Enhancing the lateral stability of the vehicle by using a fuzzy algorithm to control the active stabilizer bar

Abstract

This article studies the instability of automobiles when steering at high speeds. In this article, the model of spatial dynamics is used to simulate vehicle oscillation. Besides, the model of nonlinear double-track dynamics is also combined to determine the effects of the wheel when steering. To limit the instability when steering, the hydraulic stabilizer bar is suggested. The performance of the system depends on the previously designed controller. The FLC algorithm with two inputs is used to control the operation of the system. The membership function and fuzzy rules are determined based on the designer’s experience. The simulation is performed by MATLAB software with three specific steering cases. In each case, the speed of the vehicle will be increased gradually. The results of the article show that the value of the roll angle is greatest in the third case, corresponding to the speed of v₃ = 100 (km/h). If the vehicle does not have a stabilizer bar, the vehicle can roll over at any time. In contrast, when the active stabilizer bar was combined with the proposed FLC algorithm, the vehicle’s stability was significantly improved. The vehicle’s roll angle and the difference in vertical force at the wheels were also significantly reduced when using this algorithm for the stabilizer bar model. This result should be further verified through the experimental process.

Keywords

Rollover active stabilizer bar nonlinear dynamics model fuzzy control

Nomenclature

α Slip angle, rad

δ Steering angle, rad

φ_x Roll angle, rad

φ_y Pitch angle, rad

φ_z Yaw angle, rad

F_ce Centrifugal force, N

F_Cij Damping force, N

F_i Inertia force of the sprung mass, N

F_iij Inertia force of the un-sprung mass, N

F_HSBij Force of the hydraulic stabilizer bar, N

F_Kij Spring force, N

F_KTij Tire force, N

F_x Longitudinal force, N

F_y Lateral force, N

F₁ External longitudinal force, N

F₂ External lateral force, N

h_ij Roughness on the road, m

M_i Inertia moment, Nm

M_z Tire moment, Nm

v_x Longitudinal velocity, ms^- 1

v_y Lateral velocity, ms^- 1

z_s Displacement of the sprung mass, m

z_uij Displacement of the un-sprung mass, m

Abbreviation

FLC

Fuzzy Logic Control

LQR

Linear Quadratic Regulator

LTR

Load Transfer Ratio

PID

Proportional Integral Derivative

SSF

Static Stability Factor

TSK

Takagi Sugeno Kang

WTSUM

Weighted Sum

1 Introduction

Over the years, the field of automotive technology has developed very strongly. The issues of vehicle stability and safety when traveling on the road are very concerning. When an automobile moves on the road, many factors can affect the stability and safety of the vehicle. In particular, the phenomenon of rollover often brings extremely heavy consequences for passengers and goods in the vehicle [1]. According to Yao et al., centrifugal force will appear when the vehicle steers [2]. This force is related to the square of the vehicle’s speed when determining its magnitude. Centrifugal force will cause the body to tilt, so a difference in the vertical force at the wheels will appear [3]. If the roll angle of the vehicle body is greater, the difference in vertical force will also be greater. The vertical force of the wheel on the outside of the revolution arc will gradually decrease until this value is zero. When this happens, the wheel will become detached from the pavement. At that time, the phenomenon of rollover may occur [4, 5]. According to Nguyen et al., when the wheel is lifted off the road surface, the roll angle of the vehicle body will reach its maximum value [6]. If the value of the speed or the steering angle is increased, the centrifugal force will also increase, which makes the vehicle more prone to rollover. Other dimensions of the vehicle, such as the height of the center of gravity, track width, etc., also affect this problem [7].

Several indicators have been put in place to warn of a rollover phenomenon. In [8], Huston and Kelly introduced the concept of a Static Stability Factor (SSF). This is considered the rollover index of the vehicle in a stationary state. This indicator is true only when the vehicle is assumed to be a rigid block with a center of gravity height of h_x and a track width of t_w: $SSF = \frac{a_{ymax}}{g} = \frac{t_{w}}{h_{x}}$ (1) If the effect of the suspension system is considered, Equation (1) can be rewritten as: $SSF = \frac{t_{w}}{h_{x}} (\frac{1}{1 + R_{φ} (1 - \frac{h_{s}}{h_{x}})})$ (2) Where:

a_y: Lateral acceleration

g: Gravitational acceleration

R_φ: Function of the roll angle

However, the case of a rollover in a stationary state rarely happens. Therefore, the static rollover index needs to be replaced by the dynamic rollover index. The dynamic rollover index (DRI), also known as the load transfer ratio (LTR), measures the difference in vertical forces at the wheels [9 –11]. $LTR = \frac{F_{zr} - F_{zl}}{F_{zr} + F_{zl}} = \frac{m_{s} h_{x}}{mg t_{w}} (a_{y} + \tan φ_{x})$ (3)

