INTRODUCTION
From a mechatronic point of view, the performance of electromechanical
motion systems can be improved by changing both the mechanical design
and the controller. Flexible link robot manipulators play an important
role in modern industry. The potential advantages that arise from the
use of lightweight flexiblelink manipulators are faster operation, lower
energy consumption and less costly structures.
The control of flexible link systems is very important in some fields, for
instance the aerospace industry, mainly due to the use of lightweight materials
in large space structures and flexible space robots. Thus, modeling and controlling
of robots with elastic links, has its own problems. In addition dynamical structure
of such robots includes nonlinear and uncertainty factors in the model. So,
designing an adaptive controller that is able to provide the performance characteristics
such as tracing the reference input, eliminating the disturbance and the conditions
of response speed is needed. Dynamic equation of the system has been obtained
by Lagrange’s method (Tian and Collins, 2005; Korayem
and Basu, 1994).
In this study two hybrid control methods are used to control a flexible
manipulator with payload. The designed controllers consist of two parts,
classical controllers; PID and Linear Quadratic Regulation (LQR) and hybrid
controllers; Sliding mode control using Gaussian Radial Basis Function
Neural Network (RBFNN) and Feedback Error Learning (FEL) technique.
The FEL method was used to develop a neural network controller to learn the
inverse dynamics of the flexiblelink system (Nakanishi and
Schaal, 2004; Talebi et al., 1998). The NN
learning is done with gradient descend method and error propagation algorithm
(Miyamura and Kimura, 2002). After that the reverse of
the process has been made completely and due to the zero error, the classic
controller output becomes zero automatically.
The FEL method exploited in this paper is consisted of a PID controller
and Fuzzy Neural Network (FNN) controller which is adapted based on output
of PID controller. In closed loop system, the FNN controller is positioned
in forward path which is to learn the inverse dynamics of the flexible
manipulator system. The FNN is a fourlayer network with fast parameter
learning. So, the adaptation of controller and system conditions is online
and fast. Also, the simulation results show these points.
Cheng and Wen (1993) developed a neural network controller
for a flexiblelink manipulator. The hub position and velocity measurements
were used to stabilize the system and a neural observer/controller was proposed
to drive the flexible arm to track a desired trajectory (Torfs
et al., 1998). Donne and Ozguner (1994) proposed
a neural controller assuming partial knowledge of the dynamics of the flexiblelink.
The unknown part of the dynamics was identified by a supervised learning algorithm.
The control was constructed of two stages, an optimal controller and an unsupervised
neural network controller using modelbased predictive control.
The scheme was based on an identification stage that also requires feedback
from the states of the system. Newton and Xu (1993a,
b) considered the joint tracking control problem for
a space manipulator using feedback errorlearning technique. However, tip position
tracking cannot be guaranteed specially for highspeed desired trajectories.
MATERIALS AND METHODS
Dynamical modeling of single flexible manipulator: The manipulator
is shown in Fig. 1 and is modeled as a pinnedfree flexible
with a payload at one end.
To obtain the equations of motion from Lagrange’s formulation, we consider
these assumptions:
• 
Using the EulerBernoulli theory of beam, to analyze the robot’s
manipulator 
• 
Using the linear elastic theory 
• 
The relative deformation that exists in the motor of robot is neglected 
• 
The link is considered to be light. Therefore, the properties of
each link are assumed to be a function of its length 
• 
We neglect the effect of gravitational potential energy 
• 
The angular displacement in the elastic links of the robots in the
direction of X, Y axis is zero 
The moment of inertia of the arm about the hub O is denoted by J_{f
}and ρ is the linear mass density. The link has length L and
the payload mass is given by_{ }M_{e}.
The control torque τ is applied at the hub of the manipulator through
the rotary actuator. The angular displacement of the manipulator, moving
in the XOY plane, is denoted by θ. The parametric values are shown
in Table 1.
For an angular displacement θ and an elastic deflection y(x, t)
the total displacement u(x, t) of a point, measured at a distance x from
the hub can be described as a function of both of the above, measured
from the direction of OX.
According to Eq. 1, the total kinetic energy of the
link is:
Axis of OX is considered as the base.

