Design of ABS sliding mode control system for in-wheel motor vehicle based on road recognition

Abstract

Aiming at the problem that the current ABS control algorithm cann’t make full use of the ground braking force to complete the braking when the complex road surface is in emergency braking, the ABS sliding mode variable structure control method based on road surface identification is proposed. Combined with the in-wheel motor of in-wheel motor electric vehicle, a coordinated control method of motor hydraulic composite is designed. Based on the fuzzy logic control method, the road adhesion coefficient is estimated to realize the identification of typical roads and dynamically obtain the optimal slip rate of different roads. The ABS sliding mode variable structure controller is designed with the optimal slip ratio and the actual slip ratio as input, and the saturation function is used to replace the sign function in the traditional sliding mode variable structure control to weaken the ’ chattering ’ phenomenon in the sliding mode variable structure control, and then the ABS controller is designed. Taking the experimental prototype vehicle driven by four-wheel hub motor as the research object, an eight-degree-of-freedom dynamic simulation model of the whole vehicle is established. Compared with the traditional PID controller, the braking time is shortened by 0.2 s and the braking distance is shortened by 2.3 m, which shows the feasibility of the designed controller. Through the simulation braking experiment of the docking road, the adaptability and real-time performance of the ABS sliding mode controller are verified, and the importance of the road adhesion coefficient identification to the ABS controller is verified.

Keywords

Vehicle engineering vehicle anti-lock braking system road identification system sliding mode control slip rate

1 Introduction

As an important active safety system, Vehicle anti-lock braking system has received more and more attention. Due to the influence of many factors, the existing anti-lock braking system cannot play a very good role in braking under certain working conditions. In order to improve the stability of the anti-lock braking system, many scholars have done a lot of research. The main achievement at present is that based on the slip rate that can be precisely controlled instead of the logic threshold as the main input, the road conditions are identified in real time, and the ABS control strategy is changed in advance according to the changes of the road surface, thereby improving the stability of the anti-lock braking system and braking performance of the car [1].

The estimation of road adhesion coefficient is not only related to road conditions, but also affected by factors such as tire material, tread pattern, tire pressure, ambient temperature, sensor measurement noise, etc., which makes the real-time robust estimation of adhesion coefficient and its stability analysis extremely difficult. A lot of research has been done on the real-time estimation algorithm of pavement adhesion coefficient at home and abroad, which can be divided into two categories: Effect-based and Cause-based [2]. Cause-based estimation method is to predict the size of the current road adhesion coefficient by measuring the road surface roughness or road surface lubrication degree and combining with past experience [3]. The estimation algorithm based on Cause-based requires additional sensors, and the recognition accuracy is greatly affected by the environment and the quality of the sensor, which is not conducive to promotion and popularization [4]. The Effect-based estimation method is to estimate the road adhesion coefficient by detecting the motion response generated by the change of the road adhesion coefficient or the adhesion coefficient on the vehicle body and the wheel [5]. The estimation algorithm of the effect-based mainly includes the Kalman filter principle and its derivative estimation algorithm. When estimating the road adhesion coefficient, the observation accuracy of the process noise and the measurement noise will affect the accuracy of the controller, and there is no good solution [6]. On the basis of reference [7], this paper puts forward the identification method of five typical roads based on fuzzy control, and takes the identification result as the input of ABS controller design to design the ABS controller.

Although the control strategy based on slip rate is obviously superior to logic threshold control in continuity and control accuracy, different control strategies also have their own shortcomings [8]. PID control needs to adjust the parameters repeatedly to achieve the precise tracking of the slip rate, and with the changes of vehicle models and working conditions, the parameters need to be adjusted again [9]. Neural Network control requires a lot of debugging and training, and the cost is relatively high, which restricts its development [10]. Fuzzy control can control the system without knowing the mathematical model of the controlled system, but its control accuracy depends on the establishment of fuzzy rules, which has relatively large requirements for researchers and relatively low accuracy [11]. The control strategies for ABS are shown in Table 1. In addition, the control and fault detection of brushless DC motor is also the main part of the current ABS control system [12]. K. Vanchinathan designed a fractional order PID controller based on genetic algorithm to control the speed of the motor, and verified the superiority of the designed control algorithm by comparing with the traditional PID based on genetic algorithm [13]. Xia k et al proposed a pulse width modulation model predictive control algorithm to suppress the noise generated by the commutation torque pulse when the motor rotates. The simulation and experimental results show that the proposed control algorithm can suppress the torque ripple and improve the performance of the motor [14]. Among the different control strategies of ABS, sliding mode control has become the first choice for designing ABS controllers because of its robustness to system uncertainty and its effectiveness in nonlinear systems [15]. He [16] et al. designed two control laws for wheel deceleration and slip rate based on sliding mode control theory for pure electric vehicles, and verified the system through a single road simulation, but did not verify the applicability and real-time performance of the control system. At present, in the design of ABS sliding mode controller, most of the literatures do not consider the influence of the change of road adhesion coefficient on the control system, and set the optimal slip rate as customization, which cann’t meet the needs of braking in complex and changeable actual road conditions [17–19]. Therefore, this paper identifies different roads through the road recognition algorithm, and then calculates the optimal slip rate, and then uses the optimal slip rate as the input of the sliding mode controller to design the ABS sliding mode controller. The chattering of the sliding mode controller is weakened by using the exponential reaching law instead of the constant reaching law and the saturation function instead of the sign function.

