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Received: December 18, 2021. Revised: February 23, 2022. 103
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Fractional Order Fuzzy PID Controller Design for 2-Link Rigid Robot
Manipulator
Hadeel I. Abdulameer1* Mohamed J. Mohamed1
1Control and Systems Engineering Department, University of Technology, Baghdad, Iraq
* Corresponding author’s Email: cse.19.13@grad.uotechnology.edu.iq
Abstract: In this paper, we present four control structures for the Fractional/Integer Order Fuzzy Proportional
Integral Derivative controllers that are used for a 2-link rigid robot manipulator. The manipulator is dealing with the
trajectory tracking problem. A metaheuristic optimization technique, namely the most valuable player algorithm, is
presented to optimize the controller's parameters while minimizing the integral of time square error. Furthermore, the
proposed controllers’ robustness is examined for changing the initial condition, exterior disturbances, and parameter
variations. MATLAB code outcomes show that the Fractional Order Fuzzy Proportional Integral Derivative
controllers ensure the best trajectory tracking and also improve the system's robustness to change the initial condition,
external disturbances, and parameter variations. The best structure is the Fractional Order Fuzzy Proportional
Derivative -Fractional Order Proportional Integral Derivative controller among all structures with the minimum
integral of time square error that is equal to 7.7481×10-5 for trajectory tracking, 2.4334×10-5 for changing of the
initial position, 2.1893×10-4 for disturbances rejection and 2.4990×10-6 for parameter variation. The result showed
also that the response of the trajectory tracking for theta1 and theta2 without overshoot and it has minimum settling
time in the case of Fractional Order Fuzzy Proportional Derivative -Fractional Order Proportional Integral Derivative
controller.
Keywords: Fractional-order controller, Fuzzy logic, Most valuable player algorithm, PID controller, Robotic
manipulator.
1. Introduction
Robotics is the branch of science concerned with
the design, simulation, and control of robots.
Nowadays robots are being used in almost every
aspect of daily life. It has accompanied people in
most of the industry and daily life jobs [1]. A very
wide range of applications was found, which include
cargo loading and unloading, automatic assembly
lines, spray paint application, handling dangerous
radioactive materials, forging, and military use. It is
well known that robot arm dynamics are highly
nonlinear and require expensive computations [2].
During its operations, the robotic manipulator is
subjected to external disturbances, a variety of
uncertainties, parameter variations, and payload
modifications in addition to the complexity and
nonlinearity difficulties. As a result, traditional
proportional-integral-derivative (PID) controllers
are not capable of providing simultaneous effective
control for trajectory tracking and constant
force/twist control [3].
Various controllers have been suggested for a 2-
link rigid robot manipulator (2-LRRM) by several
authors. A fuzzy controller and different
conventional control techniques like PD, PID, and
computed torque control were proposed in [4].
When compared to conventional controllers, the
fuzzy controller provided the best performance and
the most effective and accurate trajectory tracking
capability. The main issues of this study were that
the robust concept does not achieve and the
parameters of the controller were not optimized by
any optimization method. In [5] a Fractional Order
Fuzzy Proportional-Integral-Derivative (FOFPID)
controller for a two-link planar rigid robotic
manipulator was presented for the trajectory
Received: December 18, 2021. Revised: February 23, 2022. 104
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
tracking problem. The robustness of the FOFPID
controller was tested for model uncertainties,
disturbance rejection, and noise suppression. Its
performance was compared with the other three
controllers namely Fuzzy PID (FPID), Fractional
Order PID (FOPID), and conventional PID.
Numerical simulation results showed that the
FOFPID controller outperforms competing
controllers.
The PD neural network (NN)-based adaptive
controller design for robotic manipulators trajectory
tracking, is subject to noise measurement and
external disturbances, which was presented in [6],
the results showed that the neural network
modification of the adaptation laws of the weights
gave batter performance in function approximation
and thus better performance of the controller
compared to the ideal adaptation law. The
modification includes e-modification and σ-
modification. The issue was that the parameters of
the controller were not optimized by any
optimization method.
