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238 Autonomous Mobile Robots

The purpose of the fuzzifier is to provide scale mapping of the crisp input to
corresponding linguistic forms noted as labels of fuzzy set. The fuzzy rule base
stores knowledge base for linguistic data and is expressed as a collection of
fuzzy IF–THEN rules. The typical fuzzy rule used in the Takagi–Sugeno–Kang
(TSK) model [42] is in the following form:
l
l
l
R :IF z 1 is F AND z 2 is F ··· AND z n is F l
1 2 n
l l l l
THEN y = k + k z 1 + ··· + k z n
0 1 n
l
l
where F (i = 1, 2, ... , n) are fuzzy sets, k ( j = 0, 1, ... , n) are real-valued
i j
l
T
parameters, z =[z 1 , z 2 , ... , z n ] is the system input, y is the system output
l
due to rule R , and l = 1, 2, ... , N. For the zero-order TSK-fuzzy system, we
l
l
have y = k . The fuzzy inference engine is the kernel of the fuzzy system and
0
uses the fuzzy IF–THEN rules to determine a mapping from the input universe
to the output universe based on fuzzy logic policies. The role of the defuzzifier
is the scale mapping of the linguistic value to a corresponding crisp output
value. For simple systems, the rules are fairly easy to derive with physical
insight. However, they become unreasonably difficult for systems with strong
nonlinear couplings yet without a good physical understanding.
On the other hand, the NNs can build up a very nice mapping between
system’s inputs and outputs. Due to its great learning capability, it can be
used to approximate any continuous function to any desired accuracy. Despite
the differences between the NNs and fuzzy logic systems, they can, in fact,
be unified at the level of the universal function approximator which integrates
the TSK-type fuzzy system and RBF NN into a connectionist structure. Nodes
in the first layer are called input linguistic nodes and corresponds to input
variables. These nodes only transmit input values to the next layer directly.
Nodes in the second layer play the role of membership functions specifying the
degree to which an input value belongs to a fuzzy set. The nodes in the third
layer are called rule nodes which represent fuzzy rules. The fourth layer is the
output layer. The links in the third layer act as the precondition of fuzzy rules
and the links in the fourth layer act as the consequence of fuzzy rules.
The output of the whole NF system is then given by
n i
n r µ l(x i )
i=1
A
y(x) = i (6.13)
n r n i µ k(x i )
w l
l=1 k=1 i=1 A i
T
] , µ k(x i ) is the membership function of linguistic
where x =[x 1 , x 2 , ... , x n i
A
i
variable x i with
2
(x i − c ik )
µ k(x i ) = exp − (6.14)
A 2
i σ
ik
© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 238 — #10