# Enhancing the Stability and Safety of Vehicle When Steering by Using the Active Stabilizer Bar

Authors: Duc Ngoc Nguyen; Minh Phung Le; Tuan Anh Nguyen; Thang Binh Hoang; Thi Thu Huong Tran; Ngoc Duyen Dang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Centrifugal force is what causes the vehicle to roll over. This force is generated when the vehicle suddenly changes direction when moving at high speed. The solution of using a stabilizer bar is suggested to minimize this phenomenon. There are currently three types of stabilizer bars in use: mechanical stabilizer bars, hydraulic stabilizer bars, and electronic stabilizer bars. The content of this article is aimed at introducing and reviewing the characteristics of the stabilizer bar. Besides, two oscillation simulation models of vehicles equipped with stabilizer bars are also analysed in this article. Additionally, the characteristics of the control algorithms for the stabilizer bar are clearly analysed. The half-dynamics model is suitable for algorithms that need to use oscillatory state matrices such as SMC, LQR, and LQG. The spatial dynamics model is suitable for some algorithms such as PID, fuzzy, and neural. The roll angle of the vehicle has been significantly improved when the stabilizer bar is fitted. In general, the stability and safety of the vehicle can be guaranteed if the vehicle uses a stabilizer bar.


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## Body

## 1. Introduction

### 1.1. The Problem of Vehicle Instability

The automobile is a vehicle that is widely used for many purposes. The automobile was born more than 135 years ago, and it is also considered a great achievement of the Industrial Revolution [1]. During the past time, the automobile industry has developed strongly in both quality and quantity. This development is proportional to the needs of consumers. In the coming time, automobile production can continue to increase sharply to meet the requirements of the 4.0 Science and Technology Revolution.When participating in traffic on the road, the vehicle may encounter many different situations. In particular, situations that cause instability often occur. Some of the vehicle’s instability situations can be mentioned such as sideslip, oversteering, and understeering [2]. In particular, the phenomenon of rollover is extremely dangerous. This phenomenon can greatly affect the safety of passengers and goods on the vehicle [3, 4].Rollover is defined as all four wheels turning to the same side (laterally) and the vehicle’s side body making contact with the ground. It can be understood in a simpler way that the vertical force at the wheels will approach zero. There are two reasons for rollover. First, the influence of lateral wind forces can cause this phenomenon. For large vehicles, such as containers or heavy trucks, the trunk can be rolled when there is a lateral wind force. A rollover will occur when the roll angle of the vehicle body is larger than the limited roll angle. In practice, the magnitude of the lateral wind force is usually not large. Therefore, this cause rarely occurs. Second, if the vehicle suddenly steers at high speed, a rollover can occur at any time [5]. When the vehicle steers, centrifugal force will appear. The direction of centrifugal force is directed to the outside of the arc of rotation. The centrifugal force produces a moment called the roll moment. This moment is proportional to the magnitude of the centrifugal force and the distance from the vehicle’s centre of gravity to the rollover axle. This dependence is represented by equation (1). According to (1), if the vehicle’s speed increases, the roll moment also increases. Besides, if the size of the vehicle is large, that is, the distance from the centre of gravity to the rollover axis (h) is also large, the value of the roll moment can increase rapidly. Therefore, the roll angle of the vehicle also increases. In addition, the steering angle and steering acceleration also affect the roll moment and roll angle. This influence is expressed through the value of the turning radius (R). Once the steering angle or steering acceleration increases, the value of the turning radius decreases. Therefore, the value of the roll moment and roll angle will also increase significantly.(1)MR=Fceh=mv2Rh,whereMR: Roll momentFce: Centrifugal forcem: Massv: VelocityThe consequences of the rollover phenomenon are extremely serious. It can be life threatening. To avoid the rollover phenomenon, parameters such as roll index, roll angle, and the difference of the vertical force at the wheel need to be reduced. Today, three ways are used to limit this phenomenon. The first way is that the dimensions of the vehicle need to be optimized. As analysed above, the centrifugal force is proportional to the mass and height of the vehicle’s centre of gravity. Therefore, if the magnitude of these two parameters is decreased, the centrifugal force will also decrease. However, it is not reasonable to change these two parameters. It will affect the performance of the vehicle. For the second solution, the driver needs to actively control the vehicle’s operation. This means that travel speed and steering angle need to be kept within the allowable range. In fact, some situations happen unexpectedly, and the driver cannot react in time. So, this solution does not really work either. In order to improve the stability of the vehicle when moving, the solution of using a stabilizer bar on the vehicle is proposed, and this is the third solution. This solution really provides high efficiency to users [6, 7].


### 1.2. Stabilizer Bar

#### 1.2.1. Passive Stabilizer Bar

Stabilizer bar is also known as an antiroll bar, sway bar, etc. [8]. Currently, there are three types of stabilizer bars commonly used, including mechanical stabilizer bar (passive), hydraulic stabilizer bar (active), and electronic stabilizer bar (active).The mechanical stabilizer bar is made from elastomeric steel, has a circular cross section, and is hollow inside. Usually, the stabilizer bar has a symmetrical U-shape. Several other shapes of bars are also used to suit the installation site (Figure1) [9]. The two ends of the stabilizer bar will attach to the hub of the wheel. The back of the bar can be rotated between two rubber bearings attached to the chassis (Figure 2). When the vehicle is going straight, no load is applied to the bar. Therefore, the stabilizer bar remains in the equilibrium position. When steering, under the action of centrifugal force, the body of the vehicle will roll. Because the body of the vehicle is rolled, the loads on the two wheels of the same axle will change. At this point, the stabilizer bar will take part in that changing load, and it will twist the back of the bar. Based on this principle, the roll angle of the vehicle will be smaller, and the phenomenon of rolling over can be improved. The value of the elastic moment produced by the stabilizer bar depends on the torsion angle of the bar.Figure 1
The shape of the stabilizer bar.Figure 2
The passive stabilizer bar.The passive stabilizer bar has a low cost, simple structure, and high durability and is easy to install at different locations on different vehicles. Therefore, this bar is commonly used in most vehicles today. However, the performance of the passive stabilizer bar is still not very good. Because the antiroll moment generated by the passive stabilizer bar is limited. Therefore, in many dangerous situations, a rollover phenomenon can still occur even though a mechanical stabilizer bar has been used. So, the active stabilizer bar is proposed to replace the conventional passive stabilizer [10].


