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Abstract

In this paper, the author introduces using active stabilizer bars (hydraulic anti-roll bars) for automobiles to improve the stability of rolling over when an automobile is steering. The car rollover oscillation is described by a complex dynamic model, a combination of the Pacejka tire model, the motion model, and the full oscillation model. A new fuzzy solution has been developed in order to control the active stability bars. This algorithm has three inputs related to the car’s rollover factor. The fuzzy rule (with 125 situations) and membership function are determined based on views related to the car’s stability and the antiroll system’s responsiveness. Numerical calculations and simulation are performed with two cases corresponding to two kinds of steering angles. Three situations correspond to three velocity values: v₁, v₂, and v₃, and four scenarios are simulated in each situation to facilitate the comparison of results. According to the research findings, output values such as roll angle, rollover index, and load change are drastically reduced when the new fuzzy solution is applied to the active anti-roll bars. In the last case, the rollover occurs when the automobile does not use any stability bars or uses regular mechanical bars. The roll angle peak values reach 8.79° and 10.79°, while the maximum value belonging to the Active with Fuzzy situation is 10.94° without rolling over (corresponding to a rollover index of 0.64). The results obtained from this paper are the basis for developing more complex solutions for stabilizer bars in the future.

Keywords

Active hydraulic stabilizer bar anti-roll system fuzzy control rollover phenomenon

Introduction

Rollover instability is an essential issue in automotive dynamics and control. Rollover accidents often have severe consequences for users.¹ Even the lives of passengers can be threatened once a car rollover accident occurs. According to Seyedi et al., rollover accidents accounted for a higher mortality rate than other traffic accidents.² This number is higher in developed countries than in developing countries.

There are numerous causes of car rollover, which have been discussed in numerous previously published papers. Alrejjal and Ksaibati³ pointed out that vehicle size was an essential cause of this problem. If the track width is too small or the distance from the roll axis to the center of gravity (CG) is too large, the car may more easily roll over. In addition, the car’s mass is also a cause for this issue. However, it is difficult to change these values because they are related to the transport performance of the automobile. Additionally, external causes also greatly influence the problem of rolling over. According to Alrejjal et al., crosswinds often negatively affect large vehicles.⁴ Furthermore, bad weather and road conditions also cause some adverse effects on the vehicle when driving.^5,6 According to Ding et al., if bumps on the road were too large, the vehicle could be more prone to rolling over.⁷ The view of Ding et al. was accepted by some other researchers.^8,9 The above causes are only indirect reasons for the phenomenon of rolling over, while the direct cause belongs to the driver’s behavior when operating the car. When steering, a centrifugal force occurs, referred to as the automobile’s CG.¹⁰ The centrifugal force makes the car’s body tilt and causes a shift in the load between the wheels. The vehicle will roll over once the dynamic force on the side of the wheels is reduced to zero, according to Ataei et al.¹¹ The centrifugal force is proportional to the lateral acceleration, while the lateral acceleration value depends on the velocity, the steering angle, and steering acceleration. If these values are increased, the centrifugal force will increase rapidly, making it easier for the automobile to roll over.

The rollover phenomenon might be predicted through rollover index (RI)¹² or load transfer ratio (LTR)¹³ criteria. The metrics are similar because they refer to the variation of dynamic loads between the wheels. In simpler terms, the car will roll when the value of these indicators reaches 1; that is, the dynamic force at the wheel approaches zero. Therefore, defining a safe and hazardous interval for these indicators is necessary, as mentioned by Chao et al.¹⁴ Besides, several other predictions related to car body roll angle or dynamic load reduction were also used to predict rollover phenomena.^13,15,16

Instead of abruptly steering at high speed, the most effective way to limit rollover is to steer slowly and move at a low speed. However, this depends on the driver’s ability to operate in each case. Therefore, it is difficult to guarantee the operation-related problems of car users. The second solution is to use anti-roll systems on cars equipped with many modern vehicles today. For instance, air suspension¹⁷ or active suspension¹⁸ could help enhance vehicle roll safety and stability. In addition, some other anti-roll systems pointed out in Cao et al.,¹⁹ Zhao et al.,²⁰ and Gao et al.²¹ also helped to ameliorate a car’s safety when steering at high speed. Now, cars all use stabilizer bars, which act as an anti-roll device on the vehicle.²² Using a stabilizer bar is an effective solution and provides high stability. Conventional automobiles use only passive stabilizer bars made from high-elastic alloy steel.²³ The impact force of a traditional bar depends on the roll angle and the unsprung mass displacement, according to Nguyen et al.²⁴ Therefore, we cannot control them. In many dangerous cases, the forces generated by bars are relatively small and cannot meet the requirements for the automobile’s stability. So, active anti-roll bars are used to replace conventional passive stabilizer bars. According to Nguyen et al., there were two active bars: hydraulic bars and electronic bars.²⁵ Electronic stabilizer bars have been used for the last few years. They are fitted on some luxury cars like the Audi SQ7, Audi AQ7, or Porsche Panamera 971 G2, and others, while hydraulic stabilizer bars are used on many vehicles, such as the BMW 5 Series GT F07, Range Rover L405, or Toyota Land Cruiser.

The topic of control for the automotive stabilizer bar is narrow; therefore, only a few related papers have been published in the past few years. Muniandy et al.²⁶ introduced the Proportional Integral-Proportional Derivative (PI-PD) algorithm to direct hydraulic bars. The parameters of this controller were adjusted by the fuzzy algorithm. The controller that was mentioned in Muniandy et al.²⁶ was indirect. The experimental process of hydraulic stabilizer bar control was described by Dawei et al.²⁷ The robust control algorithm H_∞ applied to the active stabilizer bars fitted to trucks was shown by Tan et al.²⁸ According to Tan et al., the rolling stability of automobiles was improved by 20% when hydraulic stabilizer bars were used. According to Nguyen, the system’s performance could be further improved by using a fuzzy algorithm.²⁹ In Nguyen,²⁹ this fuzzy algorithm used only one input. In Nguyen,³⁰ it was upgraded to two inputs. According to these results, the quality of the controller was higher once the number of inputs was increased. This was also demonstrated by Nguyen et al.³¹ Regarding the fuzzy solution, choosing the optimal parameters was very difficult. The parameters, membership functions, and fuzzy rules were often chosen based on the researcher’s experience. Therefore, it took much time to conduct simulations and tests. If the number of executions were large, the results would be more accurate, and vice versa. This was considered a limitation of the optimization of fuzzy controller parameters. The fuzzy algorithms mentioned above only involve one or two simple inputs. Therefore, it cannot sufficiently reflect the influence of other dynamic behaviors.

