Estimation of Road Friction Coefficient in Different Road Conditions Based on Vehicle Braking Dynamics

Abstract

The accurate estimation of road friction coefficient in the active safety control system has become increasingly prominent. Most previous studies on road friction estimation have only used vehicle longitudinal or lateral dynamics and often ignored the load transfer, which tends to cause inaccurate of the actual road friction coefficient. A novel method considering load transfer of front and rear axles is proposed to estimate road friction coefficient based on braking dynamic model of two-wheeled vehicle. Sliding mode control technique is used to build the ideal braking torque controller, which control target is to control the actual wheel slip ratio of front and rear wheels tracking the ideal wheel slip ratio. In order to eliminate the chattering problem of the sliding mode controller, integral switching surface is used to design the sliding mode surface. A second order linear extended state observer is designed to observe road friction coefficient based on wheel speed and braking torque of front and rear wheels. The proposed road friction coefficient estimation schemes are evaluated by simulation in ADAMS/Car. The results show that the estimated values can well agree with the actual values in different road conditions. The observer can estimate road friction coefficient exactly in real-time and resist external disturbance. The proposed research provides a novel method to estimate road friction coefficient with strong robustness and more accurate.

1 Introduction

It is a powerful means to improve vehicle driving safety and stability performances via active safety system such as emergency collision avoidance (ECA), active front steering (AFS), anti-lock braking system (ABS), direct yaw moment control (DYC) and traction control system (TCS) [1,2,3,4,5,6]. They work well only with the tire forces within the friction limit, which means knowledge of the road friction coefficient may improve the performance of the systems. For example, during a steering process, the lateral tire force is limited by the road friction coefficient. The vehicle would drift out if the vehicle steers severely at a relatively high speed because of limitation of the lateral tire force. If the active control system could estimate the friction limitation at the time driver begins to steer and initiatives to reduce the speed, the lateral dynamics of the vehicle would be improved [4]. Wheel braking under the different road condition, we usually can’t get the real-time value of road friction coefficient, which leads the instability of the whole control process [7, 8]. So road friction coefficient has an important significance in the vehicle chassis electronic control system design. The accurate estimation of the road friction coefficient can facilitate the improvement of the active safety system and attain a better performance in operating vehicle safety systems. Active safety system can automatically adjust the control strategy according to changing of road surfaces with respect to the friction properties, and which can maximize the function of the control system.

In recent years, to obtain road friction coefficient, many scholars have proposed various estimation methods [8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Among them, domestic scholars especially Liang LI and their teams used signal fusion method [8, 9], double cubature kalman filter method [10], and observer [11] to estimate the road friction coefficient. Generally speaking, they are mainly classified into two groups of special-sensor-based [12,13,14] methods and vehicle-dynamics-based methods, also known as Cause-based and Effect-based [15]. Method of Cause-based was used by optical sensors to measure light absorption and scattering of road according to the road surface shapes and physical properties. This method looks simple and direct, but has practical issue of cost, which limits its use in production vehicle. Effect- based method was presented by measuring the related response of vehicle dynamics model and applies extended kalman filtering or other algorithm to obtain its value. The vehicle dynamics model included both longitudinal and/or lateral dynamics [16, 17]. The main features of these methods could make full use of the on-board sensors and reduce costs, which has been widely used.

Two very similar studies [18, 19] used the kalman filter (KF) to estimate the longitudinal force of the vehicle first and then through the recursive least squares (RLS) method and the change of CUSUM estimated the road friction coefficient. Wenzel, et al. [20], reported another method of the dual extended kalman filter (DEKF) for road friction coefficient estimation. Comparing kalman filter algorithm and the extended kalman filtering algorithm, Ref. [21] design a extended state observer (ESO) by means of the dynamics model of 1/4 tire for braking to estimate road friction coefficient. This method can ensure high calculation accuracy and do not need to solve Jacobian trial.

