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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Simple Models of Walking and Running

Practical legged locomotion is one of the fundamental problems in robotics; we've seen amazing progress over the last few years, but there are still some fundamental problems. Much of the recent progress is due to improvements in hardware -- a legged robot must carry all of its sensors, actuators and power and traditionally this meant underpowered motors that act as far-from-ideal force/torque sources -- but Boston Dynamics and other companies have made incredible progress here. The control systems implemented on these systems, though, are still surprisingly heuristic -- they require dramatically higher bandwidth and lower latency that the human motor control system and still perform worse in challenging environments.

The control of walking robots is fundamentally underactuated -- assuming we cannot pull on the ground (and barring any aerodynamic effects!), then no matter how powerful my actuators are, there is nothing that they can do to accelerate the center of mass of the robot towards the ground faster than gravity. But beyond the underactuated systems we have studied so far, the control of walking robots faces an additional complexity: controlling through contact. The fact that we can get non-zero contact forces if and only if two bodies are in contact introduces a fundamentally discrete/combinatorial element into the problem††I've had some interesting arguments with folks on this point, because it's possible to write contact models that smooth the discontinuity, and/or to model systems that have neglible collision events. But replacing a discontinuity in the vector field with a stiff but smooth transition does not remove the need to decide whether or not your robot should make contact to accomplish a task..

Contact is fundamental for many aspects of robotics (manipulation is my other favorite example); it's sad to see so many robots going through life avoiding contact at any cost. Controlling contact means that your robot is capable of performing physical work on the environment; isn't that the point of robotics, afterall?

I like to start our discussion of contact with walking robots for a few important reasons. As you'll see in this chapter, there are a number of elegant "simple models" of walking that capture much of the essence of the problem with minimal complexity. But there is another, more subtle, reason: I want to continue to use the language of stability. At this point we've seen both finite-horizon and infinite-horizon problem formulations -- when we can say something about the behavior of a system in the limit as time goes to infinity, it's almost always cleaner than a finite-time analysis (because time drops away). Walking provides a very clean way to use the language of stability in studying behavior of systems that are sometimes in contact but sometimes out of contact in the limiting case; we just need to extend our notions from stability of a fixed point to the stability of a (periodic) cycle.

Limit Cycles

A limit cycle is an periodic orbit that is a limit set of the dynamical system; they can be stable (the limit set of neighboring trajectories as time goes to infinity) or unstable (the limit set of neighboring trajectories as time goes to negative infinity). One of the simplest models of limit cycle behavior is the Van der Pol oscillator. Let's examine that first...

Van der Pol Oscillator

The Van der Pol oscillator is described by a second-order differential equation in one variable: $$\ddot{q} + \mu (q^2 - 1)\dot{q} + q = 0, \quad \mu>0$$ One can think of this system as almost a simple spring-mass-damper system, except that it has nonlinear damping. In particular, the velocity term dissipates energy when $|q|>1$, and adds energy when $|q|<1$. Therefore, it is not terribly surprising to see that the system settles into a stable oscillation from almost any initial conditions (the exception is the state $q=0,\dot{q}=0$). This can be seen nicely in the phase portrait below.

System trajectories of the Van der Pol oscillator with $\mu =.2$. (Left) phase portrait. (Right) time domain.

However, if we examine system trajectories in the time domain (see the right panel of the figure above), then we can see that our traditional notion for stability of a trajectory, $\lim_{t\rightarrow \infty} |\bx(t) - \bx^*(t)| = 0$, is not satisfied here for any $\bx^*(t)$. Although the phase portrait clearly reveals that all trajectories converge to the same orbit, the time domain plot reveals that these trajectories do not necessarily synchronize in time.

