Controlled Symmetries and Passive Walking 1

It was shown in Spong [1999b] that the passive gaits for a planar 2DOF biped walking on a shallow slope (Goswami et al. [1995]) can be made slope invariant by a passivity based control that compensates only the gravitational torques acting on the biped. In this paper we extend these results to the general case of a 3-D, n-DOF robot. We show that if there exists a passive walking gait, i.e., if there exists a ground slope and a set of initial conditions that give rise to a stable limit cycle trajectory of the system, then there exists a passivity-based nonlinear control law that renders the limit cycle slope invariant. The result is constructive in the sense that we generate the resulting control law and initial conditions from the initial conditions of the passive biped and the ground slope. This intuitively simple result relies on some well-known symmetries in the dynamics of mechanical systems with respect to the group action of SO(3) on solution trajectories of the system. We also discuss the use of passivity-based control to increase the basin of attraction of the passive limit cycle.