In this paper, we propose a novel fuzzy solution to control the performance of anti-rollover systems. This algorithm provides two novel contributions. Firstly, the algorithm is designed based on considering the influence of more vehicle dynamic behaviors than previous studies.^29–31 Secondly, the fuzzy rules and membership functions are formed based on the inheritance and development of previously successfully applied rules. These rules are carefully selected and improved to increase adaptability. In addition, the vehicle oscillation is considered by a complex spatial dynamic model instead of just a half-dynamic model.²⁸

The rollover assessment and literature review are presented in the paper’s first section. The contents related to the setting of the control algorithm and dynamics model are presented in the next section. Then, the simulation results and conclusions are given in the last two sections.

Materials and methods

The model of the dynamic vehicle

The dynamic model is utilized to illustrate the rollover behavior of the automobile while steering. The author introduces using a combined model with 14 degrees of freedom (DOFs) in this paper. This model is made up of three models: a spatial oscillation model with 7 DOFs, a travel model with 3 DOFs, and a tire model with 3 DOFs. Compared with the half-oscillation model (only 4 DOFs), the spatial oscillation model more fully describes the car’s oscillations in many states. The schematic of a 7 DOFs model is illustrated in Figure 1.

Figure 1.

The vehicle model 7 DOFs.³⁰

The 7 DOFs model has five masses, including one sprung and four unsprung masses. D’Alembert’s principle is used to establish the equations that are used to describe the oscillating state.

\begin{array}{l} m_{s} \frac{d^{2}}{d t^{2}} z_{s} = \sum_{i, j = 1}^{2} {K_{i j} [z_{s} - z_{u i j} + {(- 1)}^{j + 1} t_{w i} ϕ + {(- 1)}^{i + 1} b_{i} θ] \\ + C_{i j} \frac{d}{d t} [z_{s} - z_{u i j} + {(- 1)}^{j + 1} t_{w i} ϕ + {(- 1)}^{i + 1} b_{i} θ] \end{array}

(1)

\begin{matrix} (J_{ϕ} + m_{s} h_{ϕ}^{2}) \frac{d^{2}}{d t^{2}} ϕ = \\ \sum_{i, j = 1}^{2} [{(- 1)}^{j - 1} {\begin{matrix} K_{ij} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] \\ + C_{ij} \frac{d}{dt} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] \end{matrix}} t_{wi}] \\ + {gsin ϕ + \frac{d}{dt} [v_{y} + (β + ψ) v_{x}] \cos ϕ} m_{s} h_{ϕ} \end{matrix}

(2)

\begin{matrix} (J_{θ} + m_{s} h_{θ}^{2}) \frac{d^{2}}{d t^{2}} θ = \\ \sum_{i, j = 1}^{2} {(- 1)}^{i - 1} {\begin{matrix} K_{ij} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] \\ + C_{ij} \frac{d}{dt} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] \end{matrix}} b_{i} \end{matrix}

(3)

\begin{matrix} m_{uij} \frac{d^{2}}{d t^{2}} z_{uij} = K_{Tij} (z_{rij} - z_{uij}) - K_{ij} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] \\ - C_{ij} \frac{d}{dt} [z_{s} - z_{uij} + {(- 1)}^{j + 1} t_{wi} ϕ + {(- 1)}^{i + 1} b_{i} θ] + {(- 1)}^{j} F_{AHSBij} \end{matrix}

(4)

The 7 DOFs model’s unknowns are represented by equations (1)–(4), while the nonlinear motion model describes the other unknowns such as ψ, v_y, and v_x. According to Figure 2, the motion model of the car has 3 DOFs. This is a nonlinear dynamics model, which fully considers the forces in all four wheels. The equations describing the motion of a car are written in the following form:

\begin{matrix} (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d}{dt} v_{x} = \sum_{i, j = 1}^{2} (F_{xij} \cos δ_{ij} - F_{yij} \sin δ_{ij}) \\ - F_{1} + (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) v_{y} \frac{d}{dt} (β + ψ) \end{matrix}

(5)

\begin{matrix} (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) \frac{d}{dt} v_{y} = \sum_{i, j = 1}^{2} (F_{xij} \sin δ_{ij} + F_{yij} \cos δ_{ij}) \\ - F_{2} - (m_{s} + \sum_{i, j = 1}^{2} m_{uij}) v_{x} \frac{d}{dt} (β + ψ) \end{matrix}

(6)

\begin{matrix} J_{ψ} \frac{d^{2}}{d t^{2}} ψ = \sum_{i, j = 1}^{2} [{(- 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}) b_{i} + F_{i} c_{i} - M_{zij}] \end{matrix}

(7)

where:

β = \arctan \frac{v_{y}}{v_{x}}

(8)

Figure 2.

Vehicle model 3 DOFs.

In equations (5)–(7), there are three unknowns, including M_z, F_x, and F_y. These are the moment and force components of the tire, which are measured by the tire model. In this research, the Pacejka complex tire model is utilized in order to describe the elastic deformation of the tires fully. According to Nguyen et al.,³¹ these forces are determined by equations (9)–(11).

F_{x} = D_{x} \sin [C_{x} artan (B_{x} κ_{x})] + S_{vx}

(9)

F_{y} = D_{y} \sin [C_{y} artan (B_{y} κ_{y})] + S_{vy}

(10)

M_{z} = D_{z} \sin [C_{z} artan (B_{z} κ_{z})] + S_{vz}

(11)

The specific calculation process for the Pacejka tire model should be referred to in Lugner et al.³² The hydraulic actuator of an active anti-roll bar generates the impact force, F_AHSB. The value of this force depends on the torque of the output shaft of the bar and r_b (the length of the arm).

F_{AHSB} = \frac{D_{m} Δ P}{r_{b}}

(12)

The pressure difference between the chambers causes the torque on the output shaft of the stabilizer bar. The relationship between them is described by equations (13) and (14).

J_{m} \frac{d^{2}}{d t^{2}} φ_{m} + B_{m} \frac{d}{dt} φ_{m} = D_{m} Δ P - T_{r}

(13)

D_{m} \frac{d}{dt} φ_{m} + k_{ce} Δ P + \frac{V}{4 β_{m}} \frac{d}{dt} Δ P = k_{qi} x_{sv}

(14)

Inside the hydraulic actuator is servo valves. When current is supplied to a hydraulic actuator, the servo valves move and cause a pressure change between the compartments. The displacement of the servo valves is described by equation (15).

\frac{d}{dt} x_{sv} τ + x_{sv} = k_{sv} i (t)

(15)

Control system

Control signal i(t) is the output of the controller. Classical conventional controllers only apply to simple linear oscillation systems, or only a few specific car oscillation states can be determined. Identifying and predicting vehicle rollover oscillation is necessary to improve the performance of anti-roll systems. Therefore, the author used the fuzzy method to control the active anti-roll bars.

Fuzzy algorithms are applied to many fields in life, such as control systems, intelligent computation, accurate forecasting, financial analysis, etc. Fuzzy algorithms are powerful in fields related to intelligent control for automotive systems.

