# A Simulation Study for Lateral Stability Control of Vehicles on Icy Asphalt Pavement

Authors: Yunsheng Zhu; Huacong Li; Kaifeng Wang; Yunhan Bao; Peng Zeng
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Black ice is an ice layer formed by freezing rain or accumulated water on the asphalt pavement surface in cold weather. This ice layer completely shields the texture structure of the pavement and destroys the original microstructure. The direct contact between the automobile tire and the ice surface leads to a sharp decrease in the adhesion coefficient, so the automobile is prone to lateral instability on the icy pavement. In this paper, the simulation model of the icy pavement is established in Matlab/Simulink to verify the control effect of the lateral stability controller based on the Electronic Stability Program under two steering limit conditions. The results show that the vehicle without a lateral stability controller will lose stability and sideslip even when it is steering at low speed on the icy pavement, and the lateral stability controller can effectively control the yaw rate of the vehicle when it is steering, which greatly reduces the offset of the sideslip angle of the centroid and inhibits the lateral acceleration exceeding the ice surface limit, which improves the maneuverability and stability of the vehicle under the freezing limit condition. The application of the controller is of great significance to improve the driving safety of the regional asphalt pavement. Due to the low adhesion coefficient of the icy pavement and the limited braking force and additional yaw moment of the tire provided by the adhesion force, the vehicle with a lateral stability controller is still likely to lose stability under the critical condition of medium or high-speed single shift line.


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## Body

## 1. Introduction

In the cold climate, pavement freezing often occurs in the mountainous area of southwest China and the humid area of South China. The lateral instability is likely to occur when the vehicle is running on icy pavements and the braking distance increases, resulting in a sharp decline in driving safety [1]. Especially, this kind of harsh road condition is more dangerous when it appears in curved sections where accidents are more frequent. Under the frozen condition of the icy pavement, the incidence of traffic accidents is about four times of normal pavement, according to statistics [2]. Therefore, traffic safety research of frozen pavement is extremely important. The existing research on icy pavement mainly focuses on the rapid detection of icy pavement, active deicing, dynamic response of icy pavement, innovation of pavement materials, and development of anticoagulants [3, 4]. There are few studies on the driving safety of the pavement that has been frozen. For icy pavements, the current mainstream application of deicing methods are manual, mechanical, chemical deicing, and other passive deicing methods and passive deicing has a long time lag between the occurrence of ice and deicing. During this interval, the pavement microstructure is completely covered by the ice layer, and the antiskid performance of the road surface is completely lost. It can only rely on the active control of the vehicle and the linear design of the road to ensure traffic safety. In fact, vehicles and pavements should be considered as a whole in driving safety. It should not only consider the mechanical properties of the pavement and road alignment characteristics but also analyze the antisliding stability of the active control measures when the vehicle is running. The driving safety includes lateral stability and longitudinal braking. The lateral stability of the vehicle refers to the ability of the vehicle to resist lateral rollover and lateral sideslip. On the low adhesion pavement, the vehicle mainly shows the risk of lateral sideslip. In recent years, with the rapid development of computer control technology and integrated circuit technology, the automotive electronic control system has been widely used in vehicle engineering [5].M Canale [6] designed a front-wheel steering controller by the nonlinear degree of membership prediction, which can better deal with the nonlinear input constraints of vehicle motion and improve the stability of the vehicle under extreme conditions. A fuzzy control algorithm was used to design a vehicle lateral stability control system with combined control of four-wheel drive and differential braking by B Li and F Yu, which improved vehicle maneuverability and stability [7]. According to the unit neural network algorithm, J Zhang [8] designed the lateral stability control, which can effectively deal with the nonlinear and time-varying of the vehicle control parameters, so that the vehicle control has good tracking and direction.Wang [9] used the yaw rate as the control variable and combined the feedback control of the PD algorithm with the feedforward control based on the feedback control output as the error to realize the adaptive control of the algorithm. Through this joint control algorithm, the active steering and yaw moment control were carried out to improve the lateral stability of the vehicle. XueNianwen [10] established a vehicle nonlinear model and designed a fuzzy controller based on fuzzy control theory, which was controlled by sideslip angle feedback control, yaw rate feedback control, and joint feedback control of these two parameters to improve vehicle stability on the wet-skid pavement. Wang Cui [11] used adaptive fuzzy PID control to control ABS. The test showed that the lateral stability of the vehicle was improved when it was running at high speed on the curved pavement.Combined with the above research and related literature, it can be seen that although there are many studies on the lateral stability control of automobiles by researchers and automobile manufacturers at home and abroad, and the research methods are diverse, the control strategies and algorithms studied are mostly for the limit conditions of general roads, and only the common snow or wet ski roads in the driving of vehicles are considered in the conditions of low adhesion roads. There are few studies on the control of icy pavement, which is a very low adhesion road. In this article, the influence of the lateral stability controller on the steering stability and maneuverability of the vehicle on icy pavement with a low adhesion coefficient is studied through computer simulation. Eight-degree-of-freedom vehicle model and magic formula model are established in Matlab/Simulink to simulate the icy pavement, and a single lane change simulation test and front wheel angle overstep simulation test are performed. The automobile lateral stability controller, which takes the yaw rate as the target variable, is based on the direct yaw moment control strategy and the sliding mode variable structure control algorithm. The controller uses unilateral double-wheel braking to obtain the larger additional yaw moment of the vehicle on the icy pavement, and the braking force distribution is designed according to the limitation of the adhesion ellipse. The application of the controller can effectively solve the handling stability problem of the vehicle after icing on asphalt pavement in southwest mountainous areas of China in winter, which is of great significance to improve the traffic safety of regional asphalt pavement and reduce the traffic accident rate.


## 2. Establishment of the Vehicle Dynamics Model

Computer simulation research is an effective means of vehicle stability control research in the early stage, and real vehicle test research is a necessary verification means of vehicle stability control research. However, the real vehicle test requires very high hardware equipment, the experimental process is complex and difficult to control and costly, and the icy pavement conditions are difficult to simulate and very dangerous. After years of development, vehicle modeling and simulation research have been very mature and reliable. In this chapter, the vehicle dynamics simulation test of the lateral stability controller designed for the icy pavement is carried out to verify its control effect on the vehicle steering on the icy pavement.


