# Study on the Influence of Road Geometry on Vehicle Lateral Instability

Authors: Yanna Yin; Huiying Wen; Lu Sun; Wei Hou
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

According to the accident analysis of vehicles in the curve, the skidding, rollover, and lateral drift of vehicles are determined as means to evaluate the lateral stability of vehicles. The utility truck of rear-wheel drive (RWD) is researched, which is high accident rate. Human-vehicle-road simulation models are established by CarSim. Through the orthogonal experiment method, the effects of different road geometries, speed, and interaction factors between road geometries on vehicle lateral stability are studied. In this paper, skidding risk of the vehicle is characterized by the Side-way Force Coefficient (SFC). Rollover risk of the vehicle is characterized by lateral acceleration and the load transfer ratio. Lateral drift risk of the vehicle is characterized by the sideslip angle of wheels. The results of orthogonal analysis reveal that the maximum tire-road friction coefficient and speed are highly significant in skidding of the vehicle. The effects of the combination of horizontal alignment and superelevation on vehicle skidding are important. The effects of horizontal alignment and speed on vehicle rollover risk are highly significant. The effects of superelevation on vehicle rollover risk are significant. The effects of the interaction of horizontal alignment and superelevation are also important on vehicles’ rollover risk. The speed and the maximum tire-road friction coefficient have highly significant effect on the vehicle’s lateral drift. The superelevation has a significant effect on the vehicle’s lateral drift. The effects of the interaction of horizontal alignment and superelevation and longitudinal slope are also important on the lateral drift of the vehicle.


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## Body

## 1. Introduction

Road accidents caused by lateral stability of the vehicle have always been a tough concern of vehicle safety in the world. In order to solve this problem, many experts and scholars work on studying how to reduce or even avoid the safety risk caused by lateral stability. The significant results have been achieved, but traffic accidents are still serious. The latest statistical data from the National Highway Traffic Safety Administration of United States [1] are shown in Table 1. Of 52,200 fatal automotive vehicle accidents, 31.5% of death toll was caused by rollover [1]. The number of rollover occurrence caused by single-vehicle crashes is greater than multiple-vehicle crashes. Therefore, there is an urgent need to investigate the accident of rollover caused by single-vehicle crashes.Table 1
United States Traffic Accident Statistics [1].
Vehicle typeRollover occurrence caused by single-vehicle crashes (%)Rollover occurrence caused by multiple-vehicle crashes (%)Death toll caused by rollover (%)Passenger car41.77.122Light truck-pickup57.121.643.3Light truck-utility64.726.648.5Light truck-van501628Other63.825Total50.912.931.5In the recent years, study on lateral stability of vehicles mainly focus on driving assistance using a vehicle stability control system, such as an antilock brake system (ABS), electronic stability controller (ESC), and acceleration slip regulation (ASR). When vehicles are in the transverse unstable state, the vehicle stability control system actively controls and corrects vehicle movement [2–5]. These systems have brought about a striking effect in antisideslip. However, most of the driving assistance systems are concerned about the antiskid risk of vehicles, paying less attention to the hidden dangers caused by vehicle rollover. Relying on the vehicle stability control system only cannot solve the road traffic safety problems completely. More attention needs to be paid to the ultimate and different causes of road traffic accidents [6].Road traffic accidents are mainly caused by three factors: roads, drivers, and vehicles. Road environment-related factors cause 27% of accidents in the United States [7]. Road geometric factors cause 34% of vehicle collisions in the United States and Britain [8]. These studies show that bad road conditions or improperly designed roads are the major causes of vehicle accidents [9].The shortcomings of the existing research on the impact of road factors on vehicle accidents caused by lateral stability are reflected in threefold.Firstly, the method of the influence of road geometry on vehicle safety is mainly the accident black spot analysis method. That is to say, accident statistics are carried out for accident-prone roads [10]. The references were seldom on the effect of road geometry on vehicle dynamics and, then, extended to vehicle lateral stability.Secondly, the existing research studies on the effect of road geometry on vehicle lateral stability mainly consider vehicle rollover or skidding. Vehicle lateral drift has not been taken into account. It has not been reported that the three indexes of rollover, skidding, and lateral drift are considered at the same time. Stergios Mavromatis focuses on the study of road factors on vehicle skidding and stopping sight distance [11–13]. Our research group has systematically studied the influence of road geometry, skid resistance, sight distance, and three-dimensional vision on vehicle rollover, skidding, and stopping sight distance [14–16].Thirdly, the existing road design parameters based on the mass point model of are designed. The radius parameters of curves are based on the minimum radius formula (1) of curves based on the classical mass point model in highway design code. The model ignores the parameter differences between vehicles, such as vehicle type, mass, and center of gravity position.(1)R=V2127fR+e,where fR—lateral friction demand of the mass point model, V(km/h)—vehicle speed, R(m)—the radius of a circular curve, and e—superelevation.Based on the NHTSA data [1], 95% of 522,000 automotive vehicle accidents are passenger cars or light trucks. As shown in Table 1, compared with passenger cars, pickup, vans, large trucks, and buses, utility trucks have the highest rollover occurrence caused by single-vehicle crashes (64.7%). Based on the analysis of vehicle accidents and the shortcomings of existing researches, this paper takes the utility trucks as the research target. A human-vehicle-road simulation model is established by CarSim. The orthogonal experimental method is adopted. Through orthogonal experiment, the effects of different road geometries, speed, and interaction factors on vehicle lateral stability are studied. Significance analysis of skidding, rollover, and lateral drift are carried out to characterize vehicle lateral stability.


## 2. Methods

### 2.1. Road Model

The road geometric alignment mainly includes horizontal alignment, longitudinal slope, superelevation, road crown, and their combination. The horizontal alignment is shown in Figure1(a), which usually consists of straight lines, transition curves, and circular curves. The radius of the circular curve is R. The longitudinal slope includes straight lines and vertical curves, as shown in Figures 1(b) and 1(c). It refers to the ratio of the height difference of two points of the same slope to its horizontal distance on the vertical section of a route [17]. Longitudinal slope is expressed by islope. In order to offset part or even all of the centrifugal force generated while vehicles negotiate a curve, the curve section is usually designed to a one-way cross slope with lateral high and medial low. It is called superelevation. As shown in Figure 1(d), the superelevation slope is isuperelevation. In order to facilitate the drainage of the road surface, the road is designed into an arch with high center and low sides, which is called the road crown. As shown in Figure 1(e), the gradient is i0. The slope is called as the composite longitudinal slope with overlapping superelevation and of longitudinal slope sections, as shown in Figure 1(f). The formula of calculating the composite longitudinal slope is [17](2)icombination=isuperelevation2+islope2.Figure 1
Road geometric design. (a) Horizontal curve. (b) Longitudinal slope. (c) Vertical curve. (d) Superelevation. (e) Road crown. (f) Composite longitudinal slope.
(a)(b)(c)(d)(e)(f)Because of the superelevation and longitudinal slope are small in road design code, the angle of the road’s cross slope is approximately equal to the superelevatione≈isuperelevation. For the same reason, the angle of the longitudinal slope is approximately equal to its longitudinal slope e≈islope. The angle of the composite longitudinal slope is approximately equal to its longitudinal slope θc≈icombination.


### 2.2. Vehicle Model

#### 2.2.1. Full Vehicle Model

In the past two decades, experts and scholars have developed various vehicle dynamics models [12]: the mass point model, bicycle model [18, 19], 8 degrees of freedom model [20], and 11 degrees of freedom model [21]. The complexity of the models increases gradually. ADAMS is a multibody dynamics simulation tool developed by MSC Software. The main function of ADAMS is to model and simulate multibody dynamics. The vehicle dynamics module of ADAMS/Car can be used for vehicle performance simulation. However, the complexity of the vehicle modeled by ADAMS/Car is very high, and the simulation speed may be very slow. The multibody vehicle simulation software CarSim developed by UMTRI is mainly used in the automobile industry. The software has been validated for many years. The simplified model includes 27 degrees of freedom. The stability and reliability of the model are very high, and the simulation speed is fast.The vehicle types studied in this paper are light truck-compact utility defined in CarSim [1, 22]. CarSim simplifies the vehicle model into 10 parts: one body part, four unsprung mass parts, four wheel parts, and one engine crankshaft part. The simplified model consists of 27 degrees of freedom, as shown in Figure 2(a): three degrees of freedom of movement (x, y, z), three degrees of freedom of rotation (X, Y, Z), four degrees of freedom of unsprung mass, four degrees of freedom of wheels, one degree of freedom of the transmission system, eight degrees of freedom of transient characteristics of tires, and four degrees of freedom of braking pressure. Specifically, vehicle modeled by CarSim is shown in Figure 2(b). The model includes seven subsystems: the body, aerodynamics, transmission assembly, brake system, steering system, tire, and suspension. In this paper, the utility truck of the rear-wheel drive is selected as the research target. The specific parameters of the utility truck are shown in Table 2 [21, 22].Figure 2
(a) Twenty-seven DOFs vehicle model. (b) CarSim vehicle structure.
(a)(b)Table 2
Vehicle simulation parameters [21, 22].
Parameter (unit)Value (utility vehicle)Sprung mass ms (kg)600Full vehicle m (kg)768Moment of inertia of sprung mass aroundX-axis Ixx (kg.m2)384.0Moment of inertia of sprung mass aroundY-axis Iyy (kg.m2)624.2Moment of inertia of sprung mass aroundZ-axis Izz (kg.m2)686.9Horizontal distance between the center of mass and front wheelsa (mm)550Horizontal distance between the center of mass and rear wheelsb (mm)1373Centroid heighth (mm)700The height between the center of mass and the center of rollhc (mm)573Front wheelbasecf (mm)1260Rear-wheelbasecr (mm)1260Wheel radiusR (mm)347Wheel moment of inertiaIw (kg.m2)0.9


