Work

Optimal Control Problems with Symmetry Breaking Cost Functions

Year: 2017

Type: article

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--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.

Source: DIGITAL.CSIC (Spanish National Research Council (CSIC))

DOI: 10.48550/arxiv.1701.06973

Authors: Bloch, Anthony, Colombo, Leonardo, Gupta, Rohit, Ohsawa, Tomoki

Institutions: University of Michigan, The University of Texas at Dallas

Language: English

Topic: Control and Stability of Dynamical Systems

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Dynamics, Integrability and Control of Mechanical and Nonholonomic Systems, Dynamics, Integrability, and Control of Mechanical and Nonholonomic Systems, INSPIRE Track 1: The Mathematics of Balance in Mechanical Systems with Impacts, Unilateral Constraints, Underactuation and Hyper-sensing: Application to Agile bipedal Locomotion