### 2.1. Eight Degrees of Freedom Vehicle Model

The premise of the vehicle dynamics simulation test is to establish a suitable vehicle dynamics model, and the rationality of the vehicle model will directly affect the accuracy of the simulation test. 2-DOF (degree of freedom) vehicle model includes only two degrees of freedom of lateral movement along they axis and yaw motion around z, which is too simple to meet the high precision requirement in vehicle simulation research. To simulate the actual situation of the vehicle more accurately, nonlinear parameters such as vehicle suspension and tire need to be introduced to establish the model. The vehicle models of four wheels and multiple degrees of freedom including five degrees of freedom, seven degrees of freedom, eight degrees of freedom, ten degrees of freedom, fifteen degrees of freedom, and twenty-nine degrees of freedom are commonly used in automobile dynamics simulation tests. In general, the higher the degrees of freedom for vehicle models are, the more the parameters are considered, and the higher the accuracy of the dynamic simulation is. However, the parameters in the complex model are hard to obtain, which makes the simulation much more difficult. Therefore, researchers usually select the appropriate vehicle model for the research object to meet the accuracy requirements and reduce the difficulty of obtaining parameters.According to the actual need of the controller simulation, an 8-DOF vehicle model is established, which is shown in Figure1. The rotational motion of the model consists of eight degrees of freedom for motion along the x-axis and y-axis, yaw motion around the z-axis, and roll motion of the body.Figure 1
A 4-wheel 8-DOF vehicle mechanical model schematic diagram.The following assumptions are made in the establishment of the 8-DOF vehicle model:(1)
There is no movement on theZ-axis;(2)
The performance parameters of each tire are identical;(3)
The steering angle change of the two pairs of front and rear wheels is the same.Through the analysis of the four-wheel and eight-degree-of-freedom vehicle mechanical model, the balance equation [12] for the lateral motion, longitudinal motion, yaw motion, and roll motion of the entire vehicle are as follows:(1)mvy−rvx+mbh1ϕ=Fyfl+Fyfrcosδ+Fxfl+Fxfrsinδ+Fyrl+Fyrr,mvx−rvy+mbh1ϕ=Fxfl+Fxfrcosδ−Fyfl+Fyfrsinδ+Fxrl+Fxrr,wr·=1IzzFyfl−Fyfrsinδ⋅bs2+Fxfr−Fxflcosδ⋅bs2+Fxfl+Fxfrsinδ⋅a+Fyfl+Fyfr⋅cosδ⋅a+Fxrr−Fxrlbs2−Fyrl+Fyrrb,Ixxϕ+Bϕϕ+Kϕϕ=mbgh1ϕ+mbh1vy+rvx.When the vehicle is running, the sideslip angle of each wheel is shown in the following equation:(2)αfl=−δf+arctanar+vyvx+0.5bsr,αfr=−δf+arctanar+vyvx−0.5bsr,αrl=arctan−br+vyvx+0.5bsr,αrr=arctan−br+vyvx−0.5bsr.The dynamic load of each wheel is affected by vehicle movements such as braking, acceleration, and steering. When the vehicle brakes, the load of the front wheels increases, and the load of the rear wheels load decreases, while the situation during acceleration is opposite to that during braking. When the vehicle turns, the vertical load of the outer wheels increases while that of the inner wheels decreases. The vertical load of the wheel affects the lateral force to a large degree. Therefore, it is necessary to consider the influence of vertical load when establishing the model. The mathematical model of load transfer is shown as follows:(3)Fzfi=mgb2a+b+max−h2a+b∓ay1bsmbhfba+b+mufhuf∓aycosϕ+gsinϕmbh1KϕfbsKϕf+Kϕr,Fzri=mga2a+b+maxh2a+b∓ay1bsmbhraa+b+murhur∓aycosϕ+gsinϕmbh1KϕrbsKϕf+Kϕr.Where I can take l or r, representing the meaning of left or right; m is the mass of the whole vehicle; mb is the sprung mass; muf is the unsprung mass of the front axle; mur is the unsprung mass of the rear axle; g is acceleration of gravity; vx is longitudinal velocity; vy is lateral velocity; ax is longitudinal acceleration; ay is lateral acceleration; ϕ is the body roll angle; r is the yaw rate; δ is the front wheel angle; Fxfl, Fxfr,Fxrl, Fxrr are the longitudinal forces of front wheel, right front wheel, left rear wheel and right rear wheel respectively; Fyf,Fyfr,Fyrl,Fyrr are the lateral force of left front wheel, right front wheel, left rear wheel, right rear wheel; Fzfi,Fzriare the vertical force of front wheel and rear wheel, respectively; Bϕ is the suspension roll damping; Kϕ is the suspension roll stiffness; Kϕf is the equivalent roll stiffness of the front suspension; Kφr is the equivalent roll stiffness of the rear suspension; Ixx、Izz are the moments of inertia; h is the height of the vehicle’s center of mass; h1 is the distance from the center of mass of the vehicle to the roll center; hf is the roll center height of the sprung mass of the front axle; hr is the roll center height of the sprung mass of the rear axle; huf is height of roll center of underspring mass of front axle; hur is height of roll center of underspring mass of rear axle; a is the distance from the front axle to the center of mass; b is the distance from the rear axle to the center of mass; bs is the wheelbase.The vehicle model is based on the common class B passenger cars. The detailed parameters of the vehicle body [13] are shown in Table 1.Table 1
Vehicle parameters.
Parameter nameParameter valueParameter nameParameter valueVehicle qualitym/kg1380Wheelbasebs/m1.48Sprung massmb /kg1200Suspension roll dampingBϕ /(Nm·rad−1)5488Distance from center of mass to front/rear axis (a/b)/m1.13/1.47Suspension roll stiffnessKϕ /(Nm·rad−1)84711Distance from the front axle to rear axleL/m2.60Body yaw rotational inertiaIzz/(kg·m2)2008Height from center of mass to roll axish/m0.28Body roll rotational inertiaIxx/(kg·m2)492According to the above mathematical model and detailed vehicle parameters, the 8-DOF vehicle model and load transfer model are established in Matlab/Simulink.


### 2.2. “Magic Formula” Tire Model

As an important part of the vehicle model, the tire parameters and mechanical properties affect the main performance of the model. To ensure that the model is closer to the real vehicle, nonlinear means should be adopted to simulate the actual working condition of the tire. The combination formula of a trigonometric function is used to fit the experimental tire data in H.B.Pacejka’s 'magic formula’ tire model, and a set of formulas in the same form can fully express the longitudinal force, lateral force, aligning torque, and the combination of longitudinal force and lateral force. This model has strong uniformity and can describe all the steady-state mechanical properties of the tire accurately. This model is easy to be programmed because of fewer fitting parameters. In addition, the magic formula is based on experimental data, which can be extrapolated to limited conditions such as freezing pavement and has good credibility [14]. Therefore, the ’magic formula’ tire model is used to establish the vehicle dynamics model in this paper.The general expression of the magic equation is as follows:(4)Y=y+Sv,y=DsinCarctanBx−EBx−arctanBx,x=X+Sh,where Y is the lateral moment, longitudinal moment, or return moment; X is the slip angle or wheel slip rate; B, C, D, E are the parameters that determine the curve stiffness, shape, peak value, and curvature, which are determined by driving environment, vehicle speed and vehicle load; SV is vertical drift; Sh is horizontal drift. Except for the shape factor C of the curve, every other parameter is the function of vertical load Fz and banking angle γ, which is obtained by parameter fitting.The lateral force formula of the tire only under the condition of lateral deviation and the formula of the longitudinal force only under slip conditions are as follows:(5)Fx=DxsinCxarctanBxλ−ExBxλ−arctanBxλ,Fy=DysinCyarctanByα−EyByα−arctanByα,where λ is the tire longitudinal slip rate; Dx=a1Fz2+a2Fz; Bx=BCDx/C⋅Dx, BCDx=a3Fz2+a4Fze−a5Fz; Ex=a6Fz2+a7Fz+a8; α is the tire slip angle， Dy=a1Fz2+a2Fz; By=BCDy/C⋅Dy, BCDy=a3sina4arctanFz/a5; Ey=a6Fz+a7; a1,a2…,a8 is the fitting parameter, specific values [15] are shown in Table 2.Table 2
Tire fitting parameters.
Ca1a2a3a4a5a6a7a8Fx1.66−21.1114549.72270.068−0.0070.0550.486Fy1.31−22.2101010771.810.2090−0.3550.706When the tire vertical load is 2.9 kN, the relationship between tire lateral force and tire sideslip angle under different pavement adhesion coefficients is shown in Figure2. It can be seen from Figure 2 that when the tire sideslip angle is constant, the greater the pavement adhesion coefficient, the greater the tire lateral force [16]. The limit of tire lateral force under different pavement adhesion coefficients is different. Especially on the low adhesion pavement, the tire lateral force is easy to break through the limit. Therefore, the influence of the pavement adhesion coefficient on tire side slid properties is an important factor affecting vehicle handling and stability. The magic tire fitting parameter is corrected according to the pavement adhesion coefficient. The correction formula is as follows:(6)Fy=Dy∗sinCy∗arctanBy∗α−EyBy∗α−arctanBy∗α,where Cy∗=5/4−μ/4Cy; Dy∗=μDy; By∗=2−μBy; μ is pavement adhesion coefficient.Figure 2
Tire cornering characteristics under different adhesion coefficients.According to the above magic formula and correction formula of adhesion coefficient, the corresponding magic formula tire model is established in Simulink.


