A new preprocessment method for road peak adhesion coefficient fusion estimation

Abstract

Accurate estimation of road peak adhesion coefficient is of great significance in vehicle active safety systems. In order to prevent the vehicle from running out of control in actual driving process, the estimation algorithm should be able to adapt to various working conditions, and transmit estimated information of road peak adhesion coefficient to Electronic Control Unit (ECU) of the vehicle timely and accurately. Therefore, a new preprocessment method for the fusion estimation algorithm of road adhesion coefficient is proposed. The core of this method is the equal ratio relationship between the longitudinal, lateral peak adhesion coefficients and the utilization adhesion coefficient under adjacent typical roads. According to this relationship, this method accomplishes normalization of the tyre model in the form of introducing parameters from the outside, so that the tyre model can be combined with the filtering algorithm to be applied to estimation, which solves the problem that the precise tyre model cannot be used for road adhesion coefficient estimation due to its complex construction. This method can adapt to most of tyre models in the field of vehicle dynamics. In addition, the existing estimation algorithms need sufficient excitation (when the comprehensive slip ratio is 0.15–0.20) to estimate the accurate road peak adhesion coefficient. This method can greatly reduce the system error caused by insufficient road excitation, and can also obtain accurate estimation results when the excitation is insufficient. Finally, in order to verify the preprocessment method, the magic formula tyre model is used to describe the tyre characteristics. After processing by the preprocessment method, the road peak adhesion coefficient is estimated in combination with Unscented Kalman Filter (UKF). The high accuracy and timeliness of the estimated results in simulation and real vehicle tests verify the effectiveness of the preprocessment method.

Introduction

With the gradual realization of Intelligent Vehicles (IV), vehicle active safety systems are also constantly developing and popularizing, and have become an important part of ensuring vehicle driving safety. Among them, real-time estimation of adhesion coefficient between tyre and road surface is of great significance to improve the performance of vehicle control systems such as anti-lock braking system (ABS), electronic stability program (ESP), and traction control system (TCS).^1–4

In recent years, scholars have carried out extensive research on road adhesion coefficient estimation methods, and basically formed two kinds of estimation algorithms: cause-based and effect-based. The cause-based algorithm predicts road adhesion coefficient by measuring road surface roughness or road lubrication and combining experience. ² Literatures^5,6 use optical sensors to measure the degree of light absorption and scattering by road surface, so as to estimate the road surface adhesion coefficient. Leng et al. ⁷ accomplished road recognition and classification by extracting road color and texture features. The effect-based method can be divided into two categories. The one is mainly to establish a mapping relationship between tyre deformation, noise characteristics measured by various sensors and adhesion coefficient.^2,8,9 Pohl et al. ¹⁰ measured the friction between tyre and road surface by monitoring the deformation and movement of tread elements during the contact between tyre and road; Alonso et al. ¹¹ proposed an asphalt state classification system based on real-time acoustic analysis of tyre/road noise, and used vector machine to classify asphalt dry and wet states. Singh et al. ¹² used the measured tyre vibration frequency response of intelligent tyre to characterize the terrain, which can realize the estimation of road friction coefficient under the condition of extremely lower levels of force utilization. The advantages of these methods are that the detection range of identification is wide, and it has a predictive effect. ⁵ However, the sensors used in literatures^5–12 are expensive and vulnerable to environmental impact,^1,5,9,13,14 which makes these methods hard for wide application in vehicles.

Therefore, scholars have shifted their attention to the other one of the effect-based method. By designing the road adhesion coefficient estimator and using the vehicle dynamic response as input, the road adhesion coefficient is estimated. Due to the practicability and high precision of this kind of algorithm, a lot of researches has emerged. From the perspective of vehicle dynamics research, this method can be divided into three categories: longitudinal, lateral, longitudinal and lateral combination conditions.

In the aspect of longitudinal condition, Yi et al. ¹⁵ established an observer-based least squares method and filtering regression method to estimate the road adhesion coefficient under longitudinal conditions. Xia et al. ¹⁶ used the improved Burckhardt tyre model to describe the longitudinal slip characteristics of tyres. By introducing a new state variable, a nonlinear estimation method for estimating the longitudinal tyre force and the maximum road friction coefficient simultaneously is proposed. Chen et al. ¹⁷ proposed a longitudinal collision avoidance safety distance model for logistics vehicles considering road slope and road adhesion coefficient. Based on the relationship between slip rate and adhesion coefficient, the least square method is used for multivariate fitting to estimate the adhesion coefficient of road. Based on the modified Burckhardt tyre model, Leng et al. ⁷ designed a disturbance observer for tyre force and tyre-road peak adhesion coefficient. A tyre-road peak adhesion coefficient estimation method based on vehicle camera is designed. The fusion strategy of dynamic estimator and visual estimator is designed based on the gain scheduling method.

In the aspect of lateral condition, Ma et al. ¹⁸ proposed a single-wheel self-aligning torque (SAT) iterative algorithm to estimate the road adhesion coefficient of the front wheel, which is only used in steering conditions. Liang et al. ¹⁹ designed a nonlinear adaptive observer to estimate the road friction coefficient by combining the estimated total alignment torque and the front axle lateral force, and proposed criteria for judging when to update the estimated road conditions. The algorithm has strong stability and robustness. Liu et al. ²⁰ proposed a method that firstly combines auxiliary particle filter with iterative extended Kalman filter (APF-IEKF) to filter non-Gaussian noise, and then uses iterative algorithm to optimize the results of the first step. The estimation speed and accuracy are improved. Although the accuracy and convergence rate can be optimized simply based on longitudinal or lateral, it cannot cover all the working conditions and the scope of application is narrow. In the actual driving process of the vehicle, there are few conditions involving only longitudinal or lateral, and most conditions are longitudinal and lateral joint conditions. In addition, in complex conditions, such as turning braking, turning driving, etc., due to its high risk, active safety control technology has greater demand, so it is more necessary to estimate the road peak adhesion coefficient. Therefore, it is particularly important and urgent to develop a road adhesion coefficient estimation algorithm for combined conditions.

