fractional-order proportional- integral (fopi) … · joe siang keek, ser lee loh and shin horng...

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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 PIλDμ 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 ( )PIu t represents controlled variable, which is also the output of the PI controller.

Notations PIpK 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 ku 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

tde t t e d

dt

λλ

λτ 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, ( )FOPIu k is

then defined as

( ) ( )( ) ( ) ( ) ( )( )FOPI FOPI FOPI

0.1f

k

p i ku k K e k K e k t k

λλ

=

= × + × ∆ Γ +

×∑ (6)

where FOPIpK 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 FOPIiK , 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 FOPIiK is merely a gain or

amplifier. Equation (7) is used to compute respective winding rates for the graphs shown in Figure

1.

( )( )( )

FOPIabsWR

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 ( )FOPIfu k


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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 1. Output of FOPI controller under different values of FOPIiK 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

WRNWR .

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 FOPIiK . Therefore, this evidently supports that FOPI

controller exhibits iso-damping properties. Other than that, the WRs show that FOPIiK is



proportional to error disregard of λ . Also, the straight-line plots of NWR show that the tuning

process of the gain FOPIiK is linear.

(a) (b) (c)

Figure 2 WRs and NWRs of FOPI controller’s integral action under variation of FOPIiK 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 FOPIiK 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

kx k

k=

+ and (9)



( )2

100cos( )sin( )( )

1 sin( )r

k ky 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 PI2.0pK = and PI 0.1iK = .

(b) with PI1.0pK = 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 PIiK , 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 FOPI2.0pK = , FOPI 0.1iK = and 0.90λ = .

(b) with FOPI1.0pK = , 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



0IAE ( ) ,

t

e dτ τ= ∫ (11)

( )

2

0ISE ( )

t

e dτ τ= ∫ and (12)

( )

2

0( )

RMSE

fk

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

PI2.0pK =

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

PI1.0pK =

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

FOPI2.0pK =

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

FOPI1.0pK =

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.


Tracking Control


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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