Vehicle dynamics model and safety analysis on mountainous road

Transcription

1 n Ø A ^ 1 32 ò1 6 Ï 2015 c 6 Control Theory & Applications Vol. 32 No. 6 Jun DOI: /CTA ì«ýäåæ. 1 S Û w, (Ü ó Æ gäzæ, ñü ÜS ) Á : é AÛ /, AO p Ý»é ý1 G K, ïá ÍÜ 8gdÝì «1 ýäåæ.±9dugoffó å.. (Ü 1GPS/IMU ÿþ&e, ) ØÓ Ó w DZ 9R 1Ö, ÏLî 1Ö= Ç(LLTR)é ý 1 ½5?1 Û. (JL²: ý1 L ý \ Ý p ݱ9 ý %pý Ý 'Çh/T k', Ý Í, h/t, ý \ Ý, ý 1 ½5, ü$ ý 1 Ý ý \ Ý Jp ý 1 ½5. ' c: ýäåæ.; î 1Ö= Ç; w Ç; Ó R 1Ö ã aò: U461 zi è: A Vehicle dynamics model and safety analysis on mountainous road WANG Hui-li, SHI Zhong-ke (School of Automation, Northwestern Polytechnical University, Xi an Shaanxi , China) Abstract: Considering the influence of road geometrical characteristics (especially the road grade angle and turning radius) on vehicle state, we build an 8 degree-of-freedoms (DOF) vehicle dynamics model and Dugoff tire force model on mountainous road. Combining with the measurement information of GPS and IMU installed on the vehicle, we estimate the wheel slips and normal tire forces for different wheels and analyze the vehicle stability based on the lateral load transfer ratio (LLTR). The experimental result shows that the vehicle lateral movement is related with the vehicle velocity and the lateral acceleration. With the increase of road slope angle, the larger the ratio of vehicle height to width h/t, the higher the lateral acceleration is, and the worse the vehicle stability will be. Reducing the lateral movement can improve the vehicle stability. Key words: vehicle dynamics model; lateral load transfer ratio; wheel slip; normal tire force 1 Introduction Vehicle rollover crashes are the leading cause of fatalities on mountainous road due to the complex terrain conditions, which have sharp turnings and up-and-down hills. Statistics show that the severity of vehicle rollover is just following the crash accidents and nearly 33% of all deaths from vehicle crashes result from rollover[1], which occurs as a direct consequence of the decreasing of lateral wheel forces and is related with vehicle driving stability[2 3]. Adaptive cruise control (ACC), collision avoidance (CA), are installed on vehicle to improve the driving safety.while most of the control method is based on the accurate vehicle dynamics model, which is critical to improve traffic safety[4]. Bicycle model, the simplest vehicle dynamics model, is widely used for its simplicity.while it is not enough to fulfill the large variation of lateral acceleration. Segel presented a three degree-of-freedom (DOF) linear vehicle model which still used for the linear operation condition[5]. Spentzas selected lateral velocity, rolling angle velocity and yaw angle velocity as state variables, and presented a three DOF non-linear model of four-wheel-steering vehicle[6]. However, the assumption that vehicle wheels are remained in contact with ground all time and neglecting of heave velocity and pitch velocity can not describe vehicle rollover very well. To improve the accuracy of dynamics model, there exists numerous papers treating different aspects of vehicle modeling. Some studies are to improve the accuracy by increasing the degrees of the dynamics model and some develop the model from linear to nonlinear[7 12]. Considering the coupling interaction of longitudinal and lateral movements, Yang and Kim presented 6 DOF and verified the results under the constant radius turn maneuver[10]. Based on seven DOF Received: 4 November 2014; Accepted: 21 April Author. zkeshi@nwpu.edu.cn; Tel.: Supported by Major Program of the National Natural Science Foundation of China ( ) and Doctorate Foundation of Northwestern Polytechnical University (CX ). Corresponding

