FRACTIONAL-ORDER PROPORTIONALINTEGRAL (FOPI) CONTROLLER FOR MECANUM-WHEELED ROBOT (MWR) IN PATH-TRACKING CONTROL

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 9, Issue 11, November 2018, pp. 325–337, Article ID: IJMET_09_11_033
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=11
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
FRACTIONAL-ORDER PROPORTIONAL-
INTEGRAL (FOPI) CONTROLLER FOR
MECANUM-WHEELED ROBOT (MWR) IN
PATH-TRACKING CONTROL
Joe Siang Keek, Ser Lee Loh* and Shin Horng Chong
Faculty of Electrical Engineering, Universiti Teknikal Malaysia Melaka, Malaysia.
*Corresponding author
ABSTRACT
This study presents experimental implementation of fractional-order proportional-
integral (FOPI) controller on a Mecanum-wheeled robot (MWR), which is a system with
nonlinearities and uncertainties, in performing tracking of a complex path i.e. ∞-shaped
path. The FOPI controller is almost as simpler as proportional-integral (PI) controller
and has supplementary advantage over PI controller due to its fractional integral. The
tracking performances of both the controllers are compared and evaluated in terms of
integral of absolute error (IAE), integral of squared error (ISE) and root-mean-square of
error (RMSE). Experimental result shows that the FOPI controller exhibits iso-damping
properties and successfully attains tracking with reduced error. Also, in this paper,
discretization of FOPI controller by using zero-order hold (ZOH) is discussed and
presented for the purpose of programming implementation on microcontroller board.
Besides that, graphical visualization of FOPI controller is presented to provide an insight
and intuitive understanding on the characteristic of the controller.
Key words: Fractional-order proportional-integral controller, Mecanum-wheeled robot,
path-tracking.
Cite this Article Joe Siang Keek, Ser Lee Loh and Shin Horng Chong, Fractional-Order
Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-
Tracking Control, International Journal of Mechanical Engineering and Technology,
9(11), 2018, pp. 325–337.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=11
1. INTRODUCTION
Integer-order proportional-integral-derivative (PID) controller is undeniably one of the most
successful control methods in control engineering. Most of the current advanced controllers have
PID controller lies within the hierarchy of the controllers. Due to the simplicity of PID controller
and the ability to control present, past and future error, the controller is well-accepted since its
introductory, and well-known even until today. However, the controller lacks robustness in
handling system uncertainties. Also, the performance of the controller is compromised when the

Joe Siang Keek, Ser Lee Loh and Shin Horng Chong
control system has nonlinearity. Thus, this leads to the proposal and emergence of adaptive or
robust controller, who may require additional scheme or parameter. Although such controllers
are certainly more advanced and have improved robustness but have relatively compromised
simplicity. As result, the structure and tuning process are more complex and time-consuming.
Therefore, fractional-order PID (FOPID) controller (or also known as PIλ
Dμ
controller) was
proposed with the intention to achieve improved robustness with minimal effort viz. by
introducing two extra degree-of-freedom (DOF) while retaining the original simplicity. With
such intention and promising results presented by many literatures, FOPID controller has
attracted much attentions of industry and researcher to nurture the controller towards maturity
and ubiquity.
Fractional calculus is the main component of FOPID controller and its realization is nothing
new as it was initiated by Leibniz and Hôpital through a letter conversation back in year 1695
[1]. Since then, many active researches on this topic begin and different approaches of
formulating the fractional calculus through generalization of integer calculus emerge. Whereas,
for fractional-order controller, it is comparably new; the effort of non-integer controller begins
as early as 1960s (see [2]) and becomes significantly active after 1990s [3]. Ever since, the
implementation of fractional-order controllers in many applications grows and the improvement
brought by the controllers is evident. One of the advantages of FOPID controller is its
contribution in system robustness, in which iso-damping properties are achieved during
parameter variation [4], [5]. Also, the fractional integral parameter, λ can be tuned to alter the
integral winding rate of the FOPID controller. Sandeep Pandey et al. implemented 2-DOF FOPID
controller on magnetic levitation system (MLS). MLS is a nonlinear system in which
conventional linear controller such as PID controller is infeasible. With the implementation of
FOPID controller, overshoot is suppressed during set-point tracking and actuator saturation is
overcame without additional scheme such as anti-windup [6]. Asem Al-Alwan et al. implemented
FOPID controller for laser beam pointing control system. Such control system is sensitive and is
subjected to disturbance and noise which are uncertain. With FOPID controller, the result shows
reduced root-mean-square error (RMSE) and peak error [7]. Meanwhile, FOPID controller is as
well applied to automatic voltage regulator (AVR) system [8] and speed control of chopper fed
DC motor drive [9]. Overall, the literatures show that FOPID controller is effective in
compensating system nonlinearity and actuator saturation, which is what the conventional PID
controller incapable of. However, the implementation of FOPID controller in Mecanum-wheeled
robot’s (MWR) path tracking and control is yet to be realized.
Mecanum wheel was invented by Bengt Ilon in 1970s. The circumference of the wheel is
made up of rollers angled at 45°. Consequently, the wheel is uniquely frictionless when it is
subjected to 45° diagonal force, thus making MWR maneuverable. However, the control of MWR
is challenging due to the presence of uncertainty. Such uncertainty includes wheel slippage and
irregularity of Mecanum wheel. As Mecanum wheel is made up of rollers, it lacks wheel tread
and tends to slip more often than conventional wheel. Asymmetric center of mass of MWR
worsens the slippage [10]. Besides that, the contact points between the wheel and floor have low
and varying friction and are shifting back and forth during motion. Such shifting may cause the
wheel to has inconsistent radius and unwanted disturbance to the robot [11]–[13]. Therefore, the
controller developed nowadays in controlling an MWR is often sophisticated, such as adaptive
robust [14] and non-singular terminal sliding mode controllers [15]. Whereas in this paper, a
FOPI controller is implemented to compensate the uncertainty and also nonlinearity of the MWR.
By comparing with the sophisticated controllers mentioned just now, FOPI controller is relatively
simpler. With auxiliary tuning parameter, FOPI controller has the potential to do beyond
conventional PI controller.

