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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, 2020<br/>
<a href="tocite.html">How to cite these notes</a> |
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<p><b>Note:</b> These are working notes used for <a
href="http://underactuated.csail.mit.edu/Spring2020/">a course being taught
at MIT</a>. They will be updated throughout the Spring 2020 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 16"><h1>Planning and
Control through Contact</h1>
<p>So far we have developed a fairly strong toolbox for planning and control
with "smooth" systems -- systems where the equations of motion are described
by a function $\dot{\bx} = f(\bx,\bu)$ which is smooth everywhere. But our
<a href="simple_legs.html">discussion of the simple
models of legged robots</a> illustrated that the dynamics of making and
breaking contact with the world are more complex -- these are often modeled
as hybrid dynamics with impact discontinuities at the collision event and
constrained dynamics during contact (with either soft or hard
constraints).</p>
<p>My goal for this chapter is to extend our computational tools into this
richer class of models. Many of our core tools still work: trajectory
optimization, Lyapunov analysis (e.g. with sums-of-squares), and LQR all have
natural equivalents.</p>
<p>Let's start with a warm-up exercise: trajectory optimization for the rimless wheel. We already have basically everything that we need for this, and it will form a nice basis for generalizing our approach throughout the chapter.</p>
<example><h1>Trajectory optmization for 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>The <a href="simple_legs.html#rimless_wheel">rimless wheel</a> was our
simplest example of a passive-dynamic walker; it has no control inputs but
exhibits a passively stable rolling fixed point. We've also <a
href="limit_cycle.html">already seen</a> that trajectory optimization can be
used as a tool for finding limit cycles of a smooth passive system, e.g. by
formulating a direct collocation problem: \begin{align*}
\find_{\bx[\cdot],h} \quad \subjto \quad & \text{collocation
constraints}(\bx[n], \bx[n+1], h), \quad \forall n \in [0, N-1] \\ & \bx[0]
= \bx[N], \\ & h_{min} \le h \le h_{max},\end{align*} where $h$ was the
timestep between the trajectory break points.</p>
<p>It turns out that applying this to the rimless wheel is quite straight
forward. We still want to find a periodic trajectory, but now have to take
into account the collision event. We can do this by modifying the
periodicity condition. Let's force the initial state to be just after the
collision, and the final state to be just before the collision, and make
sure they are related to each other via the collision equation:
\begin{align*} \find_{\bx[\cdot],h} \quad \subjto \quad & \text{collocation
constraints}(\bx[n], \bx[n+1], h), \quad \forall n \in [0, N-1] \\ &
\theta[0] = \gamma - \alpha, \\ & \theta[N] = \gamma + \alpha, \\ &
\dot\theta[0] = \dot\theta[N] \cos(2\alpha)\\ & h_{min} \le h \le h_{max}.