#### 1.2.2. Hydraulic Stabilizer Bar

The hydraulic stabilizer bar is also known as the active stabilizer bar. The structure of the hydraulic stabilizer bar is shown in Figure3. The hydraulic motor (1) is placed in the centre of the bar. Outside the motor are the valves (12). These valves are connected to the pipeline (14) via the link connector (13). The two ends of the bar are attached to the lever arm (8) via a ball joint (10) and the back of the bar rests on two bearings (2).Figure 3
The hydraulic stabilizer bar.The hydraulic stabilizer bar is controlled fully automatically through the hydraulic actuator and controller. When the vehicle body is rolled, this phenomenon is detected by the sensor and converted into a signal, which is sent back to the controller. Here, the controller will evaluate the received signal and give a control signal based on the previously established algorithm. The output signal of the controller is a current of no more than 24 volts. If a voltage signal is sent to the actuator, the hydraulic valves will perform the displacement. This displacement will provide a high-pressure oil line to the motor. When the motor is running, the two arms can rotate in opposite directions. For some types of hydraulic stabilizer bars, the arm can only be rotated on one side, and the other side is fixed. The movement of the arm will pull the unsprung mass (wheel) back to its original position. As a result, the roll angle of the vehicle body and the difference in vertical force at the two sides of the wheel will be reduced. This means the vehicle’s stability and safety can be significantly improved.A hydraulic stabilizer bar can produce a large impact force. This force will help the vehicle return to a stable state when steering. However, this system is quite complex and cumbersome. Besides, the slowness of the system is also an important issue that has not been resolved. The hydraulic stabilizer bar is often used on some mid-range vehicles such as BMW X5, BMW X6, Mercedes GLE 63S, or on some heavy trucks, etc. To solve the problem of slow-acting time, the idea of using an electronic stabilizer bar was proposed.


#### 1.2.3. Electronic Stabilizer Bar

The electronic stabilizer bar is a kind of active stabilizer bar. Instead of hydraulic motors and cumbersome pipes, the actuator of the electronic stabilizer bar is an electric motor with a planetary gearbox. The structure of the electronic stabilizer bar is shown in Figure4.Figure 4
The electronic stabilizer bar.The electronic stabilizer bar is fully controlled automatically. Based on the signal received from the sensor or from the camera [11], the current will be generated by the controller. This current is supplied to the electric motor of the bar. From there, the impact force will be generated based on the displacement of the arm. Unlike the hydraulic stabilizer bar, the voltage of the electronic stabilizer bar can be up to 48 volts or more. With this energy, the electronic stabilizer bar can easily generate enough torque to support the vehicle’s stability.The electronic stabilizer bar has a more compact structure than the hydraulic stabilizer bar. Besides, the delay of the electronic stabilizer bar is also much smaller. Therefore, the effect of the electronic stabilizer bar is extremely positive. However, the cost is one of the limitations of this bar. Nowadays, an electronic stabilizer bar is often equipped on some high-end vehicles such as the Audi Q7, Audi SQ7, and Porsche Panamera.


## 1.1. The Problem of Vehicle Instability

The automobile is a vehicle that is widely used for many purposes. The automobile was born more than 135 years ago, and it is also considered a great achievement of the Industrial Revolution [1]. During the past time, the automobile industry has developed strongly in both quality and quantity. This development is proportional to the needs of consumers. In the coming time, automobile production can continue to increase sharply to meet the requirements of the 4.0 Science and Technology Revolution.When participating in traffic on the road, the vehicle may encounter many different situations. In particular, situations that cause instability often occur. Some of the vehicle’s instability situations can be mentioned such as sideslip, oversteering, and understeering [2]. In particular, the phenomenon of rollover is extremely dangerous. This phenomenon can greatly affect the safety of passengers and goods on the vehicle [3, 4].Rollover is defined as all four wheels turning to the same side (laterally) and the vehicle’s side body making contact with the ground. It can be understood in a simpler way that the vertical force at the wheels will approach zero. There are two reasons for rollover. First, the influence of lateral wind forces can cause this phenomenon. For large vehicles, such as containers or heavy trucks, the trunk can be rolled when there is a lateral wind force. A rollover will occur when the roll angle of the vehicle body is larger than the limited roll angle. In practice, the magnitude of the lateral wind force is usually not large. Therefore, this cause rarely occurs. Second, if the vehicle suddenly steers at high speed, a rollover can occur at any time [5]. When the vehicle steers, centrifugal force will appear. The direction of centrifugal force is directed to the outside of the arc of rotation. The centrifugal force produces a moment called the roll moment. This moment is proportional to the magnitude of the centrifugal force and the distance from the vehicle’s centre of gravity to the rollover axle. This dependence is represented by equation (1). According to (1), if the vehicle’s speed increases, the roll moment also increases. Besides, if the size of the vehicle is large, that is, the distance from the centre of gravity to the rollover axis (h) is also large, the value of the roll moment can increase rapidly. Therefore, the roll angle of the vehicle also increases. In addition, the steering angle and steering acceleration also affect the roll moment and roll angle. This influence is expressed through the value of the turning radius (R). Once the steering angle or steering acceleration increases, the value of the turning radius decreases. Therefore, the value of the roll moment and roll angle will also increase significantly.(1)MR=Fceh=mv2Rh,whereMR: Roll momentFce: Centrifugal forcem: Massv: VelocityThe consequences of the rollover phenomenon are extremely serious. It can be life threatening. To avoid the rollover phenomenon, parameters such as roll index, roll angle, and the difference of the vertical force at the wheel need to be reduced. Today, three ways are used to limit this phenomenon. The first way is that the dimensions of the vehicle need to be optimized. As analysed above, the centrifugal force is proportional to the mass and height of the vehicle’s centre of gravity. Therefore, if the magnitude of these two parameters is decreased, the centrifugal force will also decrease. However, it is not reasonable to change these two parameters. It will affect the performance of the vehicle. For the second solution, the driver needs to actively control the vehicle’s operation. This means that travel speed and steering angle need to be kept within the allowable range. In fact, some situations happen unexpectedly, and the driver cannot react in time. So, this solution does not really work either. In order to improve the stability of the vehicle when moving, the solution of using a stabilizer bar on the vehicle is proposed, and this is the third solution. This solution really provides high efficiency to users [6, 7].


