Active suppression of 5DOF vehicle via acceleration-difference-feedback and fuzzy-PID control strategy

Abstract

This study investigates active control methods for vehicles aimed at reducing passenger motion sickness. A key challenge examined is the unwanted coupling between vertical and pitch dynamics in active suspension systems, which stems from traditional velocity-based feedback control in five-degree-of-freedom (5DOF) semi-vehicle models. To address this issue, a novel control scheme combining acceleration-difference-feedback (ADF) with fuzzy proportional-integral-derivative (PID) control, termed ADF-FuzzyPID control, is introduced. Through theoretical analysis, this strategy enables effective decoupling of the vertical and pitching motions of the vehicle body. By integrating ADF with fuzzy PID control, the proposed method achieves approximate independent control over vertical and pitch angular accelerations in the 5DOF system, leading to simultaneous attenuation of both response components. Simulation outcomes confirm that the approach significantly reduces vertical acceleration while markedly improving pitch angular acceleration performance under random disturbances on Class C road conditions. Compared with a passive suspension setup, the implementation of ADF-FuzzyPID control results in reductions of 63.5%, 22%, and 90.1% in the root-mean-squared (RMS) values of seat vertical acceleration, vehicle body vertical acceleration, and pitch angular acceleration, respectively.

Keywords

active suspension acceleration-difference-feedback (ADF)nearly independent regulation ADF-FuzzyPID control strategy

1. Introduction

As intelligent driving technologies advance alongside the electric vehicle sector, cars have evolved from mere transport means to multifunctional living spaces.¹ Consequently, passenger comfort has become crucial, now ranked as a key performance factor along with safety and powertrain efficiency.² However, in scenarios like heavy stop-and-go traffic or uneven roads, repeated vertical oscillations can lead to significant dynamic coupling, causing motion sickness in over 60% of passengers.³ This issue arises from mismatches among the vestibular, visual, and proprioceptive systems.⁴ Therefore, effectively reducing these multi-axis vibrations through advanced suspension system control is essential for improving ride quality and enhancing the acceptance of intelligent vehicles.

The vehicle suspension system is crucial for enhancing ride comfort and handling dynamics. These systems are typically classified into three types, that is, passive,^5,6 semi-active,⁷ and active suspensions.^8,9 Active suspensions offer superior ride quality and stability compared to their passive and semi-active counterparts. Consequently, significant research has focused on developing control strategies for active suspension systems, which combine traditional passive elements with controllable actuators to improve overall performance. Although currently limited in commercial use, active suspension systems have strong potential for future market growth due to their favorable balance of performance and efficiency.

Vehicle suspension systems have so far used a variety of actuation technologies, such as active and semi-active suspensions with magnetorheological dampers.^10–12 Because of its ability to improve ride quality and vehicle handling, the active suspension control (ASC) system, which combines a passive suspension element with an adjustable actuator, has attracted a lot of research interest. See Ref. 6–9 for a thorough summary of previous ASC research. Furthermore,¹³ provides an overview of developments in seat-mounted suspension technology.

The vertical and lateral vibrations caused by vehicles on uneven road surfaces are recognized as the primary causes of passenger motion sickness.^14,15 Consequently, enhancing vehicle dynamics through active suspension control to minimize vertical and lateral accelerations has become vital for improving riding comfort and reducing motion sickness discomfort.^16,17

For vibrational control problems, significant progress has been made in structure dynamical systems and active suspension systems. Controls of structure systems have been addressed in Refs. 18–20, the problems of resonance^21–23 and the control of energy harvesting²⁴ was provided. Recent studies utilize a four-degree-of-freedom (4DOF) semi-vehicle model to develop advanced control strategies that effectively manage vertical and pitch angular accelerations. Proposed methods include optimal control with static output feedback,^25–28 LQG control,²⁹ robust control,³⁰ fuzzy logic-based controls,³¹ fuzzy-PID combinations,³² adaptive and feedforward techniques,^33–35 nonlinear model predictive control (MPC),³⁶ siding mode control,³⁷ active disturbance rejection control,³⁸ event-triggered mechanisms,³⁹ multi-objective optimization frameworks,⁴⁰ and fractional-order MPC approaches.⁴¹

Prior studies have mostly focused on active suspension control solutions utilizing a 4DOF semi-vehicle model, generally ignoring the function of passengers in vehicle modeling. Although it is commonly acknowledged that improving ride comfort requires lowering the vertical acceleration of the sprung mass, new research suggests that both vertical and pitch angular accelerations have a substantial effect on motion sickness. The majority of studies solely assess accelerations near the sprung mass’s center of gravity. Vertical acceleration at head level must be taken into account in order to properly treat motion sickness.⁴²

The inability of conventional 4DOF models to effectively represent or reduce occupant discomfort stems from their inability to capture passenger dynamics. This constraint is the driving force behind our research, which looks into suspension control algorithms using a five-degree-of-freedom (5DOF) semi-vehicle model that includes passenger-body contact for a more accurate assessment of ride comfort.

There has been little development in this field since few studies have used a 5DOF semi-vehicle model to study active suspension control solutions. The following is a summary of the major contributions that are pertinent to this study. A hybrid multi-objective optimization approach was presented in Ref. 43 for the design of passive suspension systems with the goal of improving dynamic performance and lowering vehicle acceleration. In order to efficiently suppress vibrations transferred to the passenger seat and chassis, Ref. 44 presents a PID control technique using fuzzy logic. In Ref. 42, a 5DOF semi-vehicle active suspension system was equipped with a H∞ static output-feedback control approach to concurrently enhance maneuverability and ride quality. Furthermore,⁴⁵ proposes a fuzzy logic optimization-based robust state feedback control technique for 5DOF semi-vehicle active suspension systems.

Motive by recent works which were shown significant limitations in managing multi-degree-of-freedom vibration coupling in vehicle body dynamics and in sufficiently enhancing ride comfort. This work establishes a closed-loop connection between the vehicle system and human response by proposing the acceleration-difference-feedback with fuzzy PID (ADF-FuzzyPID) control technique focused on passenger comfort in order to close these research gaps. The novelty of this work lies in the proposed ADF-FuzzyPID control approach, where acceleration-difference-feedback is combined with fuzzy PID control to effectively nearly decouple and independently regulate the vertical and pitch movements of a 5DOF semi-vehicle system. There are two primary ways to present this study’s contributions.

1. Dynamic decoupling technique utilizing proportional feedback of vertical acceleration difference.

This work offers a nearly decoupled method based on acceleration-difference-feedback (ADF) to reduce the strong correlation between pitch and vertical motions in automobiles. This method creates a real-time proportional acceleration-difference (AD) signal by utilizing measured vertical accelerations at the vehicle’s center of mass, front suspension, and rear suspension.

The ADF component effectively for compensates the motion coupling when pitch angular movements are superimposed on vertical motion. By adding a well calibrated proportional gain to the feedback path, the compensation signal is returned to the front and rear suspension control channels. By actively lowering interference between the vertical and pitch-angular degrees of freedom, this enables the near-decoupling of these dynamic modes. Compared to prior decoupling strategies for coupled vertical and pitch accelerations, our method offers a simpler design and easier practical implementation.^46–49

Pitch and vertical angular accelerations in vehicles can be independently controlled thanks to the efficient application of this dynamic ADF-based near-decoupling approach. By addressing the primary vehicle body vibrations at their source, vibrational energy transfer to seats and passengers is significantly reduced. This strategy effectively lessens the primary causes of occupant motion sickness by emphasizing occupant protection above vehicle-centered control.

