Fuzzy PID Based Trajectory Tracking Control of Mobile Robot and its Simulation in Simulink

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1 , htt://x.o.org/ /jca Fuzzy PID Base Trajectory Trackng Control of Moble Robot an ts Smulaton n Smulnk Qng Xu, Jangmng Kan, Shanan Chen an Shengq Yan School of Technology, Bejng Forestry Unversty kanjm@bjfu.eu.cn Abstract The recse control of moble robot s an mortant ssue n robotcs fel. In ths aer, the moton moel of moble robot s establshe by mechansm analyss. Then, a fuzzy PID controller s esgne for trajectory trackng of moble robot. The controller conssts of a PID controller an a fuzzy nference unt wth two nuts an three oututs to tune the arameters of PID controller accorng to the error an error rate. Fnally, the moel of a four-wheel moble robot, fuzzy PID an tratonal PID controller are all smulate n Smulnk. Smulaton exerments are reache n fferent contons. The result shows that the moble robot wth the fuzzy PID controller can track the esre tral by about 3 secons n avance an the overshoot of the system wll ecrease by 40 ercent, comarng to the moble robot wth the tratonal PID controller. The avantages of fuzzy PID controller for trajectory trackng control of moble robot are major n ts raty, stablty, ant-nterference an trackng recson. Keywors: Moble Robot, Trajectory Trackng, Fuzzy PID Control, Smulaton n Smulnk 1. Introucton Robot s one of the most oular fels of automaton technques an artfcal ntellgence technques. It also reresents a new level of manufacturng technology eveloment. Meanwhle, the eveloment of robots asks for hgher stanar of automaton control technology, ntellgent technology, sensor technology an manufacturng technology. From brth to now, robot technology has mae a conserable rogress an eveloment. The eveloment of the robot whch can move n work sace s ncrebly fast an has become a branch of robotcs fel whch s evelong roserously. Accorng to the movng sace range, robots can be ve nto unerwater robots, groun robots, flyng robots an sace robots, etc. Accorng to rvng methos, t can be ve nto wheele moble robots, tracke moble robot, an leg moble robot (nclung humano robots), etc. [1]. Moble robot technology s a combnaton of sensng technology, controllng technology, nformaton rocessng technology, machnng rocessng technology, electronc technology, comuter technology an many other technologes [2]. In the rocess of eveloment of robotcs, controllng has always been an mortant ssue. The erformance of controllers wll be rectly relate to the level of the robot's workng ablty. Control roblems of moble robot can be ve nto two categores: stablzaton control an trackng control. Trackng control s an mortant an ractcal ssue n moble robot moton control. It can be ve nto trajectory trackng control an ath trackng control. In the trajectory trackng control, the esre trajectory requre by moble robot s gven n grah base on tme relaton. Whle Corresonng Author ISSN: IJCA Coyrght c 2014 SERSC

