A Fuzzy Control Strategy and Optimization for Four Wheel Steering System

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1 A Fuzzy Control Strtegy nd Optimiztion for Four Wheel Steering System Jie Zhng Yunqing Zhng Liping Chen Jingzhou Yng b Center for Computer-Aided Design Huzhong University of Science & Technology Wuhn Hubei P.R. Chin jiehust@gmil.com b Center for Computer-Aided Design University of Iow Iow City IA 546 USA jyng@engineering.uiow.edu Abstrct This pper presents fuzzy logic control strtegy on four-wheel steering(4ws) vehicle bsed on multi-body vehicle dynmic model. The multi-body vehicle dynmic model bsed on ADAMS cn ccurtely predict the dynmics performnce of the vehicle. Fuzzy logic is pplied to trck the yw velocity of the two degrees of freedom idel model through the co-simultion of ADAMS nd Mtlb Fuzzy control unit with the optimized membership function. The fuzzy control prmeters re optimized nd nlyzed by combined optimiztion lgorithm (Genetic Algorithm (GA) nd Nonliner Progrmg Qudrtic Line serch (NLPQL) method) combined with response surfce model (RSM). Single lne chnge test is chosen to vlidte the fuzzy control logic strtegy. Simultion result shows tht four-wheel steering vehicle with the fuzzy control logic strtegy cn improve vehicle hndling stbility gretly compring with trditionl front wheel steering. Index Terms Four-wheel steering fuzzy control multi-body genetic lgorithm I. INTRODUCTION ECENT reserch nd developments suggest tht Rfour-wheel steering systems cn effectively improve trnsient response of vehicles in corning. Mny ctul nd theoreticl 4WS controllers re such tht in high speed steering ngle of rer nd front wheels re in the sme direction to improve stbility of vehicle nd stisfy pssenger relxtion. But in low speed especilly in prking geometry the steering ngle of front nd rer wheels re chosen in the opposite direction to improve mneuverbility of vehicle. During the lst 0 yers mny different control methods hve been pplied on 4WS system. Nlecz[10-1] used proportionl controller in his nonliner 3DOF model while Furukw[13] used proportionl nd compenstor controllers in his control strtegy. Whitehed[14] investigted controller by setting the chnge rte of sideslip ngle nd sideslip ngle to zero which ultimtely resulted in n uncoupled system for the sideslip ngle. Ngi[8 15] used full stte compenstor with the use of error of stte vribles. To be extended Song[7] introduced dul steering schemes s useful method for controlling both yw rte nd lterl velocity stte vribles. H-MLv[3] studied multi-objective H optimiztion control bsed on yw velocity feedbck; Pul I. Ro nd Hoyong Kim[5] pplied sliding mode control in 4WS vehicle control indicting tht hndling performnce in high speed ws improved. Among them only Will[ 6] nd Szoslnd[4] used fuzzy logic method to investigte the performnce on controlling the wheel ngle. In this pper we choose fuzzy logic method to control the sideslip ngle nd yw velocity ccording DOF idel model. Fuzzy logic control is proved to be n efficient wy to implement engineering heuristics into control solution. The min dvntges of using fuzzy logic is to reduce the very detiled models of controller. But fuzzy logic controller is usully built bsed on the opertor s knowledge resulting in tht controllers designed by different experts my be vrious. To solve this problem optimiztion strtegy is introduced in this pper nd fuzzy logic controller is designed form the view of optimiztion which ensure the controller optiml. The membership function of the controller is optimized by integrted lgorithm (Genetic Algorithm (GA) nd Nonliner Progrmg Qudrtic Line serch (NLPQL) method) combined with RSM to mintin the controller ccurte nd stble. This pper presents multi-body vehicle dynmic model nd dynmic control strtegy by compring the multi-body model with the idel model. The control strtegy considers both yw velocity nd sideslip ngle β objectives. The method is optimized nd