Using Equilibria and Virtual Holonomic Constraints to Generate Families of Walking Gaits

We present a principled approach to generating walking gaits for pointand curved-feet, underactuated, multi-degreeof-freedom, 2D and 3D biped walkers subject to a biped’s hybrid dynamics and user-defined virtual holonomic constraints (constraints that are enforced through feedback control). Specifically, we combine numerical continuation methods (NCM) and the hybrid zero dynamics (HZD) framework to generate families of gaits starting from equilibria of the biped model that satisfy the hybrid dynamics and virtual holonomic constraints (VHC) [1], [2]. The main contribution of the NCM-HZD framework is a topological approach to gait generation. Under this formulation, gaits form connected components in a finite parameter space. As we are interested in generating gaits from equilibria, we give a sufficient condition for when the connected component of an equilibrium point contains a branch of walking gaits. We cannot prove that these conditions hold for all bipeds, but we give a fast and simple computational test that reports whether the condition holds for a given robot. We have had success generating gaits for a representative set of bipeds, including a point-foot model of Boston Dynamics’ Atlas robot.