In the aspect of longitudinal and lateral combination conditions, Li et al. ²¹ identified the actual working condition by using sensor signals, and obtained the road adhesion coefficient of braking, driving and steering conditions. The comprehensive road adhesion coefficient under complex conditions was obtained by integrating the three. According to practical situation, the estimation of road adhesion coefficient is obtained through the allocation and cooperation among ABS, TCS, and active yaw control system (AYC), which has strong engineering application significance. However, the algorithm relies on three active safety technologies for the estimation and identification of the full working conditions, and requires a wide range of measurement signals, which has high requirements for vehicle configuration. Ahn et al. ²² proposed the estimation method of road adhesion coefficient under four conditions of mid-lateral, large-lateral, small-longitudinal, and large-longitudinal excitations, and realized integration by switching logic for estimation of various working conditions. It introduces the construction of five observers covering all working conditions in detail, which have wide estimation range and strong stability. However, due to the method needs to switch between models, it may lead to the problem of large memory occupation of the core unit of the vehicle and long convergence time. Chen et al. ²³ used available signals to observe longitudinal and lateral tyre forces. By establishing a polynomial function relationship between tyre force and road adhesion coefficient, the longitudinal and lateral road adhesion coefficients were estimated respectively. The comprehensive road adhesion coefficient was obtained by weighted fusion of mean square error. Compared with the first two methods, this kind method is simple, has short calculation cycle and strong robustness, but does not consider incentive sensitivity. In the case of sufficient incentives, ideal estimated results can be achieved in short time; however, when the excitation is weak, such as uniform linear motion and small lateral motion, accurate estimation results may not be obtained. It should be pointed out that simple and practical, strong robustness, fast convergence, and strong incentive sensitivity should be the proper properties of the estimation algorithm. These three methods all consider longitudinal, lateral and the coupling of longitudinal/lateral combined conditions, covering the whole driving condition of the vehicle. However, because of the different methods used, they also show different effects, they have their own advantages and disadvantages in the above four characteristics. In summary, it is not difficult to find that the existing methods are not perfect, and a better estimation algorithm is still needed to estimate the peak adhesion coefficient of the road.

Based on the above problems, the main contribution of this paper is the proposal of the equal ratio relationship between the longitudinal and lateral peak adhesion coefficients of adjacent typical road surface and the utilization adhesion coefficient according to the Kiencke tyre model. The above relationship is verified by comparing the variation surfaces of longitudinal, lateral road adhesion coefficients with longitudinal slip rate and tyre slip angle of the typical roads. According to this relationship, a simple and easy-to-implement tyre model normalization framework is built. This is of great significance for the application of high precision and high complexity tyre model in estimation. In fact, few simple tyre models can accurately reproduce the friction performance of the tyre-road interface while maintaining a simple form. ²³ This is also the dilemma faced by the complex tyre model in application. The normalization framework can accomplish the normalization of the tyre model by introducing parameters from the outside, avoiding the complex internal parameters, So it can adapt to the most of tyre models in the field of vehicle dynamics, which means that the most of tyre models can be used to estimate the road peak adhesion coefficient. Since different tyre models have high fitting accuracy in different fields, the framework extends the scope of application of road peak adhesion coefficient estimation algorithm based on tyre model greatly. In addition, in terms of incentive sensitivity, the existing estimation algorithms often need large incentives to estimate the accurate road peak adhesion coefficient, and this large incentive has a great negative impact on vehicle handling stability and comfort. ² Because the equal ratio relation can avoid a wide range of system errors, it can get more accurate recognition results even at small slip rate, which ensures high sensitivity of the algorithm to excitation and the robustness under nearly all working conditions.

The MF tyre model is selected as the representative of the tyre model with high precision and high complexity. The longitudinal and lateral road peak adhesion coefficients of the road are estimated by combining the vehicle dynamics model and UKF. And the comprehensive road peak adhesion coefficient is obtained by fusion. The above algorithms are verified in simulation and real vehicle tests.

The other parts are as follows : The first section mainly introduces the operation framework of the fusion estimation algorithm for road peak adhesion coefficient based on preprocessment method. The second section introduces all the models involved in this paper. The third section introduces the principle, theoretical analysis and verification of preprocessment algorithm. The fourth section introduces the implementation steps of fusion estimation strategy for road peak adhesion coefficient based on preprocessment algorithm. The fifth and sixth sections are simulation and experimental verification. The seventh section is the conclusion.

Algorithm framework for road peak adhesion coefficient estimation

The logical framework of the algorithm is shown in Figure 1. The required signal input was obtained by CarSim/Simulink simulation or real vehicle test, and the signal was used to calculate the dynamic parameters. Combined with the Kiencke and Nielsen tyre model,^24,25 the equal ratio relation between the longitudinal/lateral peak adhesion coefficients of the adjacent typical roads and the utilization adhesion coefficients was derived. The normalized formula of tyre force is derived and used as the normalized framework. The MF tyre model is selected to estimate the tyre force and embedded into the normalized framework. Combined with three DOF vehicle dynamics model, the system equation is obtained. Through UKF,^26,27 the estimation results of longitudinal and lateral road peak adhesion coefficients of four wheels are obtained. Through the treatment of equation (36), the comprehensive road peak adhesion coefficient is obtained.

Figure 1.

Framework of road adhesion coefficient estimation algorithm.

Model building

Establish vehicle dynamics model

In order to maintain the rigor of the paper, this paper makes the following assumptions,

• (1) Assuming that road peak adhesion coefficient remains constant when the tire is subjected to different loads;

• (2) Assuming that road peak adhesion coefficient remains constant on the contact surface between tire and road;

• (3) The influence of pavement (macroscopic and microscopic texture), tire (rubber type, pattern, temperature, speed, pressure) on the road peak adhesion coefficient is ignored.

The three DOF vehicle model is established as shown in Figure 2.

Figure 2.

The three DOF vehicle model.

