Control and planning of 3-D dynamic walking with asymptotically stable gait primitives

In this paper, we present a hierarchical framework that enables motion planning for asymptotically stable 3-D bipedal walking in the same way that planning is already possible for zero moment point walking. This framework is based on the construction of asymptotically stable gait primitives for a class of hybrid dynamical systems with impacts. Each primitive corresponds to an asymptotically stable hybrid limit cycle that admits rules a priori for sequential composition with other primitives, reducing a high-dimensional feedback motion planning problem into a low-dimensional discrete tree search. As a constructive example, we develop this planning framework for the 3-D compass-gait biped, where each primitive corresponds to walking along an arc of constant curvature for a fixed number of steps. We apply a discrete search algorithm to plan a sequence of these primitives, taking the 3-D biped stably from start to goal in workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated five-link biped.