Table 1
ABS control algorithm comparative analysis table

Control algorithm\Compare Control effect Advantage Disadvantage

Classic control Control results depend on the selection of parameters Does not rely on mathematical models, simple control, and is easy to implement Sensitive to parameter changes, the transient response needs to be improved

Optimal control Excellent control effect High control accuracy, can be used in combination with other control algorithms An accurate mathematical model is required, and the requirements for parameter setting are very high

Sliding mode control Excellent control effect and high robustness Does not rely on precise mathematical models, with good robustness and strong stability There is inherent chattering that cannot be eliminated

Fuzzy control Control accuracy is closely related to the number of classifications designed by fuzzy control rules Does not rely on mathematical models, good robustness and good ductility Difficulty in parameter adjustment and high requirements for experience

Control algorithm\Compare	Control effect	Advantage	Disadvantage
Classic control	Control results depend on the selection of parameters	Does not rely on mathematical models, simple control, and is easy to implement	Sensitive to parameter changes, the transient response needs to be improved
Optimal control	Excellent control effect	High control accuracy, can be used in combination with other control algorithms	An accurate mathematical model is required, and the requirements for parameter setting are very high
Sliding mode control	Excellent control effect and high robustness	Does not rely on precise mathematical models, with good robustness and strong stability	There is inherent chattering that cannot be eliminated
Fuzzy control	Control accuracy is closely related to the number of classifications designed by fuzzy control rules	Does not rely on mathematical models, good robustness and good ductility	Difficulty in parameter adjustment and high requirements for experience

2 Experimental prototype and its dynamic modeling

Taking the in-wheel motor electric vehicle trial-produced by our research group as the research object, in order to facilitate the study of its kinematic characteristics, it is simplified to an eight-degree-of-freedom vehicle model, and the tire model adopts the Dugoff tire model [20]. The in-wheel motor electric vehicle is shown in Fig. 1.

Fig. 1

Hub motor electric vehicle.

2.1 Brushless DC Motor model

The in-wheel motor of the prototype is a Brushless DC Motor, which is widely used in in-wheel motor electric vehicles due to its advantages of low temperature rise, low noise, high torque, high speed, low energy consumption and long life. In order to simplify the model, the model derivation of the brushless DC motor is based on the following assumptions [21].

Ignore the motor core saturation, regardless of eddy current and hysteresis loss;

Ignoring armature reaction;

The conductivity of the permanent magnet material is zero, and the internal permeability is consistent with the air;

The air gap magnetic field of the motor is sinusoidally distributed in the air;

No damping winding on the rotor;

(1) The voltage balance equation of stator three-phase winding: $\begin{matrix} (\begin{matrix} u_{a} \\ u_{b} \\ u_{c} \end{matrix}) = (\begin{matrix} R & 0 & 0 \\ 0 & R & 0 \\ 0 & 0 & R \end{matrix}) (\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}) \\ + (\begin{matrix} L - M & 0 & 0 \\ 0 & L - M & 0 \\ 0 & 0 & L - M \end{matrix}) P (\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}) - (\begin{matrix} e_{a} \\ e_{b} \\ e_{c} \end{matrix}) \end{matrix}$ (1)

In the formula, u_a, u_b, u_c is the phase voltages at both ends of the stator winding; i_a, i_b, i_c is the phase current flowing through the stator winding coil; e_a, e_b, e_c is the stator phase winding electromotive force; R is the equivalent resistance of the motor; L is the self-inductance of each phase winding; M is the mutual inductance between each two-phase windings; P is a differential operator.