The following differential equations of motion are established:

Longitudinal equation:

$$\begin{array}{c}{m}_z({\stackrel{·}{v}}_x-v_y\gamma )=F_{\mathrm{xfl}}\cos {\delta }_f\hfill \\ -F_{\mathrm{yfl}}\sin {\delta }_f+F_{\mathrm{xfr}}\cos {\delta }_f-F_{\mathrm{yfr}}\sin {\delta }_f\hfill \\ +F_{\mathrm{xrl}}+F_{\mathrm{xrr}}\hfill \end{array}$$

(1)

Lateral equation:

$$\begin{array}{c}{m}_z({\stackrel{·}{v}}_y+v_x\gamma )=F_{\mathrm{xfl}}\sin {\delta }_f+F_{\mathrm{yfl}}\cos {\delta }_f\hfill \\ +F_{\mathrm{xfr}}\sin {\delta }_f+F_{\mathrm{yfr}}\cos {\delta }_f+F_{\mathrm{yrl}}+F_{\mathrm{yrr}}\hfill \end{array}$$

(2)

Yaw equation:

$$\begin{array}{c}\stackrel{·}{\gamma }{I}_z=a(F_{\mathrm{xfl}}\sin {\delta }_f+F_{\mathrm{yfl}}\cos {\delta }_f+F_{\mathrm{xfr}}\sin {\delta }_f\hfill \\ +F_{\mathrm{yfr}}\cos {\delta }_f)+\frac{T}{2}(F_{\mathrm{xrr}}+F_{\mathrm{xfr}}\cos {\delta }_f\hfill \\ -F_{\mathrm{yfr}}\sin {\delta }_f)-\frac{T}{2}(F_{\mathrm{xfl}}\cos {\delta }_f-F_{\mathrm{yfl}}\sin {\delta }_f\hfill \\ +F_{\mathrm{xrl}})-(F_{\mathrm{yrl}}+F_{\mathrm{yrr}})b\hfill \end{array}$$

(3)

The load of each wheel can be expressed as

$$F_{\mathrm{zfl}}=\frac{b}{2L}{m}_{z}g-\frac{b{h}_g}{\mathrm{LB}}{m}_b{a}_y-\frac{m_b{a}_x{h}_g}{2L}+k_{\psi }{\psi }_b$$

(4)

$$F_{\mathrm{zfr}}=\frac{b}{2L}{m}_{z}g+\frac{b{h}_g}{\mathrm{LB}}{m}_b{a}_y-\frac{m_b{a}_x{h}_g}{2L}+k_{\psi }{\psi }_b$$

(5)

$$F_{\mathrm{zrl}}=\frac{a}{2L}{m}_{z}g-\frac{a{h}_g}{\mathrm{LB}}{m}_b{a}_y+\frac{m_b{a}_x{h}_g}{2L}+k_{\psi }{\psi }_b$$

(6)

$$F_{\mathrm{zmr}}=\frac{a}{2L}{m}_{z}g+\frac{a{h}_g}{\mathrm{LB}}{m}_b{a}_y+\frac{m_b{a}_x{h}_g}{2L}+k_{\psi }{\psi }_b$$

(7)

The main parameters of the vehicle are set as shown in Table 1.

Table 1.

Main parameters of vehicle model.

Symbol	Value	Notes
$m_z$	880 kg	Vehicle mass
$L$	2.040 m	Wheel base
$a$	1.145 m	Distance from centroid to front axle
$b$	0.895 m	Distance from centroid to rear axle
$h_g$	0.54 m	Centroid height
$T$	1.3 m	Wheel track width
$I_z$	832.3 kg·m2	Moment of inertia about the z-axis
$K_{\psi }$	25,041 N/rad	Tyre slip angle stiffness

Tyre model

The tyre longitudinal force and lateral force on the known road are easily estimated by the tyre model. There are many tyre models available in the field of vehicle dynamics, each of which has the ability of accurately describing the tyre dynamic response in a certain field. Such as Dugoff et al., MF, LuGre, Uni-Tyre, and so on.^28–33 Therefore, in the control process, we can select the tyre model with the highest accuracy according to the field of tyre dynamics studied. The estimated road adhesion coefficient can be obtained by matching with the corresponding filtering algorithms (UKF, Extended Kalman Filter (EKF), et al.). However, due to the variety of tyre model forms and parameters, some models with high fitting accuracy usually have more parameters and more complex forms, so it is difficult to establish system equations for filtering. Considering the difficulties in this regard, the preprocessment method mentioned in this paper can easily connect the output force of tyre model with the road adhesion coefficient, which is conducive to the estimation of road adhesion coefficient by filtering algorithm.

The MF tyre model has been widely used in vehicle dynamics simulation and analysis due to its high fitting accuracy and wide application range. ⁸ However, it is difficult to estimate the road adhesion coefficient through MF tyre model due to its complex form, various parameters and close correlation. In this paper, MF tyre model will be taken as an example to study the estimation of road peak adhesion coefficient, which will verify the preprocessment method.

MF tyre model

Steady-state pure driving (braking) conditions

The longitudinal force of MF tyre model under pure linear working condition can be expressed as:

$$\begin{array}{c}{F}_{x0}=D_x\sin [C_x{\tan }^{-1}\{B_x{\kappa }_x\hfill \\ -E_x(B_x{\kappa }_x-{\tan }^{-1}(B_x{\kappa }_x))\}]+S_{\mathrm{VX}}\hfill \end{array}$$

(8)

Steady-state turning conditions

The lateral force under the condition of pure lateral deflection can be expressed as:

$$\begin{array}{c}{F}_{y0}=D_y\sin [C_y{\tan }^{-1}\{B_y{\alpha }_y\hfill \\ -E_y(B_y{\alpha }_y-{\tan }^{-1}(B_y{\alpha }_y))\}]+S_{\mathrm{Vy}}\hfill \end{array}$$

(9)

Combined steady-state driving (braking) and turning conditions

The longitudinal force and the lateral force can be expressed as

$$F_x=\frac{\left|{\sigma }_x\right|}{\sigma }{F}_{x0}$$

(10)

$$F_y=\frac{\left|{\sigma }_y\right|}{\sigma }{F}_{y0}$$

(11)

$${\sigma }_x=\frac{{\kappa }_x}{1+{\kappa }_x}$$

(12)

$${\sigma }_y=\frac{\tan \alpha }{1+{\kappa }_x}$$

(13)

$$\sigma =\sqrt{{\sigma }_x^2+{\sigma }_x^2}$$

(14)

The longitudinal slip ratio of the MF model is:

$${\kappa }_x=\frac{\omega R-v_x}{v_x}$$

(15)

Tyre slip angle can be represented as:

$$\alpha ={\delta }_f-{\tan }^{-1}\left(\frac{v_y}{\left|v_x\right|}\right)$$

(16)

Where $v_x$ is the longitudinal wheel center speed and $v_y$ is the lateral speed of the contact point between the wheel and the road surface. $R$ is rolling radius of tyre. The other parameters in the formula (8)–(16) are shown in Appendix A.

