# Road Adhesion Coefficient Estimation Based on Second-Order Linear Extended State Observer

Authors: Jian Wang; Jun Yang; Jinpeng Yu; Mengjun Wu; Nan Li
Journal: Mathematical Problems in Engineering
 (2023)

---

## Abstract

In order to estimate different road adhesion coefficients, a new method based on the second-order linear extended state observer is proposed to improve the active safety and driving stability of vehicles. The vehicle single-wheel driving dynamics model and single-wheel braking dynamics model are built with analyzing the driving and braking processes of the vehicle. The controllers of ASR and ABS are designed by using sliding mode structure control, and the chattering phenomenon of sliding mode control is eliminated by the saturation function. The road adhesion coefficient can be estimated by the second-order linear extended state observer in real time with the wheel speeds, and driving torque and braking torque are used as input variables. The simulation results show that the road adhesion coefficient estimation method based on the second-order linear extended state observer can accurately identify the adhesion coefficient of different roads in the presence of external interference and has strong robustness. At the same time, it also provides a reference for the design of the stability control system of unmanned vehicles.


---

## Body

## 1. Introduction

The real-time, fast, and accurate identification of road adhesion coefficient is one of the hot spots and difficult issues in the field of automobile intelligence, which directly affects vehicle active safety, vehicle state estimation accuracy, and vehicle handling stability [1–3]. The electronic control technology and control strategy of active safety system are developed rapidly, which can regulate the force between the tire and road surface. As known that the transmission of force between the road and the tire is constrained by the road adhesion coefficient, the advantages and disadvantages of active safety control strategy depend on whether the current road adhesion can be estimated accurately. If the road adhesion coefficient can be accurately estimated, the vehicle electronic control system of active safety system can adjust the control strategy in real time, which can improve the vehicle safety, reduce the increase the effectiveness of proactive intervention, and reduce the occurrence of traffic accident.At present, there are two methods to estimate the road adhesion coefficient, which are the cause-based method and effect-based method. The cause-based method identifies the road surface by measuring the roughness of the road and by measuring the wet smoothness of the road surface. Niu et al. used optical sensors to measure the adhesion coefficient of the road surface, but the optical sensors were very expensive [4]. Yadav and Singh identified the road surface using a laser beam with an accuracy of 98%, in which the identification calculation process was complex [5]. Ishihama et al. identified the road surface by collecting the sound wave spectrum reflected by the sound wave emitter [6]. Eom et al. used infrared sensors and temperature sensors to identify the road surface temperature by analyzing the solar radiation intensity, but the calculation accuracy was not high [7]. The cause-based method directly uses the sensor to identify the road adhesion coefficient, which is simple and easy to operate, but the sensor is expensive, so it cannot be used in mass-production vehicles.The effect-based method can identify the road adhesion coefficient based on tire noise, tire deformation, longitudinal, and lateral dynamic response of the vehicle. Yang et al. used the wavelet analysis method to analyze vibration signals under different roads to identify the road surface adhesion coefficient, which has strong robustness and poor practicability [8]. Choi et al. used the least square method combined with vehicle sensor and GPS data to estimate road adhesion coefficient through a vehicle longitudinal model, but they needed enough data points, poor real-time performance, and high accuracy of data points [9]. Han et al. proposed a road surface recognition method based on the change of road characteristic coefficient and peak adhesion coefficient, but the accuracy of the method was low [10]. Maia et al. built a braking vehicle model for a single road and used the energy method to identify the road adhesion coefficient, but there was little research on the estimation of variable road adhesion coefficient [11–13]. Acosta M used a neural network and extended Kalman filter to estimate the wheel yaw angle and combined it with the change of vehicle yaw angular velocity to estimate the road surface adhesion coefficient [14–17]. Zareian et al. used the Kalman filter to estimate the road adhesion coefficient of vehicles with a steer-by-wire system, but the estimation accuracy of vehicles under braking conditions was not well [18]. Kojima and Raksincharoensak estimated the road surface adhesion coefficient based on the deviation of yaw angular velocity and lateral acceleration, and the real vehicle test showed that the estimation method had strong real time and robustness [19].Extended state observer can estimate nonlinear system with uncertainty, which has been applied in vehicle state estimation, vehicle active suspensions, automobile speed tracking, EPS system control, and vehicle active suspensions. At present, the research on the estimation of road adhesion coefficient was mainly focused on vehicle braking conditions, and there was a lack of research under vehicle driving and braking integrated conditions. In the preliminary research, the design of the ABS sliding mode controller was completed, and the path-tracking control of the bus was realized by using an extended state observer with strong robustness [20–24]. In this paper, taking the wheel skid rate and slip rate as the control goal, the sliding mode variable structure control was adopted to design the ASR and ABS controller, and the saturation function was used to eliminate the chattering phenomenon of the sliding mode control. The input variables include the front and rear wheel speed, driving torque, and braking torque, the road adhesion coefficient can be estimated in real time by the second-order linear expansion state observer (SOLESO), and the effectiveness of the control method was verified by virtual simulation experiments.


