A novel semi-active control of an integrated chassis and seat quasi-zero stiffness suspension system for off-road vehicles

Abstract

This article introduces an innovative semi-active suspension system for off-road vehicles, incorporating a quasi-zero stiffness suspension to enhance driver comfort. The system includes a central pneumatic spring, double-acting pneumatic linear actuators for adjustable stiffness, and magnetorheological dampers for controlled damping. Using Lyapunov stability theory, a static output feedback control law is designed to address uncertainties in road profiles, driver body parameters, and actuator saturation, relying on measured variables as feedback. Performance constraints focus on driver head acceleration, suspension displacement, and dynamic tire load. The control law problem is converted into a convex optimization problem with linear matrix inequalities. The new semi-active QZSS system is theoretically tested via Matlab simulations and validated through hardware-in-the-loop testing. Comparative analysis between passive QZSS and the new semi-active QZSS on different roads, and vehicle speeds closely aligns with simulation results, with minor disparities attributable to network discrepancies and signal interruptions. Impressively, the semi-active QZSS system reduces driver head vertical acceleration between 26.53 and 35.0% compared to passive air QZSS, showing potential for significantly improving ride comfort, reducing vibration transfer, and enhancing driver wellbeing with minimal energy requirement.

Keywords

Quasi-zero-stiffness suspension off-road vehicles integrated suspension system semi-active suspension static output feedback controller

1. Introduction

Off-road vehicles, designed with robust suspensions to endure harsh conditions, often expose drivers to severe vibrations, particularly in industries such as mining, construction, and agriculture (Ning et al., 2016). These vibrations pose health risks, contributing to musculoskeletal issues, reduced productivity, and increased social costs (Qiao et al., 2022; Tiemessen et al., 2007). ISO 2631-1 1997 identifies a negative correlation between ride comfort and vertical acceleration, with low-frequency vertical vibrations (3–10 Hz) being particularly detrimental to seated individuals (Zhao and Wang, 2019). The resonant frequency of the human spine and head (around 4 Hz) amplifies these effects (Randall et al., 1997). To address these concerns, researchers have explored methods to enhance vibration isolation for off-road vehicle drivers. Some focused on improving the vehicle’s primary suspension (Huang et al., 2015; Salmani et al., 2022; Wen et al., 2017; Zhao et al., 2016), while others emphasized seat suspension as a complementary approach (Al-Ashmori and Wang, 2020; Qi et al., 2020; Tu et al., 2020). Vehicle suspensions are categorized as passive, semi-active, and active. Passive systems, while simple and cost-effective, struggle to isolate low-frequency vibrations effectively (Holtz and Van Niekerk, 2010; Jiregna and Sirata, 2020; Liu and Yu, 2020; Ozbek et al., 2023). On the other hand, active suspension systems although offer the best performance, they are considerably expensive and require more energy for implementation which has an impact on the vehicle energy consumption

This paper addresses the challenge of isolating low-frequency vibrations in off-road vehicles, which is crucial for enhancing driver comfort and well-being. Conventional passive suspension systems struggle to handle low-frequency vibrations effectively, necessitating the exploration of alternative technologies like quasi-zero stiffness suspension (QZSS). QZSS is a nonlinear system known for its high static and very low dynamic stiffness characteristics, enabling a wider displacement range around the equilibrium position (Li et al., 2021; Suman et al., 2021) Researchers have investigated its application in vehicle suspensions, including an optimized quasi-zero stiffness structure incorporated into a vehicle seat (Wang et al., 2023), semi-active quasi-zero stiffness seat performance evaluation under different vibratory roller models (Yang et al., 2022), and harmonic balance analysis of force transmissibility (Xu et al., 2014). While QZSS technology effectively reduces sprung mass acceleration, it also decreases suspension stiffness, which can affect road holding and suspension travel (Qi et al., 2020; Yan et al., 2022). To address this, suspension damping becomes a critical factor that can be adjusted in real-time. Magnetorheological (MR) dampers, controllable by varying the viscosity of magnetorheological fluid through current adjustments, have shown promise in enhancing driver comfort (Sun et al., 2015) and designing semi-controlled variable stiffness and variable damping systems (Deng et al., 2022a). MR dampers have been studied in the context of seat suspension impact protection (Deng et al., 2022b). Incorporating controllable dampers into semi-active suspension systems necessitates appropriate control algorithms. Various strategies like skyhook control, sliding mode control, adaptive control, neural network control, and hybrid damping control have been applied in the field (Aljarbouh et al., 2021; Basargan et al., 2022; Bhardawaj et al., 2020; Ghoniem et al., 2020; Liu et al., 2019). However, the literature on integrated QZSS chassis and seat suspension control often overlooks their high nonlinearity and also fails to consider the unique dynamics of off-road vehicle environments. Moreover, the conventional treatment of driver mass as a solid body neglects the uncertainties associated with driver mass and stiffness variations.

This research introduces a novel static output feedback damping control method for an integrated chassis and seat air QZSS system, specifically designed to tackle low-frequency vibrations in off-road vehicles. Building upon our previous work on optimizing QZSS driver seat design (Atindana et al., 2023), the approach considers the segmented driver mass with varying dynamic characteristics interconnected through joints experiencing different forces. The paper is structured with sections covering system description and mathematical modeling, QZSS system control, numerical simulations, results analysis, experimental validation, and conclusions.

2. Description and mathematical modeling of quasi-zero stiffness suspension

The proposed seat suspension is a one-degree-of-freedom (DOF) system featuring a pneumatic spring and a pneumatic negative stiffness mechanism using paired pneumatic linear actuators (PLAs). It relies on precise solenoid-operated valves to control air pressures in both components, ensuring a stable equilibrium position relative to the driver’s mass. This design, depicted in Figures 1 and 2, allows for seat suspension travel while maintaining stability. The one-degree-of-freedom seat design offers simplicity, cost-effectiveness, and efficiency. By requiring fewer computational and hardware resources, it provides a practical solution for enhancing comfort in seat suspension systems at a minimal cost.

The system’s instantaneous internal pressure magnitude is governed by the classic polytropic process ( $P V^{n} = P_{0} V_{0}^{n}$ ); where P₀ and V₀ are the air spring’s initial pressure and volume, P and V are the air pressure in any other working state, respectively, and n is the polytropic index. Air spring vertical force ( $F_{a s}$ ) is

F_{a s} = A_{e} [P_{a s 0} {(\frac{Z_{a s 0}}{Z_{a s 0} - q_{0}})}^{n} - P_{a}]

(1)

where A_e is the air spring effective area,

q_{0}

represents displacement (q(t)-q₀(t)), while P_as0 and Z_as0 are the air spring pressure and height in the equilibrium position, and P_a is the atmospheric pressure.

Defining the PLAs joints’ static distance as $L_{0,}$ the sprung mass in any different position produces a distance that is equal to the summation of $L_{0}$ and the PLA piston displacement d(t). Thus, establishing a geometric relationship as

L = \sqrt{L_{0}^{2} + {q_{0}}^{2}}, \sin θ = \frac{z}{\sqrt{L_{0}^{2} + {q_{0}}^{2}}}, Δ d = \sqrt{(L_{0}^{2} + {q_{0}}^{2})} - L_{0}

(2)

The QZSS system parameter values are presented in Table 2 (see appendix). When assembled to the single degree of freedom QZSS system, the PLAs forces in the horizontal plane neutralize each other in the dynamic state. In contrast, the PLAs forces acting vertically are added. Therefore, the total vertical force exerted by the negative stiffness PLAs can be determined as

F_{N S S} = 2 A_{c} [P_{c 0} {(\frac{d_{0}}{d_{0} + Δ d})}^{n_{c}} - P_{a}] \sin θ

(3)

where A_c is the effective cylinder area, P_c0 is the initial pressure, d₀ is the PLA piston position.

The QZSS system force (F_eq) required to return the sprung mass to its equilibrium position can be determined as

F_{e q} = {m_{0} g - F}_{a s} - F_{N S S}

(4)

where

m_{0}

is the sprung mass, and

g

is the acceleration due to gravity

The Taylor series expansion is employed to linearize the nonlinear force $F_{e q}$ of the QZSS system; consider $x$ to be equal to $q_{0} - q_{a}$ the Taylor series expansion of $F_{e q}$ at x = 0 gives

{F_{e q} = k}_{0} + k_{1} (x - 0) + k_{2} {(x - 0)}^{2} + \dots + k_{n} {(x - 0)}^{n} + R_{n} (x)

where

k_{n}

represents the coefficient of the Taylor expansion, and

R_{n} (x)

is the remainder of Taylor’s expansion. Given that the suspension stiffness remains relatively constant during normal vehicle operation, the coefficient

k_{1}

of Taylor expansion formula is designated as the nominal stiffness

k

2.1. Vehicle, seat, and driver integrated model

The research employs an integrated seat, chassis, and driver model, which had been validated in our earlier publication (Atindana et al., 2023). This model assumes that the truck’s sprung mass behaves as a rigid body with pitch (λ), roll (ϕ), and vehicle center of mass bounce (q_c) motions, neglecting yaw and lateral movements. The chassis sprung mass is denoted as ms, and there are four unsprung masses (m_11, m_12, m_13, and m₁₄) with one degree of freedom each, represented by q_t1, q_t2, q_t3, and q_t4. Additionally, q_1, q_2, q_3, and q₄ represent the road input excitations at the wheels. The sum of a_f and a_r defines the truck’s wheelbase, and r_w and r_e specify the seat system’s position relative to the vehicle’s center of mass and centerline. The driver mass includes thigh-pelvis mass (M1), lower and upper torso masses (M2, M3), and head mass (M4), interconnected by three joints (Jt1, Jt2, and Jt3) as shown in Figure 3. The vehicle parameters (Table 3), the driver segmented masses parameters (Table 4), details of the driver, seat, and the vehicle equations of motion (68–84) are presented in the appendix.