The number of inputs to the fuzzy controller is closely related to the control system’s quality. Usually, only one or two input signals were used.^29,30 In this paper, the author uses three input signals for a fuzzy controller, considering a new point of the paper. Three input signals to the system include unsprung mass displacement, roll angle, and the rollover index. All three parameters are interdependent; for instance, as the roll angle increases, the displacement increases, which leads to an increase in the rollover index. However, their increases are not proportional to each other. In the vehicle situation where active stabilizer bars are used, the bars exert a force to reduce the unsprung mass displacement, that is, reduce the rollover index and roll angle. However, the displacement and rollover index values can be reduced more strongly than the roll angle. This reason is evidence to reinforce the author’s point about using three input signals.

Consider a proposition with three inputs and one output satisfying the condition (16).

If X_{1} is A_{1} and X_{2} is A_{2} and X_{3} is A_{3}, then γ = B

(16)

The membership function of the system is determined according to fuzzy math terms, as shown in (17).

\begin{matrix} μ_{R} = (x_{1}, x_{2}, x_{3}, y) = MIN [μ_{R} (x_{1}, y); μ_{R} (x_{2}, y); μ_{R} (x_{3}, y)] \\ = MIN [\min (α_{1}; μ_{B} (y)); \min (α_{2}; μ_{B} (y)); (α_{3}; μ_{B} (y))] \\ = MIN [\min (α_{1}; α_{2}; α_{3}); μ_{B} (y)] \end{matrix}

(17)

The weighted average (WTAVER) formula is used to calculate the output of the fuzzy model, like (18).

y' = \frac{\sum_{i = 1}^{n} α_{i} y_{i}}{\sum_{i = 1}^{n} α_{i}} = \frac{α_{1} y_{1} + α_{2} y_{2} + α_{3} y_{3} + . . . + α_{n} y_{n}}{α_{1} + α_{2} + α_{3} + . . . + α_{n}}

(18)

Where:

y_{i} = g_{i} (x_{1}, x_{2}, x_{3})

(19)

α_{i} = MIN (α_{i, 1}, α_{i, 2}, α_{i, 3})

(20)

The output signal needs to be large enough to control the bar’s operation in the most stable way. If the output signal is too small, the system efficiency is low. Conversely, if this value is too high, it can cause some adverse effects, such as those related to inertia. Besides, the system’s response needs to be fast and suitable to improve the controller’s stability. Based on these criteria and the author’s experience through previous simulation studies, three membership functions for fuzzy systems are proposed, as illustrated in Figure 3. The membership functions are divided into five steps, while the fuzzy rules are divided into seven levels. As shown in Figure 3, two types of functions are used, including the Gaussian membership function (GAUSSMF) and trapezoidal membership function (TRAPMF). The value of the membership degree is determined by equations (21) for GAUSSMF and (22) for TRAPMF. The fuzzy rules, fuzzy surfaces, and membership functions established in this study are different from Nguyen.³³ Several optimal soft computing methods to improve the performance of automotive mechatronic systems have been proposed in Yıldız³⁴ and Bingül and Yıldız.³⁵ These methods all provide high performance and can be used as a basis for developing control algorithms for active stabilizer bars.

μ (x) = e^{- \frac{1}{2} {(\frac{x - c}{σ})}^{2}}

(21)

μ (x) = {\begin{matrix} 0 x \leq a \\ \frac{x - a}{b - a} a \leq x < b \\ 1 b \leq x \leq c \\ \frac{d - x}{d - c} c \leq x < d \\ 0 x \geq d \end{matrix}

(22)

Figure 3.

Membership functions.

The symbols a, b, c, and d are referred to in Figure 4.

Figure 4.

TRAPMF.

Fuzzy rules are illustrated based on many situations. The prediction rate will be more accurate if this number is significant, but it isn’t effortless. In contrast, it is simpler to use a small number of situations to make predictions, but the accuracy is not good. Based on the author’s point of view, there are 125 situations used to build fuzzy rules in this study (Table 1). Fuzzy surfaces are illustrated in Figure 5 with three representative graphs. After the controller design is complete, calculations and simulations should continue.

Table 1.

Fuzzy rules.

Input (1st)	Input (2nd)	Input (3rd)	Output	Input (1st)	Input (2nd)	Input (3rd)	Output
PS	NL	NL	NE	PL	NE	PS	PO
PS	NL	NS	NE	PL	NE	PL	PL
PS	NL	NE	NE	PL	PS	NL	PO
PS	NL	PS	NE	PL	PS	NS	PO
PS	NL	PL	PO	PL	PS	NE	PO
PS	NS	NL	NE	PL	PS	PS	PL
PS	NS	NS	NE	PL	PS	PL	PL
PS	NS	NE	NE	PL	PL	NL	PO
PS	NS	PS	PO	PL	PL	NS	PO
PS	NS	PL	PO	PL	PL	NE	PL
PS	NE	NL	NE	PL	PL	PS	PL
PS	NE	NS	NE	PL	PL	PL	VPL
PS	NE	NE	PO	NL	NL	NL	VNL
PS	NE	PS	PO	NL	NL	NS	NL
PS	NE	PL	PO	NL	NL	NE	NL
PS	PS	NL	NE	NL	NL	PS	NE
PS	PS	NS	PO	NL	NL	PL	NE
PS	PS	NE	PO	NL	NS	NL	NL
PS	PS	PS	PO	NL	NS	NS	NL
PS	PS	PL	PL	NL	NS	NE	NE
PS	PL	NL	PO	NL	NS	PS	NE
PS	PL	NS	PO	NL	NS	PL	NE
PS	PL	NE	PO	NL	NE	NL	NL
PS	PL	PS	PL	NL	NE	NS	NE
PS	PL	PL	PL	NL	NE	NE	NE
NS	NL	NL	NL	NL	NE	PS	NE
NS	NL	NS	NL	NL	NE	PL	NE
NS	NL	NE	NE	NL	PS	NL	NE
NS	NL	PS	NE	NL	PS	NS	NE
NS	NL	PL	NE	NL	PS	NE	NE
NS	NS	NL	NL	NL	PS	PS	NE
NS	NS	NS	NE	NL	PS	PL	PO
NS	NS	NE	NE	NL	PL	NL	NE
NS	NS	PS	NE	NL	PL	NS	NE
NS	NS	PL	NE	NL	PL	NE	NE
NS	NE	NL	NE	NL	PL	PS	PO
NS	NE	NS	NE	NL	PL	PL	PO
NS	NE	NE	NE	NE	NL	NL	NL
NS	NE	PS	NE	NE	NL	NS	NE
NS	NE	PL	PO	NE	NL	NE	NE
NS	PS	NL	NE	NE	NL	PS	NE
NS	PS	NS	NE	NE	NL	PL	NE
NS	PS	NE	NE	NE	NS	NL	NE
NS	PS	PS	PO	NE	NS	NS	NE
NS	PS	PL	PO	NE	NS	NE	NE
NS	PL	NL	NE	NE	NS	PS	NE
NS	PL	NS	NE	NE	NS	PL	PO
NS	PL	NE	PO	NE	NE	NL	NE
NS	PL	PS	PO	NE	NE	NS	NE
NS	PL	PL	PO	NE	NE	NE	NE
PL	NL	NL	NE	NE	NE	PS	PO
PL	NL	NS	NE	NE	NE	PL	PO
PL	NL	NE	NE	NE	PS	NL	NE
PL	NL	PS	PO	NE	PS	NS	NE
PL	NL	PL	PO	NE	PS	NE	PO
PL	NS	NL	NE	NE	PS	PS	PO
PL	NS	NS	NE	NE	PS	PL	PO
PL	NS	NE	PO	NE	PL	NL	NE
PL	NS	PS	PO	NE	PL	NS	PO
PL	NS	PL	PO	NE	PL	NE	PO
PL	NE	NL	NE	NE	PL	PS	PO
PL	NE	NS	PO	NE	PL	PL	PL
PL	NE	NE	PO