## 2. Dynamic Models of Driving and Braking

### 2.1. Driving Dynamic Model

The force and torque balance analysis of the 1/4 vehicle driving dynamic model is carried out, where the influences of air resistance and road roughness are ignored. The 1/4 vehicle driving model is established as shown in Figure1.Figure 1
Quarter vehicle driving model.Based on the force and moment balance analysis of Figure1, the force and moment equations can be obtained:(1)mtotv˙=Ft,(2)Jω˙=Tp−FtR−Tf,where mtot is the 1/4 vehicle model mass; v is the vehicle speed; J is the wheel moment of inertia, ω is the wheel angular speed; R is the wheel radius; Ft is the ground driving force; Fz is the ground normal load; FL is the additional vertical load; v is the vehicle speed; Tp is the driving torque; Tf is the rolling resistance moment.The relationship between ground driving forceFt, the adhesion coefficient of road surface μ(λ), and the ground normal load Fz can be written as follows:(3)Ft=μλFz.The 1/4 vehicle model massmtot can be written as follows:(4)mtot=14mcar+mtyre,where mcar is the mass of car body; mtyre is the mass of tire. The additional vertical load FL is described as follows:(5)FL=mcarhcg2Lv˙,where hcg is the height of center of gravity; L is the wheelbase. The rolling resistance moment Tf can be described as follows:(6)Tf=FzfR,where ƒ is the rolling resistance coefficient; R is the wheel radius. The ground normal load Fz is described as follows:(7)Fz=mtotg+mcarhcg2Lv˙,where g is gravitational acceleration. The slip rate of wheel λ1 can be described as follows:(8)λ1=1−vωR.By substituting the functional relation expressions ofFt, mtot, FL, Tf, and Fz into formulas (1) and (2), which can be obtained:(9)v˙=μλmtotgmtot−mcarhcg/2Lμλ,(10)ω˙=Tp−mtotg+mcarhcg/2Lv˙Rμλ+RfJ.


### 2.2. Braking Dynamic Model

Based on the force and torque balance analysis of the 1/4 vehicle braking dynamic model, the braking dynamic model ignores the influence of air resistance and road roughness. The 1/4 vehicle braking dynamic model is established as shown in Figure2.Figure 2
Quarter vehicle braking model.The force of the 1/4 vehicle braking model is analyzed:(11)mtotv˙=−Fb,Jω˙=FbR−Tb,where mtot is the 1/4 vehicle model mass; J is the wheel moment of inertia; ω is the wheel angular speed; R is the wheel radius; Fb is the ground braking force; Fz is the ground normal load; v is the vehicle speed; Tb is the braking torque.The relationship between ground braking forceFb, the adhesion coefficient of road surface μ(λ), and the ground normal load Fz can be written as follows:(12)Fb=μλFz.The 1/4 vehicle model massmtot can be written as follows:(13)mtot=14mcar+mtyre.The wheel slip rate is defined as follows:(14)λ2=v−ωRv,where mcar is the quality of the car body; mtyre is the quality of the tire; μ(λ) is the road adhesion coefficient; λ2 is the wheel slip ratio.


### 2.3. Tire Model

In the driving and braking conditions, the Burckhardt tire model is a function of road adhesion coefficient and speed. The formula is as follows:(15)μλ,v=C11−e−C2λ−C3λe−C4λv,where C1, C2, and C3 are the tire adhesion characteristic parameters; C4 is the influence parameter of vehicle speed on the adhesion characteristics; v is the speed; λ is the tire slip rate. The range of values of C1, C2, and C3 under different road conditions is shown in Table 1.Table 1
Tire characteristic parameter values.
Road conditionsC1C2C3Dry asphalt road1.280123.990.52Wet asphalt road0.85733.8220.347Dry concrete road1.197325.1680.5373Snow road surface0.194694.1290.0646Ice road0.05306.390