Passive forces acting on the chassis suspension

F_{21} = k_{21} (q_{s 1} - q_{t 1}) + c_{21} ({\dot{q}}_{s 1} - {\dot{q}}_{t 1})

(5)

F_{22} = k_{22} (q_{s 2} - q_{t 2}) + c_{22} ({\dot{q}}_{s 2} - {\dot{q}}_{t 2})

(6)

F_{23} = k_{23} (q_{s 3} - q_{t 3}) + c_{23} ({\dot{q}}_{s 3} - {\dot{q}}_{t 3})

(7)

F_{24} = k_{24} (q_{s 4} - q_{t 4}) + c_{24} ({\dot{q}}_{s 4} - {\dot{q}}_{t 4})

(8)

Passive force acting on the seat suspension

F_{p} = k_{s} (q_{0} - q_{a}) + C_{s} ({\dot{q}}_{0} - {\dot{q}}_{a})

(9)

The dynamic equations of the integrated model provide an expression for the mass, damping, and stiffness matrices

M_{u} {\ddot{x}}_{u} = B_{s} ({\dot{x}}_{s} - {\dot{x}}_{u}) + K (x_{s} - x_{u}) + K_{t} (w - x_{u}) + F

(10)

M_{s} \ddot{q} = a B_{s} ({\dot{x}}_{u} - {\dot{x}}_{s}) + a K (x_{u} - x_{s}) - a F + a_{p} (k_{s} a_{f} x_{D} - k_{s} a_{p}^{T} q + c_{s} a_{f} {\dot{x}}_{D} - c_{s} a_{p}^{T} \dot{q}) + a_{p} F_{s}

(11)

{M_{D} {\ddot{x}}_{D} = B}_{D} {\dot{x}}_{D} + K_{D} x_{D} + a_{f}^{T} k_{s} a_{p}^{T} q + a_{f}^{T} c_{s} a_{p}^{T} \dot{q} - F_{s}

(12)

where

M_{u} = [\begin{array}{l} m_{u 11} & 0 & 0 & 0 \\ 0 & m_{u 12} & 0 & 0 \\ 0 & 0 & m_{u 13} & 0 \\ 0 & 0 & 0 & m_{u 14} \end{array}], M_{s} = [\begin{array}{l} m_{s} & 0 & 0 \\ 0 & I_{φ} & 0 \\ 0 & 0 & I_{λ} \end{array}]

M_{D} = [\begin{array}{l} m_{f} & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{c u} & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & m_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & m_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{4} \end{array}], B_{s} = [\begin{array}{l} c_{21} & 0 & 0 & 0 \\ 0 & c_{22} & 0 & 0 \\ 0 & 0 & c_{23} & 0 \\ 0 & 0 & 0 & c_{24} \end{array}]

K = [\begin{array}{l} k_{21} & 0 & 0 & 0 \\ 0 & k_{22} & 0 & 0 \\ 0 & 0 & k_{23} & 0 \\ 0 & 0 & 0 & k_{24} \end{array}], K_{t} = [\begin{array}{l} k_{t 11} & 0 & 0 & 0 \\ 0 & k_{t 12} & 0 & 0 \\ 0 & 0 & k_{t 13} & 0 \\ 0 & 0 & 0 & k_{t 14} \end{array}]

B_{D} = [\begin{array}{c} - c_{s} - c_{c u} & c_{c u} & 0 & 0 & 0 & 0 \\ c_{c u} & - c_{c u} - c_{1} & c_{1} & 0 & 0 & 0 \\ 0 & c_{1} & - c_{1} - c_{2} & c_{2} & 0 & 0 \\ 0 & 0 & c_{2} & - c_{2} - c_{3} & c_{3} & 0 \\ 0 & 0 & 0 & c_{3} & - c_{3} - c_{4} & c_{4} \\ 0 & 0 & 0 & 0 & c_{4} & - c_{4} \end{array}]

K_{V} = [\begin{array}{c} - k_{s} - k_{c u} & k_{c u} & 0 & 0 & 0 & 0 \\ k_{c u} & - k_{c u} - k_{1} & k_{1} & 0 & 0 & 0 \\ 0 & k_{1} & - k_{1} - k_{2} & k_{2} & 0 & 0 \\ 0 & 0 & k_{2} & - k_{2} - k_{3} & k_{3} & 0 \\ 0 & 0 & 0 & k_{3} & - k_{3} - k_{4} & k_{4} \\ 0 & 0 & 0 & 0 & k_{4} & - k_{4} \end{array}]

a = [\begin{array}{c} 1 & 1 & 1 & 1 \\ - a_{f} & - a_{f} & a_{r} & a_{r} \\ T_{F} & - T_{F} & T_{R} & - T_{R} \end{array}], a_{p} = [\begin{array}{c} 1 \\ - r_{e} \\ r_{w} \end{array}], a_{f} = [\begin{array}{l} 1 & 0 & 0 & 0 & 0 & 0 \end{array}]

q = {[\begin{array}{l} q_{c} & φ & λ \end{array}]}^{T}

x_{s} = {[\begin{array}{l} q_{s 1} & q_{s 2} & q_{s 3} & q_{s 4} \end{array}]}^{T}, x_{u} = {[\begin{array}{l} q_{t 1} & q_{t 2} & q_{t 3} & q_{t 4} \end{array}]}^{T}

x_{D M} = {[\begin{array}{l} q_{0} & q_{c u} & q_{m 1} & q_{m 2} & q_{m 3} & q_{m 4} \end{array}]}^{T}

w = {[\begin{array}{l} q_{1} & q_{2} & q_{3} & q_{4} \end{array}]}^{T}, F_{M R} = {[\begin{array}{l} F_{C 21} & F_{C 22} & F_{C 23} & F_{C 24} \end{array}]}^{T}

By establishing a relationship between vectors as

q_{0} = a_{f} x_{D}

q_{P s} = a_{p}^{T} q

x_{s} = a^{T} q

F_{s} = k_{s} a_{f} x_{D} - k_{s} a_{p}^{T} q + c_{s} a_{f} {\dot{x}}_{D} - c_{s} a_{p}^{T} \dot{q}

A further unification of (10)–(12) can be obtained as follows

M_{p} \ddot{p} + B_{p} \dot{p} + K_{p} p = K_{w} w + K_{f} u

(13)

In which

p = {[\begin{array}{l} x_{u} & q & x_{D} \end{array}]}^{T}, M_{p} = [\begin{array}{l} M_{u} & 0 & 0 \\ 0 & M_{s} & 0 \\ 0 & 0 & M_{D} \end{array}]

B_{p} = [\begin{array}{l} B_{c} & - B_{c} a^{T} & 0 \\ - a B_{c} & a B_{c} a^{T} + a_{p} c_{c} a_{p}^{T} & - a_{p} c_{c} a_{f} \\ 0 & - a_{f}^{T} c_{c} a_{p}^{T} & - B_{D} \end{array}]

K_{p} = [\begin{array}{l} K + K_{t} & - K a^{T} & 0 \\ - a K & a K a^{T} + a_{p} k_{f s} a_{p}^{T} & - a_{p} k_{f s} a_{f} \\ 0 & - a_{f}^{T} k_{f s} a_{p}^{T} & - K_{D} \end{array}]

K_{w} = [\begin{array}{l} K_{t} \\ 0 \\ 0 \end{array}], K_{f} = [\begin{array}{l} I & 0 \\ - a & a_{p} \\ 0 & - a_{f}^{T} \end{array}], u = [\begin{array}{l} F_{M R} \\ F_{S} \end{array}]

Ultimately, the state-space equation of the nominal integrated system (vehicle, seat, and driver) is given as follows

\dot{x} = A x + B_{w} w + B u

(14)

where

A = [\begin{array}{l} 0 & I \\ - M_{p}^{- 1} K_{p} & - M_{p}^{- 1} B_{p} \end{array}], B_{w} = [\begin{array}{l} 0 \\ M_{p}^{- 1} K_{w} \end{array}], B = [\begin{array}{l} 0 \\ M_{p}^{- 1} K_{f} \end{array}]

and

x = {[\begin{array}{l} p^{T} & {\dot{p}}^{T} \end{array}]}^{T}

is the state vector,

\begin{array}{l} p = {[\begin{array}{l} x_{u} & q & x_{D M} \end{array}]}^{T} \\ q = {[\begin{array}{l} q_{c} & φ & λ \end{array}]}^{T} \\ x_{u} = {[\begin{array}{l} q_{t 1} & q_{t 2} & q_{t 3} & q_{t 4} \end{array}]}^{T} \\ x_{D M} = {[\begin{array}{l} q_{0} & q_{c u} & q_{m 1} & q_{m 2} & q_{m 3} & q_{m 4} \end{array}]}^{T} \end{array}

u = {[\begin{array}{l} F_{M R}^{T} & F_{s}^{T} \end{array}]}^{T}

represents the input vector of the controller,

F_{M R} = {[\begin{array}{l} F_{C 21} & F_{C 22} & F_{C 23} & F_{C 24} \end{array}]}^{T}

, and

w = {[\begin{array}{l} q_{w 1} & q_{w 2} & q_{w 3} & q_{w 4} \end{array}]}^{T}

represents the road excitation vector.

