Krishnaprasad, Perinkulam S.Relative Motion in a system of coupled rigid bodies can yield global reorientation (or phase shift.) We give a formula to compute such a phase shift and interpret the same in geometric terms. The theory of connections in principal bundles provides the proper setting for questions of the addressed in this paper. A related optimal control problem leads to singular riemannian geometry.en-USIntelligent ServomechanismsTechnical Report