Fractional-Order Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-
Tracking Control
This paper is organized as follows: Section 2 covers the discretization of FOPI controller,
which is necessary for the programming of the microcontroller board. Section 3 presents insight
regarding the differences between integer-order and fractional-order controllers and their
responses towards errors through graphical visualization. Then, Section 4 discusses experimental
setup and path tracking performance of the MWR. Finally, Section 5 concludes and ends this
paper with future work.
2. DISCRETIZATION OF FRACTIONAL-ORDER PROPORTIONAL-
INTEGRAL (FOPI) CONTROLLER
As fractional-order proportional-integral (FOPI) controller is at higher hierarchical level than
integer-order proportional-integral (PI) controller, understanding the discretization of PI
controller is the stepping stone of FOPI controller’s discretization. The formula of PI controller
in continuous time-domain is given as
( ) ( )( ) ( )PI PI PI
0
t
p iu t K e t K e dτ τ
 
= × + × 
 ∫ (1)
where ( )PI
u t represents controlled variable, which is also the output of the PI controller.
Notations PI
pK and PI
iK are proportional gain and integral gain of the PI controller, respectively.
Notation ( )e t represents error, which is given as
( ) ( ) ( )e t r t y t= − . (2)
Error, ( )e t is the difference between reference value (setpoint), ( )r t and processed variable,
( )y t . With the understanding that integration computes the area under the curve of error,
recursive discrete form of PI controller can then be defined as
( ) ( )( ) ( ) ( )( )PI PI PI
0
,f
k
p i k
u k K e k K e k t k
=
 = × + × ∆ ∑ (3)
where k denotes timestep. Notation ( )t k∆ is the time elapsed from previous timestep ( )1k −
; in other words,
( ) ( ) ( )1 .t k t k t k∆ = − − (4)
The conversion from Equation (1) to Equation (3) involves zero-order-hold (ZOH) with
sampling time of 15±3 ms. The rule applied for the numerical integration in Equation (3) is based
on rectangular rule. Since the sampling time of the MWR is relatively faster than the process
whose actuator’s (motor) maximum speed is rated at 19 RPM, rectangular rule is acceptable.
Whereas trapezoidal rule does not give significant difference in term of accuracy. Afterall,
sampling frequency is encouraged to be as high as possible because transport delay may cause
the process to become unstable [16].
Riemann-Liouville (RL) definition of fractional-order integration is chosen in this paper due
to its advantages in term of simplicity and usage of Euler’s gamma function [17], [18]. The
properties and values of the gamma function can be found in [19]. RL definition of fractional
integration is as shown as Equation (5).
( )
( )
( ) ( )1
0
1 td
e t t e d
dt
λ
λ
λ
τ τ τ
λ
−
−
−
= −
Γ ∫ (5)

where λ denotes fractional value of integration and is 0 1λ< < [1]. Whereas, ( ).Γ represents
the gamma function. Then, output of the FOPI controller in recursive discrete form, ( )FOPI
u k is
then defined as
( ) ( )( ) ( ) ( ) ( )( )FOPI FOPI FOPI
0
.1f
k
p i k
u k K e k K e k t k
λ
λ
=
 = × + × ∆ Γ +
 