Establishing an approximately linear relationship between the vehicle pitch $λ$ and roll $φ$ angles, $\sin λ \approx λ, \cos λ \approx 1, \sin φ \approx φ, \cos φ \approx 1$ , the following equations are realized

q_{a} = q_{c} - r_{e} φ + r_{w} λ

(15)

q_{s 1} = q_{c} - a_{f} φ + T_{L} λ

(16)

q_{s 2} = q_{c} - a_{f} φ - T_{L} λ

(17)

q_{s 3} = q_{c} + a_{r} φ + T_{R} λ

(18)

q_{s 4} = q_{c} + a_{r} φ - T_{R} λ

(19)

3. Integrated suspension system control

The integrated chassis and seat semi-active QZSS system control relies on the use of MR dampers to fine-tune the necessary damping stiffness for optimal vibration isolation. This is achieved through precise control of the electrical current governing the MR dampers. These advanced MR dampers utilize magnetorheological fluid, which can adjust its yield stress when exposed to a magnetic field.

3.1. Magnetorheological damper model

The integrated suspension system requires distinct damping forces for chassis and seat suspensions, leading to the selection of specific MR dampers for each. A bigger MR damper is chosen for the chassis suspension (Nima, 2008), while a small cylindrical MR damper is employed for the seat suspension (Choi and Han, 2003). Both types of MR dampers operate on similar principles but have customized parameter values optimized for their respective applications (Tables 5 and 6). A damping controller is designed to coordinate and control the damping forces of the integrated suspension system, considering the unique attributes of both the chassis and seat MR dampers.

As stated above, the MR dampers utilize MR fluid, which changes its yield stress when subjected to a magnetic field. The relationship between the yield stress ( $τ_{y}$ ) and the magnetic field strength (H) is defined by (20), where $α_{m}$ and $β_{m}$ are inherent characteristics of the MR fluid, established through experimental analysis

τ_{y} (H) = α_{m} H^{β_{m}}

(20)

For laminar flow occurring in the annular orifice, the flow resistance in the absence of a magnetic field is expressed as

R_{m} = \frac{12 η_{1} L_{m}}{b_{1} h_{m}^{3}}

(21)

Meanwhile, the drop in pressure of the MR fluid resulting from the magnetic field is expressed as

Δ P_{MR} = 2 \frac{L_{m}}{h_{m}} τ_{y} (H) = 2 \frac{L_{m}}{h_{m}} α_{m} H^{β_{m}}, where H = \frac{N I}{2 h_{m}}

(22)

the overall damping force

(F_{d})

of the MR damper is expressed in (40), considering parameters such as gas compliance (

C_{gm}

), piston-rod area (

A_{rm}

), piston area (

A_{pm}

), and the influence of the magnetic field on fluid viscosity

sgn (\cdot)

. The third component in equation (23) represents the controllable damping force (

F_{M R}

) influenced by the magnetic field

F_{d} = \frac{A_{rm}^{2}}{C_{gm}} x_{p} + {(A_{pm} - A_{rm})}^{2} R_{m} {\dot{x}}_{p} + (A_{pm} - A_{rm}) Δ P_{MR} sgn ({\dot{x}}_{p})

(23)

Consequently, (23) can be reformulated as follows

F_{d} = k_{e} x_{p} + c_{e} {\dot{x}}_{p} + F_{MR}, F_{M R} = (A_{p m} - A_{r m}) 2 \frac{l}{h} α H^{β} sgn ({\dot{x}}_{p})

(24)

where

k_{e}

is the effective stiffness influenced by gas pressure,

c_{e}

represents the effective damping attributed to fluid viscosity,

F_{MR}

refers to the damping force responsive to the controlled electric field,

x_{p}

and

{\dot{x}}_{p}

accordingly denotes excitation displacement and velocity.

The control input $u_{i}$ for each MR damper follows semi-active conditions (25), ensuring energy dissipation within a stable system. This control input guides the regulation of input current (I) applied to the MR dampers to achieve desired damping characteristics, as expressed in (26)

\bar{u} = {\begin{cases} {\bar{u}}_{i} & if {\bar{u}}_{i} ({\dot{q}}_{s i} - {\dot{q}}_{t i}) > 0 \\ 0 & if {\bar{u}}_{i} ({\dot{q}}_{s i} - {\dot{q}}_{t i}) \leq 0, \\ {\bar{u}}_{i} & if {\bar{u}}_{i} ({\dot{q}}_{0 i} - {\dot{q}}_{a i}) > 0 \\ 0 & if {\bar{u}}_{i} ({\dot{q}}_{0 i} - {\dot{q}}_{a i}) \leq 0 \end{cases}

(25)

I = \frac{2 h_{m}}{N} {[u \cdot \frac{h_{\dot{m}}}{8 α_{m} \cdot L_{m} (A_{pm} - A_{rm})}]}^{1 / β_{m}}

(26)

3.2. Integrated chassis and seat suspension control objectives

The control objectives for the integrated seat and vehicle suspension system aim to enhance driver comfort by reducing vertical acceleration of the driver’s head. This is expressed as the performance index in (27). Energy management is also crucial in evaluating suspension performance, as it affects the overall vehicle efficiency

q_{g 1} = {\dot{q}}_{m 4} = {\hat{C}}_{h} x

(27)

To achieve minimal driver-head acceleration, and better road holding, the vertical displacements of road wheels $q_{t i}, i \in {k_{21}, k_{22}, k_{23}, k_{24}}$ , and suspension deflections $q_{s i} - q_{t i}, i \in {k_{21}, k_{22}, k_{23}, k_{24}}$ must be minimized. Separate control inputs $q_{g 2}$ and $q_{g 3}$ are defined for these variables in (28) and (29), respectively, with corresponding matrices $C_{2}$ and $C_{3}$

q_{g 2} = {[\begin{array}{l} q_{s 1} - q_{t 1} & q_{s 2} - q_{t 2} & q_{s 3} - q_{t 3} & q_{s 4} - q_{t 4} & q_{c u} - q_{a} \end{array}]}^{T} = C_{2} x

(28)

and

q_{g 3} = {[\begin{array}{l} q_{t 1} & q_{t 2} & q_{t 3} & q_{t 4} \end{array}]}^{T} = C_{3} x

(29)

In the design of the controller, it is difficult to optimally achieve the multiple opposing control goals simultaneously. So a compromise between the control goals is sought, transforming them into a unified control problem using a weighted sum approach. The control output $C_{q}$ is expressed as a weighted sum of individual control outputs for each goal, with weights ( $α_{2}$ and $α_{3}$ ) representing goal importance

\begin{array}{l} q & = [\begin{array}{l} q_{g 1} \\ α_{2} q_{g 2} \\ α_{3} q_{g 3} \end{array}] = [\begin{array}{l} {\hat{C}}_{h} \\ α_{2} C_{2} \\ α_{3} C_{3} \end{array}] x = ([\begin{array}{l} C_{h} \\ α_{2} C_{2} \\ α_{3} C_{3} \end{array}] + [\begin{array}{l} Δ C_{h} \\ 0 \\ 0 \end{array}]) x \\ = ([\begin{array}{l} C_{h} \\ α_{2} C_{2} \\ α_{3} C_{3} \end{array}] + [\begin{array}{l} H_{c} F E_{h} \\ 0 \\ 0 \end{array}]) x = (C_{q} + Δ C_{q}) x \equiv {\hat{C}}_{q} x \end{array}

(30)

\begin{array}{l} C_{q} = {[\begin{array}{l} C_{h}^{T} & α_{2} C_{2}^{T} & α_{3} C_{3}^{T} \end{array}]}^{T} = {[\begin{array}{l} \sum_{i = 1}^{2} h_{i} C_{i}^{T} & α_{2} C_{2}^{T} & α_{3} C_{3}^{T} \end{array}]}^{T} = \sum_{i = 1}^{2} h_{i} {\hat{C}}_{i} \\ {\hat{C}}_{i} = {[\begin{array}{l} C_{i}^{T} & α_{2} C_{2}^{T} & α_{3} C_{3}^{T} \end{array}]}^{T}, Δ C_{q} = {\hat{H}}_{c} F E_{h}, {\hat{H}}_{c} = {[\begin{array}{l} H_{c}^{T} & 0 & 0 \end{array}]}^{T} \end{array}

(31)

The controller’s performance is evaluated in terms of the $L_{2}$ gain relative to road excitation w and control output q, as defined in (32). The performance metric for the asymptotically stable controller is determined to be ${‖ q ‖}_{2}^{2} = \int_{0}^{\infty} q^{T} (t) q (t) d t$ and ${‖ w ‖}_{2}^{2} = \int_{0}^{\infty} w^{T} (t) w (t) d t$ . A low $L_{2}$ gain signifies minimal driver-head acceleration, indicating optimal suspension performance and stability

{‖ T_{qw} ‖}_{\infty} = \sup_{q_{2} \neq 0} \frac{{‖ q ‖}_{2}}{{‖ w ‖}_{2}}

(32)

