Vehicle Yaw Stability Control Using the Fuzzy-Logic Controller Bin Li, Daofei Li, and Fan Yu "
Abstract-The yaw stability of a vehicle is cr ucial to vehicle safety in steer ing manoeuvr es. In this paper , a fuzzy-logic contr oller is designed for impr oving vehicle yaw stability by cor r ective yaw moment gener ated fr om differ ential br aking so that the yaw r ate and body sideslip angle can tr ace their desir ed values. An 8-DOF vehicle model with nonlinear tir e char acter istic is developed to captur e the vehicle longitudinal, later al, yaw and r oll motion, and four wheel r otation motion. The body sideslip angle is estimated by a 3-DOF vehicle model. Simulations of J -tur n and lane change ar e car r ied out and the simulation r esults show the effectiveness of the fuzzy-logic contr oller for impr oving vehicle yaw stability.
I. INTRODUCTION aw stability control is a major concern for vehicle stability control systems. Dangerous yaw motions of a vehicle may result from unexpected yaw disturbances like tire pressure loss, side wind force or braking on unilaterally icy road. The driver may not react quickly enough under these situations. So, it is necessary to develop active control system to help the driver maintain yaw stability of the vehicle. Recent studies have shown that yaw moment control is an effective approach to improve vehicle handling and stability [1]. The yaw moment control aims to make the vehicle yaw rate and sideslip angle follow their desired yaw rate and sideslip angle. In emergency, vehicle yaw stability control system generates the corrective yaw moment by steering, differential braking or traction control inputs to compensate the driver and keep the vehicle stable. Most studies on vehicle yaw stability control used the steering approach to generate yaw moment [2], [3]. A fuzzy active yaw control system based on steering for 4WD vehicles was discussed in [2]. The idea of differential braking was presented as vehicle steering intervention in [4], and applied to control vehicle yaw stability in [5]. In [6], a control system based on optimal control theory was proposed to improve vehicle handling and stability through the yaw moment generated by braking force. In this paper, a fuzzy-logic controller is proposed based on the differential braking for improving vehicle yaw stability.
Y
This work was supported by National Natural Science Foundation of China under Grant No.50575141. The authors express gratitude to NSFC for the financial support. Bin Li is with State Key Laboratory of Mechanical System and Vibration, Institute of Automotive Engineering, Shanghai Jiao Tong University (Phone: 8621-34206061, email:
[email protected]). Daofei Li is with State Key Laboratory of Mechanical System and Vibration, Institute of Automotive Engineering, Shanghai Jiao Tong University (email:
[email protected]). Fan Yu is with State Key Laboratory of Mechanical System and Vibration, Institute of Automotive Engineering, Shanghai Jiao Tong University (email:
[email protected]).
1-4244-1266-8/07/$25.00 ©2007 IEEE.
Fuzzy logic control is a knowledge-based control approach which can mimic human’s experience to control complex systems. Because of its capability of handling system nonlinearity and uncertainty, this method is a good choice for controlling a vehicle with high nonlinearity [7]. One advantage is that the controller can be described in vague linguistic terms. Another reason is that a fuzzy logic toolbox is provided in Matlab environment which simplifies both the programming and tuning of the controller. This paper is organized as follows. An 8-DOF vehicle model with nonlinear tire characteristic used for simulation is introduced in Section II; In Section III. the structure of yaw stability control system is proposed firstly; then, the 3-DOF model used to estimate the sideslip angle is given; next, differential braking control strategy is proposed; finally, fuzzy logic controller development is presented. In Section IV, the simulation results are given and discussed to demonstrate the effectiveness of the proposed controller. Conclusions are presented in Section V. II. VEHICLE AND TIRE MODELS A. Vehicle model Fig.1 shows the simulation model used in this study. The simulation model is based on an 8-DOF nonlinear vehicle model which includes the longitudinal velocity (u), lateral velocity (v), yaw rate (r), roll rate (p) of the sprung mass and the rotational velocities ( 1, 2, 3, 4) of the four wheels. In this vehicle model, the vertical and pitch motion are neglected. The load transfers are also taken into account in the model. Thus, the differential equations of vehicle motion are expressed as follows: (1) m * u$ - vr + ? Â Fxi m * v$ - ur + - ms ep$ ? Â Fyi
(2)
I z r$ - I xz p$ ? a * Fy1 - Fy 2 + /b * Fy 3 - Fy 4 + -
t * Fx 2 - Fx 4 / Fx1 / Fx3 + 2
I x p$ - I xz r$ / ms e * v$ - ur + ?
