Controlled Lagrangians and Stabilization of Euler--Poincar\'e Mechanical Systems with Broken Symmetry II: Potential Shaping

TL;DR

This paper extends the method of potential shaping via controlled Lagrangians to Euler--Poincaré systems with broken symmetry, providing a unified framework for stabilization of systems like underwater vehicles and tops.

Contribution

It introduces a representation-based approach for potential shaping in Euler--Poincaré systems with broken symmetry, simplifying the control design process.

Findings

Derived matching conditions for potential modification using advected parameters.

Unified framework reproduces previous results for heavy tops and underwater vehicles.

Simplified control problem formulation demonstrated for underwater vehicle example.

Abstract

We apply the method of controlled Lagrangians by potential shaping to Euler--Poincar\'e mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group $\mathsf{G} \ltimes V$ with a particular interest in those systems whose configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$. The key idea behind the work is the use of representations of $\mathsf{G} \ltimes V$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting the representations and advected parameters. Our motivating examples are a heavy top spinning on a movable base and an underwater vehicle with non-coincident centers of gravity and buoyancy. We consider a few different control problems for these systems, and show that our results give a general framework that reproduces our previous work on the former example and also those of Leonard on the latter. Also, in one of the latter cases, we demonstrate the advantage of our representation-based approach by giving a simpler and more succinct formulation of the problem.