A Fractional-Order Fuzzy Proportional Integral-
Fractional Order Derivative Filter (FOFPI-D) for
controlling a nonlinear two-link robotic manipulator
system was introduced in [7]. Fuzzy proportional
integral-derivative filter (FPI-D) and proportional
integral-derivative filter (PI-D) controllers were also
prepared to compare their results with that of the
FOFPI-D. The results were shown that the FOFPI-D
controller outperforms other designed controllers. In
[8] an adaptive neural network (NN) for feed-
forward compensation is used alongside a sectorial
fuzzy controller SFC in the feedback loop to control
the trajectory tracking of the robot manipulator. The
results were presented in comparison with the PD
plus feed-forward controller, feed-forward SFC, and
feed-forward adaptive neural nonlinear PD control.
The suggested controller outperforms its
competitors, the main problems were that the robust
concept does not achieve and the controller didn’t
eliminate the chattering property in the control
signal. The fuzzy-neural-network PID (FNN- PID)
control framework of the robotic manipulator was
introduced in [3]. A fuzzy neural network algorithm
was proposed to adjust the PID controller
parameters effectively and quickly. Computer
simulations were conducted and examined to
demonstrate the proposed method’s efficiency. In
[9] three types of dynamic control strategies are
used for the PUMA 560 robot manipulator. The
strategies are PID, Sliding Mode Control (SMC),
and Integral Sliding Mode Control (ISMC). The
strategies were proposed based on Particle Swarm
Optimization (PSO) Algorithm. Simulation results
showed that the proposed tuning method achieved a
high level of stability, as well as an excellent
implementation of the proposed strategy for the
PUMA robot. A robust type-2 adaptive control had
been developed for the trajectory tracking of an
industrial 3-DOF manipulator robot in faulty
conditions in [10]. The adaptation involves using
Lyapunov stability concepts to update fuzzy type-2
parameters online. The simulation results of the
proposed control strategy showed that it was capable
of delivering a small tracking error even when
payload variation and actuator defects were present.
This work will use four structures of the
Fractional/Integer Order Fuzzy PID controller
(FOFPID, IOFPID), combining the fractional-order
actions will increase the robustness of the controller,
hence a more powerful and flexible design method
could be developed to meet the specifications of the
controlled system [11]. The most valuable player
algorithm (MVPA) is adopted to find the best values
of the controller’s parameters. A comparison
between the performance of FOFPID controllers and
IOFPID controllers has been made and the obtained
results have been presented.
The main contributions of the proposed
controllers are highlighted as follows:
1- Four structures of the Fractional/Integer
Order Fuzzy PID controllers are designed at
the same work.
2- Compared with [4, 6] who did not use the
optimal values of the controller, instead,
MVPA is used to get the optimum values for
the controllers.
3- The robustness of the proposed controllers is
demonstrated by changing the initial
condition, external disturbances, and
parameter variations which are not
demonstrated in [4, 8].
4- The control signals of the proposed
controllers have no chattering while in [8]
there is chattering in the control signal.
5- In comparison with [5, 7], the results of the
proposed FOFPID controllers are better or
converge to the best values obtained from
the existing controllers.
The remainder of this work is arranged as
follows. In Section 2, the dynamical model of the 2-
LRRM is explained. In Section 3, the suggested
FOFPID, IOFPID controllers are illustrated. In
Section 4 the proposed MVPA is described in
detailed steps. The results of the simulation and the
conclusion are given in Sections 5 and 6
respectively.
Received: December 18, 2021. Revised: February 23, 2022. 105
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Figure. 1 The structure of 2-LRRM
2. The dynamic model of 2-LRRM
The 2-LRRM scheme is shown in Fig. 1. It
consists of two links of lengths and . Their
centers of masses are and respectively, that
are located at the distal ends of links. To generate
the controlling torque at points A and B, DC motors
are used to estimate the angular positions (and)
and velocities (and) of the links, encoders are
employed [7].
The manipulator’s dynamic equation of motion
is used in robotics to set up the basic control
equations. The torques generated by the actuators
are used to produce the manipulator’s arm dynamic
motion in a robotic system. The association between
the input torques and the time rates of change of the
robot arm components configurations characterizes
the robotic system dynamic modeling which is
concerned with the derivation of the equations of
motion of the manipulator as a function of the forces
and moments acting on it. As a result, dynamic
modeling of a robot manipulator includes specifying
the functions that map the forces acting on the
structures and the joint positions, velocities, and
accelerations [12].