## 2.1. Eight Degrees of Freedom Vehicle Model

The premise of the vehicle dynamics simulation test is to establish a suitable vehicle dynamics model, and the rationality of the vehicle model will directly affect the accuracy of the simulation test. 2-DOF (degree of freedom) vehicle model includes only two degrees of freedom of lateral movement along they axis and yaw motion around z, which is too simple to meet the high precision requirement in vehicle simulation research. To simulate the actual situation of the vehicle more accurately, nonlinear parameters such as vehicle suspension and tire need to be introduced to establish the model. The vehicle models of four wheels and multiple degrees of freedom including five degrees of freedom, seven degrees of freedom, eight degrees of freedom, ten degrees of freedom, fifteen degrees of freedom, and twenty-nine degrees of freedom are commonly used in automobile dynamics simulation tests. In general, the higher the degrees of freedom for vehicle models are, the more the parameters are considered, and the higher the accuracy of the dynamic simulation is. However, the parameters in the complex model are hard to obtain, which makes the simulation much more difficult. Therefore, researchers usually select the appropriate vehicle model for the research object to meet the accuracy requirements and reduce the difficulty of obtaining parameters.According to the actual need of the controller simulation, an 8-DOF vehicle model is established, which is shown in Figure1. The rotational motion of the model consists of eight degrees of freedom for motion along the x-axis and y-axis, yaw motion around the z-axis, and roll motion of the body.Figure 1
A 4-wheel 8-DOF vehicle mechanical model schematic diagram.The following assumptions are made in the establishment of the 8-DOF vehicle model:(1)
There is no movement on theZ-axis;(2)
The performance parameters of each tire are identical;(3)
The steering angle change of the two pairs of front and rear wheels is the same.Through the analysis of the four-wheel and eight-degree-of-freedom vehicle mechanical model, the balance equation [12] for the lateral motion, longitudinal motion, yaw motion, and roll motion of the entire vehicle are as follows:(1)mvy−rvx+mbh1ϕ=Fyfl+Fyfrcosδ+Fxfl+Fxfrsinδ+Fyrl+Fyrr,mvx−rvy+mbh1ϕ=Fxfl+Fxfrcosδ−Fyfl+Fyfrsinδ+Fxrl+Fxrr,wr·=1IzzFyfl−Fyfrsinδ⋅bs2+Fxfr−Fxflcosδ⋅bs2+Fxfl+Fxfrsinδ⋅a+Fyfl+Fyfr⋅cosδ⋅a+Fxrr−Fxrlbs2−Fyrl+Fyrrb,Ixxϕ+Bϕϕ+Kϕϕ=mbgh1ϕ+mbh1vy+rvx.When the vehicle is running, the sideslip angle of each wheel is shown in the following equation:(2)αfl=−δf+arctanar+vyvx+0.5bsr,αfr=−δf+arctanar+vyvx−0.5bsr,αrl=arctan−br+vyvx+0.5bsr,αrr=arctan−br+vyvx−0.5bsr.The dynamic load of each wheel is affected by vehicle movements such as braking, acceleration, and steering. When the vehicle brakes, the load of the front wheels increases, and the load of the rear wheels load decreases, while the situation during acceleration is opposite to that during braking. When the vehicle turns, the vertical load of the outer wheels increases while that of the inner wheels decreases. The vertical load of the wheel affects the lateral force to a large degree. Therefore, it is necessary to consider the influence of vertical load when establishing the model. The mathematical model of load transfer is shown as follows:(3)Fzfi=mgb2a+b+max−h2a+b∓ay1bsmbhfba+b+mufhuf∓aycosϕ+gsinϕmbh1KϕfbsKϕf+Kϕr,Fzri=mga2a+b+maxh2a+b∓ay1bsmbhraa+b+murhur∓aycosϕ+gsinϕmbh1KϕrbsKϕf+Kϕr.Where I can take l or r, representing the meaning of left or right; m is the mass of the whole vehicle; mb is the sprung mass; muf is the unsprung mass of the front axle; mur is the unsprung mass of the rear axle; g is acceleration of gravity; vx is longitudinal velocity; vy is lateral velocity; ax is longitudinal acceleration; ay is lateral acceleration; ϕ is the body roll angle; r is the yaw rate; δ is the front wheel angle; Fxfl, Fxfr,Fxrl, Fxrr are the longitudinal forces of front wheel, right front wheel, left rear wheel and right rear wheel respectively; Fyf,Fyfr,Fyrl,Fyrr are the lateral force of left front wheel, right front wheel, left rear wheel, right rear wheel; Fzfi,Fzriare the vertical force of front wheel and rear wheel, respectively; Bϕ is the suspension roll damping; Kϕ is the suspension roll stiffness; Kϕf is the equivalent roll stiffness of the front suspension; Kφr is the equivalent roll stiffness of the rear suspension; Ixx、Izz are the moments of inertia; h is the height of the vehicle’s center of mass; h1 is the distance from the center of mass of the vehicle to the roll center; hf is the roll center height of the sprung mass of the front axle; hr is the roll center height of the sprung mass of the rear axle; huf is height of roll center of underspring mass of front axle; hur is height of roll center of underspring mass of rear axle; a is the distance from the front axle to the center of mass; b is the distance from the rear axle to the center of mass; bs is the wheelbase.The vehicle model is based on the common class B passenger cars. The detailed parameters of the vehicle body [13] are shown in Table 1.Table 1
Vehicle parameters.
Parameter nameParameter valueParameter nameParameter valueVehicle qualitym/kg1380Wheelbasebs/m1.48Sprung massmb /kg1200Suspension roll dampingBϕ /(Nm·rad−1)5488Distance from center of mass to front/rear axis (a/b)/m1.13/1.47Suspension roll stiffnessKϕ /(Nm·rad−1)84711Distance from the front axle to rear axleL/m2.60Body yaw rotational inertiaIzz/(kg·m2)2008Height from center of mass to roll axish/m0.28Body roll rotational inertiaIxx/(kg·m2)492According to the above mathematical model and detailed vehicle parameters, the 8-DOF vehicle model and load transfer model are established in Matlab/Simulink.


## 2.2. “Magic Formula” Tire Model

As an important part of the vehicle model, the tire parameters and mechanical properties affect the main performance of the model. To ensure that the model is closer to the real vehicle, nonlinear means should be adopted to simulate the actual working condition of the tire. The combination formula of a trigonometric function is used to fit the experimental tire data in H.B.Pacejka’s 'magic formula’ tire model, and a set of formulas in the same form can fully express the longitudinal force, lateral force, aligning torque, and the combination of longitudinal force and lateral force. This model has strong uniformity and can describe all the steady-state mechanical properties of the tire accurately. This model is easy to be programmed because of fewer fitting parameters. In addition, the magic formula is based on experimental data, which can be extrapolated to limited conditions such as freezing pavement and has good credibility [14]. Therefore, the ’magic formula’ tire model is used to establish the vehicle dynamics model in this paper.The general expression of the magic equation is as follows:(4)Y=y+Sv,y=DsinCarctanBx−EBx−arctanBx,x=X+Sh,where Y is the lateral moment, longitudinal moment, or return moment; X is the slip angle or wheel slip rate; B, C, D, E are the parameters that determine the curve stiffness, shape, peak value, and curvature, which are determined by driving environment, vehicle speed and vehicle load; SV is vertical drift; Sh is horizontal drift. Except for the shape factor C of the curve, every other parameter is the function of vertical load Fz and banking angle γ, which is obtained by parameter fitting.The lateral force formula of the tire only under the condition of lateral deviation and the formula of the longitudinal force only under slip conditions are as follows:(5)Fx=DxsinCxarctanBxλ−ExBxλ−arctanBxλ,Fy=DysinCyarctanByα−EyByα−arctanByα,where λ is the tire longitudinal slip rate; Dx=a1Fz2+a2Fz; Bx=BCDx/C⋅Dx, BCDx=a3Fz2+a4Fze−a5Fz; Ex=a6Fz2+a7Fz+a8; α is the tire slip angle， Dy=a1Fz2+a2Fz; By=BCDy/C⋅Dy, BCDy=a3sina4arctanFz/a5; Ey=a6Fz+a7; a1,a2…,a8 is the fitting parameter, specific values [15] are shown in Table 2.Table 2
Tire fitting parameters.
Ca1a2a3a4a5a6a7a8Fx1.66−21.1114549.72270.068−0.0070.0550.486Fy1.31−22.2101010771.810.2090−0.3550.706When the tire vertical load is 2.9 kN, the relationship between tire lateral force and tire sideslip angle under different pavement adhesion coefficients is shown in Figure2. It can be seen from Figure 2 that when the tire sideslip angle is constant, the greater the pavement adhesion coefficient, the greater the tire lateral force [16]. The limit of tire lateral force under different pavement adhesion coefficients is different. Especially on the low adhesion pavement, the tire lateral force is easy to break through the limit. Therefore, the influence of the pavement adhesion coefficient on tire side slid properties is an important factor affecting vehicle handling and stability. The magic tire fitting parameter is corrected according to the pavement adhesion coefficient. The correction formula is as follows:(6)Fy=Dy∗sinCy∗arctanBy∗α−EyBy∗α−arctanBy∗α,where Cy∗=5/4−μ/4Cy; Dy∗=μDy; By∗=2−μBy; μ is pavement adhesion coefficient.Figure 2
Tire cornering characteristics under different adhesion coefficients.According to the above magic formula and correction formula of adhesion coefficient, the corresponding magic formula tire model is established in Simulink.