#### 2.2.2. Tire Model

The “Magic Formula” tire model of Pacejka [23–25] with high fitting accuracy is used in this paper. The expressions are as follows:(3)y=DsinCarctanBx−EBx−arctanBx.In the formula,y can be a longitudinal force, a lateral force, or an aligning torque, while the independent variable can represent the sideslip angle or the longitudinal slip rate of the tire in different cases. Based on our previous research [26], this paper considers the tire with good quality as the research object, ignoring the zero drift in the horizontal and vertical directions of tires. The longitudinal force formula under the pure slip condition is fitted by the algebraic polynomial method [27, 28]:(4a)Fxλ=DxsinCxarctanBxλ−ExBxλ−arctanBxλ,where Cx=PCx1, Dx=PDx1+PDx2dfz⋅Fz, Ex=PEx1+PEx2dfz, Kx=Fz⋅PKx1+PKx2dfz⋅expPKx3dfz, and Bx=Kx/CxDx.Under the condition of pure sideslip, the formula of lateral force is as follows:(4b)Fyα=DysinCyarctanByα−EyByα−arctanByα,where Cy=PCy1, Dy=PDy1+PDy2dfz⋅Fz, Ey=PEy1+PEy2dfz, Ky=PKy1⋅Fz0⋅sin2tan−1Fz/PKy2Fz0, and By=Ky/CyDy.Here,α is the wheel sideslip angle, λ is the wheel slip rate, Fxλ is the longitudinal force calculated in the pure slip condition, Fyα is the lateral force of the tire in pure sideslip movement, and Fz is the vertical force of the tire; the value of fitting parameters is shown in Table 3.Table 3
Fitting parameters of the “Magic Formula” [20].
PCx1PDx1PDx2PEx1PEx2PKx1PKx2PKx31.621.035−0.04870.5−0.12219.4−0.130.171PCy1PDy1PDy2PEy1PEy2PKy1PKy21.29−0.90.18−1.070.68−12.951.72


### 2.3. Driver Model

There are two kinds of vehicle dynamics simulation systems: the open-loop simulation system and closed-loop simulation system. The open-loop simulation system regards the vehicle itself as an independent control system. The system does not consider the driver’s feedback characteristics, as shown in Figure3(a). The closed-loop simulation system incorporates the driver’s feedback into the system. The driver continuously corrects the steering wheel angle through track estimation to form a closed-loop system, as shown in Figure 3(b). The closed-loop system can simulate the influence of road factors on vehicle dynamics more realistically. In this paper, a closed-loop simulation optimal preview model raised by Charles C. Macadam is used, which can correct the trajectory [22, 29, 30]. Drivers get information about the road ahead and vehicles through “preview” and “perception”. Drivers make judgments based on the acquired information and adjust the motion state of vehicles. This forms a closed system.Figure 3
Flow chart of simulation. (a) Open-loop simulation. (b) Closed-loop simulation.
(a)(b)The station formula of the target position [22] is(5)Starg,i=S+iVxTm.S refers to the station of the center line of the road’s horizontal geometric figure defined by coordinatesX and Y. For any station S, it corresponds to unique X and Y coordinates. Route S is defined as a function of X and Y through a linear connection point. For each pair of X–Y coordinates, the Pythagorean theorem is used to calculate the corresponding s-increment.(6)Si=Si−1+Xi−Xi−12+Yi−Yi−12.The calculation of the controller uses a special coordinate axis system, as shown in Figure4. In this coordinate axis, the center of the front axle of the vehicle should be located at x = 0 and y = 0. The yaw angle of a vehicle is defined as ψ=0, the alignment of the X and Y axles with the longitudinal and lateral axes of the vehicle, respectively. In the coordinate axes of the driver’s controller, the motion of the vehicle is predicted based on these axes. The axis is fixed in the inertial reference and rotates ψ degree based on the inertial axis.Figure 4
Axis system of the steering controller.The target translates transversely in this coordinate system. First, the inertia ofX and Y coordinates of the path are calculated as the path function of the target position (Starg). Then, applying the transformation [22],(7)Ytarg=YStarg−YVcosψ−XStarg−XVsinψ.Within the coordinate range of the steering controller, the vehicle is always at the origin of the axis shown in Figure4. The time is 0, and the target path is known from time 0 to preview time T.


## 2.1. Road Model

The road geometric alignment mainly includes horizontal alignment, longitudinal slope, superelevation, road crown, and their combination. The horizontal alignment is shown in Figure1(a), which usually consists of straight lines, transition curves, and circular curves. The radius of the circular curve is R. The longitudinal slope includes straight lines and vertical curves, as shown in Figures 1(b) and 1(c). It refers to the ratio of the height difference of two points of the same slope to its horizontal distance on the vertical section of a route [17]. Longitudinal slope is expressed by islope. In order to offset part or even all of the centrifugal force generated while vehicles negotiate a curve, the curve section is usually designed to a one-way cross slope with lateral high and medial low. It is called superelevation. As shown in Figure 1(d), the superelevation slope is isuperelevation. In order to facilitate the drainage of the road surface, the road is designed into an arch with high center and low sides, which is called the road crown. As shown in Figure 1(e), the gradient is i0. The slope is called as the composite longitudinal slope with overlapping superelevation and of longitudinal slope sections, as shown in Figure 1(f). The formula of calculating the composite longitudinal slope is [17](2)icombination=isuperelevation2+islope2.Figure 1
Road geometric design. (a) Horizontal curve. (b) Longitudinal slope. (c) Vertical curve. (d) Superelevation. (e) Road crown. (f) Composite longitudinal slope.
(a)(b)(c)(d)(e)(f)Because of the superelevation and longitudinal slope are small in road design code, the angle of the road’s cross slope is approximately equal to the superelevatione≈isuperelevation. For the same reason, the angle of the longitudinal slope is approximately equal to its longitudinal slope e≈islope. The angle of the composite longitudinal slope is approximately equal to its longitudinal slope θc≈icombination.


## 2.2. Vehicle Model

### 2.2.1. Full Vehicle Model

In the past two decades, experts and scholars have developed various vehicle dynamics models [12]: the mass point model, bicycle model [18, 19], 8 degrees of freedom model [20], and 11 degrees of freedom model [21]. The complexity of the models increases gradually. ADAMS is a multibody dynamics simulation tool developed by MSC Software. The main function of ADAMS is to model and simulate multibody dynamics. The vehicle dynamics module of ADAMS/Car can be used for vehicle performance simulation. However, the complexity of the vehicle modeled by ADAMS/Car is very high, and the simulation speed may be very slow. The multibody vehicle simulation software CarSim developed by UMTRI is mainly used in the automobile industry. The software has been validated for many years. The simplified model includes 27 degrees of freedom. The stability and reliability of the model are very high, and the simulation speed is fast.The vehicle types studied in this paper are light truck-compact utility defined in CarSim [1, 22]. CarSim simplifies the vehicle model into 10 parts: one body part, four unsprung mass parts, four wheel parts, and one engine crankshaft part. The simplified model consists of 27 degrees of freedom, as shown in Figure 2(a): three degrees of freedom of movement (x, y, z), three degrees of freedom of rotation (X, Y, Z), four degrees of freedom of unsprung mass, four degrees of freedom of wheels, one degree of freedom of the transmission system, eight degrees of freedom of transient characteristics of tires, and four degrees of freedom of braking pressure. Specifically, vehicle modeled by CarSim is shown in Figure 2(b). The model includes seven subsystems: the body, aerodynamics, transmission assembly, brake system, steering system, tire, and suspension. In this paper, the utility truck of the rear-wheel drive is selected as the research target. The specific parameters of the utility truck are shown in Table 2 [21, 22].Figure 2
(a) Twenty-seven DOFs vehicle model. (b) CarSim vehicle structure.
(a)(b)Table 2
Vehicle simulation parameters [21, 22].
Parameter (unit)Value (utility vehicle)Sprung mass ms (kg)600Full vehicle m (kg)768Moment of inertia of sprung mass aroundX-axis Ixx (kg.m2)384.0Moment of inertia of sprung mass aroundY-axis Iyy (kg.m2)624.2Moment of inertia of sprung mass aroundZ-axis Izz (kg.m2)686.9Horizontal distance between the center of mass and front wheelsa (mm)550Horizontal distance between the center of mass and rear wheelsb (mm)1373Centroid heighth (mm)700The height between the center of mass and the center of rollhc (mm)573Front wheelbasecf (mm)1260Rear-wheelbasecr (mm)1260Wheel radiusR (mm)347Wheel moment of inertiaIw (kg.m2)0.9