## 2.1. Driving Dynamic Model

The force and torque balance analysis of the 1/4 vehicle driving dynamic model is carried out, where the influences of air resistance and road roughness are ignored. The 1/4 vehicle driving model is established as shown in Figure1.Figure 1
Quarter vehicle driving model.Based on the force and moment balance analysis of Figure1, the force and moment equations can be obtained:(1)mtotv˙=Ft,(2)Jω˙=Tp−FtR−Tf,where mtot is the 1/4 vehicle model mass; v is the vehicle speed; J is the wheel moment of inertia, ω is the wheel angular speed; R is the wheel radius; Ft is the ground driving force; Fz is the ground normal load; FL is the additional vertical load; v is the vehicle speed; Tp is the driving torque; Tf is the rolling resistance moment.The relationship between ground driving forceFt, the adhesion coefficient of road surface μ(λ), and the ground normal load Fz can be written as follows:(3)Ft=μλFz.The 1/4 vehicle model massmtot can be written as follows:(4)mtot=14mcar+mtyre,where mcar is the mass of car body; mtyre is the mass of tire. The additional vertical load FL is described as follows:(5)FL=mcarhcg2Lv˙,where hcg is the height of center of gravity; L is the wheelbase. The rolling resistance moment Tf can be described as follows:(6)Tf=FzfR,where ƒ is the rolling resistance coefficient; R is the wheel radius. The ground normal load Fz is described as follows:(7)Fz=mtotg+mcarhcg2Lv˙,where g is gravitational acceleration. The slip rate of wheel λ1 can be described as follows:(8)λ1=1−vωR.By substituting the functional relation expressions ofFt, mtot, FL, Tf, and Fz into formulas (1) and (2), which can be obtained:(9)v˙=μλmtotgmtot−mcarhcg/2Lμλ,(10)ω˙=Tp−mtotg+mcarhcg/2Lv˙Rμλ+RfJ.


## 2.2. Braking Dynamic Model

Based on the force and torque balance analysis of the 1/4 vehicle braking dynamic model, the braking dynamic model ignores the influence of air resistance and road roughness. The 1/4 vehicle braking dynamic model is established as shown in Figure2.Figure 2
Quarter vehicle braking model.The force of the 1/4 vehicle braking model is analyzed:(11)mtotv˙=−Fb,Jω˙=FbR−Tb,where mtot is the 1/4 vehicle model mass; J is the wheel moment of inertia; ω is the wheel angular speed; R is the wheel radius; Fb is the ground braking force; Fz is the ground normal load; v is the vehicle speed; Tb is the braking torque.The relationship between ground braking forceFb, the adhesion coefficient of road surface μ(λ), and the ground normal load Fz can be written as follows:(12)Fb=μλFz.The 1/4 vehicle model massmtot can be written as follows:(13)mtot=14mcar+mtyre.The wheel slip rate is defined as follows:(14)λ2=v−ωRv,where mcar is the quality of the car body; mtyre is the quality of the tire; μ(λ) is the road adhesion coefficient; λ2 is the wheel slip ratio.


## 2.3. Tire Model

In the driving and braking conditions, the Burckhardt tire model is a function of road adhesion coefficient and speed. The formula is as follows:(15)μλ,v=C11−e−C2λ−C3λe−C4λv,where C1, C2, and C3 are the tire adhesion characteristic parameters; C4 is the influence parameter of vehicle speed on the adhesion characteristics; v is the speed; λ is the tire slip rate. The range of values of C1, C2, and C3 under different road conditions is shown in Table 1.Table 1
Tire characteristic parameter values.
Road conditionsC1C2C3Dry asphalt road1.280123.990.52Wet asphalt road0.85733.8220.347Dry concrete road1.197325.1680.5373Snow road surface0.194694.1290.0646Ice road0.05306.390


## 3. Driving Torque Control of ASR System

The automobile antislip regulation system (ASR) can control the actual wheel slip rate at the target slip rate, and the sliding mode control switching surface of the ASR system is defined as follows:(16)s1=λ1−λd1,where λ1 is the actual slip rate of the wheel; λd1 is the target slip rate.By calculating the first derivative of the sliding mode control switching surfaceS1, the equivalent control driving torque of the ASR system can be obtained:(17)s˙1=0,(18)λ˙1=1−λ1ω˙−1/Rv˙ω,(19)λ˙1−λ˙d1=0.The equivalent control driving torque can be obtained by combining formula (18) with formula (9) and formula (10):(20)Teq.p=JR1−λ1f2+Jf3,where(21)f2=μλmtotgmtot−mcarhcg/2Lμλ,f3=1Jmtotg+mcarhcg2Lv˙Rμλ+Rf.The driving torque of sliding mode variable structure control is defined as follows:(22)Tp=Teq.p−k1Signs1.From the reachability of sliding mode variable structure control, it is known that the switching surface of sliding mode control needs to satisfy:(23)s1s˙1≤−ηs1.From the formula (16) and the formula (17), it can be deduced that the driving torque controlled by the sliding mode controller is(24)Tp=JR1−λ1f2+Jf3−Jηω1−λ1Signs1.In order to eliminate the chattering problem of sliding mode controller, the saturation functionSat(s/ϕ) is used instead of discontinuous switching function Sign(s). Finally, the expression of driving moment Tp is obtained as follows:(25)Tp=JR1−λ1f2+Jf3−Jηω1−λ1Sats1φ,where ϕ is the design parameter.