This example shows that stability of a trajectory (in time) is not the definition we want. Instead we will define the stability of an orbit, $\bx^*(t)$, using \[ \min_\tau || \bx(t) - \bx^*(\tau) ||.\] Here the quantity of interest is the distance between our trajectory and the closest point on the orbit. Depending on exactly how this distance evolves with time, we can define orbital stability: for every small $\epsilon>0$, we can find a $\delta>0$ such that $\min_\tau||\bx(t_0) - \bx^*(\tau)|| < \delta$ implies $\forall t, \min_\tau ||\bx(t) - \bx^*(\tau)|| < \epsilon$). The definitions for asymptotic orbital stability, exponential orbital stability, finite-time orbital stability follow naturally, as does the definition of an unstable orbit. In the case of limit cycles, this $\bx^*(t)$ is a periodic solution with $\bx^*(t+t_{period}) = \bx^*(t)$.

As we begin to extend our notions of stability, allow me to make a few quick connections to the discussion of stability in the chapter on Lyapunov analysis. It's interesting to know that if we can find an invariant set in state space which does not contain any fixed points, then this set must contain a closed orbit (this is the Poincaré-Bendixson Theorem)Strogatz94. It's also interesting to note that gradient potential fields (and Lyapunov functions with $\dot{V}(\bx) \prec 0$) cannot have a closed orbitStrogatz94, and consequently we will also have to slightly modify the Lyapunov analysis we have introduced so far to handle limit cycles. We'll discuss this in some details in an upcoming chapter.

Poincaré Maps

The first tool one should know for analyzing the stability of a limit cycle, and one that will serve us quite well for the examples in this chapter, is the method of Poincaré. Let's consider a dynamical system, $\dot{\bx} = {\bf f}(\bx),$ with $\bx \in \Re^n.$ Define an $n-1$ dimensional surface of section, $S$. We will also require that $S$ is transverse to the flow (i.e., all trajectories starting on $S$ flow through $S$, not parallel to it). The Poincaré map (or return map) is a mapping from $S$ to itself: $$\bx_p[n+1] = {\bf P}(\bx_p[n]),$$ where $\bx_p[n]$ is the state of the system at the $n$th crossing of the surface of section. Note that we will use the notation $\bx_p$ to distinguish the state of the discrete-time system from the continuous-time system; they are related by $\bx_p[n] = \bx(t_c[n])$, where $t_c[n]$ is the time of the $n$th crossing of $S$.

For our purposes in this chapter, it will be useful for us to allow the Poincaré maps to also include fixed points of the original continuous-time system, and to define the return time to be zero. These points do not satisfy the transversality condition, and require some care numerically, but they do not compromise our analysis.

Return map for the Van der Pol Oscillator

Since the state space of this system is two-dimensional, a return map for the system must have dimension one. For instance, we can take $S$ to be the line segment where $q = 0$, $\dot{q} \ge 0$. It is easy to see that all trajectories that start on S are transverse to it.

One dimensional maps, like one dimensional flows, are amenable to graphical analysis.

(Left) Phase plot with the surface of section, $S$, drawn with a black dashed line. (Right) The resulting Poincaré first-return map (blue), and the line of slope one (red).

If we can demonstrate that ${\bf P}(\bx_p)$ exists for all $\bx_p \in S$, (all trajectories that leave $S$ eventually return to $S$), then something remarkable happens: we can reduce the stability analysis for a limit cycle in the original coordinates into the stability analysis of a fixed point on the discrete map. If $\bx^*_p$ is a stable (or unstable) fixed point of the map ${\bf P}$, then there exists a unique periodic orbit $\bx^*(t)$ which passes through $\bx^*_p$ that is a (stable, or unstable) limit cycle. If we are able to upper-bound the time it takes for trajectories leaving $S$ to return, then it is also possible to infer asympotitic orbital or even exponential orbital stability.

In practice it is often difficult or impossible to find ${\bf P}$ analytically, but it can be sampled quite reasonably numerically. Once a fixed point of ${\bf P}$ is obtained (by finding a solution to$\bx^*_p = {\bf P}(\bx^*_p)$), we can infer local limit cycle stability with an eigenvalue analysis. If this eigen-analysis is performed in the original coordinates (as opposed to the $n-1$ dimensional coordinates of $S$), then there will always be a at least one zero eigenvalue - corresponding to perturbations along the limit cycle which do not change the state of first return. The limit cycle is considered locally exponentially stable if all remaining eigenvalues, $\lambda_i$, have magnitude less than one, $|\lambda_i|<1$.