ms geh / * Kh f - Kh r + h / * Ch f - Ch r + p I wy$ i ? Fxi Rw / Tbi
(3)
(4) (5)
In the above equations, m denotes the complete vehicle mass, ms the sprung mass, Fxi and Fyi (i=1,2,3,4) the tire force components in x- and y-directions respectively, e the distance
of the sprung mass center gravity(C.G.) from the roll axis (fixed roll axis); a and b the distance from C.G. to the front and rear axle respectively, t the track width (assuming front track width equals to the rear track width), Iz and Ix the yaw inertia moment and roll inertia moment respectively, Ixz the sprung mass product of inertia, Iw the wheel moment of inertia, Rw the wheel radius, Tbi the brake torque applied to i-wheel, K and C the roll stiffness and roll damping respectively. The suffixes f and r in the equations refer to front and rear. f1
f2
B. Tire model In present study, the nonlinear combined tire model proposed by Pacejka [8] is introduced to calculate the longitudinal and lateral forces generated by tires. The tire model depends on the normal load Fz, wheel slip and wheel side slip angle g. The side slip angles of each tire are calculated as: Ã Ô Ä v - ar Õ (12) c1,2 ? f / arctan Ä Õ ÄÄ u a 1 tr ÕÕ Å 2 Ö
c 3,4
h
Ã Ô Ä v / br Õ ? / arctan Ä Õ ÄÄ u a 1 tr ÕÕ Å 2 Ö
The wheel slip values are given by: u / wi Rw ni ? i ui
Fig.1. 8-DOF nonlinear vehicle model
The components of tire force (Fxi and Fyi) can be calculated through the following transformation: Ç Fxi à cos f i / sin f i Ô Ç Fxwi (6) ÈF Ù ? Ä Ù , i ? 1, 2,3, 4 ÕÈ É yi Ú Å sin f i cos f i Ö É Fywi Ú Where, Fxwi and Fywi are the longitudinal and lateral tire forces in the wheel plane, respectively; hi is the steering angle (neglecting the effect of roll steer): f1 ? f 2 ? f , f 3 ? f 4 ? 0 . (7) Considering the load transfers caused by longitudinal and lateral acceleration, the normal load for each wheel can be expressed as follows: * u$ / vr + hcg Ô m * v$ - ur + hcg mg à (8) Fz1 ? Kd Äb / Ä ÕÕ 2L Å g t Ö Ô m * v$ - ur + hcg Kd ÕÕ / t Ö
* u$ / vr + hcg mg à Fz 3 ? ÄÄ a 2L Å g
Ô m * v$ - ur + hcg *1 / K d + (10) ÕÕ t Ö
* u$ / vr + hcg mg à ÄÄ a 2L Å g
Where, K d ?
Kh r
*K
hf
Ô m * v$ - ur + hcg *1 / K d + ÕÕ / t Ö
- Kh r +
(9)
(11)
; L is wheel base (L= a+b); hcg
is height of the sprung mass C.G..
(14)
Where ui is the longitudinal velocity component in wheel plane. Neglecting the self-aligning moment, the longitudinal and lateral tire forces are determined by the following equations: F , xwi ?
u yi u xi Fxi 0 , F , ywi ? Fyi 0 ui ui
(15)
In the above equations, Fxi0 and Fyi0 are the longitudinal and lateral forces in pure conditions, respectively. u xi and u yi are the longitudinal and lateral components of the theoretical slip u i which are related to side slip angle g and wheel slip , respectively. Considering the dynamic property of tire which can be expressed as standard first-order delay system, actual forces generated from the tire can be expressed as follows [9]: F , ywi F , xwi , Fywi ? Fxwi ? (16) 1 - v xi © s 1 - v yi © s
v
* u$ / vr + hcg mg à Fz 2 ? ÄÄ b / 2L Å g
Fz 4 ?