The equations for x- position and y-position of
are given by:
(1)
(2)
Similarly, the equations for x- position and the
y-position of are given by:
(3)
(4)
The kinetic energy is defined as:
(5)
And the potential energy can be written as:
(6)
Next, by Lagrange Dynamic, we form the
Lagrangian which is defined as:
(7)
The Euler Lagrange Equation is given by:
(8)
Where is the torque applied to the i’th link
Lastly, following Lagrange’s equation, the dynamics
of the arm are given by the two coupled nonlinear
differential equations [13]:
(9)
(10)
These manipulator dynamics are in the standard
form
M (11)
With is the Coriolis/centripetal vector,
is the inertia matrix, and is the gravity
vector. Note that M (θ) is symmetric.
]
V is the Coriolis and centrifugal matrix which is
given by
Received: December 18, 2021. Revised: February 23, 2022. 106
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Table 1. The parameters of 2-LRRM
parameters
Nominal value
0.1 kg
0.1 kg
0.8 m
0.4 m
9.81 m/
The gravity vector is given by:
The parameters are considered in Table 1
3. Controller design
Before describing the proposed controllers, we
will give a brief overview of the components of
these controllers, and then explain the nature and
structures of the proposed controllers for the 2-
LRRM.
3.1 The components of the proposed controllers
3.1.1. PID controller
The PID controller contains three parts as a style
of the feedback control loop, the first part is
proportional which is responsible for providing an
overall control action that is directly related to the
error signal through a gain factor. The second one is
the integral part which is used to reduce the steady-
state error by using a low-frequency compensation
or an integrator. The third and final part is the
derivative part which is responsible for improving
the transient state response by using a high-
frequency compensator or a differentiator [14, 15].
A typical PID controller is called the “three-term”
controller. Its transfer function is usually given as
Eq. (12).
(12)
Where is the proportional gain, is the
integral gain, is the derivative gain.
3.1.2. Fractional order PID controller
Fractional order PID is one of the most effective
controls that are common and useful in practical
industries. Podlubny and Oustaloup suggest the
fractional order PID controller, namely the PIλDμ or
FOPID which is the generalization form of the
classical PID controller. They used the fractional
order controller to establish the CRONE-controller
(Commande Robuste d'Ordre Non-Enterier
controller) in their series of papers and books [11].
Non-integer integration and differentiation are
described in a variety of ways. The Grünwald–
Letnikov (GL) and Riemann–Liouville (RL)
definitions, as well as the Caputo definitions, are the
most widely used[16].The GL definition is:
(13)
While the RL definition is given by:
a
(14)
For (n-1<α<n) and ℾ(x) is the well-known
Euler’s Gamma function.
(15)
where k R+.
Caputo's definition can be written as
a
(16)
for
The PIλDμ controller transfer function is given as
the ratio of the controller output U(s) and error
E(s)[11].
(17)
By adding more general control behaviors of the
PIλDμ type, more acceptable results between the
positive and negative effects of traditional PID
could be obtained. Furthermore, more flexible and
powerful design methods could be developed by
collaborating fractional-order actions, to meet the
specifications of the controlled system [11].
3.1.3. Fuzzy logic controller (FLC)
To introduce human decision-making and
experience to the plant, Fuzzy Logic Controllers
(FLCs) are represented to the system to include the
intelligence to the controller. A set of linguistic rules
or relational expressions are used to represent the
relationships between the input and the output [17].
Received: December 18, 2021. Revised: February 23, 2022. 107
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Figure. 2 The structure of the fuzzy controller
Figure. 3 The Fractional/Integer order fuzzy PD+I controller structure for the 2-LRRM
Figure. 4 The structure of the one block Fractional/Integer order fuzzy PID controller for 2-LRRM
A fuzzy logic controller block diagram is given in
Fig. 2. The fuzzy controller has four major parts: the
first one is the rule-base which contains a set of
rules for the most effective control of the system
that represents the knowledge. The second part is
the inference mechanism which decides which
control rules are applicable at the current
circumstance and then determines what should be
the output of the controller to the plant. The third
part is the fuzzification interface which simply
modifies the inputs so that they can match the rules
of the rule base. The final part is the defuzzification
interface that transforms the inference mechanism's
conclusions into the inputs to the plant [18, 19].
3.2 The structures of the proposed controller
Four structures of FOFPID and IOFPID
controllers are proposed to control the trajectory
tracking of the 2-LRRM.
