Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups

Abstract

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.

Keywords

• lagrangian systems
• symmetry reduction
• euler-poincaré equations
• multi-agent control systems
• lie-poisson integrators