## 1.2. Stabilizer Bar

### 1.2.1. Passive Stabilizer Bar

Stabilizer bar is also known as an antiroll bar, sway bar, etc. [8]. Currently, there are three types of stabilizer bars commonly used, including mechanical stabilizer bar (passive), hydraulic stabilizer bar (active), and electronic stabilizer bar (active).The mechanical stabilizer bar is made from elastomeric steel, has a circular cross section, and is hollow inside. Usually, the stabilizer bar has a symmetrical U-shape. Several other shapes of bars are also used to suit the installation site (Figure1) [9]. The two ends of the stabilizer bar will attach to the hub of the wheel. The back of the bar can be rotated between two rubber bearings attached to the chassis (Figure 2). When the vehicle is going straight, no load is applied to the bar. Therefore, the stabilizer bar remains in the equilibrium position. When steering, under the action of centrifugal force, the body of the vehicle will roll. Because the body of the vehicle is rolled, the loads on the two wheels of the same axle will change. At this point, the stabilizer bar will take part in that changing load, and it will twist the back of the bar. Based on this principle, the roll angle of the vehicle will be smaller, and the phenomenon of rolling over can be improved. The value of the elastic moment produced by the stabilizer bar depends on the torsion angle of the bar.Figure 1
The shape of the stabilizer bar.Figure 2
The passive stabilizer bar.The passive stabilizer bar has a low cost, simple structure, and high durability and is easy to install at different locations on different vehicles. Therefore, this bar is commonly used in most vehicles today. However, the performance of the passive stabilizer bar is still not very good. Because the antiroll moment generated by the passive stabilizer bar is limited. Therefore, in many dangerous situations, a rollover phenomenon can still occur even though a mechanical stabilizer bar has been used. So, the active stabilizer bar is proposed to replace the conventional passive stabilizer [10].


### 1.2.2. Hydraulic Stabilizer Bar

The hydraulic stabilizer bar is also known as the active stabilizer bar. The structure of the hydraulic stabilizer bar is shown in Figure3. The hydraulic motor (1) is placed in the centre of the bar. Outside the motor are the valves (12). These valves are connected to the pipeline (14) via the link connector (13). The two ends of the bar are attached to the lever arm (8) via a ball joint (10) and the back of the bar rests on two bearings (2).Figure 3
The hydraulic stabilizer bar.The hydraulic stabilizer bar is controlled fully automatically through the hydraulic actuator and controller. When the vehicle body is rolled, this phenomenon is detected by the sensor and converted into a signal, which is sent back to the controller. Here, the controller will evaluate the received signal and give a control signal based on the previously established algorithm. The output signal of the controller is a current of no more than 24 volts. If a voltage signal is sent to the actuator, the hydraulic valves will perform the displacement. This displacement will provide a high-pressure oil line to the motor. When the motor is running, the two arms can rotate in opposite directions. For some types of hydraulic stabilizer bars, the arm can only be rotated on one side, and the other side is fixed. The movement of the arm will pull the unsprung mass (wheel) back to its original position. As a result, the roll angle of the vehicle body and the difference in vertical force at the two sides of the wheel will be reduced. This means the vehicle’s stability and safety can be significantly improved.A hydraulic stabilizer bar can produce a large impact force. This force will help the vehicle return to a stable state when steering. However, this system is quite complex and cumbersome. Besides, the slowness of the system is also an important issue that has not been resolved. The hydraulic stabilizer bar is often used on some mid-range vehicles such as BMW X5, BMW X6, Mercedes GLE 63S, or on some heavy trucks, etc. To solve the problem of slow-acting time, the idea of using an electronic stabilizer bar was proposed.


### 1.2.3. Electronic Stabilizer Bar

The electronic stabilizer bar is a kind of active stabilizer bar. Instead of hydraulic motors and cumbersome pipes, the actuator of the electronic stabilizer bar is an electric motor with a planetary gearbox. The structure of the electronic stabilizer bar is shown in Figure4.Figure 4
The electronic stabilizer bar.The electronic stabilizer bar is fully controlled automatically. Based on the signal received from the sensor or from the camera [11], the current will be generated by the controller. This current is supplied to the electric motor of the bar. From there, the impact force will be generated based on the displacement of the arm. Unlike the hydraulic stabilizer bar, the voltage of the electronic stabilizer bar can be up to 48 volts or more. With this energy, the electronic stabilizer bar can easily generate enough torque to support the vehicle’s stability.The electronic stabilizer bar has a more compact structure than the hydraulic stabilizer bar. Besides, the delay of the electronic stabilizer bar is also much smaller. Therefore, the effect of the electronic stabilizer bar is extremely positive. However, the cost is one of the limitations of this bar. Nowadays, an electronic stabilizer bar is often equipped on some high-end vehicles such as the Audi Q7, Audi SQ7, and Porsche Panamera.