## 4. Braking Torque Control of ABS System

The automobile antilock braking system (ABS) can control the actual wheel slip rate at the target slip rate, and the sliding mode control switching surface of the ABS system can be shown as follows:(26)s2=λ2−λd2,where λ2 is the actual slip rate of the wheel; λd2 is the target slip rate.By calculating the first derivative of the sliding mode control switching surfaceS2, the equivalent control driving torque of the ABS system can be obtained:(27)s˙2=0,λ˙2=1−λ2v˙−ω˙Rv,λ˙2=1v−RJFbR−Tb+1−λ2v˙,λ˙2−λ˙d2=0.The equivalent control driving torque can be obtained:(28)Teq.b=JuλgR1−λ2+uλmtotgR.The braking torque of sliding mode variable structure control is defined as follows:(29)Tb=Teq.b−k2Signs2.From the reachability of sliding mode variable structure control, the switching surface of sliding mode control needs to be satisfied:(30)s2s˙2≤−ηs2.From formula (28) and formula (29), it can be deduced that the braking torque controlled by the sliding mode controller is(31)Teq.b=JuλgR1−λ2+uλmtotgR−k2JvRSigns2.In order to eliminate the chattering problem of the sliding mode controller, the saturation functionSat(s/ϕ) is used instead of the discontinuous switching function Sign(s). Finally, the expression of braking moment Tb is obtained:(32)Tb=JuλgR1−λ2+uλmtotgR−k2JvRSats2φ,where ϕ is the design parameter.


## 5. Design of Linear Extended State Observer

### 5.1. Basic Theory of LESO

The LESO can observe the model uncertainties and unknown disturbances, which is suitable for the input and output parameters of the known observer. For n-order nonlinear time-varying SISO dynamic systems,(33)ynt=fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt+bu,where y and u are the output and input variables of the controller; fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt is a nonlinear time-varying unknown dynamics including internal dynamics of the system and external interference; w (t) is the external disturbance, which can be the time-varying or stationary; b is the control gain. The equation (33) can be written as(34)ynt=fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt+b−b0u+b0u=f+b0u.Assuming thatf is differentiable, let f˙=h, the equation (34) can be written in its state space form:(35)x˙1=x2,x˙2=x3,⋯,x˙n=xn+1+b0u,x˙n+1=h,y=x1,where xn+1=f is the expanded state, xt=x1tx2t⋯xn+1tT∈Rn+1, u∈R, and y∈R are the state variables, inputs, and outputs of the system. The equation (35) can write its state space expression as follows:(36)X˙=An+1Xn+1+b0Bn+1u+En+1f,˙y=Cn+1Xn+1,where(37)An+1=010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0,Bn+1=00⋮10,En+1=00⋮01.The general form of the linear extended state observer corresponding to the state space is shown as follows:(38)e1=z1−y,z˙1=z2−l1e1,z˙2=z3−l2e1,⋮,z˙n=zn+1−lne1+b0u,z˙n+1=−ln+1e1,where z1z2⋯zn+1T is the estimated value of the expansion state x1x2⋯xn+1T; l1l2⋯ln+1 is the design parameter.