Figure 5.

Fuzzy surface.

Results and discussion

Simulation information

Simulation should be done to evaluate the quality of the control system. The specifications of the simulated car are referred to in Table 2.

Table 2.

Automotive specification.

Description	Unit	Value	Notation
Time coefficient	s	5 × 10⁻³	τ
Distance from the center of gravity to axles	m	1.25/1.68	b
Viscous friction coefficient	Nmsrad⁻¹	10	B _m
Flow per revolution	m³rad⁻¹	1.6 × 10⁻⁵	D _m
Distance from the center of gravity to roll axis	m	0.56	h _φ
Distance from the center of gravity to pitch axis	m	0.54	h _θ
Inertia moment of the hydraulic actuator	kgm²	3.5	J _m
Inertia moment (longitudinal)	kgm²	690	J _φ
Inertia moment (lateral)	kgm²	2810	J _θ
Inertia moment (vertical)	kgm²	2785	J _ψ
Total flow pressure coefficient	m⁵N⁻¹s⁻¹	4 × 10⁻¹¹	k _ce
Valve flow gain coefficient	m²s⁻¹	2 × 10⁻²	k _qi
Servo valve coefficient	mA⁻¹	3 × 10⁻²	k _sv
Sprung mass	kg	1780	m _s
Unsprung mass	kg	48	m _u
Half of the track width	m	0.725/0.720	t _w
Trapped oil volume	m³	1 × 10⁻³	V
Actuator modulus of the oil	Nm⁻²	6 × 10⁶	β _m

The input to this work is the driver’s control, such as steering angle and motion speed. The output of the simulation problem is the values related to the automobile’s stability, including the roll angle, the rollover index, and the force of wheels. In this study, we propose to use the sine steering angle (the first case) and the fish-hook steering angle (the second case) as the input parameters of the dynamics model. The sinusoidal steering angle (Figure 6) depicts the single-lane change of an automobile, while the fish-hook steering angle (Figure 6) shows sudden steering to avoid an obstacle while moving. These types of steering have a reasonably large amplitude and acceleration, which can cause rollovers when traveling at high speeds.

Figure 6.

Steering angle.

There are three situations mentioned in the simulation for each case, corresponding to the car’s speed values: v₁ = 65 km/h, v₂ = 75 km/h, and v₃ = 85 km/h. To make it easier to compare results, the author simulated four scenarios in each situation. In the first scenario, the car does not have any anti-roll bars. Passive stabilizer bars are equipped at rear and front axles of the vehicle in the second scenario. In the third scenario, the automobile uses active anti-roll bars directed by a conventional PID method (PID is a popular control method for automotive mechatronic systems and practical industrial applications. This algorithm has many advantages, such as a simple design process, reasonable cost, fast response speed, high reliability, and high systematicity. In this work, the parameters of the PID controller are determined by a genetic algorithm, which is formulated in seven stages: population assessment, object encoding, fitness finding, crossover, mutation, convergence evaluation, and decoding. The algorithm aims to find the ideal coefficients for decreasing the roll angle value. This algorithm is only applicable to the SISO system.). Finally, a fuzzy control solution for hydraulic anti-roll bars is used in the remaining scenario.

Simulation results

The calculation results are shown in the order corresponding to the cases.

The first case

The first situation

In the first case, the car travels at an average speed of 65 km/h. The variation of the roll angle with the simulation time is illustrated in Figure 7. According to this finding, the roll angle value within the first second is zero because the steering angle is zero. From that time, the roll angle increases (in the positive direction) and peaks at 7.25°, corresponding to the first scenario (the automobile does not use any anti-roll bars). The peak value of the second scenario (Passive) is slightly lower, at only about 6.90°. Meanwhile, the maximum roll angle value when using the active stabilizer bars is only 6.37° and 5.92° for the Active with PID and Active with Fuzzy scenarios, respectively.

Figure 7.

Roll angle (the first situation) – first case.

After reaching the peak, the roll angle value gradually decreases to zero and increases rapidly in the opposite direction (the second phase of the steering angle). Although the steering angle is sinusoidal (the first and second phases are similar but opposite in the direction), the maximum roll angle in the latter phase is smaller than in the previous phase. This happens because of the nonlinear deformation of the tire, which is described through the Pacejka tire model.

The body roll angle is the cause of the change in the dynamic force between two wheels on an axle. This change is depicted in Figure 8 by four graphs corresponding to four scenarios. The dynamic force change is most significant when the car does not use stabilizer bars (Figure 8(a)). The difference between the biggest value (F_z₁₂) and the smallest value (F_z₁₁) of the front wheels is up to 6858.93 N, while the value of rear wheels (F_z₂₂ and F_z₂₁) is 6411.77 N. The more significant the difference between values, the greater the change in dynamic force. Rollover instability can occur if the dynamic force of any wheel is declined to zero. In this scenario, the smallest dynamic force value belongs to the left rear wheel, reaching 1072.18 N. This value is sufficient for the wheel to interact well with the road. The difference in the dynamics force in the second situation (Figure 8(b)) is smaller than in the first, with only 5426.16 N in front and 5031.79 N in the rear. Because the difference is negligible, the wheel’s dynamic force reduction is also smaller. The smallest force value of the wheel in position (21) is only 1761.91 N when the automobile uses mechanical stabilizer bars.

Figure 8.

Vertical force (the first situation) – first case: (a) None, (b) Passive, (c) Active with PID, and (d) Active with Fuzzy.