The value of this index ranges from –1 to+1. If its value reaches two thresholds ±1, a rollover phenomenon can occur [12]. This index is supposedly a nonlinear function, as stated by Huang et al., and it is dependent not only on the velocity but also on the roll angle and the yaw angle. [13]. $LTR = \frac{m ({\dot{v}}_{y} + v_{x} φ_{z} - h {\ddot{φ}}_{x}) H + k φ_{x} + c {\dot{φ}}_{x}}{t_{w} mg}$ (4) Where:

m: Total mass of the vehicle

m_s: Sprung mass

k: Equivalent stiffness of spring

c: Equivalent stiffness of damper

φ_x: Roll angle

φ_z: Yaw angle

However, this value is only true for a linear single-track dynamics model. Then the left wheel and the right wheel are assumed to be one. In fact, for vehicles with 4 wheels, if a wheel is separated from the road surface, the rollover phenomenon can also occur. Therefore, determining the exact time when this phenomenon occurs is extremely complicated.

Several solutions have been used to limit the rollover phenomenon. In [14], Xiao et al. proposed the use of an active suspension system with hydraulic cylinders. This system helps to improve vehicle stability [15, 16]. Besides, the method of using the stabilizer bar is also proposed for use [17]. The stabilizer bar can be fitted on the front or rear axle, or both. As for the mechanical stabilizer bar, also known as the passive stabilizer bar, it can rely on the elastic force to limit the roll angle of the vehicle. The passive stabilizer bar has a simple structure, easy installation, and low cost [18, 19]. Despite this, the impact of a mechanical stabilizer bar is not positive. The roll angle of the vehicle body and the difference in vertical forces at the wheels are still quite large when the vehicle uses a mechanical stabilizer bar [20, 21]. To improve vehicle stability and safety, an active stabilizer bar was used to replace the conventional passive stabilizer [22].

According to Nguyen and Hoang, there are two types of active stabilizer bars: the hydraulic stabilizer bar and the electronic stabilizer bar [18]. The hydraulic stabilizer bar has a hydraulic motor, which generates torque. This motor is controlled by the voltage signal supplied from the controller [23]. The amount of liquid through the motor is pressurized by a hydraulic pump, which is an axial piston pump [24]. The valves inside the actuator are servo-valve with two or four gates [25]. The hydraulic stabilizer bar has a cumbersome and complicated structure, and the sensitivity is not high. However, the torque generated by the hydraulic actuator is very large. Therefore, it can effectively ensure the stability of the vehicle when steering. To improve the response time of the system, an electronic stabilizer bar should be used [26]. The electronic stabilizer bar has a compact structure; however, its cost is quite high.

Controlling the active stabilizer bar may be done in various methods. In [27], Her et al. used a yaw angle supervised algorithm to control the behavior of the stabilizer bar with a single-track dynamics model. Besides, the roll angle supervised algorithm is also shown with the one-degree-of-freedom model. In [28], Pourasad et al. introduced the LQR control algorithm to minimize the cost function. This algorithm is applied to single-track and roll dynamics models with one degree of freedom. In [29], Muniandy et al. used the fuzzy algorithm to adjust the parameters of the PI controller for the active stabilizer bar. Because this controller has two parameters, K_P and K_I, two Fuzzy controllers are required, respectively [30]. However, the above studies have not mentioned the influence of hydraulic actuators. The authors just assume that the impact force is generated directly by the controller. This is not entirely accurate. In [31, 32], hydraulic actuators were considered in a dynamic model of a vehicle. According to Nguyen, the model of the hydraulic actuator is nonlinear; therefore, the control algorithm needs to be determined appropriately [33]. In [34], Nguyen introduced the fuzzy algorithm with two inputs to control the active stabilizer bar. Fuzzy rules are determined based on the designer’s experience. When the vehicle is equipped with an active stabilizer bar, the roll angle of the vehicle’s body is significantly reduced. This is shown in the article by Park et al. [35]. In addition, many studies on control algorithms for the stabilizer bar have also been published [36 –38].

