Stability, scaling, and chaos in passive-dynamic gait models

In this work, we study computer simulations of simple biped models with no actuation except gravity, and no control. These so-called passive-dynamic models of human gait were first studied by McGeer (1989). Computer simulations were also used to construct two kneed walkers for demonstration purposes.; We begin our study with a simple one-parameter walking model, in 2-D, then we move to more general 2-D models with and without knees, and finally we study a 3-D model with no knees. In 2-D, we are most interested in gait efficiency, while in 3-D, we focus on gait stability. We find general rules for the one-parameter model which can be extended to understand the behavior of the more complicated models.; A summary of the main points is as follows: (1) The “simplest” walking model with only a point-mass at the hip exhibits two gait cycles, one of which is stable at small slopes. Both gait cycles extend to arbitrarily small slopes, and are therefore “perfectly efficient.” This model has a step length proportional to the cube root of the slope; power usage scales with (velocity)4. An asymptotic analysis agrees with numerical simulation results at small slopes. The long-step gait exhibits period doubling bifurcations to chaotic gait as the slope is varied. (2) More general models with and without knees can also have up to two gait cycles, one of which can be stable. In general, these models will not be able to walk at arbitrarily small slopes. We present mass distribution conditions for perfect walking efficiency. These “tuned” walkers retain one of the cube-root-scaling gaits, but the other gait, which is always unstable, has a step length proportional to the slope at very small slopes. A period-doubling route to chaos is also numerically-demonstrated for a tuned kneed walker. Some data is also presented from a working physical walker. (3) In 3-D, planar 2-D gaits still exist but are unstable. A torsional spring at the hip of a 3-D model improves its stability somewhat. Automated gradient-based parameter searches to minimize the maximum eigenvalue terminate at local minima; no stable 3-D walking gaits were found for our model. We conclude that this model is not sufficient to explain the stability of the walker of Coleman and Ruina (1998).