### 2.2.2. Tire Model

The “Magic Formula” tire model of Pacejka [23–25] with high fitting accuracy is used in this paper. The expressions are as follows:(3)y=DsinCarctanBx−EBx−arctanBx.In the formula,y can be a longitudinal force, a lateral force, or an aligning torque, while the independent variable can represent the sideslip angle or the longitudinal slip rate of the tire in different cases. Based on our previous research [26], this paper considers the tire with good quality as the research object, ignoring the zero drift in the horizontal and vertical directions of tires. The longitudinal force formula under the pure slip condition is fitted by the algebraic polynomial method [27, 28]:(4a)Fxλ=DxsinCxarctanBxλ−ExBxλ−arctanBxλ,where Cx=PCx1, Dx=PDx1+PDx2dfz⋅Fz, Ex=PEx1+PEx2dfz, Kx=Fz⋅PKx1+PKx2dfz⋅expPKx3dfz, and Bx=Kx/CxDx.Under the condition of pure sideslip, the formula of lateral force is as follows:(4b)Fyα=DysinCyarctanByα−EyByα−arctanByα,where Cy=PCy1, Dy=PDy1+PDy2dfz⋅Fz, Ey=PEy1+PEy2dfz, Ky=PKy1⋅Fz0⋅sin2tan−1Fz/PKy2Fz0, and By=Ky/CyDy.Here,α is the wheel sideslip angle, λ is the wheel slip rate, Fxλ is the longitudinal force calculated in the pure slip condition, Fyα is the lateral force of the tire in pure sideslip movement, and Fz is the vertical force of the tire; the value of fitting parameters is shown in Table 3.Table 3
Fitting parameters of the “Magic Formula” [20].
PCx1PDx1PDx2PEx1PEx2PKx1PKx2PKx31.621.035−0.04870.5−0.12219.4−0.130.171PCy1PDy1PDy2PEy1PEy2PKy1PKy21.29−0.90.18−1.070.68−12.951.72


## 2.2.1. Full Vehicle Model

In the past two decades, experts and scholars have developed various vehicle dynamics models [12]: the mass point model, bicycle model [18, 19], 8 degrees of freedom model [20], and 11 degrees of freedom model [21]. The complexity of the models increases gradually. ADAMS is a multibody dynamics simulation tool developed by MSC Software. The main function of ADAMS is to model and simulate multibody dynamics. The vehicle dynamics module of ADAMS/Car can be used for vehicle performance simulation. However, the complexity of the vehicle modeled by ADAMS/Car is very high, and the simulation speed may be very slow. The multibody vehicle simulation software CarSim developed by UMTRI is mainly used in the automobile industry. The software has been validated for many years. The simplified model includes 27 degrees of freedom. The stability and reliability of the model are very high, and the simulation speed is fast.The vehicle types studied in this paper are light truck-compact utility defined in CarSim [1, 22]. CarSim simplifies the vehicle model into 10 parts: one body part, four unsprung mass parts, four wheel parts, and one engine crankshaft part. The simplified model consists of 27 degrees of freedom, as shown in Figure 2(a): three degrees of freedom of movement (x, y, z), three degrees of freedom of rotation (X, Y, Z), four degrees of freedom of unsprung mass, four degrees of freedom of wheels, one degree of freedom of the transmission system, eight degrees of freedom of transient characteristics of tires, and four degrees of freedom of braking pressure. Specifically, vehicle modeled by CarSim is shown in Figure 2(b). The model includes seven subsystems: the body, aerodynamics, transmission assembly, brake system, steering system, tire, and suspension. In this paper, the utility truck of the rear-wheel drive is selected as the research target. The specific parameters of the utility truck are shown in Table 2 [21, 22].Figure 2
(a) Twenty-seven DOFs vehicle model. (b) CarSim vehicle structure.
(a)(b)Table 2
Vehicle simulation parameters [21, 22].
Parameter (unit)Value (utility vehicle)Sprung mass ms (kg)600Full vehicle m (kg)768Moment of inertia of sprung mass aroundX-axis Ixx (kg.m2)384.0Moment of inertia of sprung mass aroundY-axis Iyy (kg.m2)624.2Moment of inertia of sprung mass aroundZ-axis Izz (kg.m2)686.9Horizontal distance between the center of mass and front wheelsa (mm)550Horizontal distance between the center of mass and rear wheelsb (mm)1373Centroid heighth (mm)700The height between the center of mass and the center of rollhc (mm)573Front wheelbasecf (mm)1260Rear-wheelbasecr (mm)1260Wheel radiusR (mm)347Wheel moment of inertiaIw (kg.m2)0.9


## 2.2.2. Tire Model

The “Magic Formula” tire model of Pacejka [23–25] with high fitting accuracy is used in this paper. The expressions are as follows:(3)y=DsinCarctanBx−EBx−arctanBx.In the formula,y can be a longitudinal force, a lateral force, or an aligning torque, while the independent variable can represent the sideslip angle or the longitudinal slip rate of the tire in different cases. Based on our previous research [26], this paper considers the tire with good quality as the research object, ignoring the zero drift in the horizontal and vertical directions of tires. The longitudinal force formula under the pure slip condition is fitted by the algebraic polynomial method [27, 28]:(4a)Fxλ=DxsinCxarctanBxλ−ExBxλ−arctanBxλ,where Cx=PCx1, Dx=PDx1+PDx2dfz⋅Fz, Ex=PEx1+PEx2dfz, Kx=Fz⋅PKx1+PKx2dfz⋅expPKx3dfz, and Bx=Kx/CxDx.Under the condition of pure sideslip, the formula of lateral force is as follows:(4b)Fyα=DysinCyarctanByα−EyByα−arctanByα,where Cy=PCy1, Dy=PDy1+PDy2dfz⋅Fz, Ey=PEy1+PEy2dfz, Ky=PKy1⋅Fz0⋅sin2tan−1Fz/PKy2Fz0, and By=Ky/CyDy.Here,α is the wheel sideslip angle, λ is the wheel slip rate, Fxλ is the longitudinal force calculated in the pure slip condition, Fyα is the lateral force of the tire in pure sideslip movement, and Fz is the vertical force of the tire; the value of fitting parameters is shown in Table 3.Table 3
Fitting parameters of the “Magic Formula” [20].
PCx1PDx1PDx2PEx1PEx2PKx1PKx2PKx31.621.035−0.04870.5−0.12219.4−0.130.171PCy1PDy1PDy2PEy1PEy2PKy1PKy21.29−0.90.18−1.070.68−12.951.72


## 2.3. Driver Model

There are two kinds of vehicle dynamics simulation systems: the open-loop simulation system and closed-loop simulation system. The open-loop simulation system regards the vehicle itself as an independent control system. The system does not consider the driver’s feedback characteristics, as shown in Figure3(a). The closed-loop simulation system incorporates the driver’s feedback into the system. The driver continuously corrects the steering wheel angle through track estimation to form a closed-loop system, as shown in Figure 3(b). The closed-loop system can simulate the influence of road factors on vehicle dynamics more realistically. In this paper, a closed-loop simulation optimal preview model raised by Charles C. Macadam is used, which can correct the trajectory [22, 29, 30]. Drivers get information about the road ahead and vehicles through “preview” and “perception”. Drivers make judgments based on the acquired information and adjust the motion state of vehicles. This forms a closed system.Figure 3
Flow chart of simulation. (a) Open-loop simulation. (b) Closed-loop simulation.
(a)(b)The station formula of the target position [22] is(5)Starg,i=S+iVxTm.S refers to the station of the center line of the road’s horizontal geometric figure defined by coordinatesX and Y. For any station S, it corresponds to unique X and Y coordinates. Route S is defined as a function of X and Y through a linear connection point. For each pair of X–Y coordinates, the Pythagorean theorem is used to calculate the corresponding s-increment.(6)Si=Si−1+Xi−Xi−12+Yi−Yi−12.The calculation of the controller uses a special coordinate axis system, as shown in Figure4. In this coordinate axis, the center of the front axle of the vehicle should be located at x = 0 and y = 0. The yaw angle of a vehicle is defined as ψ=0, the alignment of the X and Y axles with the longitudinal and lateral axes of the vehicle, respectively. In the coordinate axes of the driver’s controller, the motion of the vehicle is predicted based on these axes. The axis is fixed in the inertial reference and rotates ψ degree based on the inertial axis.Figure 4
Axis system of the steering controller.The target translates transversely in this coordinate system. First, the inertia ofX and Y coordinates of the path are calculated as the path function of the target position (Starg). Then, applying the transformation [22],(7)Ytarg=YStarg−YVcosψ−XStarg−XVsinψ.Within the coordinate range of the steering controller, the vehicle is always at the origin of the axis shown in Figure4. The time is 0, and the target path is known from time 0 to preview time T.