\end{align*} Although it is likely not needed for this simple example
(since the dynamics are sufficiently limited), for completeness one should
also add constraints to ensure that none of the intermediate points are in
contact, $$\gamma - \alpha \le \theta[n] < \gamma + \alpha, \quad \forall n
\in [1,N-1].$$</p>
<p>The result is a simple and clean numerical algorithm for finding the
rolling limit cycle solution of the rimless wheel. Please take it for a
spin:</p>
<jupyter>examples/contact.ipynb</jupyter>
</example>
<p>The specific case of the rimless wheel is quite clean. But before we apply
it to the compass gait, the kneed compass gait, the spring-loaded inverted
pendulum, etc, then we should stop and figure out a more general form.</p>
<section><h1>(Autonomous) Hybrid Systems</h1>
<p>Recall how we modeled the dynamics of the simple legged robots. First, we
derived the equations of motion (independently) for each possible contact
configuration -- for example, in the spring-loaded inverted pendulum (SLIP)
model we had one set of equations governing the $(x,y)$ position of the mass
during the flight phase, and a completely separate set of equations written
in polar coordinates, $(r,\theta)$, describing the stance phase. Then we
did a little additional work to describe the transitions between these
models -- e.g., in SLIP we transitioned from flight to stance when the foot
first touches the ground. When simulating this model, it means that we have
a discrete "event" which occurs at the moment of foot collision, and an
immediate discontinuous change to the state of the robot (in this case we
even change out the state variables).</p>
<p>The language of <i>hybrid systems</i> gives us a rich language for
describing systems of this form, and a suite of tools for analyzing and
controlling them. The term "hybrid systems" is a bit overloaded, here we
use "hybrid" to mean both discrete- and continuous-time, and the particular
systems we consider here are sometimes called <i>autonomous</i>
hybrid systems because the internal dynamics can cause the discrete changes
without any exogeneous input†<sidenote>†This is in contrast
to, for instance, the model of a power-train where a change in gears comes
as an external input.</sidenote>. In the hybrid systems formulation, we describe a system by a set of <i>modes</i> each described by (ideally
smooth) continuous dynamics, a set of <i>guards</i> which here are
continuous functions whos zero-level set describes the conditions which
trigger an event, and a set of <i>resets</i> which describe the discrete
update to the state that is triggered by the guard. Each guard is
associated with a particular mode, and we can have multiple guards per mode.
Every guard has at most one reset. You will occasionally here guards
referred to as "witness functions", since they play that role in simulation,
and resets are sometimes referred to as "transition functions".</p>
<p>The imagery that I like to keep in my head for hybrid systems is
illustrated below for a simple example of a robot's heel striking the
ground. A solution trajectory of the hybrid system has a continuous
trajectory inside each mode, punctuated by discrete updates when the
trajectory hits the zero-level set of the guard (here the distance between
the heel and the ground becomes zero), with the reset describing the
discrete change in the state variables.</p>
<figure> <img width="90%" src="figures/hybrid.svg"/>
<figcaption>Modeling contact as a hybrid system.</figcaption>
</figure>
<p>For this robot foot, we can decompose the dynamics into distinct modes: (1) foot in the air, (2) only heel on the ground, (3) heel and toe on the ground, (4) only toe on the ground (push-off). More generally, we will write the dynamics of mode $i$ as ${\bf f}_i$, the guard which signals the transition mode $i$ to mode $j$ as ${\bf \phi}_{i,j}$ (where $\phi_{i,j}(\bx_i) > 0$ inside mode $i$), and the reset map from $i$ to $j$ as ${\bf \Delta}_{i,j}$, as illustrated in the following figure:</p>
<figure> <img width="100%" src="figures/hybrid_guard_reset.svg"/>
<figcaption>The language of hybrid systems: modes, guards, and reset maps.</figcaption>
</figure>
<subsection><h1>Hybrid trajectory optimization</h1>
<p>Using the more general language of modes, guards, and resets, we can begin to formulate the "hybrid trajectory optimization" problem. In hybrid trajectory optimization, there is a major distinction between trajectory optimization where <i>the mode sequence is known apriori</i> and the optimization is just attempting to solve for the continuous-time trajectories, vs one in which we must also discover the mode sequence.</p>
<p>For the case when the mode sequence is fixed, then hybrid trajectory
optimization is as simple as stitching together multiple individual
mathematical programs into a single mathematical program, with the
boundary conditions constrained to enforce the guard/reset constraints.
For a simple hybrid system with a given a mode sequence and using the
shorthand $\bx_k$, etc, for the state in mode in the $k$th segment of the
sequence, we can write: \begin{align*} \find_{\bx_k[\cdot],h_k} \quad
\subjto \quad & \bx_0[0] = \bx_0, \\ \forall k \quad &
\phi_{k,k+1}(\bx_k[N_k]) = 0, \\ & \bx_{k+1}[0] = {\bf
\Delta}_{i,j}(\bx_k[N_k]), \\ & \phi_{k,k'}(\bx_k[n_k]) > 0, \quad \forall
n_k \in [0, N_k], \forall k' \\ & h_{min} \le h_k \le h_{max}, \\ &
\text{collocation constraints}_k(\bx_k[n_k], \bx_k[n_k+1], h_k), \quad
\forall n_k \in [0, N_k-1]. \end{align*} It is then natural to add control
inputs (as additional decision variables), and to add an objective and any
more constraints.