### 5.2. Design of LESO Based on ASR System

In our previous research, the LESO method has been used to solve the vehicle path tracking problem, and the effectiveness of the controller was very good. In order to estimate the road coefficient, this method was the first time to be used under the driving and braking conditions [20–22].The controlled object can be transformed into an integral series system, which can be expressed as follows:(39)x˙=fx,wt+but,y=x,where w(t) is the external disturbance; f(x, w(t)) is the internal and external disturbance; b is the gain of the control quantity; u is the control quantity.Through the dynamic model of single-wheel vehicle drive, the following formulas can be obtained:(40)ω˙=Tp−mtotg+mcarhcg/2Lv˙Rμλ+RfJ.By changing the formula (40), we can get(41)ω˙=−mtotg+mcarhcg/2Lv˙Rμλ+RfJ+1JTp,where ω=x1, 1/J=b1, −mtotg+mcarhcg/2Lv˙Rμλ+Rf/J=x2, and then, the formula (41) can be further simplified to an integral series system:(42)x˙1=x2+b1u1,y1=x1.The variablesx1 and x2 in formula (42) can be observed by using a LESO, where the input of the observer is the driving torque output of the sliding mode controller and the wheel speed signal of the wheel. The design of the LESO is as follows:(43)ek=z1k−y1k,β01=2ω0,β02=ω02,z1k+1=z1k+hz2k−β01ek+b1u1k,z2k+1=z2k+h−β02ek,where ω0 is the bandwidth of the ESO; h is the simulation step; b1 is the control gain; u1, y1, z1, and z2 are input signals and output signals, respectively.The road adhesion coefficient can be estimated by the LESO, and the following formula can be obtained:(44)ω=z1,−mtotg+mcarhcg/2Lv˙Rμλ+RfJ=z2.Among them,z1 and z2 are the observed values of x1 and x2, and the road adhesion coefficient based on automobile driving antiskid system is obtained, which is as follows:(45)μλ=−z2Jmtotg+mcarhcg/2Lv˙R−f.


### 5.3. Design of Linear Extended State Observer Based on ABS System

The following formulas can be obtained from the braking dynamics model of a single-wheeled vehicle:(46)ω˙=μλFzR−TbJ.The formula (47) can be obtained by changing the formula (46):(47)ω˙=μλmtotgRJ−1JTp,where ω=x3, −1/J=b2, μλmtotgR/J=x4, and then, the formula (47) can be further simplified to an integral series system:(48)x˙3=x4+b2u2.y2=x3.The variablesx3 and x4 in formula (48) can be observed by using a LESO, where the input of the observer is the braking torque output of the sliding mode controller and the wheel speed signal of the wheel. The design of the LESO is as follows:(49)ek=z3k−y2k,β01=2ω0,β02=ω02,z3k+1=z3k+hz3k−β01ek+b2u2k,z4k+1=z4k+h−β02ek.In the formula,ω0 is the bandwidth of the LESO; h is the simulation step; b2 is the control gain; u2, y2, z3, and z4 are input signals and output signals, respectively.The road adhesion coefficient can be estimated by the LESO, and the following formula can be obtained:(50)ω=z3.μλmtotgRJ=z4,where z3 and z4 are the observed values of x3 and x4, and the road adhesion coefficient based on the automobile antilock braking system is obtained, which is as follows:(51)μλ=z4JmtotgR.


## 5.1. Basic Theory of LESO

The LESO can observe the model uncertainties and unknown disturbances, which is suitable for the input and output parameters of the known observer. For n-order nonlinear time-varying SISO dynamic systems,(33)ynt=fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt+bu,where y and u are the output and input variables of the controller; fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt is a nonlinear time-varying unknown dynamics including internal dynamics of the system and external interference; w (t) is the external disturbance, which can be the time-varying or stationary; b is the control gain. The equation (33) can be written as(34)ynt=fyt,y˙t,⋯,yn−1t,u,u˙,⋯,un−1,wt+b−b0u+b0u=f+b0u.Assuming thatf is differentiable, let f˙=h, the equation (34) can be written in its state space form:(35)x˙1=x2,x˙2=x3,⋯,x˙n=xn+1+b0u,x˙n+1=h,y=x1,where xn+1=f is the expanded state, xt=x1tx2t⋯xn+1tT∈Rn+1, u∈R, and y∈R are the state variables, inputs, and outputs of the system. The equation (35) can write its state space expression as follows:(36)X˙=An+1Xn+1+b0Bn+1u+En+1f,˙y=Cn+1Xn+1,where(37)An+1=010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0,Bn+1=00⋮10,En+1=00⋮01.The general form of the linear extended state observer corresponding to the state space is shown as follows:(38)e1=z1−y,z˙1=z2−l1e1,z˙2=z3−l2e1,⋮,z˙n=zn+1−lne1+b0u,z˙n+1=−ln+1e1,where z1z2⋯zn+1T is the estimated value of the expansion state x1x2⋯xn+1T; l1l2⋯ln+1 is the design parameter.