## 3. Safety Margin Analysis of Lateral Stability

This paper uses the safety margin from our previous study as a variable to reflect the extent of the index approaching the threshold. As shown in Figure5, the safety margin-station curve of the vehicle dynamics index is obtained from the vehicle dynamics simulation. It can be seen that the larger the value of Ijt is, the closer it is to the threshold Ij0 and the greater the possibility of rollover, skidding, and lateral drift are. Therefore, this paper uses formula (8) to define the safety margin of index J to measure the probability of accident risk caused by lateral unstability of vehicles.(8)Mj=Ij0−maxIjt,0≤Mj≤Ij0.Figure 5
The station curve of index j.


### 3.1. Safety Margin of Vehicle Skidding Risk

When a vehicle is traveling on a curve, the lateral force perpendicular to the direction of the vehicle will be produced due to the influence of centrifugal force and cross slope. When the lateral force is equal to or exceeds the maximum lateral adhesion force provided by the road surface, it will cause lateral slip of one or both axles of the vehicle, which is called skidding. Vehicle skidding includes skidding of four wheels, skidding of the front wheel, and skidding of the rear wheels. The phenomenon that the trajectory tracking ability is lost due to the skidding of the front axle in a curve is called the front-wheel skidding.The phenomenon of instability caused by skidding of four wheels is called lateral drift [31]. Lateral drift is studied separately below. According to the definition of the skidding, the lateral force coefficient of wheels is taken as the indicator of vehicle skidding in this paper. The lateral force coefficient [32] formula of the vehicle is(9)μl=maxFyiFZi,i=1,2,3,4,where Fyi is the tire lateral force, FZi is the tire vertical force, i = 1, 2, 3, 4 are the left-front wheel, left-rear wheel, right-front wheel, and right-rear wheel, respectively;When the lateral force coefficient of any wheel is greater than the lateral maximum tire-road friction coefficient, the vehicle will skid. Letμl0 be the adhesion coefficient provided by the pavement when skidding is about to occur. It also requires the lateral maximum tire-road friction coefficient to avoid skidding. When μl>μl0, skidding will occur. The tested maximum tire-road friction coefficient fp is 0.8, 0.5, and 0.2 [32], which correspond, respectively, to dry, wet, and snow conditions of the asphalt pavement. According to [33], the lateral maximum tire-road friction fRmax=0.925fp. However, the vehicle is at a high slip value in this paper, and the brake is blocked or the slip is accelerated [32, 34]. In such a situation, the maximum tire-road friction coefficient is equivalent to the sliding adhesion coefficient fs. Namely, fp=fs. Therefore, fRmax=0.925fs. Table 4 shows the maximum tire-road friction coefficients, slip friction coefficient, and lateral maximum tire-road friction coefficient of asphalt pavements under dry, wet, and snow conditions.Table 4
The maximum tire-road friction coefficients, slip friction coefficient, and lateral maximum tire-road friction coefficient [32–34].
Maximum tire-road friction coefficientsSlip friction coefficientLateral maximum tire-road friction coefficientfpfsfRmax=0.925fs0.80.750.690.50.450.410.20.150.13fp: maximum tire-road friction coefficient; fs: slip friction coefficient; the slip friction coefficient is equal to the braking force coefficient when the sliding rate is 100%. fRmax: lateral maximum tire-road friction coefficient.The safety margin of vehicle skidding on curves is defined as the difference between the lateral maximum tire-road friction coefficient provided by the available tire-road and the lateral friction demand [35]. So, the corresponding safety margin of vehicle skidding is obtained:(10)M1=μl0−maxμl,0≤M1≤μl0.


### 3.2. Safety Margin of Vehicle Rollover Risk

There are two types of vehicle rollover: nontripping rollover and tripping rollover. When vehicles’ lateral acceleration exceeds the compensation limit of lateral tire weight transfer, the vehicle rotates 90 degrees or more around its longitudinal axis, which is called nontripping rollover. Another is that when a vehicle slips, the vehicle loses control. The vehicle is impacted by the cross slope, the curb, and mollisol, so as to “trip” the vehicle, and rotates 90 degrees or more around its longitudinal axis, which is called tripping rollover. Because the risk of skidding occurs before tripping rollover and skidding has been expressed above, this paper only studies nontripping rollover. Based on the abovementioned definition of rollover, the lateral acceleration and load transfer ratio are selected as the indicators to measure vehicle rollover risk. The load transfer ratio is defined as the ratio of the amount of load transferred from the inside to the outside to the total load [36], as shown in the following equation:(11)LTR=∑i=12FZRi−FZLi∑i=12FZRi+FZLi.When the inner wheel load is zero, that is, the inner vehicle leaves the ground, at this time,LTR=0, and the threshold of the load transfer ratio is, thus, obtained. The corresponding safety margin of rollover is(12)M2=LTR0−maxLTRt,0≤M2≤LTR0.In this paper, the quasistatic model of a rigid vehicle is used. The rigid vehicle means neglecting the elastic deformation of the vehicle and tire [32]. Quasistatic refers to the vehicle steady-state steering [32]. The quasistatic model of a rigid vehicle is shown in Figure 6. The torque formula of the contact point between the outside wheel and the ground is as follows:(13)Fcosα−mgsinαhg+Nxi⋅B−Fsinα+mgcosα⋅B2=0.Figure 6
The free-body diagram of the quasistatic model of a rigid vehicle.Here,F is the centrifugal force; α is the bank angle of roads; Nyi and Ny0 are the lateral forces on the inner and outer tires, respectively; Nxi and Nx0 are the vertical forces on the inner and outer tires, respectively; ay is the lateral acceleration; and B is the wheel base.As the bank angleα of the road is generally small, sinα≈tanα=α=isuperelevation, cosα≈1. Moreover, Fcosα−mgsinα=may, which can be derived from equation (13) as follows:(14)mayhg+Nxi⋅B−Fα+mg⋅B2=0.WhenNxi becomes 0, the vehicle begins to rollover. The lateral acceleration when the vehicle begins to rollover is called the rollover threshold, which can be derived from equation (14) as follows:(15)ay=Fα+mg⋅B/2mhg.Clearly, when the bank angle isα=0, the rollover threshold is ay=B/2hgg. The parameters in Table 2 are substituted into equation (15) to obtain the resulting rollover threshold ay=0.9g. The quasistatic model of the rigid vehicle is reasonable only when the lateral acceleration changes slowly [32]. When the lateral acceleration changes rapidly, a physical model of roll is needed. However, modeling is not critical in this paper, and this paper directly applies the conclusion of [32]. For the utility truck, the rollover threshold value of the physical model of roll is lower 30% than that of the quasistatic model of rigid vehicle [32]. Therefore, the rollover threshold value of the physical model of roll is ay0=1−30%⋅ay=0.63g, which can be derived from equation (8) as follows:(16)M3=ay0−maxayt,0≤M3≤ay0.


### 3.3. Safety Margin of the Vehicle’s Lateral Drift Risk

High sideslip cornering is also called drift [37]. When vehicles are traveling in curves when the speed is too high (e.g., racing activities) or the radius of the curves is too small, their wheels will lose grip. The sideslip angle of the vehicle exceeds its control limit, and the vehicle runs at the tire-road adhesion limit [38]. At this time, the vehicle is turning in the opposite direction of bend with large sideslip angle, and the wheels are nearly saturated. This state of the vehicle is called drift [39].According to the definition of lateral drift, the sideslip angle of a vehicle is determined as the indicator of vehicle lateral drift. Sideslip angle is the angle between the direction of movement of the wheel center and the lateral velocity [34, 37]. The expression is as follows:(17)β=arctanVyVx,where β- vehicle sideslip angle; Vy- lateral speed of the tire-road contact point; and Vx- longitudinal speed of the wheel rotation center.Based on the literature of Yu [32] and the definition of lateral drift, when the lateral friction coefficient of the rear wheels of the vehicle reaches the critical value of skidding, the rear wheel is close to saturation state, and the sideslip angle at this time is defined as the threshold of lateral drift. The road model is developed by CarSim. The radius of the road is designed as 250 m, the longitudinal slope is designed as 0.06, and the superelevation is designed as 0.08 (the road model is the most unfavourable road condition on the basis of <Highway Engineering Technology Standard (JTG B 01-2014)> when the design speed is 80 km/h.). The peak adhesion coefficient of the pavement is 0.8, 0.5, and 0.2. The vehicle speed starts at 80 km/h, and increases continuously every other 5 km/h until the vehicle is about to skid. At this time, the vehicle’s sideslip angle is 3.71°, 2.34°, and 0.59 , respectively, and this value is defined as the lateral drift threshold of vehicles. The safety margin of lateral drift of the vehicle is(18)M4=β0−maxβ,0≤M4≤β0.