2 838 Control Theory & Applications Vol. 32 vehicle models, Ahmadi presented an adaptive nonlinear control scheme which regarded the steering wheel angle and wheel torque as input variables [11]. Imine developed an active steering assistance system and used a high-order sliding mode observer to estimate vehicle dynamics [12]. Li et al studied the effects of chassis integrated control-based driver-vehicle closed loop system on vehicle handling performance by the method of linear matrix inequalities [13]. Mario estimated vehicle sideslip angle by a nonlinear 5DOF single-track vehicle dynamics model with stochastic modeling of tire forces [14].Some studies presented the model for different types of vehicles, such as Li studied the coordinated control of steering and driving in off-road intelligent vehicle [15], Liu studied the trailer dynamics model [16], Pazooki proposed the dynamics model of SUV [17], Guo developed the lateral and longitudinal control of intelligent electric vehicles [18]. However, most of the presented work are primarily focused on the vehicle dynamics model on level road and ignored the effects of road longitudinal grade and bank angle, which are critical to vehicle attitude determination [19 20]. Actually,when the vehicle is driving along the turning road especially for mountainous road, the turning angles and normal force for each wheel are different due to the effects of road angle and centrifugal forces. Therefore, some phenomena on mountainous road (such as vehicle will rollover to inside) can not explained by the level vehicle dynamics model. With the goal of studying vehicle stability on mountainous road, an 8 DOF vehicle dynamics model and Dugoff tire force model is established here. The remainder of the paper is organized as follows. The coordinate systems used for vehicle dynamics model are introduced briefly in Section 2. Then the vehicle dynamics model and Dugoff tire force model are established and the vehicle stability is analyzed by the index LLTR. In Section 3, the result is verified and the lateral acceleration for vehicle safety traveling is obtained. The conclusion of the paper is presented in the final section. 2 Vehicle dynamics model Vehicle dynamics model typically consist of two components, a model describes the vehicle dynamics and a tire model describes the forces generated between the tire and road. To construct the vehicle dynamics model, three coordinate systems (vehicle coordinate system, tire coordinate system and geographic Cartesian coordinate system) are used here. 2.1 Coordinate systems Vehicle coordinate system is fixed to the gravity center of the vehicle, x-axis points from rear to front, y-axis from centerline to right, and z-axis is directed down, which combines to create right-handed coordinate system. By this coordinate system, attitude information about the vehicle can be obtained, which include longitudinal, lateral and vertical velocity, together with yaw, pitch and rollover rate. For each tire coordinate system, the original is at the center of tire contact ground, x-axis points to front along the tire, y-axis to right and z-axis is vertical to ground. The geographic Cartesian coordinate system is fixed on the earth, and the original is at the start position of experiment, the axis directions are the east, north and up. It is assumed that the directions of these coordinate systems are calibrated before the experiment. 2.2 Vehicle dynamics model Without loss of generality, the vehicle is traveling on road with grade angle θ and bank angle α. Here the road curve angle is assumed to be zero since it can be expressed by steering wheel angle. The top view of vehicle is shown in Fig.1, together with the friction forces acting on the vehicle. According to force balance and moment balance during vehicle traveling, an 8 DOF vehicle dynamics model can be established, which includes longitudinal and lateral force balance, yaw and rollover movement balance equation, as well as the rotational dynamics of four wheels. The vehicle model can be expressed as follows: V x = V y ψ {F xfl cos β l F xfr cos β r F yfl sin β l F yfr sin β r F xrl F xrr }/m g sin θ m s h s ψ γ/m, V y = V x ψ {F xfl sin β l F xfr sin β r F yfl cos β l F yfr cos β r F yrl F yrr }/m g cos θ sin α m s h s γ/m, ψ = (F xfl cos β l F xfr cos β r F yfl sin β l F yfr sin β r )T f /(2I zz ) (F xfl sin β l F xfr sin β r F yfl cos β l F yfr cos β r )l f /I zz (F xrl F xrr )T r /(2I zz ) (F yrl F yrr )l r /I zz I xz /I zz γ, I xx γ = I xz ψ ms h s ( V y V x ψ) m s h s gγ (K γf K γr )γ (D γf D γr ) γ I ω ω kj = R ω F xkj T ekj T bkj, k = f, r, j = l, r. (1) Where, V x, V y refer to longitudinal and lateral velocity of vehicle separately, ψ and γ are yaw rate and rollover rate separately, β l and β r denote steering angle of left and right for the front wheel, F xkj and F ykj refer to longitudinal and lateral force for each wheel (k = f, r means front and rear, j = l, r means left and right). l f and l r are distances from gravity center to front and rear axles, T f and T r are the front and rear track widths respectively. m is total mass of vehicle and m s is sprung mass, g is gravitation acceleration; θ and α refer to road

