A period-three passive gait tracking control for bipedal walking of a compass-gait biped robot

In this paper we interest on the tracking control subject of a period-three passive bipedal walking for a planar compass-gait biped robot. This biped robot is an impulsive mechanical system of two-degrees-of-freedom known to possess passive periodic bipedal walking patterns found to be reminiscent of human walking. It is well known that such compass-gait biped can walk steadily down an inclined slope without any control. The behavior of its gait depends closely on the ground slope as well as on its inertial parameters. We showed recently that such compass-gait robot can possess two distinct periodic stable gaits for the same ground slope: a period-1 gait and another period-3 gait which has not been fully studied yet. This paper deals with the period-3 passive stable gait. Some properties of this typical gait are discussed in this paper revealing its typical symmetry, efficiency and its importance for the control subject. We reveal that the compass-gait biped robot presents an elegant and attractive manner to walk with the period-three passive gait which can be employed in some desirable objectives for the passive bipedal walking. We show in addition that the period-3 gait is very sensitive to small perturbations and becomes unstable where the passive walk of the compass biped is found to converge to the period-1 gait. This conducts to the fact that the basin of attraction of such period-three gait is very restricted. Then, in order to stabilize the passive period-three gait, we propose in this paper an energy tracking control law. Furthermore, we apply this same control law to track the period-three passive gait when the biped robot walks down a different sloped surface.