Based on the reviews introduced above, this article focuses on the design and evaluation of the efficiency of the control algorithm for the hydraulic stabilizer bar. In this article, the model of spatial dynamics that fully describes the effects of vehicle oscillations is used. Besides, the hydraulic actuator is also mentioned in this model. In order to control the stabilizer bar’s functioning, it is proposed to use an FLC algorithm with various inputs. This article’s content consists of four sections. In the first section, the concepts of the stabilizer bar are presented. In addition, several previous studies on stabilizer bars were also analyzed. In the next section, the dynamics model of the vehicle and the dynamics model of the hydraulic actuator are established. Fuzzy theory and the FLC control algorithm are also shown in this section. In the third section, the simulation is performed. The results of the simulation are analyzed and compared in this section. In the last section, some conclusions are pointed out. The specific content of the article is presented in the following sections.

2 Model and control

2.1 Vehicle model

To simulate the vehicle’s oscillation while steering, a spatial dynamics model with 10 degrees of freedom is used. In which there are 7 degrees of freedom of oscillation and 3 degrees of freedom of motion. Considering Fig. 1, the vehicle oscillation is given as the following equation:

Fig. 1

Model of the spatial dynamics.

$F_{i} = \sum_{i, j = 1}^{2} (F_{Kij} + F_{Cij})$ (5) $\begin{matrix} M_{ix} = \sum_{i, j = 1}^{2} ({(- 1)}^{j - 1} (F_{Kij} + F_{Cij}) t_{w i}) \\ + (gsin φ_{x} + a_{y} \cos φ_{x}) m_{s} h_{x} \end{matrix}$ (6) $M_{iy} = \sum_{i, j = 1}^{2} ({(- 1)}^{i - 1} (F_{Kij} + F_{Cij}) l_{i})$ (7) $F_{ij} = F_{KTij} - F_{Kij} - F_{Cij} + {(- 1)}^{j} F_{HSBij}$ (8) Where: $F_{i} = m_{s} \frac{d^{2} z_{s}}{d t^{2}}$ (9) $F_{ij} = m_{uij} \frac{d^{2} z_{uij}}{d t^{2}}$ (10) $M_{ix} = (J_{x} + m_{s} h_{x}^{2}) \frac{d^{2} φ_{x}}{d t^{2}}$ (11) $M_{iy} = (J_{y} + m_{s} h_{y}^{2}) \frac{d^{2} φ_{y}}{d t^{2}}$ (12) The lateral acceleration (a_y) of equation (6) is determined through a nonlinear double-track dynamics model (Fig. 2). There are 3 DOFs included in this model.

Fig. 2

Model of the nonlinear double-track dynamics.

$\begin{matrix} F_{ix} = \sum_{i, j = 1}^{2} (F_{xij} \cos δ_{ij} - F_{yij} \sin δ_{ij}) \\ - F_{1} + F_{cex} \end{matrix}$ (13) $\begin{matrix} F_{iy} = \sum_{i, j = 1}^{2} (F_{xij} \sin δ_{ij} + F_{yij} \cos δ_{ij}) \\ - F_{2} - F_{cey} \end{matrix}$ (14) $M_{iz} = \sum_{i, j = 1}^{2} (\begin{matrix} {(- 1)}^{j} (F_{xij} \cos δ_{ij} - F_{yij} \sin δ_{ij}) t_{w i} \\ + {(- 1)}^{i + 1} (F_{xij} \sin δ_{ij} + F_{yij} \cos δ_{ij}) l_{i} \\ + F_{i} c_{i} - M_{zij} \end{matrix})$ (15) Where: $F_{ix} = (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d v_{x}}{dt}$ (16) $F_{iy} = (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d v_{y}}{dt}$ (17) $F_{cex} = (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d (α + φ_{z})}{dt} v_{y}$ (18) $F_{cey} = (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d (α + φ_{z})}{dt} v_{x}$ (19) $M_{iz} = J_{z} \frac{d^{2} φ_{z}}{d t^{2}}$ (20)

Equations (13)–(15) can be calculated if the forces and moments at the wheel are known in advance. Many tire models are used to determine the forces at the wheel. In particular, the Pacejka tire model has the most outstanding characteristics. This is a complex nonlinear tire model with many parameters. According to [39], the longitudinal force F_x, lateral force F_y, and moment M_z of the wheel are functions depending on the slip ratio s_x, slip angle α, vertical force F_z, velocity, etc.