Received: December 18, 2021. Revised: February 23, 2022. 108
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Figure. 5 The structure of the two-block Fractional/Integer order fuzzy PID controller for 2-LRRM
Figure. 6 The structure of the Fractional/Integer order fuzzy PD- Fractional/Integer order PID controller for 2-LRRM
3.2.1. Fractional/Integer order fuzzy PD + I controller
design (FOFPD+I/IOFPD+I)
The Fractional/Integer Order fuzzy PD+I
controller general block structure of 2-LRRM for
trajectory tracking is shown in Fig.3.
This figure represents the individual controller
for each input of 2-LRRM controlling, where the
reference trajectory is compared with the actual
trajectory for each link. The error e and its change
are the input variables for the basic fuzzy controller.
The fuzzy controller has only a fuzzy
proportional-differential control block. It does not
have a fuzzy integral control block. The Integral
Control (IC) is merged with the fuzzy PD controller
to improve the steady-state performance of the
system [20].
3.2.2. One block fractional/Integer order fuzzy PID
controller design (OBFOFPID/OBIOFPID)
The separate controller for each input of 2-
LRRM controlling of the One Block
Fractional/Integer Order Fuzzy PID controller is
shown in Fig. 4.
This controller is formed as a summation of the
fuzzy PD controller output and the fuzzy PI
controller output while the output of the fuzzy PD
will be fed to the integrator to form a fuzzy PI
controller. The standard fuzzy PID controller is
constructed by choosing the inputs to be error e and
derivative of the error and the output to be the
control signal u [21]
3.2.3. Two block Fractional/Integer order fuzzy PID
controller design (TBFOFPID/TBIOFPID)
The Two-Block Fractional/Integer Order Fuzzy
PID controller general structure is shown in Fig. 5.
This figure represents the individual controller for
each input of 2-LRRM controlling.
Since it is difficult for the fuzzy PD to remove
the steady-state error, it is known that the feasibility
of the fuzzy PI control is more than that of the fuzzy
PD control. The fuzzy PI control type is well-known
for its poor transient response performance due to
the inner integration process. A fuzzy PID is used to
retain the precise features of the PID controller
while merging the performance of fuzzy PI control
and fuzzy PD control at the same time [22].
Received: December 18, 2021. Revised: February 23, 2022. 109
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Table 2. Rule base for an error, derivative error, and FLC
output
NL
NM
NS
Z
PS
PM
PL
NL
NL
NL
NL
NL
NM
NS
Z
NM
NL
NL
NL
NM
NS
Z
PS
NS
NL
NL
NM
NS
Z
PS
PM
Z
NL
NM
NS
Z
PS
PM
PL
PS
NM
NS
Z
PS
PM
PL
PL
PM
NS
Z
PS
PM
PL
PL
PL
PL
Z
PS
PM
PL
PL
PL
PL
3.2.4. Fractional/Integer order fuzzy PD-
Fractional/Integer order PID controller design
(FOFPD-FOPID/IOFPD-IOPID)
The separate controllers for each input of the 2-
LRRM controlling of the Fractional/Integer Order
Fuzzy PD- Fractional/Integer Order PID controller
are shown in Fig. 6. The fuzzy controller uses error
e and derivative of error as input signals. This
controller is constructed from fuzzy PD and PID
controller in which the output of the fuzzy PD will
be input to the PID controller [23]
In this study seven Gaussian membership
functions (MF), as "Negative large (NL)",
"Negative Medium (NM)", "Negative Small (NS)",
"Zero (Z)", "Positive Small (PS)", "Positive Medium
(PM)" and at last "Positive Large (PL)" are used for
each input signal e, and control signal U and the
universe of discourse chosen to be [-10,10] where
the rules in the rule base as shown in Table 2.
4. Most valuable player algorithm
The Most Valuable Player Algorithm (MVPA) is
a recently generated algorithm suggested by
Bouchekara 2017. It is motivated by sport, in which
the group of players is organized into teams, then
the teams start competing to win the championship,
inside each team, individual players are trying to
win the most valuable player trophy by competing
against each other [24].
MVPA has the following characteristics: it
converges speedily, it is reliable, and it is efficient,
To develop optimality, MVPA exploits a population,
which is a group of skilled players that are similar to
the design variables in which the number of the
player's skills corresponds to the problem dimension,
a player is represented as following [25]:
(18)
The collection of players constricts a group or a
team which is given by:
(19)
Where S denotes the skill, Players Size is the
number of players in the competition, and Problem
Size is the problem dimension. Each team has its
prim player (i.e., their current most skillful player).