## 5.2. Design of LESO Based on ASR System

In our previous research, the LESO method has been used to solve the vehicle path tracking problem, and the effectiveness of the controller was very good. In order to estimate the road coefficient, this method was the first time to be used under the driving and braking conditions [20–22].The controlled object can be transformed into an integral series system, which can be expressed as follows:(39)x˙=fx,wt+but,y=x,where w(t) is the external disturbance; f(x, w(t)) is the internal and external disturbance; b is the gain of the control quantity; u is the control quantity.Through the dynamic model of single-wheel vehicle drive, the following formulas can be obtained:(40)ω˙=Tp−mtotg+mcarhcg/2Lv˙Rμλ+RfJ.By changing the formula (40), we can get(41)ω˙=−mtotg+mcarhcg/2Lv˙Rμλ+RfJ+1JTp,where ω=x1, 1/J=b1, −mtotg+mcarhcg/2Lv˙Rμλ+Rf/J=x2, and then, the formula (41) can be further simplified to an integral series system:(42)x˙1=x2+b1u1,y1=x1.The variablesx1 and x2 in formula (42) can be observed by using a LESO, where the input of the observer is the driving torque output of the sliding mode controller and the wheel speed signal of the wheel. The design of the LESO is as follows:(43)ek=z1k−y1k,β01=2ω0,β02=ω02,z1k+1=z1k+hz2k−β01ek+b1u1k,z2k+1=z2k+h−β02ek,where ω0 is the bandwidth of the ESO; h is the simulation step; b1 is the control gain; u1, y1, z1, and z2 are input signals and output signals, respectively.The road adhesion coefficient can be estimated by the LESO, and the following formula can be obtained:(44)ω=z1,−mtotg+mcarhcg/2Lv˙Rμλ+RfJ=z2.Among them,z1 and z2 are the observed values of x1 and x2, and the road adhesion coefficient based on automobile driving antiskid system is obtained, which is as follows:(45)μλ=−z2Jmtotg+mcarhcg/2Lv˙R−f.


## 5.3. Design of Linear Extended State Observer Based on ABS System

The following formulas can be obtained from the braking dynamics model of a single-wheeled vehicle:(46)ω˙=μλFzR−TbJ.The formula (47) can be obtained by changing the formula (46):(47)ω˙=μλmtotgRJ−1JTp,where ω=x3, −1/J=b2, μλmtotgR/J=x4, and then, the formula (47) can be further simplified to an integral series system:(48)x˙3=x4+b2u2.y2=x3.The variablesx3 and x4 in formula (48) can be observed by using a LESO, where the input of the observer is the braking torque output of the sliding mode controller and the wheel speed signal of the wheel. The design of the LESO is as follows:(49)ek=z3k−y2k,β01=2ω0,β02=ω02,z3k+1=z3k+hz3k−β01ek+b2u2k,z4k+1=z4k+h−β02ek.In the formula,ω0 is the bandwidth of the LESO; h is the simulation step; b2 is the control gain; u2, y2, z3, and z4 are input signals and output signals, respectively.The road adhesion coefficient can be estimated by the LESO, and the following formula can be obtained:(50)ω=z3.μλmtotgRJ=z4,where z3 and z4 are the observed values of x3 and x4, and the road adhesion coefficient based on the automobile antilock braking system is obtained, which is as follows:(51)μλ=z4JmtotgR.


## 6. Simulation Verification of Road Adhesion Coefficient Estimation

To verify the accuracy and effectiveness of the method based on the LESO, the Carsim software is used to verify the virtual simulation test. The three-dimensional virtual road provided by Carsim software is used to set up the road. Among them, the related parameters of the automobile aremcar = 1500 kg, mtyre = 40 kg, L = 2.582 m, hcg = 0.533 m, R = 0.326 m, J = 1.7 kg·m2, f = 0.03. The vehicle test model is shown in Figure 3.Figure 3
Carsim vehicle test model.Under the condition that the input signals such as driving moment, braking moment, and wheel speed are known, the adhesion coefficients of different roads can be estimated by the designed expansion state observer, and the estimated structure block diagram of road adhesion coefficient is shown in Figure4.Figure 4
Block diagram of road adhesion coefficient Estimation.


