Nonholonomic Mechanical Systems with Symmetry
Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and R. Murray [1996]
Arch. Rational Mech. Anal., 136, 21-99
Abstract:
This work develops the geometry and dynamics of mechanical systems with
nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics
and with a view to control-theoretical applications. The basic methodology
is that of geometric mechanics applied to the Lagrange-d'Alembert formulation,
generalizing the use of connections and momentum maps associated with a given
symmetry group to this case. We begin by formulating the mechanics of nonholonomic
systems using an Ehresmann connection to model the constraints, and show
how the curvature of this connection enters into Lagrange's equations. Unlike the
situation with standard configuration-space constraints, the presence of symmetries
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 group variables. This momentum equation,
which plays an important role in control problems, involves parallel transport operators
and is computed explicitly in coordinates. An alternative description using
a "body reference frame" relates part of the momentum equation to the components
of the Euler-Poincaré equations along those 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 we call the nonholonomic connection.When the nonholonomic connection
is a principal connection for the given symmetry group, we show how to perform
Lagrangian reduction in the presence of nonholonomic constraints, generalizing
previous results which only held in special cases. Several detailed examples are
given to illustrate the theory.