Smooth Approximations for Hybrid Optimal Control Problems with Application to Robotic Walking

This paper investigates optimal control problems formulated over a class of hybrid dynamical systems which display event-triggered discrete jumps. Due to the discontinuous nature of the underlying dynamics, previous approaches to solving optimal control problems over this class of systems generally rely on fixing the number and sequence of discrete jumps a priori, or search over possible mode sequences in a combinatorial manner. Employing contributions from the geometric theory of hybrid systems, we instead formulate a family of smooth approximate problems formulated over a family of smooth control systems which faithfully approximate the dynamics of the original hybrid system, in an appropriate metric. Efficient gradient-based methods can be used to solve the smooth approximations, without specifying the sequence of discrete transitions ahead of time. Under appropriate hypothesis, the gradients of the smooth problem are shown to be well-conditioned and closely approximate the gradients of the non-smooth problem (when they exist). Two cases studies demonstrate the utility of the approach, including an in-depth application to generating a stable walking motion for a bipedal robot.