Preprocessment method for road peak adhesion coefficient fusion estimation

Construction of standard $\mathbf{\mu }-$ s curve

Kiencke model is used to describe the relationship between comprehensive road utilization adhesion coefficient and comprehensive slip ratio. Kiencke tyre model optimized the Burckhardt ²⁴ model. As shown in equation (17)

$$\begin{array}{c}{\mu }_{\mathrm{Res}}(s_{\mathrm{Res}})=\hfill \\ (c_1(1-e^{-c_2{s}_{\mathrm{Res}}})-c_3{s}_{\mathrm{Res}})·\hfill \\ e^{-c_4·s_{\mathrm{Res}}·v_c}·(1-c_5{F}_Z^2)\hfill \end{array}$$

(17)

Where ${\mu }_{\mathrm{Res}}$ is the comprehensive road adhesion coefficient, $s_{\mathrm{Res}}$ is the comprehensive slip ratio, $v_c$ is the velocity of the center of gravity of the vehicle, $F_z$ is normal wheel load. Parameters $c_1$ , $c_2$ , $c_3$ vary with the road conditions, of which values under six different typical roads are shown in Table 2.^24,25 Parameter $c_4$ is between 0.002 and $0.004\mathrm{s}/\mathrm{m}$ ,and parameter $c_5$ is $0.00015(1/\mathrm{kN}{)}^2$ . ²⁵

Table 2.

$c_1$ , $c_2$ , $c_3$ fitting coefficient.

Road surface type	c 1	c 2	c 3
Dry asphalt	1.2801	23.99	0.52
Wet asphalt	0.857	33.822	0.347
Cement	1.1973	25.168	0.5373
Wet pebbles	0.4004	33.7080	0.1204
Ice	0.05	306.39	0
Snow	0.1946	94.129	0.0646

Longitudinal slip ratio $s_x$ can be expressed as

$$s_x=\{\begin{array}{c}\frac{\omega R·\cos \alpha -v_w}{v_w}\mathrm{braking}\hfill \\ \multicolumn{1}{c}{\frac{\omega R·\cos \alpha -v_w}{\omega R·\cos \alpha }\mathrm{driving}}\end{array}$$

(18)

Lateral slip ratio $s_y$ can be expressed as:

$$s_y=\{\begin{array}{c}\frac{\omega R·\sin \alpha }{v_w}\mathrm{braking}\hfill \\ \multicolumn{1}{c}{\tan \alpha \mathrm{driving}}\end{array}$$

(19)

The relationship between longitudinal and lateral slip ratio can be expressed as:

$$s_{\mathrm{Re}s}=\sqrt{s_x^2+s_y^2}\le 1$$

(20)

Where $\omega$ is wheel angular velocity, $v_w$ is tyre centroid velocity.

Longitudinal and lateral road utilization adhesion coefficients can be expressed as equations (21) and (22)

$${\mu }_x=\frac{{\mu }_{\mathrm{Re}s}{s}_x}{s_{\mathrm{Re}s}}$$

(21)

$${\mu }_y=\frac{{\mu }_{\mathrm{Re}s}{s}_y}{s_{\mathrm{Re}s}}$$

(22)

Comparative analysis

From the Kiencke tyre model, we can get Figure 3.

Figure 3.

Typical road adhesion coefficient curve.

It can be seen from Figure 3 that under six typical roads, the change trend of the relationship curve between the comprehensive road adhesion coefficient and the comprehensive slip rate is extremely similar, especially between adjacent roads, such as asphalt and cement, wet pebbles and snow road. Therefore, the relationship between the utilization adhesion coefficient of road surface and the peak adhesion coefficient can be obtained as ³⁴ :

$$\frac{{\mu }_{\mathrm{Re}s1}}{{\mu }_{\mathrm{R}max1}}=\frac{{\mu }_{\mathrm{Re}s2}}{{\mu }_{\mathrm{R}max2}}$$

(23)

Assume that roads 1 and 2 are adjacent roads, and ${\mu }_{\mathrm{Re}s1}$ , ${\mu }_{\mathrm{Re}s2}$ are the utilization adhesion coefficient of road 1 and road 2 respectively, ${\mu }_{\mathrm{R}max1}$ , ${\mu }_{\mathrm{R}max2}$ are the peak adhesion coefficient of road 1 and 2 respectively.

Similarity analysis in longitudinal and lateral road utilization adhesion coefficient

Figure 3 shows the relationship between the comprehensive slip ratio and the comprehensive road adhesion coefficient, but it cannot further describe the dynamic performance in the longitudinal and lateral directions. In longitudinal and lateral combined conditions, it cannot achieve accurate enough information. In order to elaborate the change law of the lateral and longitudinal road utilization adhesion coefficient under complex conditions, according to the Kiencke tyre model, the change characteristics of the lateral and longitudinal road utilization adhesion coefficients under six typical roads are obtained. In view of the limited space, only the variation characteristics of adhesion coefficient of asphalt and cement roads are given in Figures 4 to 7.

Figure 4.

Variation characteristics of road utilization adhesion coefficient under braking condition: (a–c) asphalt and (d–f) cement.

Figure 5.

Variation characteristics of road utilization adhesion coefficient under driving condition: (a–c) asphalt and (d–f) cement.

Figure 6.

