Synchronization on Lie Groups: Coordination of Blind Agents

TL;DR

This paper introduces a novel extremum seeking control algorithm enabling blind agents on Lie groups to synchronize without communication, leveraging geodesic dithers and Taylor expansions on Riemannian manifolds.

Contribution

It develops a communication-free synchronization method for agents on Lie groups using extremum seeking and geodesic dithers, applicable to $SO(3)$ and $SE(3)$.

Findings

Successful synchronization on $SO(3)$ and $SE(3)$.

Algorithm independent of network topology.

Effective on agents with no direct communication.

Abstract

This paper presents an algorithm for the synchronization of blind agents (agents are unable to observe other agents, i.e. no communication) evolving on a connected Lie group $G$. We employ the method of extremum seeking control for nonlinear dynamical systems defined on connected Riemannian manifolds to achieve a synchronization among the agents. This approach is independent of the underlying graph of the system and each agent updates its position on $G$ by only receiving the synchronization cost function. The results are obtained by employing the notion of geodesic dithers for extremum seeking on Riemannian manifolds and their equivalent version on Lie groups and applying Taylor expansion of smooth functions on Riemannian manifolds. We apply the obtained results to synchronization problems defined on Lie groups $SO(3)$ and $SE(3)$ to demonstrate the efficacy of the proposed algorithm.