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http://www.iaeme.com/IJMET/index.asp 325 editor@iaeme.com International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 11, November 2018, pp. 325337, 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 IAEMEPublication 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. 325337. http://www.iaeme.c
2 om/IJMET/issues.asp?JType=IJMET&VType=9&
om/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 http://www.iaeme.com/IJMET/index.asp 326 editor@iaeme.com 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 PI 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 i
3 mplementation of fractional-order contro
mplementation 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, l 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 robots (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
4 the wheel and floor have low and varyin
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 Controlhttp://www.iaeme.com/IJMET/index.asp 327 editor@iaeme.com 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 controllers discretization. The formula of PI controller in continuous time-domain is given as ()()()()PIPIPIpi utKetKed tt =´+´ (1)where ()PI ut represents controlled variable, which is also the output of the PI controller. Notations PI p K and PI i K are proportional gain and integral gain of the PI controller, respectively. Notation ( ) et represents error, which is given as ( ) ( ) ( ) etrtyt =-. (2 ) Error, ( ) et is the difference bet
5 ween reference value (setpoint), ( ) rt
ween reference value (setpoint), ( ) rt and processed variable, ( ) yt . With the understanding that integration computes the area under the curve of error, recursive discrete form of PI controller can then be defined as ()() ( ) ()() ( ) PIPIPI , piukKekKektk =´+´D \n (3)where k denotes timestep. Notation () tk D is the time elapsed from previous timestep ( ) 1 k - ; in other words, ( ) ( ) ( ) 1. tktktk D=-- (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 actuators (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 Eulers 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). ()() ()() td etted dt ttt =-(5) Joe Siang Keek, Ser Lee Loh and Shin Horng Chong http://www.iaeme.com/IJMET/index.asp 328 editor@iaeme.com where l denotes fractional value of integration and is 01 l [1]. Whereas, ( ) . G represents the gamma function. Then, output of the FOPI controller in recursive discrete form, ()FOPI uk is then defined as ()() ( ) ()()() ( ) FOPIFOPIFOPI . piukKekKektk =´+´D \r G+ \n (6 ) where FOPI p K and FOPI i K 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 utilize
6 s optimization method in tuning the cont
s 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 l equals to 1.00 is exactly the same as PI controller. Figure 1 shows the output of FOPI controller under the variation of FOPI i K , l and error. Through observation and comparison of the graphs, the most significant finding is that parameter l varies the winding rate of integral action, whereas FOPI i K is merely a gain or amplifier. Equation (7) is used to compute respective winding rates for the graphs shown in Figure 1. ( ) ( ) () FOPIabsWRuktk(7)where WR represents winding rate of the integral action and f k denotes final timestep. The winding rate is basically rate of change of controllers output ( ) FOPI f uk Fractional-Order Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-Tracking Controlhttp://www.iaeme.com/IJMET/index.asp 329 editor
7 @iaeme.com (a) (b) (c) (d)
@iaeme.com (a) (b) (c) (d) (e) (f) (g) (h) (i) \n \r\r \r\r FOPI i K \r\r\r\n \n\r \n l \n \r\r \n \n \n l \n \n \n l !in ( ) f tk seconds. To further evaluate the winding rate, normalized winding rate is used, which is simply FOPI WR NWR. i K (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 l decreases, the slopes (gradients) of the WRs decrease; smaller value of l reduces the effect of varying FOPI i K . Therefore, this evidently supports that FOPI controller exhibits iso-damping properties. Other than that, the WRs show that FOPI i K is Joe Siang Keek, Ser Lee Loh and Shin Horng Chong http://www.iaeme.com/IJMET/index.asp 330 editor@iaeme.com proportional to error disregard of l . Also, the straight-line plots of NWR show that the tuning process of the gain FOPI i K is linear. (a)(b)(c)"# $"# \n \r\r% &\r \r\r FOPI i K \r\r\r\n l \n \r\r \n\r l \n l ! (a)(b)(c) \n \r\r \r\r l FOPI i K \n \r\r\r ''\n\r \r\r\r( ''\n \r\r\r ''Next, Figure 3 depicts the output of FOPI controller under the variation of l and error, with FOPI i K equals to 0.10. One significant finding can be observed from the graphs shown in Figure 3 is the difference between slopes of l equals to 1.00 and others; the gradient between each slope is significantly different with each other, especially for 1.00 l = and 0.96 l = . This shows that the tuning process of parameter l is nonlinear. To clearly portray the nonlinearity of the variation, Figure 4 compiles and presents t
8 heir WRs and NWRs. Since the tuning proc