This article, considering load transfer of front and rear axles, the braking dynamic model of two-wheeled vehicle was built. Sliding mode control method was used to build the ideal braking torque controller, which control objective is to control the actual wheel slip ratio of front and rear wheels tracking the ideal wheel slip ratio. In order to eliminate chattering problem of the sliding mode controller, integral switching surface was used to design the sliding mode surface. Road friction coefficient can be observed by second order linear extended state based on wheel speed and braking torque of front and rear wheels. Comparing with the article discussed, this method considered the effect of axle load transfer to the road friction coefficient estimation. It has both fewer parameters taking into account and higher computational efficiency.

2 Vehicle Braking Dynamics Model

2.1 Full-Vehicle Model

The assumptions for the vehicle model are as follows: (1) Neglect the effect of road slope; (2) Ignore the load transfers by the lateral accelerations; (3) The effects of air resistance and rolling resistance of tire are neglected; (4) Ignore the effects of transmission system, steering system and suspension system on vehicle. Vehicle model presented here is a two-wheeled vehicle model with the sketch given in Fig. 1.

Fig. 1

Two-wheeled vehicle model. h _f, h _r—Height of vehicle front, rear unsprung mass, l _f, l _r ψ—Distance between centre of gravity and the front, rear axes, F _zf, F _zr—Vertical forces of vehicle front, rear tire, m _f, m _r—Front, rear unsprung mass, x—Displacement in the process, m _s—Sprung mass of vehicle, h _s—Height of vehicle sprung mass

From Fig. 1, the body motions of longitudinal and yaw can be described, respectively, by Eqs. (1)–(4) as follows:

$$\dot{x} = V,$$

(1)

$$\dot{V} = - g\frac{{\mu (\lambda_{\text{f}} )m_{1} + \mu (\lambda_{\text{r}} )m_{2} }}{{m - \mu (\lambda_{\text{f}} )m_{3} + \mu (\lambda_{\text{r}} )m_{3} }},$$

(2)

$$\dot{\omega }_{\text{f}} = \frac{1}{{2J_{\text{f}} }}( - T_{\text{bf}} + \mu (\lambda_{\text{f}} )m_{1} R_{{{\omega }}} g - \mu (\lambda_{\text{f}} )m_{3} R_{{{\omega }}} \ddot{x}),$$

(3)

$$\dot{\omega }_{\text{r}} = \frac{1}{{2J_{\text{r}} }}( - T_{\text{br}} + \mu (\lambda_{\text{r}} )m_{2} R_{{{\omega }}} g + \mu (\lambda_{\text{r}} )m_{3} R_{{{\omega }}} \ddot{x}),$$

(4)

where

$$m_{1} = \frac{{l_{\text{r}} }}{{l_{\text{f}} + l_{\text{r}} }}m\;,\;m_{2} = \frac{{l_{\text{f}} }}{{l_{\text{f}} + l_{\text{r}} }}m\;,\;m_{3} = \frac{{m_{\text{f}} h_{\text{f}} + m_{\text{s}} h_{\text{s}} + m_{\text{r}} h_{\text{r}} }}{{l_{\text{f}} + l_{\text{r}} }},$$

where µ(λ _f) and µ(λ _r) are the road friction coefficient of front and rear wheels,m is the total mass of vehicle, V is the vehicle longitudinal velocity, T _bf and T _br are the braking torque of front and rear wheels, J _f and J _r are the moment of inertia of front and rear wheels, ω _f and ω _r are the angular speed of front and rear wheels, R _ω is the wheel rolling radius.

2.2 Full-Vehicle Model

Tire force is so important that it influences the precision of simulation. The tire model should reflect the effect of tire vertical force on longitudinal and lateral forces, and the interaction of longitudinal and lateral forces. In order to predict the vehicle longitudinal force in braking conditions, Burckhardt model is introduced by theoretical deformation and simulation analyses on the basis of the Magic Formula model [22, 23]. It provides the tire-road coefficient of friction $\mu$ as a function of the wheel slip $\lambda$ and the vehicle velocity $V$. The equation can be described as

$$\mu (\lambda ,V) = \left( {C_{1} (1 - \exp (C_{2} \lambda )) - C_{3} \lambda )} \right)\exp ( - C_{4} \lambda V),$$

(5)

where C ₁, C ₂ and C ₃ are the characteristic parameters of tire adhesion; C ₄ is the influence parameter of car speed to adhesion and is in the range 0.02–0.04. Table 1 shows friction model parameters for different road conditions.