## 1.2.1. Passive Stabilizer Bar

Stabilizer bar is also known as an antiroll bar, sway bar, etc. [8]. Currently, there are three types of stabilizer bars commonly used, including mechanical stabilizer bar (passive), hydraulic stabilizer bar (active), and electronic stabilizer bar (active).The mechanical stabilizer bar is made from elastomeric steel, has a circular cross section, and is hollow inside. Usually, the stabilizer bar has a symmetrical U-shape. Several other shapes of bars are also used to suit the installation site (Figure1) [9]. The two ends of the stabilizer bar will attach to the hub of the wheel. The back of the bar can be rotated between two rubber bearings attached to the chassis (Figure 2). When the vehicle is going straight, no load is applied to the bar. Therefore, the stabilizer bar remains in the equilibrium position. When steering, under the action of centrifugal force, the body of the vehicle will roll. Because the body of the vehicle is rolled, the loads on the two wheels of the same axle will change. At this point, the stabilizer bar will take part in that changing load, and it will twist the back of the bar. Based on this principle, the roll angle of the vehicle will be smaller, and the phenomenon of rolling over can be improved. The value of the elastic moment produced by the stabilizer bar depends on the torsion angle of the bar.Figure 1
The shape of the stabilizer bar.Figure 2
The passive stabilizer bar.The passive stabilizer bar has a low cost, simple structure, and high durability and is easy to install at different locations on different vehicles. Therefore, this bar is commonly used in most vehicles today. However, the performance of the passive stabilizer bar is still not very good. Because the antiroll moment generated by the passive stabilizer bar is limited. Therefore, in many dangerous situations, a rollover phenomenon can still occur even though a mechanical stabilizer bar has been used. So, the active stabilizer bar is proposed to replace the conventional passive stabilizer [10].


## 1.2.2. Hydraulic Stabilizer Bar

The hydraulic stabilizer bar is also known as the active stabilizer bar. The structure of the hydraulic stabilizer bar is shown in Figure3. The hydraulic motor (1) is placed in the centre of the bar. Outside the motor are the valves (12). These valves are connected to the pipeline (14) via the link connector (13). The two ends of the bar are attached to the lever arm (8) via a ball joint (10) and the back of the bar rests on two bearings (2).Figure 3
The hydraulic stabilizer bar.The hydraulic stabilizer bar is controlled fully automatically through the hydraulic actuator and controller. When the vehicle body is rolled, this phenomenon is detected by the sensor and converted into a signal, which is sent back to the controller. Here, the controller will evaluate the received signal and give a control signal based on the previously established algorithm. The output signal of the controller is a current of no more than 24 volts. If a voltage signal is sent to the actuator, the hydraulic valves will perform the displacement. This displacement will provide a high-pressure oil line to the motor. When the motor is running, the two arms can rotate in opposite directions. For some types of hydraulic stabilizer bars, the arm can only be rotated on one side, and the other side is fixed. The movement of the arm will pull the unsprung mass (wheel) back to its original position. As a result, the roll angle of the vehicle body and the difference in vertical force at the two sides of the wheel will be reduced. This means the vehicle’s stability and safety can be significantly improved.A hydraulic stabilizer bar can produce a large impact force. This force will help the vehicle return to a stable state when steering. However, this system is quite complex and cumbersome. Besides, the slowness of the system is also an important issue that has not been resolved. The hydraulic stabilizer bar is often used on some mid-range vehicles such as BMW X5, BMW X6, Mercedes GLE 63S, or on some heavy trucks, etc. To solve the problem of slow-acting time, the idea of using an electronic stabilizer bar was proposed.


## 1.2.3. Electronic Stabilizer Bar

The electronic stabilizer bar is a kind of active stabilizer bar. Instead of hydraulic motors and cumbersome pipes, the actuator of the electronic stabilizer bar is an electric motor with a planetary gearbox. The structure of the electronic stabilizer bar is shown in Figure4.Figure 4
The electronic stabilizer bar.The electronic stabilizer bar is fully controlled automatically. Based on the signal received from the sensor or from the camera [11], the current will be generated by the controller. This current is supplied to the electric motor of the bar. From there, the impact force will be generated based on the displacement of the arm. Unlike the hydraulic stabilizer bar, the voltage of the electronic stabilizer bar can be up to 48 volts or more. With this energy, the electronic stabilizer bar can easily generate enough torque to support the vehicle’s stability.The electronic stabilizer bar has a more compact structure than the hydraulic stabilizer bar. Besides, the delay of the electronic stabilizer bar is also much smaller. Therefore, the effect of the electronic stabilizer bar is extremely positive. However, the cost is one of the limitations of this bar. Nowadays, an electronic stabilizer bar is often equipped on some high-end vehicles such as the Audi Q7, Audi SQ7, and Porsche Panamera.