Longitudinal force: $F_{x} = D_{x} \sin [C_{x} artan (B_{x} κ_{x})]$ (21) Where: $κ_{x} = (1 - E_{x}) λ + \frac{E_{x}}{B_{x}} \arctan (B_{x} λ)$ (22) $C_{x} = 1.65$ (23) $D_{x} = a_{1} F_{z}^{2} + a_{2} F_{z}$ (24) $BC D_{x} = \frac{a_{3} F_{z}^{2} + a_{4} F_{z}}{e^{a_{5} F_{z}}}$ (25) $B_{x} = \frac{BC D_{x}}{C_{x} D_{x}}$ (26) $E_{x} = a_{6} F_{z}^{2} + a_{7} F_{z} + a_{8}$ (27) Lateral force: $F_{y} = D_{y} \sin [C_{y} artan (B_{y} κ_{y})] + S_{vy}$ (28) Where: $\begin{matrix} κ_{y} = (1 - E_{y}) (α + S_{hy}) \\ + \frac{E_{y}}{B_{y}} \arctan [B_{y} (α + S_{hy})] \end{matrix}$ (29) $C_{y} = 1.30$ (30) $D_{y} = a_{1} F_{z}^{2} + a_{2} F_{z}$ (31) $BC D_{y} = a_{3} \sin [a_{4} artan (a_{5} F_{z})]$ (32) $B_{y} = \frac{BC D_{y}}{C_{y} D_{y}}$ (33) $E_{y} = a_{6} F_{z}^{2} + a_{7} F_{z} + a_{8}$ (34) $S_{hy} = a_{9} γ$ (35) $S_{vy} = (a_{10} F_{z}^{2} + a_{11} F_{z}) γ$ (36) Aligning moment: $M_{z} = D_{z} \sin [C_{z} artan (B_{z} κ_{z})] + S_{vz}$ (37) Where: $\begin{matrix} κ_{z} = (1 - E_{z}) (α + S_{hz}) \\ + \frac{E_{z}}{B_{z}} \arctan [B_{z} (α + S_{hz})] \end{matrix}$ (38) $C_{z} = 2.40$ (39) $D_{z} = a_{1} F_{z}^{2} + a_{2} F_{z}$ (40) $BC D_{z} = \frac{a_{3} F_{z}^{2} + a_{4} F_{z}}{e^{a_{5} F_{z}}}$ (41) $B_{z} = \frac{BC D_{z}}{C_{z} D_{z}}$ (42) $E_{z} = a_{6} F_{z}^{2} + a_{7} F_{z} + a_{8}$ (43) $S_{hz} = a_{9} γ$ (44) $S_{vz} = (a_{10} F_{z}^{2} + a_{11} F_{z}) γ$ (45)

The symbols a_i (i = 1 ÷ 11) are parameters of the tire model, and γ is the kingpin angle.

For the above system of equations, only the unknown F_HSBij cannot be calculated. This value will be determined through the dynamics model of the hydraulic actuator.

2.2 Hydraulic motor

The active stabilizer bar is controlled by a hydraulic motor. This is a hydraulic motor with two-way action. This motor operates on the voltage signal provided by the controller. Once the current is supplied, the servo valves inside the motor will move. The displacement of the valves will change the fluid flow inside the motor. This will cause the motor to work, i.e., produce the corresponding force on both sides of the arms of the stabilizer bar. The diagram of the hydraulic system is shown in Fig. 3.

Fig. 3

The diagram of the hydraulic system.

