Control Performance Assessment of Fractional-Order PID Controllers
Applied to Tracking Trajectory Control of Robotic Systems
Abstract: - Performing tracking tasks in robotic manipulators presents many challenges for the controller
design, especially in presence of external disturbances that affects the dynamical behavior of the robotic
system. This paper presents the design of a Fractional-Order PID controller with the computed torque control
strategy for the tracking control of a two degree of freedom robotic manipulator. The proposed technique is
contrasted against the classical PID controller with the computed torque control strategy. To validate the
proposed controllers, the robotic system is simulated using an MSC-ADAMS/MATLAB co-simulation model,
which is employed for identification and control tasks. The proposed model is tested in presence of external
disturbances in the applied torque, random noise in the feedback loop and payload variations. Obtained results
show that the Fractional-Order PID controller with the computed torque control strategy has a better
performance in presence of the analyzed external disturbances for tracking tasks.
Key-Words: - Fractional control, FOPID, IOPID, MSC-ADAMS/MATLAB cosimulation model.
Received: March 16, 2021. Revised: January 3, 2022. Accepted: January 22, 2022. Published: February 10, 2022.
1. Introduccion
Robotic manipulators perform position and tracking
tasks, using the dynamic model of the robotic
system for the design of the control strategy. If the
dynamic model of the manipulator is well known,
the tracking tasks control can be designed using the
computed torque control strategy. This control
strategy employs the feedback linearization to
compensate the nonlinearities of the robotic
manipulator model, allowing the application of
classic control strategies as the PID controller.
Although this control strategy is simple and
effective, rarely the complete dynamic model of the
robotic system is available due to the presence of
external disturbances and parametric uncertainness
biasing the controller tuning and affecting the
stability and performance of the system.
Control strategies like QFT [6][7], H infinity [8][9],
adaptive control [10], [11], or synthesis [10][12]
are employed for the control of robotic systems with
external disturbances and parametric uncertainness.
However, the main limitation of these control
strategies is the lack of knowledge of the robotic
system dynamics.
On the other hand, Fractional-Order control
strategies can help to reach a robust performance in
the presence of external disturbances and parametric
uncertainness even without a complete dynamic
model of the system [13], [14]–[16].
This paper presents the design of a tracking control
for a two degree of freedom robotic manipulator
employing a Fractional-Order PID controller
(FOPID) with the computed torque control strategy.
Initially, the kinematic and dynamic model of the
robotic system is calculated. Then, to MSC-
ADAMS / MATLAB cosimulation model of the
manipulator is built for the stages of identification,
control, and validation. After that, the dynamic
model of the robot is identified employing the
recursive least squares algorithm. From the
JAIRO VIOLA
Department of Mechanical Engineering
University of California, Merced
5200 Lake Rd, Merced, CA 95343, USA
LUIS ANGEL
Department of Electronic Engineering
Universidad Pontificia Bolivariana
Via Piedecuesta km 7, Bucaramanga, COLOMBIA
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identified dynamic model, the FOPID controller
with computer torque control is tuned.
An Integer-Order PID controller (IOPID) is
designed to compare the performance of the FOPID
controller. Then, a robustness analysis is performed
for the IOPID and FOPID controllers considering
the presence of external disturbances in the joints
torque, random noise in the feedback loop, and
payload variations. The results of the robustness
analyzed are quantified using a novel Control
Performance Assessment methodology, which
proposes a set of performance metrics based on the
angular position, velocity, and acceleration errors of
the manipulator.
The main contribution of this paper is the
implementation of the Computed Torque Control
strategy with the Fractional Order PID controller for
the tracking trajectory problem applied to robotic
systems and the development of a novel control
performance assessment methodology for fractional-
order controllers.
