Stabilization of Periodic Orbits for Planar Walking With Noninstantaneous Double-Support Phase

This paper presents an analytical approach to design a continuous time-invariant two-level control scheme for asymptotic stabilization of a desired period-one trajectory for a hybrid model describing walking by a planar biped robot with noninstantaneous double-support phase and point feet. It is assumed that the hybrid model consists of both single- and double-support phases. The design method is based on the concept of hybrid zero dynamics. At the first level, parameterized continuous within-stride controllers, including single- and double-support-phase controllers, are employed. These controllers create a family of 2-D finite-time attractive and invariant submanifolds on which the dynamics of the mechanical system is restricted. Moreover, since the mechanical system during the double-support phase is overactuated, the feedback law during this phase is designed to be minimum norm on the desired periodic orbit. At the second level, parameters of the within-stride controllers are updated by an event-based update law to achieve hybrid invariance, which results in a reduced-order hybrid model for walking. By these means, stability properties of the periodic orbit can be analyzed and modified by a restricted Poincaré return map. Finally, a numerical example for the proposed control scheme is presented.