According to the results obtained from Figure 8(c) and (d), the dynamic force change is less. However, this change goes through three stages instead of the two that resulted from the first two scenarios. The important reason for this is the significant effect of the inertia force caused by the active bar. However, this problem does not adversely affect the vehicle’s roll stability. The minimum force values of the rear wheel are 2608.73 N (Active with PID) and 3355.58 N (Active with Fuzzy). These ideal numbers represent the car’s safety against the risk of rolling over. After finishing the steering, the steering angle comebacks to the zero position. This causes the roll angle to decrease to zero suddenly and eliminates the difference in forces of the wheels. For the first two scenarios, the convergence of dynamic forces is faster. While the convergence of Active with PID scenarios is quite long, the fuzzy algorithm makes the active stabilizer bar work more sensitively, thereby helping to converge these values earlier than the PID algorithm.

The rollover index provides a clearer description of the risk of rolling over when the automobile is steering at high speed. According to Figure 9, the maximum vehicle rollover index, in this case, is not extensive. Their peak values are only 0.75, 0.59, 0.39, and 0.22, respectively, for four scenarios: None, Passive, Active with PID, and Active with Fuzzy. In general, there is not any possible danger when the car is driving to change lanes at a normal speed, v₁ = 65 km/h. So, increasing the velocity value is necessary to investigate the car’s oscillation in more dangerous situations.

Figure 9.

The rollover index (the first situation) – first case.

The second situation

The second situation simulates the car oscillation at a higher speed, v₂ = 75 km/h. A sound prediction is made that the output values of the simulation problem will vary more strongly than in the first situation.

Figure 10 illustrates the change of roll angle in two oscillation phases when the car steers at v₂. In this situation, the roll angle change trend is nearly identical to that in the first situation, while its value is more significant. The highest point of roll angle is 8.35° for the scenario without any stabilizer bar equipped with the automobile. Second place still belongs to the Passive scenario with 7.95°, 0.4° less than None scenario. Once the hydraulic anti-roll bars are utilized, the roll angle peak value can be reduced even more, to only 7.35° and 6.90° for the other two scenarios. Compared to the first scenario, the car’s maximum roll angle is declined by up to 1.45° when the fuzzy method is applied to direct the stability bars. This change is significant and can help improve the automobile’s roll stability when moving at dangerous speeds.

Figure 10.

Roll angle (the second situation) – first case.

The change in dynamic force in the second situation is greater than in the first situation (Figure 11). This is because the automobile is traveling at a higher velocity, v₂ > v₁. The difference in the front wheel force can be as high as 7901.22 N once the automobile is not using stability bars (Figure 11(a)), while the difference at the rear wheels is also up to 7385.12 N. The significant difference in vertical force results from a severe reduction in the dynamic forces of the wheels. The smallest force value of the wheel at position (21) is only 585.50 N, which is close to the hazard threshold. While the minimum value of the second scenario (Passive) is 1378.48 N, which is 792.98 N higher than the other scenario. It is worth noting that the active bar can help enhance the stability of the car better by balancing the load between two wheels. According to the results from Figure 11(c) and (d), the smallest force value of the rear wheel is only 2260.22 N for the Active with PID scenario and 3073.33 N for the other scenario. When comparing the two control algorithms, we find that the fuzzy algorithm helps to converge the values more effectively than PID.

Figure 11.

Vertical force (the second situation) – first case: (a) None, (b) Passive, (c) Active with PID, and (d) Active with Fuzzy.

The rollover index value in the second situation is more significant than in the first situation. According to the results plotted in Figure 12, the peaks of the rollover index are 0.86, 0.68, and 0.47 for the first three scenarios, respectively, while its maximum value is only 0.25 when the automobile has anti-roll bars controlled by the new fuzzy method. In this situation, the smallest peak rollover index (belonging to the Active with Fuzzy scenario) is only 29.07% of the maximum rollover index (belonging to the None scenario). The safety and stability of the car are always guaranteed if the car utilizes active bars. A warning indicates that if the vehicle lacks stabilizer bars, a rollover may occur as the steering angle or speed increases.

Figure 12.

The rollover index (the second situation) – first case.

The third situation

At even higher speeds, v₃ = 85 km/h, roll instability is more likely to occur than in the two situations mentioned above. According to the results illustrated in Figure 13, the rollover oscillation of the automobile is stable when the automobile uses stabilizer bars (both active and passive bars). However, rollover occurs when the automobile has no anti-roll bars at t = 3.428 s. The maximum roll angle value before rolling over reached 9.46°. This is the limited angle of a car that can be reached before rolling over. Meanwhile, the maximum roll angle of the remaining scenarios is 8.90°, 8.21°, and 7.78° for the first phase and 7.21°, 6.59°, and 6.15° for the second phase.

Figure 13.

Roll angle (the third situation) – first case.

The sudden reduction of rear wheel dynamics to zero causes a rollover phenomenon (Figure 14(a)). Meanwhile, the smallest dynamic force value of wheels when the car has passive stabilizer bars is only 1378.11 N (Figure 14(b)). Although this value still ensures the interaction between wheels and road when the automobile is in motion, it is still an unsafe threshold because its magnitude is not high. This problem is improved by using active anti-roll bars to replace conventional stability bars, which raises the minimum dynamic force value to 2163.78 N for the Active with PID scenario (Figure 14(c)) and 3073.59 N for the other scenario (Figure 14(d)). These values help confirm the effectiveness of hydraulic stabilizer bars.

Figure 14.

Vertical force (the third situation) – first case: (a) None, (b) Passive, (c) Active with PID, and Figure (d) Active with Fuzzy.

The RI in the last situation is the highest compared to the other three (Figure 15). This number reaches 1 when the vehicle rolls over (in the first scenario). While its value is only 0.28 for the scenario where a car uses a hydraulic bar with a new fuzzy solution. The car’s roll stability is guaranteed more effectively when using a new algorithm for the vehicle’s anti-roll system.

Figure 15.

The rollover index (the third situation) – first case.

The second case

The first situation

The second case uses a fish-hook steering angle with a larger amplitude and acceleration than the first case (Figure 6). This steering angle describes the emergency state of the car when steering suddenly to avoid an obstacle. Similar to the above case, this one still has three situations corresponding to three speed values. In each situation, four scenarios are used to compare the results obtained.

For the first situation (v₁), the change in the roll angle of the sprung mass is shown in Figure 16. In the first phase, these values rapidly increase from zero to the top, reaching 4.83 (None), 4.57° (Passive), 4.45° (Active with PID), and 4.18° (Active with Fuzzy). These numbers gradually decrease to zero and rise again strongly (in the opposite direction). The rollover occurs at time t = 3.62 s for None scenario with a limited roll angle is 9.23° (Figure 16). Once the car uses the stabilizer bars, the vehicle’s safety can be ensured at the medium speed range, v₁ = 65 km/h. Based on these findings, the roll angle’s maximum value corresponding to the remaining three scenarios is 9.24°, 8.91°, and 8.17°, respectively.