The working principle of the hydraulic motor is described through the following equations: $\frac{d X_{v}}{dt} τ + X_{v} = K_{v} u (t)$ (46) $K_{qi} X_{v} = D_{m} \frac{d θ_{m}}{dt} + K_{ce} Δ P + \frac{V_{t}}{4 β_{e}} \frac{d Δ P}{dt}$ (47) $D_{m} Δ P = J_{m} \frac{d^{2} θ_{m}}{d t^{2}} + B_{m} \frac{d θ_{m}}{dt} + T_{r}$ (48) Where:

X_v: Displacement of the spool valve

ΔP: Variation of pressure

θ_m: Rotor angle

T_r: Resistance moment

u(t): Voltage signal

The remaining coefficients of equations from (46) to (48) are shown in Table 2.

Table 2

The reference parameters

Symbol	Description	Value	Unit
m_s	Sprung mass	1760	kg
m_uij	Un-sprungmass	47	kg
h_x	Distance from center of gravity to roll axis	0.5	m
t_wi	Half of the track width front/rear axle	0.730/0.725	m
l_i	Distance from center of gravity to front/rear axle	1.15/1.55	m
J_x	Inertia moment of the x-axis	680	kgm²
J_y	Inertia moment of the y-axis	2750	kgm²
J_z	Inertia moment of the z-axis	2625	kgm²
τ	Time constant	0.004	s
K_v	Servo-valve gain	0.03	mA^- 1
K_qi	Valve flow gain coefficient	0.02	m²s^- 1
K_ce	Total flow pressure coefficient	4×10^- 11	m⁵N^- 1s^- 1
V_t	Total volume of trapped oil	1×10^- 3	m³
β_e	Effective bulk modulus of the oil	6×10⁶	Nm^- 2
D_m	Flow per revolution	1.6×10^- 5	m³rad^- 1
B_m	Viscous friction coefficient	10	Nmsrad^- 1
J_m	Moment of inertia of the hydraulic motor	3.5	kgm²
G	Gravitational acceleration	9.81	ms^- 2

The efficiency of the hydraulic actuator is based on the voltage signal supplied by the controller. Therefore, it is necessary to design the controller appropriately to improve the performance of the system.

2.3 FLC algorithm

In this article, the TSK control model is used. The TSK model with m inputs and r outputs has the following form:

IF γ ₁ = A_1,1 AND γ ₂ = A_2,1 . . . AND γ_m=A_m,1 THEN y₁ = f_1,1(x₁, x₂, x₃, . . . , x_m) AND y₂ = f_2,1(x₁, x₂, x₃, . . . , x_m) AND . . . AND y_r = f_r,1(x₁, x₂, x₃, . . . , x_m) OR

IF γ ₁ = A_1,2 AND γ ₂ = A_2,2 . . . AND γ_m=A_m,2 THEN y₁ = f_1,2(x₁, x₂, x₃, . . . , x_m) AND y₂ = f_2,2(x₁, x₂, x₃, . . . , x_m) AND . . . AND y_r = f_r,2(x₁, x₂, x₃, . . . , x_m) OR

. . . OR

IF γ ₁ = A_1,n AND γ ₂ = A_2,n . . . AND γ_m=A_m,n THEN y₁ = f_1,n(x₁, x₂, x₃, . . . , x_m) AND y₂ = f_2,n(x₁, x₂, x₃, . . . , x_m) AND . . . AND y_r = f_r,n(x₁, x₂, x₃, . . . , x_m)

Divide the above fuzzy system into r subsystems:

IF γ ₁ = A_1,1 AND γ ₂ = A_2,1 . . . AND γ_m=A_m,1 THEN y_q = f_q,1(x₁, x₂, x₃, . . . , x_m) OR

IF γ ₁ = A_1,2 AND γ ₂ = A_2,2 . . . AND γ_m=A_m,2 THEN y_q = f_q,2(x₁, x₂, x₃, . . . , x_m) OR

. . . OR

IF γ ₁ = A_1,N AND γ ₂ = A_2,N . . . AND γ_m=A_m,n THEN y_q = f_q,n(x₁, x₂, x₃, . . . , x_m)

The output of the q subsystem is defined as follows: $y_{q}^{'} = \frac{\sum_{i = 1}^{n} α_{i} f_{q, i} (x)}{\sum_{i = 1}^{n} α_{i}}$ (49) With: $α_{i} = α_{i, 1} α_{i, 2} . . . α_{i, m}$ (50) $x = {[\begin{matrix} x_{1} & x_{2} & . . . & x_{m} \end{matrix}]}^{T}$ (51)