### 6.1. Single Road Surface

The SOLESO based on ASR system and ABS system is simulated and verified under the high adhesion road and low adhesion road, respectively. The parameters of SOLESO areh = 0.001 s, ω0 = 100, β01 = 200, β02 = 10000, b1 = 0.5882. Figure 5 shows the simulation results on a high adhesion road with the coefficient of 0.8, and Figure 6 shows the simulation results on low adhesion road with the coefficient of 0.2.Figure 5
Road adhesion coefficient estimation (high adhesion).
(a)(b)(c)(d)Figure 6
Road adhesion coefficient estimation (low adhesion).
(a)(b)(c)(d)From Figures 5(a) and 5(b), we can see that the SOLESO based on the ASR system can observe the real road adhesion coefficient quickly and accurately, and the maximum error between the observed value and the actual value is less than ±0.1. When the simulation time is more than 1 s, the error between the observed value and the actual value is almost zero. Figure 5(c) shows the driving torque output of the ASR system, and Figure 5(d) shows the wheel slip rate.It can be seen from Figures 6(a) and 6(b) that the SOLESO based on the ASR system can quickly and accurately observe the adhesion coefficient of real road surface on low adhesion snow road, and the maximum error between the observed value and the actual value is within ±0.05. Figure 6(c) shows the driving torque output of the ASR system, and Figure 6(d) shows the wheel slip rate.From Figures 7(a) and 7(b), we can see that the SOLESO based on the ABS system can observe the real adhesion coefficient quickly and accurately in real time, and the maximum error of the estimated value is less than 0.1. When the simulation time is more than 1 s, the error between the observed value and the actual value is 0. Figure 7(c) shows the braking torque of the ABS system, and Figure 7(d) shows the wheel slip ratio.Figure 7
Road adhesion coefficient estimation based on ABS (high adhesion).
(a)(b)(c)(d)As can be seen from Figure8, the designed observer based on the ABS system can real time and quickly and accurately observe the real adhesion coefficient under the ice and snow road, and the observation error is less than 0.03.Figure 8
Road adhesion coefficient estimation based on ABS (low adhesion).
(a)(b)(c)(d)


### 6.2. Butt Joint Road

The process of vehicle driving from high adhesion road (adhesion coefficient 0.8) to low adhesion road (adhesion coefficient 0.2) is simulated. In the first 2 s, the car is driving on the dry asphalt road, and the car is driving on the snow road in the last 3 s. The simulation results are shown in Figure9.Figure 9
Road adhesion coefficient estimation (high adhesion—low adhesion).
(a)(b)(c)(d)As can be seen from Figure9, when the vehicle runs from high adhesion road to low adhesion road, the SOLESO based on ASR can observe the real road adhesion coefficient quickly and accurately, and the maximum error between the observed value and the actual value is less than ±0.01.When the vehicle runs from high adhesion road to low adhesion road under braking condition, the observation result of road surface adhesion coefficient based on the SOLESO is shown in Figure10. It can be seen from the diagram that the maximum error between the estimated value and the real value is less than 0.1, the response is rapid, and the estimation is accurate.Figure 10
Road adhesion coefficient estimation (high adhesion—low adhesion).
(a)(b)(c)(d)


### 6.3. Road with External Disturbance

There is usually existing white noise interference in the wheel speed sensor signal, the mean value of white noise interference is 0, and the variance is 1. The estimated results of the road adhesion coefficient are shown in Figures11 and 12.Figure 11
Road adhesion coefficient estimation (white noise disturbance).
(a)(b)(c)(d)Figure 12
Road adhesion coefficient estimation (white noise disturbance).
(a)(b)(c)(d)It can be seen from Figure11 that the SOLESO based on ASR designed in this paper can observe the road adhesion coefficient real time and accurately, whether from high adhesion road surface to low adhesion road surface or from low adhesion road surface to high adhesion road surface. The maximum error of the estimated value is less than 0.02.It can be seen from Figure12 that whether the vehicle runs from high adhesion road to low adhesion road or from low adhesion road to high adhesion road, the designed road adhesion coefficient observer based on ABS can observe the road adhesion coefficient real time and accurately, and the maximum error of the estimated value is less than 0.01. The statistical of road coefficient estimation errors is shown in Table 2.Table 2
Road coefficient estimation errors.
Driving cycleRoad conditionsMaximum errorRelative error (%)Driving conditionsSingle road surface0.045Butt joint road0.011.25Road with external disturbance0.011.25Braking conditionsSingle road surface0.033.75Butt joint road0.0354.34Road with external disturbance0.011.25


