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The Finite Elements with Switch Detection (FESD) is a high-accuracy methodfor the numerical simulation and solution of optimal control problems subjectto discontinuous ODEs. In this article, we extend the FESD method [Nurkanovi\'cet al., 2022] to the dynamic equations of multiple rigid bodies that exhibitstate jumps due to impacts and Coulomb friction. This new method is referred toas FESD with Jumps (FESD-J). Starting from the standard Runge-Kutta equations,we let the integration step sizes be degrees of freedom. Additional constraintsare introduced to ensure exact switch detection and to remove spurious degreesof freedom if no switches occur. Moreover, at the boundaries of each finiteelement, we impose the impact equations in their complementarity form, at boththe position and velocity level. They compute the normal and tangentialimpulses in case of contact making. Otherwise, they are reduced to thecontinuity conditions for the velocities. FESD-J treats multiple contacts,where each contact can have a different coefficient of restitution andfriction. All methods introduced in this paper are implemented in theopen-source software package NOSNOC. We illustrate the use of FESD-J in bothsimulation and optimal control examples.

FESD-J: Finite Elements with Switch Detection for Numerical Optimal Control of Rigid Bodies with Impacts and Coulomb Friction