3.3. Robust controller design considering parameter uncertainties

This subsection covers the robust controller design, accounting for uncertainties tied to driver mass variations and actuator saturation for optimal performance. Driver mass uncertainties are resolved by considering the segmented driver masses (M1–M4) to have consistent mass change ratios, ranging from $m_{\min}$ to $m_{\max}$ (33). Matrices $A_{i}$ in (34) are adjusted to account for parameter uncertainties arising from changes in driver mass

\frac{1}{m} = h_{1} \frac{1}{m_{\min}} + h_{2} \frac{1}{m_{\max}}

(33)

In which, $h_{1}$ and $h_{2}$ are $m_{\min}$ and $m_{\max}$ weighting factors, and accordingly obtained as follows

h_{1} = \frac{1 / m - 1 / m_{\max}}{1 / m_{\min} - 1 m_{\max}}, h_{2} = \frac{1 / m_{\max} - 1 / m}{1 / m_{\min} - 1 / m_{\max}}

(34)

Observably, $h_{i} ⩾ 0, i = 1,2$ , and $\sum_{i = 1}^{2} h_{i} = 1$ . Defining $m_{\min} = (1 - δ) m = δ_{\min} m = δ_{\min} \sum_{i = 1}^{4} m_{i}, m_{\max} = (1 + δ) m = δ_{\max} m = δ_{\max} \sum_{i = 1}^{4} m_{i}$ , where $0 < δ < 1, δ_{\min} = 1 - δ$ , and $δ_{\max} = 1 + δ$

\dot{x} = \sum_{i = 1}^{2} h_{i} A_{i} x + B_{w} w + B \bar{u}

(35)

Similarly, variations in driver mass affect body stiffness, which is challenging to quantify precisely. Norm-bounded matrices $Δ A_{i}$ are introduced to represent stiffness uncertainties in (36). The driver’s body damping coefficients and stiffness uncertainties are defined as $Δ A_{i} = H_{a} F E_{i}$ , where $F$ is an undetermined matrix function bounded by $F^{T} F ⩽ I$ , while $H_{a}$ and $E_{i}$ are dimensionally established constant matrices with definable positions and change domains. Equation (37) represents the system with these uncertainties, where ${\hat{A}}_{h}$ combines the nominal system and uncertainties

\dot{x} = \sum_{i = 1}^{2} h_{i} (A_{i} + Δ A_{i}) x + B_{w} w + B \bar{u}

(36)

\dot{x} = {\hat{A}}_{h} x + B_{w} w + B \bar{u}

(37)

where

\bar{u}

defines the control input saturation (sat) function. Which can be expressed as

sat (u) = {\begin{cases} - u_{\lim} & if u < - u_{\lim} \\ u & if - u_{\lim} ⩽ u ⩽ u_{\lim} \\ u_{\lim} & if u > u_{\lim} \end{cases}

(38)

u_{\lim}

defines the limit of the control input.

The following lemma is employed to solve the actuator saturation challenge:

Lemma 1(Choi et al., 2016); with reference to the saturation condition in (38), when $| u | ⩽ \frac{u_{\lim}}{ε}$ , then $∥ \bar{u} - \frac{1 + ε}{2} u ∥ ⩽ \frac{1 - ε}{2} ∥ u ∥$ .

Consequently

{[\bar{u} - \frac{1 + ε}{2} u]}^{T} [\bar{u} - \frac{1 + ε}{2} u] ⩽ {(\frac{1 - ε}{2})}^{2} u^{T} u

(39)

where

0 < ε < 1

is a specified scalar

In the subsequent step, (37) can now be expressed as (40) below when applying Lemma 1

\begin{array}{l} \dot{x} & = {\hat{A}}_{h} x + B_{w} w + \frac{1 + ε}{2} B u + B (\bar{u} - \frac{1 + ε}{2} u) \\ = {\hat{A}}_{h} x + B_{w} w + \frac{1 + ε}{2} B u + B D \end{array}

(40)

where

D = \bar{u} - \frac{1 + ε}{2} u .

Subsequently, Lemma 2 (Du et al., 2013; Sun et al., 2011) is employed for the actual result derivation: Where every matrix/vector X and Y with relevant dimensions are expressed as

X^{T} Y + Y^{T} X ⩽ ε X^{T} X + ε^{- 1} Y^{T} Y

(41)

In which $ε > 0$ is any scalar.

Finally, (42) describes the state feedback controller used to achieve the desired control objectives, accounting for uncertainties and actuator saturation

u = K C x

(42)

In control systems engineering, implementing a state feedback controller assumes that all state variables are measurable. However, this is challenging in complex systems like the human body with multiple degrees of freedom (DOF). Measuring variables like driver body segment velocities and displacements accurately is difficult due to the dynamic nature of the human body. To address this issue, a static output feedback controller is used, which relies on output variables rather than directly measurable state variables to infer the system’s internal states. The controller is presented (43), where u represents the control input, x denotes the system’s state vector, and $K_{1} C_{1}, K_{2} C_{2} \dots, K_{5} C_{5}$ is a block matrix composed of the feedback gain matrices $K_{1}$ and constant matrices $C_{1}$

u = [\begin{array}{l} K_{1} C_{1} \\ K_{2} C_{2} \\ ⋮ \\ K_{5} C_{5} \end{array}] x

(43)

The controller gain matrix K is designed based on the desired suspension system performance objectives. Hence, feedback gain matrices $K_{1}$ to $K_{5}$ are carefully selected to ensure stability and optimal performance are met. Constant matrices $C_{1}$ to $C_{5}$ define output variable combinations for estimating and controlling the system’s internal states effectively. By selecting appropriate output variables, the control system gains insight into the system’s behavior. To determine the controller gain matrices $K_{1}$ , a Lyapunov function $V (x)$ is defined in (44). By differentiating $V (x)$ and applying the system (37), the time derivative of $V (x)$ is expressed in (45). Substituting the system equation into this expression yields (46), which is crucial for stability analysis

V (x) = x^{T} P x

(44)

\dot{V} (x) = {\dot{x}}^{T} P x + x^{T} P \dot{x}

(45)

\begin{array}{l} \dot{V} (x) = {[{\hat{A}}_{h} x + B_{w} w + \frac{1 + ε}{2} B u + B D]}^{T} P x + x^{T} P [{\hat{A}}_{h} x + B_{w} w + \frac{1 + ε}{2} B u + B D] \\ = {{\hat{A}}_{h} x + B_{w} w + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} B_{i} u_{i} + B_{i} D_{i}]} P x + x^{T} P {{\hat{A}}_{h} x + B_{w} w + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} B_{i} u_{i} + B_{i} D_{i}]} \\ D_{i} = {\bar{u}}_{i} - \frac{1 + ε}{2} u_{i} \end{array}

(46)

In (46), ${\hat{A}}_{h}$ represents the system matrix, $B_{w}$ is the disturbance input matrix, $w$ is the disturbance input vector, $B$ is the control input matrix, $u$ is the control input vector, and $B D$ represents the input disturbance matrix. This equation allows for the analysis of the time derivative of the Lyapunov function $V (x)$ , providing insights into the system stability (Θ). The analysis involves selecting appropriate values for the positive definite matrix P ( $P > 0)$ to ensure stability and meet control objectives (Θ).

Applying Lemma 1, and 2, and $u_{i} = K_{i} C_{i} x$ , then, (64) is realized as follows

\dot{V} (x) ⩽ x^{T} {{\hat{A}}_{h}^{T} P + P {\hat{A}}_{h} + \sum_{i = 1}^{5} [{(\frac{1 + ε_{i}}{2} B_{i} K_{i} C_{i})}^{T} P + \frac{1 + ε_{i}}{2} P B_{i} K_{i} C_{i}]} x + w^{T} B_{w}^{T} P x + x^{T} P B_{w} w + \sum_{i = 1}^{5} (ϵ_{i} D_{i}^{T} D_{i} + ϵ_{i}^{- 1} x^{T} P B_{i} B_{i}^{T} P x) ⩽ x^{T} {{\hat{A}}_{h}^{T} P + P {\hat{A}}_{h} + \sum_{i = 1}^{5} [{(\frac{1 + ε_{i}}{2} B_{i} K_{i} C_{i})}^{T} P + \frac{1 + ε_{i}}{2} P B_{i} K_{i} C_{i}]} x + w^{T} B_{w}^{T} P x + x^{T} P B_{w} w + \sum_{i = 1}^{5} [ϵ_{i} {(\frac{1 - ε_{i}}{2})}^{2} u_{i}^{T} u_{i} + ϵ_{i}^{- 1} x^{T} P B_{i} B_{i}^{T} P x] = x^{T} Θ x + w^{T} B_{w}^{T} P x + x^{T} P B_{w} w

(47)

where

Θ = {\hat{A}}_{h}^{T} P + P {\hat{A}}_{h} + \sum_{i = 1}^{5} [{(\frac{1 + ε_{i}}{2} B_{i} K_{i} C_{i})}^{T} P + \frac{1 + ε_{i}}{2} P B_{i} K_{i} C_{s i} + ϵ_{i} {(\frac{1 - ε_{i}}{2})}^{2} C_{i}^{T} K_{i}^{T} K_{i} C_{i} + ϵ_{i}^{- 1} P B_{i} B_{i}^{T} P]

Lastly, (48) defines a performance index incorporating both sides of (47). This index helps assess and optimize controller performance for complex systems, considering uncertainties and limited measurements

\begin{array}{l} \dot{V} (x) + q^{T} q - γ^{2} w^{T} w ⩽ [\begin{array}{l} x^{T} & w^{T} \end{array}] [\begin{array}{l} Θ + {\hat{C}}_{q}^{T} {\hat{C}}_{q} & P B_{w} \\ B_{w}^{T} P & - γ^{2} I \end{array}] [\begin{array}{l} x \\ w \end{array}] \\ = [\begin{array}{l} x^{T} & w^{T} \end{array}] Π [\begin{array}{l} x \\ w \end{array}] \end{array}

(48)

where

Π = [\begin{array}{l} Θ + {\hat{C}}_{q}^{T} {\hat{C}}_{q} & P B_{w} \\ B_{w}^{T} P & - γ^{2} I \end{array}]

Deductively, if $Π < 0$ , then it can be concluded that, $\dot{V} (x) + q^{T} q - γ^{2} {q_{w}}^{T} q_{w} < 0$ , and therefore, ${∥ T_{q w} ∥}_{\infty} < γ$ and the inceptive condition $x (0) = 0$ (Boyd et al., 1994). From (48), it can be deduced that the system (37) is asymptotically stable when at zero road excitation (w = 0) $Π < 0$ , and $\dot{V} (x) < 0$ .