Figure 16.

Roll angle (the first situation) – second case.

The rollover of a car is more clearly illustrated by the dynamic force graphic shown in Figure 17. For the first scenario (the car without the anti-roll bar), the dynamic force value of the wheel in position (22) quickly drops to zero. However, the value of the minimum dynamic force in other situations remained stable, such as 908.58 N for the Passive scenario, 1163.71 N for the PID scenario, and 2230.59 N for the Fuzzy scenario. The car can still be guaranteed to be safe when traveling at the v₁ speed if it is equipped with passive or active anti-roll bars.

Figure 17.

Vertical force (the first situation) – second case: (a) None, (b) Passive, (c) Active with PID, and (d) Active with Fuzzy.

According to Figure 18, the value of RI approaches zero when the automobile does not have stability bars, while other values are smaller if the vehicle uses anti-roll bars. Their peak values that are obtained from the simulation are 1.00 (None), 0.79 (Passive), 0.73 (Active with PID), and 0.48 (Active with Fuzzy).

Figure 18.

The rollover index (the first situation) – second case.

The second situation

As the car’s speed increases, v₂ = 75 km/h, the rollover may occur sooner. Looking at Figure 19 more closely, we can see that this problem occurs at time t = 3.42 s for None scenario, 0.2 s earlier than the first situation. Therefore, the limited roll angle of the automobile without the stabilizer bar will be lower, at only 8.94°. With higher speed, the peak value of the roll angle increases sharply, reaching 10.78° (Passive), 10.37° (PID), and 9.57° (Fuzzy).

Figure 19.

Roll angle (the second situation) – second case.

The dynamic force on the wheel when the rollover occurs at time t = 3.42 s is zero (None scenario; Figure 20). There is still a reserve to ensure interaction between the road and wheels for the rest of the scenarios. For the Passive scenario, the minimum dynamic force value is relatively small, only 347.08 N. This number can easily approach zero if the wheel hits a bump or is affected by external forces, such as a crosswind. The vehicle’s roll stability is ensured more effectively with active anti-roll bars. If these bars are controlled by a conventional PID algorithm (Figure 20(c)), the minimum force value of the wheel (22) is 623.66 N, almost twice as high as in the Passive scenario. However, there is still a drawback to the PID scenario: the convergence is not good after the end of the steering process. This can be completely overcome by using the new fuzzy algorithm designed in this paper to replace conventional algorithms. According to Figure 20(d), the minimum force of the wheel is able to reach 1896.19 N, many times higher than the remaining scenarios. Besides, the convergence of values is fast.

Figure 20.

Vertical force (the second situation) – second case: (a) None, (b) Passive, (c) Active with PID, and (d) Active with Fuzzy.

The rollover index helps assess a vehicle rollover risk more effectively. According to Figure 21, the maximum value of the roll index can be as high as 0.92 if the vehicle uses passive stabilizer bars. This value drops slightly to only 0.85 once the PID algorithm controls the anti-roll bars. In particular, this number declines sharply to only 0.55 if we apply the new fuzzy algorithm to the anti-roll bars. The fuzzy algorithm can increase the effectiveness of controlling the active stabilizer bars.

Figure 21.

The rollover index (the second situation) – second case.

The third situation

The dangerous rollover is simulated in the third high-speed situation, v₃ = 85 km/h. In this situation, rollover occurs in two scenarios: Passive and None. At time t = 3.34 s, the car rolls over if it does not have any anti-roll bars. The vehicle can achieve a limited roll angle before rolling over is 8.79°. If the car only uses a mechanical anti-roll bar, the rollover occurs at time t = 3.60 s with a limited roll angle of 10.79°, as shown in Figure 22. The active stabilizer bar helps to ensure safety to a high degree. Therefore, the maximum roll angle when using the active stabilizer bar can be up to 11.75° (PID) and 10.94° (Fuzzy).

Figure 22.

Roll angle (the third situation) – second case.

The decrease in the dynamic force causes the car to roll over. According to Figure 23(a) and (b), these values decrease to zero for both scenarios: Passive and None. For the scenario using the active stabilizer bar with the PID method, the minimum value of the dynamic force is only 180.07 N. This is a minimal value, which warns of a possible rollover at any time (Figure 23(c)). A well-founded prediction is made that if the car’s velocity is increased by 5 km/h, a rollover will occur in this scenario. However, the automobile’s roll stability is always ensured when it uses active anti-roll bars directed with the new fuzzy set solution. According to Figure 23(d), the minimum force value can be up to 1558.89 N. This is a safe threshold that prevents rolling over even at higher speeds.

Figure 23.

Vertical force (the third situation) – second case: (a) None, (b) Passive, (c) Active with PID, and (d) Active with Fuzzy.

Finally, a description of the risk of rollover is shown in Figure 24. The value of the rollover index reaches 1 when rollover occurs (Passive and None). In terms of using the active anti-roll bar utilized with the PID algorithm, the rollover index value can reach 0.96, while this figure is only 0.64 for the other scenario. These results show the superiority of the new fuzzy algorithm established in this study when applied to active hydraulic bar control.

Figure 24.

The rollover index (the third situation) – second case.

If the input differs, the output will also differ. Overall, the active anti-roll bar helps to ensure the automobile’s roll stability in many situations. The values related to roll angle, rollover index, and dynamic load change are smaller when cars use hydraulic anti-roll bars utilized by the novel fuzzy algorithm. The results obtained from the simulation process are listed in Table 3 (the first case) and Table 4 (the second case).

Table 3.

Simulation results – first case.

Objects	v ₁ = 65 km/h		v ₂ = 75 km/h		v ₃ = 85 km/h
	Value	Percent	Value	Percent	Value	Percent
Maximum roll angle (°)
Fuzzy	5.92	81.66	6.90	82.63	7.78	82.24
PID	6.37	87.86	7.35	88.02	8.21	86.79
Passive	6.90	95.17	7.95	95.21	8.90	94.08
None	7.25	100	8.35	100	9.46	100
Minimum vertical force (N)
Fuzzy	3355.58	312.97	3073.33	524.91	3073.59
PID	2608.73	243.31	2260.22	386.03	2163.78
Passive	1761.91	164.33	1378.48	235.44	1378.11
None	1072.18	100	585.50	100	0
Rollover index
Fuzzy	0.22	29.33	0.25	29.07	0.28	28.00
PID	0.39	52.00	0.47	54.65	0.49	49.00
Passive	0.59	78.67	0.68	79.07	0.76	76.00
None	0.75	100	0.86	100	1	100

Table 4.

Simulation results – second case.