Variation of adhesion coefficient of six typical roads under braking condition: (a) longitudinal road utilization adhesion coefficient, (b) lateral road utilization adhesion coefficient, and (c) comprehensive road utilization adhesion coefficient.

Figure 7.

Variation characteristics of utilization adhesion coefficient of six typical roads under driving condition: (a) longitudinal road utilization adhesion coefficient, (b) lateral road utilization adhesion coefficient, and (c) comprehensive road utilization adhesion coefficient.

From Figures 4 to 7, Whether in braking conditions or driving conditions, the change trend of ${\mu }_x$ , ${\mu }_y$ , and ${\mu }_{\mathrm{Re}s}$ of six typical roads is still similar, especially between adjacent roads. Therefore, according to the similarity principle, we can get the following conclusions:

$$\frac{{\mu }_{x1}}{{\mu }_{xmax1}}=\frac{{\mu }_{x2}}{{\mu }_{xmax2}}$$

(24)

$$\frac{{\mu }_{y1}}{{\mu }_{ymax1}}=\frac{{\mu }_{y2}}{{\mu }_{ymax2}}$$

(25)

Where ${\mu }_{x1}$ and ${\mu }_{x2}$ are longitudinal utilization adhesion coefficient of road 1 and road 2 respectively, ${\mu }_{xmax1}$ and ${\mu }_{xmax2}$ are longitudinal peak adhesion coefficient of road 1 and 2 respectively, ${\mu }_{y1}$ and ${\mu }_{y2}$ are lateral utilization adhesion coefficient of road 1 and 2 respectively, ${\mu }_{ymax1}$ and ${\mu }_{ymax2}$ are lateral peak adhesion coefficient of road 1 and 2 respectively.

For the convenience of subsequent elaboration, it is assumed that the adjacent roads with the highest similarity to the target road can always be found in these six typical roads. The target road and the adjacent road with the highest similarity to the target road are set as road g and road h respectively. The relationship between longitudinal force and longitudinal utilization adhesion coefficient of road surface can be expressed as equation (26). The relationship between lateral force and lateral utilization adhesion coefficient of road surface can be expressed as equation (27). ²⁵

$$F_{\mathrm{xg}}={\mu }_{\mathrm{xg}}{F}_z$$

(26)

$$F_{\mathrm{yg}}={\mu }_{\mathrm{yg}}{F}_z$$

(27)

Where ${\mu }_{\mathrm{xg}}$ and ${\mu }_{\mathrm{yg}}$ are the longitudinal and lateral utilization adhesion coefficient of the target road, respectively.

It can be seen from formula (24)–(27) that the curve characteristics of the known adjacent roads can be used as a reference for estimating the peak adhesion coefficient of the target road. Thus equations (28) and (29) is obtained.

$$F_{\mathrm{xg}}=\frac{{\mu }_{xmaxg}}{{\mu }_{xmaxh}}{F}_{\mathrm{xh}}$$

(28)

$$F_{\mathrm{yg}}=\frac{{\mu }_{ymaxg}}{{\mu }_{ymaxh}}{F}_{\mathrm{yh}}$$

(29)

Where ${\mu }_{xmaxg}$ and ${\mu }_{ymaxg}$ are the longitudinal peak adhesion coefficient and lateral peak adhesion coefficient of the target road respectively, which are the estimated values. ${\mu }_{xmaxh}$ and ${\mu }_{ymaxh}$ are the longitudinal peak adhesion coefficient and lateral peak adhesion coefficient of the adjacent road respectively, $F_{\mathrm{xg}}$ and $F_{\mathrm{yg}}$ are the longitudinal force and lateral force of the tyre on the target road respectively. $F_{\mathrm{xh}}$ and $F_{\mathrm{yh}}$ are the longitudinal force and lateral force of the tyre on the adjacent road respectively.

Then equations (28) and (29) can be written as the following normalized form.

$$F_{\mathrm{xg}}={\mu }_{xmaxg}{F}_{\mathrm{xg}}^0$$

(30)

$$F_{\mathrm{yg}}={\mu }_{ymaxg}{F}_{\mathrm{yg}}^0$$

(31)

Where $F_{\mathrm{xg}}^0$ and $F_{\mathrm{yg}}^0$ are the longitudinal and lateral normalized tyre forces of the target road respectively.

Road utilization adhesion coefficient under driving and braking conditions

From formula (18) and (19), the curves of adhesion coefficient versus longitudinal slip ratio and tyre slip angle should be different between driving and braking conditions, as shown in Figure 8.

Figure 8.

Comparison of road utilization adhesion coefficient under driving and braking conditions: (a) longitudinal road utilization adhesion coefficient, (b) lateral road utilization adhesion coefficient, and (c) comprehensive road utilization adhesion coefficient.

It can be seen from Figure 8 that with the increase of tyre slip angle, the difference between the two conditions is gradually obvious.

But it can also be seen that when the tyre slip angle is within 15°, the gap between the two surfaces is very small, almost coincident. Therefore, it is assumed that when the tyre slip angle is within 15°, the slip rate is in $[0,1]$ , the change of the two conditions are same.

Selection of adjacent road

According to existing research and experimental data, ⁴ the value in Figures 3 to 8 is larger than the actual value of road calibration. However, it does not affect the ratio relationship between road utilization adhesion coefficient and peak adhesion coefficient.

The road peak adhesion coefficient ${\mu }_r$ is set to 0.85 and 0.9 in the simulation. In the real vehicle test, the test road is dry asphalt surface, so the adjacent road is selected as cement road.