The condition of $Π < 0$ is analogous to (49) below when $Π$ and $diag (P^{- 1}, I)$ are multiplied before and after, and accordingly transposed, while defining $Q = P^{- 1}, w_{i} C_{i} = C_{i} Q$ and $Y_{i} = K_{i} w_{i}$

[\begin{array}{l} ξ + Q {\hat{C}}_{q}^{T} {\hat{C}}_{q} Q & B_{w} \\ B_{w}^{T} & - γ^{2} I \end{array}] < 0

(49)

where

ξ = Q {\hat{A}}_{h}^{T} + {\hat{A}}_{h} Q + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{s i}^{T} Y_{i}^{T} C_{i}^{T} + \frac{1 + ε_{i}}{2} C_{i} Y_{i} C_{s i} + ϵ_{i}^{- 1} {(\frac{1 - ε_{i}}{2})}^{2} C_{s i}^{T} Y_{i}^{T} Y_{i} C_{s i} + ε_{i} C_{i} C_{i}^{T}]

And (49) is analogous to (50) below when Schur complement equivalence is applied

[\begin{array}{c} ξ_{1} & C_{1}^{T} Y_{1}^{T} & C_{2}^{T} Y_{2}^{T} & \dots & C_{5}^{T} Y_{5}^{T} & Q {\hat{C}}_{q}^{T} & B_{w} \\ * & - ϵ_{1} {(\frac{2}{1 - ε_{1}})}^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & * & - ϵ_{2} {(\frac{2}{1 - ε_{2}})}^{2} I & 0 & 0 & 0 & 0 \\ * & * & * & ⋱ & 0 & 0 & 0 \\ * & * & * & * & - ϵ_{5} {(\frac{2}{1 - ε_{5}})}^{2} I & 0 & 0 \\ * & * & * & * & * & - I & 0 \\ * & * & * & * & * & * & - γ^{2} I \end{array}] < 0

(50)

where

ξ_{1} = Q {\hat{A}}_{h}^{T} + {\hat{A}}_{h} Q + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{s i}^{T} Y_{i}^{T} C_{i}^{T} + \frac{1 + ε_{i}}{2} C_{i} Y_{i} C_{s i} + ϵ_{i} C_{i} C_{i}^{T}]

Consequently, (50) can further be defined as follows

[\begin{array}{c} ξ_{2} & C_{1}^{T} Y_{1}^{T} & C_{2}^{T} Y_{2}^{T} & \dots & C_{5}^{T} Y_{5}^{T} & Q {(C_{q} + Δ C_{q})}^{T} & B_{w} \\ * & - ϵ_{1} {(\frac{2}{1 - ε_{1}})}^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & * & - ϵ_{2} {(\frac{2}{1 - ε_{2}})}^{2} I & 0 & 0 & 0 & 0 \\ * & * & * & ⋱ & 0 & 0 & 0 \\ * & * & * & * & - ϵ_{5} {(\frac{2}{1 - ε_{5}})}^{2} I & 0 & 0 \\ * & * & * & * & * & - I & 0 \\ * & * & * & * & * & * & - γ^{2} I \end{array}] < 0

(51)

where $ξ_{2} = Q {(A_{h} + Δ A_{h})}^{T} + (A_{h} + Δ A_{h}) Q + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{i}^{T} Y_{i}^{T} B_{i}^{T} + \frac{1 + ε_{i}}{2} B_{i} Y_{i} C_{i} + ϵ_{i} B_{i} B_{i}^{T}] .$ Thus, the results are obtained according to Lemma 3 below:

Lemma 3 (Zhang et al., 2018). With properly dimensioned matrices $Σ_{1}, Σ_{2}, Σ_{3}$ , with $Σ_{1}^{T} = Σ_{1}$ , then

Σ_{1} + Σ_{3} Δ Σ_{2} + Σ_{2}^{T} Δ Σ_{3}^{T} < 0

holds for all

Δ

fulfilling

Δ^{T} Δ \leq I

if and only if for

ϵ > 0

Σ_{1} + ϵ Σ_{3} Σ_{3}^{T} + ϵ^{- 1} Σ_{2}^{T} Σ_{2} < 0

Thus, inequality (51) becomes analogous to (52) below

Σ_{1} + Σ_{3} F Σ_{2} + Σ_{2}^{T} F^{T} Σ_{3}^{T} < 0

(52)

In which

Σ_{1} = [\begin{array}{c} ξ_{3} & C_{1}^{T} Y_{1}^{T} & C_{2}^{T} Y_{2}^{T} & \dots & C_{5}^{T} Y_{5}^{T} & Q C_{q}^{T} & B_{w} \\ * & - ϵ_{1} {(\frac{2}{1 - ε_{1}})}^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & * & - ϵ_{2} {(\frac{2}{1 - ε_{2}})}^{2} I & 0 & 0 & 0 & 0 \\ * & * & * & ⋱ & 0 & 0 & 0 \\ * & * & * & * & - ϵ_{5} {(\frac{2}{1 - ε_{5}})}^{2} I & 0 & 0 \\ * & * & * & * & * & - I & 0 \\ * & * & * & * & * & * & - γ^{2} I \end{array}]

ξ_{3} = Q A_{h}^{T} + A_{h} Q + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{i}^{T} Y_{i}^{T} B_{i}^{T} + \frac{1 + ε_{i}}{2} B_{i} Y_{i} C_{i} + ϵ_{i} B_{i} B_{i}^{T}], Σ_{3}^{T} = [\begin{array}{l} H_{a}^{T} & 0 & \dots & 0 & {\hat{H}}_{c}^{T} & 0 \end{array}], Σ_{2} = [\begin{array}{l} E_{h} Q & 0 & \dots & 0 & 0 & 0 \end{array}] .

Observably, by applying Lemma 3, inequality (52) is fulfilled if the following inequality holds for

η > 0

[\begin{array}{c} Q C_{q}^{T} \\ ξ_{4} & C_{1}^{T} Y_{1}^{T} & C_{2}^{T} Y_{2}^{T} & \dots & C_{5}^{T} Y_{5}^{T} & + η H_{a} {\hat{H}}_{c}^{T} & B_{w} & Q E_{h}^{T} \\ * & - ϵ_{1} {(\frac{2}{1 - ε_{1}})}^{2} I & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & - ϵ_{2} {(\frac{2}{1 - ε_{2}})}^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & * & * & ⋱ & 0 & 0 & 0 & 0 \\ * & * & * & * & - ϵ_{5} {(\frac{2}{1 - ε_{5}})}^{2} I & 0 & 0 & 0 \\ * & * & * & * & * & - I + η H_{c} {\hat{H}}_{c}^{T} & 0 & 0 \\ * & * & * & * & * & * & - γ^{2} I & 0 \\ * & * & * & * & * & * & * & - η I \end{array}] < 0

(53)

In which $ξ_{4} = Q A_{h}^{T} + A_{h} Q + η H_{a} H_{a}^{T} + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{i}^{T} Y_{i}^{T} B_{i}^{T} + \frac{1 + ε_{i}}{2} B_{i} Y_{i} C_{i} + ϵ_{i} B_{i} B_{i}^{T}]$ .

