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

TL;DR

This paper develops a symmetry reduction framework for optimal control problems of multi-agent systems on Lie groups, addressing partial symmetry breaking costs and applying it to collision avoidance on SE(2).

Contribution

It introduces a novel reduction method for optimal control with partial symmetry breaking, applicable to multi-agent systems on Lie groups, including discrete and continuous cases.

Findings

Derived reduced optimality conditions using symmetry reduction techniques.

Applied the framework to collision and obstacle avoidance for multiple vehicles on SE(2).

Demonstrated effectiveness in handling partial symmetry breaking in multi-agent control problems.

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.