(13)
Where, v xi and yi are the time constant of the longitudinal and lateral tire forces respectively; s is the Laplace operator. III. CONTROLLER DESIGN To improve the vehicle handling and stability, a fuzzy logic controller is proposed in this research which aims to make the actual yaw rate and sideslip angle trace their desired values so that good trajectory and yaw stability can be achieved. The proposed yaw stability control system is shown in Fig.2. A. Nominal yaw rate and sideslip angle Reference model in the control system mentioned above generates the desired values of the yaw rate and sideslip angle to the steering input. The desired yaw rate is calculated as a function of the steering input and the vehicle longitudinal
velocity which considered as a constant velocity. Based on the linear 2 DOF vehicle plane model [10], it can be given by: u f (17) rd ? L *1 - Ku 2 + Where, K is the understeering coefficient (K=0.002 in this paper). The desired sideslip angle of vehicle should be controlled in a small range. In this study, the desired sideslip angle equals to zero: dd ? 0 (18)
d
dd
d
T b1 Mz
rd
Tb4
r
Fig.2. The structure of yaw stability control system
B. Sideslip angle estimator Since it is difficult to measure the actual sideslip angle, the sideslip angle is estimated by 3-DOF vehicle model [11] of which the dynamic equations are given as follows: m * u$ - vr + ? Fx1 - Fx 2 - Fx 3 - Fx 4 (19) m * v$ - ur + ? Fy1 - Fy 2 - Fy 3 - Fy 4
(20)
I z r$ ? a * Fy1 - Fy 2 + /b * Fy 3 - Fy 4 + -
t * Fx 2 - Fx 4 / Fx1 / Fx3 + 2
(21)
Where Fxi and Fyi can be calculated by the nonlinear tire model mentioned above. Yaw rate can be measured directly. The 3-DOF vehicle model is used to estimate the forward velocity and lateral velocity of the vehicle, so the sideslip angle can be easily calculated by: ÃvÔ d ? arctan Ä Õ (22) ÅuÖ C. Differential braking control Differential braking control is an effective method to enhance the vehicle yaw stability. Based on the driving conditions, the vehicle yaw and lateral motion can be effectively controlled by the yaw moment generated by controlling the brake torque applied on certain wheel. But, the capability of generating yaw moment is different for different wheels. Generally, braking front outer wheel can generate the maximum yaw moment which restrains the tendency of
vehicle over-steering whereas braking rear inner wheel is more effective in under-steering condition [12]. In this paper, according to the sign of the generated yaw moment, braking pressure is applied to the front outer wheel (wheel 1) when the sign of corrective yaw moment is negative whereas braking pressure is applied to rear inner wheel (wheel 4). Thus, vehicle yaw stability can be controlled by adjusting the front and rear wheel braking pressure. D. Fuzzy logic controller design Fuzzy logic control is a nonlinear control method and effective in handling the uncertainties and nonlinearities associated with the complex control systems. In recent years, many scholars have applied fuzzy logic control method to vehicle control systems such as automatic transmission, engine control and anti-lock braking systems. Compared with traditional control algorithms, superior characteristic can be realized through the use of fuzzy logic control. In order to obtain the desired vehicle response, we apply this method to the vehicle yaw stability control and develop the fuzzy logic controller to make the yaw rate and sideslip angle follow their desired values. To minimize the errors between the vehicle response and their desired values, in our fuzzy logic controller, the yaw rate error Fr and the sideslip angle error Fd are selected as two control input variables: Fr ? r / rd , Fd ? d / d d (23) The control output variable is the corrective yaw moment: Mz. Both the control input and output variables can be fuzzified into seven fuzzy sets: PB (positive big), PM (positive medium), PS (positive small), ZE (zero), NS (negative small), NM (negative medium), NB (negative big). Triangular member function is adopted in this study. The membership functions of variables Fr , Fd and Mz are shown in Fig. 3 and 4. The corrective yaw moment mainly influences on the yaw rate. When the actual yaw rate is smaller than its desired value, a large and positive yaw moment should be generated and braking torque will be applied onto the rear inner wheel (wheel 4) whereas a lager and negative should be generated and braking torque will be applied onto the front outer wheel (wheel 1). At the same time, considering the effect of sideslip angle, the rules can be tuned through different simulations. The fuzzy controller rules table is shown in TABLE I. In this study, we use Madmdani method for the fuzzy inference, which can be expressed as the following rule forms: IF Fd is A and Fr is B, THEN Mz is C Where A, B and C are fuzzy sets for the input and output variables defined above. In the defuzzification, the center of area (COA) method is used to calculate the inferred value of the fuzzy controller output.