Relying on the delineations $A_{h} = \sum_{j = 1}^{2} h_{j} A_{j}, C_{q} = \sum_{j = 1}^{2} h_{j} {\hat{C}}_{j}, E_{h} = \sum_{j = 1}^{2} h_{j} E_{j}$ , and given that $h_{j} ⩾ 0$ and $\sum_{j = 1}^{2} h_{j} = 1$ , (53) is analogous to (54) below as follows

[\begin{array}{c} Q {\hat{C}}_{j}^{T} \\ ξ_{5} & C_{1}^{T} Y_{1}^{T} & C_{2}^{T} Y_{2}^{T} & \dots & C_{5}^{T} Y_{5}^{T} & + η H_{a} {\hat{H}}_{c}^{T} & B_{w} & Q E_{j}^{T} \\ * & - ϵ_{1} {(\frac{2}{1 - ε_{1}})}^{2} I & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & - ϵ_{2} {(\frac{2}{1 - ε_{2}})}^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & * & * & ⋱ & 0 & 0 & 0 & 0 \\ * & * & * & * & - ϵ_{5} {(\frac{2}{1 - ε_{5}})}^{2} I & 0 & 0 & 0 \\ * & * & * & * & * & - I + η H_{c} {\hat{H}}_{c}^{T} & 0 & 0 \\ * & * & * & * & * & * & - γ^{2} I & 0 \\ * & * & * & * & * & * & * & - η I \end{array}] < 0, j = 1,2,

(54)

where $ξ_{5} = Q A_{j}^{T} + A_{j} Q + η H_{a} H_{a}^{T} + \sum_{i = 1}^{5} [\frac{1 + ε_{i}}{2} C_{i}^{T} Y_{i}^{T} B_{i}^{T} + \frac{1 + ε_{i}}{2} B_{i} Y_{i} C_{s i} + ϵ_{i} B_{i} B_{i}^{T}]$ . Furthermore, with reference to (43), the constraint $| u_{i} | ⩽ \frac{u_{\lim i}}{ε_{i}}$ is accordingly defined as follows

| K_{i} C_{s i} x | ⩽ \frac{u_{l i}}{ε_{i}}

(55)

Let $Ω (K_{i}) = {x ∥ x^{T} C_{i}^{T} K_{i}^{T} K_{i} C_{i} x ∣ ⩽ {(\frac{u_{\lim i}}{ε_{i}})}^{2}}$ , the analogous condition for an ellipsoid $Ω (P, ρ) = {x ∣ x^{T} P x ⩽ ρ}$ which is a subset of $Ω (K_{i})$ , that is $Ω (P, ρ) \subset$ $Ω (K_{i})$ is defined as (Kennouche et al., 2023)

K_{i} C_{i} {(\frac{P}{ρ})}^{- 1} C_{i}^{T} K_{i}^{T} ⩽ {(\frac{u_{l i}}{ε_{i}})}^{2}

(56)

Inequality (56) can be redefined as (57) shown below when Schur complement equivalence is applied

[\begin{array}{l} {(\frac{u_{l i}}{ε_{i}})}^{2} & K_{i} C_{i} {(\frac{P}{ρ})}^{- 1} \\ {(\frac{P}{ρ})}^{- 1} C_{i}^{T} K_{i}^{T} & {(\frac{P}{ρ})}^{- 1} I \end{array}] ⩾ 0

(57)

Similarly, relying on the delineations $Q = P^{- 1}, W_{i} C_{i} = C_{i} Q$ , and $Y_{i} = K_{i} W_{i}$ inequality (57) is analogous to (58) below

[\begin{array}{l} {(\frac{u_{li i}}{ε_{i}})}^{2} & Y_{i} C_{i} \\ C_{i}^{T} Y_{i}^{T} & ρ^{- 1} Q \end{array}] ⩾ 0

(58)

Notedly, linear matrix inequalities-LMIs (49) and (58) with equality constraint $w_{i} C_{i} = C_{s} Q$ are considered as the static output feedback controller design feasibility problem. Analogously, the equality constraint $w_{i} C_{i} = C_{i} Q$ can be transformed as follows: (Goyal et al., 2015; Wu et al., 2017)

tr [{(w_{i} C_{i} - C_{i} Q)}^{T} (w_{i} C_{i} - C_{i} Q)] = 0

(59)

Consequently, establishing condition (60)

{(w_{i} C_{i} - C_{i} Q)}^{T} (w_{i} C_{i} - C_{i} Q) ⩽ μ I

(60)

where

μ > 0

, thus, it is then analogous to

[\begin{array}{l} - μ I & {(w_{i} C_{i} - C_{i} Q)}^{T} \\ w_{i} C_{i} - C_{i} Q & - I \end{array}] ⩽ 0

(61)

Utilizing the Schur complement equivalence, (61) is equally a LMI and can be numerically determined if $μ$ is assumed to be an insignificant positive value.

Accordingly, the controller design problem can be formulated as shown below. The objective is to ensure the stability of the semi-active system (37) when coupled with the controller (43). Stability is defined as ${∥ T_{q w} ∥}_{\infty} < γ$ , where $γ$ is a positive constant. Additionally, the following given parameters are considered: $γ > 0, ε_{i} > 0, ρ > 0, μ \approx 0$ , and $u_{\lim i}, i = 1,2, \dots, 5$ .

To achieve stability, it is necessary to find matrices $Q > 0$ , $w_{i}$ , and $Y_{i}$ , as well as scalars $ε_{i} > 0,$ and $η > 0,$ for $i = 1,2, \dots, 5$ . These variables should satisfy the following Linear Matrix Inequalities (LMIs): (54), (58), and (61). If a feasible solution exists, the feedback gain matrices $K_{i}$ can be determined as $K_{i} = Y_{i} w_{i}^{- 1}, i = 1,2, \dots, 5$ .

The LMIs (54), (58), and (61) represent linear matrix inequalities that incorporate $\min γ^{2}$ . These inequalities provide a means to reframe the controller design problem as a minimization problem. The goal is to achieve the minimum value of $\min γ^{2}$ while adhering to the constraints defined by LMIs (49), (58), and (61). This can be approached as a curvilinear optimization problem, which can be solved using standard software solutions.

4. System simulation and validation

Co-simulation results analysis is performed to evaluate the vibration isolation capabilities of the semi-active QZSS system on bumpy roads. Control feedback indicators for chassis suspension included unsprung mass velocities $q_{t i}, i \in {q_{t 1}, q_{t 2}, q_{t 3}, q_{t 4}}$ , and displacements $q_{t i}$ , as well as relative velocities between unsprung and sprung masses $q_{s i} - q_{t i})$ . Seat suspension control relied on relative displacement $q_{0} - q_{a}$ , and velocity ${\dot{q}}_{0} - {\dot{q}}_{a}$ between the seat structure and the cabin floor. Controller input limits were determined by the maximum current MR dampers could handle: For the chassis suspension dampers, the maximum allowable current was set at 4A, resulting in a damping force of 1534N. On the other hand, the seat suspension damper was limited to 2A, generating a damping force of 421N. Both the chassis and seat dampers control frequency is 100 Hz. Controller gain matrices for the integrated suspension system were optimized through constrained minimization problems defined by LMIs (49), (58), and (61)

K_{1} = 10^{5} \times [\begin{array}{l} - 0.2274 x_{1} & - 0.0061 x_{14} & 0.0861 x_{18} & - 0.0871 x_{19} & 0.0655 x_{20} \end{array}]

(62)

K_{2} = 10^{5} \times [\begin{array}{l} - 0.0590 x_{2} & - 0.0024 x_{15} & 0.0544 x_{18} & - 0.0550 x_{19} & - 0.0414 x_{20} \end{array}]

(63)

K_{3} = 10^{5} \times [\begin{array}{l} - 0.1400 x_{3} & - 0.0024 x_{16} & 0.0428 x_{18} & 0.0772 x_{19} & 0.0323 x_{20} \end{array}]

(64)

K_{4} = 10^{5} \times [\begin{array}{l} - 0.0867 x_{4} & - 0.0013 x_{17} & 0.0347 x_{18} & 0.0626 x_{19} & - 0.0262 x_{20} \end{array}]

(65)

K_{5} = 10^{5} \times [\begin{array}{l} 3.9666 x_{8} & - 0.0162 x_{18} & 0.0048 x_{19} & - 0.0040 x_{20} & 0.0462 x_{21} \end{array}]

(66)

To ascertain the comprehensive vibration isolation performance of the new QZSS integrated vehicle and seat semi-active system, a brief high-intensity bump is chosen as the road disturbance input (Yazici and Sever, 2018)

z_{r} (t) = {\begin{cases} \frac{a}{2} (1 - \cos (\frac{2 π v_{0}}{l} t)), & 0 ⩽ t ⩽ \frac{l}{v_{0}} \\ 0, & t > \frac{l}{v_{0}} \end{cases}

(67)

where

a

(0.1 m) and

l

(4 m) represent the bump height and the length, respectively,

v_{0}

(30 km/h) is vehicle velocity, and t is the excitation time.

4.1. Simulation results and analysis

The research conducted a deliberate approach to simulate bumpy road conditions by inducing distinct amplitudes in the right and left road wheels, causing a controlled roll motion in the vehicle. This arrangement introduced a harmonious effect by applying identical peak amplitudes to both front and rear wheels, with a controlled time delay between them. This approach aimed to comprehensively depict the uneven terrain of the road and its impact on vehicle dynamics.

Comparing the semi-active QZSS with the passive QZSS revealed valuable insights. Visual representations of controlled dynamic tire load and suspension displacement (Figures 4 and 5) demonstrate the semi-active QZSS’s ability to maintain these critical parameters within an acceptable range. This showcased the system’s potential to provide superior vibration isolation while maintaining adequate road-holding and vehicle-handling performance, surpassing passive suspension.