Using WTSUM theory: $y^{'} = α_{1} y_{1} + α_{2} y_{2} + . . . + α_{n} y_{n}$ (52) Let: $w_{i} = \frac{α_{i}}{\sum_{i = 1}^{n} α_{i}}$ (53) Then, equation (52) is rewritten as: $y^{'} = \sum_{i = 1}^{n} w_{i} y_{i} = θ^{T} Φ (x)$ (54) Where: $θ = {[\begin{matrix} y_{1} & y_{2} & . . . & y_{n} \end{matrix}]}^{T}$ (55) $Φ (x) = {[\begin{matrix} w_{1} (x) & w_{2} (x) & . . . & w_{n} (x) \end{matrix}]}^{T}$ (56)

Combining the equation from (49) to (54), the output of the system is given as the following equation: $y_{q}^{'} = \sum_{i = 1}^{n} α_{i} f_{q, i} (x)$ (57)

The membership function used in this article has two inputs and one output. The two inputs include the signal of the roll angle and roll acceleration, which is fed to the hydraulic actuator. The selection of inputs is based on improving the roll angle and roll acceleration stability. If the roll angle is too large, the vehicle may roll over. Moreover, if the roll acceleration is too significant, the vehicle oscillation will also be larger. So, it’s perfectly appropriate to use these two values. This is considered a new point of the article. The membership function of the model is given as shown in Fig. 4.

Fig. 4

Membership function.

Fuzzy rules are determined based on many simulations that have been conducted in the past. This rule is shown in Table 1.

Table 1

Fuzzy rules

	μ₁	μ₂	μ₃	μ₄	μ₅
ψ₁	Λ₁	Λ₁	Λ₂	Λ₂	Λ₃
ψ₂	Λ₁	Λ₂	Λ₂	Λ₃	Λ₃
ψ₃	Λ₂	Λ₂	Λ₃	Λ₃	Λ₄
ψ₄	Λ₂	Λ₃	Λ₃	Λ₄	Λ₅
ψ₅	Λ₃	Λ₃	Λ₄	Λ₅	Λ₅
ψ₆	Λ₃	Λ₄	Λ₅	Λ₅	Λ₆
ψ₇	Λ₄	Λ₅	Λ₅	Λ₆	Λ₆
ψ₈	Λ₅	Λ₅	Λ₆	Λ₆	Λ₇
ψ₉	Λ₅	Λ₆	Λ₆	Λ₇	Λ₇

Based on this fuzzy rule, the rule surface is given in Fig. 5. This is a nonlinear function that can respond to the stability of the car when steering. The output voltage signal is controlled from -24 (V) to+24 (V). The limit of the input parameters is also defined within a specific range, which is entirely consistent with the actual conditions.

Fig. 5

Rule surface.

3 Result and discussion

3.1 Simulation conditions

To evaluate the effectiveness of the controller, the vehicle’s oscillation is simulated through three specific steering cases (Fig. 6). In each case, four situations are covered, including:

Fig. 6

Steering angle.

+The vehicle uses the active stabilizer bar controlled by a Fuzzy controller (FLC).

+The vehicle uses the active stabilizer bar controlled by a PID controller (PID) (The parameters of the PID controller can be referenced from [31]).

+The vehicle uses the conventional mechanical stabilizer bar (Passive).

+The vehicle does not use the stabilizer bar (None).

The speed of the vehicle will be changed from v₁ = 60 (km/h) to v₂ = 80 (km/h) and v₃ = 100 (km/h). With input parameters including steering angle and velocity, the output data of the problem will be the roll angle and vertical force of the wheel. The simulation is performed by the MATLAB-Simulink software. The reference parameters are given in Table 2.

3.2 Results

Case 1:

In the first case, a J-turn steering angle is suggested. This type of steering is often used when the driver steers the vehicle into a roundabout. The value of the steering angle of the wheel will increase from zero to eight degrees, after which it will be fixed at this threshold. The rise time of the steering angle is about 1 (s).

