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 (postscript, 972K, 67 pages) 
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Richard Murray (murray@indra.caltech.edu)
Last modified: Tue Dec 31 10:16:58 1996