## 2. Vehicle Dynamics Models

### 2.1. The Mechanical Stabilizer Bar Model

As mentioned above, the mechanical stabilizer bar has a simple structure and low cost. Therefore, it is widely used in most vehicles today. There are several methods that have been used to determine the force of a stabilizer bar. According to Zulkarnain et al., the value of the impact force is proportional to the torsional stiffness of the stabilizer bar [12]. Accordingly, if the torsion angle is larger, the value of the impact force will also be larger.(2)FA=ktfzs,zu,ϕ.Similar to the idea of Zulkarnain, Gosselin-Brisson et al. also show how to determine the impact force based on the flexural stiffness of the stabilizer bar [13]. In equation (3), the stiffness is a function of the displacements of the sprung mass and unsprung mass. The effect of the roll angle is not considered in this equation.(3)FA=kbfzs,zu,whereϕ: Roll angle of the sprung massϕu: Roll angle of the unsprung masskb: Bending stiffness of the stabilizer barkt: Torsional stiffness of the stabilizer barzs: Displacement of the sprung masszu: Displacement of the unsprung massAccording to Vu et al., the impact moment of the mechanical stabilizer bar is a complex function expressed as equation (4) [14]. The dimensional parameters used in the equation can be found in Figure 5. In addition, instead of using the displacements of the sprung mass and the unsprung mass, the roll angle of the sprung mass and unsprung mass is used instead.(4)MA=4ktaa+b+dc2ϕ−4kta+b+d2c2ϕu.Figure 5
Schematic of the passive stabilizer bar.Recently, Nguyen introduced a new model, which is used to determine the impact force of a passive stabilizer bar. According to [15], the force is a nonlinear function that depends on the displacement of the unsprung mass. This is a completely new and unique idea.(5)ξ=FAb+d33EJx+c2GJptan∫0a+bFAdz.In Nguyen’s model, the stabilizer bar will be bent and twisted. Flexion occurs at the arm, while twisting occurs in the back. Therefore, the bending stiffness and torsion stiffness are considered in the above formula.The above models each have their own characteristics. However, the performance of the passive stabilizer bar is still not very good. Therefore, the active stabilizer bar is suggested for use as an alternative to the conventional mechanical stabilizer bar.


### 2.2. The Active Stabilizer Bar Model

#### 2.2.1. Half-Dynamics Model with Hydraulic Cylinder

This dynamic model was introduced by Vu et al. in [16]. Accordingly, the authors used a half-dynamic model with four degrees of freedom, as shown in Figure 6.Figure 6
Half-dynamics model.The equations describing the oscillations of the half-dynamics model are given as follows:(6)mvβ˙+ψ˙−mshϕ¨=Fyf+Fyr,−Ixzϕ¨+Izzψ¨=Fyflf−Fyrlr,Ixx+msh2ϕ¨−Ixzψ¨=msghϕ+msvhβ˙+ψ˙−kfϕ˙−ϕ˙uf−bfϕ˙−ϕ˙ur+MARf+Tf−krϕ−ϕur−brϕ˙−ϕ˙urMARr+Tr,−rFyf=mufvr−hufβ˙+ψ˙+mufghufϕuf−ktfϕuf+kfϕ−ϕuf+bfϕ˙−ϕ˙uf+MARf+Tf,−rFyr=murvr−hurβ˙+ψ˙−murghurϕur−ktfϕur+krϕ−ϕur+brϕ˙−ϕ˙ur+MARr+Tr.whereφ: Roll angle of the sprung massψ: Yaw angleφu: Roll angle of the unsprung massFy: Lateral forceg: Gravitational accelerationI: Moment of inertiams: Sprung massmu: Unsprung massv: VelocityThe value of the impact torque of the mechanical stabilizer bar is calculated according to (4). The value of the torque generated from the hydraulic stabilizer bar is determined according to (7):(7)T=2lFA.The actuation forceFA is a complex function that depends on the displacement of the piston of the hydraulic actuator.(8)FA=fΔP,xsv,it.Using this model, state variables can be easily written in matrix form. Therefore, it is suitable for some control algorithms such as LQR, LPV, or SMC. However, this model does not take into account nonlinear tire deformation. In addition, the interaction between the front and rear axles has not been mentioned.Hydraulic actuators are also widely used in brake and differential systems [17]. Its effect is enormous. The hydraulic actuator can be considered as a semi-active control system [18].


#### 2.2.2. Spatial Dynamics Model with Hydraulic Motor

The spatial dynamics model describing vehicle’s oscillations was introduced in [19] by Nguyen. This model is shown in Figure 7.Figure 7
Spatial dynamics model.With this model, the vehicle’s oscillations are represented through seven degrees of freedom. The equations describing the oscillations of the vehicle are given as follows:(9)mz¨=FC11+FK11+FC12+FK12+FC21+FK21+FC22+FK22,Ix+mh2φ¨=FC11+FK11−FC12−FK12tw1+FC21+FK21−FC22−FK22tw2+gsinϕ+aycosϕmh,Iy+mh12θ¨=FC11+FK11+FC12+FK12a1−FC21+FK21+FC22+FK22a2,m11ξ¨11=FKT11−FC11−FK11−FA1,m12ξ¨12=FKT12−FC12−FK12+FA1,m21ξ¨21=FKT21−FC21−FK21−FA2,m22ξ¨22=FKT22−FC22−FK22+FA2.The lateral acceleration, which causes the body to roll, is calculated through the forces at the wheel.(10)Mv˙x−α˙+ψ˙vy=∑i,j=12Fxijcosδij−Fyijsinδij−F1,Mv˙y+α˙+ψ˙vx=∑i,j=12Fxijsinδij+Fyijcosδij−F2,IZψ¨=∑i,j=12−1jFxijcosδij−Fyijsinδijtwi+−1i+1Fxijsinδij+Fyijcosδijai+Fici−MZij.The use of a nonlinear double-track dynamics model with four-wheel steering is often highly effective. This has been demonstrated in [20]. The Pacejka nonlinear tire model is mentioned in Nguyen’s article. Accordingly, these values are calculated quite complicatedly [21].(11)Fx=DxsinCxarctanBx1-Exsx+Shx+ExarctanBxsx+Shx+Svx,Fy=DysinCyarctanBy1-Eyα+Shy+EyarctanByα+Shy+Svy,Mz=DzsinCzarctanBz1-Ezα+Shz+EzarctanBzα+Shz+Svz.whereθ: Pitch angle&phiv: Roll angleψ: Yaw angleξij: Displacement of the unsprung massδij: Steering angleFC: Damper forceFK: Spring forceFKT: Tire forceFx: Longitudinal forceFy: Lateral forceg: Gravitational accelerationI: Moment of inertiam: Sprung massM: Total massmij: Unsprung massMz: Moment of the wheelz: Displacement of the sprung massThe spatial dynamic model has high accuracy. Besides, nonlinear factors are also considered in this model. This model can simulate a vehicle’s oscillations under complex conditions. However, setting up the oscillatory state matrix is very difficult. Therefore, this model is suitable for control algorithms such as PID, fuzzy, and neural.


