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

TL;DR

This paper presents a novel time-freezing reformulation and numerical approach for optimal control of complementarity Lagrangian systems with state jumps, enabling high-accuracy solutions for systems with impacts and friction.

Contribution

It introduces a new time-freezing reformulation that transforms the problem into a Filippov system, facilitating the use of advanced numerical methods for systems with impacts.

Findings

The method accurately solves optimal control problems with impacts.

The approach handles boundary evolution after state jumps.

Numerical example demonstrates effectiveness on a hopping robot.

Abstract

This paper introduces a novel time-freezing reformulation and numerical methods for optimal control of complementarity Lagrangian systems (CLS) with state jumps. We cover the difficult case when the system evolves on the boundary of the dynamic's feasible set after the state jump. 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 dynamical 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. The resulting time-freezing system is a Filippov system that has jump discontinuities only in the first time derivative instead of the trajectory itself. This enables one to use the recently proposed Finite Elements with Switch Detection [Nurkanovic et al., 2022], which makes high accuracy numerical optimal control of CLS with impacts and friction possible. We detail how to recover the solution of the original system and show how to select appropriate auxiliary dynamics. The theoretical findings are illustrated on a nontrivial numerical optimal control example of a hopping one-legged robot.