A Study of the Passive Gait of a Compass-Like Biped Robot: Symmetry and Chaos

Abstract

The focus of this work is a systematic study of the passive gait of a compass-like, planar, biped robot on inclined slopes. The robot is kinematically equivalent to a double pendulum, possessing two kneeless legs with point masses and a third point mass at the "hip" joint. Three parameters, namely, the ground-slope angle and the normalized mass and length of the robot describe its gait. We show that in response to a continuous change in any one of its parame ters, the symmetric and steady stable gait of the unpowered robot gradually evolves through a regime of bifurcations characterized by progressively complicated asymmetric gaits, eventually arriving at an apparently chaotic gait where no two steps are identical. The robot can maintain this gait indefinitely. A necessary (but not sufficient) condition for the stability of such gaits is the contraction of the "phase-fluid" volume. For this frictionless robot, the volume contraction, which we compute, is caused by the dissipative effects of the ground-impact model. In the chaotic regime, the fractal dimension of the robot's strange attractor We present a novel graphical technique based on the first return map that compactly captures the entire evolution of the gait, from symmetry to chaos. Additional passive dissipative elements in the robot joint result in a significant improvement in the stability and the versatility of the gait, and provide a rich repertoire for simple control laws.