This paper is structured as follows. Section II shows
the computed torque control strategy and the design
methodology for the IOPID and FOPID controllers
with the computed torque control strategy. Section
III presents the case study, a two degree of freedom
robotic manipulator, its kinematic and the dynamic
models, and the MSC-ADAMS/MATLAB
cosimulation model and its parametric
identification. Section IV presents the IOPID and
FOPID controllers with computed torque control
design and its performance for different trajectories.
Section V shows the robustness tests and Control
Performance Assessment for the FOPID and IOPID
controllers. Finally, conclusions and future works
are presented.
2. Feedback linearization via Computed Torque
Control
Computed torque control (CTC) employs the
feedback linearization technique to obtain a
linearized and decoupled model of the robotic
system to control it with linear control strategies.
From [17], the Dynamic model of a robotic system
is given by (1).
(1)
where are the manipulator joints position,
velocity and acceleration, is the inertia matrix,
is the Coriolis matrix, and is the
gravity vector. Applying the linearization law (2)
where is the new input of the system.
.
(2)
Assuming an exact knowledge of the dynamic
model, combining (1) and (2) results (3).
(3)
If is invertible, (3) correspond to a
linearized and decoupled double integrator system
for each link of the robotic system as shown in (4).
(4)
For the design of the IOPID controller, the control
law (4) is proposed, where is the joint position
error, is the joint velocity error, and is the
desired joint acceleration.
.
(5)
Replacing (5) in (4), the control law in closed loop
for the robotic system is given by (6).
(6)
Passing (6) to the Laplace Domain, the
characteristic polynomial of the system is obtained.
(7)
In order to find the IOPID controller terms, the
desired polynomial is given by (8) where is a non-
dominant pole of the closed system, is the
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damping ratio, and is the natural frequency of
the system.
(8)
From (7) and (8), the resulting expression (9) allows
to calculate the terms of the IOPID controller for
each joint.
(9)
According to [18]–[20], FOPID controller is
defined by the following integral-differential
equation (10).
(10)
where, is the proportional term, is the integral
time term, is the derivative time term, is the
non-integer order of the integral term, and is the
non-integer order of the derivative term. Assuming
, the new control law for the linearized
system (4) with the FOPID controller (10) is:
(11)
Replacing (11) in (4):
(12)
Passing (12) to the Laplace domain, the desired
characteristic polynomial is obtained:
(13)
From (13), the equation contains the Fractional-
Order terms () that rise to five the degree of
freedom of the characteristic polynomial of the
system. Therefore, a solution based in optimization
algorithms is proposed to find the FOPID controller
terms. This methodology employs the non-linear
conditions (14)-(18) that describes the desired
behavior of the system in the frequency domain to
reach a robust performance and stability.
Phase margin ():
(14)
Gain crossover frequency ():
(15)
Robustness against plant gain variations:
(16)
High frequency noise rejection:
(17)
Output disturbance rejection:
(18)
where is the FOPID controller transfer
function, is the system transfer function,
is the phase margin, is the gain crossover
frequency, and are the maximum values for
the sensitivity and complementary sensitivity
functions. MATLAB FMINCON algorithm is
employed to solve the multiobjective optimization
problem. the cost function for the FOPID controller
tuning, is given by (14), the optimization constrains
are (15)-(18), and the ISE criterion is selected as
performance index for the algorithm.
3. CASE STUDY: TWO DEGREE OF FREEDOM
ROBOTIC MANIPULATOR
As shown in Fig.1, the two degree of freedom
robotic manipulator moves through the plane
. As can be observed, is the movement of the
first link with length , measured with reference to
the coordinate system . On the other hand,
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is the movement of the second link with length
measured with reference to the coordinate system
. Besides, the fixed reference system
is in the joint of the first link, making
the rotation axis of the manipulator. As well, the
reference system
is in the joint of the
second link, and the reference system
is
in the final effector of the manipulator. The gravity
vector is in parallel to the axis of the reference
system
.
