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192 Autonomous Mobile Robots

5.1 INTRODUCTION
In this chapter we study the stabilization problem for nonholonomic systems.
Nonholonomic control systems are becoming increasingly important in
research and industry as they present many interesting features and potenti-
alities. From the researchers’ point of view nonholonomic control systems are
a prototype of strongly nonlinear systems, requiring a fully nonlinear analysis,
since all first approximation methods are inadequate. Thus, the design of a good
control law for a given nonholonomic system is a challenging task. On the other
hand, from an industrial point of view, nonholonomic systems are extremely
appealing for their efficiency and flexibility. They can be used as means of
transport, inspection, and operation in free space and hostile environments.
Before moving to the technical discussion, it is worth pointing out why we
deal with nonholonomic control systems and why we focus on noncontinuous,
hybrid, or time-varying stabilizers. A possible answer to the first question can
be found in the words of D. Edelen [1]: “Real problems in the real world rarely
exhibit themselves in those pleasant forms wherein one can model them in terms
of systems with holonomic constraints. […] The second law of thermodynamics
tells us that such holonomic representations must ultimately degenerate from
the domain of the real into ethereal flights of fancy.”
A more practical answer comes from everyday life. Consider the problem
of parking a car, we can only drive forward or backward and steer to the left or
to the right. Observe a falling cat, an astronaut, a gymnast, or a diver: they can
change configuration requiring no contact with fixed objects. These examples
seem to be weakly related but, from a mathematical point of view, they are all
examples of nonholonomic control problems.
Consider now the second question. In the earlier examples an experienced
operator is able to perform the proper succession of operations in order to drive
a nonholonomic system from an initial configuration to a final one. However,
when the ability of the operator is not enough or when we desire to automatic-
ally reconfigure a nonholonomic system, it is necessary to design a regulator.
Hence the birth of the theory of nonholonomic control or nonholonomic motion
planning. Unfortunately, one of the first results of such a theory was a negative
one [2]: there exists no continuously differentiable, time invariant, control law
able to asymptotically stabilize a controllable nonholonomic system. There-
fore, many researchers have proposed and studied discontinuous, hybrid, or
time varying control laws.
We now briefly review some of the existing results on the control of non-
holonomic systems (see References 3 and 4 for further detail). The control
strategies for nonholonomic systems can be divided into two main groups:
open loop strategies and closed loop (feedback) strategies. In the latter group
we can further distinguish between continuous and discontinuous control laws. 1
1
A third possible approach is the one based upon sampled-data control laws.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 192 — #6