2. Multi-objective hybrid cooperative control framework.

(1) PI control strategy for seat suspension.

The primary role of the seat suspension is to reduce medium- to low-frequency residual vibrations transmitted through the vehicle body. Given its straightforward dynamic model and requirement for high steady-state precision, a proportional-integral (PI) controller is employed. This configuration offers rapid dynamic response and eliminates static errors, ensuring improved ride quality by maintaining smoothness between the occupant and the seat during high-frequency excitations.

(2) Fuzzy PID control strategy for vehicle body suspension.

The vehicle body suspension faces significant disturbances across multiple frequency bands as well as uncertainties due to varying road surfaces. To address these challenges, a fuzzy PID control scheme is adopted.^32,45,50 This method allows real-time tuning of PID gains through fuzzy reasoning mechanisms based on inputs like vertical velocity deviations of the vehicle body. By integrating a conventional PID structure with fuzzy logic’s adaptive robustness, this controller achieves excellent vibration damping performance while maintaining stability in various unpredictable operating conditions.

The following sections comprise the structure of this study. The 5DOF semi-vehicle model utilized in this work with an active suspension system is explained in Section 2. The new ADF-FuzzyPID control technique is introduced in Section 3. The decoupling dynamics and separation in managing pitch and vertical angular accelerations are explained by two new theorems. Section 4 comprises performing numerical simulations to validate the suggested ADF-FuzzyPID control strategy’s applicability and efficiency for the stated vehicle system. Finally, some last observations round up the study.

2. Five DOF semi-vehicle model with active suspension

A 5DOF semi-vehicle model is widely used to analyze and improve ride comfort while reducing motion sickness. Its main goal is to reduce human sensitivity to vibration by addressing primary sources of discomfort. These five degrees of freedom relate directly to key comfort aspects, such as chassis vertical displacement affects overall bounce, pitch rotation reflects tilt during acceleration and braking, and front and rear axle motions capture road-induced disturbances.

To ensure computational efficiency and analytical clarity, standard simplifications are made as follows. The chassis is treated as rigid, suspension and seat systems are modeled as spring-damper elements, and the occupant is represented as a point mass. The focus remains on vertical vibration dynamics. Based on Newton’s second law, the governing equations include inertial, elastic, and damping forces, providing a clear description of how vibrations travel and attenuate from the road through the wheels, suspension, body, seat, and finally to the occupant. This approach allows effective tuning of suspension and seating parameters to minimize harmful oscillations, enhancing ride comfort and reducing motion-induced discomfort.

A vehicle with an active suspension system is modeled using a 5DOF semi-vehicle model, as shown in Figure 1. The vertical displacement of the seat mass $m_{p}$ is represented by $z_{p}$ in this model, and the position of the seat on the vehicle body is indicated by $z_{c p}$ . The variable $z_{c}$ represents the vertical displacement of the sprung mass $m_{c}$ , while $θ_{c}$ represents the pitch angle of the vehicle. Furthermore, the vertical displacement at the connection points between the front and rear suspensions and the vehicle body are represented by the variables $z_{2 f}$ and $z_{2 r}$ . $z_{1 f}$ and $z_{1 r}$ describe the vertical motion of the front and rear unsprung components. $z_{0 f}$ and $z_{0 r}$ signify the vertical excitation imposed by the road profile at the front and rear tire–road contact locations.

Figure 1.

Schematics of 5DOF semi-vehicle model.

m_1f and m_1r stand for the masses of the front and rear unsprung elements.

k_{1 f}

and

k_{1 r}

quantify the elastic resistance offered by the front and rear suspension springs.

c_{f}

and

c_{r}

characterize the energy dissipation capacity of the front and rear dampers.

k_{2 f}

and

k_{2 r}

reflect the vertical stiffness of the front and rear pneumatic tires. The geometric parameters

l_{f}

l_{r}

, and

l_{p}

denote the longitudinal distances measured from the vehicle’s center of mass to the front axle, rear axle, and occupant seat location, respectively.

U_{f}

U_{r}

, and

U_{p}

are the net electromagnetic control forces generated at the front suspension, rear suspension, and seat-mounted actuator, respectively. Table 1 lists the relevant physical characteristics and measurements.⁵¹

Table 1.

List of physical parameters.

Parameter	Description	Value	Unit
$m_{c}$	Sprung mass	690	kg
$I_{c}$	Pitch inertia	1222	kg-m²
$m_{1 f}$	Front unsrung mass	40	kg
$m_{1 r}$	Rear unsrung mass	45	kg
$m_{p}$	Mass of passenger and seat	60	kg
$k_{2 f}$	Front tire stiffness	200	kN/m
$k_{2 r}$	Rear tire stiffness	200	kN/m
$k_{1 f}$	Front spring stiffness	17	kN/m
$k_{1 r}$	Rear spring stiffness	22	kN/m
$c_{f}$	Front damping coefficient	1.5	kNs/m
$c_{r}$	Rear damping coefficient	1.5	kNs/m
$k_{p}$	Seat spring stiffness	130	kN/m
$c_{p}$	Seat damping coefficient	0.9	kNs/m
$l_{f}$	CG to front axle	1.3	m
$l_{r}$	CG to rear axle	1.5	m
$l_{p}$	CG to seat	1.4	m

The variables associated to displacement are satisfied by

z_{c p} = z_{c} - l_{p} θ_{c}, z_{2 f} = z_{c} - l_{f} θ_{c}, z_{2 r} = z_{c} + l_{r} θ_{c}

(1)

Time derivatives can be used to successively determine the variables of acceleration and velocity.

{\dot{z}}_{c p} = {\dot{z}}_{c} - l_{p} {\dot{θ}}_{c}, {\dot{z}}_{2 f} = {\dot{z}}_{c} - l_{f} {\dot{θ}}_{c}, {\dot{z}}_{2 r} = {\dot{z}}_{c} + l_{r} {\dot{θ}}_{c}

(2)

{\ddot{z}}_{c p} = {\ddot{z}}_{c} - l_{p} {\ddot{θ}}_{c}, {\ddot{z}}_{2 f} = {\ddot{z}}_{c} - l_{f} {\ddot{θ}}_{c}, {\ddot{z}}_{2 r} = {\ddot{z}}_{c} + l_{r} {\ddot{θ}}_{c}

(3)

The dynamic equations of the 5DOF semi-vehicle model can be obtained as follows for the forces, $F_{p}$ , $F_{f}$ , and $F_{r}$ , applied to the vehicle by the seat, front, and rear suspensions as shown in Figure 2, respectively.

Figure 2.

Force diagram of 5DOF semi-vehicle model.

The equation for the vertical motion of the vehicle body,

m_{c} {\ddot{z}}_{c} = F_{f} + F_{r} - F_{p}

(4)

The equation for the body pitch angle,

I_{c} \ddot{θ_{c}} = {- l_{f} F}_{f} {+ l_{r} F}_{r} {- l_{p} F}_{p}

(5)

The equation for the vertical motion of the seat system,

m_{p} {\ddot{z}}_{p} = F_{p}

(6)

Besides, the equation of motion for the unsprung mass of the front suspension,

m_{1 f} {\ddot{z}}_{1 f} = - k_{2 f} (z_{1 f} - z_{0 f}) {- F}_{f}

(7)

Equation of motion for the unsprung mass of the rear suspension,

m_{2 f} {\ddot{z}}_{1 r} = - k_{2 r} (z_{1 r} - z_{0 r}) {- F}_{r}

(8)

In this case, the variables are defined as follows. $z_{1 f}$ represents the vertical movement of the front suspension system, $z_{1 r}$ indicates the vertical movement of the rear suspension system, $z_{0 f}$ corresponds to the vertical position change of the road surface associated with the front suspension; and $z_{0 r}$ refers to the vertical position change of the road surface linked to the rear suspension.

