LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 379where TJ(t) is a curve in g vanishing at the endpoints.- The Euler-Poincare equations hold on g x V
d 8l 8l 8l
dt8~ = ad~ 8~ + 8a o a
where Pv : g ___, V is given by~ r--> ~v, the infinitesimal action, p~ : V* ___, g*is its dual, and v o a= p~a.
This is proved by a straightforward adaptation of the proof of Euler-Poincare
reduction that was presented in the first lecture. The main extra feature is to keeptrack of the "constraint" ao = constant. Since a 0 = ga, we get 0 = (8g)a + goa,
i.e., 8a = -(g-^1 og)a = -TJa.
Nonholonomic reduced equations. Reduction procedures were applied to the
Lagrange d'Alembert principle in Bloch, Krishnaprasad, Marsden and Murray
[1996]. The form of the resulting reduced equations is
g-^19 -A(r)r + B(r)p,
p rT a(r )r +TT /J(r )p +PT "f(r )p
M(r)r -C(r,r) + N(r,r,p) + T.
The first equation describes the motion in the group variables as the flow of a left-
invariant vector field on G determined by the internal shape r, the internal
velocity r, as well as the nonholonomic momentum p , which is, roughly speak-
ing, the component of momentum in the symmetry directions compatible with the
constraints. The second equation is the momentum equation. Notice that the
momentum equation has terms that are
• quadratic in r '
- linear in r and p and
- quadratic in p.
The coefficients f3(r) define a connection and this term is called the transport part of
the momentum equation. The curvature of this connection plays an important role
in stability theory. The third equation describes the motion in the shape variables
r. The term M(r) is the mass matrix of the system, C is the Coriolis term
and T represent internal control forces. This framework has proven to be useful for
controllability, gait selection, and locomotion of systems like the snakeboard. We
will come back to this below.
A nice example illustrating many features of nonholonomic systems is the bicy-
cle (see Figure 3.4). This example (under drastically simplified modeling assump-
tions) is studied in Koon and Marsden [1998a]. We also mention that Koon and
Marsden [1998c] study the above reduced Lagrange d'Alembert equations from the
Poisson point of view.
The snakeboard. The snakeboard is an interesting example of a system in which
there is a nontrivial interaction between the forces of constraint and the momentum
laws that arise due to symmetries. · As such, it has played an important role in
understanding the many subtleties of nonholonomic systems.
The snakeboard is similar to a skateboard, with one exception: the front and
back pairs of wheels can be rotated independently about their vertical axes. Thus,
a rider stands with one foot above each of the wheel bases, and can couple twisting
motions of the torso with turning of the feet. The most interesting aspect of this