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Adaptive Neural-Fuzzy Control of Mobile Robots 235

in this chapter is the m-input, (m−1)-chain, single generator chained form given
by Walsh and Bushnell [38]

˙ x 1 = u 1

˙ x j,i = u 1 x j,i+1 (2 ≤ i ≤ n j − 1)(1 ≤ j ≤ m − 1) (6.11)

˙ x j,n j = u j+1

T n
Note that in Equation (6.11), X =[x 1 , X 2 , ... , X m ] ∈ R with X j =
T
] (2 ≤ j ≤ m) are the states and u =[u 1 , u 2 , ... , u m ]
[x j−1,2 , ... , x j−1,n j−1
are the inputs of the kinematic subsystem.
The class of nonholonomic systems in chained form was first introduced
in Reference 7 and has been studied as a benchmark example in the literature.
It is the most important canonical form that is commonly used in the study
of nonholonomic control systems. The necessary and sufficient conditions for
transforming system (6.5) into the chained form are given in Reference 39.
Theoretical challenges and practical interests have provided substantial motiv-
ation for the extensive study of nonholonomic systems in chained form. The
following assumption is made.

Assumption 6.1 The kinematic model of the nonholonomic system given by
(6.5) can be converted into the chained form (6.11) by some diffeomorphic
coordinate transformation X = T 1 (q) and state feedback v = T 2 (q)u where
u is a new control input.

The existence and construction of these systems have been established in
References 38 and 40. For the notations on the differential geometry used below,
readers are referred to Reference 41.

Proposition 6.1 Consider the drift-free nonholonomic system

˙ q = r 1 (q)˙z 1 + ··· + r m (q)˙z m

where r i (q) are smooth, linearly independent input vector fields. There exist
state transformation X = T 1 (q) and feedback ˙z = T 2 (q)u on some open set
n
U ⊂ R to transform the system into an (m−1)-chain, single-generator chained
form, if and only if there exists a basis f 1 , ... , f m for 0 := span{r 1 , ... , r m }

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 235 — #7