According to the winding voltage equation, the equivalent circuit of the Brushless DC Motor is shown in Fig. 2:

(2) Electromagnetic torque equation

Fig. 2

Large equivalent DC circuit diagram of Brushless DC motor.

The electromagnetic torque of brushless DC motor is: $T_{e} = (e_{a} i_{a} + e_{b} i_{b} + e_{c} i_{c}) / ω_{d}$ (2) $T_{e} = K_{t} i_{abc}$ (3) $ω_{d} = \frac{2 π}{60} n$ (4)

In the formula, T_e is electromagnetic torque; ω _d is the mechanical angular velocity of the motor; K_t is the motor torque coefficient; i_abc is the winding phase current in steady state; n is the motor rotation speed.

(3) Motor motion equation

The motion equation of Brushless DC Motor is: $T_{e} = J_{d} \frac{d ω_{d}}{dt} + B ω_{d} + T_{l}$ (5)

In the formula, J_d is the moment of inertia of the motor; B is the damping coefficient; T_l is load torque;

2.2 Brake model

The brake hydraulic transfer system [22] can be simplified into an electromagnetic proportional link, a first-order inertia link and an integral link, and the simplified model transfer function is shown in Formula (6): $G (s) = \frac{K}{s \cdot (T_{p} s + 1)}$ (6)

In the formula, s is the Laplace operator; K is the system gain; T_p is the time constant.

3 Estimation of road adhesion coefficient based on fuzzy logic algorithm

3.1 Single-wheel tire model

The first stage of the controller design process is to obtain an accurate dynamic model of the physical system or device to be controlled. Use the equivalent simple model of the quarter vehicle dynamic motion described in [23]. Figure 3 shows the schematic diagram of the two-degree-of-freedom dynamics model of a quarter vehicle, which can simulate linear braking conditions on flat roads. In order to derive the nonlinear differential equation of ABS, some common assumptions are considered in order to obtain the appropriate control law. First, only the longitudinal dynamics of the vehicle are considered, ignoring vertical and lateral motion. Second, there is no interaction between the four wheels of the vehicle. Second, there is no interaction between the four wheels of the vehicle. Finally, it is assumed that rolling resistance and air resistance are ignored because rolling resistance and wind resistance are relatively small during braking.

According to Newton’s second law, the dynamics equation of vehicle and wheel can be expressed as: $m {\dot{v}}_{x} = - F_{x}$ (7) $I \dot{ω} = F_{x} r - T_{μ}$ (8)

In the formula, m is 1/4 vehicle mass; v_x represents the longitudinal speed; I is the moment of inertia of the vehicle; ω is the wheel angular velocity; T_μ is the torque of the brake; g is the acceleration of gravity; F_z is the normal support force of the tire by the ground; r is the wheel radius. According to the sliding condition, the longitudinal friction force F_x can be expressed by formula (9): $F_{x} = μ_{x} (λ) F_{z}$ (9)

Fig. 3

Single-wheel vehicle model.

Without considering the axle load transfer during braking, F_z is expressed as: $F_{z} = mg$ (10)

The variable used to express the difference between vehicle speed and wheel speed during braking is called wheel slip rate λ and is defined as follows: $λ = \frac{v_{x} - r ω}{v_{x}}$ (11)

It can be known from formula (11), the slip rate λ ranges from 0 to 1. When the slip rate is λ=0, it indicates that the wheel is doing pure rolling motion at this time, and the speed is equal to the wheel speed without braking. When the slip rateλ is between (0,1), it indicates that the wheel is in the braking state, and the wheel is rolling and sliding, and the wheel is not locked completely. When the slip rate isλ=1, it indicates that the wheel is doing pure sliding motion at this time, and the wheel is completely locked and dragged sliding. Taking the derivative of formula (11) with respect to time, and substituting formula (7), (8), (9) and (10) into simplification, the dynamic change rate of slip rate can be obtained as follows: $\dot{λ} = \frac{1}{v_{x}} [(1 - λ) \frac{μ_{x} (λ) F_{z}}{m} - \frac{r}{I ω} μ_{x} (λ) F_{z} r + \frac{r}{I ω} T_{μ}]$ (12)

The adhesion coefficient of road surface was determined by analyzing the relationship between longitudinal force and slip rate of tire. Burckhardt tire longitudinal force model was selected as the research object, and various typical road surface (μ-λ) (adhesion coefficient – slip rate) was fitted through a large number of road surface tests, and the expression is as follows [24]: $μ_{x} (λ) = c_{1} (1 - exp (- c_{2} λ)) - c_{3} λ$ (13)

In the formula, c₁ is the maximum value of the friction curve, c₂ is the shape of the friction curve, and c₃ is the friction curve difference between the maximum value and the corresponding value. Table 2 shows the friction model parameters of various road surfaces. According to the friction model parameters of different road surfaces in Table 2 and equation (13), the road adhesion coefficient and slip rate (μ-λ) curves can be obtained, as shown in Fig. 4.