The change in roll angle over the investigation time is shown in Fig. 7. At an average speed of v₁ = 60 (km/h), the maximum value of the roll angle is not large, only reaching 5.62°, 5.35°, 4.71°, and 4.47° for the four situations, respectively. When the vehicle’s speed increases up to v₂ = 80 (km/h), the value of the roll angle also increases accordingly. The roll angle of the vehicle still reaches its maximum when the vehicle does not use the stabilizer bar, reaching 7.49°. This value decreases slightly to 6.91° when the mechanical stabilizer bar is installed at the two axles of the vehicle. The value of the roll angle drops more drastically when the active stabilizer bar is used. It is only 6.38° and 6.02°, respectively, with two control algorithms, PID and FLC. If the vehicle’s speed continues to increase, v₃ = 100 (km/h), the value of the roll angle will continue to increase. Their peak values can go up to nearly 9° if the vehicle does not have the stabilizer bar. This is a rather dangerous state because it can cause the vehicle to roll over. After reaching the maximum value, the roll angle of the vehicle tends to decrease gradually, although the steering angle remains the same. The cause of this phenomenon is the nonlinear deformation of the tire. Under the influence of nonlinear tires, the slip angle will gradually decrease even though the steering angle does not change. This results in a gradual decrease in the vehicle body’s roll angle.

Fig. 7

Roll angle (a –v₁; b –v₂; c –v₃).

Figure 8 shows the change of the vertical force at the wheels corresponding to the four situations examined at a speed of v₁ = 60 (km/h). The tendency of these values to change in different situations is different. According to this result, the difference in vertical force between the two wheels on the same axle is the smallest when the vehicle uses the active stabilizer bar controlled by the FLC algorithm. Meanwhile, the disparity between the other three situations is larger.

Fig. 8

Vertical force –v₁ (a –FLC; b –PID; c –Passive; d –None).

The change in the vertical force at the wheel will increase as the vehicle’s speed increases. This is demonstrated in Fig. 9. This changing trend remains like the one shown above. If the speed continues to increase, the wheel is likely to separate from the road surface (Fig. 10). If a vehicle does not use a stabilizer bar, the vertical force’s minimum value at the rear wheel is only F_z21 = 220.35 (N). This is a very small value. When it reaches zero, the wheel will be lifted off the road. At this point, a rollover may occur. However, this situation can be significantly improved if the vehicle uses a hydraulic stabilizer bar with the FLC algorithm. The minimum value of the vertical force at the wheel is only 3081.42 (N), which is a very stable level. An FLC algorithm used for an active stabilizer bar helps the vehicle move more steadily at high speeds.

Fig. 9

Vertical force –v₂ (a –FLC; b –PID; c –Passive; d –None).

Fig. 10

Vertical force –v₃ (a –FLC; b –PID; c –Passive; d –None).

Case 2:

In the second case, double change lane steering is used. After the steering is finished, the vehicle will return to its original (or nearly original) lane. The change in the vehicle’s roll angle will follow the steering signal.

According to the results shown in Fig. 11, the maximum value of the roll angle in this case is only like that of the first case. Since the steering angle and the steering acceleration in the two cases are similar, the results obtained are therefore similar. The active stabilizer bar still helps maintain the stability of the vehicle at all three speeds.

Fig. 11

Roll angle (a –v₁; b –v₂; c –v₃).

The main consideration, in this case, is the variation of the vertical force of the wheel. This result is shown in Figs. 12 –14. According to these results, the variation of the vertical force of the wheels is sine but not cyclic because of the deformation of the elastic tires. After reversing the steering wheel, the vehicle body will return to the equilibrium position, i.e., the roll angle of the vehicle body is zero. Therefore, the variation in the vertical force of the wheel will also cease to exist. This is only true in situations where the vehicle uses an active stabilizer bar controlled by the FLC algorithm, a vehicle using a passive stabilizer bar, or a vehicle that does not use a stabilizer bar. In the other situation, this is not true. Due to the influence of the controller, the difference in vertical force on the two sides of the wheel still exists. However, the difference is not much. This can still affect the vehicle’s stability.

Fig. 12

Vertical force –v₁ (a –FLC; b –PID; c –Passive; d –None).

Fig. 13

Vertical force –v₂ (a –FLC; b –PID; c –Passive; d –None).

Fig. 14

Vertical force –v₃ (a –FLC; b –PID; c –Passive; d –None).

