Passivity-based control of underactuated biped robots within hybrid zero dynamics approach

The concept of hybrid zero dynamics is a promising approach for designing exponentially stabilizing controllers for dynamic walking with some degrees of underactuation. By this approach a feedback controller is designed such that a stable periodic orbit, within an invariant submanifold for the hybrid closed-loop system is created. This is usually achieved through an exponentially fast dynamics transverse to the zero dynamics manifold and the stability properties of such periodic orbit is then transferred to the full-order dynamic system. In this paper a passivity-based controller for a planar biped with one degree of underactuation is designed. By this approach we aim to preserve the natural dynamics of the system in the transverse dynamics (i.e. the dynamics transverse to the zero dynamics manifold) in contrast to the common input-output linearization method which cancels these dynamics. A Lyapunov stability analysis of the full-order system based on the conditional stability theorem is presented. By this analysis, the asymptotic stability of the periodic orbit in lower dimensional state space is extended to the full dimensional space. The results of the analysis are verified by simulation on a seven-link biped robot walking with zero ankle torque in sagittal plane.