## 3. Simulation Analysis of Vehicle Lateral Stability Controlleron Icy Pavement

To verify the control effect of the designed vehicle lateral stability controller, this paper analyzes the difference in the motion state of the vehicle when there is a lateral stability control system and when there is no stability control system on the Matlab/Simulink simulation platform built. The stability control system in this paper is specially designed for ice-freezing roads, and the simulation test will be carried out under the condition of extremely low ice-freezing adhesion. In the development of the lateral stability system of the vehicle, the simulation tests of two steering limit conditions are usually carried out on the vehicle to verify the effect of the controller. If the lateral stability controller can have a good stability control effect under the steering limit condition, the lateral safety of the normal driving of the vehicle is guaranteed. The two steering limit conditions are general single-line-shifting conditions and front-wheel angle oversteps conditions. The road adhesion coefficient is set according to the most dangerous smooth ice surface. From the above, the road adhesion coefficient is 0.1 [17].


### 3.1. Lateral Stability Controller Structure

The lateral stability controller is based on the automotive ESP system. The overall structure of the lateral stability controller is shown in Figure3. First, the driver inputs the steering wheel angle of the car. The car runs on the pavement and the vehicle motion parameters are collected through sensors. The actual motion parameters output by the car are used as the ideal reference model for the car. In the ideal model, the ideal value of the target variable in the current vehicle motion state is output by calculation. Then the steady-state of the vehicle is judged according to the difference between the actual value of the target variable output by the vehicle model and the ideal value of the target variable. When the state of the vehicle does not meet the lateral stability condition, the lateral stability controller starts to work. The automatic control algorithm of the controller determines the specific control measures for vehicle instability by analyzing the parameters such as the difference of yaw rate and the differential of difference/time and combining it with the stability control strategy of the controller. The control signal is input into the automotive electronic system to form a closed-loop control to ensure the lateral stability of the vehicle. The corresponding control model is established according to the specific design of each module in Matlab/Simulink for the simulation verification of the lateral stability controller.Figure 3
The overall structure of the vehicle lateral stability controller.After determining the design idea of vehicle lateral stability control, the yaw rate is determined as the target variable of lateral stability control, and the ideal yaw rate is modified according to the freezing condition. The lateral stability controller uses direct yaw moment control (DYC) as the control strategy, and its characteristics can be well explained by the following examples: In Figure4, when the vehicle steering shows excessive understeer, the positive additional yaw moment can be generated for the brake of inner wheel ①③ to correct the understeer trend; when the vehicle steering shows excessive steering, the lateral wheel ②④ brake can produce negative additional yaw moment to correct excessive steering trend. This is the differential braking control mode of direct yaw moment. In order to obtain a better control effect, this paper adopts unilateral two-wheel braking to obtain a greater additional yaw moment on the ice surface. According to the limitation of the ice surface attachment ellipse, the braking force distribution and calculation model are designed.Figure 4
Car steering diagram.In this paper, the sliding mode variable structure control algorithm is designed to control the additional yaw moment of the output. It can still have good control accuracy without a large number of test data debugging parameters. Moreover, the sliding mode control is essentially a nonlinear control, with the nature of uncertain system structure, and it will change rapidly with the change of the current state of the control system. Therefore, it has strong tracking, adaptability, and robustness. The ultralow adhesion characteristics of the iced pavement and the strong nonlinearity of the vehicle driving make the lateral stability control of the vehicle need to have the ability to track the target variable quickly and have a strong anti-interference ability. So sliding mode variable structure control is very suitable for vehicle lateral stability control under freezing conditions.The input of the sliding mode variable structure control algorithm module is the differencee obtained by the difference between the actual yaw rate ωr of the vehicle curve motion output and the ideal yaw rate ω0 of the ideal model output; the output is the additional yaw moment Mz calculated by the control algorithm. The design of the sliding mode controller is mainly divided into sliding mode control surface design and sliding mode control law design.The controller is designed based on a linear two-degree-of-freedom model with an additional yaw momentMz.According to the linear two-degree-of-freedom vehicle motion differential equation:(7)x·=Ax+Bδf+CMz,where A=k1+k2/mvak1−bk2/mv2−1ak1−bk2/Iza2k1+b2k2/Izv; B=−k1/mv−ak1/Iz; C=01/Iz; x=β,ωdT , Vehicle state variables, k1 and k2, are the lateral stiffness of front and rear wheels; v is centroid velocity; IZ is the rotational inertia of the car around the Z axis; ωd is steady yaw rate.According to the sliding mode control theory and the nonlinear system of vehicle control, the following switching functions are selected to further reduce the tracking error:(8)s=e+λ0∫0tetdt,where e=ωr−ω0; λ0 is a positive weighting factor; ∫0tetdt is mainly used to limit the steady-state error.Taking the derivation of (8), we can get the following:(9)s·=e·+λe=ωr·−ω0·+λωr−ω0.From formula (7), we can get the following:(10)ωr·=ak1−bk2Izβ+a2k1+b2k2Izvωr−ak1Izδf+MzIz.Lets·=0 to get the equivalent control input:(11)Mz_equ=−Izbk2−ak1Izβ−a2k1+b2k2Izvωr+ak1Izδf+λ0ωr−ω0.In order to ensure the sliding mode condition of the system under various conditions, the equal-speed reaching law design method is adopted, and the control law is defined as follows:(12)Mz=Mz_equ−ksgns.where k is the rate of the moving point of the system approaching the switching surface, the greater the k is, the faster the approaching speed is, but the jitter is also large; sgn· is the symbolic function.The choice ofk needs to satisfy the sliding mode reachability condition:(13)12ddts2=s⋅s·≤−η0s,where η0 is a positive real number.The discontinuous sign functionsgn· is replaced by a continuous saturation function to further weaken the chattering that may occur in the control system:(14)satx=sgnx,x>1,x,x≤1.Therefore, the final sliding mode control law is as follows:(15)Mz=Mz_equ−ksats/ε,where ε is the sliding mode variable structure thickness, the parameter value selection affects the smoothness of the control system.


### 3.2. Adhesion Coefficient of Frozen Pavement

The changes in the adhesion coefficient are one of the main reasons that affect the pavement condition [8]. In the double exponential model, the pavement surface adhesion coefficient is expressed as a nonlinear function containing three parameters of pavement condition, vehicle speed, and load. Therefore, the variation characteristics of the pavement adhesion coefficient can be well satisfied. The pavement adhesion characteristic parameters under various pavement conditions and driving conditions in the double exponential model are shown as follows:(16)μxp=0.92×0.1304σ+0.002eσ64−v−0.0426u,μxs=0.677×0.1553σ+0.002eσ64−v,Sxm=0.1532σ3−0.47801σ2+0.256σ+0.1693+0.105lg64/v,where μxp, Sxm are the peak adhesion coefficient and the corresponding wheel slip rate; μxs is the pavement adhesion coefficient when the wheels are completely slipped; v is the actual vehicle speed; u is the wheel load factor = actual wheel load/calibrated wheel load; σ is the influencing factor of the adhesion characteristic that characterizes the pavement condition, based on a large number of experimental data, the specific values can be seen in Table 3 Pavement-tire double exponential model is as follows:(17)μ=Aexp−bSxm−exp−aSxm.Table 3
Characteristic factor of common asphalt pavementsσ.
Pavement situationAsphalt (dry)Asphalt (wet)Snow (compaction)Icy pavement (dry)Icy pavement (wet)σ00.1340.751.01.2The characteristic adhesion parameter values required in (17) are given in the following formulas, where equation (18) is iteratively solved for a, and iteration is stopped when ak+1−ak/ak≤10−3 and a=ak+1.(18)ak+1=1Sxmlnμxp/μxs−Sxmak−1μxp/μxs−1,A=μxs1−aexp−aSxm,b=−ln1−aexpaSxm.According to the above double-exponential model [9], it can be known that the adhesion coefficient of asphalt pavement decreases sharply to about 0.1 when the pavement is frozen. Therefore, the adhesion coefficient of the smooth icy surface is 0.1 in the study of vehicle lateral stability [10, 11, 14].


