LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 389then gives Wong's equations by the following simple calculations:8l - ·f3. 8l 1 ogf3r ·{3 ., al
ora - gapr ' ora = 2 ora r r ; [)[la = >-a.
The constraints are fl= 0 and so the reduced Euler-Lagrange equations become
d al al
----
dt [)fa ora
d- ,\b
dt
But
d al al
dt ora ora
and so we have the desired equations. •
R e m ark. There is a rich literature on Wong's equations and it was an important
ingredient in the development of reduction theory. Some references are Sternberg
[1977], Guillemin and Sternberg [1978], Weinstein [1978], Montgomery, Marsden
and Ratiu [1984], Montgomery [1984], Koon and Marsden [1997] and Cendra, Holm,
Marsden, and Ratiu [1 998 ].
Nonh o lo nomic optimal control. Using a synthesis of the techniques used above
for the Heisenberg system and the falling cat problem, Koon and Marsden [1997]
generalized these problems to the nonholonomic case. In addition, these meth-
ods allow one to treat the falling cat problem even in the case that the angular
momentum is not zero.
In this process the momentum equation plays the role of the constraint. It is
inserted as a first order differential constraint on the nonholonomic momentum.
S t a bilizing th e inverte d pendulum. We next illustrate the stabilization method
of Bloch, Leonard and Ma rsden [1997] for a pendulum on a cart. T he inverted
spherical pendulum is a little more complicated, but the methods also work in this
case. T hey also work in many other cases, such as satellites with internal rotors,
underwater vehicles with rotors, etc. (See Bloch, Leonard and Marsden [1998]).
Our methods are similar in spirit to energy methods that are proving very effective
(see, e.g., Astri::im and Furuta [1997]).
Our idea for stabilization is to use the mechanical structure and to create an en-
ergy extrema. Dissipation (real or control) can then convert stability to asymptotic
stability. The main complication for more complex systems such as the inverted
spherical pendulum, the satellit e and the underwater vehicle are gyroscopic forces.
Reduction techniques and in particular, magenetic terms that arise from reduction
are perfect tools for investigating such problems.
Lagra n g ia n for t h e cart-pendulum system. Let s denote the position of the
cart on the s-axis and let e denote the angle of the pendulum from t he upright
vertical, as in Figure 4.2.