Table 2

Friction model parameters of different road conditions

Surface conditions	c ₁	c ₂	c ₃
Dry asphalt	1.2801	23.99	0.52
Wet asphalt	0.857	33.822	0.347
Dry concrete	1.1973	25.168	0.5373
Snow	0.1946	94.129	0.0646
Ice	0.05	306.39	0

Fig. 4

Relationship between adhesion coefficient and slip rate of road surface.

3.2 Design of fuzzy rules

Taking a single wheel as the object of road identification, the longitudinal force F_x and slip rateλ of the tire were obtained according to the vehicle model and Dugoff tire model established above. Longitudinal force F_x and slip rateλ were used as input of road identification module by fuzzy logic control. Simulation experiments were carried out on typical adhesion road surface under different working conditions respectively. The approximate degree of fuzzy adhesion coefficient was compared with that of theoretical adhesion coefficient, and then the recognition of current road surface was realized.

When identifying models, fuzzy logic control does not need to know the specific system model, but only needs to establish the corresponding fuzzy rule base, and good identification effects can be obtained by later adjustment of fuzzy rules according to experiments [25]. The input of fuzzy controller for road surface identification is longitudinal force F_x and brake slip rateλ. Through the process of fuzzification, logical reasoning and defuzzification, the similarity between the output result and typical road surface is compared, so as to realize road surface identification.

(1) Parameter fuzzification

As shown in Fig. 5, the slip rate is blurred into two fuzzy subsets [0,0.01) and (0.01,1]. When the slip rate is greater than 0.15, the membership degree of the fuzzy subset with large slip rate is set to 1. It can be seen from Fig. 5 that when the wheel slip rate is small, the discernability of road surface is low, and it is difficult to identify the current road surface. Moreover, identification of road surface when the wheel slip rate is small will not affect the work of chassis safety system. According to experience, the longitudinal force of tire is normalized and fuzzy, as shown in Fig. 6.

Fig. 5

Wheel slip rateλ fuzzification.

Fig. 6

Fuzzy normalization of longitudinal forces of wheels.

Through continuous debugging and identification of different pavement results to make them conform to the actual situation, the fuzzy rules are shown in Table 3. The fuzzy wheel slip rate and the normalized and fuzzy wheel longitudinal force are taken as the input of the fuzzy reasoning system. The output is the fuzzy weight coefficients c₁, c₂ and c₃ in the (μ-λ) relation under the current road surface.

Table 3

Fuzzy rules table

Fuzzy rules	Input		Output
	λ	F_x	c ₁	c ₂	c ₃
1	ZN	None	ZM	ZN	ZM
2	PN	ZN	ZN	PM	ZN
3	PN	ZM	ZM	PN	ZM
4	PN	ZE	ZM	ZE	ZE
5	PN	PM	PM	ZM	PN
6	PN	PN	PN	ZN	PM

(3) Fuzzy solution

After the wheel slip rate and tire longitudinal force are fuzzified, the desired output can be obtained only by de-fuzzifying. Gaussian membership function is used to allocate the output weight coefficient by virtue of its good distinction. Combined with Fig. 7, the center of gravity method is used to defuzzify the fuzzy quantity.

Fig. 7

Solving fuzzy fuzzy subset and membership function.

3.3 Calculate the optimal slip rate

According to formula (13), let ∂μ/∂λ = 0, and get the best slip rate: $λ_{d} = \frac{1}{c_{2}} ln \frac{c_{1} c_{2}}{c_{3}}$ (14)

Where c_1, c₂ and c₃ are the weight coefficients obtained by fuzzy logic algorithm.

4 Design of sliding mode controller

The main problems in the design of ABS controller are the highly nonlinear model dynamics, uncertainty caused by parameter changes and unknown external interference [26]. Among the control methods of ABS, sliding mode variable structure control is widely studied because of its robustness to model uncertainty and applicability to nonlinear systems. A sliding mode controller based on road recognition is designed. In this controller, exponential approach law is used instead of constant approach law and saturation function is used instead of sign function to reduce chattering. Figure 8 is the design scheme of sliding mode controller based on road surface recognition. The design of sliding mode controller mainly includes the following three parts:

Design of sliding mode switching surface;

Design of equivalent braking torque and switching control law;

Chattering suppression design.