## 6.1. Single Road Surface

The SOLESO based on ASR system and ABS system is simulated and verified under the high adhesion road and low adhesion road, respectively. The parameters of SOLESO areh = 0.001 s, ω0 = 100, β01 = 200, β02 = 10000, b1 = 0.5882. Figure 5 shows the simulation results on a high adhesion road with the coefficient of 0.8, and Figure 6 shows the simulation results on low adhesion road with the coefficient of 0.2.Figure 5
Road adhesion coefficient estimation (high adhesion).
(a)(b)(c)(d)Figure 6
Road adhesion coefficient estimation (low adhesion).
(a)(b)(c)(d)From Figures 5(a) and 5(b), we can see that the SOLESO based on the ASR system can observe the real road adhesion coefficient quickly and accurately, and the maximum error between the observed value and the actual value is less than ±0.1. When the simulation time is more than 1 s, the error between the observed value and the actual value is almost zero. Figure 5(c) shows the driving torque output of the ASR system, and Figure 5(d) shows the wheel slip rate.It can be seen from Figures 6(a) and 6(b) that the SOLESO based on the ASR system can quickly and accurately observe the adhesion coefficient of real road surface on low adhesion snow road, and the maximum error between the observed value and the actual value is within ±0.05. Figure 6(c) shows the driving torque output of the ASR system, and Figure 6(d) shows the wheel slip rate.From Figures 7(a) and 7(b), we can see that the SOLESO based on the ABS system can observe the real adhesion coefficient quickly and accurately in real time, and the maximum error of the estimated value is less than 0.1. When the simulation time is more than 1 s, the error between the observed value and the actual value is 0. Figure 7(c) shows the braking torque of the ABS system, and Figure 7(d) shows the wheel slip ratio.Figure 7
Road adhesion coefficient estimation based on ABS (high adhesion).
(a)(b)(c)(d)As can be seen from Figure8, the designed observer based on the ABS system can real time and quickly and accurately observe the real adhesion coefficient under the ice and snow road, and the observation error is less than 0.03.Figure 8
Road adhesion coefficient estimation based on ABS (low adhesion).
(a)(b)(c)(d)


## 6.2. Butt Joint Road

The process of vehicle driving from high adhesion road (adhesion coefficient 0.8) to low adhesion road (adhesion coefficient 0.2) is simulated. In the first 2 s, the car is driving on the dry asphalt road, and the car is driving on the snow road in the last 3 s. The simulation results are shown in Figure9.Figure 9
Road adhesion coefficient estimation (high adhesion—low adhesion).
(a)(b)(c)(d)As can be seen from Figure9, when the vehicle runs from high adhesion road to low adhesion road, the SOLESO based on ASR can observe the real road adhesion coefficient quickly and accurately, and the maximum error between the observed value and the actual value is less than ±0.01.When the vehicle runs from high adhesion road to low adhesion road under braking condition, the observation result of road surface adhesion coefficient based on the SOLESO is shown in Figure10. It can be seen from the diagram that the maximum error between the estimated value and the real value is less than 0.1, the response is rapid, and the estimation is accurate.Figure 10
Road adhesion coefficient estimation (high adhesion—low adhesion).
(a)(b)(c)(d)


## 6.3. Road with External Disturbance

There is usually existing white noise interference in the wheel speed sensor signal, the mean value of white noise interference is 0, and the variance is 1. The estimated results of the road adhesion coefficient are shown in Figures11 and 12.Figure 11
Road adhesion coefficient estimation (white noise disturbance).
(a)(b)(c)(d)Figure 12
Road adhesion coefficient estimation (white noise disturbance).
(a)(b)(c)(d)It can be seen from Figure11 that the SOLESO based on ASR designed in this paper can observe the road adhesion coefficient real time and accurately, whether from high adhesion road surface to low adhesion road surface or from low adhesion road surface to high adhesion road surface. The maximum error of the estimated value is less than 0.02.It can be seen from Figure12 that whether the vehicle runs from high adhesion road to low adhesion road or from low adhesion road to high adhesion road, the designed road adhesion coefficient observer based on ABS can observe the road adhesion coefficient real time and accurately, and the maximum error of the estimated value is less than 0.01. The statistical of road coefficient estimation errors is shown in Table 2.Table 2
Road coefficient estimation errors.
Driving cycleRoad conditionsMaximum errorRelative error (%)Driving conditionsSingle road surface0.045Butt joint road0.011.25Road with external disturbance0.011.25Braking conditionsSingle road surface0.033.75Butt joint road0.0354.34Road with external disturbance0.011.25


## 7. Conclusions

In this paper, the wheel driving and braking processes are analyzed, the dynamic models of wheel driving and braking are established, and the sliding mode variable structure control method is used to design ASR and ABS controllers. The SOLESO is designed with the wheel speed signal, driving moment and braking moment as the input signal, and the road adhesion coefficient as the output signal, and the cosimulation experiments under different roads are carried out. The simulation results show that the designed method can estimate the road adhesion coefficient accurately and quickly, which has strong robustness. The SOLESO method designed in this paper can be used to estimate the road adhesion coefficient under the conditions of vehicle steering, driving, and braking, which can raise the vehicle stability by big percentages.



---

*Source: 5953102-2023-02-23.xml*