×∑ (6)
where FOPI
pK and FOPI
iK are proportional and integral gains of FOPI controller, respectively.
Finally, Equation (6) is converted into C++ programming language for the microcontroller board
to understand and control the MWR.
3. GRAPHICAL VISUALIZATION OF FRACTIONAL-ORDER
PROPORTIONAL-INTEGRAL (FOPI) CONTROLLER
Nowadays, the tuning of fractional-order controller often utilizes optimization method in tuning
the controller. Genetic Algorithm (GA) and Ant Colony Optimization (ACO) based on cost
function of integral of time-weighted absolute error (ITAE) are used to tune a FOPID controller
for automatic voltage regulator system can be seen in [8]. FOPID controller tuning by using a
combination of GA and nonlinear optimization for laser beam pointing system can be seen in [7].
Whereas, Artificial Bee Colony (ABC) algorithm is used for speed control of chopper fed DC
motor [9]. Tuning method by using the optimization is straightforward and produces promising
result. However, the downside is it iterates based on the mathematical model given and therefore,
the accuracy of the model needs to be as accurate with the actual system as possible. Moreover,
since the computation of the optimization is iterative and automatic, such approach or process
does not provide much intuitive understanding about the fractional parameters; parametric values
of FOPID controller is displayed at the end of iteration or when local minimal is achieved. As
result, manual tuning of FOPID controller is unlikely while PID controllers nowadays can be
tuned manually based on intuition. Therefore, this section intends to provide some insight
regarding the properties of FOPI controller.
It is important to take note that a FOPI controller with λ equals to 1.00 is exactly the same
as PI controller. Figure 1 shows the output of FOPI controller under the variation of FOPI
iK , λ
and error. Through observation and comparison of the graphs, the most significant finding is that
parameter λ varies the winding rate of integral action, whereas FOPI
iK is merely a gain or
amplifier. Equation (7) is used to compute respective winding rates for the graphs shown in Figure
1.
( )( )
( )
FOPI
abs
WR
f
f
u k
t k
= (7)
where WR represents winding rate of the integral action and fk denotes final timestep. The
winding rate is basically rate of change of controller’s output ( )FOPI
fu k

Tracking Control
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 1. Output of FOPI controller under different values of FOPI
iK and error with:
(a) & (b) & (c) λ equals to 1.00 (i.e. PI controller).
(d) & (e) & (f) λ equals to 0.96.
(g) & (h) & (i) λ equals to 0.92.
in ( )ft k seconds. To further evaluate the winding rate, normalized winding rate is used,
which is simply
FOPI
WR
NWR .
iK
= (8)
For better comparison and observation, Figure 2 compiles and depicts the winding rates
(WRs) and normalized winding rates (NWRs) in a series of graphs. By comparing the WRs shown
in Figure 2, we can notice that as λ decreases, the slopes (gradients) of the WRs decrease; smaller
value of λ reduces the effect of varying FOPI
iK . Therefore, this evidently supports that FOPI
controller exhibits iso-damping properties. Other than that, the WRs show that FOPI
iK is

proportional to error disregard of λ . Also, the straight-line plots of NWR show that the tuning
process of the gain FOPI
iK is linear.
(a) (b) (c)
Figure 2 WRs and NWRs of FOPI controller’s integral action under variation of FOPI
iK and error:
(a) with λ equals to 1.00 (i.e. PI controller).
(b) with λ equals to 0.96.
(c) with λ equals to 0.92.
(a) (b) (c)
Figure 3 Output of FOPI controller under different values of λ and FOPI
iK equals to 0.10:
(a) with error equals to 100 mm.
(b) with error equals to 80 mm.
(c) with error equals to 60 mm.
Next, Figure 3 depicts the output of FOPI controller under the variation of λ and error, with
FOPI
iK equals to 0.10. One significant finding can be observed from the graphs shown in Figure
3 is the difference between slopes of λ equals to 1.00 and others; the gradient between each slope
is significantly different with each other, especially for 1.00λ = and 0.96λ = . This shows that the
tuning process of parameter λ is nonlinear. To clearly portray the nonlinearity of the variation,
Figure 4 compiles and presents their WRs and NWRs.
Since the tuning process of λ is nonlinear, this may be the reason why optimization method
is often preferred in the literatures reviewed. Also, example of situation where fractional integral
controller is applicable for a nonlinear system viz. magnetic levitation system (MLS) can be seen
in [6]. Since MWR involves multiple axes of controls which are nonlinear as well, FOPI
controller is hopeful for the control system of the MWR in this paper, and is expected to perform
beyond the conventional PI controller which is ineffective for motor system with nonlinear

