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Adaptive Neural-Fuzzy Control of Mobile Robots 231

architecture that uses fuzzy algorithms to represent the knowledge in a natural
and interpretable manner, while preserving the learning ability of NNs as well as
the associated convergence and stability. The neuro-fuzzy (NF) system is a
NN-based fuzzy logic control and decision system, and is suitable for online
systems identification and control.
For adaptive NF control system design, the parameterized NF approx-
imators are generally expressed as a series of the commonly used radial
basis function (RBF) because of its nice approximation properties, that is,

y = w j φ(σ j , x − c j ), where w j is the connection weight, and c j and
j
σ j are the center and width respectively that decide the shape of the function
φ. The major challenge in the RBF approximation problem lies in the selection
of the receptive center and width, that is, c j and σ j as they both appear non-
linearly. In general, there are three kinds of methods to determine c j and σ j .
The first is the grid-type partition method, which uses a grid partitioning of the
multidimensional space and defines a number of fuzzy sets or nodes for each
variable. This is the most intuitive approach but the problem is the exponential
growth of fuzzy rules or nodes in relation to the dimension of the input space.
The second kind is the clustering algorithm, such as fuzzy C-means (FCM) [30]
and the nearest-neighborhood cluster algorithm [31]. These methods are found
tobeusefulinchoosingparameters, butrequireoff-linelearning. Inaddition, the
gradient descent method is usually employed for fine tuning the parameters c j
and σ j by clustering algorithm so that the approximation accuracy is improved.
The last type consists of optimization approaches such as genetic algorithms
(GA). However, the problem with either the gradient descent method or GA is
that the learning and the adaptation speeds are slow. On the other hand, most
of the adaptive control schemes using RBF as an approximator only consider
the updating law of weights w j to simplify the design [32]. However, it is obvi-
ous that the parameters, c j and σ j are important in capturing the fast-changing
system dynamics, reducing the approximation error, and improving the control
performance [33]. An adaptive scheme of tuning both the weights w j and the
center and width, c j and σ j , was presented in Reference 34.
Motivated by previous works on the control of nonholonomic constrained
mechanical systems and the approximation-based adaptive control of nonlinear
systems, adaptive NF control is developed in this chapter for nonholonomic con-
strained mobile robotic systems using Lyapunov stability analysis in a unified
procedure. Despite the differences between the NNs and fuzzy logic systems,
they actually can be unified at the level of the universal function approxim-
ator, termed as the NF networks which are multilayer feedforward networks
that integrate the TSK-type fuzzy system and RBF NN into a connectionist
structure. Indeed, 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. Because
of the difficulty in deriving the rules in fuzzy systems for systems with little

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