Nonholonomic passive decomposition: Weak decomposability, controllability and control design

We extend the results of [1] to general non-holonomic mechanical systems with weak decomposability (as opposed to strong decomposability [1]). That is, for a nonholonomic mechanical system (or for multiple of them) with weak decomposability, given a submersion h defined on their configuration space and specifying a certain motion coordination aspect (e.g. internal formation shape), we can decompose their Lagrange-D'Alembert dynamics into: 1) shape system, describing the motion of h(q) (i.e. formation aspect); 2) locked system, describing the dynamics on the level set of h with the formation aspect h(q) fixed (i.e. maneuver aspect); 3) quotient system, whose motion affects both the locked and shape aspects simultaneously; and 4) energetically conservative (or skew-symmetric) coupling among them. Moreover, the locked, shape, and quotient systems all inherit Lagrangian like structure and passivity. We also study the issues of controllability and present control design examples with their application for three wheeled mobile robots maneuvering while changing formation shape.