Case 3:

In the last situation, fishhook steering is introduced. This is considered one of the most dangerous situations [40]. The value of the steering angle is very large. Besides, the steering acceleration is also large. This increases the risk of rolling over when traveling at high speeds.

The roll angle of the vehicle when steering in this way is larger, especially in the second phase (Fig. 15). At a speed of v₁ = 60 (km/h), the maximum roll angle that the vehicle can reach is 6.88°, which corresponds to the situation where the vehicle does not use the stabilizer bar. For the other three situations, the maximum roll angle values are only 6.38°, 6.20°, and 5.91°, respectively. A rollover phenomenon has occurred if the vehicle’s speed is increased to a higher value, v₂ = 80 (km/h). This phenomenon occurs at time t = 3.6 (s), which corresponds to the situation where the vehicle does not have a stabilizer bar. When the speed increases to v₃ = 100 (km/h), the roll angle of the vehicle is in two situations: the vehicle using the active stabilizer bar with PID controller and the vehicle using the passive stabilizer bar is very high, reaching 10.62° and 10.84°. Meanwhile, this value is only 9.99° if the FLC algorithm is used.

Fig. 15

Roll angle (a –v₁; b –v₂; c –v₃).

Warning signs of vehicle rollover are shown more clearly through the change in vertical force at the wheels. For the average speed, v₁, the minimum value of vertical force received by the wheel is only 1033.45 (N), corresponding to the situation where the vehicle does not use the stabilizer bar (Fig. 16). Meanwhile, this value can be up to 2498.45 (N) if the vehicle uses a hydraulic stabilizer bar.

Fig. 16

Vertical force –v₁ (a –FLC; b –PID; c –Passive; d –None).

At higher speeds, the change in vertical force at the wheel will also be larger. This is shown in Figs. 17 18. When the vehicle was traveling at a very high speed, v₃, the vehicle would have rolled over if the stabilizer bar was not used. This occurs at t = 3.4 (s). If the vehicle only uses a conventional mechanical stabilizer bar, the value of the vertical force at the wheel can be reduced to only 740.82 (N), which is a very low and dangerous threshold. If the speed continues to increase, or the steering angle becomes larger, the vehicle can also roll over, even though it has been equipped with stabilizer bars on both the front and rear axles. In this case, the PID controller is not able to promote its performance. The value of the vertical force at the wheel is similar to the above situation. This is very dangerous. In contrast, stability is always guaranteed when the FLC algorithm is used to control the active stabilizer bar. The hydraulic stabilizer bar controlled by the FLC algorithm ensures the vehicle moves stably and safely in many different situations.

Fig. 17

Vertical force –v₂ (a –FLC; b –PID; c –Passive; d –None).

Fig. 18

Vertical force –v₃ (a –FLC; b –PID; c –Passive; d –None).

4 Conclusions

Vehicle lateral instability can occur when the vehicle is steering at a high speed. In some special cases, the vehicle may be rolled over. In this article, a spatial dynamics model with 10 degrees of freedom is considered to simulate vehicle oscillations when steering. This is a nonlinear dynamic model that considers the influence of the wheel and other factors. Besides, the FLC algorithm to control the active stabilizer bar is also applied. This algorithm uses two input signals, including an original signal and a signal that has undergone differential twice. The membership function is designed based on the author’s experience, which has been accumulated through many previous simulations. The vehicle’s oscillation is described through three specific steering cases. The vehicle’s speed is also gradually increased through three steps. In each case, four situations are investigated.

The results of this research showed that the vehicle’s stability was improved when the vehicle used an active stabilizer bar controlled by the FLC algorithm. According to this result, both the roll angle and the difference in vertical force at the wheels have also been significantly reduced compared to the other three situations. This is proven by all cases examined, and at all different velocity values. When the vehicle moved at a very high speed and steered in a fishhook style, the vehicle would roll over if the stabilizer bar was not used. Instability may also occur if the vehicle is only equipped with a mechanical stabilizer bar or a hydraulic stabilizer bar controlled by the PID algorithm. If the speed continues to increase, the vehicle can roll over at any time. However, the stability of the vehicle is still ensured when the FLC algorithm is applied to the active stabilizer bar. This helps the vehicle to move more safely at high speeds. The FLC algorithm can be used in conjunction with other intelligent control algorithms in the future to increase system efficiency.

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