Objects	v ₁ = 65 km/h		v ₂ = 75 km/h		v ₃ = 85 km/h
	Value	Percent	Value	Percent	Value	Percent
Maximum roll angle (°)
Fuzzy	8.17	88.52	9.57	107.05	10.94	124.46
PID	8.91	96.53	10.37	116.00	11.75	133.67
Passive	9.24	100.11	10.78	120.58	10.79	122.75
None	9.23	100.00	8.94	100.00	8.79	100.00
Minimum vertical force (N)
Fuzzy	2230.59		1896.19		1558.89
PID	1163.71		623.66		180.07
Passive	908.58		347.08		0
None	0		0		0
Rollover index
Fuzzy	0.48	48.00	0.55	55.00	0.64	64.00
PID	0.73	73.00	0.85	85.00	0.96	96.00
Passive	0.79	79.00	0.92	92.00	1	100.00
None	1	100.00	1	100.00	1	100.00

Conclusions

Rollover accidents lead to dire consequences for passengers. In this study, the author proposes using the active anti-roll bar to limit rollover. A new combination of a nonlinear travel dynamics model, the Pacejka tire model, and a spatial oscillation model is made to describe the oscillation of a car when rolling over. The fuzzy control solution with many inputs for the anti-roll bar is determined based on the views related to vehicle rollover stability, which is the second novelty of the paper.

This study uses a numerical simulation method with two cases. The simulation input includes the steering angle and speed of the car, and output is the RI and the car roll angle. According to the study’s findings, the output values decrease sharply when using active stabilizer bars, compared with mechanical stabilizer bars or no anti-roll bars. Besides, the interaction between the road and wheels is always ensured more effectively when using a new fuzzy set solution to the stability bars.

This paper’s results help demonstrate the efficiency of the hydraulic bar, which is controlled by the new fuzzy algorithm. However, this algorithm still has some disadvantages, such as that the convergence is not high (although better than the PID algorithm), the influence of inertia has not been eliminated, and others. This problem can be overcome by combining many other intelligent control algorithms with the fuzzy algorithm to improve the system performance. Additionally, conducting some related experiments is necessary to determine the influence of external factors fully.

Footnotes

Appendix

Notation

θ	Pitch angle	rad
ψ	Yaw angle	rad
φ	Roll angle	rad
δ	Steering angle	rad
ϕ _m	Motor shaft rotation angle	rad
ΔP	Differential pressure	Nm⁻²
F _AHSB	Hydraulic stabilizer bar force	N
F _x	Wheel longitudinal force	N
F _y	Wheel lateral force	N
i(t)	Control signal	A
M _z	Wheel moment	Nm
T _r	Resistance moment	Nm
v _x	Longitudinal velocity	ms⁻¹
v _y	Lateral velocity	ms⁻¹
x _sv	Servo valve displacement	M
z _s	Sprung mass displacement	M
z _u	Unsprung mass displacement	M
β	Heading angle	rad
C	Damper coefficient	Nsm⁻¹
K	Spring coefficient	Nm⁻¹
K _T	Tire coefficient	Nm⁻¹

Handling Editor: Sharmili Pandian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tuan Anh Nguyen

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Input (1st)	Input (2nd)	Input (3rd)	Output	Input (1st)	Input (2nd)	Input (3rd)	Output
PS	NL	NL	NE	PL	NE	PS	PO
PS	NL	NS	NE	PL	NE	PL	PL
PS	NL	NE	NE	PL	PS	NL	PO
PS	NL	PS	NE	PL	PS	NS	PO
PS	NL	PL	PO	PL	PS	NE	PO
PS	NS	NL	NE	PL	PS	PS	PL
PS	NS	NS	NE	PL	PS	PL	PL
PS	NS	NE	NE	PL	PL	NL	PO
PS	NS	PS	PO	PL	PL	NS	PO
PS	NS	PL	PO	PL	PL	NE	PL
PS	NE	NL	NE	PL	PL	PS	PL
PS	NE	NS	NE	PL	PL	PL	VPL
PS	NE	NE	PO	NL	NL	NL	VNL
PS	NE	PS	PO	NL	NL	NS	NL
PS	NE	PL	PO	NL	NL	NE	NL
PS	PS	NL	NE	NL	NL	PS	NE
PS	PS	NS	PO	NL	NL	PL	NE
PS	PS	NE	PO	NL	NS	NL	NL
PS	PS	PS	PO	NL	NS	NS	NL
PS	PS	PL	PL	NL	NS	NE	NE
PS	PL	NL	PO	NL	NS	PS	NE
PS	PL	NS	PO	NL	NS	PL	NE
PS	PL	NE	PO	NL	NE	NL	NL
PS	PL	PS	PL	NL	NE	NS	NE
PS	PL	PL	PL	NL	NE	NE	NE
NS	NL	NL	NL	NL	NE	PS	NE
NS	NL	NS	NL	NL	NE	PL	NE
NS	NL	NE	NE	NL	PS	NL	NE
NS	NL	PS	NE	NL	PS	NS	NE
NS	NL	PL	NE	NL	PS	NE	NE
NS	NS	NL	NL	NL	PS	PS	NE
NS	NS	NS	NE	NL	PS	PL	PO
NS	NS	NE	NE	NL	PL	NL	NE
NS	NS	PS	NE	NL	PL	NS	NE
NS	NS	PL	NE	NL	PL	NE	NE
NS	NE	NL	NE	NL	PL	PS	PO
NS	NE	NS	NE	NL	PL	PL	PO
NS	NE	NE	NE	NE	NL	NL	NL
NS	NE	PS	NE	NE	NL	NS	NE
NS	NE	PL	PO	NE	NL	NE	NE
NS	PS	NL	NE	NE	NL	PS	NE
NS	PS	NS	NE	NE	NL	PL	NE
NS	PS	NE	NE	NE	NS	NL	NE
NS	PS	PS	PO	NE	NS	NS	NE
NS	PS	PL	PO	NE	NS	NE	NE
NS	PL	NL	NE	NE	NS	PS	NE
NS	PL	NS	NE	NE	NS	PL	PO
NS	PL	NE	PO	NE	NE	NL	NE
NS	PL	PS	PO	NE	NE	NS	NE
NS	PL	PL	PO	NE	NE	NE	NE
PL	NL	NL	NE	NE	NE	PS	PO
PL	NL	NS	NE	NE	NE	PL	PO
PL	NL	NE	NE	NE	PS	NL	NE
PL	NL	PS	PO	NE	PS	NS	NE
PL	NL	PL	PO	NE	PS	NE	PO
PL	NS	NL	NE	NE	PS	PS	PO
PL	NS	NS	NE	NE	PS	PL	PO
PL	NS	NE	PO	NE	PL	NL	NE
PL	NS	PS	PO	NE	PL	NS	PO
PL	NS	PL	PO	NE	PL	NE	PO
PL	NE	NL	NE	NE	PL	PS	PO
PL	NE	NS	PO	NE	PL	PL	PL
PL	NE	NE	PO