The applying forces, $F_{p}$ , of the seat suspension, $F_{f}$ and $F_{r}$ , of the front and rear suspension systems are composed by

F_{p} = F_{p b} + U_{p}

(9)

F_{f} = F_{f b} + U_{f}

(10)

F_{r} = F_{r b} + U_{r}

(11)

The actuating forces generated by the actuator in the seat, front, and rear suspensions are indicated by the variables $U_{p}$ , $U_{f}$ , and $U_{r}$ , respectively. The directions of these forced variables are also shown in Figure 2. The passive forces of the front, rear, and seat suspension systems are denoted by $F_{p b}$ , $F_{f b}$ , and $F_{r b}$ , respectively, and represented in the following.

F_{p b} = - k_{p} (z_{p} - z_{c p}) - c_{p} ({\dot{z}}_{p} - {\dot{z}}_{c p})

(12)

F_{f b} = - k_{1 f} (z_{2 f} - z_{1 f}) - c_{f} ({\dot{z}}_{2 f} - {\dot{z}}_{1 f})

(13)

F_{r b} = - k_{1 r} (z_{2 r} - z_{1 r}) - c_{r} ({\dot{z}}_{2 r} - {\dot{z}}_{1 r})

(14)

3. Acceleration-difference-feedback with fuzzy-PID control strategy

The active control algorithm design aims to enhance the vehicle’s performance. In a five-degree-of-freedom semi-vehicle model, the vertical acceleration of the seat, the vertical acceleration of the vehicle body, and the pitch angular acceleration of the vehicle body are crucial indicators of vehicle performance. Notably, the pitch angular acceleration is caused by a force imbalance between the front and rear suspensions. Figure 3 depicts the architecture of the ADF-FuzzyPID control strategy and describes an ADF control associated with PI and fuzzy PID control approaches. The suggested approach consists of two distinct control strategy components.

Figure 3.

Architecture of ADF-FuzzyPID control strategy.

3.1. Design of active control strategy for suspensions

The total control forces exerted by the actuators in the seat, as well as in the front and rear suspensions, are represented as $U_{p}$ , $U_{f}$ , and $U_{r}$ in Equations (9) to (11). Each of these forces is decomposed into two components.

U_{p} = U_{p 1} + U_{p 2}

(15)

U_{f} = U_{f 1} + U_{f 2}

(16)

U_{r} = U_{r 1} + U_{r 2}

(17)

The terms $U_{p i}, U_{f i}, U_{r i}, i = 1, 2$ in Equations (15) to (17) are defined as follow.

(1) Decoupling control strategy based on ADF signals.

Only the currently sensed vehicle acceleration data is utilized by the ADF signals, ${Δ a}_{p} = {\ddot{z}}_{c p} - {\ddot{z}}_{c}$ , ${Δ a}_{f} = {\ddot{z}}_{2 f} - {\ddot{z}}_{c}$ , and ${Δ a}_{r} = {\ddot{z}}_{2 r} - {\ddot{z}}_{c}$ . Additionally, decoupling control techniques based on ADF control have been developed for the forces of $U_{p 1}$ , $U_{f 1}$ , and $U_{r 1}$ .

U_{p 1} = {- Ω}_{p} m_{c} ({\ddot{z}}_{c p} - {\ddot{z}}_{c}) = Ω_{p} m_{c} l_{p} {\ddot{θ}}_{c}

(18)

U_{f 1} = {- Ω}_{f} m_{c} ({\ddot{z}}_{2 f} - {\ddot{z}}_{c}) = Ω_{f} m_{c} l_{f} {\ddot{θ}}_{c}

(19)

U_{r 1} = {- Ω}_{r} m_{c} ({\ddot{z}}_{2 r} - {\ddot{z}}_{c}) = - Ω_{r} m_{c} l_{r} {\ddot{θ}}_{c}

(20)

By utilizing Equation (3), the proportional coefficients in the ADF control for the seat, front, and rear suspensions are denoted by $Ω_{p}$ , $Ω_{f}$ , and $Ω_{r}$ , respectively

(2) PI control strategy for the seat suspension.

The difference between the expected zero speed and the actual vertical speed measurement obtained from the sensors on each suspension is what defines the velocity errors at the interface between the seat, front, and rear suspension systems and the vehicle body, ${Δ e}_{p}$ ${Δ e}_{f}$ , ${Δ e}_{r}$ .

{Δ e}_{p} = 0 - {\dot{z}}_{c p}, {Δ e}_{f} = 0 - {\dot{z}}_{2 f}, {Δ e}_{r} = 0 - {\dot{z}}_{2 r}

(21)

Compared to the vehicle body, the seat’s vertical motion characteristics are rather straightforward. Because of its straightforward structure, ease of parameter tweaking, and cheap computational complexity, a PI control technique is used when the conditions for control precision and stability of seat vertical movement are met. This study employs the PI control technique for $U_{p 2}$ , which accounts for the velocity disparity at the points where the vehicle frame and seat suspension systems meet.

U_{p 2} = k_{p p} {Δ e}_{p} + k_{i p} \int {Δ e}_{p} d t

(22)

where the proportional and integral coefficients of the PI method for the seat suspension are denoted by the variables

k_{p p}

and

k_{i p}

(3) Fuzzy PID control strategies for the front and rear suspensions.

For the front and rear suspensions, two fuzzy PID control strategies are employed to simultaneously prevent vertical acceleration and regulate pitch angular acceleration. By adaptively altering the control gains, these methods can enhance the resilience and anti-disturbance characteristics of the control system. The fuzzy PID control strategy is used for $U_{f 2}$ and $U_{r 2}$ by considering the velocity differential at the points where the suspension systems and the vehicle frame meet.

U_{f 2} = k_{p f} {Δ e}_{f} + k_{i f} \int {Δ e}_{f} d t + k_{d f} {Δ \dot{e}}_{f}

(23)

U_{r 2} = k_{pr} {Δ e}_{r} + k_{i r} \int {Δ e}_{r} d t + k_{d r} {Δ \dot{e}}_{r}

(24)

The proportional, integral, and differential coefficients of the fuzzy PID algorithm for the front suspension are denoted by the variables $k_{p f}$ , $k_{i f}$ , and $k_{d f}$ , respectively. Similarly, the proportional, integral, and differential coefficients of the fuzzy PID algorithm for the rear suspension are represented by $k_{pr}$ , $k_{i r}$ , and $k_{d r}$ .

