Yaw-rate Control for Electric Vehicle with Active Front/Rear Steering and Driving/Braking Force Distribution of Rear Wheels

Transcription

1 The th IEEE International Workshop on Advanced Motion Control March -4,, Nagaoka, Japan Yaw-rate Control for Electric Vehicle with Active Front/Rear Steering and Driving/Braking Force Distribution of Rear Wheels Naoki Ando and Hiroshi Fujimoto Yokohama National University 79-5, Tokiwadai, Hodogaya-ku, Yokohama, 4 85 Japan Phone: , ando@hfl.dnj.ynu.ac.jp, hfuji@ynu.ac.jp Abstract Direct yaw-moment control (DYC) is an effective method to make vehicle motion stable. In the DYC of vehicles having in-wheel motors and both active steering system and active steering system, the control input generally has redundancy. The input variables can not be decided uniquely to generate the same yaw-moment. The equalization of workload for each wheel by the longitudinal force and lateral force distribution enhances vehicle cornering performance. Theore, we propose the longitudinal force and lateral force distribution method based on least squares solution of equations of longitudinal, lateral and yawing motion in this paper. Moreover, we propose a method of lateral force control and a method of yaw moment compensation to attenuate tracking error in this controller. we show that the equalization of workload for each wheel and quick yawrate response are achieved in the proposed methods. Simulations and experiments are carried out to confirm effectiveness of the proposed methods. I. INTRODUCTION A characteristic of electric vehicles (EVs) is that they are driven by motors. Because of this, from the viewpoint of vehicle stability motion control, the EVs have advantages as follows []. Motor torque is generated accurately and its response is quick. Actual motor torque can be calculated accurately with the detected current values. As motors are small and light-weight, they can be attached in each wheel and driven independently. Vehicle stability motion control techniques such as the direct yaw moment control (DYC) and anear wheel steering system (4WS) have been researched and developed. As regards vehicles having four in-wheel motors which can be driven independently or vehicles with active steering and active steering, control input generally has redundancy. This means that input variables can not be decided uniquely to control each longitudinal, lateral, and yawing motion. Theore, distribution methods have been proposed in [] [6]. In [], the least-square method of tire workload which is defined as the rate of resultant force in friction circle of each wheel was proposed. Moreover, the minimax optimization method of tire workload was proposed in [3], and the equalization method of tire workload based on sequential quadratic programing and steepest gradient algorithm was proposed in [4]. Additionally, distribution method without considering tire workload was discussed in [5] and [6]. In [6], yaw stability control on a wheel-drive based 4WD vehicle using torque-biasing devices was discussed. The problem of these methods was the lateral force control method based on the non-linear tire model which caused modeling error. Authors group owns two experimental vehicles. One is the four wheel independent drive vehicle named FPEV3 4WD COMS LONG. The base vehicle is the COMS LONG which is the wheel independent drive WD vehicle produced by TOYOTA Auto Body Co., Ltd. The other is the wheel independent drive vehicle with active steering anear steering named FPEV-Kanon which was produced by authors laboratory. Additionally, we have proposed the driving/braking force distribution method applied to the FPEV3 4WD COMS LONG and the driving/braking force and / wheel steering angle distribution method applied to the FPEV-Kanon in [7]. In this paper, we show longitudinal force and lateral force distribution method applied to the FPEV-Kanon based on least squares solution of equations of motion for achieving the equalization of the workload for each wheel. Additionally, we show a lateral force control method, and a method of yaw moment compensation while