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

A Lifting method for analyzing distributed synchronization on the unit sphere