A Differentiable Formulation of a Set‐Valued Force Law Approximating Planar Coulomb Friction and Its use in Optimal Control

ABSTRACT In this paper, we present a regularization of the set‐valued Coulomb friction law based on a regularized projection. The regularized projection is used to formulate differentiable equations encoding a set‐valued friction law that can be used in typical optimal control frameworks. The regularized friction law is compared to the popular sigmoid friction law regularization of set‐valued Coulomb friction in the context of an optimal control problem for a pendulum driven via a frictional clutch. The numerical experiments show that the regularized projection‐based friction law leads to a better approximation of the set‐valued Coulomb friction law than the sigmoid friction law while even achieving faster convergence of the underlying optimization problem. In particular, the regularized projection‐based friction law, being set‐valued itself, leads to a solution of the optimal control problem that can satisfy the sticking condition of the set‐valued Coulomb friction law exactly, whereas the sigmoid friction law is unable to predict sticking motions.