Figure 1. Two degree of freedom robotic
manipulator
Applying the homogeneous transformation
matrices to the robotic system links, the direct
kinematic model of the system is given by (19). The
inverse kinematics of the robotic manipulator is
presented in (20), which is obtained from the
algebraic analysis of (19).
(19)
(20)
From [17], the Dynamic model of any robotic
system is given by (1), where this model can be
obtained by the Euler-Lagrange method, the virtual
work principle, and the Newton-Euler recursive
method. Applying the Euler-Lagrange method, the
dynamic model of the two degree of freedom
robotic manipulator is given by (21), where and
are the applied torque to the joints and ,
and are the links mass, and indicate the
position of the gravity center of each link, is the
gravity vector, and are the links inertia,
and are the position, velocity
and acceleration of each link respectively. Finally,
and are the viscous and Coulomb
frictions of each link
(21)
. For the identification process, the dynamic model
of the robotic manipulator (21) is parametrized in
the matrix form (1) as shown in (22).
(22)
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where the parameters are:
;
;
;
; ; ; ;
;
Notice parameters will be obtained employing
the identification algorithm. Notice that
parameters (for ) are in function of the
physical dimensions of the manipulator. However,
the remaining parameters depends of the viscous
and Coulomb frictions of each joint, and changes
according to the operating condition of the robotic
system. A dynamic cosimulation model of the
robotic manipulator is built to analyze its dynamical
behavior, which incorporates the physical effects
acting on the robotic manipulator as the Viscous and
Coulomb frictions and body inertias. The
cosimulation model is shown in Fig.2. As can be
observed, the manipulator is parallel to the gravity
vector, which should be compensated by the
controller. The robotic system turns around the axis,
and the applied torque is perpendicular to the
rotation axis. This MSC-ADAMS model is exported
to MATLAB using the ADAMS-controls toolbox to
perform the identification and control stages.
Figure 2. Two degree of freedom robotic
manipulator built in MSC-ADAMS
4. DYNAMIC MODEL PARAMETRIC
IDENTIFICATION
The recursive least squares algorithm is employed
for the parametric identification of the robotic
manipulator shown in Fig.2. Notice that this step is
required because although the physical dimensions
of the robot are well known, the Viscous and
Coulomb frictions of each joint are unknown. The
parametrized model of the robotic manipulator
given by (22) shows the parameters, which are
associated to non-linear functions dependent of the
joint position, velocity, and acceleration of the
system. However, this parameters can be
calculated using linear regression matrices. For the
multivariable case, the recursive least squares
algorithm employs the linear regressor (23)
described by [21] where is the measured
output vector, is the regression matrix, and
is the vector of estimated parameters.
(23)
The recursive least squares descripted in (24)
employs samples from joint position, velocity, and
acceleration as the applied joint torques to the
robotic system, which are taken from the MSC-
ADAMS/MATLAB cosimulation model, where
is the vector of estimated parameters, is
the regression matrix, is the error for each
iteration, and is the covariance matrix.
(24)
The identification trajectory selection is an
important task for the parametric identification of
the dynamic model of a robotic manipulator since
the quality of the estimation depends of the
regressor persistent excitation [22]. The selected
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trajectories (25) and (26) are shown in Fig.3 for the
joints respectively, composed by the
superposition of sinusoidal functions that ensure the
regressor persistent excitation.
(25)
(26)
Figure 3. Proposed identification trajectories
for and
Applying the trajectories (25) and (26) to the
MSC-ADAMS / MATLAB co-simulation model of
the manipulator, the joint position, velocity,
acceleration, and joint applied torque data are
obtained. These data are entered in the recursive
least squares algorithm (24), obtaining the
parameters of the dynamic model (22), which are
, , , ,
, , ,
, , .
