A Unified Framework for Consensus and Synchronization on Lie Groups admitting a Bi-Invariant Metric

TL;DR

This paper develops a unified mathematical framework for consensus and synchronization algorithms applicable to agents evolving on Lie groups with bi-invariant metrics, generalizing existing models like Euclidean space and the circle.

Contribution

It introduces a general theory that extends consensus and synchronization methods to Lie groups with bi-invariant metrics, revealing underlying geometric properties for convergence.

Findings

Unified framework for Lie group consensus and synchronization

Generalization of Euclidean and circle cases

Insights into geometric conditions for convergence

Abstract

For a finite number of agents evolving on a Euclidean space and linked to each other by a connected graph, the Laplacian flow that is based on the inter-agent errors, ensures consensus or synchronization for both first and second-order dynamics. When such agents evolve on a circle (the Kuramoto oscillator), the flow that depends on the sinusoid of the inter-agent error angles generalizes the same. In this work, it is shown that the Laplacian flow and the Kuramoto oscillator are special cases of a more general theory of consensus on Lie groups that admit bi-invariant metrics. Such a theory not only enables generalization of these consensus and synchronization algorithms to Lie groups but also provide insight on to the abstract group theoretic and differential geometric properties that ensures convergence in Euclidean space and the circle.