It is sometimes possible to infer more global stability properties of the return map by examining ${\bf P}$. We will study computational methods in the later chapter. Koditschek91 describes some less computational stability properties known for unimodal maps which are useful for the simple systems we study in this chapter.

A particularly graphical method for understanding the dynamics of a one-dimensional iterated map is the "staircase method": Sketch the Poincaré map -- a plot of $x_p[n]$ vs $x_p[n+1]$ -- and also the line of slope one. In order to "iterate" the map, take an initial condition on the plot as a point $\left( x_p[k], x_p[k] \right)$, and move vertically on the plot to the point $\left( x_p[k], x_p[k+1] = f(x_p[k]) \right)$. Now move horizontally to the next point $\left( x_p[k+1], x_p[k+1] \right)$ and repeat.

The "staircase" on the return map of the Van der Pol Oscillator with $\mu = 0.2$, starting from two initial conditions ($\dot{q}_p[0] = .5$ and $\dot{q}_p[0] = 3.5$).

We can quickly recognize the fixed points as the points where the return map crosses this line. We can infer local exponential stability if the absolute value of the slope of return map at the fixed point is less than one ($|\lambda| < 1$). The figure above illustrates the stability of the Van der Pol limit cycle; it also illustrates the (one-sided) instability of the fixed point at the origin. Understanding the regions of attraction of fixed points on an iterated map (or any discrete-time system) graphically requires a little more care, though, than the continuous-time flows we examined before. The discrete maps can jump a finite distance on each step, and will in fact oscillate back and forth from either side of the fixed point when $\lambda < 0.$

Simple Models of Walking

Much of fundamental work in the dynamics of legged robots was done originally in the context of studying passive-dynamic walkers (often just "passive walker"). We've already seen my favorite example of a passive walker, in the first chapter, but here it is again:

A 3D passive dynamic walker by Steve Collins and Andy RuinaCollins01.

This amazing robot has no motors and no controllers... it walks down a tiny ramp and is powered entirely by gravity. We can build up our understanding of this robot in a series of steps. As we will see, what is remarkable is that even though the system is passive, we believe that the periodic gait you see in the video is actually a stable limit cycle!

Well before the first passive walkers, one of the earliest models of walking was proposed by McMahonMcMahon80, who argued that humans use a mostly ballistic (passive) gait. He observed that muscle activity (recorded via EMG) in the "stance leg" is relatively high, but muscle activity in the "swing leg" is very low, except for at the very beginning and end of swing. He proposed a "ballistic walker" model -- a three-link pendulum in the sagittal plane, with one link from the ankle to the hip representing a straight "stance leg", the second link from the hip back down to the "swing leg" knee, and the final link from the knee down to the swing foot -- and argued that this model could largely reproduce the kinematics of gait. This model of "walking by vaulting" is considered overly simplistic today, as we now understand much better the role of compliance in the stance leg during walking. His model was also restricted to the continuous portion of gait, not the actual dynamics of taking a step. But it was the start. Nearly a decade later, McGeerMcGeer90 followed up with a series of similarly inspired walking machines, which he coined "passive-dynamic walkers".

The Rimless Wheel

The rimless wheel. The orientation of the stance leg, $\theta$, is measured clockwise from the vertical axis.

Perhaps the simplest possible model of a legged robot, introduced as the conceptual framework for passive-dynamic walking by McGeerMcGeer90, is the rimless wheel. This system has rigid legs and only a point mass at the hip as illustrated in the figure above. To further simplify the analysis, we make the following modeling assumptions:

  • Collisions with ground are inelastic and impulsive (only angular momentum is conserved around the point of collision).
  • The stance foot acts as a pin joint and does not slip.
  • The transfer of support at the time of contact is instantaneous (no double support phase).
  • $0 \le \gamma < \frac{\pi}{2}$, $0 < \alpha < \frac{\pi}{2}$, $l > 0$.

Note that the coordinate system used here is slightly different than for the simple pendulum ($\theta=0$ is at the top, and the sign of $\theta$ has changed).

update the coordinates (here and in drake)

The most comprehensive analysis of the rimless wheel was done by Coleman98a.