3 No. 6 WANG Hui-li et al: Vehicle dynamics model and safety analysis on mountainous road 839 grade and bank angle, γ is rollover angle; h is the height of gravity center, h s is the distance from gravity center to roller center. K γf and K γr are rolling stiffness for front and rear suspension; D γf andd γr are rolling damp for front and rear suspension. I zz, I xx, and I xz are second moment of mass about z, x, and y axis separately. I ω is the moment of inertia of tire, ω kj is angular velocity for different wheel, R ω is wheel effective radius, T ekj and T bkj are traction torque and braking torque for wheel kj separately. Fig. 1 Vehicles moving on a slope road 2.3 Dugoff tire force model The complexity of vehicle model given in (1) depends on the model of longitudinal and lateral tire forces F x, F y. The lateral tire force F y can be considered to have a linear relationship with respect to sideslip angle when it is less than 0.4 g. While for the actual turning driving, the tire force has highly nonlinear characteristics due to a variety of factors. Here the wellknown Dugoff tire force model is used to express the F zfl = mgl r cos θ cos α 2(l f l r ) F zfr = mgl r cos θ cos α 2(l f l r ) F zrl = mgl f cos θ cos α 2(l f l r ) F zrr = mgl f cos θ cos α 2(l f l r ) tire force, which expresses as follows [21] : { Fxkj = C xkj s kj /(1 s kj ) f(λ kj ), F ykj = C ykj tan δ kj /(1 s kj ) f(λ kj ), (2) where, C xkj, C ykj are longitudinal slip stiffness and cornering stiffness for different tires separately. The function f( )is expressed by { (2 λ)λ, λ < 1, f(λ) = 1, λ 1, λ kj = µf zkj (1 s kj ) 2 (C xkj s kj ) 2 (C ykj tan δ kj ) 2, µ is coefficient of adhesion between road and tire, and F zkj is normal tire forces. Sideslip angle δ kj and wheel slip s kj are defined as δ fl =β l arctan(v y l f ψ)/(v x 0.5T f ψ), δ fr =β r arctan(v y l f ψ)/(v x 0.5T f ψ), (3) δ rf = arctan(v y l r ψ)/(v x 0.5T r ψ), δ rr = arctan(v y l r ψ)/(v x 0.5T r ψ), s kj = R ωω kj V xkj, k = f, r, j = l, r, (4) max{r ω ω kj, V xkj } V xkj is longitudinal velocity for different wheels and it is significant in calculating wheel slip. Considering characteristics of vehicle dynamics, longitudinal velocity for different tires can be given as the following: V xfl = (V x 0.5T f ψ) cos β l (V y l f ψ) sin β l, V xfr = (V x 0.5T f ψ) cos β r (V y l f ψ) sin β r, V xrf = V x 0.5T r ψ, V xrr = V x 0.5T r ψ. (5) Taking into consideration the load transfer due to longitudinal, lateral, yaw and roller accelerations together with gravity components on sloped road, normal tire force can be expressed by static and dynamic load transfer as follows: 2(l f l r ) mg cos θ sin αl rh ma xh (l f l r )T f 2(l f l r ) ma yl r h K γfγ D γf γ, (l f l r )T f T f 2(l f l r ) mg cos θ sin αl rh ma xh (l f l r )T f 2(l f l r ) ma yl r h K γfγ D γf γ, (l f l r )T f T f 2(l f l r ) mg cos θ sin αl fh ma xh (l f l r )T r 2(l f l r ) ma yl f h K γrγ D γr γ, (l f l r )T r T r 2(l f l r ) mg cos θ sin αl fh ma xh (l f l r )T r 2(l f l r ) ma yl f h K γrγ D γr γ. (l f l r )T r T r (6) Where, a x = V x V y ψ g sin θ, a y = V y V x ψ g cos θ sin α are longitudinal and lateral acceleration for vehicle separately. Combining formula (1) through (6), the whole vehicle dynamics model and Dugoff tire force model on mountainous road can be obtained. 2.4 Lateral load transfer ratio and desired yaw rate An important index in the study of vehicle rollover is lateral load transfer ratio (LLTR), which can be defined as the difference of normal tire forces between left and right over the total normal force:

