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synchronization between neural cells can result in a rhythm generator that controls
joints motion in quadruped, multi-legged, and biped robots.
Dynamical models of coupled neural oscillators can serve as Central Pattern
Generators (CPGs) [
70
,
212
]. This means that they stand for higher level control
elements in a multi-layered control scheme which provide the activation frequency
and rhythm for controllers operating at the lower level, e.g. controllers that provide
motion to robot's joints. CPG methods have been used to control various kinds of
robots, such as crawling robots and legged robots and various modes of locomotion
such as basic gait control, gait transitions control, dynamic adaptive locomotion
control, etc. [
85
]. CPG models have been used with hexapod and octopod robots
inspired by insect locomotion [
11
]. CPGs have been also used for controlling swim-
ming robots, such as lamprey robots [
40
]. Quadruped walking robots controlled with
the use of CPGs have been studied in [
62
]. Models of CPGs are also increasingly
used for the control of biped locomotion in humanoid robots [
79
].
The problem of synchronization of coupled neural oscillators becomes more
difficult when there is uncertainty about the parameters of the dynamical model
of the neurons [
8
]. Thus, it is rather unlikely that coupled neurons will have
identical dynamical models and that these models will be free of parametric
variations and external disturbances. To synchronize neurons subject to model
uncertainties and external disturbances several approaches have been proposed.
In [
130
] by using the Lyapunov function method and calculating the largest
Lyapunov exponent, respectively, a sufficient condition and a necessary condition
of the coupling coefficient for achieving self-synchronization between two coupled
FitzHugh-Nagumo neurons are established. In [
5
] matrix inequalities on the basis
of Lyapunov stability theory are used to design a robust synchronizing controller
for the model of the coupled FitzHugh-Nagumo neurons. In [
217
] robust control
system combining backstepping and sliding mode control techniques is used to
realize the synchronization of two gap junction coupled chaotic FitzHugh-Nagumo
neurons under external electrical stimulation. In [
207
] a Lyapunov function-based
control law is introduced, which transforms the FitzHugh-Nagumo neurons into
an equivalent passive system. It is proved that the equivalent system can be
asymptotically stabilized at any desired fixed state, namely, synchronization can be
succeeded. In [
144
] synchronizing control for coupled FitzHugh-Nagumo neurons
is succeeded using a Lyapunov function approach. The control signal is based on
feedback of the synchronization error between the master and the slave neurons.
Finally, the use of differential geometric methods in the modelling and control of
FitzHugh-Nagumo neuron dynamics has been studied in [
47
,
220
].
In this chapter, differential flatness theory has been proposed for the synchro-
nization of coupled nonlinear oscillators of the FitzHugh-Nagumo type. Differential
flatness theory is currently a main direction in nonlinear dynamical systems and
enables linearization for a wider class of systems than the one succeeded with Lie-
algebra methods [
157
,
173
,
177
]. To find out if a dynamical system is differentially
flat, the following should be examined: (1) the existence of the so-called flat output,
i.e. a new variable which is expressed as a function of the system's state variables.
The flat output and its derivatives should not be coupled in the form of an ordinary