Stance Dynamics

The dynamics of the system when one leg is on the ground are given by $$\ddot\theta = \frac{g}{l}\sin(\theta).$$ If we assume that the system is started in a configuration directly after a transfer of support ($\theta(0^+) = \gamma-\alpha$), then forward walking occurs when the system has an initial velocity, $\dot\theta(0^+) > \omega_1,$ where $$\omega_1 = \sqrt{ 2 \frac{g}{l} \left[ 1 - \cos \left (\gamma-\alpha \right) \right]}.$$ $\omega_1$ is the threshold at which the system has enough kinetic energy to vault the mass over the stance leg and take a step. This threshold is zero for $\gamma = \alpha$ and does not exist for $\gamma > \alpha$ (because when $\gamma > \alpha$ the mass is always ahead of the stance foot, and the standing fixed point disappears). The next foot touches down when $\theta(t) = \gamma+\alpha$, at which point the conversion of potential energy into kinetic energy yields the velocity $$\dot\theta(t^-) = \sqrt{\dot\theta^2(0^+) + 4\frac{g}{l} \sin\alpha \sin\gamma }.$$ $t^-$ denotes the time immediately before the collision.

Foot Collision

Angular momentum is conserved around the point of impact

The angular momentum around the point of collision at time $t$ just before the next foot collides with the ground is $$L(t^-) = -ml^2\dot\theta(t^-) \cos(2\alpha).$$ The angular momentum at the same point immediately after the collision is $$L(t^+) = -ml^2\dot\theta(t^+).$$ Assuming angular momentum is conserved, this collision causes an instantaneous loss of velocity: $$\dot\theta(t^+) = \dot\theta(t^-) \cos(2\alpha).$$

Forward simulation

Given the stance dynamics, the collision detection logic ($\theta = \gamma \pm \alpha$), and the collision update, we can numerically simulate this simple model. Doing so reveals something remarkable... the wheel starts rolling, but then one of two things happens: it either runs out of energy and stands still, or it quickly falls into a stable periodic solution. Convergence to this stable periodic solution appears to happen from a huge range of initial conditions. Try it yourself.

Numerical simulation of the rimless wheel

examples/simple_legs.ipynb

I've setup the notebook to make it easy for you to try a handful of interesting initial conditions. And even made an interactive widget for you to watch the simulation phase portrait as you change those initial conditions. Give it a try! (I recommend using Binder instead of Colab so you get the interactive features)

The rimless wheel model actually uses a number of the more nuanced features of in order to simulate the hybrid system accurately (as do many of the examples in this chapter). It actually registers the collision events of the system -- the simulator uses zero-crossing detection to ensure that the time/state of collision is obtained accurately, and then applies the reset map.

Phase portrait of the rimless wheel ($m=1$, $l=1$, $g=9.8$, $\alpha=\pi/8$, $\gamma=0.08$).

One of the fantastic things about the rimless wheel is that the dynamics are exactly the undamped simple pendulum that we've thought so much about. So you will recognize the phase portraits of the system -- they are centered around the unstable fixed point of the simple pendulum.

Poincaré Map

We can now derive the angular velocity at the beginning of each stance phase as a function of the angular velocity of the previous stance phase. First, we will handle the case where $\gamma \le \alpha$ and $\dot\theta_n^+ > \omega_1$. The "step-to-step return map", factoring losses from a single collision, the resulting map is: $$\dot\theta^{+}_{n+1} = \cos(2\alpha) \sqrt{ ({\dot\theta_n}^{+})^{2} + 4\frac{g}{l}\sin\alpha \sin\gamma}.$$ where the $\dot{\theta}^{+}$ indicates the velocity just after the energy loss at impact has occurred.