The competition’s MVP is the finest player in the
league (the player that has the best solution so far).
4.1 The most valuable player algorithm
1- Initialization: a population of Players Size,
players are arbitrarily created in the search space.
2- Teams formation: once the players’ population
has been created, they are distributed randomly
to form teams of Teams Size.
3- Competition phase: in this phase, each player is
trying to enhance his skills separately to be the
best player and then compete as teams, they play
against one another. There are two steps:
individual competition and team competition.
• Individual competition: it is genuine that any
player wants to be the best player for the
team and the competition MVP. So, the
player attempts to develop his abilities.
Therefore, the players’ skills of TEAMi are
reassessed as follows:
(20)
Where a rand is a random number distributed
randomly in the range [0 1], Franchise Player is the
best player in the team.
• Team competition: in this stage, another
team TEAMj is randomly selected,
where then TEAMi and TEAMj
compete against each other to decide the
best team.
The Franchise Player fitness represents the team
fitness and it is normalized in the MVPA, by
assessment as follows:
(21)
Then, to calculate the probability that
beats , the following formula is used:
Received: December 18, 2021. Revised: February 23, 2022. 110
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
(22)
Finally, in the team competition phase if
is chosen to play against , and wins,
then player skills are reassessed by:
(23)
Otherwise, the TEAMi players’ skills are
calculated by:
(24)
It should be noted that the skills of the player
have upper and lower limits, and all of the players
are actively trying to enhance their skills in each
competition. At any updating phase, if the player's
skills go beyond their bound then it must be limited
to the bound of skills. The checking of the player
skills bound is performed regularly in a special stage
that is called the bound checking stage.
4- Application of greediness: an assessment is
done after each step of competition (individual
and team competition). The population before
and after the competition phase is compared in
this assessment. If the results are better than that
of the initial stage, the solution is accepted.
5- Application of elitism: the poorest players are
substituted with the better ones in this phase.
The number of elite players is chosen as the
third of the Players Size.
6- Remove duplicates: in the population, if two
consecutive players are the same, the subsequent
player will be substituted by another player.
7- Termination criterion: the algorithm repeats
several times. The number of iterations is
specified by MaxNFix (maximum number of
fixtures) [24, 25].
5. Simulation and result
The performance of trajectory tracking and the
robustness of the FOFPID and IOFPID controllers
are discussed in this section. The proposed
controllers, the 2-LRRM, and test trajectory are
simulated using MATLAB code. The simulation
time is taken as 4s, while the sampling time is taken
as 1ms. The orders of the proposed FOFPID
controller can be adjusted to meet the design
specifications and give flexibility in choosing the
control constraints. Additionally, Grunewald’s
approximation of the 5th order (N=5) is used for the
fraction operator design. Frequency range [0.001,
1000] rad/s is used with the approximation for the
fractional operator design.
The trajectory tracking of each link is calculated
so that the manipulator can follow it. The results
then are used as a function of the performance index
for each controller. The Integral of Time Square
Error (ITSE) performance index is used in the test.
The MVPA was employed to regulate the
constraints of FOFPID and IOFPID controllers
according to the tracking error between the 2-LRRM
real path and the reference path using two initial
positions (0.1745, 0.1745) and (-0.1745,-0.1745) for
and respectively. The MVPA setting is as
follows; the population size=40, team size=5, team
players =8, and the maximum number of
iterations=300. The best solution resulting in the last
iteration is taken as the result of MVPA. The
performance assessment of FOFPID and IOFPID
controllers is based on the computation of the ITSE,
the best controller is the one that has the less value.
The ITSE can be calculated using the following
formula:
(25)
Where and are the difference
between the desired and real trajectories for link1
and link2 respectively. The desired trajectories
and for link1 and link2 have been given in
Eqs. (26) and (27), respectively as follows:
(26)
(27)
The best values for all proposed controllers
gains resulting from the last iteration of MVPA are
shown in Table 3, and the corresponding ITSE of
these controllers as in Table 4. In general, the ITSE
values for the FOFPID controllers are lower than
that of IOFPID controllers in all structures and the
FOFPD-FOPID gives the less value of ITSE among
all FOFPID controllers. The drawing of trajectory
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International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
(a)
(b)
(c)
(d)
(e)
Figure. 7 (a) Desired and actual theta1, (b) Desired and
actual theta2, (c) Desired and actual paths, (d) Controller
output (torqu1), and (e) Controller output (torqu2)
tracking of theta1 and theat2, the path traced by the
end-effector of the 2-LRRM, and controller output
are presented in Fig. 7.