Fusion strategy

The fusion strategy can be expressed as:

$$k_x=\frac{{\mu }_{\mathrm{xh}0}}{{\mu }_{xmaxh}}$$

(32)

$$k_y=\frac{{\mu }_{\mathrm{yh}0}}{{\mu }_{ymaxh}}$$

(33)

$${\mu }_{\mathrm{xg}0}=k_x{\mu }_{xmaxg}$$

(34)

$${\mu }_{\mathrm{yg}0}=k_y{\mu }_{ymaxg}$$

(35)

$${\mu }_{Rmaxg}=\sqrt{{({\mu }_{\mathrm{xg}0})}^2+{({\mu }_{\mathrm{yg}0})}^2}$$

(36)

Where ${\mu }_{Rmaxg}$ is the comprehensive road peak adhesion coefficient of the target road. When the comprehensive road utilization adhesion coefficient of the adjacent road ${\mu }_{\mathrm{Re}\mathrm{sh}}$ reaches the peak value, the corresponding longitudinal and lateral road utilization adhesion coefficients are ${\mu }_{\mathrm{xh}0}$ , ${\mu }_{\mathrm{yh}0}$ respectively. Accordingly, When the comprehensive road utilization adhesion coefficient of the target road ${\mu }_{\mathrm{Re}\mathrm{sg}}$ reaches the peak value, the corresponding longitudinal and lateral road utilization adhesion coefficients are ${\mu }_{\mathrm{xg}0}$ and ${\mu }_{\mathrm{yg}0}$ , respectively. The values of ${\mu }_{\mathrm{xh}0}$ and ${\mu }_{\mathrm{yh}0}$ can be determined by referring to the following. Figure 9 shows the comparison of the variation curves of the longitudinal, lateral, and comprehensive road utilization adhesion coefficient under cement road.

Figure 9.

Variation of longitudinal, lateral, and comprehensive utilization road adhesion coefficient of cement road.

As shown in Figure 9, the pure white area is the peak area of comprehensive road utilization adhesion coefficient. In addition, based on the actual running process of the vehicle, it is infeasible to change the tyre slip angle of the vehicle in order to reach the peak area, but it is feasible and realistic to change the longitudinal slip rate of the vehicle by ABS to reach the peak area. Therefore, taking the tyre slip angle $\alpha$ as the abscissa, and taking the longitudinal slip rate $s_{x0}$ which can make the comprehensive road utilization adhesion coefficient reach the peak area as the ordinate. The corresponding ${\mu }_{\mathrm{xh}0}$ and ${\mu }_{\mathrm{yh}0}$ can be obtained by substitute the coordinate into the equation (17).

Estimation method of road peak adhesion coefficient based on preprocessment algorithm

Normalization of tyre model

MF tyre model can be expressed as:

$$F_0=D\sin [C\arctan \{B\kappa -E(B\kappa -\arctan (B\kappa ))\}]+S_V$$

(37)

According to formulas (24)–(31), the normalized form of MF can be obtained as follows:

$$\{\begin{array}{c}{F}_{\mathrm{xg}}={\mu }_{xmaxg}{F}_{\mathrm{xg}}^0\hfill \\ F_{\mathrm{xg}}^0=\frac{\left|{\sigma }_x\right|}{{\mu }_{x0max}\sigma }\{D_x\sin [C_x{\tan }^{-1}\{B_x{\kappa }_x-E_x(B_x\kappa -{\tan }^{-1}(B_x{\kappa }_x))\}]+S_{\mathrm{Vx}}\}\hfill \end{array}$$

(38)

$$\{\begin{array}{c}{F}_{\mathrm{yg}}={\mu }_{ymaxg}{F}_{\mathrm{yg}}^0\hfill \\ F_{\mathrm{yg}}^0=\frac{\left|{\sigma }_y\right|}{{\mu }_{y0max}\sigma }\{D_y\sin [C_y{\tan }^{-1}\{B_y{\alpha }_y-E_y(B_y{\alpha }_y-{\tan }^{-1}(B_y{\alpha }_y))\}]+S_{\mathrm{Vy}}\}\hfill \end{array}$$

(39)

Establishment of system equation

In order to accomplish the estimation of the longitudinal and lateral peak adhesion coefficient of the road surface, the formulas (1), (2), and (3) are deformed to obtain

$$\begin{array}{c}{\stackrel{·}{v}}_x-v_y\gamma =\hfill \\ {\mu }_{\mathrm{xfl}}\frac{F_{\mathrm{xfl}}^0\cos {\delta }_f}{m_z}+{\mu }_{\mathrm{xfr}}\frac{F_{\mathrm{xfr}}^0\cos {\delta }_f}{m_z}+{\mu }_{\mathrm{xrl}}\frac{F_{\mathrm{xrl}}^0}{m_z}\hfill \\ +{\mu }_{\mathrm{xrr}}\frac{F_{\mathrm{xrr}}^0}{m_z}-{\mu }_{\mathrm{yfl}}\frac{F_{\mathrm{yfl}}^0\sin {\delta }_f}{m_z}-{\mu }_{\mathrm{yfr}}\frac{F_{\mathrm{yfr}}^0\sin {\delta }_f}{m_z}\hfill \end{array}$$

(40)

$$\begin{array}{c}{\stackrel{·}{v}}_y+v_x\gamma =\hfill \\ {\mu }_{\mathrm{xfl}}\frac{F_{\mathrm{xfl}}^0\sin {\delta }_f}{m_z}+{\mu }_{\mathrm{xfr}}\frac{F_{\mathrm{xfr}}^0\sin {\delta }_f}{m_z}+{\mu }_{\mathrm{yfl}}\frac{F_{\mathrm{yfl}}^0\cos {\delta }_f}{m_z}\hfill \\ +{\mu }_{\mathrm{yfr}}\frac{F_{\mathrm{yfr}}^0\cos {\delta }_f}{m_z}+{\mu }_{\mathrm{yrl}}\frac{F_{\mathrm{yrl}}^0}{m_z}+{\mu }_{\mathrm{yrr}}\frac{F_{\mathrm{yrr}}^0}{m_z}\hfill \end{array}$$