Using the same analysis for the remaining cases, we can complete the return map. The threshold for taking a step in the opposite direction is $$\omega_2 = - \sqrt{2 \frac{g}{l} [1 - \cos(\alpha + \gamma)]}.$$ For $\omega_2 < \dot\theta_n^{+} < \omega_1,$ we have $$\dot\theta_{n+1}^{+} = -\dot\theta_n^{+} \cos(2\alpha).$$ Finally, for $\dot\theta_n^{+} < \omega_2$, we have $$\dot\theta_{n+1}^{+} = - \cos(2\alpha)\sqrt{(\dot\theta_n^{+})^2 - 4 \frac{g}{l} \sin\alpha \sin\gamma}.$$ Altogether, we have (for $\gamma \le \alpha$) $$\dot\theta_{n+1}^{+} = \begin{cases} \cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 + 4 \frac{g}{l} \sin\alpha \sin\gamma}, & \text{ } \omega_1 < \dot\theta_n^{+} \\ -\dot\theta_n^{+} \cos(2\alpha), & \text{ } \omega_2 < \dot\theta_n^{+} < \omega_1 \\ -\cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 - 4\frac{g}{l} \sin\alpha \sin\gamma}, & \dot\theta_n^{+} < w_2 \end{cases}.$$

Notice that the return map is undefined for $\dot\theta_n = \{ \omega_1, \omega_2 \}$, because from these configurations, the wheel will end up in the (unstable) equilibrium point where $\theta = 0$ and $\dot\theta = 0$, and will therefore never return to the map.

This return map blends smoothly into the case where $\gamma > \alpha$. In this regime, $$\dot\theta_{n+1}^{+} = \begin{cases} \cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 + 4 \frac{g}{l} \sin\alpha \sin\gamma}, & \text{ } 0 \le \dot\theta_n^{+} \\ -\dot\theta_n^{+} \cos(2\alpha), & \text{ } \omega_2 < \dot\theta_n^{+} < 0 \\ -\cos(2\alpha) \sqrt{(\dot\theta_n^{+})^2 - 4\frac{g}{l} \sin\alpha \sin\gamma}, & \dot\theta_n^{+} \le w_2 \end{cases}.$$ Notice that the formerly undefined points at $\{\omega_1,\omega_2\}$ are now well-defined transitions with $\omega_1 = 0$, because it is kinematically impossible to have the wheel statically balancing on a single leg.

Fixed Points and Stability

For a fixed point, we require that $\dot\theta_{n+1}^{+} = \dot\theta_n^{+} = \omega^*$. Our system has two possible fixed points, depending on the parameters: $$ \omega_{stand}^* = 0,~~~ \omega_{roll}^* = \cot(2 \alpha) \sqrt{4 \frac{g}{l} \sin\alpha\sin\gamma}.$$ The limit cycle plotted above illustrates a state-space trajectory in the vicinity of the rolling fixed point. $\omega_{stand}^*$ is a fixed point whenever $\gamma < \alpha$. $\omega_{roll}^*$ is a fixed point whenever $\omega_{roll}^* > \omega_1$. It is interesting to view these bifurcations in terms of $\gamma$. For small $\gamma$, $\omega_{stand}$ is the only fixed point, because energy lost from collisions with the ground is not compensated for by gravity. As we increase $\gamma$, we obtain a stable rolling solution, where the collisions with the ground exactly balance the conversion of gravitational potential to kinetic energy. As we increase $\gamma$ further to $\gamma > \alpha$, it becomes impossible for the center of mass of the wheel to be inside the support polygon, making standing an unstable configuration.

Limit cycle trajectory of the rimless wheel ($m=1,l=1,g=9.8,\alpha=\pi/8,\gamma=0.15$). All hatched regions converge to the rolling fixed point, $\omega_{roll}^*$; the white regions converge to zero velocity ($\omega_{stand}^*$).

Stability of standing still

I opened this chapter with the idea that the natural notion of stability for a walking system is periodic stability, and I'll stand by it. But we can also examine the stability of a fixed-point (of the original coordinates; no Poincaré this time) for a system that has contact mechanics. For a legged robot, a fixed point means standing still. It's a little subtle. [Coming soon.]

call out to the fact that I left the origin OUT of the poincare map on the van der pol.