Table 1 Friction model parameters of different road

Fig. 2 illustrates the relationship of road friction coefficient vs. slip ratio at different road conditions.

Fig. 2

Relationship of road friction coefficient vs. slip ratio

As can be seen from the figure, Burckhardt tire model describes the nonlinear change law of road friction coefficient vs. wheel slip rate is good.

3 Braking Torque Controller Design

3.1 Braking Torque Design

The front and rear wheel longitudinal slip ratio can be described as.

$$\lambda_{\text{f}} = \frac{{V - \omega_{\text{f}} R_{\omega } }}{V},$$

(6)

$$\lambda_{\text{r}} = \frac{{V - \omega_{\text{r}} R_{\omega } }}{V},$$

(7)

where λ _f and λ _r denotes the front and rear wheel longitudinal slip ratio respectively and the derivative of which with respect to time are, respectively, given by.

$$\dot{\lambda }_{\text{f}} = \frac{{\dot{V}(1 - \lambda_{\text{f}} ) - \dot{\omega }_{\text{f}} R_{\omega } }}{V},$$

(8)

$$\dot{\lambda }_{\text{r}} = \frac{{\dot{V}(1 - \lambda_{\text{r}} ) - \dot{\omega }_{\text{r}} R_{\omega } }}{V}.$$

(9)

Substituting Eqs. (8), (9) into vehicle dynamics equation, then

$$\dot{V} = f_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} ),$$

(10)

$$\dot{\lambda }_{\text{f}} = \frac{{f_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{f}} ) - R_{\omega } f_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + u_{\text{f}} }}{V},$$

(11)

$$\dot{\lambda }_{\text{r}} = \frac{{f_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{r}} ) - R_{\omega } f_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + u_{\text{r}} }}{V},$$

(12)

where

$$f_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = - g\frac{{\mu (\lambda_{\text{f}} )m_{1} + \mu (\lambda_{\text{r}} )m_{2} }}{{m - \mu (\lambda_{\text{f}} )m_{3} + \mu (\lambda_{\text{r}} )m_{3} }},$$

$$f_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = \frac{1}{{2J_{\text{f}} }}(\mu (\lambda_{\text{f}} )m_{1} R_{\omega } g - \mu (\lambda_{\text{f}} )m_{3} R_{\omega } f_{2} ),$$

$$f_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = \frac{1}{{2J_{\text{r}} }}(\mu (\lambda_{r} )m_{2} R_{\omega } g + \mu (\lambda_{\text{r}} )m_{3} R_{\omega } f_{2} ),$$

$$u_{\text{f}} = \frac{{T_{\text{bf}} R_{\omega } }}{{2J_{\text{f}} }},\;\;\;\;u_{\text{r}} = \frac{{T_{\text{br}} R_{\omega } }}{{2J_{\text{r}} }}.$$

It can be assumed that the front and rear wheel friction coefficient change between 0 and 1 and vehicle total mass changes within a certain range, namely

$$m_{1}^{ - } \le m_{1} \le m_{1}^{ + } ,$$

(13)

$$m_{2}^{ - } \le m_{2} \le m_{2}^{ + } ,$$

(14)

$$m_{3}^{ - } \le m_{3} \le m_{3}^{ + } ,$$

(15)

$$m^{ - } \le m \le m^{ + } ,$$

(16)

$$0 \le \mu (\lambda_{\text{f}} ),\;\;\mu (\lambda_{\text{r}} ) \le 1,$$

(17)

where the variation range of f ₂, f ₃ and f ₄ can be, respectively, expressed as

$$- g \le f_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \le 0,$$

(18)

$$0 \le f_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \le \frac{{R_{\omega } g}}{{2J_{\text{f}} }}(m_{1}^{ + } + m_{3}^{ + } ),$$

