Optimal Control of Piecewise-Smooth Control Systems via Singular Perturbations

This paper investigates optimal control problems formulated over a class of piecewise-smooth controlled vector fields. Rather than optimizing over the discontinuous system directly, we instead formulate optimal control problems over a family of regularizations which are obtained by "smoothing out" the discontinuity in the original system using tools from singular perturbation theory. Standard, efficient derivative-based algorithms are immediately applicable to solve these smooth approximations to local optimally. Under standard regularity conditions, it is demonstrated that the smooth approximations provide accurate derivative information about the non-smooth problem in the limiting case. The utility of the technique is demonstrated in an in-depth example, where we utilize recently developed reduced-order modeling techniques from the dynamic walking community to generate motion plans across contact sequences for a 18-DOF model of a lower-body exoskeleton.