# Nonholonomic Mechanical Systems and Symmetry

### A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. M. Murray

CDS Technical Report 94-013

To appear *Archive for Rational Mechanics and Analysis*

### Abstract

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the point of view of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics emphasizing the formulation of Lagrange d'Alembert with the use of connections and momentum maps associated with the given symmetry group. We begin by recalling and extending the results of Koiller from the case of principal connections to the general Ehresmann case. Unlike the situation with standard configuration space constraints, the symmetry in the nonholonomic case may or may not lead to conservation laws. In any case, the momentum map determined by the symmetry group satisfies a useful differential equation that decouples from the group variables. This momentum equation is shown to have the form of a covariant derivative of the momentum equal to a component of the internal generalized force. An alternative description using a "body reference frame" realizes part of the momentum equation as those components of the Euler-Poincare equations along the symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those 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 snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object, the nonholonomic connection. Under conditions that include the Chaplygin case (we use the terminology "purely kinematic") and the case in which the momentum is conserved, it is known that one can perform a reduction similar to Lagrangian reduction, which includes the Routh procedure. We generalize this reduction procedure to the case in which the nonholonomic connection is a principal connection for the given symmetry group; this case includes all of the examples considered in the paper and many others as well, such as the wobblestone, the nonvertical disk and the bicycle. Another purpose of this work is to lay the foundation for future work on mechanical systems with control so that one can adapt well developed techniques from holonomic systems, such as constructive controllability and geometric phases. Although this will be the subject of future work, the methodology of the present paper is developed with these goals in mind.

- CDS Technical Report: http://www.cds.caltech.edu/~murray/preprints/cds94-013.pdf
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