## 2.1. The Mechanical Stabilizer Bar Model

As mentioned above, the mechanical stabilizer bar has a simple structure and low cost. Therefore, it is widely used in most vehicles today. There are several methods that have been used to determine the force of a stabilizer bar. According to Zulkarnain et al., the value of the impact force is proportional to the torsional stiffness of the stabilizer bar [12]. Accordingly, if the torsion angle is larger, the value of the impact force will also be larger.(2)FA=ktfzs,zu,ϕ.Similar to the idea of Zulkarnain, Gosselin-Brisson et al. also show how to determine the impact force based on the flexural stiffness of the stabilizer bar [13]. In equation (3), the stiffness is a function of the displacements of the sprung mass and unsprung mass. The effect of the roll angle is not considered in this equation.(3)FA=kbfzs,zu,whereϕ: Roll angle of the sprung massϕu: Roll angle of the unsprung masskb: Bending stiffness of the stabilizer barkt: Torsional stiffness of the stabilizer barzs: Displacement of the sprung masszu: Displacement of the unsprung massAccording to Vu et al., the impact moment of the mechanical stabilizer bar is a complex function expressed as equation (4) [14]. The dimensional parameters used in the equation can be found in Figure 5. In addition, instead of using the displacements of the sprung mass and the unsprung mass, the roll angle of the sprung mass and unsprung mass is used instead.(4)MA=4ktaa+b+dc2ϕ−4kta+b+d2c2ϕu.Figure 5
Schematic of the passive stabilizer bar.Recently, Nguyen introduced a new model, which is used to determine the impact force of a passive stabilizer bar. According to [15], the force is a nonlinear function that depends on the displacement of the unsprung mass. This is a completely new and unique idea.(5)ξ=FAb+d33EJx+c2GJptan∫0a+bFAdz.In Nguyen’s model, the stabilizer bar will be bent and twisted. Flexion occurs at the arm, while twisting occurs in the back. Therefore, the bending stiffness and torsion stiffness are considered in the above formula.The above models each have their own characteristics. However, the performance of the passive stabilizer bar is still not very good. Therefore, the active stabilizer bar is suggested for use as an alternative to the conventional mechanical stabilizer bar.


## 2.2. The Active Stabilizer Bar Model

### 2.2.1. Half-Dynamics Model with Hydraulic Cylinder

This dynamic model was introduced by Vu et al. in [16]. Accordingly, the authors used a half-dynamic model with four degrees of freedom, as shown in Figure 6.Figure 6
Half-dynamics model.The equations describing the oscillations of the half-dynamics model are given as follows:(6)mvβ˙+ψ˙−mshϕ¨=Fyf+Fyr,−Ixzϕ¨+Izzψ¨=Fyflf−Fyrlr,Ixx+msh2ϕ¨−Ixzψ¨=msghϕ+msvhβ˙+ψ˙−kfϕ˙−ϕ˙uf−bfϕ˙−ϕ˙ur+MARf+Tf−krϕ−ϕur−brϕ˙−ϕ˙urMARr+Tr,−rFyf=mufvr−hufβ˙+ψ˙+mufghufϕuf−ktfϕuf+kfϕ−ϕuf+bfϕ˙−ϕ˙uf+MARf+Tf,−rFyr=murvr−hurβ˙+ψ˙−murghurϕur−ktfϕur+krϕ−ϕur+brϕ˙−ϕ˙ur+MARr+Tr.whereφ: Roll angle of the sprung massψ: Yaw angleφu: Roll angle of the unsprung massFy: Lateral forceg: Gravitational accelerationI: Moment of inertiams: Sprung massmu: Unsprung massv: VelocityThe value of the impact torque of the mechanical stabilizer bar is calculated according to (4). The value of the torque generated from the hydraulic stabilizer bar is determined according to (7):(7)T=2lFA.The actuation forceFA is a complex function that depends on the displacement of the piston of the hydraulic actuator.(8)FA=fΔP,xsv,it.Using this model, state variables can be easily written in matrix form. Therefore, it is suitable for some control algorithms such as LQR, LPV, or SMC. However, this model does not take into account nonlinear tire deformation. In addition, the interaction between the front and rear axles has not been mentioned.Hydraulic actuators are also widely used in brake and differential systems [17]. Its effect is enormous. The hydraulic actuator can be considered as a semi-active control system [18].


