheel model, it is rarely possible for more complex models. As we will see below, this approach will also be more compatible with designing continuous feedback controller that stabilize the limit cycle.

Transverse linearization

In the case of Lyapunov analysis around a fixed point, there was an important special case: for stable linear systems, we actually have a recipe for constructing a Lyapunov function. As a result, for nonlinear systems, we often found it convenient to begin our search by linearizing around the fixed point and using the Lyapunov candidate for the linear system as an initial guess. That same approach can be extended to limit cycle analysis.

Region of attraction estimation using sums-of-squares

Coming soon. If you are interested, see Manchester10a+Manchester10b.

Feedback design

For underactuation degree one.

It turns out many of our simple walking models -- particulary ones with a point foot that are derived in the minimal coordinates -- are only short one actuator (between the foot and the ground). One can represent even fairly complex robots this way; much of the theory that I'll elude to here was originally developed by Jessy Grizzle and his group in the context of the bipedal robot RABBIT. Jessy's key observation was that limit cycle stability is effectively stability in $n-1$ degrees of freedom, and you can often achieve it easily with $n-1$ actuators -- he called this line of work "Hybrid Zero Dynamics" (HZD). We'll deal with the "hybrid" part of that in the next chapter, but here is a simple example to illustrate the "zero dynamics" concept. [Coming sooon...]

Add a simple example here.

Stabilizing the zero dynamics is synonymous with achieving orbital stability. Interestingly, it doesn't actually guarantee that you will go around the limit cycle. One "feature" of the HZD walkers is that you can actually stand in front of them and stop their forward progress with your hand -- the control system will continue to act, maintaining itself on the cycle, but progress along the cycle has stopped. In some of the robots you could even push them backwards and they would move through the same walking cycle in reverse. If one wants to certify that forward progress will be achieved, then the tasks is reduced to inspecting the one-dimensional dynamics of the phase variable along the cycle.

The notion of "zero dynamics" is certainly not restricted to systems with underactuation of degree one. In general, we can easily stabilize a manifold of dimension $m$ with $m$ actuators (we saw this in the section on task-space partial feedback linearization), and if being on that manifold is sufficient to achieve our task, or if we can certify that the resulting dynamics on the manifold are sufficient for our task, then life is good. But in the "underactuation degree one" case, the manifold under study is a trajectory/orbit, and the tools from this chapter are immediately applicable.

Transverse LQR

Orbital stabilization for non-periodic trajectories

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