A Consensus Algorithm for Second-Order Systems Evolving on Lie Groups

TL;DR

This paper introduces a novel consensus algorithm for multi-agent systems modeled as mechanical control systems on Lie groups, extending Euclidean space algorithms to more general geometric settings, with proven stability and practical application to attitude consensus.

Contribution

It extends Laplacian flow consensus algorithms to systems on Lie groups, providing a stability proof and a practical control design for attitude consensus.

Findings

Proven stability of the consensus algorithm on Lie groups.

Successful numerical validation with attitude consensus example.

Abstract

In this paper, a consensus algorithm is proposed for interacting multi-agents, which can be modeled as simple Mechanical Control Systems (MCS) evolving on a general Lie group. The standard Laplacian flow consensus algorithm for double integrator systems evolving on Euclidean spaces is extended to a general Lie group. A tracking error function is defined on a general smooth manifold for measuring the error between the configurations of two interacting agents. The stability of the desired consensus equilibrium is proved using a generalized version of Lyapunov theory and LaSalle's invariance principle applicable for systems evolving on a smooth manifold. The proposed consensus control input requires only the configuration information of the neighboring agents and does not require their velocities and inertia tensors. The design of tracking error function and consensus control inputs are demonstrated through an application of attitude consensus problem for multiple communicating rigid bodies. The consensus algorithm is numerically validated by demonstrating the attitude consensus problem.