nlyzed by combined optimiztion lgorithm (Genetic Algorithm (GA) nd Nonliner Progrmg Qudrtic Line serch (NLPQL) method) combined with RSM. Under extreme motion stte the simultion results indicte tht the proposed method cn gin dynmic stbility nd improve the ccurcy of fuzzy control. The pper consists of seven prts. The second section presents the multi-body vehicle dynmics model nd DOF idel reference model. In the third prt we integrted the dynmics model nd controller for 4WS co-simultion using fuzzy logic control method. Fuzzy control strtegy is the next prt. In the fifth prt we optimize the membership function of fuzzy controller by combined lgorithm GA nd NLPQL. Finlly the simultion result nd conclusion re presented. II. VEHICLE DYNAMICS MODEL AND CONTROL OBJECTIVES FOR 4WS A. Liner vehicle model for 4WS Becuse 4WS system syntheticlly tkes both trnsversl nd yw dynmic problem into ccount we consider the multi-body dynmic model s whole vehicle model nd DOF model (show in Fig. 1) s n idel reference model. The

2 vehicle is four-wheel steering system. Fig. 1 A DOF idel vehicle model The dynmic equtions for the idel DOF reference model re formulted s: mv(β & + ) =Y 1 +Y (1) I & = Y1 - by Lterl force on wheels cn be expressed s: Y 1 = -k1β 1 =-k 1(β + - δ 1) v () b Y = -k β =-k (β - - δ ) where velocity 1 m v denotes the mss of vehicle v is vehicle forwrd represents the yw velocity β is side-slip ngle K1 nd K represent the front nd rer wheels cornering stiffness respectively. δ 1 nd δ represent front nd rer wheel steering ngle respectively. Y1 nd Y represent the lterl force of front nd rer wheel. I is moment of inerti nd b is length of front nd rer xle to CG.. Then Considering the vehicle velocity to be time invrint the dynmic equtions cn be described in stte spce form: X& = AX + BU Y = CX + DU k+k 1 bk-k 1 k1 k - -1 Where mv mv mv mv A= B= bk-k 1 k+bk 1 k 1 bk - - Iv Iv I I 1 0 C= 0 0 T T D= X=(β & & ) & U=(δ 1 δ ) The prmeters of the idel DOF vehicle re shown in Tble 1. Tble 1 Idel vehicle model prmeters m( kg) 1400 I (Kgm^) 1993 (m) b(m) K 1 (kn/rd) 5.48 K ( kn/rd) B. Multi-body vehicle dynmic model for 4WS simultion Multi-body dynmic simultion is used to successfully (3) simulte wide vriety of vehicles nd predict the sfety mobility stbility nd operting lods of the complete system. The theoreticl bsis of multi-body vehicle dynmic model for 4WS simultion is multi-body dynmics nd the kinetic eqution built with Lgrnge Multiplier Method is presented s: d T T T T T T ( ) ( ) +Φ qρ+ θqμ = Q (4) dt q& q Integrity constrint equtions: Φ ( qt ) = 0 Nonholonomic constrint equtions: θ ( qqt & ) = 0 Where T represents kinetic energy of the system q is generlized coordinte vector Q denotes generlized force vector ρ nd μ represent the vector of Lgrnge multipliers corresponding the Integrity constrints nd Nonholonomic constrins. The multi-body vehicle dynmic model ws built in ADAMS/CAR environment. This model includes seven subsystems-front suspension model rer suspension model brke system powertrin system steering system tire nd bodywork model s shown in Fig.. During the building of the multi-body model we hve considered the joint constrints nd the force element such s springs dmpers bushing nd so on. We hve lso considered the nonlinerity of tire nd the flexibility of certin prts which ccurtely reflects the prcticl vehicle system. Fig. Assemble vehicle model C. Tire model In most time 4WS hs high nonlinerity nd we should dopt nonliner tire model. Thus it introduces Pcejk's Mgic Formul Model [9] which hs high precision for longitudinl force of wheel nd side-force nd lso hs better confidence level out of rnge of limit vlue. It cn be expressed in the following form. Y1 = y+ Sv y = G sin( F rctn( Ex H ( Ex rctn( Ex)))) (5) x = X1 + Sh Where Y1(x) represents lterl-force opposite rotry moment or longitudinl force X1 is sideslip ngle ( β ) or wheel slip rtio S. The coefficients E F G H re detered