(41)

$$\begin{array}{l}\stackrel{.}{\gamma }={\mu }_{xfl}(\frac{a}{I_z}{F}_{xfl}^0\sin {\delta }_f-\frac{T}{2I_z}{F}_{xfl}^0\cos {\delta }_f)+{\mu }_{xfr}(\frac{a}{I_z}{F}_{xfr}^0\sin {\delta }_f\\ +\frac{T}{2I_z}{F}_{xfr}^0\cos {\delta }_f)-{\mu }_{xrl}\frac{T}{2I_z}{F}_{xrl}^0+{\mu }_{xrr}\frac{T}{2I_z}{F}_{xrr}^0\\ +{\mu }_{yfl}(\frac{a}{I_z}{F}_{yfl}^0\cos {\delta }_f+\frac{T}{2I_z}{F}_{yfl}^0\sin {\delta }_f)+{\mu }_{yfr}(\frac{a}{I_z}{F}_{yfr}^0\cos {\delta }_f\\ -\frac{T}{2I_z}{F}_{yfr}^0\sin {\delta }_f)-{\mu }_{yrl}\frac{b}{I_z}{F}_{yrl}^0-{\mu }_{yrr}\frac{b}{I_z}{F}_{yrr}^0\end{array}$$

(42)

The state equation can be written as:

$$\left(\begin{array}{c}\multicolumn{1}{c}{{\mu }_{\mathrm{xfl}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xfr}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xrl}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xrr}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yfl}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yfr}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yrl}}(n+1)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yrr}}(n+1)}\end{array}\right)=I_{8\times 8}·\left(\begin{array}{c}\multicolumn{1}{c}{{\mu }_{\mathrm{xfl}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xfr}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xrl}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{xrr}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yfl}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yfr}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yrl}}(n)}\\ \multicolumn{1}{c}{{\mu }_{\mathrm{yrr}}(n)}\end{array}\right)+w(t)$$

(43)

The measurement equation can be written as:

$$O_1=O_2·O_3+v(t)$$

(44)

$$O_1=\left(\begin{array}{c}({\stackrel{·}{v}}_x-v_y\gamma )\hfill \\ \multicolumn{1}{c}{v_y+v_x\gamma }\\ \multicolumn{1}{c}{\stackrel{·}{\gamma }}\end{array}\right)$$

(45)

$$O_2=\left(\begin{array}{cccccccc}\frac{F_{\mathrm{xfl}}^0\cos {\delta }_f}{m_z}\hfill & \frac{F_{\mathrm{xfr}}^0\cos {\delta }_f}{m_z}\hfill & \frac{F_{\mathrm{xrl}}^0}{m_z}\hfill & \frac{F_{\mathrm{xrr}}^0}{m_z}\hfill & -\frac{F_{\mathrm{yfl}}^0\sin {\delta }_f}{m_z}\hfill & -\frac{F_{\mathrm{yfr}}^0\sin {\delta }_f}{m_z}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{F_{\mathrm{xfl}}^0\sin {\delta }_f}{m_z}}& \frac{F_{\mathrm{xfr}}^0\sin {\delta }_f}{m_z}\hfill & 0\hfill & 0\hfill & \frac{F_{\mathrm{yfl}}^0\cos {\delta }_f}{m_z}\hfill & \frac{F_{\mathrm{yfr}}^0\cos {\delta }_f}{m_z}\hfill & \frac{F_{\mathrm{yrl}}^0}{m_z}\hfill & \frac{F_{\mathrm{yrr}}^0}{m_z}\hfill \\ \multicolumn{1}{c}{H(3,1)}& H(3,2)\hfill & H(3,3)\hfill & H(3,4)\hfill & H(3,5)\hfill & H(3,6)\hfill & H(3,7)\hfill & H(3,8)\hfill \end{array}\right)$$

(46)

$$O_3={\left(\begin{array}{cccccccc}{\mu }_{\mathrm{xfl}}\hfill & {\mu }_{\mathrm{xfr}}\hfill & {\mu }_{\mathrm{xrl}}\hfill & {\mu }_{\mathrm{xrr}}\hfill & {\mu }_{\mathrm{yfl}}\hfill & {\mu }_{\mathrm{yfr}}\hfill & {\mu }_{\mathrm{yrl}}\hfill & {\mu }_{\mathrm{yrr}}\hfill \end{array}\right)}^T$$

(47)

Among them, ${\mu }_{\mathrm{xij}}(\mathrm{ij}=\mathrm{fl},\mathrm{fr},\mathrm{rl},\mathrm{rr})$ means longitudinal peak adhesion coefficient between four wheels and road. ${\mu }_{\mathrm{yij}}(\mathrm{ij}=\mathrm{fl},\mathrm{fr},\mathrm{rl},\mathrm{rr})$ means lateral peak adhesion coefficient between four wheels and road. Random variables $w(t)$ and $v(t)$ are process noise and measurement noise respectively.

UKF

The estimation flow of UKF can be expressed as the Figure 10. The initial value in the filtering process, measurement noise covariance $R=0.17\times I_{3\times 3}$ , process noise covariance is $Q=0.000038\times I_{8\times 8}$ , the corresponding covariance matrix is $P_0=0.21\times I_{8\times 8}$ .

Figure 10.

UKF estimation process.

Simulation analysis and verification

The algorithm is verified under two working conditions by the cosimulation of CarSim and Matlab/Simulink.

Linear braking condition

The road peak adhesion coefficient is 0.85, the initial speed is 100 km/h, and the straight-line braking condition ³⁵ is set. Taking the left front wheel as an example, the simulation results are shown in Figure 11:

Figure 11.

Simulation results under linear braking condition: (a) tyre slip angle, (b) longitudinal slip ratio, (c) longitudinal acceleration, and (d) estimated value of road adhesion coefficient.

It can be seen from Figure 11 that the road excitation under this condition is weak, and the tyre slip angle and longitudinal slip rate cannot make the comprehensive road utilization adhesion coefficient reach the peak area. However, from the estimation effect, ${\mu }_{Rmaxg}$ converges to 0.85 before 0.4 s, and then maintains a stable fluctuation. ${\mu }_{\mathrm{xg}0}$ is the best utilization adhesion coefficient which can be achieved by longitudinal utilization of adhesion coefficient under current working conditions, its change trend is almost coincident with ${\mu }_{Rmaxg}$ , and there is only a slight gap in numerical value. Similarly, as the best value that can be achieved by the lateral road utilization adhesion coefficient under the current working condition, ${\mu }_{\mathrm{yg}0}$ is stable at a very low level, because the demand for lateral force under this working condition is indeed relatively small.

