Spatio-temporal optimization of multi-phase movements: Dealing with contacts and switching dynamics

We address the optimal control problem of robotic systems with variable stiffness actuation (VSA) including switching dynamics and discontinuous state transitions. Our focus in this paper is to consider tasks that have multiple phases of movement, contacts and impacts with the environment. By modelling such tasks as an approximate hybrid dynamical system with time-based switching, we develop a systematic methodology to simultaneously optimize control commands, stiffness profiles and temporal aspect of the movement such as switching instances and total movement duration. Numerical evaluations on a simple switching system, a realistic brachiating robot model with VSA, and a hopper with variable stiffness springs demonstrate the effectiveness of the proposed approach.