### 3.3. Evaluation Standard of Vehicle Stability Control System

In the simulation test of vehicle stability control, due to the sharp steering of the wheel on the icy pavement with low adhesion, the vehicle will inevitably slip and cause the controller to intervene. The controller effect is evaluated by the simulation results according to the following standards:(1)
When the controller intervenes and applies the additional yaw moment control, it should be able to effectively control the sideslip of the vehicle, that is, to curb the sideslip angle of the centroid of the vehicle within a stable range and to make the yaw rate track the ideal yaw rate under the same working condition as far as possible, so that the vehicle is easy to control. It can be known from [1] that on the ice and snow pavement, the limit eigenvalue of the sideslip angle of the vehicle mass center is about ±2°. If it exceeds this eigenvalue, the vehicle will be difficult to control and lead to a radical sideslip.(2)
The lateral acceleration of the car during steering should not exceed the adhesion limit of the pavementu⋅g. The lateral acceleration on the smooth icy surface is 0.1 g.(3)
The response of each parameter is fast, the stable adjustment time is short, and the overshoot cannot be too large, which meets the requirements of rapidity and stability of the control system response.


### 3.4. Simulation Analysis of Single Lane Change for Icy Pavement

There is the sinusoidal input of the steering wheel angle in the single lane change condition, which is corresponded with rapid glide lane change for emergency obstacle avoidance. The simulation condition setting is shown in Figure5: Input a cycle sine steering wheel angle signal at 2s, the maximum input angle is 1.44 rad, the frequency is 0.4HZ, the ratio of steering wheel angle to front wheel angle is 18 : 1, and the road adhesion coefficient is 0.1 (icy pavement). When the car is turning, its driving stability is related to its front wheel angle, pavement adhesion coefficient, and driving speed. The initial speeds of 30 km/h and 50 km/h are selected to analyze the control effect of the stability controller under different speeds according to the experimental results; variation trend is shown in Figures 6-7.Figure 5
Sine input of steering wheel angle.Figure 6
Simulation results of single lane change at an initial velocity of 30 km/h (a) Change in yaw velocity (b) Change in sideslip angle (c) Vehicle trajectory (d) Change in lateral acceleration.
(a)(b)(c)(d)Figure 7
Simulation results of single lane change at an initial velocity of 50 km/h. (a) Change in yaw velocity. (b) Changes in sideslip angle. (c) Vehicle trajectory. (d) Change in lateral acceleration.
(a)(b)(c)(d)The simulation results of the single lane change on the icy pavement at the initial speed of 30 km/h are shown in Figure 6.It can be seen from Figure6 that the yaw velocity, sideslip angle, and lateral acceleration of the vehicle equipped with the controller are in good agreement with the ideal value. It means that the controller has a good correction effect on the vehicle trajectory. Figure 6(a) shows that the vehicle yaw rate with the controller only has a small viscous response in its mutation stage, and the maximum stability deviation is only 0.003 rad/s. The vehicle yaw rate without a controller has a large deviation, which is 66% higher than the maximum ideal yaw rate. It can be seen from Figure 6(b) that the maximum centroid sideslip angle of the vehicle without the lateral stability controller reaches 0.034 rad (1.97°), which is close to the eigenvalue of the centroid sideslip angle on the icy pavement. The centroid sideslip angle of the vehicle equipped with the stability controller is limited to 0.007 rad (0.42°), which is far less than the centroid sideslip angle of the vehicle without the control. It is shown in Figure 6(c) that the vehicle without the controller has slipped at the first corner peak and flew directly out of the desired trajectory at the second corner peak, indicating that the vehicle without a stable controller can no longer travel stably on the icy pavement at a speed of 30 km/h. The allowable adhesion limit of the tire on the icy surface is much smaller than that on the high adhesion pavement, and the tire is more likely to get in a nonlinear state. Therefore, it is difficult for the vehicle to follow the desired trajectory of the driver when making a single lane change on the icy surface. The vehicle equipped with the controller has completed the single lane change. Although the direction has slightly deviated, it can be easily corrected by turning the steering wheel based on vehicle stability. It can be seen from Figure 6(d) that the lateral acceleration of the vehicle without a controller is in a nonlinear region, and the numerical change is uncertain. The maximum value has reached the allowable maximum lateral acceleration 0.1 g of the adhesion limit, and the vehicle is in an unstable state. The lateral acceleration of the vehicle with the controller changes linearly, which is consistent with the change of yaw velocity and does not reach the maximum acceleration limit.The simulation results of the single lane change on the icy pavement at the initial speed of 50 km/h are shown in Figure 7.It can be seen from Figure7 that the vehicle equipped with the controller has a certain control effect on the yaw velocity, which still has a certain offset relative to the ideal yaw rate, especially at the peak value of the second corner. The maximum offset reaches 0.08 rad/s, 133% higher than the ideal yaw velocity, and the maximum value of the sideslip angle reaches 0.016 rad (0.92°). The vehicle trajectory is slipped obviously, and the lateral acceleration changes nonlinearly at the peak value of the second corner. The maximum value is close to the limit of 0.1 g, and the vehicle stability is reduced.The above analysis shows that when the car avoids obstacles and changes lanes on the icy surface (adhesion coefficient 0.1) at a low speed (30 km/h), the controller has a good control effect on the car; When the speed reaches 50 km/h, under extreme condition of single lane change, the effect of the controller decreases and the vehicle slips.By analyzing the driving control process at a speed of 50 km/h, the braking force of each wheel provided by the controller and the maximum braking force of each wheel provided by the pavement adhesion coefficient during the stability control process can be obtained, which are shown in Figure8.Figure 8
The braking force of each wheel applied by the controller in the state of 50 km/h single lane change. (a) Left front wheel braking force. (b) Left rear wheel braking force. (c) Right front wheel braking force. (d) Right rear wheel braking force.
(a)(b)(c)(d)In Figure8, the dotted line represents the maximum braking force provided by the pavement adhesion coefficient calculated by the adhesion ellipse, and the solid line represents the actual braking force output by the controller. It can be seen that the braking force of each wheel reaches the upper limit value of the braking force provided by the icy surface adhesion. The correction effect of the controller on the vehicle attitude has reached the limit. Therefore, the vehicle can only be stabilized by increasing the road adhesion coefficient. When the pavement adhesion coefficient increases to 0.2, the simulation results of single lane change at the initial speed of 50 km/h are shown in Figure 9.Figure 9
Simulation results of single lane change with adhesion coefficient of 0.2 and initial velocity of 50 km/h. (a) Change in yaw velocity. (b) Changes in sideslip angle. (c) Vehicle trajectory. (d) Change in lateral acceleration.
(a)(b)(c)(d)It can be seen from Figure 9 that after the adhesion coefficient increases from 0.1 to 0.2, the yaw velocity of the car equipped with the yaw controller can rematch the ideal value, the sideslip angle is also stabilized in a small range, and the car’s trajectory shows a perfect single lane change, the controller has an excellent effect on car stability control.