## 3.1. Safety Margin of Vehicle Skidding Risk

When a vehicle is traveling on a curve, the lateral force perpendicular to the direction of the vehicle will be produced due to the influence of centrifugal force and cross slope. When the lateral force is equal to or exceeds the maximum lateral adhesion force provided by the road surface, it will cause lateral slip of one or both axles of the vehicle, which is called skidding. Vehicle skidding includes skidding of four wheels, skidding of the front wheel, and skidding of the rear wheels. The phenomenon that the trajectory tracking ability is lost due to the skidding of the front axle in a curve is called the front-wheel skidding.The phenomenon of instability caused by skidding of four wheels is called lateral drift [31]. Lateral drift is studied separately below. According to the definition of the skidding, the lateral force coefficient of wheels is taken as the indicator of vehicle skidding in this paper. The lateral force coefficient [32] formula of the vehicle is(9)μl=maxFyiFZi,i=1,2,3,4,where Fyi is the tire lateral force, FZi is the tire vertical force, i = 1, 2, 3, 4 are the left-front wheel, left-rear wheel, right-front wheel, and right-rear wheel, respectively;When the lateral force coefficient of any wheel is greater than the lateral maximum tire-road friction coefficient, the vehicle will skid. Letμl0 be the adhesion coefficient provided by the pavement when skidding is about to occur. It also requires the lateral maximum tire-road friction coefficient to avoid skidding. When μl>μl0, skidding will occur. The tested maximum tire-road friction coefficient fp is 0.8, 0.5, and 0.2 [32], which correspond, respectively, to dry, wet, and snow conditions of the asphalt pavement. According to [33], the lateral maximum tire-road friction fRmax=0.925fp. However, the vehicle is at a high slip value in this paper, and the brake is blocked or the slip is accelerated [32, 34]. In such a situation, the maximum tire-road friction coefficient is equivalent to the sliding adhesion coefficient fs. Namely, fp=fs. Therefore, fRmax=0.925fs. Table 4 shows the maximum tire-road friction coefficients, slip friction coefficient, and lateral maximum tire-road friction coefficient of asphalt pavements under dry, wet, and snow conditions.Table 4
The maximum tire-road friction coefficients, slip friction coefficient, and lateral maximum tire-road friction coefficient [32–34].
Maximum tire-road friction coefficientsSlip friction coefficientLateral maximum tire-road friction coefficientfpfsfRmax=0.925fs0.80.750.690.50.450.410.20.150.13fp: maximum tire-road friction coefficient; fs: slip friction coefficient; the slip friction coefficient is equal to the braking force coefficient when the sliding rate is 100%. fRmax: lateral maximum tire-road friction coefficient.The safety margin of vehicle skidding on curves is defined as the difference between the lateral maximum tire-road friction coefficient provided by the available tire-road and the lateral friction demand [35]. So, the corresponding safety margin of vehicle skidding is obtained:(10)M1=μl0−maxμl,0≤M1≤μl0.


## 3.2. Safety Margin of Vehicle Rollover Risk

There are two types of vehicle rollover: nontripping rollover and tripping rollover. When vehicles’ lateral acceleration exceeds the compensation limit of lateral tire weight transfer, the vehicle rotates 90 degrees or more around its longitudinal axis, which is called nontripping rollover. Another is that when a vehicle slips, the vehicle loses control. The vehicle is impacted by the cross slope, the curb, and mollisol, so as to “trip” the vehicle, and rotates 90 degrees or more around its longitudinal axis, which is called tripping rollover. Because the risk of skidding occurs before tripping rollover and skidding has been expressed above, this paper only studies nontripping rollover. Based on the abovementioned definition of rollover, the lateral acceleration and load transfer ratio are selected as the indicators to measure vehicle rollover risk. The load transfer ratio is defined as the ratio of the amount of load transferred from the inside to the outside to the total load [36], as shown in the following equation:(11)LTR=∑i=12FZRi−FZLi∑i=12FZRi+FZLi.When the inner wheel load is zero, that is, the inner vehicle leaves the ground, at this time,LTR=0, and the threshold of the load transfer ratio is, thus, obtained. The corresponding safety margin of rollover is(12)M2=LTR0−maxLTRt,0≤M2≤LTR0.In this paper, the quasistatic model of a rigid vehicle is used. The rigid vehicle means neglecting the elastic deformation of the vehicle and tire [32]. Quasistatic refers to the vehicle steady-state steering [32]. The quasistatic model of a rigid vehicle is shown in Figure 6. The torque formula of the contact point between the outside wheel and the ground is as follows:(13)Fcosα−mgsinαhg+Nxi⋅B−Fsinα+mgcosα⋅B2=0.Figure 6
The free-body diagram of the quasistatic model of a rigid vehicle.Here,F is the centrifugal force; α is the bank angle of roads; Nyi and Ny0 are the lateral forces on the inner and outer tires, respectively; Nxi and Nx0 are the vertical forces on the inner and outer tires, respectively; ay is the lateral acceleration; and B is the wheel base.As the bank angleα of the road is generally small, sinα≈tanα=α=isuperelevation, cosα≈1. Moreover, Fcosα−mgsinα=may, which can be derived from equation (13) as follows:(14)mayhg+Nxi⋅B−Fα+mg⋅B2=0.WhenNxi becomes 0, the vehicle begins to rollover. The lateral acceleration when the vehicle begins to rollover is called the rollover threshold, which can be derived from equation (14) as follows:(15)ay=Fα+mg⋅B/2mhg.Clearly, when the bank angle isα=0, the rollover threshold is ay=B/2hgg. The parameters in Table 2 are substituted into equation (15) to obtain the resulting rollover threshold ay=0.9g. The quasistatic model of the rigid vehicle is reasonable only when the lateral acceleration changes slowly [32]. When the lateral acceleration changes rapidly, a physical model of roll is needed. However, modeling is not critical in this paper, and this paper directly applies the conclusion of [32]. For the utility truck, the rollover threshold value of the physical model of roll is lower 30% than that of the quasistatic model of rigid vehicle [32]. Therefore, the rollover threshold value of the physical model of roll is ay0=1−30%⋅ay=0.63g, which can be derived from equation (8) as follows:(16)M3=ay0−maxayt,0≤M3≤ay0.


## 3.3. Safety Margin of the Vehicle’s Lateral Drift Risk

High sideslip cornering is also called drift [37]. When vehicles are traveling in curves when the speed is too high (e.g., racing activities) or the radius of the curves is too small, their wheels will lose grip. The sideslip angle of the vehicle exceeds its control limit, and the vehicle runs at the tire-road adhesion limit [38]. At this time, the vehicle is turning in the opposite direction of bend with large sideslip angle, and the wheels are nearly saturated. This state of the vehicle is called drift [39].According to the definition of lateral drift, the sideslip angle of a vehicle is determined as the indicator of vehicle lateral drift. Sideslip angle is the angle between the direction of movement of the wheel center and the lateral velocity [34, 37]. The expression is as follows:(17)β=arctanVyVx,where β- vehicle sideslip angle; Vy- lateral speed of the tire-road contact point; and Vx- longitudinal speed of the wheel rotation center.Based on the literature of Yu [32] and the definition of lateral drift, when the lateral friction coefficient of the rear wheels of the vehicle reaches the critical value of skidding, the rear wheel is close to saturation state, and the sideslip angle at this time is defined as the threshold of lateral drift. The road model is developed by CarSim. The radius of the road is designed as 250 m, the longitudinal slope is designed as 0.06, and the superelevation is designed as 0.08 (the road model is the most unfavourable road condition on the basis of <Highway Engineering Technology Standard (JTG B 01-2014)> when the design speed is 80 km/h.). The peak adhesion coefficient of the pavement is 0.8, 0.5, and 0.2. The vehicle speed starts at 80 km/h, and increases continuously every other 5 km/h until the vehicle is about to skid. At this time, the vehicle’s sideslip angle is 3.71°, 2.34°, and 0.59 , respectively, and this value is defined as the lateral drift threshold of vehicles. The safety margin of lateral drift of the vehicle is(18)M4=β0−maxβ,0≤M4≤β0.


## 4. Numerical Analysis

### 4.1. Orthogonal Experimental Design with Interaction

In this paper, the method orthogonal experimental design is used to analyze the influence of the road’s horizontal alignment (A), longitudinal slope (B), superelevation (C), maximum tire-road friction coefficient (D), and vehicle speed (E) on vehicle lateral stability. The interaction of horizontal alignment and the longitudinal slope (AB), the interaction of horizontal alignment and superelevation (AC), and the interaction of horizontal alignment and the pavement friction coefficient (AD) are also considered. The levels of factors are shown in Table 5, and each factor contains three levels. The orthogonal table of L27313 (this form can be seen in the supplementary document (Experimental data). The supplementary document is the selected orthogonal table and the experimental data for orthogonal analysis) is selected in this paper. Based on literature [40], we can get the location of the interaction factors column. The header’s design of the table is shown in Table 6, with blank columns as error items.Table 5
Factors and level table [40].
LevelABCDEHorizontal alignment (m)Grade (%)Superelevation (%)Maximum tire-road friction coefficientSpeed (km/h)1700−440.28021000060.510031500480.8120Table 6
Header’s design with factors interaction and mixed levels.
FactorABAB1AB2CAC1AC2DAD1AD2EColumn12345678910111213