To validate the identified dynamic model of the
robotic manipulator, the applied joint torque from
the MSC-ADAMS/MATLAB cosimulation model is
compared with the applied torque from the dynamic
model (22) built in SIMULINK, which employs the
estimated parameters presented in Table I. Figure
4 Shows the applied joint torque of the cosimulation
model versus the applied joint torque of the
identified dynamic model. As can be observed, the
applied torque from the identified dynamic model is
like the MSC-ADAMS/MATLAB cosimulation
model. The RMSE value given by (27) is employed
as fit measure between the MSC-
ADAMS/MATLAB cosimulation model and the
identified dynamic model, where is the total
amount of samples, and is the error between
the applied torques. The RMSE values for are
0.0182 and 0.0018. So that, it can be said that the
identified model approaches the dynamic model of
the robotic manipulator.
(27)
(a)
(b)
Figure 4. Applied torque validation from (a) MSC-
ADAMS/MATLAB cosimulation model (b)
Identified dynamic model
5. FOPID AND IOPID CONTROLLER DESIGN
The IOPID and FOPID controllers with the
computed torque control are tuned to reach a robust
performance and stability. Hence, the desired
operating conditions consider a time response
without overshoot () or a , and a
settling time of the system of , which
requires a . Applying the tuning
methodologies presented in Section II and Section
III, the obtained terms for the IOPID and FOPID
controllers are presented in Table I.
TABLE I IOPID AND FOPID CONTROLLER
CONSTANTS
Controller
IOPID
440
400
40
1
1
FOPID
729
0.2
100
0.85
0.84
The two degrees of freedom robotic manipulator
is tested for tracking tasks in order to evaluate the
IOPID and FOPID controllers performance. Fig.5
shows the proposed desired trajectory for a specific
task, which employs a cartesian trajectory planner to
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set the trajectory for each joint of the robot. Thus,
the final effector of the robot can be carried from an
initial point to a final point through a straight line
under certain velocity, acceleration, and Jerk
conditions, considering the physical actuators
restrictions.
The trajectory planner is formed by three
segments descripted by a 6-1-6 polynomial [23].
The first segment is the acceleration section and is
represented for a sixth order polynomial. The
second segment is the constant velocity section and
is represented with a first order polynomial. The
third segment belongs to the deceleration segment
and is represented by a sixth order polynomial.
Notice the first and third segments employs a high
order polynomial to reach a smooth jerk variation.
The spatial trajectory produced by the final
effector employing the IOPID and FOPID
controllers with the computed torque control
strategy is presented in Fig.5. As can be observed,
the trajectory that corresponds to a triangular shape,
shows that the robotic manipulator has a good
performance for tracking tasks using the IOPID and
FOPID controllers.
The position error for each joint employing the
IOPID and FOPID controllers is presented in Fig.6,
which shows that the FOPID controller with
computed torque control has a less joint error than
the IOPID controller with the computed torque
control. The control action is presented in Fig.7,
where the magnitude of the control action is similar
for the IOPID and FOPID controllers, indicating
that both the FOPID controller and the IOPID
controller requires the same energy to reach the
desired trajectory.
6. CONTROL PERFORMANCE ASSESSMENT
Three tests are performed to evaluate the IOPID
and FOPID controllers robustness. On the first test,
a step-type external disturbance of is added to
each joint torque while the robotic manipulator
performs the desired trajectory presented in Fig.6.
For the joint the disturbance magnitude is
and for the joint . Figures
9 to 10 present the spatial performance and the joint
error of the robotic system. As can be observed, the
external disturbances affect the desired trajectory;
however, the IOPID and FOPID controllers return
the robotic system to the desired path.
Figure 6. Tracking performance of the IOPID and
FOPID
(a)
(b)
Figure 6. Joint error using the IOPID and FOPID
controllers for (a) and (b) .
(a)
(b)
Figure 7. Joint applied torque using the IOPID and
FOPID controllers for (a) and (b) .
Figure 8. Tracking performance of the IOPID and
FOPID controllers in presence of external
disturbance.
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(a)
(b)
Figure 9. Joint error in presence of external
disturbances using the IOPID and FOPID
controllers for (a) and (b) .
