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

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 parameters 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 indeenitely. A necessary (but not suucient) condition for the stability of such gaits is the contraction of the \phase uid" volume. For this frictionless robot the volume contraction, which we compute, is caused by the dissipative eeects of the ground impact model. In the chaotic regime the fractal dimension of the robot's strange attractor (2.07) compared to its state-space dimension (4) also reveals strong contraction. We present a novel graphical technique based on the rst 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 signiicant improvement in the stability and the versatility of the gait and provides a rich repertoire for simple control laws. Biped robots and other legged robots are potentially better suited than wheeled vehicles to the maintenance of hazardous environments (nuclear and chemical reactors), exploration of unstruc-tured and unpaved terrains (ocean oors, polar regions, lunar and Martian surfaces), deep-forest logging, fruit harvesting etc. At present, one of the main obstacles to a wider application of legged robots is their lack of energy eeciency. In comparison, their biological analogues demonstrate impressive energy economy during normal walking gait. In fact, EMG studies have shown 32]]27] relative muscle inactivity during the swing phase of human walk makes it almost passive. This illustrates the superiority of the biological control strategy which functions in harmony with the natural inertial dynamics of the body in the gravitational eld. The long-term motivation behind the current study is to formulate a simple biologically-inspired active control law for a 17-dof biped robot 5] being built in Project BIP coordinated by the INRIA laboratory in Grenoble, France. …