A Simplified Stability Study for a Biped Walk with Underactuated and Overactuated Phases

This paper is devoted to a stability study of a walking gait for a biped. The walking gait is periodic and it is composed of a single-support phase, a passive impact, and a double-support phase. The reference trajectories are described as a function of the shin orientation versus the ground of the stance leg. We use the Poincaré map to study the stability of the walking gait of the biped. We only study the stability of dynamics not controlled during the single-support phase, i.e., the dynamics of the shin angle. We then suppose there is no perturbation in the tracking of the references of the other joint angles of the biped. The studied Poincaré map is then of dimension one. With a particular control law in double support, it is shown theoretically and in simulation that a perturbation error in the velocity of the shin angle can be eliminated in one step only. The zone of convergence in one step is determined. The condition of existence of a cyclic gait is given, and for a given cyclic gait, the stability condition is also given. It is shown that due to the given control law for the overactuated double-support phase, a cyclic motion is practically guaranteed to be stable. It should be noted it is possible for the biped to reach a periodic regime from a stopped position in one step. KEY WORDS—walking biped, orbital stability, passive impact, dynamically stable gait, Poincaré return map, simulation results