Almost Global Consensus on the $n$ -Sphere

This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the $n$ -dimensional unit sphere or $n$ -sphere. For any connected, undirected graph and all $n\in \mathbb{N} \backslash \lbrace 1\rbrace$ , 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 $n\in \mathbb{N}\backslash \lbrace 1\rbrace$ differs from the circle and $\mathsf {SO}(3)$ , where the corresponding protocols fail to generate almost global consensus. Moreover, we derive a novel consensus protocol on $\mathsf {SO}(3)$ by combining two almost globally convergent protocols on the $n$ -sphere for $n\in \lbrace 1,2\rbrace$ . Theoretical and simulation results suggest that the combined protocol yields almost global consensus on $\mathsf {SO}(3)$ .