The Compass Gait

The rimless wheel models only the dynamics of the stance leg, and simply assumes that there will always be a swing leg in position at the time of collision. To remove this assumption, we take away all but two of the spokes, and place a pin joint at the hip. To model the dynamics of swing, we add point masses to each of the legs. I've derived the equations of motion for the system assuming that we have a torque actuator at the hip - resulting in swing dynamics equivalent to an Acrobot (although in a different coordinate frame) - but let's examine the passive dynamics of the system first ($\bu = 0$).

The compass gait

In addition to the modeling assumptions used for the rimless wheel, we also assume that the swing leg retracts in order to clear the ground without disturbing the position of the mass of that leg. This model, known as the compass gait, is well studied in the literature using numerical methods Goswami96a+Goswami99+Spong03, but relatively little is known about it analytically.

The state of this robot can be described by 4 variables: $\theta_{st},\theta_{sw},\dot\theta_{st}$, and $\dot\theta_{sw}$. The abbreviation $st$ is shorthand for the stance leg and $sw$ for the swing leg. Using $\bq = [ \theta_{st}, \theta_{sw} ]^T$ and $\bu = \tau$, we can write the dynamics as $$\bM(\bq)\ddot\bq + \bC(\bq,\dot\bq)\dot\bq = \btau_g(q) + \bB\bu,$$ with \begin{gather*} \bM = \begin{bmatrix} (m_h+m)l^2 + ma^2 & -mlb\cos(\theta_{st}-\theta_{sw}) \\ -mlb\cos(\theta_{st}-\theta_{sw}) & mb^2 \end{bmatrix}\\ \bC = \begin{bmatrix} 0 & -mlb\sin(\theta_{st}-\theta_{sw})\dot\theta_{sw} \\ mlb\sin(\theta_{st}-\theta_{sw})\dot\theta_{st} & 0 \end{bmatrix} \\ \btau_g(q) = \begin{bmatrix} (m_hl + ma + ml)g\sin(\theta_{st}) \\ -mbg\sin(\theta_{sw}) \end{bmatrix},\\ \bB = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \end{gather*} and $l=a+b$.

The foot collision is an instantaneous change of velocity caused by a impulsive force at the foot that brings the foot to rest. The update to the velocities can be calculated using the derivation for impulsive collisions derived in the appendix. To use it, we proceed with the following steps:

  • Add a "floating base" to the system by adding a free (x,y) joint at the pre-impact stance toe, $\bq_{fb} = [x,y,\theta_{st},\theta_{sw}]^T.$
  • Calculate the mass matrix for this new system, call it $\bM_{fb}$.
  • Write the position of the (pre-impact) swing toe as a function $\bphi(\bq_{fb})$. Define the Jacobian of this function: $\bJ = \frac{\partial \bphi}{\partial \bq_{fb}}.$
  • The post-impact velocity that ensures that $\dot\bphi = 0$ after the collision is given by $$\dot\bq^+ = \left[ \bI - \bM_{fb}^{-1}\bJ^T[\bJ\bM_{fb}^{-1}\bJ^T]^{-1}\bJ\right] \dot\bq^-,$$ noting that $\dot{x}^- = \dot{y}^- = 0$, and that you need only read out the last two elements of $\dot{\bq}^+$. The velocity of the post-impact stance foot will be zero by definition, and the new velocity of the pre-impact stance foot can be completely determined from the minimal coordinates.
  • Switch the stance and swing leg positions and velocities.

Numerical simulation of the compass gait

You can simulate the passive compass gait using:

examples/simple_legs.ipynb

Try playing with the initial conditions. You'll find this system is much more sensitive to initial conditions than the rimless wheel. It actually took some work to find the initial conditions that make it walk stably.

Numerical integration of these equations reveals a stable limit cycle, plotted below. Notice that when the left leg is in stance (top part of the plot), the trajectory quite resembles the familiar pendulum dynamics of the rimless wheel. But this time, when the leg impacts, it takes a long arc around as the swing leg before returning to stance. The impacts are also clearly visible as discontinuous changes (decreases) in velocity. The dependence of this limit cycle on the system parameters has been studied extensively in Goswami96a.

