Synchronization on Lie Groups: Coordination of Blind Agents

This paper presents an algorithm for 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 synchronization among the agents. In this approach, 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. Due to geometrical properties of the synchronization set, we employ the method of quotient manifolds to prove the convergence of the proposed algorithm. The obtained results are applied to synchronization problems on SE(3) to demonstrate the efficacy of the proposed algorithm.