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differential equation (ODE), (2) the components of the system (i.e., state variables
and control input) should be expressed as functions of the flat output and its
derivatives [
58
,
101
,
101
,
109
,
125
,
172
]. In certain cases the differential flatness
theory enables transformation to a linearized form (canonical Brunovsky form) for
which the design of the controller becomes easier. In other cases by showing that a
system is differentially flat one can easily design a reference trajectory as a function
of the so-called flat output and can find a control law that assures tracking of this
desirable trajectory [
58
,
200
].
This chapter is concerned with proving differential flatness of the model of
the coupled FitzHugh-Nagumo neural oscillators and its resulting description in
the Brunovksy (canonical) form [
125
]. By defining specific state variables as flat
outputs an equivalent description of the coupled FitzHugh-Nagumo neurons in the
Brunovksy (linear canonical) form is obtained. It is shown that for the linearized
model of the coupled neural oscillators it is possible to design a feedback controller
that succeeds their synchronization. At a second stage, a novel Kalman Filtering
method, the Derivative-free nonlinear Kalman Filter, is proposed for estimating the
non-directly measurable elements of the state vector of the linearized system. With
the redesign of the proposed Kalman Filter as a disturbance observer, it becomes
possible to estimate disturbance terms affecting the model of the coupled FitzHugh-
Nagumo neurons and to use these terms in the feedback controller. By avoiding
linearization approximations, the proposed filtering method improves the accuracy
of estimation and results in smooth control signal variations and in minimization of
the synchronization error [
158
,
159
,
170
].
4.7.2
Coupled Neural Oscillators as Coordinators of Motion
Coupled neural oscillators can provide as output signals that maintain a specific
phase difference and which can be used for the coordination of robot's motion.
These coupled neural oscillators are also known as CPGs and can synchronize
the motion performed by the legs of quadruped and biped robots. The concept of
CPGs comes from biological neurons, located at the spinal level, and which are able
of generating rhythmic commands for the muscles. CPGs receive commands from
higher levels of the central nervous system and also from peripheral afferents. Thus
their functioning is the result of the interaction between central commands and local
reflexes.
Neuronal mechanisms in the brain select which CPGs will be activated at
every time instant. The basal ganglia play an important role in this. Under resting
conditions the output layer of the basal ganglia (the pallidum) maintains different
CPG neurons under inhibition. To achieve CPG activation, striatial neurons, the
input layer of the basal ganglia, inhibit cells in the pallidum which keep under
inhibition the CPG neurons. The striatial neurons can, in turn, be activated from
either neocortex or directly from the thalamus. The responsiveness of striatial
neurons to activation can be facilitated by dopaminergic inputs (Fig.
4.6
). On