Combined turning and braking condition

The road is set as a circular road with a radius of 30 m, the road peak adhesion coefficient is 0.9, the initial speed is 60 km/h, and combined turning braking condition ³⁶ is set. Taking the left front wheel as an example, the simulation results are shown in Figure 12:

Figure 12.

Simulation results under the combined turning and braking condition: (a) tyre slip angle, (b) longitudinal slip ratio, (c) longitudinal acceleration, (d) lateral acceleration, (e) steering angle, and (f) estimated value of road adhesion coefficient.

From Figure 12(a) and (b), this condition is severe, road excitation is large and complex, but still not enough to reach the peak area. It can be seen from Figure 12(f) that ${\mu }_{Rmaxg}$ converges to 0.9 at 0.8 s, and then maintains a stable fluctuation. Due to the large longitudinal road excitation, ${\mu }_{\mathrm{xg}0}$ converges to about 0.62 in a very short time, and there is a small decline at 0.5 s, and a rebound at 2.5 s, the overall situation is stable. ${\mu }_{\mathrm{yg}0}$ stabilized to about 0.67 at 0.8 s. It can be seen that when both longitudinal and lateral excitations exist, both longitudinal and lateral utilization adhesion coefficients play an important role in the friction process between tyre and road.

Experimental verification

Calibration test of road sliding adhesion coefficient

BM-III pendulum friction coefficient tester ³⁷ was used to test the sliding adhesion coefficient ⁴ of the experimental road. As shown in Figure 13, the 100 m straight asphalt road was selected for calibration test. The calibration result is shown in Figure 14.

Figure 13.

Calibration test of road sliding adhesion coefficient: (a) overall and (b) local.

Figure 14.

Road sliding adhesion coefficient calibration results.

It can be seen from Figure 14 that the variation range of sliding adhesion coefficient for experimental road is $[0.8,0.92]$ .

Real vehicle test

As shown in Figure 15, the test platform is a wire-controlled modified UTV (Utility Vehicle), and the driving mode is four-wheel independent drive. The vehicle is equipped with a variety of sensors, including GPS, inertial navigation, steering wheel angle sensors, and wheel speed sensor, which can check the test results. The vehicle controller is used to sample data and estimate it offline.

Figure 15.

Real vehicle test platform.

Straight line test

The test road is dry asphalt road as shown in Figure 13(b). The working condition is set as linear driving. ³⁵ The test speed is about 35 km/h, taking the left front wheel as an example, and the test results are shown in Figure 16.

Figure 16.

Test results for linear test: (a) tyre slip angle, (b) longitudinal slip ratio, (c) longitudinal acceleration, and (d) estimated value of road adhesion coefficient.

It can be seen from Figure 16 that the road excitation is weak in the linear drive vehicle test. ${\mu }_{Rmaxg}$ converges to 0.8 before 0.6 s, the maximum fluctuation can reach 0.98, and the minimum fluctuation can reach 0.78, but most of them are located in [0.8, 0.92] and are basically higher than the calibrated average value 0.859. ${\mu }_{\mathrm{xg}0}$ and ${\mu }_{Rmaxg}$ are almost completely overlapped, and ${\mu }_{\mathrm{yg}0}$ remains at a very low level. The negative value of the longitudinal slip rate in Figure 16(b) before 0.8 s has been noted. This may be caused by the incomplete synchronization of four wheels’ rotation speed of the electrical vehicle and the inertial navigation fluctuation error, but it does not affect the final estimation results.

Curved test

The test road surface is a dry asphalt ring road surface, and the test section is a semi-ring steady-state turning condition with a radius of 33 m ³⁶ with a test speed of about 40 km/h, and the test results are shown in Figure 17:

Figure 17.

Test results for curved test: (a) tyre slip angle, (b) longitudinal slip ratio, (c) longitudinal acceleration, (d) lateral acceleration, (e) steering angle, and (f) estimated value of road adhesion coefficient.

It can be seen from Figure 17(a) that under the driving condition of the curved test, the maximum tyre slip angle is 2.4°, and the maximum longitudinal slip rate can reach 0.049. It can be seen from Figure 17(f) that ${\mu }_{Rmaxg}$ converges to 0.8 at 0.6 s, then stabilizes at [0.8, 0.88], and is basically higher than the calibrated average value 0.859. The change trend of ${\mu }_{\mathrm{xg}0}$ is similar to ${\mu }_{Rmaxg}$ , but there is a significant difference in the numerical value. ${\mu }_{\mathrm{xg}0}$ converges to 0.8 in 0.7 s, and then stabilizes at about 0.83. ${\mu }_{\mathrm{yg}0}$ converges to about 0.2 at 0.6 s, and then remains stable, which is also corresponding to a certain amount of lateral force in this working condition.

Conclusion

The conclusions drawn are as follows.

• (1) From the analysis of the equal ratio relationship, a normalized framework with great inclusiveness to the tyre model is derived, which makes almost all tyre models can be applied into the estimation method based on tyre models.

• (2) The real vehicle and simulation experiments show that when the vehicle is driving on the dry asphalt road, the longitudinal slip rate and tyre slip angle cannot reach the value which can map the comprehensive road peak adhesion coefficient under the general braking or acceleration conditions. This leads to that the estimation algorithm based on vehicle dynamic response is difficult to accurately estimate the maximum adhesion capacity of asphalt road.

• (3) With the proposed preprocessment method, the estimation algorithm can accurately and timely estimate the comprehensive road peak adhesion coefficient under insufficient road excitation, which verifies high sensitivity to road excitation and the universality to complex tyre models of the preprocessment algorithm.

• (4) The optimal values of longitudinal and lateral road utilization adhesion coefficients under actual working conditions are also given, which provides corresponding reference signals for vehicle active safety systems.

In summary, based on the assumptions proposed above, this algorithm has many advantages, such as simple idea, high precision, certain robustness, and wide application range to working conditions of vehicle, which will promote development of vehicle active safety systems.

Finally, although accurate estimation results have been obtained in this paper, the performance of the estimation algorithm can still be further improved by improving the fusion method of the lateral and longitudinal road adhesion coefficients, which is also the focus of the next research.