It is clear from previous results that the response
of the trajectory tracking for theta1 and theta2
without overshoot and it has minimum settling time
in case of FOFPD-FOPID controller, while it has
maximum overshoot and maximum settling time in
case of IOFPD+I.
To check the robustness of the FOFPID and
IOFPID controllers, another primary position such
as [0.15, 0.15] for [,], is taken to test the ability
of the suggested controllers to track the two-link
robot on the chosen path. The obtained result is
shown in Table 5. Fig. 8 shows the trajectory
tracking of theta1 and theta2 and the path tracked by
the end-effector of the 2-LRRM with changing the
initial position for all controllers.
Despite changing the initial positions, the
FOFPD-FOPID controller remains the best
performer than the rest, since there is no overshoot
in the response of trajectory tracking of theta1 and
theta2 and the settling time is the minimum. While
the IOFPD+I is the worst because the response of
theta1 and theta2 has maximum overshoot and
maximum settling time.
Another test for the robustness of the FOFPID and
IOFPID controllers by adding disturbance term [sin
(50t), sin (50t)] to the control action [,], and
making the initial position as [0, 0], without
retraining the parameters (gains) of FOFPID and
IOFPID controllers to confirm the robustness and
the ability of each controller.
The obtained result is shown in Table 6. The
trajectory tracking of theta1 and theta2 and the path
Received: December 18, 2021. Revised: February 23, 2022. 112
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Table 3. The gains of the FOFPID and IOFPID controllers
Controller
Link
NO
Kp
Kd
Ki
Ko
FOFPD-
FOPID
L1
Kp1= -10.1724
Kp2= 9.9166
Kd1= -11.4738
Kd2= 5.8532
65437.9
-
0.0329
= 0.7617
0.0888
L2
Kp1= -15.0553
Kp2= 47.7082
Kd1= -12.4056
Kd2= 9.7917
34.2747
-
0.0488
0.9997
0.0179
IOFPD-
IOPID
L1
Kp1= 28.2710
Kp2= -23.0416
Kd1= -25.5490
Kd2= -1.1712
0.2352
-
-
-
L2
Kp1= -20.8094
Kp2= 28.0464
Kd1= 26.5808
Kd2= 1.8930
-0.0294
-
-
-
TBFOFPID
L1
Kp1= -6.4796
Kp2= -10.1531
Kd1= -2.8004
Kd2= -15.5898
18.6482
11.0377
0.4778
0.0933
L2
Kp1= 10.7877
Kp2= -4.2935
Kd1= 0.1559
Kd2= -6.3977
-12.7727
19.4736
0.7859
0.2110
TBIOFPID
L1
Kp1= - 4.0141
Kp2= -2.3489
Kd1= -52.2225
Kd2= -96.6613
- 2.6777
94.8572
-
-
L2
Kp1= 8.2145
Kp2= 8.1349
Kd1= 95.1769
Kd2= 97.9022
31.6260
-90.1372
-
-
OBFOFPID
L1
15.3138
4.6533
6.7374
-15.2112
83330.
06340.
L2
15.5876
30.1463
12.2027
-30.3515
.44190
47720.