tracking error exists in the lateral force controller. Simulations and experiments are carried out to confirm effectiveness of the proposed methods. II. VEHICLE MODELING In this section, a general modeling of vehicles including two EVs of authors laboratory is developed. Theore, EVs having four in-wheel motors which can be driven independently and with active steering and active steering are modeled. On the assumption that each anear wheel steering angle defined as δ f and δ r is small enough to disregard, equations of each longitudinal, lateral and yawing motion of these vehicles as shown in Fig. are written as //$6. IEEE 76

2 follows[8], F xfl + F xfr + F xrl + F xrr = F x, () F yfl + F yfr + F yrl + F yrr = F y, () d f F xfl + d f F xfr F xrl + F xrr = N z, (3) l f (F yfl + F yfr ) l r (F yrl + F yrr ) = N t, (4) N z + N t = M z, (5) where longitudinal force of the vehicle is defined as F x, longitudinal force of each wheel as F xfl, F xfr, F xrl and F xrr, lateral force of the vehicle as F y, lateral force of each wheel as F yfl, F yfr, F yrl and F yrr, yaw moment generated by longitudinal force of each wheel as N z, yaw moment generated by lateral force of each wheel as N t, yaw moment of the vehicle as M z, tread base of anear axle as d f and, and distance from the center of gravity to anear axle tread as l f and l r. Then vertical load of each wheel becomes as follows[8], F zfl = l r l Mg ρ f a y M h g a x M h g d f l, (6) F zfr = l r l Mg + ρ f a y M h g a x M h g d f l, (7) F zrl = l f l Mg ρ ra y M h g + a x M h g l, (8) F zrr = l f l Mg + ρ ra y M h g + a x M h g l, (9) where vertical load of each wheel are defined as F zfl, F zfr, F zrl and F zrr, vehicle weight as M, wheel base as l, roll stiffness distribution as ρ f and ρ r, the height of the center of gravity as h g, and acceleration of each longitudinal direction and lateral direction as a x and a y. Moreover the relation between longitudinal force F x, lateral force F y and vertical load F z has to satisfy the following equation in any case[8]. Fx + Fy μf z, () where a coefficient of friction is defined as μ. That is, magnitude of force generated between tire anoad in all directions can not exceed a product of the coefficient of friction μ and vertical load F z, and then this vector has to be inside of a circle μf z in radius. This circle is called friction circle, which is shown in Fig.. Moreover, if each tire s side slip angle is small enough to disregard, vehicles can be modeled as the linear two-wheel vehicle[8]. Provided this, the relation between lateral force of each wheel, cornering force Y f and Y r, tire side slip angle α f and α r, vehicle side slip angle β, yaw-rateγ and steering angle of anear wheel δ f and δ r are approximated as follows[8], F yfl F yfr F yf Y f = C f α f = C f ( β+ l f V γ δ f F yrl F yrr F yr Y r = C r α r = C r ( β l r V γ δ r ), () ), () d f F xfl δ f F x F xfr δ f F yfl F yfr l f F y l l δ r M z r Fxrl δ r F xrr F yrl F yrr Fig.. Vehicle model. μf z F y Fig.. F x δ Friction circle. Fig. 3. FPEV-Kanon. Fig. 4. In-wheel motor. where cornering stiffness are defined as C f and C r. III. EXPERIMENTAL VEHICLE The main specifications of the experimental electric vehicle FPEV-Kanon produced by authors laboratory are as follows. Outer rotor type in-wheel motors are installed. Electric power steering systems are adopted. Li-ion batteries are equipped. Multi-sensing Hub Unit Bearings are installed. Outer rotor type in-wheel motors made by Toyo Electric Mfg. Co., Ltd. are installed in each wheel. Their maximum torque is 34 Nm, and their maximum power is.7 kw. Moreover active steering and active steering systems are available. Each of these active steering is ized by the electric power steering (EPS) system which is composed of the 5 W DC motor made by maxon motor AG. Additionally, Li-ion batteries made by Litcel Co., Ltd. are equipped for driving the in-wheel motors. Moreover Multi-sensing Hub Unit Bearings (MSHubs) developed by NSK Ltd. are attached to each wheel. This hub unit bearing has the function of sensing the wheel s lateral force. Fig.3 shows this EV, and Fig.4 shows this motor. IV. CONTROL SYSTEM DESIGN A. Yaw Moment Observer (YMO)[9] From (3) (5), yawing dynamics is formulated as follow, I γ = N z + N t + N d, (3) where vehicle yaw inertia moment is defined as I and disturbance yaw moment as N d. Then, by composing a disturbance observer shown in Fig.5, this observer compensates the lumped disturbance N td := N t + N d by the control yaw moment N z which is generated by the torque difference between left and right motors and nominalizes the system as follow, γ(s) I n s N in(s). (4) 77