Fig. 1: 
Schematic representation of the flexible manipulator
system 
Table 1: 
Parametric value for the flexible manipulator 

The potential energy, caused from elasticity of the system can be written
as:
E is the modulus of elasticity for the beam material and we denote the
second moment of area of the beam crosssection. The lagrangian of the
system can be written as:
It is possible to obtain the dynamic equation of the system, using the
Lagrange equation:
where, Q_{i} is the generalized force related to the generalized
coordinate q_{i}.
The generalized force can be specified, according to the virtual work,
done by no conservative force that acts on the system.
According to Eq. 2 and 5, the equation
of manipulator motion can be obtained as follows:
The dynamic equation of the manipulator is described as:
Where:
The corresponding boundary and initial conditions are given by:
Note that the displacement y(x, t) can be considered according to the
hermitic polynomial terms.
φ_{i} (x) are the function of hermitic figures and indicate
the displacement and flexural slope in the beam.
q_{i} (t) indicates the degrees of freedom of manipulator.
Substituting Eq. 9 into Eq. 6 and
7 by applying boundary and initial conditions Eq.
8, the following ordinary differential equations can be derived.
If X is assumed as state space variable
and y is assumed as the output, .
The form of the state equation is:
Table 2: 
Mode dependent parameters 

Designing the PID controller: Here, we are going to design a PID
controller in order to improve the step response of the system. Different
methods are suggested for PID controller designing. All these methods,
notice the process of choosing the control parameters in order to provide
the desired operation characteristics. The method that we have used here
is ZieglerNichols approximate method.
Considering the numerical quantities of Table 2 and
the determined state equations for two modeling modes in the previous
section, the poles of the opened loop system will be:
Thus, the existence of mixed coupled poles in the system transformation
function in second mode, make the ZieglerNichols design method, impossible.
In order to solve the problem and make the system stable, we have assigned
the system eigen value.
This problem leads to designing a state feedback for the system and then
a PID controller is designed by state feedback for the stabilized system.
The transformation function of the PID controller for both modes is obtained
as follows:
Linear quadratic regulation and nonzero regulation: This method
determines the state feedback gain matrix that minimizes J in order to
achieve some compromise between the use of control effort, the magnitude
and the speed of response that together guarantee a stable system.
After linearization of the system:
Determine the matrix k of the LQR vector:
So in order to minimize the performance index:
where, Q and R are the positive definite hermetical or real symmetric
matrix.
Note that the second term on the right hand side, accounts for the expenditure
of the energy on the control efforts. The matrix of Q and R determine
the relative importance of the error and the expenditure of this energy.
According to the full rank of controllability matrix (A, B) and
matrix
, the LQR is designed with integral control and then u (t) is applied
to the nonlinear system in order to track of the unit step function.
Structure of fuzzy neuro network: Here, we will construct a feed
forward four layers fuzzy neural network. To implement the fuzzy control
rules stated in Eq. 16, first layer accepts input variables.
Its nodes represent input linguistic variables. Second layer is used to
calculate Gaussian Membership values. Nodes in this layer represent the
terms of the receptive linguistic variables. Nodes at the third layer
represent fuzzy rules, the layer where each node is related to an individual
output of the system. The links between third layer and fourth layer are
connected by the weighing values .
For a multiinput single output FNN system, let x be the input linguistic variable
and α_{j} as the firing strength of rule j, which is obtained by the
product of the grades of the membership function
in the antecedent. The proposed FNN is shown in Fig. 2. If
w_{j} represents the jth consequence link weight, the inferred value
y, is then obtained by taking the weighted sum of its input (Lee
and Teng, 2000).
The proposed FNN realizes the following fuzzy control rules:
where, for i = 1, 2,...,n, u_{ij} = x_{i}, A_{1j},...,A_{nj}
are fuzzy sets, w_{j }is a fuzzy singleton and n is the
number of inputs. Finally, the output of FNN is obtained:
Where:

Fig. 2: 
The configuration of the proposed FNN 
Layered operation of the FNN: Here, we shall indicate the signal
propagation and operation functions of the nodes in each layer. In the
following description
denotes the ith input of a node in the kth layer,
denotes the ith node output in layer k.
Layer 1: Input layer: The nodes in this layer only transmit input
value to the next layer directly.
From Eq. 19, the link weight at first layer
is unity.
Layer 2: Membership layer: In this layer, each node performs a
membership function and acts as a unit of memory. The Gaussian function
is adopted here as a membership function, thus we have:
where, m_{ij} and σ_{ij} are the center (or mean)
and width (or Standard DeviationSTD) of the Gaussian membership function.
The subscript ij indicates the jth term of the ith input x_{i}.
Layer 3: Rules layer: The nodes in this layer are called rate
nodes. The following AND operation is applied to each rule node to integrate
this fanin values:
Where:
The output
of a rule node represents the firing strength of its corresponding rule.
Layer 4: Output layer: Each node in this layer is called an output
linguistic node. This layer performances the defuzzification operation.
The node output is a linear combination of consequences obtained from
each rule, That is:
where,
(the link weight) is the output action strength of the jth output associated
with the ith rule. The
are the tuning factors of this layer.
Finally, the overall representation of input x and the mth output y
is:
where, m_{ij}, σ_{ij} and w_{mj }are tuning
parameters.
In Fig. 3, the signals that are fed into the FNN can be calculated
as (Zhou et al., 2002):
e_{k} = u_{r}–y_{m,
k} 
_{}(24) 
eck = y_{m, k}–y_{m, k–1} 
During the on line learning process, The fuzzy control rule stated in
Eq. 13 and the input command of the plant V_{com}
have been tuned. From Fig. 3, we have the following
equation.
V_{com} = V_{KT}+ΔV_{K} 
_{}(25) 
The method used in Fig. 4 is called Feedback Error
learning.
FEL structure: From controlling point of view, this method is an adaptive
method. Something that propounded earlier as FEL is Fig. 4
(Topalov et al., 1998):
As it is shown in Fig. 4 reverse process is built on the
way of feedforward. Something that is very important in this case is the error
that is used after the PID classic controller as a signal for training the reverse
process.
After making the reverse process, completely, because the error is zero,
the classic controller output will exit the circuit by itself. Now, if
every kind of disturbance or parameter changing occurs, the classic controller
will come in again and will take the control and at this moment, new reverse
process will be built. So, by this method we would have a dynamic control
strategy. To express this method, below formulation is presented that
is shown as a feedback adaptive controller in Fig. 5.

Fig. 3: 
Block diagram of the fuzzy neural network control system
for Flexible Manipulator. 

