# Optimal Control Problems with Symmetry Breaking Cost Functions

## TL;DR

This paper explores symmetry reduction in optimal control problems on Lie groups with partial symmetry breaking costs, using variational principles to derive continuous and discrete equations, and demonstrates applications like obstacle-aware motion planning.

## Contribution

It introduces a variational framework for symmetry breaking in optimal control on Lie groups, deriving new continuous and discrete Lie-Poisson equations and applying them to practical motion planning.

## Key findings

- Derived Euler--Poincaré equations from variational principles.
- Obtained discrete-time Lie-Poisson equations via time discretization.
- Demonstrated application in obstacle-aware motion planning.

## Abstract

We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincar\'e equations from a variational principle. By applying a Legendre transformation to it, we recover the Lie-Poisson equations obtained by A. D. Borum [Master's Thesis, University of Illinois at Urbana-Champaign, 2015] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.