Input (1st)	Input (2nd)	Input (3rd)	Output	Input (1st)	Input (2nd)	Input (3rd)	Output
PS	NL	NL	NE	PL	NE	PS	PO
PS	NL	NS	NE	PL	NE	PL	PL
PS	NL	NE	NE	PL	PS	NL	PO
PS	NL	PS	NE	PL	PS	NS	PO
PS	NL	PL	PO	PL	PS	NE	PO
PS	NS	NL	NE	PL	PS	PS	PL
PS	NS	NS	NE	PL	PS	PL	PL
PS	NS	NE	NE	PL	PL	NL	PO
PS	NS	PS	PO	PL	PL	NS	PO
PS	NS	PL	PO	PL	PL	NE	PL
PS	NE	NL	NE	PL	PL	PS	PL
PS	NE	NS	NE	PL	PL	PL	VPL
PS	NE	NE	PO	NL	NL	NL	VNL
PS	NE	PS	PO	NL	NL	NS	NL
PS	NE	PL	PO	NL	NL	NE	NL
PS	PS	NL	NE	NL	NL	PS	NE
PS	PS	NS	PO	NL	NL	PL	NE
PS	PS	NE	PO	NL	NS	NL	NL
PS	PS	PS	PO	NL	NS	NS	NL
PS	PS	PL	PL	NL	NS	NE	NE
PS	PL	NL	PO	NL	NS	PS	NE
PS	PL	NS	PO	NL	NS	PL	NE
PS	PL	NE	PO	NL	NE	NL	NL
PS	PL	PS	PL	NL	NE	NS	NE
PS	PL	PL	PL	NL	NE	NE	NE
NS	NL	NL	NL	NL	NE	PS	NE
NS	NL	NS	NL	NL	NE	PL	NE
NS	NL	NE	NE	NL	PS	NL	NE
NS	NL	PS	NE	NL	PS	NS	NE
NS	NL	PL	NE	NL	PS	NE	NE
NS	NS	NL	NL	NL	PS	PS	NE
NS	NS	NS	NE	NL	PS	PL	PO
NS	NS	NE	NE	NL	PL	NL	NE
NS	NS	PS	NE	NL	PL	NS	NE
NS	NS	PL	NE	NL	PL	NE	NE
NS	NE	NL	NE	NL	PL	PS	PO
NS	NE	NS	NE	NL	PL	PL	PO
NS	NE	NE	NE	NE	NL	NL	NL
NS	NE	PS	NE	NE	NL	NS	NE
NS	NE	PL	PO	NE	NL	NE	NE
NS	PS	NL	NE	NE	NL	PS	NE
NS	PS	NS	NE	NE	NL	PL	NE
NS	PS	NE	NE	NE	NS	NL	NE
NS	PS	PS	PO	NE	NS	NS	NE
NS	PS	PL	PO	NE	NS	NE	NE
NS	PL	NL	NE	NE	NS	PS	NE
NS	PL	NS	NE	NE	NS	PL	PO
NS	PL	NE	PO	NE	NE	NL	NE
NS	PL	PS	PO	NE	NE	NS	NE
NS	PL	PL	PO	NE	NE	NE	NE
PL	NL	NL	NE	NE	NE	PS	PO
PL	NL	NS	NE	NE	NE	PL	PO
PL	NL	NE	NE	NE	PS	NL	NE
PL	NL	PS	PO	NE	PS	NS	NE
PL	NL	PL	PO	NE	PS	NE	PO
PL	NS	NL	NE	NE	PS	PS	PO
PL	NS	NS	NE	NE	PS	PL	PO
PL	NS	NE	PO	NE	PL	NL	NE
PL	NS	PS	PO	NE	PL	NS	PO
PL	NS	PL	PO	NE	PL	NE	PO
PL	NE	NL	NE	NE	PL	PS	PO
PL	NE	NS	PO	NE	PL	PL	PL
PL	NE	NE	PO

Rollover prevention by designing a new fuzzy set solution for automotive active stabilizer bars based on a complex dynamic model

Abstract

Keywords

Introduction

Materials and methods

The model of the dynamic vehicle

Control system

Results and discussion

Simulation information

Simulation results

The first case

The first situation

The second situation

The third situation

The second case

The first situation

The second situation

The third situation

Conclusions

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References

Input (1st)	Input (2nd)	Input (3rd)	Output	Input (1st)	Input (2nd)	Input (3rd)	Output
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PS	NE	NS	NE	PL	PL	PL	VPL
PS	NE	NE	PO	NL	NL	NL	VNL
PS	NE	PS	PO	NL	NL	NS	NL
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PS	PS	NL	NE	NL	NL	PS	NE
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PS	PS	NE	PO	NL	NS	NL	NL
PS	PS	PS	PO	NL	NS	NS	NL
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PS	PL	PL	PL	NL	NE	NE	NE
NS	NL	NL	NL	NL	NE	PS	NE
NS	NL	NS	NL	NL	NE	PL	NE
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NS	NL	PS	NE	NL	PS	NS	NE
NS	NL	PL	NE	NL	PS	NE	NE
NS	NS	NL	NL	NL	PS	PS	NE
NS	NS	NS	NE	NL	PS	PL	PO
NS	NS	NE	NE	NL	PL	NL	NE
NS	NS	PS	NE	NL	PL	NS	NE
NS	NS	PL	NE	NL	PL	NE	NE
NS	NE	NL	NE	NL	PL	PS	PO
NS	NE	NS	NE	NL	PL	PL	PO
NS	NE	NE	NE	NE	NL	NL	NL
NS	NE	PS	NE	NE	NL	NS	NE
NS	NE	PL	PO	NE	NL	NE	NE
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NS	PS	NS	NE	NE	NL	PL	NE
NS	PS	NE	NE	NE	NS	NL	NE
NS	PS	PS	PO	NE	NS	NS	NE
NS	PS	PL	PO	NE	NS	NE	NE
NS	PL	NL	NE	NE	NS	PS	NE
NS	PL	NS	NE	NE	NS	PL	PO
NS	PL	NE	PO	NE	NE	NL	NE
NS	PL	PS	PO	NE	NE	NS	NE
NS	PL	PL	PO	NE	NE	NE	NE
PL	NL	NL	NE	NE	NE	PS	PO
PL	NL	NS	NE	NE	NE	PL	PO
PL	NL	NE	NE	NE	PS	NL	NE
PL	NL	PS	PO	NE	PS	NS	NE
PL	NL	PL	PO	NE	PS	NE	PO
PL	NS	NL	NE	NE	PS	PS	PO
PL	NS	NS	NE	NE	PS	PL	PO
PL	NS	NE	PO	NE	PL	NL	NE
PL	NS	PS	PO	NE	PL	NS	PO
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