Almost global consensus on the n-sphere

TL;DR

This paper proves that a broad class of consensus protocols on n-spheres achieve almost global consensus, providing new insights into multi-agent systems on Riemannian manifolds and introducing a novel protocol for SO(3).

Contribution

It establishes almost global convergence of consensus protocols on n-spheres for all dimensions except one, and introduces a new protocol for SO(3) by combining existing ones.

Findings

Protocols achieve almost global consensus on n-spheres.

The new SO(3) protocol combines two n-sphere protocols.

Simulation results support theoretical convergence claims.

Abstract

This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit sphere or n-sphere. For any connected, undirected graph and all n in N\{1}, each protocol in said class is shown to yield almost global consensus. The feedback laws are negative gradients of Lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol. This convergence result sheds new light on the general problem of consensus on Riemannian manifolds; the n-sphere for n in N\{1} differs from the circle and SO(3) where the corresponding protocols fail to generate almost global consensus. Moreover, we derive a novel consensus protocol on SO(3) by combining two almost globally convergent protocols on the n-sphere for n in {1,2}. Theoretical and simulation results suggest that the combined protocol yields almost global consensus on SO(3).