Optimal Control Problems with Symmetry Breaking Cost Functions
This work provides a variational framework for optimal control with partial symmetry breaking, extending known results to a broader class of cost functions, but is incremental as it builds on prior work by Borum.
The authors derive Euler-Poincaré and discrete-time Lie-Poisson equations for optimal control problems on Lie groups with symmetry-breaking cost functions, and validate the theory with motion planning examples including obstacle avoidance.
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--Poincaré 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.