Figure 6 analyzed bump sensitivities in terms of seat suspension displacement and driver head acceleration, highlighting the semi-active system’s benefits. It achieved a smaller peak acceleration and shorter equilibrium time, indicating its ability to absorb shocks promptly, emphasizing its superior emergency response compared to the passive system. The new semi-active suspension consistently outperformed the optimized passive system, enhancing driver ride comfort significantly. Considering variations in driver masses, and damping coefficients, a comprehensive evaluation was performed for both the semi-active and passive suspension systems. Figure 7 illustrates the new semi-active suspension’s consistent superiority in enhancing driver ride comfort across various parameters and vehicle speeds. Figure 8 depicts the control current and the corresponding damping force of MR dampers in the seat and chassis suspension subsystems. It revealed that the seat suspension’s MR damper required significantly less current than the chassis suspension’s damper, emphasizing the seat suspension’s role as an auxiliary system to enhance ride comfort without straining the vehicle’s power requirements.

4.2. Hardware-in-the-Loop test validation

Simulation confirms the control algorithm’s efficiency theoretically but has practical limitations. Validating the semi-active system in a real vehicle is also costly. To overcome these challenges, the Hardware-in-the-Loop (HiL) test offers a dependable solution. Hardware-in-the-Loop seamlessly integrates diverse hardware interfaces using VeriStand software. It establishes a connection between the host computer and the NI real-time simulator through Ethernet, while the NI real-time simulator interfaces with a TDK power supply via COM. The HiL test framework involves three main components: creating the QZSS model, developing the D2P fast prototyping control technique, and configuring VeriStand engineering files. MATLAB/Simulink was crucial for crafting the QZSS system. The VeriStand Interface connects to the model’s inputs and outputs. After construction, the model is compiled into a dll file, containing all relevant data for the negative stiffness air suspension. This comprehensive framework is shown in Figure 9. Meanwhile, ISO 8608:1995 road grades D, E, and F, were adopted for the suspension validation. Especially of interest is the semi-active QZSS performance evaluation on the F-road profile at a speed of 30 km/h, which approximately simulates the real driving environment and speed of off-road vehicles such as construction, mining, and agriculture vehicles.

4.3. Test results

The presented results, depicted in Figures 10 and 11, offer a comprehensive comparison between the dynamic tire load and suspension displacement performance indices of the passive, and the innovative semi-active QZSS. Additionally, Figure 12 expands this visual narrative by providing an overview of the seat suspension deflection and displacement characteristics of both the passive and semi-active QZSS. Importantly, the test results closely matched the simulation outcomes, indicating a strong correlation. However, minor discrepancies were observed, possibly due to network variations and occasional signal interruptions. A deeper RMS analysis, as evidenced in Table 1, confirmed the superiority of the integrated semi-active QZSS system. Notably, at a speed of 30 km/h, the new semi-active suspension significantly reduced driver head vertical acceleration between 26.53% and 35.0% compared to the passive QZSS system. This substantial reduction directly translated into enhanced ride comfort and reduced vibration transfer to the driver, as vividly depicted in Figure 13.

Table 1.

Comparative performance evaluation of passive and semi-active QZSS system using the driver head vertical acceleration as the performance indicator.

Road profile type	Driver mass (Kg)	Passive QZSS RMS-values (m/s²)	Semi-active QZSS RMS-values (m/s²)	Variation (%)
D-road	60	0.050	0.036	−28.85
	75	0.059	0.041	−30.51
	95	0.052	0.037	−31.80
E-road	60	0.049	0.039	−26.53
	75	0.041	0.030	−26.83
	95	0.061	0.040	−34.43
F-road	60	0.040	0.029	−27.50
	75	0.051	0.035	−31.37
	95	0.060	0.039	−35.00

Figures 14(a) and (b) offered a closer look at the control current demand for operating the integrated chassis and seat semi-active suspension, revealing minimal current demand for the new semi-active QZSS system. This suggests that the system’s operation imposes a negligible burden on the vehicle’s overall power requirements, indicating energy efficiency and practical viability. This energy efficiency not only enhances sustainability but also reduces strain on the vehicle’s power supply, making it more suitable for real-world applications. The comprehensive analysis of these results underscores the transformative potential of the new semi-active QZSS system and validates the effectiveness of the static output feedback controller. The empirical evidence, in line with simulations, supports improved ride comfort, vibration isolation, and energy efficiency.

5. Conclusion

This research introduced a novel static output feedback damping control method applied to an integrated chassis and seat quasi-zero stiffness suspension (QZSS) system. The design of the control law is formulated as a convex optimization problem with linear matrix inequalities, and the varying driver body stiffness due to the variation of driver mass was treated as norm-bounded matrices. The integrated controller handles five individual inputs, including one seat MR damper and four chassis MR dampers, allowing for diverse feedback signals and corresponding control inputs. Extensive simulations and analysis highlight the superior performance of the semi-active QZSS system over the passive QZSS system. High-fidelity Hardware-in-the-Loop (HiL) testing confirms a significant reduction in suspension displacement, and driver head vertical acceleration, ranging from 26.53 to 35.0%, under various speeds and road conditions. This improvement enhances ride comfort and reduces vibration transfer. The research not only demonstrates the transformative potential of the static output feedback semi-active QZSS system but also substantiates its effectiveness through quantitative analysis and empirical evidence.

Footnotes

Author contributions

Vincent Akolbire Atindana: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Prof. Xing Xu: Supervision, Resources, Project administration, Funding acquisition, Jacob Kwaku Nkrumah: Investigation, Data curation, Anthoney Akayeti: Writing - review & editing, and Xinwei Jiang: Visualization.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant No. 51875256.

ORCID iDs

Vincent Akolbdsire Atindana

Xing Xu

Data Availability Statement

Research data can be made available by the corresponding author upon request.

Appendix

Figure A1.

QZSS seat schematic diagram (Supplement of Figure 2).

Table 2.

Quasi-zero stiffness suspension parameter values for optimal vibration isolation and stability.

Parameters	Driver mass (Kg)	Values
P_c0 (bar)	60	3.0
	75	3.4
	95	4.0
d₀ (m)	60	0.07
	75	0.06
	95	0.05
A_c (m²)	60	0.0015
	75	0.0014
	95	0.0012

Table 3.

Full-truck and seat model parameters.

Parameter	Unit	Value
Front wheel mass	(Kg)	350
Rear wheel mass	(Kg)	450
Front tire stiffness	(N/m)	1 × 10⁶
Rear tire stiff	(N/m)	1.8 × 10⁶
Front wheel spring stiffness	(N/m)	3 × 10⁵
Rear wheel spring stiffness	(N/m)	4 × 10⁵
Front suspension damper coefficient	(N.s/m)	1 × 10⁴
Rear suspension damper coefficient	(N.s/m)	1.5 × 10⁴
Rolling inertia of vehicle body mass	(Kg.m²⁾	1 × 10⁵
Pitch inertia	(Kg.m²⁾	2 × 10⁴
Distance from the front wheel axle to the center of gravity	(m)	2.37
Distance from the rear wheel axle to the center of gravity	(m)	1.33
Longitudinal distance from the seat position to the center of gravity	(m)	1.0
Lateral distance from the seat position to the vehicle center line	(m)	0.64
Vehicle body mass	(Kg)	1.89 × 10⁴
Vehicle velocity	(Km/h)	20,30, 50
Seat frame mass	(Kg)	16
Seat cushion mass	(Kg)	1.5
Seat air spring stiffness	(N/m)	2000
Seat frame stiffness	(N/m)	31,000
Seat cushion stiffness	(N/m)	18,000
Seat suspension damping coefficient	(N.s/m)	493.56
Seat cushion damping	(N.s/m)	200
Seat frame damping	(N.s/m)	830
Pneumatic cylinder diameter	(m)	0.04

Table 4.

Driver body geometric parameters.

Whole body (kg)	Parameter	Segmented Mass (Kg)	Inertia (Kg-m²)	L-distance (mm)	Position w (mm)	Position q (mm)
60	Thighs-pelvis	19.78	0.58	290	95	83.5
	Lower torso	8.04	0.45	155	−48	247
	Upper torso	18.28	0.36	170	21	540
	Head	4.68	0.022	105	100	790
75	Thighs-pelvis	23.48	0.99	290	95	83.5
	Lower torso	9.53	0.71	155	−48	247
	Upper torso	21.78	0.51	170	21	540
	Head	5.63	0.027	105	100	790
95	Thighs-pelvis	29.7	1.97	290	95	83.5
	Lower torso	12.08	1.27	155	−48	247
	Upper torso	27.53	0.82	170	21	540
	Head	7.13	0.05	105	100	790
	Joint 1				−44	139
	Joint 2				−17	386
	Joint 3				53	706
	C_b1				173	0
	C_b2				0	0
	C_b3				−80	230

Driver segmented masses equations of motion

Segmented mass 1 (thigh-pelvis)

m_{1} {\ddot{w}}_{1} = - f_{1 w} - f_{C_{b 1} w} - f_{C_{b 2} w}

m_{1} {\ddot{q}}_{1} = - f_{1 q} - f_{C_{b 1} q} - f_{C_{b 2} q}

(68)

I_{1} {\ddot{δ}}_{1} = - f_{1 w} L_{1 J_{1} V} - f_{1 q} L_{1 J_{1} H} - f_{C_{b 1} q} L_{1 C_{b 2} H} - f_{C b 1 w} L_{1 C_{b 1} V} - f_{C_{b 2} q} L_{1 C_{b 2} H} - f_{C b 2 w} L_{1 C_{b 2} V} - f_{r 1}

where w and q denote horizontal and vertical body motion, L represents the distance between the body’s center of gravity (GC) and seat contact points, and δ signifies the mass’s translational angle.

