NONHOLONOMIC MECHANICAL SYSTEMS WITH SYMMETRY

This work develops the geometry and dynamics of mechanical systems wit h nonholonomic constraints and symmetry from the perspective of Lagran gian mechanics and with a view to control-theoretical applications. Th e basic methodology is that of geometric mechanics applied to the Lagr ange-d'Alembert formulation, generalizing the use of connections and m omentum maps associated with a given symmetry group to this case. We b egin by formulating the mechanics of nonholonomic systems using an Ehr esmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situa tion with standard configuration-space constraints, the presence of sy mmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the grou p variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is compute d explicitly in coordinates. An alternative description using a ''body reference frame'' relates part of the momentum equation to the compon ents of the Euler-Poincare equations along those symmetry directions c onsistent with the constraints. One of the purposes of this paper is t o derive this evolution equation for the momentum and to distinguish g eometrically and mechanically the cases where it is conserved and thos e where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboar d, a modified version of the skateboard which uses momentum coupling f or locomotion generation. We construct a synthesis of the mechanical c onnection and the Ehresmann connection defining the constraints, obtai ning an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given s ymmetry group, we show how to perform Lagrangian reduction in the pres ence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to ill ustrate the theory.