### 4.2. Analysis of Experimental Results

Tables7–10 are the variance analysis table of skidding’s safety margin, rollover’s safety margin, and lateral drift’s safety margin. The influence of some factors on the experimental results is not obvious. Namely, mean square value Sj¯<Se¯. Therefore, the sum of squares of row of these factors’ location is incorporated into the error line e, and a new error line is obtained. As shown in Table 7, SA¯<Se¯, SB¯<Se¯, SA×B¯<Se¯, SA×D¯<Se¯; therefore, we should incorporate SA, SB, SA×B, SA×D into Se. New sum of squares of errors and degrees of freedom are obtained. SeΔ=Se+SA+SB+SA×B+SA×D and feΔ=fe+fA+fB+fA×B+fA×D. Analysis of variance is carried on by the formula F=Sj/fj/SeΔ/feΔ. If F≥F1−αfj,feΔ, the significance of this factor on the experimental results is inferred from the significance α. If Fj≥F1−0.01fj,feΔ, the influence of this factor on the experimental results is highly significant. If F1−0.01fj,feΔ>Fj≥F1−0.05fj,feΔ, the influence of this factor on the experimental results is significant. If F1−0.05fj,feΔ>Fj≥F1−0.1fj,feΔ, the influence of this factor on the experimental results is commonly significant. If F1−0.1fj,feΔ>Fj, the influence of this factor on the experimental results is not significant. For the same reason, Tables 8–10 are available.Table 7
The table of variance analysis of skidding’s safety marginM1.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△3.63021.815F1−0.052,16=3.63F1−0.012,16=6.23F1−0.14,16=2.33F1−0.054,16=3.07B△4.51922.259C22.296211.1484.704SignificantD14507.63027253.8153060.68Highly significantE132.741266.37028.004Highly significant(A × B)△7.70341.926A × C19.92644.9822.102Nonsignificant(A × D)△3.92640.982e18.14844.537e△37.926162.370Table 8
The table of variance analysis of rollover’s safety marginM2.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△7.35623.678F1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△6.58623.293C55.352227.6763.455SignificanceD△15.27827.639E76.289238.1444.761Significance(A × B)△21.51445.379(A × C)△49.634412.406(A × D)△11.62342.906E64.258416.065E△176.249228.011Table 9
The table of variance analysis of rollover’s safety marginM3.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA16360.9328180.03799.669Highly significantF1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△0.06920.345C0.02320.0115D△0.04120.0205E19248.21729624.108117.264Highly significant(A × B)△0.7140.1775(A × C)△0.11740.029(A × D)△0.09240.023E1804.5474451.137e△1805.5992282.072Table 10
The table of variance analysis of the safety margin of vehicle lateral driftM4.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△213.5042106.752F1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△40.831220.416C442.2582221.1294.48SignificantD4479872223993.54540.07Highly significantE2207.72521103.86222.374Highly significant(A × B)△38.04149.510(A × C)△275.292468.823(A × D)△43.575410.894e474.1724118.543e△1085.4152249.337For the safety margin of skidding M1, analysis of variance based on orthogonal Table6 is carried on. Because the skidding safety margin obtained is small, the original value is multiplied by 1000 in the calculation process. Table 7 is FD>F1−0.012,16 obtained by the analysis of variance. In the table, F1−0.012,16>FC≥F1−0.052,16, FD>F1−0.012,16, FE>F1−0.012,16, and FAC<F1−0.12,16, it shows that the maximum tire-road friction coefficient and vehicle speed have a highly significant influence on the vehicle’s lateral force coefficient. That is to say, the impact on vehicle skidding is highly significant. The superelevation has a significant influence on the vehicle’s lateral force coefficient. That is to say, the impact on vehicle skidding is significant. Other factors have no significant effect on vehicle skidding.In this paper, the rangeRj of the safety margin of skidding is calculated to determine the order of primary and secondary factors affecting skidding. From main to secondary, RD>RE>RAC>RC>RAB>RA>RB>RAD. Obviously, except for a single factor, the maximum tire-road friction coefficient, vehicle speed, and superelevation, the interaction of horizontal alignment and superelevation has an important effect on vehicle skidding. It cannot be ignored.The results from two analyses suggest that the maximum tire-road friction coefficient and vehicle speed have a highly significant effect on vehicle skidding. The superelevation has a significant influence on the vehicle’s skidding. The interaction of horizontal alignment and superelevation has an important influence on vehicle’s skidding.Figure7 shows the effect of the significant single factor on the safety margin of vehicle skidding. Figure 7(a) shows the effect of superelevation on the safety margin of vehicle skidding. The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m−1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen in the Figure 7(a), with the increase in superelevation values, the safety margin of vehicle skidding increases. Therefore, it is more difficult for the vehicle to skid.Figure 7
The effect of the significant single factor on the safety margin of the vehicle’s skidding. (a) Effect of superelevation on the safety margin of skidding. (b) Effect of the pavement friction coefficient on the safety margin of skidding. (c) Effect of speed on the safety margin of skidding.
(a)(b)(c)Figure7(b) shows the effect of the maximum tire-road friction coefficient on the safety margin of vehicle skidding. The speed v=80km/h, radius of the circular curve R = 700 m, and the superelevation e=0.06. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The maximum tire-road friction coefficients are fp=0.2,0.5,0.8. As can be seen from the Figure 7(b), with the increase in the maximum tire-road friction coefficient, the safety margin of vehicle skidding increases. Therefore, it is more difficult for the vehicle to skid.Figure7(c) shows the effect of vehicle speed on the safety margin of vehicle skidding. Radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from Figure 7(c), with the increase in vehicle speed, the safety margin of vehicle skidding reduces. Therefore, vehicles are more prone to skidding.For the safety margin of vehicle rolloverM2, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of vehicle rollover obtained is small, the original value is multiplied by 100 in the calculation process. Table 8 is obtained by variance analysis. In the table, FC>F1−0.052,22, FE>F1−0.052,22, and it shows that superelevation and vehicle speed have a significant effect on the lateral load transfer ratio of vehicles, that is, the superelevation and vehicle speed have a significant effect on the vehicle’s rollover.In this paper, the rangeRj of the safety margin of vehicle rollover is calculated to determine the order of primary and secondary factors affecting rollover. From main to secondary, RAC>RE>RC>RAB>RAD>RD>RA>RB. Obviously, except for a single factor, superelevation and vehicle speed, the interaction of horizontal alignment and superelevation has a significant effect on vehicle rollover. It cannot be ignored.For the safety margin of vehicle rolloverM3, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of vehicle rollover obtained is small, the original value is multiplied by 1000 in the calculation process. Table 9 is obtained by variance analysis. In the table, FA>F1−0.012,22, FE>F1−0.012,22, and it shows that horizontal alignment and vehicle speed have a highly significant effect on the vehicle lateral acceleration. That is, the horizontal alignment and vehicle speed have a highly significant effect on the vehicle’s rollover.The variance analysis fromM2 and M3 suggests that the horizontal alignment and vehicle speed have a highly significant effect on the vehicle’s rollover. The superelevation has a significant influence on the vehicle’s rollover. The influence of the interaction of horizontal alignment and superelevation cannot be neglected and also an important factor.Figure8 shows the effect of the significant single factor on the safety margin of vehicle rollover. Figure 8(a) shows the effect of superelevation on the safety margin of vehicle rollover (M2). The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 8(a), with the increase of superelevation values, the safety margin of the vehicle’s rollover increases. Therefore, vehicles are less prone to rollover.Figure 8
The effect of the significant single factor on the safety margin of vehicle rollover. (a) Effect of superelevation on the safety margin of rollover. (b) Effect of speed on the safety margin rollover. (c) Effect of the radius of the horizontal curve on the safety margin of rollover (M3). (d) Effect of speed on the safety margin of rollover (M3).
(a)(b)(c)(d)Figure8(b) (M2) and Figure 8(d) (M3) show the effect of vehicle speed on the safety margin of vehicle rollover. The radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from the Figures 8(b) and 8(d), with the increase of vehicle speed, the safety margin of vehicle rollover reduces. Therefore, vehicles are more prone to rollover. As can be seen from Figure 8, the safety margin of vehicle rollover expressed by the load transfer ratio (M2) is greatly affected by the clothoid curve section, and the rollover safety margin expressed by the lateral acceleration (M3) is less affected by the clothoid curve section.Figure8(c) shows the effect of horizontal alignment on the safety margin of vehicle rollover. The speed v = 80 km/h, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The radii of the circular curve are R=700m,1000m,1500m. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 8(c), with the increase in the radius of the circular curve values, the safety margin of vehicle rollover increases. Therefore, vehicles are less prone to rollover.For the safety margin of the vehicle’s lateral driftM4, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of the vehicle’s lateral drift obtained is small, the original value is multiplied by 100 in the calculation process. Table 10 is obtained by variance analysis. In the table, FC>F1−0.052,22, FD>F1−0.012,22, FE>F1−0.012,22, and it shows that the superelevation has a significant effect on the vehicle sideslip angle. Therefore, the superelevation has a significant effect on the vehicle’s lateral drift. The vehicle speed and maximum tire-road friction coefficient have a highly significant effect on the vehicle sideslip angle. Therefore, the vehicle speed has a highly significant effect on the vehicle’s lateral drift.In this paper, the rangeRj of the safety margin of the vehicle’s lateral drift is calculated to determine the order of primary and secondary factors affecting the vehicle’s lateral drift. From main to secondary, RD>RB>RE>RAC>RC>RAD>RA>RAB. Obviously, except for a single factor, superelevation, vehicle speed, and maximum tire-road friction coefficient, the interaction of horizontal alignment and superelevation also have an important effect on the vehicle’s lateral drift. It cannot be ignored.The results from two analyses suggest that the vehicle speed and maximum tire-road friction coefficient have a highly significant effect on the vehicle’s lateral drift. The superelevation has a significant effect on the vehicle’s lateral drift. The interaction of horizontal alignment and superelevation and longitudinal slope has an important effect on the vehicle’s lateral drift. It cannot be ignored.Figure9 shows the effect of the significant single factor on the safety margin of the vehicle’s lateral drift. Figure 9(a) shows the effect of superelevation on the safety margin of the vehicle’s lateral drift. The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 9(a), with the increase in superelevation, the safety margin of the vehicle’s lateral drift increases. Therefore, vehicles are less prone to lateral drift.Figure 9
The effect of the significant single factor on the safety margin of the vehicle’s lateral drift. (a) Effect of superelevation on the safety margin of lateral drifting. (b) Effect of the peak friction coefficient on the safety margin of lateral drifting. (c) Effect of speed on the safety margin of lateral drifting.
(a)(b)(c)Figure9(b) shows the effect of the maximum tire-road friction coefficient on the safety margin of lateral drift. The speed v=80km/h, radius of the circular curve R=700m, and the superelevation e=0.06. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The maximum tire-road friction coefficients are fp=0.2,0.5,0.8. As can be seen from Figure 9(b), with the increase in the maximum tire-road friction coefficient, the safety margin of lateral drift increases. Therefore, it is more difficult for the vehicle to drift.Figure9(c) shows the effect of vehicle speed on the safety margin of the vehicle’s lateral drift. Radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from the Figure 9(c), with the increase in vehicle speed, the safety margin of the vehicle’s lateral drift reduces. Therefore, vehicles are more prone to lateral drift.