2 n ath trackng control, the esre trajectory s escrbe by geometrc arameters whch can be easly acheve (such as the arc of the ath). Trajectory trackng control s necessary when the robot s aske to arrve at a certan locaton n a certan tme. Path trackng control s arorate when the robot s aske to track the ath gven by geometrc arameters at a certan see [3]. Among several control methos, PID control s the most common one an use wely, also n robot fel. By usng PID control, both stablty control an trackng control can be acheve. But statc error exts n trackng control, an ts erformance s relate to scalng factor K [4]. Wu esgne a PD controller that the velocty s the control objectve to control a guance moble robot [5]; Qjun Chen analyze the stablty an robustness of three commonly use robot trackng algorthm base on PD, an comare ther control erformance [6]. In aton, there are control methos combnng PID control an some other control methos. For examle, Xnxn roose an aatve control metho whch combne the feeforwar an PID feeback, makng use of the roertes of oerator ynamc moel an can be ale to trackng control robot [7]. Whle stuyng the trajectory trackng control for the uncertan robot an usng the combnaton of PD controller an feeforwar control, Dayng esgne a Robust Aatve Controller to ensure stablty of the overall stuaton [8]. Jafarov esgne a PID controller wth a new varable structure for mechancal han system of stablzaton moel wth sturbance of arameters [9]. Moble robot base on PID controller uses the course angle error as the nut of the controller, an outut of the controller s the robot's rvng angle. But n fact, the robot s course angle s also affecte by ts velocty, rotatonal nerta, centre-of-gravty oston an cornerng coeffcent of the front an rear wheel (cause by the wheel whch s not strctly erencular to the groun). An the fference between ameter of rvng wheels wth frcton an the number of changes an other factors whch are ffcult to etermne n the ractcal roa contons make t ffcult to obtan global coornaton of PID controller. The arameters gane from the small turnng angle exerment are narorate when usng n a large turnng angle exerment, an vce versa [10]. At the same tme, ue to varous envronmental factors, settng the sutable arameters of PID controller s extremely har to reach. What s more, t s less ntellgent [11]. Fuzzy control s useful for the control object whch has a comlex mathematc moel, nonlnearty, lag an coulng. Generally fuzzy relaton s fuzzfcate concet of relatonsh n mathematcs [12]. Whle fuzzy controller can well aat to the nonlnear an tme varablty of controlle object an exhbts goo robust roertes, ts stable control accuracy s oor an not elcate enough. It s ffcult to acheve hgh control recson, esecally near the equlbrum ont. Meanwhle, t s lack of ntegral control an har to elmnate the system statc error. To eal wth these shortcomngs, the basc fuzzy controller s often combne wth PID controller n ractce n orer to use ther own character an make the effect more erfect to meet the nees of a varety of nustres. A fuzzy control base PID arameter auto settng controller s eveloe n ths aer. Accorng to the feeback error, arameters of the PID controller such as K, K, K can be tune. The smulaton results show that the controller can reuce the ffculty of PID arameter tunng, mrove the accuracy of trajectory trackng control, an mrove the aatablty, raty an ant-jammng erformance of the controlle system. The avalablty of the metho can be ensure by smulaton an exermental results. 234 Coyrght c 2014 SERSC

3 2. Knematcs Moel of Moble Robots In ths stuy, t s assume that the robot was movng on eal lane, the groun can be consere as regular, the robot s a rg boy that the eformaton of t can be gnore; wheels an the groun can meet the ure rollng contons wthout relatve slng. The oston of the robot on a Cartesan lane can be llustrate by Fgure 1. Pont P s the reference ont of the robot boy, the coornate system of the boy s efne as XvPYv, oston an orentaton of the robot can be efne through [x y θ]t, where (x, y) s the coornate oston of the robot s center n the Cartesan lane, an θ s ts course angle [13]. The matrx whch use to transform rectangular lane coornate system to robot s coornate system s as formula (1). R ( ) cos sn 0 sn cos Fgure 1. Knematcs Moels of Moble Robots As the Fgure 1 shows, γ s the rvng angle, R s the turnng raus, L s the stance between the front an rear lne, v s the rate [14]. Thus, knematcs moels of moble robots can be establshe as formula (2) shown below: x v cos y v sn v tan L 3. Desgn of Fuzzy PID Controller Structure of the Fuzzy self-tunng PID Controller 0 Combnng fuzzy auto tunng PID control an fuzzy control, achevng nonlnear functon between error e an error rate ec an arameters k k k through fuzzy nference ( 2 ) (1) Coyrght c 2014 SERSC 235