For the second test, the robotic system payload is
varied to evaluate the performance of the IOPID and
FOPID controllers. Hence, an payload is
placed in the final effector of the robotic
manipulator while it performs the desired trajectory,
which response is presented in Fig.10. As can be
observed, despite the payload variation, the FOPID
controller has a better tracking response than the
IOPID controller. Likewise, Fig.11 presents the
joint error in and for the IOPID and the
FOPID controllers. As can be observed, the highest
joint error in and coincide with the trajectory
point (0.8, 0) where and are perpendicular to
the gravity vector. However, the joint error is less
when the FOPID controller is employed, indicating
a better performance for payload variations. The
third test introduces a random noise with
degrees amplitude in the feedback loop of each joint
of the robotic system. Fig.13 shows the effect of the
random noise while the robotic manipulator
performs the desired trajectory employing the
IOPID and FOPID controllers. As can be observed,
both the IOPID and the FOPID controllers are
affected by the presence random noise in the
feedback loop. Fig.14 presents the joint error in
and when the IOPID and FOPID controllers are
employed in presence of random noise in the
feedback loop. As can be appreciated, the joint error
rises significantly for and , especially when the
IOPID controller is employed.
The tracking norm (28) proposed by [24] is
employed to stablish a quantitative comparison
between the IOPID and the FOPID controllers
performance for each joint of the robotic system
Figure 10. Tracking performance of the IOPID and
FOPID controllers with payload variations
(a)
(b)
Figure 11. Joint error in presence of payload
variations employing the IOPID and FOPID
controllers for (a) q1 and (b) q2
Figure 13. Tracking performance of the IOPID and
FOPID controllers in presence of random noise in
the feedback loop
(28)
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(a)
(b)
Figure 14. Joint error in presence of random noise in
the feedback loop using the IOPID and FOPID
controllers for (a) and (b) .
where is the total time of the trajectory, is the
joint position error and is the joint velocity
error. In addition, to obtain a general index of the
tracking tasks performances, the tracking
norm (29) is proposed, which is based in (28) and
consider the position and velocity errors of the
joints of the robotic system.
(29)
Table III summarizes the performance norms (28)
and (29) for the IOPID and FOPID controllers on
nominal operation conditions, presence of external
disturbances in the joint torque, payload variations,
and random noise in the feedback loop. As can be
observed, for the nominal operation conditions, the
norm and is less for the FOPID controller
than for the IOPID controller, indicating FOPID
controller has a better performance for tracking
tasks with an improvement of 34.7%.
In presence of external disturbances, Table IV
shows that for the FOPID controller, the norm
and is less regarding to the IOPID controller
with an 46.12% improve. Therefore, it can be said
that the FOPID Controller is more robust than the
IOPID controller in presence of external
disturbances, which proves that the FOPID
controller makes a better trajectory correction when
a disturbance appears.
Analyzing the robotic system performance in
presence of random noise in the feedback loop, the
norm is similar for the IOPID and FOPID
controllers. However, for the norm a
significantly improvement of 20.19% is reached.
Besides, the norm indicates that the FOPID
controller is 11.32% better than the IOPID
controller. For this reason, although the presence of
random noise in the feedback loop has a great effect
over the tracking tasks, the FOPID controller has a
less position and velocity errors, so employing the
FOPID controller makes more robust the robotic
system against the presence of random noise in the
feedback loop.
Also, Table IV shows the RMS value of the
applied torque of each joint of the robotic system
when the IOPID and FOPID controllers with the
computed torque control strategy are employed. As
can be observed, the RMS value on and has
similar values for the nominal operation, presence
external disturbance, and payload variation tests. In
contrast, for the random noise in the feedback loop
test, the magnitude of the applied torque is bigger
than in the previous tests because the IOPID and
FOPID controllers must make a greater trajectory
correction rising the applied torque in and .
