A Lifting method for analyzing distributed synchronization on the unit sphere

TL;DR

This paper presents a novel lifting method that transforms the analysis of distributed synchronization on the unit sphere into a higher-dimensional Euclidean space, enabling global convergence analysis for various switching network topologies.

Contribution

The paper introduces a global lifting approach for analyzing convergence of distributed consensus on the unit sphere, extending previous hemisphere-based results to the entire sphere.

Findings

The lifting method applies globally, unlike previous hemisphere-restricted approaches.

Consensus on the unit sphere is shown to be asymptotically stable under switching directed graphs.

The method generalizes convergence results for a broad class of protocols and topologies.

Abstract

This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence.