Design of Active Suspension Controllers for 8 × 8 Armored Combat Vehicles

Abstract

This paper presents a method to design an active suspension controller for 8 × 8 armored combat vehicles, which is called corner damping control (CDC). It is assumed that the target vehicle with 8 × 8 drive mechanisms and 8 suspensions has active actuators on each suspension for vertical, roll and pitch motion control on a sprung mass. A state-space model with 22 state variables is derived from the target vehicle. With the state-space model, a linear quadratic (LQ) cost function is defined. The control objective is to reduce the vertical acceleration, pitch and roll angles of a sprung mass for ride comfort, durability and turret stabilization. To avoid full-state feedback of LQR, a static output feedback control (SOF) is selected as a control structure for CDC. The vertical velocity, roll and pitch rates of a sprung mass, and vertical velocities at each corner, are selected as a sensor output. With those sensor outputs and LQ cost function, four LQ SOF controllers are designed. To validate the effectiveness of the LQ SOF controllers, simulation is carried out on a vehicle simulation package. From the simulation results, it is shown that the proposed CDC with LQ SOF controllers with a much smaller number of sensor outputs and controller gains can reduce the vertical acceleration, pitch and roll angles of a sprung mass and, as a result, improve ride comfort, durability and turret stabilization.

1. Introduction

To date, a large number of papers have been published on suspension control for a passenger car [1,2]. The main objectives of suspension control for a passenger car are ride comfort, road holding and motion sickness [2]. On the contrary, there have been a small number of papers published on suspension control for military vehicles [3]. In particular, there are few papers on suspension control for multi-wheeled military vehicles (MWMVs) such as 6 × 6 or 8 × 8 armored combat vehicles (ACVs) [4,5]. Generally, the objectives of suspension control for MWMVs are much stricter than those of passenger cars because those have been developed assuming to run on off-roads. For example, besides ride comfort and road holding, durability and turret stabilization are needed for MWMVs or ACVs [6,7]. In this paper, 8 × 8 ACV is chosen as the target vehicle [8,9,10,11,12,13].

Generally, it has been well known that ride comfort is related to the vertical acceleration, a_z, of a sprung mass (SPM) for passenger cars or commercial vehicles [14,15]. This is also valid for MWMVs. Moreover, this measure can be interpreted as durability because it is evaluated with an acceleration [16,17]. Motion sickness has been considered for MWMVs [18]. However, motion sickness mitigation is not considered a control objective in this paper. Turret stabilization is related to firing accuracy, which has been measured with the pitch angle and its rate of the SPM. For continuous fire, the settling times of the pitch and roll angles (θ and ϕ) of an SPM after firing are important [4,6,13]. Among them, the pitch angle response is much more important than the roll angle one because the firing accuracy heavily depends on the pitch angle [6]. For this reason, ϕ and θ are selected as a measure of turret stabilization in this paper [11,13]. In summary, a_z, ϕ and θ of an SPM are selected as a measure of ride comfort, durability, and turret stabilization.

To date, active suspension and magneto-rheological (MR) damper have been selected as an actuator for vehicle suspension control [2,5]. Among them, active suspension has been mainly selected because it is systematic and effective when controlling the vibration of an SPM related to ride comfort and road holding. For this reason, a lot of papers have been published on active suspension control [1,2,3,4,5,19,20,21,22,23]. However, there are few papers on active suspension control for 8 × 8 ACVs because the target vehicle is too heavy for active suspension to be realized in real 8 × 8 ACVs [8]. Instead, most of the papers on suspension control for 8 × 8 ACVs have selected a semi-active actuator such as MR dampers [9,10,11,12,13]. In this paper, the active suspension is used as an actuator when controlling the vertical, roll and pitch (VRP) motions of the SPM. The active suspension control for 8 × 8 ACVs can give insight to actuator capability which can be realizable in real vehicles. For example, the maximum force and the bandwidth of an actuator which are needed for suspension control can be specified for real 8 × 8 ACVs. In other words, the active suspension control for 8 × 8 ACVs can be used as a baseline when designing a suspension controller with another actuator such as an MR damper.

To improve ride comfort, durability and turret stabilization, the VRP motions of an SPM should be controlled by using actuators in suspensions. For this purpose, it is necessary to build a vehicle model describing the VRP motions of the SPM. As a vehicle model for suspension control, quarter-, half- and full-car models (FCMs) have been used. However, the first two models can correspondingly describe only the vertical motion and the vertical and pitch motions of an SPM. On the contrary, an FCM can describe the VRP motions of an SPM. For this reason, an 11-DOF FCM is used as a vehicle model for 8 × 8 ACVs in this paper [8,10,13]. From the FCM, a linear state-space equation (SSE) is derived.

For suspension controller design, linear quadratic optimal control (LQOC) including LQR and LQG, H_∞ control, sliding mode control and adaptive control have been applied [24,25,26,27]. Among them, LQOC, i.e., LQR is the most widely used for suspension control because it can provide a clear relationship between tuning parameters and regulated outputs through the LQ cost function (LQCF) [1,2,19]. Generally, LQCF is defined by the SSE for control objectives such as ride comfort, durability and turret stabilization. In view of the control structure, LQR uses full-state feedback (FSF), which requires all state variables to be measured. For example, the 11-DOF FCM for 8 × 8 ACVs has 22 state variables and 8 control inputs. As a consequence, the gain matrix of LQR for the 11-DOF FCM has the dimension of 8 × 22 or 96 gain elements, which is impossible to implement in real 8 × 8 ACVs. Moreover, it is necessary to measure 22 state variables for FSF. This is the reason why LQR has not been applied to active suspension control for 8 × 8 ACVs to date. To resolve the problem, this paper presents corner damping control (CDC), which adopts a static output feedback (SOF) control instead of FSF [28,29,30].

As sensor outputs for SOF control, the vertical velocity (v_zc), roll and pitch rates ($\dot{\varphi }$ and $\dot{\theta }$) of SPM, and the vertical velocities at eight corners of the SPM are chosen. Compared to other variables such as ϕ and θ, it is relatively easy to obtain those signals in real vehicles [31,32,33,34]. With those outputs, four SOF controllers are proposed, which have 2 ~ 5 gains and need 3 ~ 8 sensor outputs. Those controllers are designed with LQOC, i.e., SSE and LQCF. Due to the much smaller number of sensor outputs and controller gains, it is easy to implement the SOF controllers in real 8 × 8 ACVs, compared to LQR [35]. For the same reason, it is easy to optimize these controllers with a LQCF. Let the SOF controllers be optimized with LQCF, called LQ SOF. To optimize the SOF controller with LQCF defined for ride comfort, durability and turret stabilization, a meta-heuristic algorithm is applied [36]. To verify the control performance of the LQ SOF controllers, frequency response analysis is carried out with SSE, and simulation is carried out on the simulation software, TruckSim^®. From analysis and simulation results, it is discussed whether the proposed CDC with LQ SOF controllers is effective or not for ride comfort, durability and turret stabilization, and which is the best LQ SOF controller for the purpose of improving ride comfort, durability and turret stabilization.

The goal of this paper is to design an SOF controller with LQOC for 8 × 8 ACV so as to improve ride comfort, durability and turret stabilization. The contributions of this paper are condensed as follows:

• To date, there have been few approaches to active suspension control for 8 × 8 ACVs. Moreover, there have been rear approaches to apply LQOC to suspension control for 8 × 8 ACVs. This paper presents the CDC, the method to design LQ SOF controllers with active suspension for 8 × 8 ACVs.
• To avoid full-state feedback and simplify the control structure, a corner damping control (CDC) is proposed. In the CDC, four SOF controllers are presented. Available sensor outputs used for the SOF controllers in CDC are easily measured or calculated from on-board sensors in real vehicles. Those SOF controllers have 2 ~ 5 gains and need 3 ~ 8 sensor outputs. When designing those SOF controllers, a much smaller number of gains are needed to be optimized and implemented.
• With the LQ SOF controllers designed in the CDC framework, frequency response is carried out with the SSE of 11-DOF FCM, and simulation is conducted on four different road profiles in TruckSim. Based on simulation results, it is discussed whether the proposed CDC with LQ SOF controllers is effective or not for ride comfort, durability and turret stabilization, and which LQ SOF controller is the best for ride comfort, durability and turret stabilization in 8 × 8 ACVs.

This paper has four sections. In Section 2, the controller design procedure is presented with an 11-DOF FCM for an 8 × 8 ACV and LQOC. In Section 3, frequency response analysis and simulation are carried out on TruckSim on four different road conditions. The conclusions are given in Section 4.

2. Design of LQ SOF Controller for 8 × 8 ACV

In this section, an SSE for 11-DOF FCM is derived. With the SSE, LQCF is defined and LQR is designed. Based on the SSE, the vectors of sensor outputs are defined for SOF control. With the vectors, the LQ SOF controller is designed.

2.1. Derivation of State-Space Model

Figure 1 shows a free-body diagram of the 11-DOF FCM for an 8 × 8 ACV [8,10,13]. Figure 1 shows that 11-DOF FCM consists of four axles, eight suspensions, a single SPM and eight unsprung masses (USPMs). As shown in Figure 1, the SPM has eight corners, which are indexed as ➊, ➋, …, ➐ and ➑, and has eight suspensions, S₁, …, S₇ and S₈, where S_i has a spring with stiffness, k_si, a damper with damping coefficient, b_si, and an active actuator used to generate the control input u_i [35]. A suspension controller calculates the eight control inputs, u₁, …, u₇ and u₈, by using the state variables, which describe the VRP motions of the SPM and the vertical motions of USPMs. In this model, the eight USPMs are excited by eight road profiles or elevations, z_r1, …, z_r7 and z_r8, which are applied as a disturbance.

The model describes eleven motions: three and eight motions of the SPM and the USPM, respectively. The VRP motions of the SPM are described by three state variables, i.e., the vertical displacement, z_c, the roll angle, ϕ, and the pitch angle, θ. In the SPM, z_si is the vertical displacement at the i-th corner of the SPM, which is closely related to z_c, ϕ and θ through the geometry of the SPM.

In the suspension S_i, the linear suspension force, f_i, is calculated as (1). With these forces and the geometric information given in Figure 1, the equations of motions (EoMs) on the SPM and USPMs are correspondingly obtained as (2) and (3) [30]. Equation (2) is arranged into the matrix-vector form of (4). In (4), f is the vector of the suspension forces, f₁, f₂, …, f₇ and f₈, and the matrix G stands for the transform from eight forces into vertical force, roll and pitch moments.

$$f_i=-k_{si}(z_{si}-z_{ui})-b_{si}({\dot{z}}_{si}-{\dot{z}}_{ui})+u_i,\hspace{1em}i=1\cdots 8$$

(1)

$$\left\{\begin{array}{l}{m}_s{\ddot{z}}_c=f_1+f_2+f_3+f_4+f_5+f_6+f_7+f_8\hfill \\ I_x\ddot{\varphi }=t_1\cdot \left(f_1-f_2\right)+t_2\cdot \left(f_3-f_4\right)+t_3\cdot \left(f_5-f_6\right)+t_4\cdot \left(f_7-f_8\right)\hfill \\ I_y\ddot{\theta }=-l_1\cdot \left(f_1+f_2\right)-l_2\cdot \left(f_3+f_4\right)+l_3\cdot \left(f_5+f_6\right)+l_4\cdot \left(f_7+f_8\right)\hfill \end{array}\right.$$

(2)

$$m_{ui}{\ddot{z}}_{ui}=-k_{ti}\left(z_{ui}-z_{ri}\right)-f_i,\hspace{1em}i=1,\cdots ,8$$

(3)

$$\left[\begin{array}{c}{m}_s{\ddot{z}}_c\\ I_x\ddot{\varphi }\\ I_y\ddot{\theta }\end{array}\right]=\underset{G}{\underbrace{\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ t_1& -t_1& t_2& -t_2& t_3& -t_3& t_4& -t_4\\ -l_1& -l_1& -l_2& -l_2& l_3& l_3& l_4& l_4\end{array}\right]}}\underset{f}{\underbrace{\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_4\\ f_5\\ f_6\\ f_7\\ f_8\end{array}\right]}}=Gf$$

(4)

In (1), z_si of the SPM is not a state variable. So, it is necessary to represent z_si with the state variables, z_c, ϕ and θ. Figure 2 shows the geometric relationship among z_si and z_c, ϕ and θ [30]. From the geometric information of the SPM given in Figure 2, z_s1, …, z_s7 and z_s8 are derived as (5) from z_c, ϕ and θ. In (5), the nonlinear sine functions are approximated as sinθ ≈ θ and sinϕ ≈ ϕ assuming that ϕ and θ are less than 10°. (5) is arranged into the matrix-vector form of (6) [8,10,13,30]. In (6), the matrix G^T indicates the geometric transform from z_c, ϕ and θ to z_s1, …, z_s7 and z_s8.