3 r v δ y u β V δ1 by vehicle velocity nd drive sitution nd Sv S the horizontl nd verticl drift. h denotes D. Control objectives for 4WS At the present time there re two min control objectives in the study on 4WS. The sideslip ngle control strtegy reduces the lterl motion nd trnsporttion of vehicle while it improves hndling mneuverbility nd reduces the dely of response of the vehicle. A lrge number of reserchers hve used this strtegy to control the 4WS vehicle such s Nikzd [17] Nlecz [10-1] Furukw [13] Cho [16] Whitehed [14] nd Xi [18]. The yw velocity control strtegy imizes the rottionl motion of vehicle nd leds the vehicle to lterl side trcking the desired trjectory. Reserchers like Nikzd [17] Song [7] nd Furukw [13] used this strtegy in their ppers. Also some reserchers re performed under reference models such s Nikzd [17] Song [7] Ngi [8 15] nd Plkovics [19]. In this pper fuzzy logic controller is used to trck the yw velocity of the DOF idel model which is detiled in the lter section. Fig. 4 -input nd 1-output fuzzy controller B. The choice of fuzzy controller prmeters Input nd output vribles cn be divided into 5 levels. They re defined s negtive big (NB) negtive medin (NM) medin zero (ZE) positive medin (PM) nd positive big (PB). In order to improve defuzzifiction speed Trimf is chosen s membership function nd the weight is equl to one. Figs. 5-7 show the membership functions of input nd output vribles the rnge of yw velocity error nd yw velocity error chnge rte re [-6 6] nd [-10 10] respectively. The rnge of output vrible is normlized within 0 to 1. III. INTEGRATED THE DYNAMICS MODEL AND CONTROLLER FOR 4WS CO-SIMULATION The simultion system is estblished by the combintion of ADAMS nd MATLAB. The structure of co-simultion is showed in Fig. 3. The control objectives hve been mentioned in the former prt nd we intend to mke sideslip ngle closest to zero nd yw velocity to trck the DOF model. The ECU sends control instructions to the rer wheel steering system bsed on yw velocity error to djust the rer wheel ngle. In the co-simultion fuzzy logic method is pplied on the control strtegy. Fig. 5 membership function of yw velocity error Fig. 6 membership function of yw velocity error chnge rte Fig. 3 Structure grph of Fuzzy control IV. FUZZY CONTROL STRATEGY DESIGN A. Fuzzy controller design The fuzzy controller present in this pper chooses yw velocity error (E) nd yw velocity error chnge rte (EC) s input vribles which cn reflect the dynmic chrcteristics of output vrible strictly. The input nd output chnnels re presented in Fig. 4. Fig. 7 membership function of rer steering ngle C. Estblishing fuzzy controlling rules We dopt verge grvity center method for defuzzifiction. When error nd error chnge rte re both negtive big the error will hve n incresing trend nd it requires the output is positive big to trck the setting vlue. Thus the rer steering wheel ngle cn be controlled intelligently ccording to the fuzzy control rules estblished bsed on control strtegy showed in Tble. The response surfce of rer steering ngle is showed in Figs. 8.

4 EC U Tble Fuzzy control rulers E PB PM ZE NM NB PB NB NB NB NS ZE PM NB NB NM ZE PS ZE NB NM ZE PM PB NM NS ZE PM PB PB NB ZE PS PB PB PB We dopt Ltin Hypercube method with uniform spce smpling nd rndom combintion. Fig. 10 shows the normlized Preto grph of ll fctors. We choose the first 10 fctors which hve gret contribution to the result nd optimize these fctors. The verticl xis represents fuzzy control vribles A represents the center of Trimf B represents width of Trimf nd numericl vlue fter the lphbet represent 5 levels of vribles. Fig. 8 Response surfce of Yw velocity V. OPTIMIZATION OF THE FUZZY CONTROLLER For the reson tht fuzzy control rules re detered by expert s experience it is difficult to optiml fuzzy control rules. However we cn consider to build fuzzy controller from the view of optimiztion. Fuzzy control rules detered by the control strtegy is relible nd the objective