OBIOFPID
L1
-1.1310
-51.7489
-31.1856
195.3759
-
-
L2
5.9723
55.5988
16.1857
-134.7754
-
-
FOFPD+I
L1
9.7843
6.9423
45.2943
-17.7013
0.3997
0.0615
L2
15.4913
5.5418
106.0475
-53.5378
0.5381
0.0240
IOFPD+I
L1
-2.5888
-57.3065
17.0627
100.7910
-
-
L2
5.6015
57.3065
36.7680
-109.9999
-
Table 4. The ITSE of the FOFPID and IOFPID
controllers
controller
ITSE
controller
ITSE
FOFPD-
FOPID
7.7481
×10-5
IOFPD-
IOPID
1.0129
×10-4
TBFOFPID
1.5261
×10-4
TBIOFPID
1.6378
×10-3
OBFOFPID
2.5903
×10-4
OBIOFPID
2.4
2013×1
0-3
FOFPD+I
6.9568
×10-4
IOFPD+I
4.4573
×10-3
Table 5. The ITSE of the FOFPID and IOFPID with
initial position (0.15, 0.15)
controller
ITSE
controller
ITSE
FOFPD-
FOPID
2.4334 5-
10×
IOFPD-
IOPID
4.2157 5-
10×
TBFOFPID
5.1637 5-
10×
TBIOFPID
6.1976 4-
10×
OBFOFPID
1.1887 4-
10×
OBIOFPID
8.5812 4-
10×
FOFPD+I
1.8352 4-
10×
IOFPD+I
1.8 3-
10×
(a)
(b)
Received: December 18, 2021. Revised: February 23, 2022. 113
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
tracked by the end-effector of the 2-LRRM using
disturbance of sin50t N-m in both links are
presented in Fig. 9.
From the results, we conclude that the FOFPID
controller functions better for the disturbances
rejection also when comparing it to the other
IOFPID controllers, where the FOFPD-FOPID is the
best one since it has the smallest ITSE.
(c)
Figure. 8 (a) Desired and actual thata1, (b) Desired and
actual theta2, and (c) Desired and actual paths with initial
positions (0.15,0.15)
Table 6. The ITSE of the FOFPID and IOFPID with
disturbances sin (50t) for both links and Initial
position(0,0)
controller
ITSE
controller
ITSE
FOFPD-
FOPID
2.1893
×10-4
IOFPD-
IOPID
1.2
×10-3
TBFOFPID
2.3
×10-3
TBIOFPID
5.1
×10-3
OBFOFPID
1.3
×10-3
OBIOFPID
6
×10-3
FOFPD+I
4.5
×10-3
IOFPD+I
6.4
×10-3
(a)
(b)
(c)
Figure. 9 (a) Desired and actual theta1, (b) Desired and
actual theta2, and (c) Desired and actual paths with
disturbance term [sin (50t), sin (50t)] and initial position
(0, 0)
Table 7. The ITSE of the FOFPID and IOFPID for 5%
increase in both masses and initial position (0,0)
controller
ITSE
controller
ITSE
FOFPD-
FOPID
2.4990
×10-6
IOFPD-
IOPID
8.4663
×10-6
TBFOFPID
7.3864
×10-5
TBIOFPID
1.2744
×10-4
OBFOFPID
8.6453
×10-5
OBIOFPID
1.0589
×10-4
FOFPD+I
3.5313
×10-5
IOFPD+I
5.9906
×10-5
Received: December 18, 2021. Revised: February 23, 2022. 114
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
(a)
(b)
(c)
Figure. 10 (a) Desired and actual theta1, (b) Desired and
actual theta2, and (c) Desired and actual paths for 5%
increase in both masses and initial position (0, 0).
The parameter variation is also investigated for
the FOFPID and IOFPID controllers, by increasing
the masses of two links 5%, the results as in Table 7,
and the trajectory tracking of theta1 and theta2 and
the path tracked by the end-effector of the 2-LRRM
for mass changes for all controllers are presented in
Fig. 10.
From the results presented, it can be deduced
that in general FOFPID controllers outperforms the
IOFPID controllers for parameter variation and the
best controller is FOFPD-FOPID among them.
The effect of adding disturbance and parameter
variation as well as changing the initial positions
together on the FOFPID and IOFPID controllers is
presented in Table.8. Fig.11 shows the trajectory
tracking of theta1 and theta2 and the path tracked by
the end-effector of the 2-LRRM for disturbance,
parameter variation as well as changing the initial
positions for all controllers.