### 3.5. Simulation Analysis of front Wheel Angle Overstep on Freezing Pavement

In the vehicle dynamics simulation, the steering wheel angle is input step by step, which is corresponded with the driving condition that the car changes from a straight line to a curve with a certain angle of the front wheel. In order to be more suitable for the continuous steering characteristics of the vehicle in the actual driving from the straight to the curved road, and taking into account the danger of the icy pavement, this paper makes an appropriate correction on the front wheel angle. The front-wheel angle is no longer directly overstepped to the peak but has a short continuous steering process. Specific simulation conditions of steering wheel angle input are shown in Figure10. The initial velocity of 40 km/h and 100 km/h are taken, respectively. The ice surface adhesion coefficient is 0.1. The peak value of the front wheel angle is set to 4°, and the vehicle steering radius is far less than the minimum radius of general Figure 11 route Figure 12 design.Figure 10
Steering wheel angle step input.Figure 11
Simulation results of front-wheel angle overstep at an initial speed of 100 km/h. (a) Change of yaw rate. (b) Change of sideslip angle of centroid. (c) Change of lateral acceleration.
(a)(b)(c)Figure 12
Simulation results of front-wheel angle overstep at an initial speed of 40 km/h. (a) Change of yaw rate. (b) Change of sideslip angle of centroid. (c) Change of lateral acceleration.
(a)(b)(c)The simulation results of front wheel angle overstep at an initial speed of 40 km/h are shown in Figure12The simulation results of front-wheel angle overstep at an initial speed of 100 km/h are shown in Figure11It can be seen from Figure12 that when the front wheel angle step input is made at 40 km/h on the ice surface without the vehicle controller, both the yaw rate and the sideslip angle of the center of mass oscillate greatly, and the vehicle slips and loses its stability.The yaw rate of a vehicle equipped with a lateral stability controller has a good tracking effect on the ideal yaw rate, which coincides with the ideal curve. The slip angle of the center of mass and lateral acceleration are suppressed obviously, and the amplitude is reduced. The controller ensures the stability of the car when turning on icy pavement.Although the sideslip angle of the vehicle is limited to a small amplitude of 0.014 rad(0.8°) to keep the vehicle in a stable state, the sideslip angle of the vehicle is not completely stable. One of the reasons is that the control target variable of the lateral stability controller is yaw rate. A more important reason is that the ideal yaw rate of the controller is not necessarily the steady yaw rate of the vehicle’s circular motion. The ideal yaw rate is limited by the road adhesion coefficient, and the value is the maximum value of the yaw rate allowed by the road adhesion coefficient and the minimum value in the steady-state yaw rate. When the car does the front wheel angle overstep simulation on the icy pavements with minimal adhesion, the steady-state yaw rate exceeds the limit value of the ice surface throughout. Therefore, the ideal yaw rate is not the yaw rate of the steady-state circular motion when the vehicle is in a circular motion close to the steady-state on the icy pavements and the corresponding sideslip angle of the mass center naturally cannot converge well. Nevertheless, the sideslip angle of the centroid is still in the stable region, and the controller still has a good control effect.It can be seen from Figure11 that when the car without a controller performs an angle step simulation at high speed of 100 km/h, the yaw rate, the center of mass slip angle, and the lateral acceleration all exhibit large irregular oscillations, and the vehicle cannot be controlled correctly.The yaw rate of a car equipped with a lateral stability controller quickly approaches the ideal yaw rate curve after an overshoot occurs. The amplitude fluctuations of the center of mass slip angle and lateral acceleration are reduced and are suppressed in a reasonable interval so that the vehicle is in a stable region.When the vehicle equipped with the lateral stability controller is turning on the icy pavement at low speed (40 km/h) or high speed (100 km/h), the output response parameters are well controlled and the lateral stability of the vehicle is improved. When the car is running at low speed on the ice (adhesion coefficient 0.1), the controller also has a good control effect on the vehicle, which is avoiding obstacles and changing lanes. When the vehicle speed continues to increase, even if there is a controller, it may not be able to keep the vehicle stable under the limited working condition of the single lane change. At this moment, the control has reached the limit of direct yaw moment control under freezing conditions. Antisliding measures should be taken to increase the pavement adhesion coefficient to ensure the safety of vehicles running on the frozen pavement at medium speed.


## 3.1. Lateral Stability Controller Structure

The lateral stability controller is based on the automotive ESP system. The overall structure of the lateral stability controller is shown in Figure3. First, the driver inputs the steering wheel angle of the car. The car runs on the pavement and the vehicle motion parameters are collected through sensors. The actual motion parameters output by the car are used as the ideal reference model for the car. In the ideal model, the ideal value of the target variable in the current vehicle motion state is output by calculation. Then the steady-state of the vehicle is judged according to the difference between the actual value of the target variable output by the vehicle model and the ideal value of the target variable. When the state of the vehicle does not meet the lateral stability condition, the lateral stability controller starts to work. The automatic control algorithm of the controller determines the specific control measures for vehicle instability by analyzing the parameters such as the difference of yaw rate and the differential of difference/time and combining it with the stability control strategy of the controller. The control signal is input into the automotive electronic system to form a closed-loop control to ensure the lateral stability of the vehicle. The corresponding control model is established according to the specific design of each module in Matlab/Simulink for the simulation verification of the lateral stability controller.Figure 3
The overall structure of the vehicle lateral stability controller.After determining the design idea of vehicle lateral stability control, the yaw rate is determined as the target variable of lateral stability control, and the ideal yaw rate is modified according to the freezing condition. The lateral stability controller uses direct yaw moment control (DYC) as the control strategy, and its characteristics can be well explained by the following examples: In Figure4, when the vehicle steering shows excessive understeer, the positive additional yaw moment can be generated for the brake of inner wheel ①③ to correct the understeer trend; when the vehicle steering shows excessive steering, the lateral wheel ②④ brake can produce negative additional yaw moment to correct excessive steering trend. This is the differential braking control mode of direct yaw moment. In order to obtain a better control effect, this paper adopts unilateral two-wheel braking to obtain a greater additional yaw moment on the ice surface. According to the limitation of the ice surface attachment ellipse, the braking force distribution and calculation model are designed.Figure 4
Car steering diagram.In this paper, the sliding mode variable structure control algorithm is designed to control the additional yaw moment of the output. It can still have good control accuracy without a large number of test data debugging parameters. Moreover, the sliding mode control is essentially a nonlinear control, with the nature of uncertain system structure, and it will change rapidly with the change of the current state of the control system. Therefore, it has strong tracking, adaptability, and robustness. The ultralow adhesion characteristics of the iced pavement and the strong nonlinearity of the vehicle driving make the lateral stability control of the vehicle need to have the ability to track the target variable quickly and have a strong anti-interference ability. So sliding mode variable structure control is very suitable for vehicle lateral stability control under freezing conditions.The input of the sliding mode variable structure control algorithm module is the differencee obtained by the difference between the actual yaw rate ωr of the vehicle curve motion output and the ideal yaw rate ω0 of the ideal model output; the output is the additional yaw moment Mz calculated by the control algorithm. The design of the sliding mode controller is mainly divided into sliding mode control surface design and sliding mode control law design.The controller is designed based on a linear two-degree-of-freedom model with an additional yaw momentMz.According to the linear two-degree-of-freedom vehicle motion differential equation:(7)x·=Ax+Bδf+CMz,where A=k1+k2/mvak1−bk2/mv2−1ak1−bk2/Iza2k1+b2k2/Izv; B=−k1/mv−ak1/Iz; C=01/Iz; x=β,ωdT , Vehicle state variables, k1 and k2, are the lateral stiffness of front and rear wheels; v is centroid velocity; IZ is the rotational inertia of the car around the Z axis; ωd is steady yaw rate.According to the sliding mode control theory and the nonlinear system of vehicle control, the following switching functions are selected to further reduce the tracking error:(8)s=e+λ0∫0tetdt,where e=ωr−ω0; λ0 is a positive weighting factor; ∫0tetdt is mainly used to limit the steady-state error.Taking the derivation of (8), we can get the following:(9)s·=e·+λe=ωr·−ω0·+λωr−ω0.From formula (7), we can get the following:(10)ωr·=ak1−bk2Izβ+a2k1+b2k2Izvωr−ak1Izδf+MzIz.Lets·=0 to get the equivalent control input:(11)Mz_equ=−Izbk2−ak1Izβ−a2k1+b2k2Izvωr+ak1Izδf+λ0ωr−ω0.In order to ensure the sliding mode condition of the system under various conditions, the equal-speed reaching law design method is adopted, and the control law is defined as follows:(12)Mz=Mz_equ−ksgns.where k is the rate of the moving point of the system approaching the switching surface, the greater the k is, the faster the approaching speed is, but the jitter is also large; sgn· is the symbolic function.The choice ofk needs to satisfy the sliding mode reachability condition:(13)12ddts2=s⋅s·≤−η0s,where η0 is a positive real number.The discontinuous sign functionsgn· is replaced by a continuous saturation function to further weaken the chattering that may occur in the control system:(14)satx=sgnx,x>1,x,x≤1.Therefore, the final sliding mode control law is as follows:(15)Mz=Mz_equ−ksats/ε,where ε is the sliding mode variable structure thickness, the parameter value selection affects the smoothness of the control system.


