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Collisions are common in many dynamical systems with real applications. Theycan be formulated as hybrid dynamical systems with discontinuitiesautomatically triggered when states transverse certain manifolds. We present analgorithm for the optimal control problem of such hybrid dynamical systemsbased on solving the equations derived from the hybrid minimum principle (HMP).The algorithm is an iterative scheme following the spirit of the method ofsuccessive approximations (MSA), and it is robust to undesired collisionsobserved in the initial guesses. We analyze the discontinuities in the systemand propose a stable collision condition, which is crucial for the convergenceof iterative algorithms in systems experiencing collisions. Subsequently, weestablish a convergence theorem demonstrating linear convergence for the MSAalgorithm when collisions are present. We also address several numericalchallenges introduced by the discontinuities. The algorithm is tested on disccollision problems whose optimal solutions exhibit one or multiple collisions.Linear convergence in terms of iteration steps and asymptotic first-orderaccuracy in terms of time discretization are observed when the algorithm isimplemented with the forward-Euler scheme. The numerical results demonstratethat the proposed algorithm has better accuracy and convergence than directmethods based on gradient descent. Furthermore, the algorithm is also simpler,more accurate, and more stable than a deep reinforcement learning method.

Solving optimal control of rigid-body dynamics with collisions using the hybrid minimum principle