Segmented mass 2 (lower torso)

m_{2} {\ddot{w}}_{2} = f_{1 w} - f_{2 w} - f_{C_{b 3} w}

m_{2} {\ddot{q}}_{2} = f_{1 q} - f_{2 q} - f_{C_{b 3} q}

(69)

I_{2} {\ddot{δ}}_{2} = - f_{1 w} L_{2 J_{t 1 V}} - f_{1 q} L_{2 J_{t 1 H}} - f_{2 w} L_{2 {J t}_{2 V}} - f_{2 q} L_{2 J_{t 2 H}} - f_{C_{b 3} w} L_{2 C_{b 3} V} - f_{C_{b 3} q} L_{2 C_{b 3} H} + f_{r 1} - f_{r 2}

Segmented mass 3 (upper torso)

m_{3} {\ddot{w}}_{3} = f_{2 w} - f_{3 w}

m_{3} {\ddot{q}}_{3} = f_{2 q} - f_{3 q}

(70)

I_{3} {\ddot{δ}}_{3} = - f_{2 w} L_{3 J_{t 2 V}} - f_{2 q} L_{3 J_{t 2 H}} - f_{3 w} L_{3 J_{t 3 V}} - f_{3 q} L_{3 J_{t 3 H}} + f_{r 2} - f_{r 3}

Segmented mass 4 (Head)

m_{4} {\ddot{w}}_{4} = f_{3 w}

m_{4} {\ddot{q}}_{4} = f_{3 q}

(71)

I_{4} {\ddot{δ}}_{4} = - f_{3 w} L_{4 J_{t 3 V}} - f_{3 q} L_{4 J_{t 3 H}} + f_{r 3}

where w₄ and q₄ define the fore-aft and vertical motion of the head.

Thus (72)

\begin{array}{l} q_{s 1} = q_{c} - a_{f} φ - r_{t} λ & q_{a} = q_{c} - r_{w} φ - r_{e} λ \\ q_{s 2} = q_{c} + a_{r} φ - r_{t} λ & L_{1 J_{t 1 H}} = J_{t 1 w} - G_{c 1 w} \\ q_{s 3} = q_{c} + a_{r} φ + L_{t} λ & L_{1 J_{t 1 V}} = J_{t 1 q} - G_{c 1 q} \\ q_{s 4} = q_{c} - a_{f} φ + L_{t} λ & M_{D C} = m_{1} + m_{2} + m_{3} + m_{4} + m_{c u} \end{array}

where M_DC is the overall driver and cushion mass.

The equation of motion for the seat and cushion mass is (73)

m_{0} {\ddot{q}}_{0} - F_{e q} {+ F}_{C u} = M_{D C} g

(74)

m_{C u} {\ddot{q}}_{C u} = F_{C u} + f_{C_{b 1} q} + f_{C_{b 2} q} + f_{C_{b 3} q} - M_{D C} g

(75)

F_{c u, e} = {\begin{cases} 0 & m_{0} - q_{C u} < 0 \\ K_{L} (m_{0} - q_{C u}) & 0 \leq m_{0} - q_{C u} < q_{C u 1} \\ K_{L} (m_{0} - q_{C u}) + K_{n - L} {(m_{0} - q_{C u} - q_{C u 1})}^{3} & q_{C u 1} \leq m_{0} - q_{C u} \end{cases}

(76)

F_{d c} = C_{C u} ({\dot{q}}_{a} - {\dot{q}}_{C u})

Then, the seat cushion’s force is (77)

F_{C u} = F_{C u, e} + F_{d c}

The full vehicle suspension governing equations of motion for the various DoFs are as follows (78)

m_{11} {\ddot{q}}_{t 1} + k_{11} (q_{t 1} - q_{1}) + k_{21} (q_{t 1} - q_{s 1}) + c_{21} ({\dot{q}}_{t 1} - {\dot{q}}_{s 1}) = 0

(79)

m_{12} {\ddot{q}}_{t 2} + k_{12} (q_{t 2} - q_{2}) + k_{22} (q_{t 2} - q_{s 2}) + c_{22} (q_{t 2} - {\dot{q}}_{s 2}) = 0

(80)

m_{13} {\ddot{q}}_{t 3} + k_{13} (q_{t 3} - q_{3}) + k_{23} (q_{t 3} - q_{s 3}) + c_{23} ({\dot{q}}_{t 3} - {\dot{q}}_{s 3}) = 0

(81)

m_{14} {\ddot{q}}_{t 4} + k_{14} (q_{t 4} - q_{4}) + k_{24} (q_{t 4} - q_{s 4}) + c_{24} ({\dot{q}}_{t 4} - {\dot{q}}_{s 4}) = 0

The vehicle’s center of mass bounce (q_c) pitch (λ), and roll (φ), equations of motion can be defined as (82)

m_{c} {\ddot{q}}_{s} + k_{21} (q_{s 1} - q_{t 1}) + c_{21} ({\dot{q}}_{s 1} - {\dot{q}}_{t 1}) + k_{22} (q_{s 2} - q_{t 2}) + c_{22} ({\dot{q}}_{s 2} - {\dot{q}}_{t 2}) + k_{23} (q_{s 3} - q_{t 3}) + c_{23} ({\dot{q}}_{s 3} - {\dot{q}}_{t 3}) + k_{24} (q_{s 4} - q_{t 4}) + c_{24} ({\dot{q}}_{s 4} - {\dot{q}}_{t 4}) + F_{s} + F_{c} = 0

(83)

I_{c w} \ddot{λ} - (k_{21} (q_{s 1} - q_{t 1}) + c_{21} ({\dot{q}}_{s 1} - {\dot{q}}_{t 1})) r_{t} - (k_{22} (q_{s 2} - q_{t 2}) + c_{22} ({\dot{q}}_{s 2} - {\dot{q}}_{t 2})) r_{t} + (k_{23} (q_{s 3} - q_{t 3}) + c_{23} ({\dot{q}}_{s 3} - {\dot{q}}_{t 3})) l_{t} + (k_{24} (q_{s 4} - q_{t 4}) + c_{24} ({\dot{q}}_{s 4} - {\dot{q}}_{t 4})) l_{t} - F_{s} r_{e} - F_{c} r_{e} = 0

(84)

I_{c e} \ddot{φ} - (k_{21} (q_{s 1} - q_{t 1}) + c_{21} ({\dot{q}}_{s 1} - {\dot{q}}_{t 1})) a_{f} + (k_{22} (q_{s 2} - q_{t 2}) + c_{22} ({\dot{q}}_{s 2} - {\dot{q}}_{t 2})) a_{r} + (k_{23} (q_{s 3} - q_{t 3}) + c_{23} ({\dot{q}}_{s 3} - {\dot{q}}_{t 3})) a_{r} - (k_{24} (q_{s 4} - q_{t 4}) + c_{24} ({\dot{q}}_{s 4} - {\dot{q}}_{t 4})) a_{f} - F_{s} r_{w} - F_{c} r_{w} = 0

Table 5.

Chassis suspension MR damper parameter values.

Parameter	Symbol	Unit	Value
Piston and cylinder gap height	$h_{m}$	mm	3.6
Piston passage effective length	$l_{m}$	mm	80
Piston area	$A_{p m}$	mm²	710
Piston rod area	$A_{r m}$	mm²	250
Number of coil turns	N	Kv/mm	200
Coil diameter	$D_{c}$	mm	1.60
Maximum damper current	I	A	4
Maximum controlled damping force	$F_{M R}$	N	1534

Table 6.

Seat suspension MR damper parameter values.

Parameter	Symbol	Unit	Value
Piston and cylinder gap height	$h_{m}$	mm	1.8
Piston passage effective length	$l_{m}$	mm	52
Piston area	$A_{p m}$	mm²	227
Piston rod area	$A_{r m}$	mm²	95
Number of coil turns	N	Kv/mm	60
Coil diameter	$D_{c}$	mm	0.75
Maximum damper current	I	A	2
Maximum controlled damping force	$F_{M R}$	N	421

Figure A2.

Dynamic tire load on a bumpy road at 30 km/h (Supplement of Figure 4).

Figure A3.

Suspension displacement on a bumpy road at 30 km/h (Supplement of Figure 5).

Figure A4.

HiL test results of dynamic tire load and suspension displacement response on F-road at 30 km/h (Supplement of Figure 10).

Table A1.

Performance comparison of the dynamic tire load, chassis suspension displacement, and seat suspension displacement RMS values h (Supplement of Table 6).

Driving condition	Performance indicators (RMS)	Passive QZSS	Semi-active QZSS	Variation (%)
Bumpy road (30 km/h)	Dynamic tire load (N)	1979	1670	−15.65
	Suspension displacement (m)	0.0092	0.0069	−24.78
	Seat displacement (m)	0.0070	0.0051	−27.14
F-road (30 km/h)	Dynamic tire load (N)	1299	1106	−14.88
	Suspension displacement (m)	0.0069	0.0053	−23.19
	Seat displacement (m)	0.0038	0.0030	−21.05

Table A2.

Semi-active QZSS simulation and HiL test results comparison (Supplement of Table 6).

Driving condition	Performance indicators (RMS)	QZSS simulation	QZSS HiL test	Variation (%)
F-road (30 km/h)	Dynamic tire load (N)	1106	1180	6.69
	Suspension displacement (m)	0.0053	0.0060	13.21
	Seat displacement (m)	0.0030	0.0034	13.33

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