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242 Autonomous Mobile Robots

∗
ˆ
Since every element of the vector (S −S ) is bounded in [−1, +1],we have
n r
T
∗ ˆ ∗ ∗ ∗
W (S − S ) ≤ |w | = W 1
i
i=1
T
Considering W S c = tr{W S c }≤ W S F · c F = S W · c ,
∗
ˆ T ˆ
ˆ ˆ
∗
ˆ T ˆ
ˆ T ˆ
∗
∗
c
c
c
c
we have
T T
∗ ˆ ˆ ∗ ˆ ˆ ∗ ˆ
|d u |≤ c · S W + σ · S W + W · S ˆc
c
c
σ
∗ ˆ ∗
+ W · S ˆσ + W 1
σ
Thus, we have shown that (6.18) holds.
6.4 ADAPTIVE NF CONTROL DESIGN
In this section, the adaptive NF control is presented for nonholonomic mobile
robots with uncertainties and external disturbances.
The following lemmas are useful in the controller design.
Lemma 6.2 Let e = H(s)r with H(s) representing an (n × m)-dimensional
strictly proper exponentially stable transfer function, r and e denoting its input
n
and output, respectively. Then r ∈ L m L m implies that e, ˙e ∈ L n L ,e is
2 ∞ 2 ∞
continuous, and e → 0 as t →∞. If, in addition, r → 0 as t →∞, then
˙ e → 0 [32].
+
Lemma 6.3 Given a differentiable function φ(t): R → R,if φ(t) ∈ L 2 and
˙ φ(t) ∈ L ∞ , then φ(t) → 0 as t →∞, where L ∞ and L 2 denote bounded and
square integrable function sets, respectively.
Consider the constrained dynamic equation (6.1) together with (n−m) inde-
pendent nonholonomic constraints (6.2). For simplicity of design, the following
assumptions are made throughout this section.
T
Assumption 6.3 Matrix R (q)B(q) is of full rank, which guarantees all m
degrees of freedom can be (independently) actuated.

It has been proven that the nonholonomic system (6.1) and (6.2) cannot be
stabilized to a single point using smooth state feedback [18]. It can only be
stabilized to a manifold of dimension (n − m) due to the existence of (n − m)
nonholonomic constraints. Though the nonsmooth feedback laws [44] or time-
varying feedback laws [4] can be used to stabilize these systems to a point,

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 242 — #14