</p>
<example><h1>A basketball trick shot.</h1>
<p>As a simple example of this hybrid trajectory optimization, I thought it would be fun to see if we can formulate the search for initial conditions that optimizes a basketball "trick shot". A quick search turned up <a href="https://youtu.be/Mayx9OrXWIM">this video</a> for inspiration.</p>
<p>Let's start simpler -- with just a "bounce pass". We can capture the
dynamics of a bouncing ball (in the plane, ignoring spin) with some very
simple dynamics: $$\bq = \begin{bmatrix}x \\ z\end{bmatrix}, \qquad
\ddot{\bq} = \begin{bmatrix} 0 \\ -g \end{bmatrix}.$$ During any time
interval without contact of duration $h$, we can actually integrate
these dynamics perfectly: $$\bx(t+h) = \begin{bmatrix} x(t) +
h\dot{x}(t) \\ z(t) + h \dot{z}(t) - \frac{1}{2}gh^2 \\ \dot{x}(t) \\
\dot{z}(t) - hg \end{bmatrix}.$$ With the bounce pass, we just consider
collisions with the ground, so we have a guard, ${\bf \phi}(\bx) = z,$
which triggers when $z=0$, and a reset map which assumes an elastic
collision with <a
href="https://en.wikipedia.org/wiki/Coefficient_of_restitution">coefficient
of restitution</a> $e$: $$\bx^+ = {\bf \Delta(\bx^-)} = \begin{bmatrix}
x^- & z^- & \dot{x}^- & - e \dot{z}^- \end{bmatrix}^T.$$</p>
<p>We'll formulate the problem as this: given an initial ball position
$(x = 0, z = 1)$, a final ball position 4m away $(x=4, z=1)$, find the
initial velocity to achieve that goal in 5 seconds. Clearly, this
implies that $\dot{x}(0) = 4/5.$ The interesting question is -- what
should we do with $\dot{z}(0)$? There are multiple solutions -- which
involve bouncing a different number of times. We can find them all with
a simple hybrid trajectory optimization, using the observation that
there are two possible solutions for each number of bounces -- one that
starts with a positive $\dot{z}(0)$ and one with a negative
$\dot{z}(0)$.</p>
<jupyter>examples/contact.ipynb</jupyter>
<figure>
<img width="100%" src="figures/bounce_pass.svg"/>
<figcaption>Trajectory optimization to find solutions for a "bounce
pass". I've plotted all solutions that were found for 2, 3, or 4 bounces... but I think it's best to stick to a single bounce if you're using this on the court.</figcaption>
</figure>
<p>Now let's try our trick shot. I'll move our goal to $x_f = -1m, z_f
= 3m,$ and introduce a vertical wall at $x=0$, and move our initial
conditions back to $x_0=-.25m.$ The collision dynamics, which now must
take into account the spin of the ball, are <a
href="multibody.html#spinning_bouncing_ball">in the appendix</a>. The
first bounce is against the wall, the second is against the floor. I'll
also constrain the final velocity to be down (have to approach the hoop
from above). Try it out.</p>
<jupyter>examples/contact.ipynb</jupyter>
<figure>
<img width="100%" src="figures/trick_shot.svg"/>
<figcaption>Trajectory optimization for the "trick shot". Nothing but
the bottom of the net! The crowd is going wild!</figcaption>
</figure>
<p>In this example, we could integrate the dynamics in each segment
analytically. That is the exception, not the rule. But you can take
the same steps with a little more code to use, e.g. direct transcription
or collocation with multiple break points in each segment.</p>
</example>
</subsection>
<subsection><h1>Stabilizing hybrid models.</h1></subsection>
<subsection><h1>Deriving hybrid models: minimal vs floating-base
coordinates</h1>
<p>There is some work to do in order to derive the equations of motion in
this form. Do you remember how we did it for the <a
href="simple_legs.html">rimless wheel and compass gait</a> examples? In
both cases we assumed that exactly one foot was attached to the ground and
that it would not slip, this allowed us to write the Lagrangian as if