3 N in N t N d + N z + + Is + I n s ˆN td Fig. 5. YMO. ωc s+ωc γ Block diagram of γ + Fig. 6. Ins + Nin + Kp + ˆNd Fx Fy Mz ωc s+ωc Tire Force Distribution Controller ˆFzij Vertical Load Estimator ˆMz Yaw- Moment Calculator + i =f, r,j =l, r DFO: Driving Force Observer δf δr Vehicle Trl Plant Trr ˆFxrj DFO Ins ωrj γ ax ay Fyij Block diagram of yaw-rate control. F x F y M z Tire Force Distribution average Fyf + τfcfns F yr + τrcrns Fxrl r Fxrr r ˆFyf average ˆFyr δ f γ δr ay Trl Vehicle Fyfl,Fyfr Plant Trr Fyrl,Fyrr Fig. 7. Block diagram of tire force distribution controller. This specific disturbance observer is called yaw moment observer (YMO)[9]. B. Yaw-rate Control Method for Electric Vehicle with Driving/Braking Force Distribution In this section, a yaw-rate control method with driving/braking force distribution of wheels is explained. This yaw-rate controller is composed of a feedback (FB) controller, a feedforword (FF) controller and the YMO mentioned in IV-A. Moreover sum of FB and FF controller outputs generates N in. By the disturbance yaw moment N dt compensation, control yaw moment N z is generated. Although this N z is used as input of both vehicle and YMO in IV-A, input of YMO to estimate disturbance yaw moment N dt is substituted for estimated control yaw moment ˆNz which is calculated with (3) and longitudinal force estimated by Driving Force Observer (DFO)[]. It is regurded to avoid the divergence of estimated disturbance yaw moment because of saturated torque of an in-wheel motor. As a result, an anti-windup controller is composed for this reason. In the case of a wheel drive vehicle, that is, longitudinal force of each wheel F xfl and F xfr can not be generated by in-wheel motors, they are substituted for F xfl = F xfr =. Theore, longitudinal force of each wheel can be solved from () and (3) as follow, ] [ [ Fxrl F xrr = ][ Fx N z ]. (5) In order to generate distributed force, torque erences are given the in-wheel motors as follows, T = rf x. (6) C. Yaw-rate Control Method for Electric Vehicle with Active Front/Rear Steering and Driving/Braking Force Distribution of Rear Wheels[7] In this section, the yaw-rate control method with active anear steering and driving/braking force distribution of wheels is explained. Yaw-rate controller is composed the same as IV-B except that N z is regarded as vehicle yaw moment M z, and ˆN dt is regarded as disturbance yaw moment N d. Furthermore, input of YMO to estimate disturbance yaw moment ˆNd is similarly substituted for estimated vehicle yaw moment ˆMz which is calculated with (3) and longitudinal force estimated with DFO, (4) and the value of the lateral force sensors, and (5) for the same reason. Block diagram of yaw-rate control is shown in Fig. 6. In the case of a wheel drive vehicle and that the vehicle can be regarded as linear two-wheel vehicle which satisfies () and (), () (5) can be approximated as follows, l f l r dr F yf F yr F xrl F xrr = F x F y M z, (7) where the left-hand side coefficient matrix is defined as A, the vector of lateral force and longitudinal force of wheels as x = [F yf F yr F xrl F xrr ] T, and the right-hand vector as b =[F x F y M z ] T. The workload of each wheel which is the rate of resultant force in friction circle is defined as follow, η fl = η rl = F xfl +F yfl F µmaxf yf zfl F xrl +F yrl µmaxf zrl µmaxf zfl,η fr = F xrl +F yr µmaxf,η zrl rr= F xfr +F yfr F µmaxf yf zfr F xrr +F yrr µmaxfzrr µmaxf zfr, F xrr +F yr µmaxfzrr, (8) where maximum value of friction coefficient is defined as μ max. On the condition that μ max