In Equations (23) and (24), the coefficients in the fuzzy PID algorithms are defined by

k_{p f} = k_{p f}^{r e f} + {Δ k}_{p f}, k_{i f} = k_{i f}^{r e f} + {Δ k}_{i f}, k_{d f} = k_{d f}^{r e f} + {Δ k}_{d f}

(25)

k_{pr} = k_{pr}^{r e f} + {Δ k}_{pr}, k_{i r} = k_{i r}^{r e f} + {Δ k}_{i r}, k_{d r} = k_{d r}^{r e f} + {Δ k}_{d r}

(26)

where the initial reference values of the control coefficients are indicated by

k_{p f}^{r e f}

k_{i f}^{r e f}

k_{d f}^{r e f}

k_{pr}^{r e f}

k_{i r}^{r e f}

, and

k_{d r}^{r e f}

Δ k_{p f}

Δ k_{i f}

Δ k_{d f}

Δ k_{pr}

Δ k_{i r}

, and

Δ k_{d r}

are the dynamic adjustment values that the fuzzy inference system produces. Figure 4 shows how the fuzzy PID control algorithms are configured for the front and rear suspensions. Based on the operational conditions of the vehicle, these dynamic adjustment values are used to update the initial reference values in real time.

Figure 4.

Configuration of fuzzy PID control algorithms.

For the front suspension, the inputs of the fuzzy inference systems are ${Δ e}_{f}$ , $Δ {\dot{e}}_{f}$ , and for the rear suspension, ${Δ e}_{r}$ , $Δ {\dot{e}}_{r}$ . Each of the input and output variables ( $Δ k_{p f}$ , $Δ k_{i f}$ , $Δ k_{d f}$ , $Δ k_{pr}$ , $Δ k_{i r}$ , and $Δ k_{d r}$ ) is separated into seven fuzzy sets, [NB, NM, NS, ZO, PS, PM, PB], where B, M, and S stand for big, medium, and small, respectively, and P, Z, and N for positive, zero, and negative.

The dynamic adjustment parameters,

Δ k_{p f}

Δ k_{i f}

Δ k_{d f}

Δ k_{pr}

Δ k_{i r}

, and

Δ k_{d r}

, are adaptively tuned based on the following fuzzy reasoning rule tables. The parameter pairs (

Δ k_{p f}

Δ k_{i f}

), (

Δ k_{d f}

Δ k_{pr}

), and (

Δ k_{i r}

Δ k_{d r}

) are presented in Tables 2–4, respectively. The membership functions for the inputs and outputs of the fuzzy inference system are depicted in Figure 5.

Table 2.

Fuzzy reasoning rule table of $Δ k_{p f}$ or $Δ k_{pr}$ .

${Δ e}_{f}$ or ${Δ e}_{r}$	$Δ {\dot{e}}_{f}$ or $Δ {\dot{e}}_{r}$
${Δ e}_{f}$ or ${Δ e}_{r}$	NB	NM	NS	Z	PS	PM	PB
NB	PB	PB	PM	PM	PS	Z	Z
NM	PB	PB	PM	PS	PS	Z	NS
NS	PM	PM	Z	PS	Z	NS	NS
Z	PM	PM	PS	Z	NS	NM	NM
PS	PS	PM	Z	NS	NS	NM	NM
PM	PS	Z	NS	NM	NM	NM	NB
PB	Z	Z	NM	NM	NM	NB	NB

Table 3.

Fuzzy reasoning rule table of $Δ k_{i p} o r Δ k_{i r}$ .

${Δ e}_{f}$ or ${Δ e}_{r}$	$Δ {\dot{e}}_{f}$ or $Δ {\dot{e}}_{r}$
${Δ e}_{f}$ or ${Δ e}_{r}$	NB	NM	NS	Z	PS	PM	PB
NB	NB	NB	NM	NM	NS	Z	Z
NM	NB	NB	NM	NS	NS	Z	NS
NS	NB	NM	NS	NS	Z	PS	PS
Z	NM	NM	NS	Z	PS	PM	PM
PS	NM	NS	Z	PS	PS	PM	PB
PM	Z	Z	PS	PS	PM	PB	PB
PB	Z	Z	PS	PM	PM	PB	PB

Table 4.

Fuzzy reasoning rule table of $Δ k_{d p}$ $o r Δ k_{d r}$ .

${Δ e}_{f}$ or ${Δ e}_{r}$	$Δ {\dot{e}}_{f}$ or $Δ {\dot{e}}_{r}$
${Δ e}_{f}$ or ${Δ e}_{r}$	NB	NM	NS	Z	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NS	Z
NS	Z	NS	NM	NM	NS	NS	Z
Z	Z	NS	NS	NS	NS	NS	Z
PS	Z	Z	Z	Z	Z	Z	Z
PM	PB	NS	PS	PS	PS	PS	PB
PB	PB	PM	PM	PM	PS	PS	PB

Figure 5.

Membership functions of input and output variables.

3.2. Process of suppressing pitch movement

The decoupled and independent control properties related to the pitch and vertical motions of the 5DOF semi-vehicle model as affected by the ADF control technique are thoroughly examined in the following section.

Theorem 1.

When the following conditions are satisfied, $Ω_{p} = α Ω_{r}$ , $α = m_{p} / m_{c}$ , $β = l_{p} / l_{r}$ , and

Ω_{f} = Ω_{r} (1 + α β) l_{r} / l_{f}

(27)

The resulting ADF control force has no effect on the vertical acceleration of the vehicle body when the ADF control forces given in Equations (18) and (20) are applied solely to the 5DOF semi-vehicle model described by Equations (1) to (14).

Proof.

Substituting Equations (9) to (17) into Equation (4) with conditions set such that $U_{p 2} = U_{f 2} = U_{r 2} = 0$ , the vertical motion equation for the 5DOF semi-vehicle model at this stage can be written as

m_{c} {\ddot{z}}_{c} = F_{f b} + F_{r b} - F_{p b} + U_{f 1} + U_{r 1} - U_{p 1}

(28)

By incorporating Equations (18) to (20) into Equation (28) and simplifying the outcome, the following expression is obtained

m_{c} {\ddot{z}}_{c} = F_{f b} + F_{r b} - F_{p b} + m_{c} {\ddot{θ}}_{c} (Ω_{f} l_{f} - Ω_{r} l_{r} - Ω_{p} l_{p})

(29)

Assuming that

Ω_{p} = (m_{p} / m_{c}) Ω_{r} = α Ω_{r}, l_{p} = β l_{r}

(30)

and substituting these relationships into Equation (29), the equation can be reformulated after algebraic manipulation as

m_{c} {\ddot{z}}_{c} = F_{f b} + F_{r b} - F_{p b} + m_{c} {\ddot{θ}}_{c} (Ω_{f} l_{f} - (1 + α β) Ω_{r} l_{r})

(31)

When the feedback coefficients follow the proportional relationship given in Equation (27), Equation (31) is lead to the following result.

m_{c} {\ddot{z}}_{c} = F_{f b} + F_{r b} - F_{p b}

(32)

The front and rear suspension control strategies based on ADF control, as explained in Equations (18) to (20), guarantee that the resulting control force obtained from ADF control strategy has no effect on the vertical movement of the vehicle body, as Equation (32) makes evident. The proof has come to an end.

Theorem 2.

The ADF control strategy outlined in Equations (18) to (20) can be used to lower the pitch angular acceleration of the vehicle body to zero for the 5DOF semi-vehicle model depicted in Equations (1) to (14) by raising the coefficient, $Ω_{r}$ , of ADF control towards infinity. The pitch and vertical movements of the vehicle body can also be independently controlled.

Proof.

Replacing Equations (9) to (11), (15) to (20), and (27) into Equation (5), we proceed to reorganize and condense the expressions in order to derive the following result.

