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Adaptive Neural-Fuzzy Control of Mobile Robots 237

Let T 1 (q) : q → X as

x 1 = h 1
x 2 = L n−2 h 2
r 1
.
.
.
h
x n−1 = L r 1 2
x n = h 2

It may be verified that T 1 (q) is a valid change of coordinates by evaluating the
Jacobian of T 1 (q) at the origin.
L n−2 h 2 = 0, let T 2 (q) : ˙z → u as
Since L r 2 r 1

˙ z 1 := u 1
1 n−1
˙ z 2 := n−2 [u 2 − (L h 2 )u 1 ]
L r 1
L r 2 r 1 h 2
Then, the local coordinate transformation X = T 1 (q) and state feedback
˙ z = T 2 (q)u render system (6.12) into the chained form

˙ x 1 = u 1
˙ x 2 = u 2
˙ x 3 = x 2 u 1
.
.
.
˙ x n = x n−1 u 1

Remark 6.1 Under certain conditions which has been stated in
Proposition 6.1, the kinematic model (6.5) can be converted into a chained
form driven by integrators.

6.3 MULTI-LAYER NF SYSTEMS
Despite the differences between the NNs and fuzzy logic systems, they actually
can be unified at the level of the universal function approximator, which are
multilayer feedforward networks that integrate the TSK-type fuzzy system and
RBF NN into a connectionist structure.
Typically, fuzzy logic systems are rule-based systems, which consists of the
fuzzifier, the fuzzy rule base, the fuzzy inference engine, and the defuzzifier.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 237 — #9