Nonsmooth optimal value and policy functions in mechanical systems subject to unilateral constraints

State-of-the-art approaches to optimal control use smooth approximations of value and policy functions and gradient-based algorithms for improving approximator parameters. Unfortunately, we show that value and policy functions that arise in optimal control of mechanical systems subject to unilateral constraints -- i.e. the contact-rich dynamics of robot locomotion and manipulation -- are generally nonsmooth due to the underlying dynamics exhibiting discontinuous or piecewise-differentiable trajectory outcomes. Simple mechanical systems are used to illustrate this result and the implications for optimal control of contact-rich robot dynamics.