(I_{c} + A (Ω_{r})) \ddot{θ_{c}} = B + C

(33)

In this context,

A (Ω_{r}) = Ω_{r} m_{c} l_{r} [(1 + α β) l_{f} + (1 + α β^{2}) l_{r}]

(34)

represents the tuning coefficient for the ADF control force,

A \ddot{θ_{c}}

, governed by

Ω_{r}

. The term

B = l_{r} F_{r b} - l_{f} F_{f b} - l_{p} F_{p b}

denotes the action component associated with passive driving forces, while

C = l_{r} U_{r 2} - l_{f} U_{f 2} - l_{p} U_{p 2}

signifies the action term within velocity error feedback mechanisms employing PI and fuzzy PID control strategies. Furthermore, the differential equation of motion for

θ_{c}

is obtained as follows.

By incorporating Equations (1) to (3), (12) to (14) into Equation (33), the expression can be streamlined to derive the outcome $N =$

(I_{c} + A (Ω_{r})) \ddot{θ_{c}} + M {\dot{θ}}_{c} + N θ_{c} = C + δ_{c} + δ_{t} = f (t)

(35)

where

M = l_{f}^{2} c_{f} + l_{r}^{2} (c_{r} - b^{2} c_{p})

and

N = l_{f}^{2} k_{1 f} + l_{r}^{2} (k_{1 r} - b^{2} k_{p})

are system parameters,

δ_{c} = (l_{f} c_{f} - l_{r} c_{r} - l_{p} c_{p}) {\dot{z}}_{c} + (l_{f} k_{1 f} - l_{r} k_{1 r} - l_{p} k_{p}) z_{c}

is the effective passive force from the center of mass, and

δ_{t} = - l_{f} c_{f} {\dot{z}}_{1 f} + l_{r} c_{r} {\dot{z}}_{1 r} - l_{p} c_{p} {\dot{z}}_{p} - l_{f} k_{1 f} z_{1 f} + l_{r} k_{1 r} z_{1 r} + l_{p} k_{p} z_{p}

is the effective passive force from tires and seat.

The sufficient condition for the stability of the second-order differential equation for $θ_{c}$ , as described by Equation (35) based on the Routh criterion, is that all coefficients in the characteristic differential equation are positive. In practice, the coefficient $Ω_{r}$ in the $A (Ω_{r})$ is generally assumed to be positive, resulting in $I_{c} + A (Ω_{r}) > 0$ . Furthermore, the damping coefficient $c_{p}$ in $M$ and stiffness coefficient $k_{p}$ in $N$ of the seat’s passive suspension system are significantly smaller than those of both front and rear suspension systems. Thus, positive values for $M$ and $N$ are achieved. Consequently, the stability of the system governed by Equation (35) can be effectively ensured.

The transfer function of the pitch angular acceleration $\ddot{θ_{c}}$ is given by

T (s) = \frac{s^{2}}{(I_{c} + A (Ω_{r})) s^{2} + M s + N}

(36)

The magnitude of transfer function approaches zero in the low-frequency band, when $s \to 0$ . The magnitude asymptotically approaches $1 / (I_{c} + A (Ω_{r}))$ in the high-frequency range, when $s \to \infty$ . The gain over the whole high-frequency range is immediately decreased by an increase in $A (Ω_{r})$ . This suggests that the resulting pitch angular acceleration $\ddot{θ_{c}}$ will be significantly dampened for input signals $f (t)$ that contain strong high-frequency components, such as transient surges or the first phase of a step response. The outcomes of the simulation validate it.

The natural frequency $ω_{n}$ and damping ratio $ξ$ corresponding to Equation (36) are expressed as

ω_{n} = \sqrt{N / (I_{c} + A (Ω_{r}))}, ξ = M / \sqrt{4 N (I_{c} + A (Ω_{r}))}

(37)

An increase in $A (Ω_{r})$ leads to a reduction in the system’s natural frequency $ω_{n}$ , thereby slowing down the dynamic response. Additionally, it increases the damping ratio $ξ$ , which helps suppress overshoot and results in a smoother transient response.

In conclusion, adding $A (Ω_{r})$ not only lowers the high-frequency gain of the transfer function $T (s)$ given in Equation (36), but it also makes the system less responsive while improving its damping properties. Together, these effects guarantee that the 5DOF semi-vehicle model’s peak value of the pitch angular acceleration response $\ddot{θ_{c}}$ is decreased within the time domain and that the response shows smoother changes under the influence of ADF control. It is clear that raising the feedback coefficient $Ω_{r}$ towards infinity causes the pitch angular acceleration $\ddot{θ_{c}}$ of the vehicle body to approach zero. Since the nearly decoupling between the pitch and vertical motions is achieved, the proposed ADF control strategy can independently regulate the pitch and vertical motions of the vehicle body The proof has come to an end.

Remark.

Theorem 2 is established by applying time-domain and frequency-domain response methods tailored to linear systems, aiming to evaluate how pitch angular acceleration is mitigated by raising the feedback coefficient $Ω_{r}$ . Despite the exclusion of nonlinear components in the dynamic model, the analytical outcomes align well with theoretical expectations. The validity of these findings is further supported by numerical simulations presented later. Although incorporating nonlinear dynamics, through methodologies like He’s frequency formulation,^52–54 is feasible, such an extension falls outside the present scope. Since the 5DOF semi-vehicle model adopted herein inherently neglects nonlinearities, no in-depth treatment of nonlinear behavior is provided.

4. Simulation tests and discussions on bump road and random Class C road excitation

Theorems 1 and 2 state that ADF control can successfully decouple and independently control the pitch and vertical motions of the vehicle body. In this study, the vehicle body’s vertical motion is actively controlled within the ADF control framework using PI and Fuzzy PID control techniques. As a result, the vehicle body’s pitch angular acceleration is greatly diminished.

This section performs the simulation in order to confirm the efficacy of the suggested closed-loop control system. The MATLAB-SIMULINK software environment is used to design and run the program. The integrated Runge-Kutta ode45 solver in the program is used for numerical integration. Using a predetermined tolerance value of $10^{- 6}$ , a variable step-size technique is used. Table 1 lists the vehicle system parameters used in the numerical simulations.

For greater pitching angular acceleration effectiveness, the feedback coefficient in the ADF control is set to $Ω_{r} = 10$ . $k_{p p} = 700$ and $k_{p i} = 0.1$ specify the design parameters for the PI active control of the seat suspension system. Equations (23) to (26) suggest fuzzy PID control schemes for the front and rear suspensions, whereas Tables 2–4 present the feedback gain tuning procedures. The fuzzy PID control gains for the front and rear suspension systems were initially set to the following reference values $k_{p f}^{r e f} = k_{pr}^{r e f} = 7000$ , $k_{i f}^{r e f} = k_{i r}^{r e f} = 2$ , $k_{d f}^{r e f} = k_{d r}^{r e f} = 5$ . Additionally, the control gains’ permissible dynamic adjustment ranges were established as $Δ k_{p f}, Δ k_{pr} \in [- 3000, 3000]$ ; $Δ k_{i f}, Δ k_{i r} \in [- 0.3, 0.3]$ ; $Δ k_{d f}, Δ k_{d r} \in [- 3, 3]$ .

4.1. Results on one bump road

A circular convex platform with dimensions of 5 meters in width and length and 0.1 meters in height serves as the road surface model that the vehicle system uses as its initial test input. During testing, the 5DOF semi-vehicle’s speed is set at 10 m/s, and the time lag between the front and rear axle inputs is determined to be 0.27 seconds.