4 system, the PID controller can tune ts arameters aatvely when error an error rate change. The rncle of ths control system s shown n Fgure 2. Fgure 2. The Prncle Dagram of the Moble Robot Controller Durng the oeraton of control system, contnuously etectng e an ec, mofy arameters of PID n real tme accorng to the fuzzy control rules, the controlle object can acheve a goo ynamc an statc erformance [15]. Combne wth the control objectves of the moton control system, the esgn of PID arameter auto settng controller s structure s shown n Fgure 3. The nuts of the controller are the evaton between the moble robot s oston an the target oston e an the evaton rate ec. The oututs of the controller are the arameter varables k k k of PID controller at the next tme. Wth usng the PID controller, the velocty v an rvng angular γ of the moble robot can be controlle. Fgure 3. Structure of the Controller Desgne for the System k k k are correcton arameter. The arameters of the PID controller k k k can begotten from the formula (3): k k k k k k k k k Thus, base on the ncremental PID control algorthm, we can acheve the transfer functon of actve settng PID controller by formula (4). (3) ( k k u ( k ) u ( k 1) ( k ) e ( k ) ( k k k )[ e ( k ) e ( k 1)] )[ e ( k ) 2 e ( k 1) e ( k 2 )] ( 4 ) 236 Coyrght c 2014 SERSC

5 3.2. Membersh Desgn of Fuzzy Controller In ths system, nut an outut of the fuzzy controller varables are the same: {NB (negatve bg), NM (negatve mle), NS (negatve small), ZO (zero), PS (ostve small), PM (ostve mle), PB (ostve bg)}. The basc oman of error e s {-180,180}, of error rate ec s {-30,30}, of k s {0,6}, of k s{0,1.8 }, of k s {0,6}. After formulate language varable an oman, we must etermne membersh of the fuzzy language varables. Gaussan membersh functons an trangular membersh functon are commonly use as Membersh functons. Conserng the esgn shoul be smle an meet real-tme requrements, the system aots trangular membersh functon. Fgure 4 shows the membersh functon scatter gram of nut varable e. Fgure 4. Membersh Functon Scatter Gram of nut Varable e 3.3. Desgn of Fuzzy Rules The most mortant art of esgnng a fuzzy controller s to summarze the technology an ractcal oeratng exerences of the engneerng staffs an obtanng fuzzy rule table of the three arameters K, K, K whch are tune searately [16]. e Table 1. Engneerng Staffs an Obtanng Fuzzy Rule Table NB NM NS ZO PS PM PB NB PB/NB/PS PB/NB/NS PM/NM/NB PM/NM/NB PS/NS/NB ZO/ZO/NM ZO/ZO/PS NM PB/NB/PS PB/NB/NS PM/NM/NB PS/NS/NM PS/NS/NM ZO/ZO/NS NZ/ZO/ZO NS PB/NB/ZO PB/NM/NS PS/NS/NM PS/NS/NM ZO/ZO/NS NS/PS/NS NS/PS/ZO ZO PM/NM/ZO PM/NM/NS PS/NS/NS ZO/ZO/NS NS/PS/NS NS/PM/ZO NM/PM/ZO PS PS/NM/ZO PS/NM/ZO ZO/ZO/ZO NS/PS/ZO NS/PS/ZO NM/PM/ZO NM/PB/ZO PM PS/ZO/PB ZO/ZO/PS NS/PS/PS NM/PM/PS NM/PM/PS NM/PM/PS NB/PB/PB PB ZO/ZO/PB ZO/ZO/PM NM/PM/PM NM/PM/PM NM/PM/PS NB/PB/PS PB/PB/PB The format of the fuzzy reasonng s: f the error e s NB, the error change rate ec s NB, then s PB, k s NB, k s PS. k 3.4. Defuzzfcaton What we gane from the fuzzy PID controller s only a fuzzy set. In fact, we must use an exact value to control the controlle objectve. Thus, the fuzzy set nees to be efuzzfe. The metho of efuzzfcaton aote by the system s area centro [17]. ec Coyrght c 2014 SERSC 237