Table 8. The ITSE of the FOFPID and IOFPID with
initial position (0.15, 0.15), disturbances sin (50t) for both
links, and a 5% increase in both masses
controller
ITSE
controller
ITSE
FOFPD-
FOPID
2.4470
×10-4
IOFPD-
IOPID
1.1
×10-3
TBFOFPID
2.2
×10-3
TBIOFPID
5.6
×10-3
OBFOFPID
1.7
×10-3
OBIOFPID
6.4
×10-3
FOFPD+I
4.5
×10-3
IOFPD+I
7.8
×10-3
(a)
(b)
Received: December 18, 2021. Revised: February 23, 2022. 115
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
(c)
Figure. 11 (a) Desired and actual theta1, (b) Desired and
actual theta2, (c) Desired and actual paths with initial
position (0.15, 0.15), disturbance term [sin (50t), sin
(50t)] and 5% increasing in both masses
Table 9. Comparison between proposed FOFPID
controllers and existing controllers for trajectory tracking
Type of
controller
IAE1
IAE2
FOFPD-FOPID
1.018710-5
2.043310-5
TBFOFPID
2.829510-5
1.638810-4
OBFOFPID
5.197610-5
7.115710-5
FOFPD+I
3.996210-5
5.012310-5
FOFPID [5]
4.63210-4
4.18010-5
2 DOF FOFPI-D
[7]
9.08610-4
5.74810-4
It is found that the ITSE for FOFPD-FOPID
controller remains the smallest among proposed
controllers despite changing initial conditions,
adding disturbance, and parameter variation, and we
can show from the response of theta1 and theta2
there is no overshoot and it has a minimum settling
time while the worst controller is the IOFPD+I.
Another comparison is done among FOFPID
controllers proposed in this work and the FOFPID,
2DOF FOFPI-D controllers that were used to
control the 2- link robot manipulator in [5] and [7].
The comparison was done in the case of trajectory
tracking by using Integral Absolute Error (IAE) for
link1 and lonk2 which are given in Eqs. (28) and
(29), respectively, and the results listed in Table 9.
The results show that the IAE for the proposed
FOFPID controllers are better or converge with the
best data obtained from the existing controllers.
(28)
(29)
6. Conclusion
In this paper, four structures of FOFPID and
IOFPID controllers were proposed for a 2-LRRM
for trajectory tracking problems. The addition of the
fractional operator to the FPID controller has given
control engineers more design flexibility because it
adds two more variables to tune. A metaheuristic
MVPA is used to tune controllers’ parameters.
Furthermore, the robustness of these controllers has
been explored for initial conditions, disturbance
rejection, and model uncertainty. The results show
that the FOFPID controllers have a respectable
capability to reduce the variance between real and
desired paths speedily and then to track the wanted
path with good accurateness and without chattering
in control signals, where the best controller is the
FOFPD-FOPID, the next is the TBFOFPID, after
that the OBFOFPID and at last the FOFPD+I.
It can be concluded that the FOFPID controllers
are better and more robust than the IOFPID
controllers in all structures, whereas the ITSE for
FOFPD-FOPID, TBFOFPID, OBFOFPID, and
FOFPD+I for trajectory tracking task equal to
7.7481×10-5, 1.5261×10-4, 2.5903×10-4, and
6.9568×10-4 respectively, while the ITSE for
IOFPD-IOPID, TBIOFPID, OBIOFPID, IOFPD+I
equal to 1.0129×10-4, 1.6378×10-3, 2.42013×10-3
and 4.4573×10-3 respectively. FOFPD-FOPID
controller is the best among all studied controllers
for trajectory tracking , disturbance rejection , and
parameter variation with superior trajectory tracking
and the smallest ITSE. Also the result showed that
the response of the trajectory tracking for theta1 and
theta2 without overshoot and it has minimum
settling time. This work also demonstrates the
capability of MVPA for tuning the parameters of 2-
LRRM controllers.
Received: December 18, 2021. Revised: February 23, 2022. 116
International Journal of Intelligent Engineering and Systems, Vol.15, No.3, 2022 DOI: 10.22266/ijies2022.0630.10
Finally, as future work, other optimization
techniques can be used instead of MVPA such as ant
colony optimization (ACO), genetic algorithm (GA),
and differential search algorithm (DSA) to tune the
parameters of the controllers. Besides,
Implementing the suggested controllers practically
by using a real robot manipulator with all necessary
hardware.
Conflicts of Interest
The authors declare that they have no known
competing financial interests or personal
relationships that could have appeared to influence
the work reported in this paper
Author Contributions
Conceptualization,1st author; methodology,1st
author; software, 2nd author; validation, 1st author;
formal analysis, 1st author; investigation, 1st author;
resources, 1st author; data curation, 1st author;
writing—original draft preparation, 1st author;
writing—review and editing, 1st author;
visualization, 1st author; supervision, 2nd author;
project administration, 2nd author.
Acknowledgments
This work was supported by the Department of
Control and Systems Engineering /University of
technology-Iraq.
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