Extending equilibria to periodic orbits for walkers using continuation methods

We present a strategy for generating period-one, open-loop walking gaits for multi-degree-of-freedom, planar biped walkers. Our approach uses equilibria of the dynamics as templates, which we connect to a family of period-one walking motions using numerical continuation methods. We define a gait as a fixed point of the walker´s hybrid dynamics which resides in a state-time-control space consisting of the robot´s post-impact state, switching time (the time at which the swing leg impacts the ground), and a finite set of design or control parameters. We demonstrate our approach on several physically-symmetric biped walkers. In particular, we prove that our approach reduces the search space for an initial gait in the state-time-control space to a one-dimensional search in switching time. We show that we can generates periodic motion without resorting to splines or reference trajectories. Finally, we compare our method to generating gaits with virtual holonomic constraints.