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198 Autonomous Mobile Robots

starting point is completely different. First of all we show that, under some
technical assumptions, a nonholonomic system admits a smooth stabilizer only
if a subset of the differential equations describing the system are not defined
on a certain hyperplane passing through the origin of the coordinates system.
Hence, we focus on such a class of systems and we give sufficient conditions for
the existence of stabilizing control laws. Finally, we show that any smooth non-
holonomic system can be always transformed into a system which is not defined
on a certain hyperplane, say P, passing through the origin of the coordinates
system. Using the above results, we will propose in Section 5.3.4 a general
procedure to design discontinuous almost asymptotically stabilizers for non-
holonomic control systems. Such a procedure yields a control law which is not
defined on P; hence the closed loop system is not defined on P. However, we
will prove that every initial condition which lies outside P yields trajectories
which converge asymptotically to the origin.

5.3.1 Stabilization of Discontinuous Nonholonomic
Systems

In this section we discuss the issue of smooth asymptotic stabilizability for
n
systems described by equations of the form (5.1) with x ∈ R and u ∈ R m+p .
We exploit a few basic facts from geometric control theory, as presented in
Reference 44. Note however that proper care has to be taken as we deal with
discontinuous functions.

Lemma 5.1 [20] Consider the system

˙ x 1 = g 11 (x 1 , x 2 )u 1
(5.12)
˙ x 2 = g 21 (x 1 , x 2 )u 1 + g 22 (x 1 , x 2 )u 2

p
m
p
n
with x 1 ∈ R , x 2 ∈ R n−p , x = col(x 1 , x 2 ) ∈ R , u 1 ∈ R , u 2 ∈ R ,
u = col(u 1 , u 2 ) ∈ R m+p , and m + p < n(g ij (x 1 , x 2 ) are matrix functions of
appropriate dimensions). Assume that the matrix function g 21 (x 1 , x 2 ) is smooth
in an open and dense set U, that the matrix functions g 11 (x 1 , x 2 ) and g 22 (x 1 , x 2 )
¯ 10
are smooth in U, and that the distribution
G = span{g 1 (x 1 , x 2 ), ... , g m+p (x 1 , x 2 )}

10 Let U be an open and dense set. We denote with ¯ U the smallest simply connected open set
properly containing U.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 198 — #12