6 4. Smulnk Moel 4.1. Control Objectve Moelng Establsh a moel of the control objectve n Smulnk, nut nclues velocty v an rvng angular, outut nclues the oston of the moble robot (x, y) an the azmuth angle θ. In orer to make t easer to set u the control system, encasulate the moel n the moble robot moule, the structure s shown n Fgure Trajectory Moelng Fgure 5. Smulnk Moel of Moble Robots The nut of the system s a real-tme movng ont; moble robot can reach the track by tracng ths ont. The target trajectory wll be set nto crcle, ellse an straght lne. The sze of the crcles an ellses, oston of the straght lne, see of the trajectory onts can be set arbtrarly. Moel of the trajectory n Smulnk s shown n Fgure 6. Fgure 6. Moel of Target Trajectory n Smulnk By changng values of K1 an K2, fferent trajectory shaes such as ellse an straght lne can be set. In target trackng, the robot shoul kee a stance wth the movng ont, whch s set to 0.01n ths system. An the followng formula (5) s efne as the see evaton: e ( x x ) 2 ( y y ) 2 (5) ( x, y s the coornate of the movng ont n trajectory.) The evaton whch s azmuth error of the robot s efne as formula (6): tan 1 y x y x ( 6 ) 238 Coyrght c 2014 SERSC

7 4.3. PID base Moble Robot Moelng After the varous arts of the mathematcal moel an smulaton moel s set u, ut them together to obtan a Smulnk smulaton moel. The evaton of oston an azmuth shoul be feeback to the nut of PID controller, the controller s outut controls velocty an course angle of robots at next secon. Its structure s shown n Fgure 7. Fgure 7. PID base Smulaton Moel of Moble Robot 4.4. Fuzzy PID base Moble Robot Moelng As for a control system base on fuzzy PID, a fuzzy controller n the front of PID controller. The nuts of fuzzy controller nclues feeback of locaton, azmuth error an error rate, the oututs are the results of arameters K, K, K n real tme tunng. After the arameters are tune, PID controller begns to control the velocty an course angle for the next secon angle of the moble robot. Its structure s shown n Fgure 8. Fgure 8. Fuzzy PID base Smulaton Moel of Moble Robot 5. Analyss of Smulaton an Results n Smulnk 5.1. Dfferent Intal Postons A crcle wth a raus of 2 s set as trackng trajectory. Set the ntal state: the robot s oston s ont (0, 0), course angle s 0 (efne as the 1st conton) an oston s (1, 1), course angle s (efne as the 2n conton). Observe robot s trackng erformance of the two controllers. The trackng trajectory grahs are shown: Coyrght c 2014 SERSC 239

8 (a) PID base (a) PID base Fgure 9. 1st Contons Fgure 10. 2n Contons (b) Fuzzy PID base (b) Fuzzy PID base 5.2. Comare ant-nterference Performance of each Control System For the two control systems, let us kee the same ntal state of the robot (both of them are the 1st conton), an track the crcular trajectory. After 10 secons, artfcally a a jammng sgnal wth a strength of 30, wth of 1 secon n the nut of the robot s course angle, an wth a strength of 3, wth wth 1 secon n the nut of the robot s see. By smulatng robot s trackng erformance of the two controllers wth jammng sgnal n see an course angle, the trackng trajectory grahs are shown: (a) PID base (b) Fuzzy PID base Fgure 11. Ant-jammng of Course Angle 240 Coyrght c 2014 SERSC

9 (a) PID base 5.3. Dfferent Shae of Trackng Trajectory Fgure 12. Ant-jammng of Velocty (b) Fuzzy PID base Shae of the Trackng Trajectory Is Ellse: Change the trackng trajectory from crcle nto ellse. The formula of ellse s x /(1.5 ) y 1. Uner the same conton of the ntal state, observe trajectory of the two systems: (a) PID base (b) Fuzzy PID base Fgure 13. Trackng Trajectores n an Ellse Path Shae of the Trackng Trajectory s a Straght Lne: Change the trackng trajectory from crcle nto straght lne. The formula of ellse s y x 2. Uner the same conton of the ntal state, observe trajectory of the two systems: (a) PID base (b) Fuzzy PID base Fgure 14. Trackng Trajectores n a Straght Lne Path 5.4. Analyss of the Exermental Results ( x Suose that the coornates of the target locus s (x,y), coornates of robot s locaton s, y ), the stance 2 2 x ) ( y ) between two onts s efne as error e. ( x y Defne tme t as abscssa, error e as vertcal axs, raw trackng error curves of the two control system n fferent conton of smulaton. Coyrght c 2014 SERSC 241

