Design and simulation of the sliding mode controller ...

Int. J. Vehicle Safety, Vol. 6, No. 4, 2013

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Design and simulation of the sliding mode controller for the vehicle blow-out process control Taiping Mo* School of Mechano-Electronic Engineering, Xidian University, Xi’an, China Email: [email protected] *Corresponding author

Xiangwen Zhang, Kefeng Fan, Wei Mo and Yaqin Qiu School of Electric Engineering and Automation, Guilin University of Electronic Technology, Guilin, China Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Abstract: During vehicle driving process, the tyre security is very important, but the blow-out is inevitable sometimes, so it is very necessary to control the vehicle stability in order to avoid serious traffic accidents caused by the blowout. In this paper, with the tyre brush model and the seven degree of freedom vehicle model, the vehicle driving system model is modelled, and the front and rear tyres blow-out process was simulated and analysed. The results show that the vehicle will become unstable after the blow-out. In order to ensure the vehicle back to the stable range, the sliding mode controller is designed with the differential braking method, and the yaw rate is controlled by adjusting the braking torque at the opposite side of the blow-out tyre. With simulation, the sliding mode controller can guarantee the vehicle stability, shorten the braking distance and realise the safety brake during the tyre blow-out process. Keywords: tyre blow-out; brush model; sliding mode controller; differential brake. Reference to this paper should be made as follows: Mo, T., Zhang, X., Fan, K., Mo, W. and Qiu, Y. (2013) ‘Design and simulation of the sliding mode controller for the vehicle blow-out process control’, Int. J. Vehicle Safety, Vol. 6, No. 4, pp.333–346.

Copyright © 2013 Inderscience Enterprises Ltd.

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Introduction

During the vehicle driving process, it is easy to cause the tyre blow-out when the tyre is at the high speed, the large load, the high pressure or long work time, and the tyre blow-out will make the vehicle out of control and cause the serious traffic accident (Zhang et al., 2004; Martin and Laumon, 2005; Ge et al., 2009). In order to solve the problem of tyre blow-out, the Tyre Pressure Monitor System (TPMS) has been researched and developed (Ge et al., 2009). For the TPMS, the tyre pressure and temperature can be monitored with the sensors in the tyre. If there is abnormal tyre pressure, the alarm will be given to the driver, and the driver can treat the problem timely so as to avoid the tyre blow-out accident. Although the TPMS has been produced in many countries, it cannot control the vehicle stability. Confronted with the tyre blow-out suddenly, the drivers may not response timely or panic. If the vehicle stability control system can control the vehicle actively, the driver can brake the vehicle safely and quickly when the tyre blow-out, so it is very useful and meaningful to research the vehicle stability control system when the tyre blow-out, but there are few results in the present. Blythe et al. (1998) simulated the tyre blow-out process and analysed the influence of the driver’s different operation on the vehicle state (Blythe et al., 1998). The simulation result shows that the driver’s wrong operation can cause vehicle tilt, and even the serious traffic accident. Guo et al. (2007) simulated the tyre blow-out property with the simple linear tyre model (Guo et al., 2007). In addition, they designed the linear quadratic optimal controller in the Carsim and simulated the control effect (Wang, 2007; Huang et al., 2009). The simulated result shows that the controller can ensure the vehicle stability in the tyre blow-out process, but the linear tyre model is too simple and it cannot reflect the non-linear variation during the tyre blow-out process. In this paper, the brush model was used to simulate the tyre blow-out property, and the sliding mode controller was designed to control the vehicle stability during the vehicle straight linear driving process. For the front tyre blow-out and the rear tyre blowout, the controller was simulated, and the control effect and the vehicle stability during the tyre blow-out is analysed and researched.

2

Seven degrees of freedom model of the vehicle

During the tyre blow-out process, the vehicle may change direction. In order to reflect the vehicle dynamics process comprehensively, the seven degrees of freedom model is adopted in the paper. Ignoring the tyre rolling resistance, the wind resistance, and not considering the suspension system and vehicle vertical movement, and the vehicle pitch and roll movement, using the body coordinate system located at the centre of the vehicle mass, the vehicle seven degrees of freedom model can be built, as shown in Figure 1. The model includes the seven degrees of freedom which are the vehicle longitudinal velocity vx, the vehicle lateral velocity vy, the yaw rate r, and the angular velocity of the four wheels fl, fr, rl, rr.