of all wheels are equal, if the performance index J is defined as square sum of the product of μ max and the workload of each wheel, the performance index J and weighted least squares solution x opt of (7) can be written as follows, J = (μ max η ij ) i=f,r j=l,r = F yf F + F yf F + F xrl +F yr F + F xrr +F yr F zfl zfr zrl zrr = x T Wx, (9) x opt = W A T (AW A T ) b, () where weighting matrix W is defined as follow, ( W = diag + F zfl F zfr, F + F, zrl zrr F, F zrl zrr ). () Block diagram of tire force distribution controller is shown in Fig.7. In order to generate the distributed longitudinal force of each wheel, torque erence value is also calculated from (6). In order to generate lateral force distributed by (), lateral force feedback control method was proposed in [7]. Provided that cornering stiffness is defined as C s, steer angle is 78

4 defined as δ and vehicle plant output is defined as C s δ,integral controller C(s) = τc ss is composed. Provided that relation between lateral force and tire side slip angle satisfies (), () and both yaw-rate and vehicle side slip angle are small enough not to infect, a closed-loop lateral force controller, time constant of which is τ, can be composed of this controller. As there are physical limiters of steer angle, anti-windup control is conducted as controller is C(s)= ɛs+ τc ss, where ɛ is minute constant. D. Direct yaw-moment control method considering yawmoment caused by tracking error of lateral force controller[7] Although the proposed method mentioned in IV-C is designed to achieve the equalization of the workload for each wheel, yaw moment can not be generated ideally when tracking error exists in the lateral force controller. This causes tracking error in yaw-rate. Theore, a method to compensate yaw moment caused by tracking error of lateral force controller is proposed based on quick response of differential torque yaw moment. From (4), () and (), the yaw moment which has to be compensated is written as follows, e Nt =N t N t =l f (F yf F yf ) l r (F yr F yr ), () where yaw moment calculated with lateral force erence is defined as Nt := l f Fyf l rfyr, the yaw moment which has to be compensated is defined as e Nt. In order to distribute this yaw moment as longitudinal force of vehicle is not generated by this distribution, from (5), the longitudinal force erence is decided as follow, [ ] [ Fxerl = ][ ], (3) F xerr e Nt where the distributed longitudinal force erence is defined as F xerl and F xerr. Then, each torque erence is calculated as follow, T = r ( F xopt + F xe). (4) On the one hand, while no tracking error exists in lateral force controller, each longitudinal force erence decided by (3) is N, and distribution by () is conducted. On the other hand, if tracking error exists, it is practicable to obtain quick yaw-rate response with this method. V. SIMULATION A. yaw-rate control with braking on low friction road In order to confirm effectiveness of equalization of workload for each wheel, simulations of cornering with braking on low friction road were carried out. Each constants of the EV were given below. The vehicle weight was defined as M = 76 kg, the vehicle yaw inertia moment as I = 549. kgm, the wheel base as l =.7 m, the tread base of anear axle as d f = =.3m, the distance from the center of gravity to anear axle tread as l f =.38 m and l r =.66 m and height of the center of gravity as h g =.454 m. The roll stiffness distribution was simply assumed as ρ f = ρ r =.5. Moreover, each constant value between the vehicle anoad is shown as follwing. The friction coefficient was defined as μ max =.3, and the anear nominal cornering stiffness as C fn =34 and C rn = 94. The proportional gain in the yaw-rate feedback controller which was defined as K p was decided as the pole of the system of which input was N in, output was γ and plant was I ns became 5 rad/sec. Then the parameters of lateral force controller were decided as τ f =.5 sec,τ r =.3 sec and ɛ =.5. Simulations without yaw-rate control, with the method mentioned in IV-B defined as conventional method and with the method mentioned in IV-D