NM
NB
NS
ZE
PS
PM
PB
1
0.8
0.6
0.4
0.2
0 -3
-2
-1
0
1
2
3
Fig.3. Membership function of input Fd and Fr
NB
NM
NS
ZE
PS
PM
PB
1
for controlled and uncontrolled vehicles, respectively. We can see from the figures that the response of vehicle with controller is better than the uncontrolled vehicle. Both yaw rate and sideslip angle can follow the reference values almost. J-turn is generated from the ramp steer input shown in the Fig.7. The simulation results for the J-turn manoeuvre are shown in Fig.8. It can be seen that the response of yaw rate and sideslip angle for the controlled vehicle can trace their desired values and the vehicle is stable. However, for the uncontrolled vehicle, the sideslip angle is apparently larger than that with controller after 2 seconds and increases with the time. As shown by the above simulation results, the proposed fuzzy logic controller can keep the vehicle stability and make the system respond quickly.
0.8
TABLE II PARAMETER OF VEHICLE MODEL
0.6
0.2
0 -3
-2
-1
0
1
2
3
Fig.4. Membership function of output Mz. TABLE I FUZZY CONTROLLER RULES TABLE
Fr / Fd
PB
PM
PS
ZE
NS
NM
NB
PB
NB
NB
NB
NM
NB
NB
NB
PM
NB
NB
NM
NM
NM
NB
NB
PS
NM
NS
NS
NS
NM
NM
NM
ZE
PM
PS
PS
ZE
NS
NS
NM
NS
PM
PM
PS
PS
PS
PM
PM
NM
PB
PB
PM
PM
PM
PB
PB
NB
PB
PB
PB
PM
PB
PB
PB
IV. SIMULATION RESULTS AND ANALYSIS To verify the effectiveness of the proposed fuzzy controller, computer simulations are carried out using 8- DOF vehicle model and nonlinear tire model in Matlab/Simulink. The responses of vehicle with and without the controller are compared. The fuzzy logic controller is designed by Fuzzy Logic Toolbox in Matlab. All the simulation tests are based on the dry road with road adhesion 0.8, neglecting air drag and road rolling resistance forces. The parameters corresponding to the vehicle model are given in TABLE II. In the simulation tests, two different steering inputs are considered: sinusoidal steering (lane change) and J-turn (shown in Fig.5 and 7, respectively). The initial vehicle velocity is 20m/s. Fig.5 shows the sinusoidal steering input. Fig.6 gives the time response of yaw rate and sideslip angle
Description Total vehicle mass Sprung mass Front axle to C.G. Rear axle to C.G. Yaw inertia Roll inertia Wheel inertia C.G. to roll axis Height of C.G Track width Wheel radius Roll stiffness Roll damping
Notation m ms a b Iz Ix Iw e hcg t Rw K C
Values 1764 1600 1.09 1.530 2400 400 2.1 0.198 0.520 1.440 0.3 113600 3770
Units Kg Kg m m Kgm2 Kgm2 Kgm2 m m m m N/m N/m/s
V. CONCLUSIONS In this research, a fuzzy logic controller is designed to make the yaw rate and sideslip angle of vehicle trace their desired values. An 8-DOF vehicle model with nonlinear tire characteristic is built for simulation tests under two different steering manoeuvres. The simulation shows that the response for the controlled vehicle is improved compared with the uncontrolled vehicle. The actual yaw rate and sideslip angle can follow their desired values to a satisfactory degree. Therefore, the vehicle yaw stability can be enhanced using the fuzzy logic controller. Future work will include a further analysis of the controller under different road conditions and driving conditions. Steering angle(deg)
0.4
4 2 0 -2 -4
0
0.5
1
1.5
2
2.5
3
Time(sec) Fig.5. Sinusoidal Steering Angle Input
3.5
4
Y a w ra te (ra d /s )
[2]
0.4
Uncontrolled
0.2 0
Controlled
-0.2 -0.4
Reference 0
0.5
1
1.5
2
2.5
3
3.5
4
S id e s lip a n g le (d e g )
Time(sec) 2
Uncontrolled
Controlled
1 0 -1 -2
Reference 0
0.5
1
1.5
2
2.5
3
3.5
4
Time(sec)
Steering angle(deg)
Fig.6. Simulation Results of Vehicle Response at Sinusoidal Steering 5 4 3 2 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
3.5
4
Time(sec)
Yaw rate(rad/s)
Fig.7. Steering Angle Input at J-turn 0.4
Reference
0.3
Uncontrolled 0.2
Controlled
0.1 0
0
0.5
1
1.5
2
2.5
3
Sideslip angle(deg)
Time(sec) 0.5
Reference
0
Controlled
-0.5 -1 -1.5
Uncontrolled 0
0.5
1
1.5
2
2.5
3
3.5
4
Time(sec) Fig.8. Simulation Results of Vehicle Response at J-turn
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