Fig. 8

Design scheme of ABS controller.

4.1 Design of sliding mode switching surface

Anti-lock braking system with sliding mode controller takes wheel slip rate as the control object and braking torque as the control object. The optimal road slip rateλ _d is calculated by identifying the road surface condition to eliminate the tracking error between the actual slip rate and the optimal slip rate, so as to control the actual slip rate of the wheel to keep close to the optimal slip rate, that is, the design goal of the controller is to make the system state ( $λ, \dot{λ}$ ) tends to (λ _d , ${\dot{λ}}_{d}$ ). The sliding mode controller changes the control law of braking torque T_μ by using the switching function, which is defined as follows: $e (t) = λ (t) - λ_{d} (t)$ (15)

Under the condition of generalized sliding mode control, exponential function approaching law is used to design sliding mode controller. The switching function should meet the sliding mode existence and accessibility conditions, as: $e (t) \frac{de (t)}{dt} ⩽ 0$ , at the same time, the stability of sliding mode motion should be satisfied.

4.2 Design of equivalent control brake torque

In the design of equivalent braking torque, exponential approaching law is adopted to obtain: $\frac{de (t)}{dt} = - ɛ sign (e) - ke ɛ > 0, k > 0$ (16)

In the formula, ɛ, k is a parameter greater than 0. Appropriate selection of ɛ, k can make the phase trajectory have different approaching laws at different positions of state control. If the ɛ value is small and the k value is quite large, the approaching velocity can be guaranteed to be large away from the sliding hyperplane and asymptotically small near the sliding hyperplane. If the value of ɛ is small enough, the approaching velocity is small, the distance of the system across the sliding hyperplane is small, and the jitter defect of the system can be improved appropriately. When k = 0, it is the isokinetic approach law commonly used in designing sliding mode controllers.

According to formula (12), (15) and (16): $\begin{matrix} \dot{e} (t) = \dot{λ} (t) - {\dot{λ}}_{d} (t) = \frac{1}{v_{x}} \\ [(1 - λ) \frac{μ (λ) F_{z}}{m} - \frac{r}{I ω} μ (λ) F_{z} r + \frac{r}{I ω} T_{μ}] \\ - {\dot{λ}}_{d} = - ɛ sign (e) - ke \end{matrix}$ (17) In formula: $sign (e) = {\begin{matrix} + 1, & e > 0 \\ 0, & e = 0 \\ - 1, & e < 0 \end{matrix}$ (18)

According to formula (17) and (18), the equivalent braking torque Tμ _eq can be deduced: $\begin{matrix} T_{μ eq} = \frac{I ω}{r} v_{x} {\dot{λ}}_{d} - \frac{I ω}{mr} μ (λ) F_{z} (1 - λ) \\ + μ (λ) F_{z} r - v_{x} (- ɛ sign (e) - ke) \end{matrix}$ (19)

According to equation (19), under the influence of Tμ _eq , the system meets the conditions of existence and accessibility of sliding mode.

Lyapunov function is defined as: $V = \frac{1}{2} e^{2}$ (20)

Take the derivative of Equation (20) and substitute (19) into: $\dot{V} = e \dot{e} = - ɛ | e | - k e^{2} ⩽ 0$ (21)

According to Equation (21), according to Lyapunov stability principle, the system is asymptotically stable.

4.3 System chattering suppression design

Theoretically, the sliding mode variable structure control has discontinuous control item sign(e), which forces the motion state point of the system to remain on the sliding surface through instantaneous switching. However, in practical application, due to the influence of calculation delay, actuator limitation, transmission delay and other factors, it is difficult to realize the ideal sliding. The system state can only reach the origin by frequently crossing the sliding mode surface, which leads to chattering phenomenon.

There are many methods to suppress chattering of the control system, but at present, saturation function is often used to replace sign function to suppress chattering of the control system.

The saturation function is defined as follows: $sat (e) = {\begin{matrix} + 1, & e > Δ \\ 1 / Δ, & | e | < Δ \\ - 1, & e < - Δ \end{matrix}$ (22)

Substitute Formula (22) for sign(e) and substitute formula (19) to obtain braking force T_μ of hub EV brake: $\begin{matrix} T_{μ} = \frac{I ω}{r} v_{x} {\dot{λ}}_{d} - \frac{I ω}{mr} μ_{x} (λ) F_{z} (1 - λ) \\ + μ_{x} (λ) F_{z} r - v_{x} (- ɛ sat (e) - ke) \end{matrix}$ (23)

Formula (23) is the braking force T_μ of the wheel ev brake under the control of the sliding mode controller under the condition that the hydraulic system completes its work.