Passive limit cycle trajectory of the compass gait. ($m=5$kg, $m_h=10$kg, $a=b=0.5$m, $\phi=0.0525$rad. $\bx(0)=[0,0,0.4,-2.0]^T$). Drawn is the phase portait of the left leg angle, which is recovered from $\theta_{st}$ and $\theta_{sw}$ in the simulation with some simple book-keeping.

The basin of attraction of the stable limit cycle is a narrow band of states surrounding the steady state trajectory. Although the simplicity of this model makes it analytically attractive, this lack of stability makes it difficult to implement on a physical device.

The Kneed Walker

To achieve a more anthropomorphic gait, as well as to acquire better foot clearance and ability to walk on rough terrain, we want to model a walker that includes a kneeHsu07. The mathematical modeling would be simpler if we used a single link for the stance leg, but we'll go ahead and use two links for each leg (and each with a point mass) because this robot can actually be built!

The Kneed Walker

At the beginning of each step, the system is modeled as a three-link pendulum, like the ballistic walkerMcMahon80+Mochon80+Spong03. The stance leg is the one in front, and it is the first link of a pendulum, with two point masses. The swing leg has two links, with the joint between them unconstrained until knee-strike. Given appropriate mass distributions and initial conditions, the swing leg bends the knee and swings forward. When the swing leg straightens out (the lower and upper length are aligned), knee-strike occurs. The knee-strike is modeled as a discrete inelastic collision, conserving angular momentum and changing velocities instantaneously.

After this collision, the knee is locked and we switch to the compass gait model with a different mass distribution. In other words, the system becomes a two-link pendulum. Again, the heel-strike is modeled as an inelastic collision. After the collision, the legs switch instantaneously. After heel-strike then, we switch back to the ballistic walker's three-link pendulum dynamics. This describes a full step cycle of the kneed walker, which is shown above.

Limit cycle trajectory for kneed walker

By switching between the dynamics of the continuous three-link and two-link pendulums with the two discrete collision events, we characterize a full point-feed kneed walker walking cycle. After finding appropriate parameters for masses and link lengths, a stable gait is found. As with the compass gait's limit cycle, there is a swing phase (top) and a stance phase (bottom). In addition to the two heel-strikes, there are two more instantaneous velocity changes from the knee-strikes as marked in the figure. This limit cycle is traversed clockwise.

Limit cycle (phase portrait) of the kneed walker
reproduce this model in drake.

Curved feet

The region of attraction of the kneed-walking limit cycle can be improved considerably by the addition of curved feet...

also knee "de-bouncers" (suction cups). Simulation model?
Tad McGeer's kneed walker. Here is Matt Haberland's guide to launching it successfully. It's nontrivial!

And beyond...

The world has seen all sorts of great variations on the passive-dynamic walker theme. Almost certainly the most impressive are the creations of Dutch artist Theo Jansen -- he likes to say that he is creating "new forms of life" which he calls the Strandbeest. There are many variations of these amazing machines -- including his beach walkers which are powered only by the wind (I've actually been able to visit Theo's workshop once; it is very windy there).

These results are very real. Robin Deits (an exceptionally talented student who felt particularly inspired once on a long weekend) once reproduced one of his most early creations in .

Robin Deits' simulation of the Strandbeest.

Simple Models of Running

Just like walking, our understanding of the dynamics and control of running has evolved by a nice interplay between the field of biomechanics and the field of robotics. But in running, I think it's fair to say that the roboticists got off the starting blocks first. And I think it's fair to say that it started with a series of hopping machines built by Marc Raibert and the Leg Laboratory (which was first at CMU, but moved to MIT) in the early 1980's. At a time where many roboticsts were building relatively clumsy walking robots that moved very slowly (when they managed to stay up!), the Leg Laboratory produced a series of hopping machines that threw themselves through the air with considerable kinetic energy and considerable flair.

To this day, I'm still a bit surprised that impressive running robots (assuming you accept hopping as the first form of running) appeared before the impressive walking robots. I would have thought that running would be a more difficult control and engineering task than walking. But these hopping machines turned out to be an incredibly clever way to build something simple and very dynamic.

