Geometric phases, anholonomy, and optimal movement

In the search for useful strategies for movement of robotic systems (e.g., manipulators, platforms) in constrained environments (e.g., in space, underwater), there appear to be new principles emerging from a deeper geometric understanding of optimal movements of nonholonomically constrained systems. The authors have exploited some new formulas for geometric phase shifts to derive effective control strategies. The theory of connections in principal bundles provides the proper framework for questions of the type addressed. An outline is presented of the essentials of this theory. A related optimal control problem and its localizations are also considered