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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Highly-articulated Legged Robots

The passive dynamic walkers and hopping robots described in the last chapter capture the fundamental dynamics of legged locomotion -- dynamics which are fundamentally nonlinear and punctuated by impulsive events due to making and breaking contact with the environment. But if you start reading the literature on humanoid robots, or many-legged robots like quadrupeds, then you will find a quite different set of ideas taking center stage: ideas like the "zero-moment point" and footstep planning. The goal of this chapter is to penetrate that world.

I'd like to start the discussion with a model that might seem quite far from the world of legged robots, but I think it's a very useful way to think about the problem.

Center of Mass Dynamics

A hovercraft model

flip the directions of the force vector in the image

Imagine you have a flying vehicle modeled as a single rigid body in a gravity field with some number of force "thrusters" attached. We'll describe the configuration of the vehicle by its orientation, $\theta$ and the location of its center of mass $(x,z)$. The vector from the center of mass to thruster $i$ is given by $r_i$, yielding the equations of motion: \begin{align*} m\ddot{x} =& \sum_i F_{i,x},\\ m\ddot{z} =& \sum_i F_{i,z} - mg,\\ I\ddot\theta =& \sum_i \left[ r_i \times F_i \right]_y, \end{align*} where I've used $F_{i,x}$ to represent the component of the force in the $x$ direction and have taken just the $y$-axis component of the cross product on the last line.

Our goal is to move the hovercraft to a desired position/orientation and to keep it there. If we take the inputs to be $F_i$, then the dynamics are affine (linear plus a constant term). As such, we can stabilize a stabilizable fixed point using a change of coordinates plus LQR or even plan/track a desired trajectory using time-varying LQR. If I were to add additional linear constraints, for instance constraining $F_{min} \le F_i \le F_{max}$, then I can still use linear model-predictive control (MPC) to plan and stabilize a desired motion. By most accounts, this is a relatively easy control problem. (Note that it would be considerably harder if my control input or input constraints depend on the orientation, $\theta$, but we don't need that here yet).

Now imagine that you just a small number of thrusters (let's say two) each with severe input constraints. To make things more interesting, let us say that you're allowed to move the thrusters, so $\dot{r}_i$ becomes an additional control input, but with some important limitations: you have to turn the thrusters off when they are moving (e.g. $|F_i||\dot{r}_i| = 0$) and there is a limit to how quickly you can move the thrusters $|\dot{r}|_i \le v_{max}$. This problem is now a lot more difficult, due especially to the fact that constraints of the form $u_1 u_2 = 0$ are non-convex.

I find this a very useful thought exercise; somehow our controller needs to make an effectively discrete decision to turn off a thruster and move it. The framework of optimal control should support this - you are sacrificing a short-term loss of control authority for the long-term benefit of having the thruster positioned where you would like, but we haven't developed tools yet that deal well with this discrete plus continuous decision making. We'll need to address that here.

Unfortunately, although the problem is already difficult, we need to continue to add constraints. Now let's say additionally, that the thrusters can only be turned on in certain locations, as cartooned here:

The union of these regions need not form a convex set. Furthermore, these locations could be specified in the robot frame, but may also be specified in the world frame, which I'll call ${\bf p}_i$: $${\bf p}_i = {\bf r}_i - \begin{bmatrix} x \\ 0 \\ z \end{bmatrix}.$$ This problem still feels like it should be tractable, but it's definitely getting hard.

Robots with (massless) legs

In my view, the hovercraft problem above is a central component of the walking problem. If we consider a walking robot with massless legs, then the feet are exactly like movable thrusters. As above, they are highly constrained - they can only produce force when they are touching the ground, and (typically) they can only produce forces in certain directions, for instance as described by a "friction cone" (you can push on the ground but not pull on the ground, and with Coulomb friction the forces tangential to the ground must be smaller than the forces normal to the ground, as described by a coefficient of friction, e.g. $|F_{\parallel}| < \mu |F_{\perp}|$).

The constraints on where you can put your feet / thrusters will depend on the kinematics of your leg, and the speed at which you can move them will depend on the full dynamics of the leg -- these are difficult constraints to deal with. But the actual dynamics of the rigid body are actually still affine, and still simple!