Tracking Control
characteristic [20]. Afterall and as conclusion, this section presents an insight of FOPI controller
in which iso-damping and nonlinear properties of the controller are graphically visualized.
Figure 4 WRs and NWRs of FOPI controller’s
integral action under variation of λ and error,
with FOPI
0.10iK = .
Figure 5 Experimental setup of the MWR in this
paper.
4. EXPERIMENTAL SETUP AND PATH TRACKING PERFORMANCE
In this paper, a Mecanum-wheeled robot (MWR) is designed and developed. The MWR is
equipped with four Mecanum wheels with radius of 30 mm and are driven by 12 V 19 RPM
Cytron SPG50-180K brushed DC geared motors. Two computer ball mice are used as sensors to
obtain fast positioning and orientation feedbacks. The sensor is not coupled with the Mecanum
wheels and therefore, wheel slippage has no effect on the robot. Cytron 32-bit ARM Cortex-M0
microcontroller board is used to process and execute commands. Figure 5 shows the physical
structure and experimental setup of the MWR.
As the actuations of the MWR are nonlinear, a simple linearization method by using inverse
of the process is applied. The inverse is obtained through open-loop step responses of the
actuators. Also, since the inverse is only an approximate, thus the nonlinearities can not be
eliminated completely, especially the nonlinearities at low speed actuations. As the literatures
show that fractional-order controller is suitable for nonlinear system, fractional-order
proportional-integral (FOPI) controller is implemented for the path tracking experiment in this
paper. The tracking performance is compared with proportional-integral (PI) controller. Figure 6
generally shows the positioning control system of the MWR in block diagram.
In Figure 6, the compensator, Q(s) is an algorithm that is derived based on the common
dynamics of MWR, in which it linearly maps the summation of heading angle and angle between
immediate and desired positions of the MWR into gains that control the Mecanum wheels
accordingly. Notation d denotes disturbance caused by the uncertainties during motion.
Controller, C(s) is either PI controller or FOPI controller which are based on Equations (3) and
(6), respectively. Both the PI and FOPI controllers are fine-tuned experimentally to obtain
satisfactory tracking performances for comparisons. The tasks of the controllers are to control the
MWR in tracking a ∞-shaped path and the path is generated based on formulae
( )2
100cos( )
( )
1 sin( )
r
k
x k
k
=
+
and (9)

( )2
100cos( )sin( )
( )
1 sin( )
r
k k
y k
k
=
+
, (10)
where ( )rx k and ( )ry k represent reference displacements in lateral (X) and longitudinal (Y)
directions, respectively.
(a)
(b)
Figure 7. ∞-shaped path tracking performance by using PI controller:
(a) with PI
2.0pK = and PI
0.1iK = .
(b) with PI
1.0pK = and PI
0.2iK = .
Figure 6 Block diagram of the MWR positioning control system.

Tracking Control
Figure 7 compiles and shows the tracking result by using PI controller with different tuning
parameters. Significant large error can be observed at rx ranges from −80 mm to −100 mm and
ry ranges from 0 mm to −20 mm. Such maximum peak error can be identified at time ranges
between 10 s and 20 s. By increasing the value of PI
iK , the error is reduced, and its duration is
slightly reduced as well. Next, the tracking performance by using FOPI controller with different
tuning parameters is compiled and shown in Figure 8. By comparing Figure 8 and Figure 7, FOPI
controllers significantly result smaller maximum peak error. FOPI controller with FOPI
1.0pK = ,
FOPI
0.2iK = and 0.90λ = produces better result than FOPI controller with
FOPI
2.0pK =
,
FOPI
0.1iK = and 0.90λ = .
(a)
(b)
Figure 8 ∞-shaped path tracking performance by using FOPI controller:
(a) with FOPI
2.0pK = , FOPI
0.1iK = and 0.90λ = .
(b) with FOPI
1.0pK = , FOPI
0.2iK = and 0.90λ = .
To validate the result numerically and statistically, the tracking experiment of each controller
is repeated five times. Each experiment is evaluated based on integral of absolute error (IAE),
integral of squared error (ISE) and root-mean-square of error (RMSE). The formula of IAE, ISE
and RMSE are