### 2.2.2. Spatial Dynamics Model with Hydraulic Motor

The spatial dynamics model describing vehicle’s oscillations was introduced in [19] by Nguyen. This model is shown in Figure 7.Figure 7
Spatial dynamics model.With this model, the vehicle’s oscillations are represented through seven degrees of freedom. The equations describing the oscillations of the vehicle are given as follows:(9)mz¨=FC11+FK11+FC12+FK12+FC21+FK21+FC22+FK22,Ix+mh2φ¨=FC11+FK11−FC12−FK12tw1+FC21+FK21−FC22−FK22tw2+gsinϕ+aycosϕmh,Iy+mh12θ¨=FC11+FK11+FC12+FK12a1−FC21+FK21+FC22+FK22a2,m11ξ¨11=FKT11−FC11−FK11−FA1,m12ξ¨12=FKT12−FC12−FK12+FA1,m21ξ¨21=FKT21−FC21−FK21−FA2,m22ξ¨22=FKT22−FC22−FK22+FA2.The lateral acceleration, which causes the body to roll, is calculated through the forces at the wheel.(10)Mv˙x−α˙+ψ˙vy=∑i,j=12Fxijcosδij−Fyijsinδij−F1,Mv˙y+α˙+ψ˙vx=∑i,j=12Fxijsinδij+Fyijcosδij−F2,IZψ¨=∑i,j=12−1jFxijcosδij−Fyijsinδijtwi+−1i+1Fxijsinδij+Fyijcosδijai+Fici−MZij.The use of a nonlinear double-track dynamics model with four-wheel steering is often highly effective. This has been demonstrated in [20]. The Pacejka nonlinear tire model is mentioned in Nguyen’s article. Accordingly, these values are calculated quite complicatedly [21].(11)Fx=DxsinCxarctanBx1-Exsx+Shx+ExarctanBxsx+Shx+Svx,Fy=DysinCyarctanBy1-Eyα+Shy+EyarctanByα+Shy+Svy,Mz=DzsinCzarctanBz1-Ezα+Shz+EzarctanBzα+Shz+Svz.whereθ: Pitch angle&phiv: Roll angleψ: Yaw angleξij: Displacement of the unsprung massδij: Steering angleFC: Damper forceFK: Spring forceFKT: Tire forceFx: Longitudinal forceFy: Lateral forceg: Gravitational accelerationI: Moment of inertiam: Sprung massM: Total massmij: Unsprung massMz: Moment of the wheelz: Displacement of the sprung massThe spatial dynamic model has high accuracy. Besides, nonlinear factors are also considered in this model. This model can simulate a vehicle’s oscillations under complex conditions. However, setting up the oscillatory state matrix is very difficult. Therefore, this model is suitable for control algorithms such as PID, fuzzy, and neural.


## 2.2.1. Half-Dynamics Model with Hydraulic Cylinder

This dynamic model was introduced by Vu et al. in [16]. Accordingly, the authors used a half-dynamic model with four degrees of freedom, as shown in Figure 6.Figure 6
Half-dynamics model.The equations describing the oscillations of the half-dynamics model are given as follows:(6)mvβ˙+ψ˙−mshϕ¨=Fyf+Fyr,−Ixzϕ¨+Izzψ¨=Fyflf−Fyrlr,Ixx+msh2ϕ¨−Ixzψ¨=msghϕ+msvhβ˙+ψ˙−kfϕ˙−ϕ˙uf−bfϕ˙−ϕ˙ur+MARf+Tf−krϕ−ϕur−brϕ˙−ϕ˙urMARr+Tr,−rFyf=mufvr−hufβ˙+ψ˙+mufghufϕuf−ktfϕuf+kfϕ−ϕuf+bfϕ˙−ϕ˙uf+MARf+Tf,−rFyr=murvr−hurβ˙+ψ˙−murghurϕur−ktfϕur+krϕ−ϕur+brϕ˙−ϕ˙ur+MARr+Tr.whereφ: Roll angle of the sprung massψ: Yaw angleφu: Roll angle of the unsprung massFy: Lateral forceg: Gravitational accelerationI: Moment of inertiams: Sprung massmu: Unsprung massv: VelocityThe value of the impact torque of the mechanical stabilizer bar is calculated according to (4). The value of the torque generated from the hydraulic stabilizer bar is determined according to (7):(7)T=2lFA.The actuation forceFA is a complex function that depends on the displacement of the piston of the hydraulic actuator.(8)FA=fΔP,xsv,it.Using this model, state variables can be easily written in matrix form. Therefore, it is suitable for some control algorithms such as LQR, LPV, or SMC. However, this model does not take into account nonlinear tire deformation. In addition, the interaction between the front and rear axles has not been mentioned.Hydraulic actuators are also widely used in brake and differential systems [17]. Its effect is enormous. The hydraulic actuator can be considered as a semi-active control system [18].