4.4 Braking force distribution control strategy of compound braking system

In order to give full play to the advantages of quick response and accurate control of hub motor on the premise of ensuring braking stability, the braking force distribution control strategy provided in literature [27] is adopted:

On the road surface with low adhesion coefficient, the hydraulic brake provides the basic braking torque, and the hub motor realizes anti-lock coordination.

On the road surface with moderate adhesion coefficient, the hub motor provides the basic braking torque, and the slippage rate is coordinated by adjusting the hydraulic braking torque, so as to prevent the wheels from locking.

On high adhesion road surface, the wheel motor does not participate in the braking control, and the hydraulic braking system alone completes the anti-lock control.

5 Vehicle ABS simulation and result analysis

Taking the four-wheel hub motor electric vehicle prototype as the research object, the vehicle parameters are described in Table 4. Using the vehicle and controller model established by Matlab/Simulink software, the established sliding mode controller based on road recognition is compared with the traditional PID controller under the condition of road adhesion coefficient of 0.8. The simulation results are shown in Fig. 9. In order to verify the control accuracy, adaptability and real-time performance of the ABS sliding mode controller, and to verify the influence of the road recognition system on the ABS controller, the braking simulation experiment is carried out on the docking road, and the simulation results are shown in Figs. 10 to 13.

Table 4
Vehicle parameters

Parameters name Symbols Numerical value Unit

1/4Vehicle mass m 426.5 kg

Sprung mass m_s 1546 kg

The distance from the center of mass to the front axis a 1.09 m

The distance from the center of mass to the back axis b 1.53 m

Vehicle wheelbase of front and rear T_f,T_r 1.535 m

Tire radius r 0.35 m

Height of the center of mass h 0.30 m

Moment of inertia of sprung mass around X axis I_x 288 kg·m²

Moment of inertia of wheel and its components I_w 3 kg·m²

Moment of inertia of sprung mass around the Z axis I_z 288 kg·m²

Longitudinal stiffness of tire C_x 70000 N·m/rad

Lateral stiffness of tire C_y 55000 N·m/rad

Roll daming C _φ 40000 N·m·s/rad

Roll stiffness K _φ 50000 N·m/rad

Parameters name	Symbols	Numerical value	Unit
1/4Vehicle mass	m	426.5	kg
Sprung mass	m_s	1546	kg
The distance from the center of mass to the front axis	a	1.09	m
The distance from the center of mass to the back axis	b	1.53	m
Vehicle wheelbase of front and rear	T_f,T_r	1.535	m
Tire radius	r	0.35	m
Height of the center of mass	h	0.30	m
Moment of inertia of sprung mass around X axis	I_x	288	kg·m²
Moment of inertia of wheel and its components	I_w	3	kg·m²
Moment of inertia of sprung mass around the Z axis	I_z	288	kg·m²
Longitudinal stiffness of tire	C_x	70000	N·m/rad
Lateral stiffness of tire	C_y	55000	N·m/rad
Roll daming	C _φ	40000	N·m·s/rad
Roll stiffness	K _φ	50000	N·m/rad

Fig. 9

Braking performance of two controllers during emergency braking under high adhesion road condition.

Fig. 10

Identification diagram of road adhesion coefficient under docking road condition.

Fig. 11

Comparison of braking speed and displacement of docking pavement with and without pavement identification.

Fig. 12

Comparison of braking torque of docking road with and without road recognition.

Fig. 13

Comparison diagram of braking slip rate of docking pavement with and without pavement identification.

5.1 Comparison of sliding mode control and traditional PID control

Traditional PID control is a widely used and mature control method. It does not need to understand the mathematical model of the controlled object. It only needs to adjust the parameters online according to the system situation, match the appropriate proportion, integral and differential coefficients, and control the controlled object. The simulation model of PID controller is built as a reference for the control effect of ABS sliding mode controller, and the simulation braking experiment is carried out under the condition of high road adhesion coefficient. The simulation condition is: the road adhesion coefficient is set to 0.8, and the speed is 100 km/h. The simulation results of the comparison between the traditional PID controller and the sliding mode controller of the ABS are shown in Fig. 9. The explanation of the abbreviations in the figure is shown in Table 5.