## 4.1. Orthogonal Experimental Design with Interaction

In this paper, the method orthogonal experimental design is used to analyze the influence of the road’s horizontal alignment (A), longitudinal slope (B), superelevation (C), maximum tire-road friction coefficient (D), and vehicle speed (E) on vehicle lateral stability. The interaction of horizontal alignment and the longitudinal slope (AB), the interaction of horizontal alignment and superelevation (AC), and the interaction of horizontal alignment and the pavement friction coefficient (AD) are also considered. The levels of factors are shown in Table 5, and each factor contains three levels. The orthogonal table of L27313 (this form can be seen in the supplementary document (Experimental data). The supplementary document is the selected orthogonal table and the experimental data for orthogonal analysis) is selected in this paper. Based on literature [40], we can get the location of the interaction factors column. The header’s design of the table is shown in Table 6, with blank columns as error items.Table 5
Factors and level table [40].
LevelABCDEHorizontal alignment (m)Grade (%)Superelevation (%)Maximum tire-road friction coefficientSpeed (km/h)1700−440.28021000060.510031500480.8120Table 6
Header’s design with factors interaction and mixed levels.
FactorABAB1AB2CAC1AC2DAD1AD2EColumn12345678910111213


## 4.2. Analysis of Experimental Results

Tables7–10 are the variance analysis table of skidding’s safety margin, rollover’s safety margin, and lateral drift’s safety margin. The influence of some factors on the experimental results is not obvious. Namely, mean square value Sj¯<Se¯. Therefore, the sum of squares of row of these factors’ location is incorporated into the error line e, and a new error line is obtained. As shown in Table 7, SA¯<Se¯, SB¯<Se¯, SA×B¯<Se¯, SA×D¯<Se¯; therefore, we should incorporate SA, SB, SA×B, SA×D into Se. New sum of squares of errors and degrees of freedom are obtained. SeΔ=Se+SA+SB+SA×B+SA×D and feΔ=fe+fA+fB+fA×B+fA×D. Analysis of variance is carried on by the formula F=Sj/fj/SeΔ/feΔ. If F≥F1−αfj,feΔ, the significance of this factor on the experimental results is inferred from the significance α. If Fj≥F1−0.01fj,feΔ, the influence of this factor on the experimental results is highly significant. If F1−0.01fj,feΔ>Fj≥F1−0.05fj,feΔ, the influence of this factor on the experimental results is significant. If F1−0.05fj,feΔ>Fj≥F1−0.1fj,feΔ, the influence of this factor on the experimental results is commonly significant. If F1−0.1fj,feΔ>Fj, the influence of this factor on the experimental results is not significant. For the same reason, Tables 8–10 are available.Table 7
The table of variance analysis of skidding’s safety marginM1.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△3.63021.815F1−0.052,16=3.63F1−0.012,16=6.23F1−0.14,16=2.33F1−0.054,16=3.07B△4.51922.259C22.296211.1484.704SignificantD14507.63027253.8153060.68Highly significantE132.741266.37028.004Highly significant(A × B)△7.70341.926A × C19.92644.9822.102Nonsignificant(A × D)△3.92640.982e18.14844.537e△37.926162.370Table 8
The table of variance analysis of rollover’s safety marginM2.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△7.35623.678F1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△6.58623.293C55.352227.6763.455SignificanceD△15.27827.639E76.289238.1444.761Significance(A × B)△21.51445.379(A × C)△49.634412.406(A × D)△11.62342.906E64.258416.065E△176.249228.011Table 9
The table of variance analysis of rollover’s safety marginM3.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA16360.9328180.03799.669Highly significantF1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△0.06920.345C0.02320.0115D△0.04120.0205E19248.21729624.108117.264Highly significant(A × B)△0.7140.1775(A × C)△0.11740.029(A × D)△0.09240.023E1804.5474451.137e△1805.5992282.072Table 10
The table of variance analysis of the safety margin of vehicle lateral driftM4.
Sources of variationSquare sumSjFreedom fMean squareSj¯F valueSignificanceCritical valueA△213.5042106.752F1−0.12,22=2.56F1−0.052,22=3.44F1−0.012,22=5.72F1−0.14,22=2.22F1−0.054,22=2.82F1−0.014,22=4.31B△40.831220.416C442.2582221.1294.48SignificantD4479872223993.54540.07Highly significantE2207.72521103.86222.374Highly significant(A × B)△38.04149.510(A × C)△275.292468.823(A × D)△43.575410.894e474.1724118.543e△1085.4152249.337For the safety margin of skidding M1, analysis of variance based on orthogonal Table6 is carried on. Because the skidding safety margin obtained is small, the original value is multiplied by 1000 in the calculation process. Table 7 is FD>F1−0.012,16 obtained by the analysis of variance. In the table, F1−0.012,16>FC≥F1−0.052,16, FD>F1−0.012,16, FE>F1−0.012,16, and FAC<F1−0.12,16, it shows that the maximum tire-road friction coefficient and vehicle speed have a highly significant influence on the vehicle’s lateral force coefficient. That is to say, the impact on vehicle skidding is highly significant. The superelevation has a significant influence on the vehicle’s lateral force coefficient. That is to say, the impact on vehicle skidding is significant. Other factors have no significant effect on vehicle skidding.In this paper, the rangeRj of the safety margin of skidding is calculated to determine the order of primary and secondary factors affecting skidding. From main to secondary, RD>RE>RAC>RC>RAB>RA>RB>RAD. Obviously, except for a single factor, the maximum tire-road friction coefficient, vehicle speed, and superelevation, the interaction of horizontal alignment and superelevation has an important effect on vehicle skidding. It cannot be ignored.The results from two analyses suggest that the maximum tire-road friction coefficient and vehicle speed have a highly significant effect on vehicle skidding. The superelevation has a significant influence on the vehicle’s skidding. The interaction of horizontal alignment and superelevation has an important influence on vehicle’s skidding.Figure7 shows the effect of the significant single factor on the safety margin of vehicle skidding. Figure 7(a) shows the effect of superelevation on the safety margin of vehicle skidding. The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m−1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen in the Figure 7(a), with the increase in superelevation values, the safety margin of vehicle skidding increases. Therefore, it is more difficult for the vehicle to skid.Figure 7
The effect of the significant single factor on the safety margin of the vehicle’s skidding. (a) Effect of superelevation on the safety margin of skidding. (b) Effect of the pavement friction coefficient on the safety margin of skidding. (c) Effect of speed on the safety margin of skidding.
(a)(b)(c)Figure7(b) shows the effect of the maximum tire-road friction coefficient on the safety margin of vehicle skidding. The speed v=80km/h, radius of the circular curve R = 700 m, and the superelevation e=0.06. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The maximum tire-road friction coefficients are fp=0.2,0.5,0.8. As can be seen from the Figure 7(b), with the increase in the maximum tire-road friction coefficient, the safety margin of vehicle skidding increases. Therefore, it is more difficult for the vehicle to skid.Figure7(c) shows the effect of vehicle speed on the safety margin of vehicle skidding. Radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from Figure 7(c), with the increase in vehicle speed, the safety margin of vehicle skidding reduces. Therefore, vehicles are more prone to skidding.For the safety margin of vehicle rolloverM2, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of vehicle rollover obtained is small, the original value is multiplied by 100 in the calculation process. Table 8 is obtained by variance analysis. In the table, FC>F1−0.052,22, FE>F1−0.052,22, and it shows that superelevation and vehicle speed have a significant effect on the lateral load transfer ratio of vehicles, that is, the superelevation and vehicle speed have a significant effect on the vehicle’s rollover.In this paper, the rangeRj of the safety margin of vehicle rollover is calculated to determine the order of primary and secondary factors affecting rollover. From main to secondary, RAC>RE>RC>RAB>RAD>RD>RA>RB. Obviously, except for a single factor, superelevation and vehicle speed, the interaction of horizontal alignment and superelevation has a significant effect on vehicle rollover. It cannot be ignored.For the safety margin of vehicle rolloverM3, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of vehicle rollover obtained is small, the original value is multiplied by 1000 in the calculation process. Table 9 is obtained by variance analysis. In the table, FA>F1−0.012,22, FE>F1−0.012,22, and it shows that horizontal alignment and vehicle speed have a highly significant effect on the vehicle lateral acceleration. That is, the horizontal alignment and vehicle speed have a highly significant effect on the vehicle’s rollover.The variance analysis fromM2 and M3 suggests that the horizontal alignment and vehicle speed have a highly significant effect on the vehicle’s rollover. The superelevation has a significant influence on the vehicle’s rollover. The influence of the interaction of horizontal alignment and superelevation cannot be neglected and also an important factor.Figure8 shows the effect of the significant single factor on the safety margin of vehicle rollover. Figure 8(a) shows the effect of superelevation on the safety margin of vehicle rollover (M2). The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 8(a), with the increase of superelevation values, the safety margin of the vehicle’s rollover increases. Therefore, vehicles are less prone to rollover.Figure 8