## 3.2. Adhesion Coefficient of Frozen Pavement

The changes in the adhesion coefficient are one of the main reasons that affect the pavement condition [8]. In the double exponential model, the pavement surface adhesion coefficient is expressed as a nonlinear function containing three parameters of pavement condition, vehicle speed, and load. Therefore, the variation characteristics of the pavement adhesion coefficient can be well satisfied. The pavement adhesion characteristic parameters under various pavement conditions and driving conditions in the double exponential model are shown as follows:(16)μxp=0.92×0.1304σ+0.002eσ64−v−0.0426u,μxs=0.677×0.1553σ+0.002eσ64−v,Sxm=0.1532σ3−0.47801σ2+0.256σ+0.1693+0.105lg64/v,where μxp, Sxm are the peak adhesion coefficient and the corresponding wheel slip rate; μxs is the pavement adhesion coefficient when the wheels are completely slipped; v is the actual vehicle speed; u is the wheel load factor = actual wheel load/calibrated wheel load; σ is the influencing factor of the adhesion characteristic that characterizes the pavement condition, based on a large number of experimental data, the specific values can be seen in Table 3 Pavement-tire double exponential model is as follows:(17)μ=Aexp−bSxm−exp−aSxm.Table 3
Characteristic factor of common asphalt pavementsσ.
Pavement situationAsphalt (dry)Asphalt (wet)Snow (compaction)Icy pavement (dry)Icy pavement (wet)σ00.1340.751.01.2The characteristic adhesion parameter values required in (17) are given in the following formulas, where equation (18) is iteratively solved for a, and iteration is stopped when ak+1−ak/ak≤10−3 and a=ak+1.(18)ak+1=1Sxmlnμxp/μxs−Sxmak−1μxp/μxs−1,A=μxs1−aexp−aSxm,b=−ln1−aexpaSxm.According to the above double-exponential model [9], it can be known that the adhesion coefficient of asphalt pavement decreases sharply to about 0.1 when the pavement is frozen. Therefore, the adhesion coefficient of the smooth icy surface is 0.1 in the study of vehicle lateral stability [10, 11, 14].


## 3.3. Evaluation Standard of Vehicle Stability Control System

In the simulation test of vehicle stability control, due to the sharp steering of the wheel on the icy pavement with low adhesion, the vehicle will inevitably slip and cause the controller to intervene. The controller effect is evaluated by the simulation results according to the following standards:(1)
When the controller intervenes and applies the additional yaw moment control, it should be able to effectively control the sideslip of the vehicle, that is, to curb the sideslip angle of the centroid of the vehicle within a stable range and to make the yaw rate track the ideal yaw rate under the same working condition as far as possible, so that the vehicle is easy to control. It can be known from [1] that on the ice and snow pavement, the limit eigenvalue of the sideslip angle of the vehicle mass center is about ±2°. If it exceeds this eigenvalue, the vehicle will be difficult to control and lead to a radical sideslip.(2)
The lateral acceleration of the car during steering should not exceed the adhesion limit of the pavementu⋅g. The lateral acceleration on the smooth icy surface is 0.1 g.(3)
The response of each parameter is fast, the stable adjustment time is short, and the overshoot cannot be too large, which meets the requirements of rapidity and stability of the control system response.


## 3.4. Simulation Analysis of Single Lane Change for Icy Pavement

There is the sinusoidal input of the steering wheel angle in the single lane change condition, which is corresponded with rapid glide lane change for emergency obstacle avoidance. The simulation condition setting is shown in Figure5: Input a cycle sine steering wheel angle signal at 2s, the maximum input angle is 1.44 rad, the frequency is 0.4HZ, the ratio of steering wheel angle to front wheel angle is 18 : 1, and the road adhesion coefficient is 0.1 (icy pavement). When the car is turning, its driving stability is related to its front wheel angle, pavement adhesion coefficient, and driving speed. The initial speeds of 30 km/h and 50 km/h are selected to analyze the control effect of the stability controller under different speeds according to the experimental results; variation trend is shown in Figures 6-7.Figure 5
Sine input of steering wheel angle.Figure 6
Simulation results of single lane change at an initial velocity of 30 km/h (a) Change in yaw velocity (b) Change in sideslip angle (c) Vehicle trajectory (d) Change in lateral acceleration.
(a)(b)(c)(d)Figure 7
Simulation results of single lane change at an initial velocity of 50 km/h. (a) Change in yaw velocity. (b) Changes in sideslip angle. (c) Vehicle trajectory. (d) Change in lateral acceleration.
(a)(b)(c)(d)The simulation results of the single lane change on the icy pavement at the initial speed of 30 km/h are shown in Figure 6.It can be seen from Figure6 that the yaw velocity, sideslip angle, and lateral acceleration of the vehicle equipped with the controller are in good agreement with the ideal value. It means that the controller has a good correction effect on the vehicle trajectory. Figure 6(a) shows that the vehicle yaw rate with the controller only has a small viscous response in its mutation stage, and the maximum stability deviation is only 0.003 rad/s. The vehicle yaw rate without a controller has a large deviation, which is 66% higher than the maximum ideal yaw rate. It can be seen from Figure 6(b) that the maximum centroid sideslip angle of the vehicle without the lateral stability controller reaches 0.034 rad (1.97°), which is close to the eigenvalue of the centroid sideslip angle on the icy pavement. The centroid sideslip angle of the vehicle equipped with the stability controller is limited to 0.007 rad (0.42°), which is far less than the centroid sideslip angle of the vehicle without the control. It is shown in Figure 6(c) that the vehicle without the controller has slipped at the first corner peak and flew directly out of the desired trajectory at the second corner peak, indicating that the vehicle without a stable controller can no longer travel stably on the icy pavement at a speed of 30 km/h. The allowable adhesion limit of the tire on the icy surface is much smaller than that on the high adhesion pavement, and the tire is more likely to get in a nonlinear state. Therefore, it is difficult for the vehicle to follow the desired trajectory of the driver when making a single lane change on the icy surface. The vehicle equipped with the controller has completed the single lane change. Although the direction has slightly deviated, it can be easily corrected by turning the steering wheel based on vehicle stability. It can be seen from Figure 6(d) that the lateral acceleration of the vehicle without a controller is in a nonlinear region, and the numerical change is uncertain. The maximum value has reached the allowable maximum lateral acceleration 0.1 g of the adhesion limit, and the vehicle is in an unstable state. The lateral acceleration of the vehicle with the controller changes linearly, which is consistent with the change of yaw velocity and does not reach the maximum acceleration limit.The simulation results of the single lane change on the icy pavement at the initial speed of 50 km/h are shown in Figure 7.It can be seen from Figure7 that the vehicle equipped with the controller has a certain control effect on the yaw velocity, which still has a certain offset relative to the ideal yaw rate, especially at the peak value of the second corner. The maximum offset reaches 0.08 rad/s, 133% higher than the ideal yaw velocity, and the maximum value of the sideslip angle reaches 0.016 rad (0.92°). The vehicle trajectory is slipped obviously, and the lateral acceleration changes nonlinearly at the peak value of the second corner. The maximum value is close to the limit of 0.1 g, and the vehicle stability is reduced.The above analysis shows that when the car avoids obstacles and changes lanes on the icy surface (adhesion coefficient 0.1) at a low speed (30 km/h), the controller has a good control effect on the car; When the speed reaches 50 km/h, under extreme condition of single lane change, the effect of the controller decreases and the vehicle slips.By analyzing the driving control process at a speed of 50 km/h, the braking force of each wheel provided by the controller and the maximum braking force of each wheel provided by the pavement adhesion coefficient during the stability control process can be obtained, which are shown in Figure8.Figure 8
The braking force of each wheel applied by the controller in the state of 50 km/h single lane change. (a) Left front wheel braking force. (b) Left rear wheel braking force. (c) Right front wheel braking force. (d) Right rear wheel braking force.
(a)(b)(c)(d)In Figure8, the dotted line represents the maximum braking force provided by the pavement adhesion coefficient calculated by the adhesion ellipse, and the solid line represents the actual braking force output by the controller. It can be seen that the braking force of each wheel reaches the upper limit value of the braking force provided by the icy surface adhesion. The correction effect of the controller on the vehicle attitude has reached the limit. Therefore, the vehicle can only be stabilized by increasing the road adhesion coefficient. When the pavement adhesion coefficient increases to 0.2, the simulation results of single lane change at the initial speed of 50 km/h are shown in Figure 9.Figure 9
Simulation results of single lane change with adhesion coefficient of 0.2 and initial velocity of 50 km/h. (a) Change in yaw velocity. (b) Changes in sideslip angle. (c) Vehicle trajectory. (d) Change in lateral acceleration.
(a)(b)(c)(d)It can be seen from Figure 9 that after the adhesion coefficient increases from 0.1 to 0.2, the yaw velocity of the car equipped with the yaw controller can rematch the ideal value, the sideslip angle is also stabilized in a small range, and the car’s trajectory shows a perfect single lane change, the controller has an excellent effect on car stability control.