(19)

$$\hbox{min} \left[ {\frac{{R_{\omega } g}}{{2J_{\text{r}} }}(m_{2}^{ - } - m_{3}^{ + } ),0} \right] \le f_{4} \le \frac{{R_{\omega } g}}{{2J_{\text{r}} }}m_{2}^{ + } .$$

(20)

The approximate values of $f_{2}$, $f_{3}$ and $f_{4}$ are expressed as

$$\hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = - 0.5g,$$

(21)

$$\hat{f}_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = \frac{{R_{\omega } g}}{{4J_{\text{f}} }}(m_{1}^{ + } + m_{3}^{ + } ),$$

(22)

$$\hat{f}_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = \frac{1}{2}\left\{ {\hbox{min} \left[ {\frac{{R_{\omega } g}}{{2J_{\text{r}} }}(m_{2}^{ - } - m_{3}^{ + } ),0} \right] + \frac{{R_{\omega } g}}{{2J_{\text{r}} }}m_{2}^{ + } } \right\}.$$

(23)

Define $\left| {f_{2} - \hat{f}_{2} } \right| \le F_{2} ,\;\left| {f_{3} - \hat{f}_{3} } \right| \le F_{3} ,\;\;{\text{and}}\left| {f_{4} - \hat{f}_{4} } \right| \le F_{4},$ Eqs. (21), (22) and (23) can be converted into the following forms:

$$F_{2} = 0.5g,$$

(24)

$$F_{3} = \frac{{R_{\omega } g}}{{4J_{\text{f}} }}(m_{1}^{ + } + m_{3}^{ + } ),$$

(25)

$$F_{4} = \frac{{R_{\omega } g}}{{2J_{\text{r}} }}m_{2}^{ + } - \frac{1}{2}\left\{ {\hbox{min} \left[ {\frac{{R_{\omega } g}}{{2J_{\text{r}} }}(m_{2}^{ - } - m_{3}^{ + } ),0} \right] + \frac{{R_{\omega } g}}{{2J_{\text{r}} }}m_{2}^{ + } } \right\}.$$

(26)

Define the difference between the actual and target slip ratio of front and rear wheels as the switching surface of sliding mode. The equations can be described as

$$S_{1} = \tilde{\lambda }_{\text{f}} = \lambda_{\text{f}} - \lambda_{\text{fd}} ,$$

(27)

$$S_{2} = \tilde{\lambda }_{\text{r}} = \lambda_{\text{r}} - \lambda_{\text{rd}} ,$$

(28)

where λ _f and λ _r denotes the actual slip ratio of front and rear wheels respectively, λ _fd and λ _rd denotes the target slip ratio of front and rear wheels respectively. To attain the equivalent control torque, the derivative of Eqs. (27), (28) with respect to time are, respectively, given by

$$T_{{{\text{eq}} . {\text{bf}}}} = \frac{{2J_{\text{f}} }}{{R_{{{\omega }}} }}\left[ {\dot{\lambda }_{\text{fd}} V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{f}} ) + R_{{{\omega }}} \hat{f}_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} )} \right],$$

(29)

$$T_{{{\text{eq}} . {\text{br}}}} = \frac{{2J_{\text{r}} }}{{R_{{{\omega }}} }}\left[ {\dot{\lambda }_{\text{rd}} V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{r}} ) + R_{{{\omega }}} \hat{f}_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} )} \right],$$

(30)

The brake torque of front and rear wheels is given as

$$T_{\text{b}} = T_{{{\text{eq}} . {\text{b}}}} - k\text{sgn} (S).$$

(31)

By accessibility conditions of switching surface, inequality must to be satisfied as follows:

$$S\dot{S} \le 0.$$

(32)

