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This paper introduces a novel time-freezing reformulation and numericalmethods for optimal control of complementarity Lagrangian systems (CLS) withstate jumps. We cover the difficult case when the system evolves on theboundary of the dynamic's feasible set after the state jump. In nonsmoothmechanics, this corresponds to inelastic impacts. The main idea of thetime-freezing reformulation is to introduce a clock state and an auxiliarydynamical system whose trajectory endpoints satisfy the state jump law. Whenthe auxiliary system is active, the clock state is not evolving, hence bytaking only the parts of the trajectory when the clock state was active, we canrecover the original solution. The resulting time-freezing system is a Filippovsystem that has jump discontinuities only in the first time derivative insteadof the trajectory itself. This enables one to use the recently proposed FiniteElements with Switch Detection [Nurkanovic et al., 2022], which makes highaccuracy numerical optimal control of CLS with impacts and friction possible.We detail how to recover the solution of the original system and show how toselect appropriate auxiliary dynamics. The theoretical findings are illustratedon a nontrivial numerical optimal control example of a hopping one-leggedrobot.

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