10 Draw trackng error curves of the moble robot n fferent ntal state, as shown n Fgure 15. From fgure (a), the control system can track the target trajectory n 7 secons after ang fuzzy controller, whle t takes 10 secons for a PID controller to fully track the target trajectory. From fgure (b), t s clear that through the analyss that after ang fuzzy controller to the control system, overshoot has reuce by 40%. Therefore, from fferent ntal oston an course angle, moble robot controlle by fuzzy PID control can track target trajectory faster wth a smaller overshoot an steay-state error an hgher trackng accuracy. (a) 1st conton Fgure 15. Trackng Error Curves (b) 2n conton Smlarly, wth the nterference n course angle an velocty, trajectory trackng errors of the moble robot can be rawn n Fgure 16. Accorng to the two fgures, t s clear that the ant-nterference erformance s very oor when the moble robot s controlle only by a PID controller, esecally n the effect of jammng sgnals of velocty. At ths tme, the robot s out of control, an the system s no longer stable. On the contrary, moble robot has a strong ablty to suress nterference an can return to the steay trajectory a short tme when usng fuzzy PID controller, even f there are nterference of velocty an course angle. (a) Interference of course angle (b) Interference of velocty Fgure 16. Trackng Error Curves wth Interference (a) Ellse ath (b) Straght lne ath Fgure 17. Trackng Error Curves n a non-crcular Path 242 Coyrght c 2014 SERSC

11 In the exerment of trackng elltcal trajectory an lnear trajectory, raw a trajectory trackng error curve of the moble robot as shown n Fgure 17. It s obvous that for noncrcular trajectory, trackng effect of the two control systems s really fferent. Barely usng PID controller to control a system wll have some trouble n tracng a straght lne trajectory, an wll have a serous error when tracng ellse trajectory. But n the control system whch s usng fuzzy PID controller, moble robot can stll track the non-crcular trajectory even f there are errors. 6. Conclusons From the grous of exerments above, after changng ntal state of moble robot, ang jammng sgnal n velocty an course angle of moble robot artfcally an changng shaes of the target trajectory, effect of each factor can be acheve by comarng each result. The results show that moble robot can track the target trajectory faster an more effectve wth smaller trackng error an hgher trackng accuracy, comarng wth PID control base moble robot. Meanwhle, moble robot whch base on fuzzy PID controller has better stablty, an mroves a lot n ant-nterference ablty an aatablty to the system, whch ncates that fuzzy PID control base moble robot has many aarent merts n velocty control, stablty of system an ant-nterference erformance n trackng trajectory controllng area. Acknowlegements Ths work was suorte by the Funamental Research Funs for the Central Unverstes (Grant No. BLYX200905) an Natonal Natural Scence Founaton of Chna (Grant No ). The authors also want to thank Yanong Zhao an Janhu Ln for the nterestng scusson on fuzzy control system. References [1] G. Xu an M. Tan, Develoment Status an Tren of Moble Robot, Robot Technque an Alcaton, vol. 3, no. 5, (2001), [2] C. C. e Wt an R. Roskam, Path followng of a 2-DOF wheele moble robot uner ath an nut torque constrants, IEEE Transactons on Robotcs an Automaton, vol. 2, no. 1, (1991), [3] K. Watanabe, Control of an omnrectonal moble robot, IEEE Transactons on Knowlege-Base Intellgent Electronc Systems, vol. 1, no. 3, (1998), [4] A. A. Pervozvansk an L. B. Freovch, Robust stablzaton of robotc manulators by PID controllers, Dynamcs an Control, vol. 9, no. 3, (1999), [5] S. F. Wu, J. S. Me an P. Y. Nu, Path Gue an Control of a Gue Wheele Moble robot, Control Engneerong, vol. 9, no. 3, (2001), [6] S. Thongcha, S. Suksakulcha an D. M. Wlkes, Sonar behavor-base fuzzy control for a moble robot, IEEE Transactons on Systems, Man, an Cybernetcs, vol. 5, no. 2, (2000), [7] X. Xn, H. Ye an C. Feng, The PID Aatve Control of Oerator, Robots, vol. 12, no. 2, (1990), [8] D. Yng an S. Songshu, Global Stablty of the PD + Feeforwar Robot Robust Aatve control, ACTA Automatc Snca, vol. 28, no. 1, (2002), [9] E. M. Jafarov, M. N. A. Parlakc an Y. Istefanoulos, A new varable structure PID-controller esgn for robot manulators, IEEE Transactons on Control Systems Technology, vol. 13, no. 1, (2005), [10] T. Takeuch, Y. Naga an N. Enomoto, Fuzzy control of a moble robot for obstacle avoance, Informaton Scences, vol. 45, no. 2, (1988), [11] S. Bentalba, A. El Hajjaj an A. Rach, Fuzzy control of a moble robot: a new aroach, IEEE Transactons on Control Alcatons, vol. 1, no. 3, (1997), [12] C. D. Jung, J. Mook, Kang an C. H. Park, Stuy of Engne Ol Relacement Tme Estmate Metho usng Fuzzy an Neural Network Algorthm n Ubqutous Envronment, Internatonal Journal of Control an Automaton (IJCA), vol. 6,no. 3, (2013), Coyrght c 2014 SERSC 243

