Lecture 3. Systems with Rolling Constraints and Locomotion
In this lecture we consider the basic mechanics of systems with rolling con-
straints and related questions of locomotion. Some of the exposition follows the
development in Bloch, Krishnaprasad, Marsden and Murray [1996] and Marsden
and Ostrowski [1998].
For systems with rolling constraints or more generally nonholonomic sys-
tems, one finds the equations of motion and properties of the solutions (such as
the fate of conservation laws) using the Lagrange-d'Alembert principle. These sys-
tems are not literally variational but the basic mechanics still comes down to
F=ma.
Consider a configuration space Q and a distribution D that describes thekinematic constraints; D is a collection of linear subspaces: Dq c TqQ, for q E Q.
We say that q(t) E Q satisfies the constraints if q(t) E Dq(t)· This distribution
is, in general, nonintegrable in the sense of Frobenius' theorem; i. e., the constraints
can be nonholonomic. Anholonomy is measured by the curvature of D.
A simple example of a nonholonomic system is the rolling disk (see Fig-
ure 3.1). Here, the constraints of rolling without slipping define the distribution
D:
x -~Reos¢
y -~Rsin</J.
These equations for the velocities may be interpreted as defining a linear subspace
of the tangent space to the configuration space. These linear spaces then make up
the constraint distribution D.
The Lagrange-d'Alembert principle. The system dynamics is determined by
a Lagrangian L : TQ -+ JR, usually the kinetic minus the potential energy. The
basic equations of motion are given by requiring that q(t) satisfy the constraints
and that
b lb L(q,q)dt=O,
for all variations satisfying bq(t) E D q(t), a::::; t::::; b.
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