Exploring Passive-Dynamic Walking - Semantic Scholar

Exploring Passive-Dynamic Walking Gordon Berman, Cornell University Jo-Anne Ting, University of Southern California Complex Systems Summer School September 15, 2005

1.

Introduction

Bipedal walking has been demonstrated through physical and computer simulations by McGeer [1] to be possible without actuation or control within a range of shallow slopes. Results from such passive dynamic walkers have led many researchers to believe the mechanics of walking should be understood, in addition to the activation and control issues, for biped locomotion to be realized. Recent work by Collins et al [2] has revived the interest in passive-dynamic walking, after several years of dormancy. This paper examines, in greater detail, experiments and results from a passive-dynamic walking model and attempts to evaluate the potential of passive-dynamic walking for biped locomotion.

2.

Passive-Dynamic Walker

2.1

Model Description

We examine the passive-dynamic walking model introduced by Garcia et al. in [3]. Figure 1 below illustrates the point-foot model, where masses exist at the hip and the feet. The hip hinge is assumed to be frictionless, and the hip mass M is assumed to be much larger than the foot mass m such that M>>m. This ensures that leg movement does not interfere with hip movement. The biped mechanism moves down a rigid sloped ramp in two phases. When the foot strikes the ramp surface, its velocity goes to zero since it has a plastic collision. While the other foot swings, this foot stays on the surface until the other, in turns, hits the ramp surface. The laws of classical rigid-body mechanics can be used to derive the equations of motion. For non-physical simulations done with this model, it is assumed that the swing foot can pass through the ramp when the stance leg is almost vertical (i.e. we ignore scuffing). This prevents foot-scuffing problems that occur with straight-leg (knee-less) walkers.

M l

# " m !

Figure 1: Passive-Dynamic Walking Model from Garcia et al. (where M is the hip mass, m is the leg mass, l is the leg length, ! is the ramp angle, " is the angle between the stance leg and the slope normal, and # is the angle between the stance leg and the swing leg )

2.2

Model Motion

Garcia et al. apply a nonlinear dynamics approach to analyze the passive-dynamic model by interpreting one step as a Poincaré map, where the gait limit cycles are fixed points on the map. To evaluate the stability of a gait cycle, we can examine the eigenvalues of the linearized map at the fixed point. The model consists of three components: two equations of motion and a transition rule at the heel strike condition. The two equations of motion can be derived from the swing phase and are expressions for angular momentum balance about the foot and about the hip (i.e. a simple double pendulum), as Equation 1 shows.

Equation 1: Equations of Motion for Double Pendulum

$1 + 2 ! (1 " cos # ) " ! (1 " cos # ) ' $*!!' $ " ! sin # (#! 2 " 2*!#! ) ' )+ & ! (1 " cos # ) ) &#!!) + & "! !*! 2 sin # % (% ( % ( $ !lg [sin(* " # " + ) " sin(* " + )] " gl sin(* " + ) ' $ 0 ' & )=& ) !g sin( * " # " + ) &% )( % 0 ( l ($ = m/M and ", # are functions of time t) Assuming that the foot is much smaller than the body (m .019, the walker no longer could sustain a gait.

Figure 4: Bifurcation Diagram of the Walker Model

4.3

Decay of Periodic Orbits

In order to get a better feel for how the system transitions into chaos, we took slices of the steady-state solution at various points in the bifurcation diagram. This was done by making histograms of the values of "%, generated at every heel strike (following a transient of about 100 time-steps). These are shown in Figure 5. In the first row, the histograms show the periodic solutions, which then widen out into more ergodic-looking solutions near ! = .0184. The solution then goes through periodic windows (such as ! = .018575), as then returns to the chaotic solutions with a large peak at the far right end of the spectrum. This peak represents a stable periodic orbit that has now decayed into an unstable one.

Figure 5: Histograms of "* at Various ! 4.4

Basins of Attraction

The analysis presented in the previous three portions of this section all rely on finding an initial condition that will lead to the attractor. This simple assumption, however, proved difficult in implementing our simulations. Often, when trying to reproduce the results from the original paper, we found that running the simulation with initial conditions in most regions of phase space would result in the walker falling down. Eventually, we discovered a small region in which the results in [3] were generated, but this result led us to systematically study the basins of attraction which lead to the attractor. For our purposes here, an initial condition is said to lead to the attractor if the walker does not fall over after integrating the equations for a long time (a few hundred steps) starting from that point. This definition for belonging to the basin of attraction works for our system because for a particular value of !, we have empirically observed that there is only one gait type. Shown in Figure 6 and Figure 7 are these basins for ! = .014, which is in the stable period-1 region, and for ! = .019, which is within the chaotic region. Only portions of the total basin are shown due to limitations in computational power. However, what is noticeable especially is the fractal shape of the basins in both cases. Also note the relatively small region of the possible phase space that these pictures represent, though experience trying initial conditions have lead us to believe that the basins of attraction do

not extend out significantly further than presented here. Also of interest is the fact that the location of the basin shifts as a function of !.Along with the fractal shape of the basin, this facet would pose difficulties in terms of designing a controller for the system.

Figure 6: Section of the Basin of Attraction for ! = .014

Figure 7: Section of the Basin of Attraction for ! = .019

5.

Discussion

Although the analysis of Garcia et al is correct in predicting stable periodic motions for small ramp angles, and even continuously walking chaotic trajectories, the prospects for control in this system are difficult. Not only do the periodic orbits decay with increasing slope, but so do the basins which attract them. As a result, one needs to address control issues regarding the periodicity of the motion, and, additionally, ensure that the system is in a state that can lead to an attractor (since this is not guaranteed). This difficulty does not mean that some sort of control on the walker is impossible, but it does move such a problem outside the scope of this paper.

6.

Future Work

The first obvious task related to this system would be to map out the basins of attraction in more detail – as we have only explored the phase space in detail at one location (however, anecdotal experiences tell us that the basins are not much larger than the pictures presented here). One question that could be asked in future work is why exactly the system’s basin of attraction exhibits the fractal structure seen in previous sections, as no analytic result to that effect is presented here. Also interesting would be to predict how the basin moves with !. This latter topic may lead to insights if one wanted to devise some control for the system, as was our original intent here. Finally, a large topic for future explorations would be to attempt to harness the chaotic trajectories the system exhibits in order to provide control to the system.

References [1] McGeer, T. Passive Dynamic Walking. International Journal of Robotics Research, No 2, pp62-82, 1990. [2] Collins, S., Ruina, A., Tedrake, R. and Wisse, M. Efficient Bipedal Robots Based on Passive-Dynamic Walkers. Science 307, 1082, February 18, 2005. [3] Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M. The Simplest Walking Model: Stability, Complexity, and Scaling. ASME Journal of Biomechanical Engineering, February 10, 1998. [4] McGeer, T. Passive Walking With Knees. Proceedings of the IEEE Conference on Robotics and Automation, No 2, pp1640-1645, 1990. [5] Lattanzio, H. Kuenzler, G. and Reading, M. Passive Dynamic Walking. Project Report, Human Power Lab, Cornell University, May 1992. [6] Coleman, M. and Ruina, A. An Uncontrolled Toy That Can Walk But Cannot Stand Still. Physical Review Letters, Vol. 80, Issue 16, pp3658-3661, April 1998. [7] Alexander, R. M. Simple Models of Human Motion. Applied Mechanics Review, No 48, pp461-469, 1995. [8] Goswami, A., Espiau, B. and Keramane, A. Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws. Journal of Autonomous Robots. 4(3), pp273-286, 1997.