12 [13] M N. U. Laskar, H. H. Vet an T. C. Chung, EKF an K-means to Generate Otmze Paths of a Moble Robot, Internatonal Journal of Control an Automaton, vol. 6, no. 2, (2013), [14] T. H. S. L an S. J. Chang, Autonomous fuzzy arkng control of a car-lke moble robot, IEEE Transactons on Systems, Man an Cybernetcs, vol. 33, no. 4, (2003), [15] Q. Gong an X. Xang, Desgn of Self-tunng Fuzzy PID Controller wth Smulnk Smulaton, Chongqng Electrc Power College Journals, vol. 17, no. 2, (2012), [16] B. Pan an Z. Jang, Base on the Aatve Fuzzy PID Servo Control of Moble Robot Research, n Proceengs of Chna Intellgent Automaton Conference, (2009). [17] X. L, G. Lu, S. L, Y. Zhou an Q. L, The Desgn of Wheele Robot Controller Base on Fuzzy PID Agorthm, Small & Secal Electrcal Machnes, vol. 11, no. 5, (2011), Authors Qng Xu, s an unergrauate major n Automaton at Bejng Forestry Unversty, P.R. Chna. Currently, he s ursung the bachelor egree at Bejng Forestry Unversty. Hs research nterests nclue boelectrcty an Sngle ch mcrocomuter system. Hs e-mal aress s bjfuxq@foxmal.com. Jangmng Kan, corresonng author of ths aer, receve n PhD egree n forestry engneerng from Bejng Forestry Unversty, P.R. Chna n Currently, he s an assocate rofessor n Bejng Forestry Unversty. Hs research nterests nclue comuter vson an ntellgent control. Hs e-mal aress s kanjm@bjfu.eu.cn. Shan an Chen, Shan an Chen s an unergrauate stuent n Automaton at Bejng Forestry Unversty, P.R. Chna. Hs research nterests nclue Smarthome an Automatc Control. Currently, he s ursung the bachelor egree at Bejng Forestry Unversty. Hs e- mal aress s csabjfu@163.com. Shengq Yan, s an unergrauate major n Automaton at Bejng Forestry Unversty, P.R. Chna. Currently, she s ursung the bachelor egree at Bejng Forestry Unversty. Her research nterests nclue mage entfcaton an robotcs. Her e-mal aress s ysq @gmal.com. 244 Coyrght c 2014 SERSC