defined as proposed method in this section were conducted. Simulations starteunning at 3 km/h. Then, at sec, wheel steering angle erence δ =.87 rad was input, and moreover, at 4sec, vehicle longitudinal force erence Fx = 7 N was input. Yawrate control and tire force distribution were begun when wheel steering angle erence δ was input. Each yaw-rate and vehicle lateral force erence value was decided as wheel steering angle erence δ was input into a neutral steer (NS) vehicle. That is, yaw-rate erence was decided as a NS vehicle steady-state value γ = V l δ and in order to make vehicle lateral acceleration a NS vehicle steady-state value a y = V l δ, the vehicle lateral force erence value was decided as Fy =Ma y. Simulation results are shown in Figs.8. Figs.8(a) (c) show simulation results of yaw-rate with the three methods. Though yaw-rate diverges in both cases of without yaw-rate control and the conventional method, it can be verified that yaw-rate converges to yaw-rate erence in the proposed method. Furthermore, according to Fig. 8(d) which shows simulation results of vehicle trajectories with the three methods, it can be verified that equalization of workload for each wheel by such as proposed method makes vehicle trajectory stable. B. yaw-rate control with braking on high friction road In order to confirm effectiveness of compensation yaw moment mentioned in IV-D, simulations of yaw-rate control with braking were carried out on the road assumed authors experimental place. In this section, each constants and parameters were the same as previous section except the constants between the tire anoad, and also the time constant value of lateral force controller. A friction coefficient was defined as μ max =.7, the anear nominal cornering stiffness as C fn = 36 and C rn = 56 which are assumed as the experimental place. Moreover the parameters of lateral force controller were decided as τ f =.3 secand τ r =. sec. Simulations with the method mentioned in IV-B defined as conventional method, with the method mentioned in IV-C defined as proposed method and with the method mentioned in IV-D defined as proposed method were conducted. Simulations starteunning at 3 km/h, then, at sec wheel steering angle erence δ =.6 rad was input, and moreover, at 4sec, vehicle longitudinal force erence Fx = N was input. Yaw-rate control and tire force 79

5 y[m] YMO with tire force distribution YMO without tire force distribution without control (a) Yaw-rate (without control). (b) Yaw-rate (conventional) (c) Yaw-rate (proposed) (d) Vehicle trajectory Fig. 8. Simulation results (low µ) Steer Angle [rad].5..5 x[m] Fig Simulation results (conventional method) Steer Angle [rad] Fig Simulation results (proposed method : with tire force distribution) Steer Angle [rad] Fig.. Simulation results (proposed method : with tire force distribution and compensation yaw moment). distribution were begun when wheel steering angle erence δ was input. Each yaw-rate and vehicle lateral force erence value was the same values as previous section. Figs.9 show simulation results of yaw-rate, each wheel s workload, torque and steer angle erence in three methods. According to Figs.9 (a), it can be verified that yawrate response got more quick than the others in the proposed method. This reason is thought that compensation yaw moment ized to generate yaw moment ideally while tracking error exists in lateral force controller. Moreover, according to Figs.9 (b), it can be verified that equalization of workload for each wheel was achieved in the both proposed methods similarly. According to Figs.9 (c), (d), it can be verified that torque erence input into in-wheel motor locateear and inner during cornering became small, and the distributed wheel steering angle became larger than wheel steering angle erence in order to reduce wheels workload. VI. EXPERIMENT In order to confirm effectiveness of compensation yaw moment mentioned in IV-C, experiments of yaw-rate control with braking were carried out. In this section, each constants, parameters, and yaw-rate control and tire force distribution method were the