0
IAE ( ) ,
t
e dτ τ= ∫ (11)
( )2
0
ISE ( )
t
e dτ τ= ∫ and (12)
( )2
0
( )
RMSE
f
k
k
f
e k
k
=
=
∑ , (13)
respectively. Table 1 shows the IAE, ISE and RMSE of each of the experiments in tabulated
form whereas Figure 9 plots the tabulated data in graphical form.
Table 1 IAE, ISE and RMSE of each experiment
Controller Type
Integral of Absolute
Error, IAE (mm)
Integral of Square
Error, ISE (mm2
)
Root-mean-square of
error, RMSE (mm)
PI Controller
PI
2.0pK =
PI
0.1iK =
13766.34 96841.83 4.4263
13723.77 118667.14 4.8962
17279.61 162409.66 5.7309
13289.70 105717.91 4.7377
15936.72 142121.43 5.4937
12250.36 86032.98 4.2734
PI Controller
PI
1.0pK =
PI
0.2iK =
11831.90 97760.43 4.6150
15239.21 123735.33 5.0073
12195.33 82250.50 4.1784
11667.63 81989.42 4.1691
13593.23 103391.01 4.6897
13231.08 89981.19 4.3713
FOPI Controller
FOPI
2.0pK =
FOPI
0.1iK =
0.90λ =
8698.25 32013.56 2.6347
10600.37 43429.42 3.0717
9192.20 32678.13 2.6601
8323.20 25349.11 2.3444
9727.69 43955.51 3.0936
10616.75 41946.41 3.0103
FOPI Controller
FOPI
1.0pK =
FOPI
0.2iK =
0.90λ =
7910.68 22883.92 2.2278
7520.04 22122.46 2.1901
10113.96 43970.00 3.0870
7528.17 18118.11 1.9825
9194.33 40490.29 2.9653
8372.97 27317.22 2.4285
Generally, based on Figure 9, the FOPI controllers performs better tracking than the PI
controllers, with smaller IAE and RMSE as final result. Also, the FOPI controllers overall display
smaller value of ISE, which means that peak errors during path tracking are reduced. Among the
PI controllers, PI controller with PI
2.0pK = and PI
0.1iK = is significantly better than with
PI
1.0pK = and PI
0.2iK = . However, among the FOPI controllers, both the FOPI controllers
produce almost similar performances, even though the tuning parameters for both the controllers
are different. Therefore, the conclusions that can be drawn from the results are FOPI controller
exhibits iso-damping properties and reduces both path tracking error and peak error.

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Finally, to evaluate the precision of the repeated path tracking experiments, coefficient of
variation (COV) of the IAEs are calculated. For PI controller with PI
2.0pK = and PI
0.1iK = , its
COV equals to 0.1296 whereas for PI controller with PI
1.0pK = and PI
0.2iK = , its COV equals
to 0.1046. For FOPI controller with FOPI
2.0pK = , FOPI
0.1iK = and 0.90λ = , its COV equals to
0.1010 whereas for FOPI controller with FOPI
1.0pK = , FOPI
0.2iK = and 0.90λ = , its COV equals
to 0.1224.
(a) (b)
(c)
Figure 9. Evaluation of path tracking by using
(a) IAE,
(b) ISE and
(c) RMSE.
5. CONCLUSION
This paper started by presenting simple discretization of PI and FOPI controllers. Then, the
properties and characteristic of FOPI controller was studied and analysed in order to provide
insight and intuitive understanding on the tuning parameters. Next, in the experimental section,
PI and FOPI controllers were implemented on a Mecanum-wheeled robot (MWR) in tracking a
∞-shaped path. The result shows that under the presence of nonlinearity and uncertainty in the
MWR, the FOPI controller managed to produce improved tracking performance than PI
controller, and successfully reduces error and peak error. Besides that, two FOPI controllers with

different controller gains produce almost similar tracking performance, in which such
characteristic is known as iso-damping. For future work, the properties and effectiveness of
fractional derivative can be studied and implemented for the MWR as well.
ACKNOWLEDGEMENT
The authors would like to thank ‘Skim Zamalah UTeM’ and UTeM high impact PJP grant
(PJP/2017/FKE/HI11/S01536) for the financial support in this research.
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FRACTIONAL-ORDER PROPORTIONALINTEGRAL (FOPI) CONTROLLER FOR MECANUM-WHEELED ROBOT (MWR) IN PATH-TRACKING CONTROL

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FRACTIONAL-ORDER PROPORTIONALINTEGRAL (FOPI) CONTROLLER FOR MECANUM-WHEELED ROBOT (MWR) IN PATH-TRACKING CONTROL