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194 Autonomous Mobile Robots

n
m
with x ∈ U ⊂ R , u ∈ R , and m < n. Despite its simple formulation this
problem does not possess a simple solution, as can be inferred from the Theorem
of Brockett [2]. This theorem, yielding necessary conditions for smooth sta-
bilizability for general nonlinear systems, provides necessary and sufficient
conditions for feedback stabilizability of nonholonomic systems.

n m
Theorem 5.1 [2] Let ˙q = g(q)u be given, with q ∈ R , u ∈ R , g(q 0 )u 0 =
0, g(·) continuously differentiable in a neighborhood of q 0 . Assume, moreover,
that span{g(q)} is a nonsingular distribution of dimension m in a neighborhood
of q 0 . Then:

1. There exists a continuously differentiable control law which makes
(q 0 , u 0 ) asymptotically stable iff m ≥ n.
2. There exists a continuously differentiable and dynamic feedback law
2
which makes (q 0 , ξ 0 , u 0 ) asymptotically stable iff m ≥ n.

The first part of Theorem 5.1 is due to Brockett [2], while the second
one to Pomet [10] (see also the work of Ryan [35] for a more general result in
the framework of nonsmooth stabilizability). We will not present the proof of
the above theorem, which can be found in the literature [2,10,36]; we simply
mention that the provided obstruction to stabilizability has a topological nature.
The essence of Theorem 5.1 is that the only interesting nonholonomic systems
are those for which the distribution g(q) drops dimension precisely at q 0 ,is
not continuously differentiable at q 0 , or is not defined at q 0 . In such cases
1
we cannot infer anything about the existence of C (smooth), time invariant,
static or dynamic, asymptotically stabilizing control laws. Motivated by the
conclusions of Brockett’s Theorem we focus on:

• State feedback control laws described by equations of the form

u = a(x) (5.2)

m
n
where α : R → R is a discontinuous function of its arguments.
• State feedback, hybrid, control laws described by equations of
the form

u = k(x, s d ), s d = k d (x, ¯s d ) (5.3)

2 ξ denotes the state of the dynamic controller.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 194 — #8