A Variational Approach to Optimal Control of Underactuated Mechanical Systems with Collisions

In real machines and mechanisms, almost all joints are limited. A common "hard limiter" in mechanical systems is a collision between bodies, or between bodies and system boundaries. Often these collisions are modelled by impulse-like contact forces. During such a collision of two or more bodies, their positions go on continuously, but their velocities jump. We consider this non-smoothness and its consequences for simulation and control in a variational setting, from which a discretization follows in a systematic way. The resulting variational integrator (VI) for collisions is known from the literature, but the incorporation of the collision equations into the Discrete Mechanics and Optimal Control (DMOC) approach follows a new idea. As many technical systems are underactuated, we take this into account too. The feed-forward control for the swing-up of a pendulum-on-cart-system, a.k.a. inverted pendulum, proves the concept. The sequence of collisions is not optimized yet, but for a given sequence, we find the cart force to optimally steer the pendulum into its final state.