1. Introduction
A vehicle will encounter various unexpected operating conditions, which bring a severe challenge to the stability of the vehicle [
1,
2]. The braking system plays a very important role in ensuring the safety of the vehicle, which can enable the vehicle to slow down and stop based on its needs and protect the life and safety of the people in the vehicle at critical moments. With the development of automotive technology, brake-by-wire technology is combined with the braking system of the vehicle to form the brake-by-wire [
3,
4]. Brake-by-wire is the development direction of the automotive braking system, which has the advantages of a compact structure, easy integration, robust scalability and rapid response [
5,
6,
7,
8].
The authors in [
9] modeled and analyzed three braking systems, the electrohydraulic brake (EHB), electric mechanical brake (EMB) and electronic wedge brake (EWB), and optimized these models using a linear transfer function. EHB, EMB and EWB energy consumption can be reduced by approximately 10%, 3% and 20%, respectively. In [
10], the author proposed a brake-by-wire combined with a controller integrated into chassis, which provided active safety functions, and it could improve vehicle stability. The authors in [
11] proposed a new brake-by-wire according to the braking requirements, and the key components of the system were selected and designed, and a new brake-by-wire structure was designed that used the motor to drive the master cylinder to generate pressure. Based on the real vehicle platform, static tests were carried out, and the results demonstrated that the control strategy can respond quickly and accurately. In order to improve the handling stability and ride comfort of the vehicle, a full-by-wire chassis coordinated control was proposed in [
12], and its effectiveness was verified by a hardware-in-the-loop test. The authors in [
13] aimed at the automatic emergency steering function of an intelligent-assisted driving system, and this paper proposed a brake-by-wire system that met its functional requirements and was verified by the simulation.
Brake-by-wire realizes the decoupling of people and vehicles, and it is easy to integrate functions such as an anti-lock braking system and automatic emergency braking to improve the active safety and handling stability of the vehicle. However, due to the issues of system reliability and functional safety, it has not yet been mass-produced. In order to obtain a better braking effect, many scholars start from the hierarchical control strategy.
A hierarchical cooperative control for the electromechanical brake-by-wire system (EBW) could achieve better braking safety and energy regeneration performances in [
14]. In [
15], a hierarchical architecture with the aim of improving vehicle safety was proposed, based on the direct yaw moment control method, which combined two different controllers to calculate the globally stable yaw moment to control the yaw angle speed and vehicle side declination. The authors in [
16,
17] designed a yaw stability control strategy, in which the upper layer used the fuzzy control to calculate the additional yaw moment that helped maintain the stability of the driving vehicle, and the lower layer distributed the movement of each wheel based on the calculated yaw moment. The hierarchical control strategy of automobile yaw stability was proposed in [
18], and the upper controller used three controls of PID, fuzzy and PID + fuzzy to calculate the additional yaw moment, and the lower controller assigned the moment to the wheel and then controlled the wheel by a motor control command. To improve the vehicle stability, a controller with a combined sliding mode variable structure and direct yaw moment was proposed in [
19]. A controller combining the fuzzy and sliding mode was proposed in [
20]. The vehicle stability under the cornering braking condition was studied in [
21]. To address the issue of vehicle instability after a single-wheel braking failure, a control strategy for redistributing the braking force was proposed [
22]. The authors in [
23] proposed a new braking dynamics model and control system verified by HIL. The vehicle stability was significantly improved under fuzzy PID control compared to no control.
Most of the above studies only consider the redistribution of the yaw moment during braking to improve vehicle stability. Although this method does not have such high accuracy and requirements for the control algorithm and it is easy to implement, its response speed is slow, and the system robustness is not strong enough. This paper uses the fuzzy control to calculate the additional yaw moment required to keep stability when braking and applies it to the corresponding wheel, so as to improve the vehicle stability.
2. Vehicle Stability Control Model
This paper establishes an ideal model of a vehicle with two degrees of freedom (DOF) as a reference to compare the actual state of the vehicle with the ideal state. When the road surface conditions are good, and the lateral acceleration is within 0.4 g, the basic motion state of the vehicle can be better described, and the rationality can be guaranteed when describing the vehicle state parameters such as yaw rate and sideslip angle. We make the following assumptions: the front wheel angle is taken as input; the influence of the suspension is ignored; the thinking vehicle is always moving parallel to the ground; the longitudinal speed of the vehicle does not change and does not consider the effect of load changes on the system; the role of air resistance is also not taken into account. The established 2-DOF model is shown in
Figure 1.
