Motion on Lie groups and its applications in Control Theory

TL;DR

This paper explores the geometric theory of motion on Lie groups and homogeneous spaces, demonstrating its relevance and application in control systems through specific examples like rigid bodies and kinematic cars.

Contribution

It introduces and discusses two methods for control systems on Lie groups, including a reduction procedure that simplifies complex equations to subgroups.

Findings

Methods are effective for control systems on Lie groups and homogeneous spaces.

Reduction procedure simplifies equations by focusing on subgroups.

Applications demonstrated with rigid body and kinematic car examples.

Abstract

The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group $G$ to that on a subgroup $H$, provided a particular solution of an associated problem in $G/H$ is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.