So that, although the applied torque required by
the IOPID and FOPID controllers is similar, the
FOPID controller has the better results for tracking
tasks with a less position and velocity errors in
presence of the analyzed disturbances.
CONCLUSIONS
This paper presented the design of a FOPID
controller with the computed torque control strategy
for the tracking control of a two degrees of freedom
robotic manipulator. The kinematic and dynamic
model of the robotic system were obtained, and the
dynamic model was identified employing the
recursive least squares algorithm. The proposed
control strategy was contrasted with a IOPID
controller with the computed torque control strategy
in presence of external disturbances, payload
variations and random noise in the feedback loop.
Obtained results from the robustness analysis
shows that the FOPID controller has a better
performance for tracking tasks since FOPID
controller has less position error and velocity error
than the IOPID controller. In addition, the FOPID
controller has a better external disturbance rejection
with a 46.12% improvement, a better performance
for payload variations with a 31.59% improvement,
and a better response to the presence of random
noise in the feedback loop with 11.32%
improvement. Also, there is not significantly
differences for the proposed tests regarding to the
applied joint torque. For these reasons, it can be said
that the FOPID controller is more robust than the
IOPID controller to perform tracking tasks.
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TABLE II TRACKING NORMS N OF THE ROBOTIC SYSTEM JOINTS IN TRACKING TASKS FOR NOMINAL
OPERATION CONDITIONS AND THE PRESENCE OF EXTERNAL DISTURBANCES
Tracking
norm
Nominal operation
External disturbance
Payload variation
Random noise
FOPID
IOPID
Improve
(%)
FOPID
IOPID
Improve
(%)
FOPID
IOPID
Improve
(%)
FOPID
IOPID
Improve
(%)
0.23
0.41
43.82
0.98
1.76
44.32
0.90
1.41
36.50
14.90
14.83
-0.47
0.57
0.84
32.12
2.22
4.15
46.51
4.74
6.92
31.50
14.47
18.13
20.19
0.61
0.93
34.70
2.43
4.51
46.12
4.83
7.06
31.59
20.77
23.42
11.32
TABLE III RMS VALUE OF THE ROBOTIC SYSTEM JOINTS IN TRACKING TASKS FOR NOMINAL OPERATION
CONDITIONS AND THE PRESENCE OF EXTERNAL DISTURBANCES
Joint
Nominal operation
External disturbance
Payload variation
Random noise
FOPID
IOPID
FOPID
IOPID
FOPID
IOPID
FOPID
IOPID
40.05
39.92
43.3
43.6
44.52
44.49
659
623
6.12
6.11
6.73
6.8
13.14
12.98
132
124
CONCLUSIONS
This paper presented the design of a FOPID
controller with the computed torque control strategy
for the tracking control of a two degrees of freedom
robotic manipulator. The kinematic and dynamic
model of the robotic system were obtained, and the
dynamic model was identified employing the
recursive least squares algorithm. The proposed
control strategy was contrasted with a IOPID
controller with the computed torque control strategy
in presence of external disturbances, payload
variations and random noise in the feedback loop.
Obtained results from the robustness analysis
shows that the FOPID controller has a better
performance for tracking tasks since FOPID
controller has less position error and velocity error
than the IOPID controller. In addition, the FOPID
controller has a better external disturbance rejection
with a 46.12% improvement, a better performance
for payload variations with a 31.59% improvement,
and a better response to the presence of random
noise in the feedback loop with 11.32%
improvement. Also, there is not significantly
differences for the proposed tests regarding to the
applied joint torque. For these reasons, it can be said
that the FOPID controller is more robust than the
IOPID controller to perform tracking tasks.
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Luis Angel, Jairo Viola
E-ISSN: 2224-2856
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Jairo Viola and Luis Angel performed the controller
design, robotic system modelling, simulation and
performance assessment, as well as the writing for
this paper.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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DOI: 10.37394/23203.2022.17.8
Luis Angel, Jairo Viola
E-ISSN: 2224-2856
73
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