Table 5
Legend explanation table

Sign Signified

PID_f, PID_r The front and rear wheel slip ratio of PID controller braking

SMC_f, SMC_r Sliding Mode controller braking front and rear wheel slip ratio

PID_Vx, PID_Vf,PID_Vr PID controller braking longitudinal speed, front wheel speed and rear wheel speed

SMC_Vx,SMC_Vf, SMC_Vr Sliding Mode controller braking longitudinal speed, front wheel speed and rear wheel speed

PID_s PID controller brake braking distance

SMC_s Braking distance of Sliding Mode controller braking

PID_Tf, PID_Tr PID controller brake front and rear wheel braking torque

SMC_Tf, SMC_Tr Sliding Mode controller braking front and rear wheel braking torque

Sign	Signified
PID_f, PID_r	The front and rear wheel slip ratio of PID controller braking
SMC_f, SMC_r	Sliding Mode controller braking front and rear wheel slip ratio
PID_Vx, PID_Vf,PID_Vr	PID controller braking longitudinal speed, front wheel speed and rear wheel speed
SMC_Vx,SMC_Vf, SMC_Vr	Sliding Mode controller braking longitudinal speed, front wheel speed and rear wheel speed
PID_s	PID controller brake braking distance
SMC_s	Braking distance of Sliding Mode controller braking
PID_Tf, PID_Tr	PID controller brake front and rear wheel braking torque
SMC_Tf, SMC_Tr	Sliding Mode controller braking front and rear wheel braking torque

From the comparison of the change of the slip rate of the front and rear wheels of the two controllers shown in Fig. 9(a), it can be seen that the transient response of the slip rate of the front and rear wheels of the designed sliding mode controller is obviously better than that of the traditional PID, the convergence speed is relatively faster, and the whole braking process is stable near the optimal slip rate. From Fig. 9(b), it can be seen that the braking time of the sliding mode controller is significantly lower than that of the PID control, which is 0.1 s lower than the PID control, and the braking distance is also significantly smaller than the traditional PID control. The designed sliding mode controller has more accurate control of the front and rear wheel speed, more uniform distribution of braking force, and better braking efficiency than the PID controller. It can be seen from Fig. 9(c) that the two controllers have better control effects on the braking torque of the front and rear wheels, but the transient response of the designed sliding mode controller is better than that of the PID control, and it can maintain stability faster.In a word, the designed sliding mode controller is obviously superior to the traditional PID control in braking time, braking distance, tracking control of slip rate and stability of braking torque, which proves the superiority of the designed sliding mode controller based on road recognition.

5.2 Simulation of ABS Sliding Mode controller based on road recognition

Under braking conditions, the adaptability and real-time performance of the designed ABS sliding mode controller based on road adhesion coefficient identification are verified, and the influence of road adhesion coefficient identification on braking performance is verified by comparison of braking conditions. The initial braking speed is 120 km/h, and the road adhesion coefficient is 0.8. When the speed drops to 80 km/h, the road adhesion coefficient mutates to 0.4, that is, from the dry asphalt pavement to the wet asphalt pavement. When the speed drops to 40 km/h, the road adhesion coefficient mutates to 0.2, from the wet asphalt pavement to the ice and snow pavement until the braking is completed. The results of road surface recognition and braking effect comparison are shown in Figs. 10–13. The explanation of the abbreviations in the figure is shown in Table 6.

Table 6
Interpreting the simulation comparison diagram of docking road surface

Sign Signified

Tμ,Eμ Real and estimated values of road adhesion coefficient

Pr_Vx, Pr_Vf, Pr_Vr, Pr_s The controller braking speed, front wheel speed, rear wheel speed and braking distance based on road recognition.

Npr_Vx, Npr_Vf, Npr_Vr, Npr_s The vehicle speed, front wheel speed, rear wheel speed and braking distance of the controller not based on road recognition during braking

Pr_Tf, Pr_Tr The braking torque of front wheel and rear wheel when the controller brakes based on road recognition

Npr_Tf, Npr_Tr The braking torque of the front wheel and the rear wheel when the controller is not based on the road surface identification.