When traversing a bumpy road at a steady speed, the suspension experiences dynamic conditions that are similar to an abrupt step disturbance. The pitch angular acceleration and vertical acceleration responses of the vehicle body and seat are shown in Figure 6, which shows a significant decrease in vibration levels. The technology successfully reduces acceleration disturbances, according to the collected data.

Figure 6.

Time responses seat and vehicle accelerations for one bump road.

In accordance with the theoretical analysis stated in Theorem 2, the ADF-FuzzyPID control system presented in this study significantly reduces the pitch angular acceleration ${\ddot{θ}}_{c}$ of the vehicle body when subjected to such high-frequency transient inputs, as shown in Figure 6(a). Additionally, as shown in Figure 6(b) and (c), the vertical accelerations of the seat and the vehicle body, ${\ddot{z}}_{p}$ and ${\ddot{z}}_{c}$ , exhibit improved vibration isolation characteristics due to the improved regulation of pitch motion.

4.2. Results on random Class C road excitation

This subsection examines how well the ADF-FuzzyPID active control suspension mitigates pitch and vertical angular accelerations in the 5DOF semi-vehicle model system shown in Table 1 applying a random Class C road profile.⁵⁵

The random Class C road profile is described by the following differential equations

{\dot{z}}_{0 f} (t) = 2 π f_{0} V z_{0 f} (t) + \sqrt{G_{q} (n_{0}) V} W (t)

(38)

{\dot{z}}_{0 r} (t) = {\dot{z}}_{0 f} (t - (a + b) / V)

(39)

where

z_{0 f} (t)

represents the road excitation and

z_{0 r} (t)

denotes the time lag of the rear-wheel road input relative to that of the front-wheel. The term

W (t)

signifies white noise with a zero mean value and an intensity power spectral density of 0.1. The low cut-off frequency is defined as

f_{0} = 0.011 Hz

, while

n_{0} = 0.1 m^{- 1}

indicates the reference spatial frequency. Additionally, the road roughness coefficient is given by

G_{q} (n_{0}) = 256 \times 10^{- 6} m^{3}

. In this scenario, the vehicle speed is set at

V = 40 m / s

Figure 7 shows the time response curves of the vehicle body’s vertical and pitch angular accelerations, ${\ddot{z}}_{c}$ and ${\ddot{θ}}_{c}$ , under excitation from the Class C road profile, as well as the vertical acceleration of the seat, ${\ddot{z}}_{p}$ . The following discusses the major performances.

(1) With an oscillation amplitude ranging from -0.82035 to 0.83943, the pitch angular acceleration ${\ddot{θ}}_{c}$ of the vehicle body frequently fluctuates in Figure 7(a) when the suspension system is not actively controlling it. Passengers’ heads oscillate quickly as a result of this high-frequency pitching motion, which keeps the visual system constantly concentrating and adjusting. The ADF-FuzzyPID active control suspension system, on the other hand, greatly reduces the amplitude of ${\ddot{θ}}_{c}$ and confines it to the range of -0.080156 to 0.093231. This result is in line with the theoretical assertion made in Theorem 2. Passenger head posture stays relatively consistent as a result, reducing the requirement for frequent visual recalibration and successfully stopping the harmful loop of visual fatigue and sensory signal conflict.

(2) With an amplitude ranging from -1.0381 to 1.6555, the vertical acceleration ${\ddot{z}}_{p}$ of the seat shows significant changes in Figure 7(b) as the vehicle travels over the Class C road surface without active suspension control. The waveform exhibits high-frequency oscillations that are sustained and not much attenuated. Motion sickness symptoms may worsen as a result of these ongoing dynamic disruptions, which may continually shock passengers’ internal organs and cause a gradual accumulation and exacerbation of sensory conflicts between somatic and vestibular signals. On the other hand, with the ADF-FuzzyPID active control suspension system, the amplitude of ${\ddot{z}}_{p}$ is effectively restricted within the range [-0.51713, 0.48396], exhibiting low oscillation and greatly stabilized waveform features.

Figure 7.

Time responses seat and vehicle accelerations for Class C road surface.

Motion sickness is effectively prevented by this real-time suppression of mechanical excitation, which significantly lessens the recurrent stimulation of internal organs and reduces the buildup of sensory conflict signals by over 90%.

(3) In Figure 7(c), in the absence of active control in the suspension system, the amplitude range of the ${\ddot{z}}_{c}$ acceleration waveform of the vehicle body is observed within [-1.1108, 1.0400]. This oscillatory behavior is continuously transmitted to the seat through the suspension, inducing a secondary fluctuation in the ${\ddot{z}}_{p}$ acceleration of the seat. Under the ADF-FuzzyPID active control strategy, the amplitude range of ${\ddot{z}}_{c}$ is reduced to [-0.95261, 0.79441], with no resonant oscillations present, thereby providing a smoother vibration input to the seat and effectively minimizing additional disturbances in ${\ddot{z}}_{p}$ caused by vehicle body vibrations.

Table 5 shows the root-mean-squared (RMS) values of seat and vehicle accelerations for both the ADF-FuzzyPID active control suspension system with

Ω_{r} = 10

and a passive suspension system without active control, operating at V = 40 m/s on a Class C road surface. When paired with the sensitive bandwidth versus physiological response method, the RMS value, which directly represents the intensity of vibration energy, allows for a clear evaluation of how each performance index affects the likelihood of motion sickness. The outcomes show that the ADF-FuzzyPID control technique outperforms the passive suspension system in terms of overall performance.

Table 5.

RMS values of seat and vehicle accelerations.

Control strategy	${\ddot{z}}_{p}$ (m/s²)	${\ddot{z}}_{c}$ (m/s²)	${\ddot{θ}}_{c}$ (rad/s²)
Non-active suspension	0.4922	0.3502	0.3033
ADF-FuzzyPID control ( $Ω_{r} = 10$ )	0.1795	0.2732	0.0301

The technique reduces the RMS value from 0.3502 m/s² to 0.2732 m/s², or around 22%, with regard to the vertical acceleration ${\ddot{z}}_{c}$ of the vehicle body. Additionally, the seat’s vertical displacement ${\ddot{z}}_{p}$ is significantly reduced, as seen by a 63.5% decrease in its RMS value. This implies that even with randomly fluctuating road conditions, the approach is still quite efficient in enhancing ride comfort.

The ADF-FuzzyPID control technique produces impressive results when it comes to pitch motion regulation. In other words, the algorithm’s great capacity to maintain vehicle body stability is demonstrated by the 90.1% reduction in the RMS value of pitch angular acceleration ${\ddot{θ}}_{c}$ , from 0.3033 rad/s² to 0.0301 rad/s². All of these results point to the ADF-FuzzyPID control strategy’s remarkable adaptability and control efficacy under random road excitation. It effectively reduces vehicle body vibrations and has significant robustness, making it an appropriate suspension control solution in the challenging driving conditions characteristic of actual automotive applications.

5. Conclusions

This study develops an innovative active suspension control strategy (ADF-FuzzyPID) for 5DOF semi-vehicle systems, integrating acceleration-difference-feedback (ADF) with fuzzy PID control to achieve dynamic decoupling of vertical and pitch motions. Theoretical analysis demonstrates that the proposed framework successfully decouples these coupled motions through ADF control. The ADF-FuzzyPID architecture enables nearly independent regulation of vertical and pitch angular accelerations, resulting in simultaneous suppression of both vibration modes. Numerical simulations under Class C road excitation reveal that the strategy reduces vertical acceleration of seat, and vertical and pitch angular accelerations of the vehicle body. The results provide both theoretical guidance and numerical validation for active suspension design, particularly in addressing motion sickness mitigation through multi-axis vibration control.

