The Time-Freezing Reformulation for Numerical Optimal Control of Complementarity Lagrangian Systems with State Jumps

This paper introduces a novel reformulation and numerical methods for optimal control of complementarity Lagrangian systems with state jumps. The solutions of the reformulated system have jump discontinuities in the first time derivative instead of the trajectory itself, which is easier to handle theoretically and numerically. We cover not only the easier case of elastic impacts, but also the difficult case, when after the state jump the system evolves on the boundary of the dynamic’s feasible set. In nonsmooth mechanics this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamic system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. We detail how to recover the solution of the original system, show how to select appropriate auxiliary dynamics and give practical numerical methods to handle discontinuous ODEs with nonunique sliding motions. Moreover, we introduce a novel auxiliary ODE for time-freezing for elastic impacts and overcome some drawbacks of [22]. The theoretical findings are illustrated on the nontrivial numerical optimal control example of a hopping one-legged robot.