is to optimize membership functions. Assug tht membership function of ech vrible is centrosymmetric we cn identify ech membership function with only two prmeters-position nd width. Thus we need seven prmeters to identify vrible's universe nd we hve totl 3 vribles so we need 1 fctors to detere the membership of the whole fuzzy control. This number is so lrge tht we use design of experiment (DOE) nlysis to select the min fctors then optimize these fctors. The flow chrt of the optimiztion is shown in Fig. 9. we choose single lne chnge test tht to study the function of dynmicl prmeters nd hndling stbility. Fig. 9 Co-optimiztion flow chrt of GA nd RSM+NLPQL Fig. 10 Preto grph nlyzed by DOE There re four objectives f or the optimiztion nd they re: (1)sideslip ngle; ()yw velocity; (3) the time lg between steering wheel ngle nd yw velocity; nd (4)the time lg between yw velocity nd lterl ccelertion. It cn be formulted s follows. To find: Design vribles to imize F t = Ψ β where Ψ= λ 1 + λ + λ 3 + λ 4 B B B B β β β T T + λ5 + λ6 + λ 7 + λ8 B β B B β T B T subject to: 0 Re rb YwRteA YwErrorA 00 RerB Re ra 0.50 YwRteB 10 0 YwErrorB 6 10 YwRteA 0 0 YwErrorB4 60 YwRteB4 10 Where B B β B B B β B β B T r B T re the imum vlues of β β β T T respectively. λ1... λ 8 re the corresponding we ights. T is the time lg between steering wheel ngle nd yw velocity. T is the time lg between yw velocity nd lterl ccelertion. β is the men vlues of sideslip ngle. β nd β re the imum nd imum vlues of sideslip

5 ngle respectively. is the men vlue of yw velocity. nd re the imum nd imum vlues of the yw velocities. To use genetic lgorithm the initil point should be in fesible region first. Then we stochsticly choose some code rry from fesible region s the first code rry of the beginning of evolution nd compute the objective function of ech solution. In ddition some codes re selected rndomly s code smple of pre-production nd produce the next genertion code rry through crossover nd vrince. Repet the process bove until obtining the optiml solution of the lst genertion the solution is the finl solution of genetic lgorithm. This lgorithm cn prevent to trp into locl optimiztion in the process of serching optiml point. Progrmg Qudrtic Line serch is clled numericl optimiztion nd mthemticl progrmg estblished by qudrtic objective function nd liner constrined function. It cn serch the optiml solution in the continuous design spce with single-pek. The combintion of two optimiztion methods cn prevent to trp in locl optimiztion nd improve optimiztion speed using RSM pproximtion model. Fig show the membership function fter optimiztion. nd results using single lne chnge experiment. The vehicle velocity is 100km/h nd the steering wheel ngle is 0 degree. Figs show the simultion result of 4WS vehicle before optimiztion fter optimiztion nd Front Wheel Steering with the sme condition. The results show tht with fuzzy control yw velocity of 4WS vehicle cn trck the idel reference model ccurtely while it is lso reduces compring with tht of front wheel steering. In ddition the sideslip ngle of 4WS vehicle cn decrese lot compring with the Front wheel steering vehicle nd the performnce of fuzzy controller fter optimiztion is better thn tht before optimiztion. Wht is more the lg phse between steering wheel ngle nd yw velocity nd tht between yw velocity nd lterl ccelertion reduces fter optimiztion (shown in Tble 3). Tble 3 Lg phse compring Lg phse Between steering wheel ngle nd yw velocity Between yw velocity nd lterl ccelertion Before optimiztion After optimiztion 0.07s 0.065s 0.155s s Fig. 11 Membership function of yw velocity error fter optimiztion Fig. 14 Comprison of rer steering ngle (Dsh nd Blue: before optimiztion; Dotdsh nd Mgent: fter optimztion) Fig. 1 Membership function of yw velocity error chnge rte fter optimiztion Fig. 13 Membership function of rer steering ngle fter optimiztion Fig. 15 Comprison of sideslip ngle (Solid nd Red: front wheel steering; Dsh nd blue: four wheel steering before optimiztion; Dotdsh nd Mgent: four wheel steering fter optimiztion) VI. SIMULATION RESULTS AND DISCUSSION Bsed on the bove formultion we illustrte the procedure

