Coordinated motion design on Lie groups

TL;DR

This paper introduces a unified geometric framework for designing coordinated motion control laws on Lie groups, enabling systematic analysis and synthesis for fully actuated and underactuated systems, with applications to SO(3), SE(2), and SE(3).

Contribution

It provides a systematic geometric approach to coordinated motion on Lie groups, linking control design with Lie group geometry and existing control laws.

Findings

Framework applies to SO(3), SE(2), SE(3)

Connects control laws with Lie group geometry

Includes analysis of fully actuated and underactuated systems

Abstract

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. This allows among others to retrieve control laws in the literature for particular examples. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3), SE(2) and SE(3).