The ideal brake torque of front and rear wheels is defined as

$$T_{\text{bf}} = \frac{{2J_{\text{f}} }}{{R_{{{\omega }}} }}\left[ \begin{aligned} & \dot{\lambda }_{\text{fd}} V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{f}} ) + R_{{{\omega }}} \hat{f}_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \\ & - (F_{\text{f}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + \eta_{1} )\text{sgn} (S_{1} ) \\ \end{aligned} \right],$$

(33)

$$T_{\text{br}} = \frac{{2J_{\text{r}} }}{{R_{{{\omega }}} }}\left[ \begin{aligned} & \dot{\lambda }_{\text{rd}} V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{r}} ) + R_{{{\omega }}} \hat{f}_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \\ & - (F_{\text{r}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + \eta_{2} )\text{sgn} (S_{2} ) \\ \end{aligned} \right].$$

(34)

Respectively

$$F_{\text{f}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = F_{2} (1 - \lambda_{\text{f}} ) + R_{{{\omega }}} F_{3} ,$$

$$F_{\text{r}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) = F_{2} (1 - \lambda_{\text{r}} ) + R_{{{\omega }}} F_{4} ,$$

where η ₁, η ₂ are positive constants.

3.2 Eliminate chattering

Chattering phenomena is one of the undesirable effects of sliding mode control. In order to eliminate the chattering phenomena in sliding mode control, a saturated function sat(S/φ) was introduced and the sliding mode controller was redesigned using integral switching surface to make the control law smooth [24]. Defining integral switching surface as

$$S_{1} = \lambda_{\text{f}} - \lambda_{\text{fd}} + \xi_{1} \int {(\lambda_{\text{f}} - \lambda_{\text{fd}} } ){\text{d}}t,$$

(35)

$$S_{2} = \lambda_{\text{r}} - \lambda_{\text{rd}} + \xi_{2} \int {(\lambda_{\text{r}} - \lambda_{\text{rd}} } ){\text{d}}t,$$

(36)

where ξ ₁ and ξ ₂ are the constant.

Using the method of integral switching surface, the ideal brake torque of front and rear wheels are, respectively, given by

$$\begin{aligned} T_{\text{bf}} = \hfill \\ \;\;\;\;\frac{{2J_{\text{f}} }}{{R_{{{\omega }}} }}\left[ \begin{aligned} & (\dot{\lambda }_{\text{fd}} - \xi_{1} \tilde{\lambda }_{\text{f}} )V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{f}} ) + R_{{{\omega }}} \hat{f}_{3} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \hfill \\ & - (F_{\text{f}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + \eta_{1} )\text{sat} \left( {\frac{{\tilde{\lambda }_{\text{f}} + \xi_{1} \int {\tilde{\lambda }_{\text{f}} {\text{d}}t} }}{{\varphi_{1} }}} \right) \hfill \\ \end{aligned} \right], \hfill \\ \end{aligned}$$

(37)

$$\begin{aligned} T_{\text{br}} = \hfill \\ \;\;\;\frac{{2J_{\text{r}} }}{{R_{{{\omega }}} }}\left[ \begin{aligned} & (\dot{\lambda }_{\text{rd}} - \xi_{2} \tilde{\lambda }_{\text{r}} )V - \hat{f}_{2} (\lambda_{\text{f}} ,\lambda_{\text{r}} )(1 - \lambda_{\text{r}} ) + R_{{{\omega }}} \hat{f}_{4} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) \\ & - (F_{\text{r}} (\lambda_{\text{f}} ,\lambda_{\text{r}} ) + \eta_{2} )\text{sat} \left( {\frac{{\tilde{\lambda }_{\text{r}} + \xi_{2} \int {\tilde{\lambda }_{\text{r}} {\text{d}}t} }}{{\varphi_{2} }}} \right) \\ \end{aligned} \right], \hfill \\ \end{aligned}$$

(38)

where φ ₁ and φ ₂ are the constant, φ ₁and φ ₂ are the boundary layer thickness which is made varying to take advantage of the system bandwidth. How to get the values are introduced by Ref. [23].