same as V-B. Experiment started accelerating to 3 km/h, then, wheel steering angle erence δ =.6 rad was input, and moreover, vehicle longitudinal force erence F x = N was input. Yaw-rate control and tire force distribution were begun when wheel steering angle erence δ was input. Each yaw-rate and vehicle lateral force erence value was the same as simulations. Figs. 4 show experimental results of yaw-rate, each wheel s workload, torque and steer angle erence. According to Figs. 4(a), it can be verified that yawrate response got more quick than the others in the proposed method. This reason is thought that compensation 73

6 left right 3 9 Steer angle [rad] Fig.. Exprimental results (conventional method). 3 left right 3 Steer angle [rad] Fig Exprimental results (proposed method : with tire force distribution). 3 left right 3 Steer angle [rad] Fig. 4. Exprimental results (proposed method : with tire force distribution and compensation yaw moment). yaw moment ized to generate yaw moment ideally while tracking error exists in lateral force controller. Moreover, according to Figs. 4(b), it can be verified that equalization of workload for each wheel was achieved in both the proposed methods similarly. According to Figs. 4(c), (d), it can be verified that torque erence input into in-wheel motor locateear and inner during cornering became small, and the distributed wheel steering angle became larger than wheel steering angle erence in order to reduce wheels workload like simulation. VII. CONCLUTION In this paper, we proposed the lateral force control method with the longitudinal force and lateral force distribution method based on least squares solution of equations of motion. Moreover, we confirmed that equalization of workload for each wheel was archived in simulations and experiments. Additionally, we proposed the method of yaw moment compensation to attenuate tracking error in the lateral force controller. We comfirmed that not only equalization of workload for each wheel but also yaw-rate quick response were archived by this method in simulation and experiment. The future work is to consider the nonlinearity of cornering stiffness in lateral force control method. ACKNOWLEDGMENT This study was partly supported by Industrial Technology Research Grant Program from New Energy and Industrial Technology Development Organization (NEDO) of Japan. REFERENCES [] Y. Hori, Future Vehicle Driven by Electricity and Control-Research on Four-Wheel-Motored: UOT Electric March II, IEEE Trans. IE, Vol. 5, No.5, 4. [] O. Mokhiamar and M. Abe, Effects of An Optimum Cooperative Chassis Control From The View Points of Tire Workload, Proc. of JSAE 3 Annual Congress, No.33-3, pp.5, 3. [3] O. Nishihara, T. Hiraoka and H. Kumamoto, Optimization of Lateral and Driving/Braking Force Distribution of Independent Steering Vehicle (Minimax Optimization of Tire Workload), Trans. JSME. C, Vol. 7, No.74, pp , 6 (in Japanese). [4] E. Ono, Y. Hattori, Y. Muragishi and K. Koibuchi, Vehicle dynamics integrated control for four-wheel-distributed steering and four-wheeldistributed traction/braking systems, Vehicle System Dynamics vol. 44, No., pp.39 5, 6. [5] J. He, D.A.Crolla, M.C.Levesley, W.J.Manning, Integrated active steering and variable torque distribution control for improving vehicle handling and stability, SAE transactions vol. 3, No. 6, pp , 4. [6] D. Piyabongkarn, J. Grogg, Q. Yuan, J. Lew and R. Rajamani, Dynamic Modeling of Torque-Biasing Devices for Vehicle Yaw Control, SAE TECHNICAL PAPER SERIES, No , 6. [7] N. Ando and H. Fujimoto, Direct Yaw-moment Control for Electric Vehicle with All-wheel Lateral-force Sensors and Active Front/Rear Steering and Driving/Braking Force Distribution of Rear Wheels, IEE of JIASC, Vol., pp , 9 (in Japanese). [8] R. Rajamani, Vehicle Dynamics and Control, Springer Science & Business Media, Inc., 6. [9] H. Fujimoto, T. Saito and T. Noguchi, Motion stabilization control of electric vehicle under snowy conditions based on yaw-moment observer, IEEE Int. Workshop Advanced Motion Control, pp. 35 4, 4. [] S.Sakai, D-TireModel ver., j.html, (in Japanese). 73