The effect of the significant single factor on the safety margin of vehicle rollover. (a) Effect of superelevation on the safety margin of rollover. (b) Effect of speed on the safety margin rollover. (c) Effect of the radius of the horizontal curve on the safety margin of rollover (M3). (d) Effect of speed on the safety margin of rollover (M3).
(a)(b)(c)(d)Figure8(b) (M2) and Figure 8(d) (M3) show the effect of vehicle speed on the safety margin of vehicle rollover. The radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from the Figures 8(b) and 8(d), with the increase of vehicle speed, the safety margin of vehicle rollover reduces. Therefore, vehicles are more prone to rollover. As can be seen from Figure 8, the safety margin of vehicle rollover expressed by the load transfer ratio (M2) is greatly affected by the clothoid curve section, and the rollover safety margin expressed by the lateral acceleration (M3) is less affected by the clothoid curve section.Figure8(c) shows the effect of horizontal alignment on the safety margin of vehicle rollover. The speed v = 80 km/h, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The radii of the circular curve are R=700m,1000m,1500m. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 8(c), with the increase in the radius of the circular curve values, the safety margin of vehicle rollover increases. Therefore, vehicles are less prone to rollover.For the safety margin of the vehicle’s lateral driftM4, analysis of variance based on orthogonal Table 6 is carried out. Because the safety margin of the vehicle’s lateral drift obtained is small, the original value is multiplied by 100 in the calculation process. Table 10 is obtained by variance analysis. In the table, FC>F1−0.052,22, FD>F1−0.012,22, FE>F1−0.012,22, and it shows that the superelevation has a significant effect on the vehicle sideslip angle. Therefore, the superelevation has a significant effect on the vehicle’s lateral drift. The vehicle speed and maximum tire-road friction coefficient have a highly significant effect on the vehicle sideslip angle. Therefore, the vehicle speed has a highly significant effect on the vehicle’s lateral drift.In this paper, the rangeRj of the safety margin of the vehicle’s lateral drift is calculated to determine the order of primary and secondary factors affecting the vehicle’s lateral drift. From main to secondary, RD>RB>RE>RAC>RC>RAD>RA>RAB. Obviously, except for a single factor, superelevation, vehicle speed, and maximum tire-road friction coefficient, the interaction of horizontal alignment and superelevation also have an important effect on the vehicle’s lateral drift. It cannot be ignored.The results from two analyses suggest that the vehicle speed and maximum tire-road friction coefficient have a highly significant effect on the vehicle’s lateral drift. The superelevation has a significant effect on the vehicle’s lateral drift. The interaction of horizontal alignment and superelevation and longitudinal slope has an important effect on the vehicle’s lateral drift. It cannot be ignored.Figure9 shows the effect of the significant single factor on the safety margin of the vehicle’s lateral drift. Figure 9(a) shows the effect of superelevation on the safety margin of the vehicle’s lateral drift. The speed v=100km/h, radius of the circular curve R=700m, and the maximum tire-road friction coefficient fp=0.8. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The values of superelevation are e=0.04,0.06,0.08. As can be seen from Figure 9(a), with the increase in superelevation, the safety margin of the vehicle’s lateral drift increases. Therefore, vehicles are less prone to lateral drift.Figure 9
The effect of the significant single factor on the safety margin of the vehicle’s lateral drift. (a) Effect of superelevation on the safety margin of lateral drifting. (b) Effect of the peak friction coefficient on the safety margin of lateral drifting. (c) Effect of speed on the safety margin of lateral drifting.
(a)(b)(c)Figure9(b) shows the effect of the maximum tire-road friction coefficient on the safety margin of lateral drift. The speed v=80km/h, radius of the circular curve R=700m, and the superelevation e=0.06. The 600 m–1200 m road section is selected as the research object. The 600–800 m road section is the transition section from a straight line to a circular curve (clothoid curve section). The 800–1200 m road section is a circular curve section. The maximum tire-road friction coefficients are fp=0.2,0.5,0.8. As can be seen from Figure 9(b), with the increase in the maximum tire-road friction coefficient, the safety margin of lateral drift increases. Therefore, it is more difficult for the vehicle to drift.Figure9(c) shows the effect of vehicle speed on the safety margin of the vehicle’s lateral drift. Radius of the circular curve R=700m, the superelevation e=0.06, and the maximum tire-road friction coefficient fp=0.8. The road section of time at 16 s–40 s is selected as the research object, which includes two sections: the transition section (clothoid curve section) and circular curve section. The vehicle speeds are v=80km/h,100km/h,120km/h. As can be seen from the Figure 9(c), with the increase in vehicle speed, the safety margin of the vehicle’s lateral drift reduces. Therefore, vehicles are more prone to lateral drift.


## 5. Conclusions

Based on the analysis of vehicles accident in a curve, the skidding, rollover, and lateral drift of vehicles are determined as the means to evaluate the lateral stability of vehicles. Aiming at the high accident rate of utility trucks with rear-wheel drive (RWD), simulation models of Human-Vehicle-Road are established by CarSim. Through the orthogonal analysis design method, the effects of road geometry, speed factors, and interaction between road geometries on the lateral stability of vehicles are studied.In this paper, the lateral force coefficient is used to characterize the safety margin of vehicle skidding. Synthesizing the results of orthogonal analysis and range analysis of the safety margin of vehicle skidding, it is suggested that the maximum tire-road friction coefficient and vehicle speed have a highly significant effect on vehicle skidding. The superelevation has a significant influence on vehicle’s skidding. The interaction of horizontal alignment and superelevation has an important effect on vehicle skidding. It cannot be ignored.Significant single factor analysis shows that, with the increase of superelevation and the maximum tire-road friction coefficient, the safety margin of vehicle skidding increases, and the vehicle is less prone to skidding. With the increase of vehicle speed, the safety margin of vehicle skidding decreases, and the vehicle is more prone to skid.In this paper, lateral acceleration and the load transfer ratio are used to characterize the safety margin of vehicle rollover. The results of orthogonal analysis and range analysis show that horizontal alignment and vehicle speed have a highly significant impact on vehicle rollover risk. The superelevation has a significant impact on vehicle rollover risk, and the interaction between horizontal alignment and superelevation is also an important factor.Significant single factor analysis shows that, with the increase of superelevation and the radius of horizontal curves, the safety margin of vehicle rollover increases, and the rollover of vehicles is less prone to occur. With the increase of vehicle speed, the safety margin (M2, M3) of vehicle rollover decreases, and the vehicle is more prone to rollover.In this paper, the safety margin of the vehicle’s lateral drift is characterized by the sideslip angle. From the results of orthogonal analysis and range analysis, we can get these conclusions. The vehicle speed and maximum tire-road friction coefficient have a highly significant effect on the vehicle’s lateral drift. The superelevation has a significant effect on the vehicle’s lateral drift. The interaction of horizontal alignment and superelevation and longitudinal slope has an important effect on the vehicle’s lateral drift. It cannot be ignored.Significant single factor analysis shows that, with the increase in superelevation, the safety margin of vehicles’ lateral drift increases, and the lateral drift of vehicles is less likely to occur. With the increase in vehicle speed and the maximum tire-road friction coefficient, the safety margin of the vehicle’s lateral drift decreases, and the lateral drift of vehicles is more likely to occur.The results of the abovementioned analysis show that the vehicle traveling risk can be reduced through improving and limiting the significant adverse factors for the lateral stability of vehicles. In our future research, besides considering the influence of a single factor on vehicle lateral stability, we should also consider the interaction among multiple factors.



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*Source: 7943739-2020-10-07.xml*