4 Road Friction Coefficient Observer Design

Linear extended state observer can expand the uncertainties and unknown perturbation controlled object model into new state observation and it is very suitable for road friction coefficient estimation problems which only have the measured output and control input [25, 26]. Using the linear extended state observer, road friction coefficient can be observed, where friction coefficient between tire and road as output of the second order linear extended state and angular speed and braking torque of front and rear wheels as the input.

By section 2.1 two-wheeled vehicle braking dynamics model we can obtain

$$\left\{ \begin{aligned} \dot{\omega }_{\text{f}} = \frac{1}{{2J_{\text{f}} }}( - T_{\text{bf}} + \mu (\lambda_{\text{f}} )m_{1} R_{{{\omega }}} g - \mu (\lambda_{\text{f}} )m_{3} R_{{{\omega }}} \ddot{x})\;, \hfill \\ \dot{\omega }_{\text{r}} = \frac{1}{{2J_{\text{r}} }}( - T_{\text{br}} + \mu (\lambda_{\text{r}} )m_{2} R_{{{\omega }}} g + \mu (\lambda_{\text{r}} )m_{3} R_{{{\omega }}} \ddot{x})\;. \hfill \\ \end{aligned} \right.$$

(39)

Rewriting Eq. (39), then

$$\left\{ \begin{aligned} \dot{\omega }_{\text{f}} = \frac{1}{{2J_{f} }}\mu (\lambda_{\text{f}} )(m_{1} R_{{{\omega }}} g - m_{3} R_{{{\omega }}} \ddot{x}) + \frac{ - 1}{{2J_{\text{f}} }}T_{\text{bf}} \;, \hfill \\ \dot{\omega }_{\text{r}} = \frac{1}{{2J_{r} }}\mu (\lambda_{\text{r}} )(m_{2} R_{{{\omega }}} g + m_{3} R_{{{\omega }}} \ddot{x}) + \frac{ - 1}{{2J_{\text{r}} }}T_{\text{br}} \;. \hfill \\ \end{aligned} \right.$$

(40)

Contained the term of the road friction coefficient of Eq. (40) were regarded as the perturbation of system, and for the expansion state variables of the system, we defined

$$\begin{aligned} \omega_{\text{f}} = x_{1} ,\;\;\frac{1}{{2J_{\text{f}} }}\mu (\lambda_{\text{f}} )(m_{1} R_{{{\omega }}} g - m_{3} R_{{{\omega }}} \ddot{x}) = x_{2} ,\;\;\omega_{\text{r}} = x_{3} ,\, \hfill \\ \frac{ - 1}{{2J_{\text{f}} }} = b_{1} ,\;\frac{1}{{2J_{\text{r}} }}\mu (\lambda_{\text{r}} )(m_{2} R_{{{\omega }}} g + m_{3} R_{{{\omega }}} \ddot{x}) = x_{4} ,\;\frac{ - 1}{{2J_{\text{r}} }} = b_{2} ,\; \hfill \\ T_{\text{bf}} = u_{1} ,\;\;T_{\text{br}} = u_{2} . \hfill \\ \end{aligned}$$

Rewriting Eq. (40) into two integrator series system are, respectively, given by

$$\left\{ \begin{aligned} \dot{x}_{1} = x_{2} + b_{1} u_{1} \;, \hfill \\ y_{1} = x_{1} \;. \hfill \\ \end{aligned} \right.$$

(41)

$$\left\{ \begin{aligned} \dot{x}_{3} = x_{4} + b_{2} u_{2} , \hfill \\ y_{2} = x_{3} \;. \hfill \\ \end{aligned} \right.$$

(42)

Using integrator series system of Eq. (41) as an example, second order linear extended state observer was designed as follows to observe the x ₁ and x ₂.