6 Fig. 16 Comprison of yw velocity (Solid nd Red: front wheel steering; Dsh nd blue: four wheel steering before optimiztion; Dotdsh nd Mgent: four wheel steering fter optimiztion; Dot nd Blck: DOF reference model) Fig. 17 Comprison of lterl ccelertion (Solid nd Red: front wheel steering; Dsh nd blue: four wheel steering before optimiztion; Dotdsh nd Mgent: four wheel steering fter optimiztion) VII. CONCLUSIONS The results show tht 4WS system cn enhnce the vehicle stbility nd hndling. We lso see tht fuzzy logic mintin good chrcter in the control of sideslip ngle nd yw velocity. For fuzzy logicl control nd high nonliner model like multi-body model this pper dopts combined optimiztion method of Genetic Algorithm globl serch RSM pproximtion model nd Nonliner Progrmg Qudrtic Line serch. The results show tht using the optimiztion method of co-simultion of multi-body dynmicl model nd fuzzy control model combined with the experimentl method for the 4WS system cn generte desired solutions. The results lso show tht fuzzy control becomes more ccurte fter optimiztion nd the 4WS cn effectively provide more stble responses. [4] Szoslnd A.. Fuzzy Logic Approch to Four Wheel Steering of Motor Vehicle. Int. J. of Vehicle Design (4): 350~359. [5] Pul I. Ro nd Hoyong Kim. Improvement of high speed 4-WS vehicle hndling performnce by sliding mode control. In: Proceedings of the Americn Control Conference. 1994: 1974~1978. [6] Anthony B. Will nd Mrcelo C.M. Teixeir. Four Wheel Steering Control System Design Using Fuzzy Models. In: Proceedings of the 1997 IEEE Interntionl Conference on Control Applictions. 1997: 73~78. [7] G.J. Song Y.S. Yoon. Feedbck Control of Four Wheel Steering Using Time Dely Control. Interntionl Journl of Vehicle Design (3): 8~98. [8] M. Ngi Y. Hirno nd S. Ymnk. Integrted Control of Active Rer Wheel Steering nd Direct Yw Moment Control. Vehicle System Dynmics : 357~370. [9] Pcejk H B nd Besselink. Mgic Formul Tyre Model with Trnsient Properties. Vehicle System Dynmics 1997 (7): 34~49. [10] A.G. Nlecz A.C. Bindemnn. Hndling Properties of Four-Wheel Steering Vehicles. SAE Pper No [11] A.G. Nlecz A.C. Bindemnn. Investigtion into the stbility of four wheel steering vehicles. Interntionl Journl of Vehicle Design (): 159~178. [1] A.G. Nlecz A.C. Bindemnn. Anlysis of the Dynmic Response of Four Wheel Steering Vehicles t High Speed. Interntionl Journl of Vehicle Design (): 179~0. [13] Y. Furukw. A Review of Four-Wheel Steering Studies from the Viewpoint of Vehicle Dynmics nd Control. Vehicle System Dynmics : 151~186 [14] J.C. Whitehed. Four Wheel Steering: Mneuverbility nd High Speed Stbiliztion. SAE Pper No [15] M. Ngi M. Ohki. Theoreticl Study on Active Four-Wheel-Steering System by Virtul Vehicle Model Following Control. Interntionl Journl of Vehicle Design (1): 16~33. [16] Y.H. Cho J. Kim. Design of Optiml Four-Wheel Steering System. Vehicle System Dynmics : 661~68. [17] V. Nikzd S. M. Nrghi. Feedbck Optiml Control of A 4WS Vehicle Using A Model Reference Trcker. Proceeding of ISME. 001: 477~484. [18] X. Xi E.H. Lw. Nonliner Anlysis of Closed Loop Driver Automobile Performnce with Four Wheel Steering Control. SAE Pper No [19] L. Plkovics. Effect of Controller Prmeters on the Steerbility of the Four Wheel Steered Cr. Vehicle System Dynmics 199 1: 109~18. REFERENCES [1] Sno S Furukw Y Shlrlshi S. Four Wheel Steering System with Rer Wheel Steer Angle Controlled s Function of Steering Wheel Angle. SAE Pper No [] Anthony B. Will. Intelligent Vehicle Brking nd Steering Control Systems. Mster s Thesis Purdue University [3] Lv H MChen NLi P.Multi-objective H optiml control for four-wheel steering vehicle bsed on yw rte trcking I Mech. E. Prt D. Journl of Automobile Engineering (D10): 1117~114.