  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.
