Controller Design for Systems on Manifolds in Euclidean Space

TL;DR

This paper develops a method for designing controllers for systems on manifolds embedded in Euclidean space by modifying dynamics outside the manifold, simplifying controller design for stabilization and tracking.

Contribution

It introduces a novel approach to controller design on manifolds using a single Euclidean coordinate system, including a technique to modify system dynamics outside the manifold.

Findings

Effective stabilization of rigid body systems on manifolds.

Unified control design using ambient Euclidean coordinates.

Modification of dynamics outside the manifold enhances control performance.

Abstract

Given a control system on a manifold that is embedded in Euclidean space, it is sometimes convenient to use a single global coordinate system in the ambient Euclidean space for controller design rather than to use multiple local charts on the manifold or coordinate-free tools from differential geometry. In this paper, we develop a theory about this and apply it to the fully actuated rigid body system for stabilization and tracking. A noteworthy point in this theory is that we legitimately modify the system dynamics outside its state-space manifold before controller design so as to add attractiveness to the manifold in the resulting dynamics.