4 840 Control Theory & Applications Vol. 32 LLTR = (F zfl F zrl ) (F zfr F zrr ). (7) F zfl F zrl F zfr F zrr Where, F zfl, F zfr, F zrl, F zrr are the normal forces for the left front wheel, right front wheel, left rear wheel and right rear wheel respectively. Substituting the expression of (6) into formula (7), then LLTR = 2(a y cos θ sin α)h (l f l r )g cos θ cos α ( l r T f l f 2Kγ 2D γ ) T r mg cos θ cos α. (8) Where, K = K γf /T f K γr /T r and D = D γf /T f D γr /T r. Expression (8) shows that LLTR is related with the lateral movement of the vehicle (a y ), roll motion (γ), road geometry information (θ, α) and the structural parameters of the vehicle (m, T, h, K, D). If we neglecting the width variation of the front and rear wheel, that is T f = T r = T, the formula (8) can be simplified as LLTR = 2(a y cos θ sin α)h T g cos θ cos α 2Kγ 2D γ mg cos θ cos α. Above formula shows that the index is related with road geometrical characteristics, vehicle parameters, lateral acceleration and vehicle side slip angle. When the vehicle parameters and road information are determined, for example, road design speed is 60 km/h, turning radius is 280 meters, the longitudinal slope is and the lateral slope is 0.05, the relationship between LLTR and vehicle velocity is shown in Fig.2. It shows that when the vehicle is traveling with the designed speed, the lateral transfer ratio is smallest. The higher speed, the larger lateral transfer ratio is and the less vehicle stability will be. It is interesting that when the driving speed is less than the designed speed, the lateral transfer ratio will increase to the opposite direction and this is the phenomena that the vehicles will rollover to inside driving on mountainous road with too small speed. It can also be seen that the stability of different vehicle is different even though they travel on the same road with the same speed. The smaller the ratio of T /(2h), the less stability of the vehicle will be. Fig. 2 Lateral transfer ratio and vehicle speed When the vehicle is driving on the straight level road, that is θ = α = 0, then the above formula can be written as LLTR = 2a yh 2Kγ 2D γ. T g mg Where 2h/T is the ratio of height to width, which is also defined as static stability factor (SSF) and is usually to express the vehicle stability with static condition. Therefore the lateral acceleration can be regarded a function of roll angle. The relationship between a y /g and γ for the level road and mountainous road is illustrated in Fig. 3. a 0 is the critical value of rollover for the static condition, a 1, a 2 are the critical value for driving on level road and mountainous road separately, b is the parameter related with the vehicle structure. Fig. 3 Rollover characteristic of the vehicle Figure 3 shows that the critical value of vehicle rollover on mountainous road is a little larger than the level road due to the gravity component on the sloped road counteracts part of the centrifugal force. While the critical roll angle for mountainous road is smaller than the level road due to the road bank angle. Formula (8) also shows that the stability of vehicle can be improved by reducing the lateral and rollover movement which is associated with vehicle velocity and acceleration. Moreover, vehicle rollover is mostly decided by the lateral acceleration and vehicle stability can be improved by controlling side slip angle. The limit state of vehicle instability is brought by the decrease of restoring yaw moment at a large side slip angle. Desired yaw rate is the expected state of the vehicle with certain speed and steering angle, and can be calculated by [22] V x δ ψ = (l f l r ) m(l rc r l f C f )Vx 2. (9) 2(l f l r )C f C r Where δ is the sideslip angle. The stability can be enhanced by controlling sideslip angle after comparing the vehicle state and desired state. 3 Model validation and safety analysis To verify the model established above, some sensors such as GPS/IMU are installed on the test vehi-

5 No. 6 WANG Hui-li et al: Vehicle dynamics model and safety analysis on mountainous road 841 cle 2000 Volkswagen Santana and the corresponding structure parameters are illustrated in Table 1. Table 1 Parameters and values used in the model Parameters Values Mass of the entire vehicle m 1070 kg The sprung mass m s 900 kg Wheel effective radius R ω m Moment of inertiai xx 500 kg/m 2 Moment of inertia I zz 2100 kg/m 2 Moment of inertia I xz 47.5 kg/m 2 Longitudinal stiffness C xkj N/rad Cornering stiffness C ykj N/rad Height of gravity center h 0.6 m Rear track widths T r 1.41 m Front Track widths T f 1.4 m Gravitation acceleration g 9.8 m/s 2 Distance from gravity center to roll center h s 0.55 m Wheel mass