Consensus and Formation Control on SE(3) for Switching Topologies

TL;DR

This paper develops control laws for achieving consensus and formation in multi-agent systems on SE(3), including special cases like attitude synchronization on SO(3), with conditions for convergence and switching topologies.

Contribution

It introduces unified control laws for SE(3) and SO(3) that work under switching topologies and provides convergence conditions for various local rotation representations.

Findings

Control laws ensure consensus on SE(3) and SO(3) under switching topologies.

Conditions for exponential convergence are established.

A coordinate transformation links different rotation representations to a common structure.

Abstract

This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the consensus problem. By a simple transformation, it is shown that the proposed control laws can be used for the formation problem. The design is first conducted on the kinematic level, where the velocities are the control laws. Then, for rigid bodies in space, the design is conducted on the dynamic level, where the torques and the forces are the control laws. On the kinematic level, first two control laws are introduced that explicitly use Euclidean transformations, then separate control laws are defined for the rotations and the translations. In the special case of purely rotational motion, the consensus problem is referred to as consensus on SO(3) or attitude synchronization. In this problem, for a broad class of local representations or parameterizations of SO(3), including the Axis-Angle Representation, the Rodrigues Parameters and the Modified Rodrigues Parameters, two types of control laws are presented that look structurally the same for any choice of local representation. For these two control laws we provide conditions on the initial rotations and the connectivity of the graph such that the system reaches consensus on SO(3). Among the contributions of this paper, there are conditions for when exponential rate of convergence occur. A theorem is provided showing that for any choice of local representation for the rotations, there is a change of coordinates such that the transformed system has a well known structure.