Shortly after Raibert built his machines, Dan Koditschek and Martin Buehler started analyzing them with simpler models Koditschek91. Years later, in collaboration with biologist Bob Full, they started to realize that the simple models used to describe the dynamics of Raibert's hoppers could also be used to describe experimental data obtained from running animals. In fact, they described an incredible range of experimental results, for animals ranging in size from a cockroach up to a horseFull99 (I think the best overall summary of this line of work is Holmes06). The most successful of the simple models was the so-called "spring-loaded inverted pendulum" (or "SLIP", for short).

The Spring-Loaded Inverted Pendulum

The model is a point mass, $m$, on top of a massless, springy leg with rest length of $l_0$, and spring constant $k$. The state of the system is given by the $x,y$ position of the center of mass, and the length, $l$, and angle $\theta$ of the leg. Like the rimless wheel, the dynamics are modeled piecewise - with one dynamics governing the flight phase, and another governing the stance phase.

Flight Phase. State variables: $\bx = [x,y,\dot{x},\dot{y}]^T.$ Dynamics are $$\dot{\bx} = \begin{bmatrix} \dot{x} \\ \dot{y} \\ 0 \\ - g \end{bmatrix}.$$

Stance Phase. State variables: $\bx = [r, \theta, \dot{r}, \dot\theta]^T.$ Kinematics are $$x = \begin{bmatrix} - r \sin\theta \\ r \cos\theta \end{bmatrix},$$ Energy is given by $$T = \frac{m}{2} (\dot{r}^2 + r^2 \dot\theta^2 ), \quad U = mgr\cos\theta + \frac{k}{2}(r_0 - r)^2.$$ Plugging these into Lagrange yields: \begin{gather} m \ddot{r} - m r \dot\theta^2 + m g \cos\theta - k (r_0 - r) = 0 \\ m r^2 \ddot{\theta} + 2mr\dot{r}\dot\theta - mgr \sin\theta = 0 \end{gather}

Simulation of the SLIP model

You can simulate the spring-loaded inverted pendulum using:

examples/spring_loaded_inverted_pendulum.ipynb

Make sure that you take time to interpret the apex-to-apex map that is plotted in the notebook.

Hopping Robots from the MIT Leg Laboratory

The Planar Monopod Hopper

A three-part control strategy: decoupling the control of hopping height, body attitude, and forward velocity Raibert86a.

Running on four legs as though they were one

bipeds, quadrupeds, and backflips...Raibert86

A Simple Model That Can Walk and Run

Exercises

Searching for Limit Cycles via Trajectory Optimization

In this exercise we use trajectory optimization to identify a limit cycle for the compass gait. We use a rather general approach: the robot dynamics is described in floating-base coordinates and frictional contacts are accurately modeled. In this notebook, you are asked to code many of the constraints this optimization problem requires:

  1. Enforce the contact between the stance foot and the ground at all the break points.
  2. Enforce the contact between the swing foot and the ground at the initial time.
  3. Prevent the penetration of the swing foot in the ground at all the break points. (In this analysis, we will neglect the scuffing between the swing foot and the ground which arises when the swing leg passes the stance leg.)
  4. Ensure that the contact force at the stance foot lies in the friction cone at all the break points.
  5. Ensure that the impulse generated by the collision of the swing foot with the ground lies in the friction cone.

One-Dimensional Hopper

In this exercise we implement a one-dimensional version of the controller proposed in Raibert84. The goal is to control the vertical motion of a hopper by generating a stable resonant oscillation that causes the system to hop off the ground at a given height. We accomplish this via a careful energy analysis of the hopping cycle: all the details are in this notebook. Besides understanding in detail the dynamics of the hopping system, in the notebook, you are asked to write two pieces of code:

  1. A function that, given the state of the hopper, returns its total mechanical energy.
  2. The hopping controller class. This is a VectorSystem that, at each sampling time, reads the state of the hopper and returns the preload of the hopper spring necessary for the system to hop at the desired height. All the necessary information for the synthesis of this controller are given in the notebook.
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