moment of inertia I ω 0.9 kg/m 2 Distances from gravity center to front axles l f Distances from gravity center to rear axles l r Rolling stiffness for front suspension K γf Rolling stiffness for rear suspension K γr Rolling damp for front suspension D γf Rolling damp for rear suspension D γr 1.1 m 1.3 m (N m)/rad (N m)/rad 2100 N/rad 2100 N/rad The experiment is done on the asphalt concrete pavement and the coefficient of friction between tire and ground is assumed as 0.8. Combing the established model and measurement information from the sensors, the responses of wheel slips for different wheels are calculated and the corresponding curves are shown in Fig. 4. Figure 4 shows that the trends of wheel slip response for the two front or rear wheels are similar, except for the amplitude. Due to the effects of road angle and uncertainty factors during the traveling, the response curves are not too smooth. Fig. 4 Response of the wheel slip for different wheels Above analysis shows that normal tire force for front and rear wheel are mainly affected by road longitude grade and distribution of normal tire force, which caused by road bank angle and centrifugal forces. The normal tire force for different wheels are varied during the traveling and the responses is shown in Fig. 5.

6 842 Control Theory & Applications Vol. 32 (mg cos θ sin α mv 2 R cos α)t 2. (10) Then the critical velocity for vehicle rollover is V h = (T 2h tan α)gr cos θ/(2h T tan α)and the critical lateral acceleration of vehicle safety driving is a y /g = (T 2h tan α) cos θ/(2h T tan α). Above formula shows that safety velocity and acceleration is related with road geometrical and vehicle parameters. For the same vehicle, the larger road grade, the smaller safety velocity should be; for the same road geometrical, the larger ratio of h/t, the smaller safety velocity is. Therefore, the driver should adjust the vehicle velocity and acceleration according to the structure parameter of the vehicle even with the same traffic conditions. Fig. 6 Lateral load transfer ratio Fig. 5 Normal tire forces for different wheels The definition of LLTR in formula (7) explains that the value of LLTR varies from 1 to 1 and the value expresses the vehicle stability. The variation of lateral load transfer is different with the time due to the effects of turning radius, road angle, vehicle velocity, and lateral acceleration. The smaller the index value, the more stable the vehicle is. The value of LLTR equals to ±1 means one side of the wheel load is zero and the vehicle will rollover at this time. By formula (7) and the normal tire force, the lateral transfer ratio is illustrated in Fig. 6, which shows that the vehicle is within the stable state and the result is consistent with the above analysis. The vehicle stability can be improved by reducing the lateral movement and rollover movement which is associated with the velocity and acceleration. The following will discuss the critical velocity and acceleration for vehicle safety driving on mountainous road, which should be satisfied ( mv 2 sin α mg cos θ cos α)h = R 4 Conclusion This paper presented an 8 DOF vehicle dynamics model on mountainous road and analyzed vehicle stability by LLTR. Then the wheel slip, lateral force and normal tire force for different wheels were estimated from the proposed model together with measurement information from GPS and IMU installed on the vehicle. The results show that the vehicle stability can be improved by reducing the lateral movement and rollover which are associated with vehicle velocity and lateral acceleration. The larger road grade, the larger ratio of h/t, the larger velocity and acceleration will be and the less vehicle stability is. The safety driving is also related with other factors such as traffic volume and weather condition, et al. Therefore, the concrete control method to enhance vehicle stability by adjusting vehicle velocity and lateral acceleration will be studied in the further. References(References): [1] National Highway Traffic Safety Administration. Motor vehicle traffic crash injury and fatality estimates, 2002 early assessment [R]. Washington DC: National Center for Statistics and Analysis Advanced Research and Analysis, 2003.

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