The force situation of a 2-DOF model can be obtained from the figure above as shown in Equation (1):
where
(rad) is the front wheel steering angle;
(N) is the lateral force of each wheel;
and
b (mm) are the front and rear wheelbases, respectively. Assuming that the lateral force is linearly related to the lateral angle, and the front wheel steering angle is small, Equation (1) can be written as:
where
(N/rad) is the lateral stiffness of each wheel;
(rad) is the lateral angle of each wheel, and their values are as follows:
where
(rad) is the sideslip angle. Combining Equations (1)–(3) can obtain the 2-DOF differential equation for automobiles:
where
m (kg) is the mass of the vehicle;
(rad/s) is the yaw rate of the vehicle;
I (kg·m
2) is the vehicle yaw moment of inertia;
is the longitudinal speed;
is the lateral speed.
When the vehicle is in motion, the yaw rate can reflect the yaw motion state of the vehicle, and the sideslip angle can reflect the deviation state from the desired path. Therefore, the vehicle stability can be characterized by them. In addition, the value of the heading angle is the sum of the sideslip angle and yaw angle, which can be described by Formula (5):
The yaw rate and sideslip angle can reflect the most basic stability characteristics of vehicle motion. The nominal values of their corresponding control variables can be determined based on the steady-state response of the automobile when driving in a constant velocity circumference. Under the constant velocity circumferential driving condition, the acceleration is zero, the yaw rate is fixed, and
, so Equation (4) can be rewritten as:
The elimination of
v from Equation (6) can obtain the yaw rate
ω in the steady-state circumferential case, and the yaw rate under ideal conditions:
where
(
) is the stability factor; due to the limitation of the attachment conditions, the maximum ideal yaw rate is:
In order to improve the control accuracy, this paper takes the smaller value of the absolute value as the ideal value for the yaw rate, namely:
By eliminating
ω from Equation (6), we can obtain the sideslip angle in the ideal case:
When the vehicle is in motion, stability control intervention needs to occur at the right time. It should intervene when the vehicle is unstable or has an instability tendency to avoid frequent control by the system. Therefore, the stability control also needs to have the function of judging instability. The two parameters of yaw rate and sideslip angle determine the steady state of the vehicle. When the sideslip angle is very small, only the yaw rate can be used to determine whether the vehicle has lost stability. The determination formula is as follows:
When the sideslip angle is large, only the yaw rate cannot completely judge whether the vehicle has lost stability. At this time, the formula for judging whether the vehicle is unstable by the sideslip angle can be expressed as:
where C, C
1 and C
2 are constants, and C = 0.165, C
1 = 4.386 and C
2 = 2.562 [
24]. If Equations (11) and (12) hold at the same time, the vehicle is stable. If either Equation (11) or (12) does not hold, it indicates that the vehicle needs stability control.
From the automobile theory, it can be seen that the lateral force will gradually decrease when the braking force is changed.
Figure 2 illustrates an example of a vehicle turning left to analyze the corresponding forces on the wheels when the braking force is applied. It demonstrates that the yaw moment of the vehicle is positive when turning left.
When the vehicle turns left, if the vehicle understeers, the positive yaw moment is required to reduce the tendency of understeering; at this time, it is necessary to apply braking to the left wheel. When the vehicle turns left, if the vehicle oversteers, at this time, a negative yaw moment is required to reduce the tendency of oversteering, and brakes need to be applied to the right wheel. In the same way, the situation when the vehicle turns right can be analyzed.
Through the analysis, it is concluded that applying the brake control to the front outer wheel and rear inner wheel produces a higher yaw moment efficiency and a more obvious effect. However, the front wheel is generally a steering wheel, and braking the front wheels can impact the steering performance. In order to avoid interference with the steering of the front wheels by brake control during the driving of the vehicle, this paper only explores the rear wheel of the vehicle. Combined with the above analysis, the specific selection of the controlled wheel in this paper is shown in
Table 1.
The additional yaw moment to restore stability when the vehicle is destabilized, named
, is calculated by the Simulink model.
and
are the braking forces of the front and rear wheels; the
and
are the axel track of the front and rear wheels, respectively. They are expressed as:
The vertical load
and
on the front and rear wheels are:
where
is the tire radius, and when the wheel is not locked, the braking torque assigned to the wheels is:
Then, the calculated braking torque is distributed to the brake wheels by applying braking pressure. In order to simplify the calculation, ignoring the influence of the driver pressing the brake pedal and slip rate, the relationship between the braking pressure and braking torque is as follows:
where
is the total braking torque, and
is the braking pressure of the corresponding wheels.
The specific values
in this paper are calculated by the data obtained by setting various braking conditions after obtaining the detailed parameters of the vehicle in CarSim. After setting any initial speed, several sets of arbitrary braking pressures are given, and the front and rear braking torque are calculated. The sideslip angle and yaw rate are output variables, and brake force is the control parameter. Furthermore, all parameters are available and known. This paper uses asymptotic stabilization. According to the tire model,
takes 0.287 m.
Table 2 shows detailed data from during the simulation.
The values calculated according to the four sets of tests are 951.22, 954.70, 955.87 and 956.4, and the average value of 954.56 is taken as the when converting the braking torque and braking pressure.