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In Figure 1, Fxfl, Fxfr, Fxrl, Fxrr,, Fyfl, Fyfr, Fyrl, Fyrr denote the longitudinal forces and lateral forces of the four tyres, respectively; Tf, Tr denote the distance between the two front wheel axles and the distance between the two driving wheel axles, respectively, Lf, Lr denote the distance between the centre of the vehicle gravity and the front wheel axle and distance between the centre of the vehicle gravity and the driving wheel axle, respectively;  denotes the vehicle steering angle. Figure 1

Seven degrees of freedom model of the vehicle (Cho et al., 2010)

According to Figure 1, the vehicle dynamics equations can be established as follows: vx  rv y 

1  Fxfl +Fxfr  cos δ   Fyfl +Fyfr  sin δ+Fxrl  Fxrr   m

(1)

v y  rvx 

1  Fxfl +Fxfr  sin δ   Fyfl +Fyfr  cos δ+Fyrl  Fyrr   m

(2)

r 

1  L f  Fxfl +Fxfr  sin δ  L f  Fyfl +Fyfr  cos δ  Lr  Fyrl +Fyrr  Iz  Tf

Tf

T    Fxfl  Fxfr  cos δ   Fyfl  Fyfr  sin δ  r  Fxrl  Fxrr   2 2 2 

(3)

ω fl 

1   RFxfl  Tbfl  Iω

(4)

ω fr 

1   RFxfr  Tbfr  Iω

(5)

ω rr 

1   RFxrr  Tbrr  Iω

(6)

ω rl 

1   RFxrl  Tbrl  Iω

(7)

where m denotes the vehicle mass, Iz and I denote the vehicle and tyre inertia vector, respectively, R denotes the tyre radius, Tbfl, Tbfr, Tbrl, Tbrr denote the braking torques of the four tyres, respectively.

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According to Huang et al. (2009) and Wang (2007), the vehicle longitudinal acceleration ax, the front tyre lateral acceleration ayf and the rear tyre lateral acceleration ayr can be defined as follows: ax  vx  rv y

(8)

a yf  v y  rvx  L f  r

(9)

a yr  v y  rvx  L f  r

(10)

With the equations (1)–(10), the vehicle dynamics movement can be simulated, and the tyre longitudinal and lateral forces can be solved with the tyre brush model. In the follows, the brush model will be discussed and the tyre parameter variation during the tyre blow-out process is analysed.

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Tyre brush model

During the tyre blow-out process, the tyre stiffness, the tyre radius and the contact length will change, and the tyre blush model can reflect the tyre longitudinal force and the tyre lateral force variation with the longitudinal slip ratio, the side slip angle, the contact length and the tyre road friction coefficient, so the brush model is used to analyse the tyre blow-out process in this paper. According to Liu and Guo (2000), the most commonly used brush model can be described as follows:  μx Fzx  1  1  1   ,      4  2 Fx   1    μx Fzx , 2

(11)

 μ y Fzy  1  1  1   ,      4  2 Fy   1    μ y Fzy , 2

(12)

where x =

2kty s y a 2 2ktx sx a 2 denotes dimensionless longitudinal slip ratio; y = denotes μx Fz μ y Fz

dimensionless total slip ratio; ktx denotes the longitudinal stiffness of the tyre tread, kty denotes the lateral stiffness of the tyre tread; µx denotes the tyre longitudinal friction coefficient; µy denotes the tyre lateral friction coefficient; Fz denotes the tyre load; a denotes half of the tyre contact length; sx and sy denote the tyre longitudinal slip ratio and the lateral slip ratio, respectively. During the vehicle braking process, they can be defined as: Sx 

Sy 

vx  ωR vx

vy vx

(13)

(14)

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With equations (11)–(14), the tyre longitudinal force and the tyre lateral force can be solved, but the tyre vertical load should be determined beforehand. For the tyre vertical load can be transferred by the tyre longitudinal acceleration and the tyre lateral acceleration during the vehicle driving process, the tyre vertical load will change dynamically, and the detailed vertical loads of each tyre can be described as follows: Fzfl 

Fzfr 

Fzrl 

Fzrr 

mgLr  mvx h 2  L f  Lr 



mgLr  mvx h 2  L f  Lr 

mgLr  mvx h 2  L f  Lr 

mgL f  mvx h 2  L f  Lr 





mv y hLr

T f  L f  Lr 

mv y hLr

T f  L f  Lr  mv y hL f

Tr  L f  Lr 



mv y hL f

Tr  L f  Lr 

(15)

(16)

(17)

(18)

where g denotes the gravity acceleration and h denotes the height of the vehicle gravity centre. During the tyre blow-out process, the tyre property is changing quickly. Guo et al. (2007) gave the low pressure tyre test results, which show that the tyre longitudinal stiffness reduces to 34.28% of the normal value, and the tyre lateral stiffness reduces to 37.63% of the normal value, and the friction coefficient increases 20 times of the normal value (Guo et al., 2007). For the test was done at low speed bench, the real tyre blow-out may cause bigger changes of the tyre property than the test results. Considered this condition, the longitudinal stiffness of the tyre tread ktx is given as 8% of the normal value after the tyre blow-out, and the lateral stiffness of the tyre tread kty is given as 10% of the normal value after the tyre blow-out in this paper. According to the actual measurement and analysis, the rim contacts with the road, and the tyre effective radius R becomes 2/3 of the normal value, and the contact length doubles, and the longitudinal friction coefficient and the lateral friction coefficient increase 20 times after the tyre blow-out. Assume the tyre blow-out process lasts 0.5 s, and the tyre parameters change linearly, the change relation of the tyre parameters among the tyre blow-out processes can be established with equations as follows: t  t0 ktx 0 ,  ktx  ktx 0  1.84ktx 0  t  t0  , t0  t