Fig. 4: 
Feedback error learning scheme 

Fig. 5: 
Nonlinear adaptive control and FEL with an adaptive
state feedback controller for a class of norder nonlinear SISO systems 
In this representation, real states are used to feed the reverse system,
as an input. By considering this structure and by using the existing principles
in adaptive control, equations for updating of value can be written.
Learning algorithm: Consider the single output case for simplicity.
Present goal is to minimize the following cost function:
where, y (k) is the desired output and ŷ (k) = O^{4 }(k)
is the current output for each discrete time k, in each training
cycle starting at the input nodes in the current output ŷ (k).
By using BP learning algorithm, the weighing vector of the FNN is adjusted
such that the error defined in Eq. 26 is less than
a designed threshold value after a given number of training cycles. The
wellknown algorithm may be written briefly as:
where, in this case η and w represent the learning rate and tuning
parameters of the FNN. Let e (k) = y (k)–ŷ (k) and W = [m, σ,
w]^{T} be the training error and weighting vector of the FNN,
then the gradient of error E(.) in Eq. 26 with respect
to an arbitrary weighting vector w is:
By recursive application of the chain rule, the error term for each layer
is first calculated and the parameters in the corresponding layers are
adjusted. With the FNN Eq. 23 and cost function defined
in Eq. 26, the update rule of w_{ij }are derived:
Where:
Similarly, the update laws of m_{ij} and σ_{ij}
are:
Where:
The BP algorithm is a widely used algorithm for training multilayer network
by means of error propagation via variation calculus. But its success
depends upon the quality of the training data.
Sliding mode control: In the design of sliding mode controller for flexible
manipulator, the control objective is to drive the output to the desired unit
step function. So by defining the tracking error to be in the following from
(Slotine and Li, 1991):
The sliding surface can be written as:
where, δ is a positive definite constant and
Structure of radial basis function neural network: Radial Basis
Function Neural Network (RBFNN) can be used to control the flexible joint.
Control structure is shown in Fig. 6.
The input of RBFNN is a sliding surface (Onder Efe et
al., 2001). The architecture of radial basis function consists of three
layers, the input, the hidden and the output layer as shown in Fig.
7.
Learning algorithm: The goal is to minimize the following cost
function:
where, S(k) is the sliding surface that was described earlier.
By using BP algorithm, the weighting vector of the RBFNN is adjusted
such that the cost function defined in Eq. 40 is less
than a designed value. The wellknown algorithm may be written briefly
as:

Fig. 6: 
The general block diagram of control 
where, η and w represent the learning rate and tuning parameter
of RBFNN. The gradient of E(.) in Eq. 41 with respect
to a weighting w is:
Where:
RESULTS AND DISCUSSION
Simulation of PID controller: By exerting this controller to the
system, the step response and the closed loop response in both modes
will be as in Fig. 8.
It is clear that the PID controller doesn’t create a suitable step response
for the system. Intense transient oscillations and high overshoot are
shortcomings of such controller, also the parameters of this controller
are constant, no adaptation with system dynamical changes, occur. In the
next section, by designing a compound controller, we try to improve the
system response.
Simulation of LQR and nonzero regulation: The result of simulations
after using LQR controllers for mode 1 and 2 are to be shown in Fig.
9 and 12.
The amplitude of oscillations around zero for mode 1 is about 0.001.
These oscillations will damp after some times. And also the amplitude
of oscillations around zero for mode 2 is about 0.0015.

Fig. 8: 
Step response by exerting PID controller, mode 1 and
mode 2 
Comparing these
two damped oscillations, shows that the frequency of oscillations in mode
2 is more than mode 1 but the amplitude of its damped oscillation is less
than mode1.
We used non zero regulation controller for regulating the unit step response.
The result of simulations is to be shown in Fig. 10
and 13. After using this controller, the oscillations
have to be done around 1 and the amplitude of damped oscillations is not
high. Figure 11 and 14 represent
the amplitude of oscillations.
Simulation of hybrid control (FNN and FEL): Here, we present the
results of the performed simulations for system hybrid control. We consider
the step torque as system’s input and will determine the system’s output
based on the state equations in modeling.
By applying a PD controller and FNN gird that was designed in the previous
section, on plant, we examine the system step response.

Fig. 9: 
Output of flexible manipulator with LQR controller (mode
1) 

Fig. 10: 
Output of flexible manipulator with LQR and nonzero
regulation controllers (mode 1) 

Fig. 11: 
Zoom of Output of flexible manipulator with LQR and
nonzero regulation controllers (mode 1) 
The designed controller
simulation has done in two modes. The reason for choosing these two modes
is to show the controller conformance with different states of the plant.
In second mode, frequency of vibration is twice as much of first mode
as shown in Fig. 15.