$$\left\{ \begin{aligned} & e(k) = z_{1} (k) - y_{1} (k)\;, \hfill \\ & \beta_{01} { = }2\omega_{0} \;,\;\;\;\beta_{02} { = }\omega_{0}^{2} \;\;, \hfill \\ z_{1} (k + 1) = z_{1} (k) + h\left[ {z_{2} (k) - \beta_{01} e(k) + b_{0} u_{1} (k)} \right], \hfill \\ & z_{2} (k + 1) = z_{2} (k) + h\left[ { - \beta_{02} e(k)} \right], \hfill \\ \end{aligned} \right.$$

(43)

where ω ₀ is the bandwidth of linear extended state observer by pole assignment, u ₁ and y ₁ are the input signal, respectively, z ₁ and z ₂ are the output signal of linear extended state observer, which are the observations of the x ₁ and x ₂, b ₀ is the estimation value of control gain b ₁. From what has been discussed above, the observations of x ₃ and x ₄ can also be formulated as Eq. (42).

To estimate the road friction coefficient, substituting Eq. (40) into Eq. (41), a linear extended state observer was built as follows in detail:

$$\left\{ \begin{aligned} & \omega_{f} = z_{1} , \hfill \\ & \frac{1}{{2J_{\text{f}} }}\mu (\lambda_{\text{f}} )(m_{1} R_{{{\omega }}} g - m_{3} R_{{{\omega }}} \ddot{x}) = z_{2} , \hfill \\ \end{aligned} \right.$$

(44)

where z ₁ and z ₂ are the observations of the x ₁ (front wheel angular speed) and x ₂ (contain the term of the road friction coefficient). Likely,

$$\left\{ \begin{aligned} & \omega_{\text{r}} = z_{3} , \hfill \\ & \frac{1}{{2J_{\text{r}} }}\mu (\lambda_{\text{r}} )(m_{2} R_{{{\omega }}} g + m_{3} R_{{{\omega }}} \ddot{x}) = z_{4} , \hfill \\ \end{aligned} \right.$$

(45)

where z ₃ and z ₄ are the observations of the x ₃ (rear wheel speed) and x ₄. Combining the Eqs. (44), (45), road friction coefficient of front and rear wheels can be formulated as

$$\left\{ \begin{aligned} \mu (\lambda_{\text{f}} ) = \frac{{2J_{\text{f}} z_{2} }}{{m_{1} R_{{{\omega }}} g - m_{3} R_{{{\omega }}} \ddot{x}}}, \hfill \\ \mu (\lambda_{\text{r}} ) = \frac{{2J_{\text{r}} z_{4} }}{{m_{2} R_{{{\omega }}} g + m_{3} R_{{{\omega }}} \ddot{x}}}\;. \hfill \\ \end{aligned} \right.$$

(46)

If the estimation scheme of rear wheel is the same with front wheel, there is no need to show the estimation scheme of rear wheel.

5 Simulation Results

In this section, a number of simulations are carried out on the simulation software of ADAMS/Car simulating a vehicle in the virtual simulation environment to analyze and evaluate the estimation scheme proposed in this paper. While for road friction coefficient estimation, set vehicle model parameters in simulation as follows: m = 1500 kg, m _s = 1285 kg, m _f = 96 kg, m _r = 119 kg, l _f = 1.186 m, l _r = 1.258 m, h _f = 0.3 m, h _r = 0.3 m, R _ω = 0.326 m, J _f = 1.7 kg m², J _r = 1.7 kg·m². The whole vehicle model in ADAMS/Car is shown in Fig. 3. A test environment for road with different friction coefficient was constructed using the compiler road builder in ADAMS/Car and the proposed road friction coefficient estimation method was tested in the constructed virtual environment.

Fig. 3

Whole vehicle test model in ADAMS/Car

The estimation method is constructed based on braking torque and wheel speed sensor. From what has been discussed above, the block diagram of road friction coefficient estimation scheme of front wheel is illustrated in Fig. 4.