Pr_f, Pr_r The slip ratio of front wheel and rear wheel during braking of controller based on road recognition

Npr_f, Npr_r Slip ratio of front and rear wheels in controller braking without road recognition

Sign	Signified
Tμ,Eμ	Real and estimated values of road adhesion coefficient
Pr_Vx, Pr_Vf, Pr_Vr, Pr_s	The controller braking speed, front wheel speed, rear wheel speed and braking distance based on road recognition.
Npr_Vx, Npr_Vf, Npr_Vr, Npr_s	The vehicle speed, front wheel speed, rear wheel speed and braking distance of the controller not based on road recognition during braking
Pr_Tf, Pr_Tr	The braking torque of front wheel and rear wheel when the controller brakes based on road recognition
Npr_Tf, Npr_Tr	The braking torque of the front wheel and the rear wheel when the controller is not based on the road surface identification.
Pr_f, Pr_r	The slip ratio of front wheel and rear wheel during braking of controller based on road recognition
Npr_f, Npr_r	Slip ratio of front and rear wheels in controller braking without road recognition

It can be seen from Fig. 10 that under the braking condition of the docking road, the road adhesion coefficient identifier based on fuzzy control can accurately identify the changing road surface, with relatively small error and high stability.

From Fig. 11, it can be seen that the ABS sliding mode controller can complete the braking on the docking road surface while ensuring that the wheels are not saturated. Without road surface recognition, the braking time is 10.07 seconds, the braking distance is 118.29 meters, and when the road surface changes, the speed and wheel speed will change obviously, but the wheel does not lock during the whole braking process; In the case of considering the road surface recognition, the braking time is 9.65 seconds, the braking distance is 115.01 meters, there is no mutation in the speed and wheel speed during the braking process, and the braking process is soft. Therefore, it can be seen from Fig. 11 that the ABS sliding mode controller can complete the braking requirements on complex roads, and the road adhesion coefficient identification can improve the control effect and shorten the braking time and distance.

Figures 12 and 13 are the comparison diagrams of braking torque and slip rate considering road adhesion coefficient. It can be seen from Figs. 12 and 13 that the output braking torque of the sliding mode controller considering the change of road adhesion coefficient is relatively small. Therefore, the slip rate is relatively accurate, so that the wheel can make full use of the braking force provided by the ground, and then shorten the braking distance. When the road adhesion coefficient changes abruptly, the ABS sliding mode controller without considering the change of road adhesion coefficient has a relatively small relative deceleration, mainly because the increase of braking force leads to the increase of vehicle slip rate. When the slip rate is greater than the optimal slip rate, the road utilization adhesion coefficient will decrease with the increase of slip rate, which will lead to the decrease of ground braking force and the relative decrease of braking deceleration. Therefore, it can show the importance of road adhesion coefficient identification. In short, the designed ABS sliding mode controller can complete the braking of complex roads. At the same time, the braking distance of the ABS sliding mode controller based on road adhesion coefficient recognition is higher than that of the ABS controller without road recognition, and the braking time and braking distance are shorter.

6 Conclusion

This paper mainly designs the ABS sliding mode controller and the road adhesion coefficient identifier. The main contents include:

Design of road recognition system. On the basis of the in-wheel motor electric prototype, the eight-degree-of-freedom model of the vehicle and the modeling of the Brushless DC Motor are completed. Based on the Burckhardt tire model, F_x/F_z and slip rateλ are used as the input of the fuzzy controller to identify the road adhesion coefficient, and the optimal slip rate is calculated by the identified parameters, which paves the way for the design of ABS sliding mode controller.

Design of ABS sliding mode controller. The difference between the optimal slip rate and the actual slip rate is used as the input of the system to design the sliding mode surface. At the same time, the exponential reaching law is used to replace the traditional constant reaching law to determine the braking torque, reduce the chattering of the system, and then realize the design of ABS sliding mode variable structure.

The superiority and feasibility of ABS sliding mode controller are proved by comparing with ABS traditional PID controller. Through the simulation experiment of docking road surface, the importance of road adhesion coefficient identification to ABS controller is proved, and the real-time performance and applicability of the designed ABS sliding mode variable structure controller are proved.

Although the designed controller has passed the simulation verification of single road and docking road, it has not passed the real vehicle experiment, so the later work mainly focuses on the real vehicle verification.

Footnotes

Consent for publication

The authors provided consent to publish the manuscript.

Availability of data and materials

Not applicable.

funding

This research was supported by the Open Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (No.2022GKLACVTKF10), the Fundamental Research Funds for the Universities of Henan Province (NSFRF220435) and Key Scientific and Technological Project of Henan Province (222102220024).

Conflict of interest

The authors declare no conflict of interest, financial or otherwise.

Acknowledgments

Xiaobin Fan conducted data analysis and interpretation. Shuaiwei Zhu wrote this paper. Pan Wang studied the concept or design. Gengxin Qi collated the data.

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