Several potential avenues for future research can be derived from this study. Firstly, the incorporation of road preview capabilities using MEMS sensors^52,56,57 holds promise for improving the control system’s ability to anticipate and counteract disturbances in advance. Secondly, a more effective approach to mitigating motion sickness could be achieved by adopting advanced vehicle dynamics models with multiple degrees of freedom, including 6DOF and 7DOF configurations, which warrant further investigation. Thirdly, the application of data-driven techniques, such as machine learning and reinforcement learning,⁵⁸ to refine the fuzzy rule base offers a viable path toward enhancing the adaptability of the control strategy, thereby supporting higher accuracy and greater robustness in challenging operational conditions.

Footnotes

Acknowledgments

The authors also acknowledge the support from the School of Mechanical and Electric Engineering, Sanming University.

ORCID iD

Chi-Hsin Yang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by Projects for Department of Science and Technology of Fujian Province (Grant nos. 2022HZ026025 and 2023T5001), Operational Funding of the Advanced Talents for Scientific Research (Grant no. 19YG04) of Sanming University, the Natural Science Foundation of Sanming University (Project Leader: Prof. Hai-Lian, Hong, Grant no. KD23003P), the Program for Innovative Research Team in Science and Technology in Fujian Province University, and Fujian Provincial Engineering Research Center for Modern Mechanical Design and Manufacturing Technology.

Declaration of conflicting interests

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

References

Htike

Papaioannou

Siampis

, et al. Fundamentals of motion planning for mitigating motion sickness in automated vehicles. IEEE Trans Veh Technol 2022; 71(3): 2375–2384. https://doi.org/10.1109/TVT.2021.3138722

Zheng

Shyrokau

Keviczky

. Mitigating motion sickness with optimization-based motion planning. IEEE Trans Intell Veh 2024; 9(1): 2553–2563. https://doi.org/10.1109/TIV.2023.3289854

Diels

Bos

, et al. Motion sickness in automated vehicles: principal research questions and the need for common protocols. SAE Int J Connected Autom Veh 2022; 5(2): 121–134. https://doi.org/10.4271/12-05-02-0011

Lackner

. Motion sickness: more than nausea and vomiting. Exp Brain Res 2014; 232: 2493–2510. https://doi.org/10.1007/s00221-014-4008-8

Karnopp

. Active damping in road vehicle suspension systems. Veh Syst Dyn 1983; 12(6): 291–311. https://doi.org/10.1080/00423118308968758

Sharp

Crolla

. Road vehicle suspension system design—A review. Veh Syst Dyn 1987; 16: 167–192. https://doi.org/10.1080/00423118708968877

Tseng

Hrovat . State of the art survey: Active and semi-active suspension control. Veh Syst Dyn 2015; 53: 1034–1062. https://doi.org/10.1080/00423114.2015.1037313

Wei

Han

. Study on modeling and control for a novel active suspension. Int J Automot Technol 2024; 25: 533–540. https://doi.org/10.1007/s12239-024-00042-6

Samaroo

Awan

Marimuthu

, et al. Performance investigation of active, semi-active and passive suspension using quarter car model. Algorithms 2025; 18(2): 100. https://doi.org/10.3390/a18020100

10.

Chen

. A new state observer-based vibration control for a suspension system with magnetorheological damper. Veh Syst Dyn 2021; 60(9): 3127–3150. https://doi.org/10.1080/00423114.2021.1942508

11.

Wang

Zhang

Lou

, et al. Asymptotic saturation magic formula model and time delay compensation control for a magnetorheological semi-active suspension. Veh Syst Dyn 2025; 1–21. https://doi.org/10.1080/00423114.2025.2512037

12.

Wei

Tan

, et al. Vibration control and energy-regenerative performance analysis of an energy-regenerative magnetorheological semi-active suspension. World Electr Veh J 2025; 16(8): 455. https://doi.org/10.3390/wevj16080455

13.

Al-Ashmori

Wang

. A systematic literature review of various control techniques for active seat suspension systems. Appl Sci 2020; 10(3): 1148. https://doi.org/10.3390/app10031148

14.

Kim

Yim

. Design of a suspension controller with human body model for ride comfort improvement and motion sickness mitigation. Actuators 2024; 13(12): 520. https://doi.org/10.3390/act13120520

15.

Kim

Yim

. Design of static output feedback suspension controllers for ride comfort improvement and motion sickness reduction. Processes 2024; 12(5): 968. https://doi.org/10.3390/pr12050968

16.

Feng

, et al. Time-delay compensation control and stability analysis of vehicle semi-active suspension systems. Mech Syst Signal Process 2025; 228(1): 112414. https://doi.org/10.1016/j.ymssp.2025.112414

17.

Shi

Zhang

, et al. Adaptive suspension state estimation based on IMMAKF on variable vehicle speed, road roughness grade and sprung mass condition. Sci Rep 2024; 14: 1740. https://doi.org/10.1038/s41598-023-49766-y

18.

Shen

. Intelligent Bi-State Control for the Structure with Magnetorheological Dampers. J Intell Mater Syst Struct 2003; 14: 35–42. https://doi.org/10.1177/1045389X03014001004

19.

Guo

. Neuro-fuzzy control strategy for earthquake-excited nonlinear magnetorheological structures. Soil Dyn Earthquake Eng 2008; 28: 717–727. https://doi.org/10.1016/j.soildyn.2007.10.013

20.

Lua

Masri

. Studies of the performance of particle dampers under dynamic loads. J Sound Vib 2010; 329: 5415–5433. https://doi.org/10.1016/j.jsv.2010.06.027

21.

Ghanem

Amer

, et al. Analyzing the motion of a forced oscillating system on the verge of resonance. J Low Freq Noise Vibr Act Control 2023; 42(2): 563–578. https://doi.org/10.1177/14613484221142182

22.

Amer

Gala

. Vibrational dynamics of a subjected system to external torque and excitation force. J Vib Control 2025; 31(11-12): 2086–2099. https://doi.org/10.1177/10775463241249618

23.

Amer

El-Sabaa

Moatimid

, et al. On the Stability of a 3DOF Vibrating System Close to Resonances. J Vib Eng Technol 2024; 12: 6297–6319. https://doi.org/10.1007/s42417-023-01253-4

24.

Amer

Arab

Galal

. On the influence of an energy harvesting device on a dynamical system. J Low Freq Noise Vibr Act Control 2024; 43(2): 669–705. https://doi.org/10.1177/14613484231224588

25.

Wilson

Sharp

Hassan

. The application of linear optimal control theory to the design of active automobile suspensions. Veh Syst Dyn 1986; 15: 105–118. https://doi.org/10.1080/00423118608968846

26.

Hac

. Optimal linear preview control of active vehicle suspension. Veh Syst Dyn 1992; 21: 167–195. https://doi.org/10.1080/00423119208969008

27.

Jeong

Shon

Chang

, et al. Design of static output feedback controllers for an active suspension system. IEEE Access 2022; 10: 26948–26964. https://doi.org/10.1109/ACCESS.2022.3157326

28.