there was a pin joint attaching the foot to the ground to obtain the
equations of motion. For the SLIP model, we derived the flight phase and
stance phase using separate Lagrangian equations each with different state
representations. I would describe this as the <i>minimal coordinates</i>
modeling approach -- it is elegant and has some important computational
advantages that we will come to appreciate in the algorithms below. But
it's a lot of work! For instance, if we also wanted to consider friction
in the foot contact of the rimless wheel, we would have to derive yet
another set of equations to describe the sliding mode (adding, for
instance, a prismatic joint that moved the foot along the ramp), plus the
guards which compute the contact force for a given state and the distance
from the boundary of the friction cone, and on and on.</p>
<!-- But, actually, it could get more complicated still: in each of the contact modes the foot could be in the static friction regime or could be sliding. And what if there is rough terrain, so we could make contact at more points. And what if we -->
<p>Fortunately, there is an alternative modeling approach for deriving the
modes, guards, and resets for contact that is more general (though
admittedly also more complex). We can instead model the robot in the
<i>floating-base coordinates</i> -- we add a fictitious six
degree-of-freedom "floating-base" joint connecting some part of the robot
to the world (in planar models, we use just three degrees-of-freedom, e.g.
$(x,z,\theta)$). We can derive the equations of motion for the
floating-base robot once, without considering contact, then add the
additional constraints that come from being in contact as contact forces
which get applied to the bodies. The resulting manipulator equations take
the form \begin{equation}\bM({\bq})\ddot{\bq} + \bC(\bq,\dot{\bq})\dot\bq
= \btau_g(\bq) + \bB\bu + \sum_i \bJ_i^T(\bq) \blambda_i,\end{equation}
where $\blambda_i$ are the constraint forces and $\bJ_i$ are the
constraint Jacobians. Conveniently, if the guard function in our contact
equations is the signed distance from contact, $\phi_i(\bq)$, then this
Jacobian is simply $\bJ_i(\bq) = \pd{\phi_i}{\bq}$. I've written the
basic derivations for the common cases (position constraints, velocity
constraints, impact equations, etc) in the
<a href="multibody.html">appendix</a>. What is important to understand
here is that this is an alternative formulation for the equations
governing the modes, guards, and resets, but that is it no longer a
minimal coordinate system -- the equations of motion are written in $2N$
state variables but the system might actually be constrained to evolve
only along a lower dimensional manifold (if we write the rimless wheel
equations with three configuration variables for the floating base, it
still only rotates around the toe when it is in stance and is inside the
friction cone). This will have implications for our algorithms.</p>
</subsection>
<todo>Grandia 2019 - Feedback MPC for Torque-controlled Legged Robots looks
at a relaxation of the friction cone via a barrier certificate for iLQR.
"In particular, the friction cone is implemented through a perturbed
second-order cone constraint. This for- mulation adds a convex penalty to
the cost function and avoids numerical ill-conditioning at the origin of the
cone." Could go in the playbook, if not here.</todo>
</section>
<!--
<section><h1>Randomized Motion Planning</h1></section>
<section><h1>Stabilizing a Fixed-Point</h1>
</section>
<section><h1>Stabilizing a Trajectory or Limit Cycle</h1>
<p>Transverse stabilization (build off limit cycle chapter)</p>
</section>
-->
</chapter>
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