heir WRs and NWRs. Since the tuning process of l 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 Fractional-Order Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-Tracking Controlhttp://www.iaeme.com/IJMET/index.asp 331 editor@iaeme.com 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. "# $"# \n \r\r% &\r \r\r l \r\r\r)FOPI 0.10 *+\r' ,"# \r4. EXPERIMENTAL SETUP AND PATH TRACKING PERFORMANCEIn 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-
9 order controller is suitable for nonline
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() () 1sin() r k xk and (9) Joe Siang Keek, Ser Lee Loh and Shin Horng Chong http://www.iaeme.com/IJMET/index.asp 332 editor@iaeme.com () 2 100cos()sin() ()1sin() kk yk, (10 ) where () r xk and () r yk represent reference displacements in lateral (X) and longitudinal (Y) directions, respectively. (a) (b) -.\r / &\r\r' 01 & \n \r\r\n PI 2.0 PI 0.1 \n\r PI 1.0 PI 0.2 iK= 2 /&\r',"# & \r1' Fractional-Order Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-Tracking Controlhttp://www.iaeme.com/IJMET/index.asp 333 editor@iaeme.com Figure 7 compiles and shows the tracking result by using PI controller with different tuning parameters. Significant large error can be observed at r x ranges from 80 mm to 100 mm and r y 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 i K , the error is reduced, and its duration is slightly redu
10 ced as well. Next, the tracking performa
ced 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 1.0 , FOPI 0.2 and 0.90 l = produces better result than FOPI controller with FOPI 2.0 , FOPI 0.1 and 0.90 l = . (a) (b) -.\r / &\r\r' 01 & \n \r\r\n FOPI 2.0 )FOPI 0.1 0.90 l = \n\r FOPI 1.0 )FOPI 0.2 0.90 l = 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 Joe Siang Keek, Ser Lee Loh and Shin Horng Chong http://www.iaeme.com/IJMET/index.asp 334 editor@iaeme.com 0 IAE() , t ed tt (11)() ISE() t ed tt and (12 ) () 2 0 () RMSEek, (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 (mm) Root-mean-square of error, RMSE (mm) PI Controller PI 2.0 PI 0.1 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 1.0 PI 0.2 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 2.0 FOPI 0.1 iK= 0.90 l = 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 1.0 FOPI 0.2 iK= 0.90 l = 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 General
11 ly, based on Figure 9, the FOPI controll
ly, 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 2.0 and PI 0.1 is significantly better than with PI 1.0 and PI 0.2 . 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. Fractional-Order Proportional-Integral (Fopi) Controller for Mecanum-Wheeled Robot (Mwr) in Path-Tracking Controlhttp://www.iaeme.com/IJMET/index.asp 335 editor@iaeme.com 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 2.0 and PI 0.1 , its COV equals to 0.1296 whereas for PI controller with PI 1.0 and PI 0.2 , its COV equals to 0.1046. ForFOPI controller with FOPI 2.0 , FOPI 0.1 and 0.90 l = , its COV equals to 0.1010 whereas for FOPI controller with FOPI 1.0 , FOPI 0.2 and 0.90 l = , its COV equals to 0.1224. (a) (b) (c) * \r / &01 &\n \n3*)\n\r \n4* \n #,4*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 Joe Siang Keek, Ser Lee L
12 oh and Shin Horng Chong http://www.iaeme
oh and Shin Horng Chong http://www.iaeme.com/IJMET/index.asp 336 editor@iaeme.com 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. REFERENCES [1]D. P. A. Dingyu Xue, YangQuan Chen, Chapter 8 Fractional-Order Controller: An Introduction, in Linear Feedback Control - Analysis and Design with MATLAB, Philadelphia: Society for Industrial and Applied Mathematics, 2007, p. 354. [2]S. Manabe, The Non-Integer Integral and its Application to Control Systems, ETJ Japan, vol. 6, no. 34, pp. 8387, 1961. [3]Y. Q. Chen, I. Petrá, and D. Xue, Fractional order control - A tutorial. Hyatt Regency Riverfront, St. Louis, MO, USA. [4]Y. C. Ying Luo, Fractional Order Motion Controls. John Wiley & Sons, Ltd., 2013. [5]Y. C. Dingyu Xue, Chunna Zhao, Fractional Order PID Control of A DC-Motor with Elastic Shaft: A Case Study, in Proceedings of the 2006 American Control Conference, 2006. [6]S. Pandey, P. Dwivedi, and A. S. Junghare, A novel 2-DOF fractional-order PI-Dcontroller with inherent anti-windup capability for a magnetic levitation system, AEU - Int. J. Electron. Commun., vol. 79, pp. 158171, 2017. [7]A. Al-alwan, X. Guo, and I. N. Doye, Laser Beam Pointing and Stabilization by Fractional-Order PID Control: Tuning Rule and Experiments, in IEEE Conference on Control Technology and Applications (CCTA), 2017, pp. 16851691. [8]A. G. S. Babu and B. T. Chiranjeevi, Implementation of fractional order PID controller for an AVR system using GA and ACO optimization techniques, in IFAC-PapersOnLine, 2016, vol. 49, no. 1, pp. 456461. [9]R. C. Division, B. Atomic, and C. Republic, Fractional Order PID Controller Design for Speed Control of Chopper Fed DC Motor Drive Using Artificial Bee Colony Algorithm, 2013 World Congr. Nat. Biol. Inspired Comput., pp. 259266
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