Yim

. Comparative study on active suspension controllers with parameter adaptive and static output feedback control. Actuators 2025; 14(3): 150. https://doi.org/10.3390/act14030150

29.

Abut

Salkım

Demosthenous

. Performance Improvement in a vehicle suspension system with FLQG and LQG control methods. Actuators 2025; 14: 137. https://doi.org/10.3390/act14030137

30.

Sina Kuseyri

. Robust control and performance evaluation of slow and fully active electromechanical suspension systems. Aust J Mech Eng 2025; 23: 932–943. https://doi.org/10.1080/14484846.2025.2478676

31.

Yatak

Hisar

Sahin

. Fuzzy logic controller for half vehicle active suspension system: An assessment on ride comfort and road holding. Int J Automot Sci Technol 2024; 8(2): 179–187. https://doi.org/10.30939/ijastech..1372001

32.

Saruchi

Ariff

MHM

Zamzuri

, et al. Novel motion sickness minimization control via fuzzy-PID controller for autonomous vehicle. Appl Sci 2020; 10(14): 4769. https://doi.org/10.3390/app10144769

33.

Kim

Yim

. Design of parameter adaptive suspension controllers with Kalman filter for ride comfort enhancement and motion sickness reduction. Appl Sci 2025; 15(9): 4977. https://doi.org/10.3390/app15094977

34.

Kim

Yim

. Design of a suspension controller with an adaptive feedforward algorithm for ride comfort enhancement and motion sickness mitigation. Actuators 2024; 13(8): 315. https://doi.org/10.3390/act13080315

35.

Jeong

Shon

Chang

, et al. Design of virtual reference feedforward controller for an active suspension system. IEEE Access 2022; 10: 65671–65684. https://doi.org/10.1109/ACCESS.2022.3184417

36.

Ricco

Alshawi

Gruber

, et al. Nonlinear model predictive control for yaw rate and body motion control through semi-active and active suspensions. Veh Syst Dyn 2023; 62(6): 1587–1620. https://doi.org/10.1080/00423114.2023.2251615

37.

Zhang

Zhao

Huang

. Optimal sliding mode hierarchical decoupled control of vehicle semi-active suspension. Veh Syst Dyn 2025; 63(11): 2086–2116. https://doi.org/10.1080/00423114.2024.2398750

38.

Nguyenm

. Active disturbance rejection control for reducing vehicle body displacement based on an extended state observer and dynamic fuzzy techniques. J Low Freq Noise Vibr Act Control 2025; 44(2): 1256–1271. https://doi.org/10.1177/14613484251315522

39.

Jia

Pan

Liang

, et al. Event-based adaptive fixed-time fuzzy control for active vehicle suspension systems with time-varying displacement constraint. IEEE Trans Fuzzy Syst 2022; 30(8): 2813–2821. https://doi.org/10.1109/tfuzz.2021.3075490

40.

Han

Zeng

, et al. Vehicle attitude control of magnetorheological semi-active suspension based on multi-objective intelligent optimization algorithm. Actuators 2024; 13(12): 466. https://doi.org/10.3390/act13120466

41.

Liu

, et al. A fractional-order model predictive control strategy with Takagi–Sugeno fuzzy optimization for vehicle active suspension system. Fractal Fract 2024; 8(10): 610. https://doi.org/10.3390/fractalfract8100610

42.

Wei

Zhang

Cai

, et al. A new method of static output-feedback H∞ controller design for 5 DOF vehicle active suspension system. J Braz Soc Mech Sci Eng 2018; 40: 132. https://doi.org/10.1007/s40430-018-1054-3

43.

Mahmoodabadi

Safaie

Bagheri

, et al. A novel combination of particle swarm optimization and genetic algorithm for pareto optimal design of a five-degree of freedom vehicle vibration model. Appl Soft Comput 2013; 13(5): 2577–2591. https://doi.org/10.1016/j.asoc.2012.11.028

44.

Demir

Keskin

Cetin

. Modeling and control of a nonlinear half-vehicle suspension system: a hybrid fuzzy logic approach. Nonlinear Dyn 2012; 67: 2139–2151. https://doi.org/10.1007/s11071-011-0135-y

45.

Mahmoodabadi

Nejadkourki

Ibrahim

. Optimal fuzzy robust state feedback control for a five DOF active suspension system. Results Control Optim 2024; 17: 100504. https://doi.org/10.1016/j.rico.2024.100504

46.

Zhang

Min

, et al. Coupling mechanism and decoupled suspension control model of a half car. Math Probl Eng 2016; 2016: 1932107–1932113. https://doi.org/10.1155/2016/1932107

47.

Ding

Zhang

, et al. Robust adaptive backstepping sliding mode control for motion mode decoupling of two-axle vehicles with active kinetic dynamic suspension systems. Int J Robust Nonlinear Control 2020; 30: 1–24. https://doi.org/10.1002/rnc.4927

48.

Zuo

Wang

. Novel method of forecasting performance of full vehicle suspension by decoupling analysis and vibration sensing experiments. Sens Mater 2020; 32(5): 1577–1592. https://doi.org/10.18494/SAM.2020.2682

49.

Peng

Shi

. Robust decoupled sliding mode control for active suspension systems with prescribed tracking performance. Control Theory Technol 2025; 23: 310–320. https://doi.org/10.1007/s11768-025-00248-8

50.

Zhang

Long

Lin

, et al. Particle swarm optimized fuzzy proportional-integral-derivative controller-based transverse leaf spring active suspension for vibration control. J Low Freq Noise Vibr Act Control 2024; 43(2): 979–996. https://doi.org/10.1177/14613484231221953

51.

Wang

Ren

Zhou

, et al. The double-delay reducing vibration control for five-degree-of-freedom half-vehicle model in idle condition. J Low Freq Noise Vibr Act Control 2020; 39(1): 203–215. https://doi.org/10.1177/1461348419828605

52.

Cui

, et al. Nonlinear dynamics in MEMS systems: Overcoming pull-in challenges and exploring innovative solutions. J Low Freq Noise Vibr Act Control 2025; 14613484251385061. https://doi.org/10.1177/14613484251385061

53.

, et al. A modified frequency formulation for nonlinear mechanical vibrations. FACTA UNIVERSITATIS Ser Mech Eng 2025; 2(2): 197–210. https://doi.org/10.22190/FUME250526022H

54.

Feng

. A circular sector vibration system in a porous medium: A fractal-fractional model and He’s frequency formulation. FACTA UNIVERSITATIS Ser Mech Eng 2025; 23(2): 377–385. https://doi.org/10.22190/FUME230428025F

55.

Wei

. Modeling and simulation of active half-vehicle suspension based on a new output-feedback H∞ controller. Int J Control Autom Syst 2024; 22: 775–784. https://doi.org/10.1007/s12555-022-0829-6

56.

. Periodic solution of a micro-electromechanical system. FACTA UNIVERSITATIS Ser Mech Eng 2024; 22(2): 187–198. https://doi.org/10.22190/FUME240603034H

57.

Niu

Feng

. A mini-review on ancient mathematics’ modern applications with an emphasis on the old Babylonian mathematics for MEMS systems. Front Phys 2024; 12: 1532630. https://doi.org/10.3389/fphy.2024.1532630

58.

Wei

, et al. Research on stick-slip vibration suppression method of drill string based on machine learning optimization. Sound Vib 2023; 57: 97–117. https://doi.org/10.32604/sv.2023.043734