## 2.2.2. Spatial Dynamics Model with Hydraulic Motor

The spatial dynamics model describing vehicle’s oscillations was introduced in [19] by Nguyen. This model is shown in Figure 7.Figure 7
Spatial dynamics model.With this model, the vehicle’s oscillations are represented through seven degrees of freedom. The equations describing the oscillations of the vehicle are given as follows:(9)mz¨=FC11+FK11+FC12+FK12+FC21+FK21+FC22+FK22,Ix+mh2φ¨=FC11+FK11−FC12−FK12tw1+FC21+FK21−FC22−FK22tw2+gsinϕ+aycosϕmh,Iy+mh12θ¨=FC11+FK11+FC12+FK12a1−FC21+FK21+FC22+FK22a2,m11ξ¨11=FKT11−FC11−FK11−FA1,m12ξ¨12=FKT12−FC12−FK12+FA1,m21ξ¨21=FKT21−FC21−FK21−FA2,m22ξ¨22=FKT22−FC22−FK22+FA2.The lateral acceleration, which causes the body to roll, is calculated through the forces at the wheel.(10)Mv˙x−α˙+ψ˙vy=∑i,j=12Fxijcosδij−Fyijsinδij−F1,Mv˙y+α˙+ψ˙vx=∑i,j=12Fxijsinδij+Fyijcosδij−F2,IZψ¨=∑i,j=12−1jFxijcosδij−Fyijsinδijtwi+−1i+1Fxijsinδij+Fyijcosδijai+Fici−MZij.The use of a nonlinear double-track dynamics model with four-wheel steering is often highly effective. This has been demonstrated in [20]. The Pacejka nonlinear tire model is mentioned in Nguyen’s article. Accordingly, these values are calculated quite complicatedly [21].(11)Fx=DxsinCxarctanBx1-Exsx+Shx+ExarctanBxsx+Shx+Svx,Fy=DysinCyarctanBy1-Eyα+Shy+EyarctanByα+Shy+Svy,Mz=DzsinCzarctanBz1-Ezα+Shz+EzarctanBzα+Shz+Svz.whereθ: Pitch angle&phiv: Roll angleψ: Yaw angleξij: Displacement of the unsprung massδij: Steering angleFC: Damper forceFK: Spring forceFKT: Tire forceFx: Longitudinal forceFy: Lateral forceg: Gravitational accelerationI: Moment of inertiam: Sprung massM: Total massmij: Unsprung massMz: Moment of the wheelz: Displacement of the sprung massThe spatial dynamic model has high accuracy. Besides, nonlinear factors are also considered in this model. This model can simulate a vehicle’s oscillations under complex conditions. However, setting up the oscillatory state matrix is very difficult. Therefore, this model is suitable for control algorithms such as PID, fuzzy, and neural.


## 3. Several Control Methods

There are many methods used to control the active stabilizer bar. In [22], Varga et al. designed the LQ controller for the hydraulic actuator. This actuator is a hydraulic motor controlled by the spool valve. Similarly, Zulkarnain et al. also use the LQR algorithm for the active stabilizer bar equipped with the half-dynamics model [23]. This algorithm is combined with the CNF algorithm. Therefore, the effect it provides is more positive. In addition, Zulkarnain et al. also integrated a Gaussian filter for the control model. This algorithm is called LQG [24]. The PID control algorithm has also been used to control the operation of the stabilizer bar equipped with the model of spatial dynamics [19]. For this algorithm, three coefficients correspond to the three stages, including proportional (KP), integral (KI), and derivative (KD). The selection of these values is extremely important. It affects the stability of the controller. In [25], Muniandy et al. used an intelligent control method with the Fuzzy PI-PD algorithm to determine the value of these parameters. This combination was also shown in [26] by Dawei et al. The difference between simulation and experimental results is not large. In [27], Khalil et al. used the fuzzy logic algorithm to determine the parameters for the controller of the electronic stabilizer bar. In general, the control algorithm for the electronic stabilizer is similar to that of the hydraulic stabilizer. According to Sintipsomboon et al., this algorithm is called fuzzy hybrid. Its main purpose is to provide current to drive the servo valves inside the actuator [28].Fuzzy control algorithms are commonly used for hydraulic stabilizer bars. In [30], Muniandy et al. used a double in-loop model with the Fuzzy algorithm. Accordingly, the roll angle of the vehicle has been significantly improved when this method is used. The FLC algorithm was developed by Marzbanrad et al. and proposed with two control modes. The dynamics model used in Marzbanrad’s study has only eight degrees of freedom [31]. With two input parameters, namely the roll angle and displacement of the unsprung mass, Nguyen proposed a two-input fuzzy controller [29]. According to this result, the rollover index of the vehicle was greatly reduced when the hydraulic stabilizer bar was used (Figure 8). Also, the value of the roll angle and the change in the vertical force at the wheels were also significantly reduced when using this algorithm (Figure 9). This is caused by the active stabilizer bar being able to produce a larger impact force than the passive stabilizer bar (Figure 10). Therefore, the phenomenon of vehicle rollover can be limited. In addition, several other algorithms have also been used to control the hydraulic stabilizer bar, such as H∞ and LPV [33–35]. Modern autonomous vehicles are also recommended to use a stabilizer bar [36, 37]. In general, the efficiency of these methods is very high.Figure 8
Roll index [29].Figure 9
Roll angle [32].Figure 10
Impact force [32].


## 4. Conclusion

A rollover phenomenon occurs when the driver makes a sudden turn at a high speed. At that time, under the influence of centrifugal force, the vehicle body will be rolled. If the value of the roll angle exceeds the limit, the rollover phenomenon can occur at any time. The consequences of rollover accidents are extremely catastrophic, causing great damage to passengers and cargo.The stabilizer bar is used to limit the rollover phenomenon of the vehicle. If a stabilizer bar is equipped with the vehicle, the roll angle and the difference in vertical forces on both sides of the wheel will be reduced when the vehicle steers. Because of that, the vehicle’s stability has been significantly improved. The passive stabilizer bar is cheap and has a long life, so it is used on most vehicle models today. In some dangerous situations, the passive stabilizer bar can still not guarantee the stability and safety requirements of the vehicle. To overcome this problem, an active stabilizer bar is proposed to be used to replace the conventional mechanical stabilizer bar. There are two types of active stabilizer bars: the hydraulic stabilizer bar and the electronic stabilizer bar. The performance of the active stabilizer bar depends entirely on its controller.This article analyses and evaluates the characteristics of the stabilizer bar. Besides, several dynamic models of the vehicle, which use the stabilizer bar, are also introduced. The characteristics of the above models are pointed out in detail. In addition, control algorithms for active stabilizer bars are also introduced in this article. The control methods are suitable for each type of dynamics model. Overall, the effect of these control methods is very positive. In the coming time, control models for active stabilizer bars will continue to be introduced.



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*Source: 5617167-2022-06-27.xml*