Capturing the full robot dynamics

We don't actually need to have massless legs for this discussion to make sense. If we use the coordinates $x,z$ to describe the location of the center of mass (CoM) of the entire robot, and $m$ to represent the entire mass of the robot, then the first two equations remain unchanged. The center of mass is a configuration dependent point, and does not have an orientation, but one important generalization of the orientation dynamics is given by the centroidal momentum matrix, $A(\bq)$, where $A(\bq)\dot{\bq}$ captures the linear and angular momentum of the robot around the center of mass Orin13. Note that the center of mass dynamics are still affine -- even for a big complicated humanoid -- but the centroidal momentum dynamics are nonlinear.

Impact dynamics

In the previous chapter we devoted relatively a lot of attention to dynamics of impact, characterized for instance by a guard that resets dynamics in a hybrid dynamical system. In those models we used impulsive ground-reaction forces (these forces are instantaneously infinite, but doing finite work) in order to explain the discontinuous change in velocity required to avoid penetration with the ground. This story can be extended naturally to the dynamics of the center of mass.

For an articulated robot, though, there are a number of possible strategies for the legs which can effect the dynamics of the center of mass. For example, if the robot hits the ground with a stiff leg like the rimless wheel, then the angular momentum about the point of collision will be conserved, but any momentum going into the ground will be lost. However, if the robot has a spring leg and a massless toe like the SLIP model, then no energy need be lost.

One strategy that many highly-articulated legged robots use is to keep their center of mass at a constant height, $$z = c \quad \Rightarrow \quad \dot{z} = \ddot{z} = 0,$$ and minimize their angular momentum about the center of mass (here $\ddot\theta=0$). Using this strategy, if the swinging foot lands roughly below the center of mass, then even with a stiff leg there is no energy dissipated in the collision - all of the momentum is conserved. This often (but does not always) justify ignoring the impacts in the center of mass dynamics of the system.

The special case of flat terrain

While not the only important case, it is extremely common for our robots to be walking over flat, or nearly flat terrain. In this situation, even if the robot is touching the ground in a number of places (e.g., two heels and two toes), the constraints on the center of mass dynamics can be summarized very efficiently.

External forces acting on a robot pushing on a flat ground

First, on flat terrain $F_{i,z}$ represents the force that is normal to the surface at contact point $i$. If we assume that the robot can only push on the ground (and not pull), then this implies $$\forall i, F_{i,z} \ge 0 \Rightarrow \sum_i F_{i,z} \ge 0 \Rightarrow \ddot{z} \ge -g.$$ In other words, if I cannot pull on the ground, then my center of mass cannot accelerate towards the ground faster than gravity.

Furthermore, if we use a Coulomb model of friction on the ground, with friction coefficient $\mu$, then $$\forall i, |F_{i,x}| \le \mu F_{i,z} \Rightarrow \sum_i |F_{i,x}| \le \mu \sum_i F_z \Rightarrow |\ddot{x}| \le \mu (\ddot{z} + g).$$ For instance, if I keep my center of mass at a constant height, then $\ddot{z}=0$ and $|\ddot{x}| \le \mu g$; this is a nice reminder of just how important friction is if you want to be able to move around in the world.

Even better, let us define the "center of pressure" (CoP) as the point on the ground where $$x_{cop} = \frac{\sum_i p_{i,x} F_{i,z}}{\sum_i F_{i,z}},$$ and since all $p_{i,z}$ are equal on flat terrain, $z_{cop}$ is just the height of the terrain. It turns out that the center of pressure is a "zero-moment point" (ZMP) -- a property that we will demonstrate below -- and moment-balance equation gives us a very important relationship between the location of the CoP and the dynamics of the CoM: \[ (m\ddot{z} + mg) (x_{cop} - x) = (z_{cop} - z) m\ddot{x} - I\ddot\theta. \] If we use the strategy proposed above for ignoring collision dynamics, $\ddot{z} = \ddot{\theta} = 0$, then we have $z - z_{cop}$ is a constant height $h$, and the result is the famous "ZMP equations": \[ \ddot{x} = -\frac{g}{h} (x_{cop}-x). \] So the location of the center of pressure completely determines the acceleration of the center of mass, and vice versa! What's more, this relationship is affine -- a property that we can exploit in a number of ways.