Fig. 4

Block diagram of road friction coefficient estimation scheme of front wheel

5.1 High Friction Coefficient Road Surface

Ideal braking torque controller can make full use of road friction coefficient when vehicle braking. The braking torque of front and rear wheels is shown in Fig. 5 which is conducted on high friction coefficient of 0.8 and the initial longitudinal velocity of 30 m/s.

Fig. 5

Braking torque of front and rear wheels µ = 0.8

Fig. 6 shows the estimated road friction coefficient by second order linear extended state based on wheel speed and braking torque of front and rear wheels. As we can see from the figure, estimated road friction coefficient of front and rear wheels is very close to the real values of road friction coefficient. The largest difference between the estimated and the real values occurs about 0.3 s ago. It also can be seen that the front wheels of road friction coefficient estimation values is better than rear wheel. However, when vehicle braking, wheel speed signal has mixed with measurement noise and it assumed to be independent white Gaussian process with zero mean.

Fig. 6

Estimation of road friction coefficient µ = 0.8

Fig. 7 shows the estimated road friction coefficient with noise interference. It is observed that the linear extended state observer estimated road friction coefficient with noise interference is also very close to the real values and exactly with strong robustness.

Fig. 7

Estimated road friction coefficient with noise interference µ = 0.8

5.2 Low Friction Coefficient Road Surface

Fig. 8 shows the braking torque of front and rear wheels conducted on low friction coefficient of 0.2 and the initial longitudinal velocity of 30 m/s. The estimated values and real values of the road friction are presented in Fig. 9 which indicates that the proposed estimator works well in low friction coefficient road surface and Fig. 10 shows the estimated road friction coefficient with noise interference respectively.

Fig. 8

Braking torque of front and rear wheels µ = 0.2

Fig. 9

Estimation of road friction coefficient µ = 0.2

Fig. 10

Estimated road friction coefficient with noise interference µ = 0.2

It can be easily seen that the proposed linear extended state observer can estimate the road friction coefficient, with good accuracy in comparison with the measurements in vehicle braking on a single high or low friction coefficient road surface. Even in noise interference, it also can estimate road friction coefficient efficiently. However, the road friction coefficient estimation values of using the front wheels are better than rear wheels.

Next, simulations in uneven friction coefficient road conditions are discussed in detail.

5.3 Uneven Friction Coefficient Road

For road friction coefficient estimation in uneven friction road, the road surface with friction coefficients ranged from high to low and low to high are designed.

The braking torque of front and rear wheels is shown in Fig. 11(a) which is conducted on friction coefficients ranged from 0.8 to 0.2 and the initial longitudinal velocity of 30 m/s. Where, in 0 to 2 s, the vehicles are driven in high friction coefficient road surface and 2 to 7 s in the low friction coefficient road surface. Similarly, the braking torque of front and rear wheels conducted on friction coefficients ranged from 0.2 to 0.8 is shown in Fig. 11b. Where, in 0 to 2 s, the vehicles are driven in low friction coefficient road surface and 2 to 5 s in the high friction coefficient road surface. Respectively, the variations of estimated road friction coefficient in ideal condition are, with noise interference, given in Fig. 12.

Fig. 11

Braking torque of front and rear wheels

Fig. 12

Simulated and estimated road friction coefficient comparison

We can easily find that the estimated road friction coefficient in ideal condition (Fig. 12(a), (c)) or measurement with noise interference (Fig. 12(b), (d)) is close to the reference values and the estimated values are less influenced by noise interference.

Though the road coefficients change greatly, the simulation results show that the proposed estimate method can still estimate road friction coefficient exactly with strong robustness, which can resist external disturbance.

6 Conclusions

• (1) According to the vehicle braking dynamics, the linear extended state observer to estimate the road friction coefficient is presented, which has a good accuracy when vehicle drive on the road of different friction coefficient.

• (2) Using the method of saturation function and integral switching surface can eliminate chattering of sliding mode control.

• (3) Simulation results using the front wheels of road friction coefficient estimation values are better than rear wheels.

• (4) The proposed method has strong robustness in different road conditions, which can resist external disturbance.