Fig. 12: 
Output of flexible manipulator with LQR controller (mode
2) 

Fig. 13: 
Output of flexible manipulator with LQR and nonzero
regulation controllers (mode 2) 

Fig. 14: 
Zoom of output of flexible manipulator with LQR and
nonzero regulation controllers (mode 2) 
It is obvious that the response overshoot has appeared at 1.3 sec and
the system response involves an overshoot of 28%. But at 3.7 sec the system
is in its stable state and error is less than 1%. It is clear that amplitude
level of output vibration has gone down to 0.00001 mm.

Fig. 15: 
Step response and error for mode 1 

Fig. 16: 
Step response and error for mode 2 
The second mode response is shown in Fig. 16. System
response in this mode has achieved its stable state and the error is less
than 2%. The difference between this mode and the first mode is more obvious.
The system response involves an overshoot of 40%. In this mode the vibration
amplitude level is reached to 0.0001 mm.
It is observed that the hybrid controller has a good adaptation with
the dynamic of the system. In the second mode, frequency of vibration,
with respect to first mode is doubled, but amplitude of the tip deflective
is limited to 0.0001 mm.
As you have seen, at first in order to make the system step response
stable and improve it, we used the PID controller, but in this part by
combining the PD controller and FNN network, it achieved the demanded
state.
Simulation of sliding mode control using gaussian radial basis function
neural networks: We applied Sliding Mode Control Using Gaussian Radial
Basis Function Neural Networks which is designed in material and methods
part to our system. The output of the system after using this controller
is the unit step. Figure 17 and 19
shows the various states of system’s response in modes 1 and 2. Regarding
these figures we found that after using this controller all sates are
stable. The oscillations in output will be removed by using this controller.
Figure 18 and 20 show the output
of the system and the control signal that was applied to the system for
mode 1 and 2. The amplitude of output without any oscillation is one and
the amount of controlling signal is finite.

Fig. 17: 
State of flexible joint with sliding mode control using
Gaussian RBFNN (mode 1) 

Fig. 18: 
Output of flexible joint (y) and control effort (u)
with sliding mode control using Gaussian RBFNN (mode 1) 

Fig. 20: 
Output of flexible joint (y) and control effort (u)
with sliding mode control using Gaussian RBFNN (mode 2) 
During 1 sec the output reaches to unit step without any overshoot. The
other controller’s output is consisted of overshoot and the output reaches
to desired value during more than 1 sec.
After comparing all of applied controllers, we found that using sliding
mode controller for flexible is more appropriate.
CONCLUSION
The designed controller consists of FNN network and conventional controller,
PID. The FNN parameters are being trained by FEL technique. Also, LQR controller
and sliding mode control using Gaussian radial basis function neural network
are designed. The PID and FNN controllers were applied for the first two modes
of vibration of the flexible arm. This controller has the ability to adapt with
system dynamic and it can also produce a robust system output, against every
kind of disturbance. The results of simulation show that system tracks the step
input, in a good way. If we want to compare the estimated results derived in
previous section with one of the main references, best of them is Tian
and Collins (2005). In this study, tip deflection is controlled at about
0.0004 mm, thus, in previous section tip deflection was derived, 0.00001 mm.
also, overshoot is reduced up to 28%, but in this reference, overshoot is reduced
to 35%. The system steady state error is limited in about 1%. Other point is
that, state feedback in PID controller designing is omitted and the controller
is converted to a simple PD. All above results indicate that, the hybrid controller
design in FEL figuration is one of the most practical methods in controller
designing.
Sliding controller which is applied for this system is appropriate. Because,
the internal dynamics in this system will be stable and also the output
would not have any oscillations and there is no overshoot in the system’s
response and during 1.5 sec the output reaches to the desired value.
Comparing this controller with other controllers like PID, FNN, LQR, which
are applied in this paper and last methods (Tian and Collins,
2005), shows that this controller has the best performance.
ACKNOWLEDGMENTS
The help of M. Naimi, V. Namazikhah and A. Vosolipour is gratefully acknowledged.