The Simplest Walking Model: Stability, Complexity, and Scaling

Abstract

We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope gamma. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < gamma < 0.015 rad. With increasing gamma, stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to gamma 1/3, and that the gravitational power used is proportional to v4 where v is the velocity along the slope.

The strengths and weaknesses of inverted pendulum models of human walking

“…The IP model of walking has been described as the "simplest walking model" [24] . Despite this only one previous paper has set out to describe the biomechanical characteristics of the IP and then only at a single walking speed [18] .…”

Humans exploit the biomechanics of bipedal gait during visually guided walking over complex terrain

“…This approach to the study of locomotion has its origins in research on passive dynamic walking robots. These simple, mechanical bipedal robots capitalize on the physical dynamics of bipedal gait in order to walk with no actuation down a shallow slope [4][5][6][7]. Minimal actuation in the form of footsprings or hip actuators allows these robots to walk over flat ground with an energetic cost of transport that is comparable with a human and is an order of magnitude less than the cost of transport of traditional walking robots [8].…”

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