As an example, we can easily relate constraints on the CoP to constraints on $\ddot{x}$. It is easy to verify from the definition that the CoP must live inside the convex hull of the ground contact points. Therefore, if we use the $\ddot{z}=\ddot\theta=0$ strategy, then this directly implies strict bounds on the possible accelerations of the center of mass given the locations of the ground contacts.

An aside: the zero-moment point derivation

The zero-moment point (ZMP) is discussed very frequently in the current literature on legged robots. It also has an unfortunate tendency to be surrounded by some confusion; a number of people have defined the ZMP is slightly different ways (see e.g. Goswami99 for a summary). Therefore, it makes sense to provide a simple derivation here.

First let us recall that for rigid body systems I can always summarize the contributions from many external forces as a single wrench (force and torque) on the object -- this is simply because the $F_i$ terms enter our equations of motion linearly. Specifically, given any point in space, $r$, in coordinates relative to $(x,z)$:

update diagram to have r instead of p

I can re-write the equations of motion as \begin{align*} m\ddot{x} =& \sum_i F_{i,x} = F_{net,x},\\ m\ddot{z} =& \sum_i F_{i,z} - mg = F_{net,z} - mg,\\ I\ddot\theta =& \sum_i \left[ r_i \times F_i \right]_y = ({\bf r} \times {\bf F}_{net})_y + \tau_{net}, \end{align*} where ${\bf F}_{net} = \sum_i {\bf F}_i$ and the value of $\tau_{net}$ depends on the location ${\bf r}$. For some choices of ${\bf r}$, we can make $\tau_{net}=0$. Examining \[ ({\bf r} \times {\bf F}_{net})_y = r_z F_{net,x} - r_x F_{net,z} = \left[ r_i \times F_i \right]_y, \] we can see that when $F_{net,z} > 0$ there is an entire line of solutions, $r_x = a r_z + b$, including one that will intercept the terrain. For walking robots, it is this point on the ground from which the external wrench can be described by a single force vector (and no moment) that is the famous "zero-moment point" (ZMP). Substituting the back in to replace $F_{net}$, we can see that \[ r_x = \frac{r_z m \ddot{x} - I \ddot\theta}{m\ddot{z} + mg}. \] If we assume that $\ddot{z}=\ddot{\theta}=0$ and replace the relative coordinates with the global coordinates ${\bf r} = {\bf p} - [x,0,z]^T$, then we arrive at precisely the equation presented above.

Furthermore, since \[\left[ r_i \times F_i \right]_y = \sum_i \left( r_{i,z} F_{i,x} - r_{i,x} F_{i,z} \right), \] and for flat terrain we have \[ r_z F_{net,x} = \sum_i r_{i,z} F_{i,x}, \] then we can see that this ZMP is exactly the CoP: \[ r_x = \frac{\sum_i r_{i,x} F_{i,z}}{ F_{net,z} }. \]

In three dimensions, we solve for the point on the ground where $\tau_{net,y} = \tau_{net,x} = 0$, but allow $\tau_{net,z} \ne 0$ to extract the analogous equations in the $y$-axis: \[ r_y = \frac{r_z m \ddot{y} - I \ddot\theta}{m\ddot{z} + mg}. \]

ZMP planning

From a CoM plan to a whole-body plan

Whole-Body Control

Coming soon. For a description of our approach with Atlas, see Kuindersma13+Kuindersma14.

Footstep planning and push recovery

Coming soon. For a description of our approach with Atlas, see Deits14a+Kuindersma14. For nice geometric insights on push recovery, see Koolen12.

Beyond ZMP planning

Coming soon. For a description of our approach with Atlas, see Dai14+Kuindersma14.

Exercises

Footstep Planning via Mixed-Integer Optimization

In this exercise we implement a simplified version of the footstep planning method proposed in Deits14a. You will find all the details in this notebook. Your goal is to code most of the components of the mixed-integer program at the core of the footstep planner:

  1. The constraint that limits the maximum step length.
  2. The constraint for which a foot cannot be in two stepping stones at the same time.
  3. The constraint that assigns each foot to a stepping stone, for each step of the robot.
  4. The objective function that minimizes the sum of the squares of the step lengths.
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