## 3.5. Simulation Analysis of front Wheel Angle Overstep on Freezing Pavement

In the vehicle dynamics simulation, the steering wheel angle is input step by step, which is corresponded with the driving condition that the car changes from a straight line to a curve with a certain angle of the front wheel. In order to be more suitable for the continuous steering characteristics of the vehicle in the actual driving from the straight to the curved road, and taking into account the danger of the icy pavement, this paper makes an appropriate correction on the front wheel angle. The front-wheel angle is no longer directly overstepped to the peak but has a short continuous steering process. Specific simulation conditions of steering wheel angle input are shown in Figure10. The initial velocity of 40 km/h and 100 km/h are taken, respectively. The ice surface adhesion coefficient is 0.1. The peak value of the front wheel angle is set to 4°, and the vehicle steering radius is far less than the minimum radius of general Figure 11 route Figure 12 design.Figure 10
Steering wheel angle step input.Figure 11
Simulation results of front-wheel angle overstep at an initial speed of 100 km/h. (a) Change of yaw rate. (b) Change of sideslip angle of centroid. (c) Change of lateral acceleration.
(a)(b)(c)Figure 12
Simulation results of front-wheel angle overstep at an initial speed of 40 km/h. (a) Change of yaw rate. (b) Change of sideslip angle of centroid. (c) Change of lateral acceleration.
(a)(b)(c)The simulation results of front wheel angle overstep at an initial speed of 40 km/h are shown in Figure12The simulation results of front-wheel angle overstep at an initial speed of 100 km/h are shown in Figure11It can be seen from Figure12 that when the front wheel angle step input is made at 40 km/h on the ice surface without the vehicle controller, both the yaw rate and the sideslip angle of the center of mass oscillate greatly, and the vehicle slips and loses its stability.The yaw rate of a vehicle equipped with a lateral stability controller has a good tracking effect on the ideal yaw rate, which coincides with the ideal curve. The slip angle of the center of mass and lateral acceleration are suppressed obviously, and the amplitude is reduced. The controller ensures the stability of the car when turning on icy pavement.Although the sideslip angle of the vehicle is limited to a small amplitude of 0.014 rad(0.8°) to keep the vehicle in a stable state, the sideslip angle of the vehicle is not completely stable. One of the reasons is that the control target variable of the lateral stability controller is yaw rate. A more important reason is that the ideal yaw rate of the controller is not necessarily the steady yaw rate of the vehicle’s circular motion. The ideal yaw rate is limited by the road adhesion coefficient, and the value is the maximum value of the yaw rate allowed by the road adhesion coefficient and the minimum value in the steady-state yaw rate. When the car does the front wheel angle overstep simulation on the icy pavements with minimal adhesion, the steady-state yaw rate exceeds the limit value of the ice surface throughout. Therefore, the ideal yaw rate is not the yaw rate of the steady-state circular motion when the vehicle is in a circular motion close to the steady-state on the icy pavements and the corresponding sideslip angle of the mass center naturally cannot converge well. Nevertheless, the sideslip angle of the centroid is still in the stable region, and the controller still has a good control effect.It can be seen from Figure11 that when the car without a controller performs an angle step simulation at high speed of 100 km/h, the yaw rate, the center of mass slip angle, and the lateral acceleration all exhibit large irregular oscillations, and the vehicle cannot be controlled correctly.The yaw rate of a car equipped with a lateral stability controller quickly approaches the ideal yaw rate curve after an overshoot occurs. The amplitude fluctuations of the center of mass slip angle and lateral acceleration are reduced and are suppressed in a reasonable interval so that the vehicle is in a stable region.When the vehicle equipped with the lateral stability controller is turning on the icy pavement at low speed (40 km/h) or high speed (100 km/h), the output response parameters are well controlled and the lateral stability of the vehicle is improved. When the car is running at low speed on the ice (adhesion coefficient 0.1), the controller also has a good control effect on the vehicle, which is avoiding obstacles and changing lanes. When the vehicle speed continues to increase, even if there is a controller, it may not be able to keep the vehicle stable under the limited working condition of the single lane change. At this moment, the control has reached the limit of direct yaw moment control under freezing conditions. Antisliding measures should be taken to increase the pavement adhesion coefficient to ensure the safety of vehicles running on the frozen pavement at medium speed.


## 4. Conclusion

Through the simulation analysis of two kinds of limit angle conditions on the freezing pavement, the control effect of the lateral stability controller on vehicle driving safety is verified. The lateral stability controller can effectively control the parameters of the vehicle steering on the freezing pavement and improve the maneuverability and stability of the vehicle, which ensures the lateral safety of the vehicle in normal driving on the freezing pavement. By analyzing the simulation results, the following conclusions can be drawn as follows:(1)
When the vehicle without the lateral stability controller makes large changes of the front wheel angle on icy pavements, such as obstacle avoidance and lane change or cornering. The vehicle is likely to lose stability and skid even if the speed is low (30–40 km/h).(2)
The lateral stability controller used in this paper can effectively control the yaw rate of the vehicle steering on the icy pavement so that it can well track the ideal yaw rate. At the same time, it also greatly reduces the offset of the sideslip angle of the centroid and inhibits the lateral acceleration exceeding the ice limit. This improves the maneuverability and stability of the vehicle under the freezing limit condition and ensures the lateral safety of the vehicle when driving normally. The application of this controller is of great significance to solving the traffic safety problems in southwest mountainous areas and southern humid areas of China and reducing the traffic accident rate.(3)
Due to the low adhesion coefficient of the icy pavement, the braking force and additional yaw moment of the tire provided by the adhesion are limited. When a car with a lateral stability control system is running on the icy pavement with a single lane change, the vehicle may be in a state of instability if the speed is too high. At this time, the stability control performance of the direct yaw moment has reached its limit. Measures such as deicing and assembling antiskid devices can increase the road adhesion coefficient and improve the vehicle’s lateral antiskid ability to ensure driving safety on icy pavements.(4)
When the vehicle performs the circular motion after the front wheel angle step-by-step input on the ice surface with minimal adhesion, the ideal yaw rate determined by the controller is almost the upper limit of the yaw rate limited by the freezing condition, rather than the steady-state yaw rate of the vehicle circular motion.


### 4.1. Future Work

Due to the limited experimental conditions and the limited level of the author, based on the current work, the following aspects can be further studied:(1)
Further improvement of the vehicle model. The driver model and road condition recognition module are added to provide more accurate data based on stable control, which makes the vehicle model closer to the real vehicle, and the simulation results and performance analysis have better practicability.(2)
Vehicle stability control can try a variety of joint control methods. The direct yaw moment control in this paper reaches the limit on the icy pavements. It can be considered to be combined with active steering and other control methods to give the vehicle a stronger control ability on the ice surface.


## 4.1. Future Work

Due to the limited experimental conditions and the limited level of the author, based on the current work, the following aspects can be further studied:(1)
Further improvement of the vehicle model. The driver model and road condition recognition module are added to provide more accurate data based on stable control, which makes the vehicle model closer to the real vehicle, and the simulation results and performance analysis have better practicability.(2)
Vehicle stability control can try a variety of joint control methods. The direct yaw moment control in this paper reaches the limit on the icy pavements. It can be considered to be combined with active steering and other control methods to give the vehicle a stronger control ability on the ice surface.



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*Source: 7361881-2022-04-09.xml*