$$\left\{\begin{array}{l}{z}_{s1}=z_c+t_1\cdot \sin \varphi -l_1\cdot \sin \theta \approx z_c+t_1\cdot \varphi -l_1\cdot \theta \hfill \\ z_{s2}=z_c-t_1\cdot \sin \varphi -l_1\cdot \sin \theta \approx z_c-t_1\cdot \varphi -l_1\cdot \theta \hfill \\ z_{s3}=z_c+t_2\cdot \sin \varphi -l_2\cdot \sin \theta \approx z_c+t_2\cdot \varphi -l_2\cdot \theta \hfill \\ z_{s4}=z_c-t_2\cdot \sin \varphi -l_2\cdot \sin \theta \approx z_c-t_2\cdot \varphi -l_2\cdot \theta \hfill \\ z_{s5}=z_c+t_3\cdot \sin \varphi +l_3\cdot \sin \theta \approx z_c+t_3\cdot \varphi +l_3\cdot \theta \hfill \\ z_{s6}=z_c-t_3\cdot \sin \varphi +l_3\cdot \sin \theta \approx z_c-t_3\cdot \varphi +l_3\cdot \theta \hfill \\ z_{s7}=z_c+t_4\cdot \sin \varphi +l_4\cdot \sin \theta \approx z_c+t_4\cdot \varphi +l_4\cdot \theta \hfill \\ z_{s8}=z_c-t_4\cdot \sin \varphi +l_4\cdot \sin \theta \approx z_c-t_4\cdot \varphi +l_4\cdot \theta \hfill \end{array}\right.$$

(5)

$$\underset{z_s}{\underbrace{\left[\begin{array}{c}{z}_{s1}\\ z_{s2}\\ z_{s3}\\ z_{s4}\\ z_{s5}\\ z_{s6}\\ z_{s7}\\ z_{s8}\end{array}\right]}}=\underset{G^T}{\underbrace{\left[\begin{array}{ccc}1& t_1& -l_1\\ 1& -t_1& -l_1\\ 1& t_2& -l_2\\ 1& -t_2& -l_2\\ 1& t_3& l_3\\ 1& -t_3& l_3\\ 1& t_4& l_4\\ 1& -t_4& l_4\end{array}\right]}}\underset{p}{\underbrace{\left[\begin{array}{c}{z}_c\\ \varphi \\ \theta \end{array}\right]}}$$

(6)

Figure 3 shows the geometric relationship among z_s1, z_s2, z_s7 and z_s8 and the state variables, z_c, ϕ and θ. From the geometric information of the SPM given in Figure 3, z_s1, z_s2, z_s7 and z_s8 can be obtained as (7) from z_c, ϕ and θ. Similar to (6), (7) is arranged into the matrix-vector form of (8) [30]. In (8), the matrix H^T indicates the geometric transform from z_c, ϕ and θ to z_s1, z_s2, z_s7 and z_s8.

$$\left\{\begin{array}{l}{z}_{s1}=z_c+t_1\cdot \sin \varphi -l_1\cdot \sin \theta \approx z_c+t_1\cdot \varphi -l_1\cdot \theta \hfill \\ z_{s2}=z_c-t_1\cdot \sin \varphi -l_1\cdot \sin \theta \approx z_c-t_1\cdot \varphi -l_1\cdot \theta \hfill \\ z_{s7}=z_c+t_4\cdot \sin \varphi +l_4\cdot \sin \theta \approx z_c+t_4\cdot \varphi +l_4\cdot \theta \hfill \\ z_{s8}=z_c-t_4\cdot \sin \varphi +l_4\cdot \sin \theta \approx z_c-t_4\cdot \varphi +l_4\cdot \theta \hfill \end{array}\right.$$

(7)

$$\underset{z_f}{\underbrace{\left[\begin{array}{c}{z}_{s1}\\ z_{s2}\\ z_{s7}\\ z_{s8}\end{array}\right]}}=\underset{H^T}{\underbrace{\left[\begin{array}{ccc}1& t_1& -l_1\\ 1& -t_1& -l_1\\ 1& t_4& -l_4\\ 1& -t_4& -l_4\end{array}\right]}}\underset{p}{\underbrace{\left[\begin{array}{c}{z}_c\\ \varphi \\ \theta \end{array}\right]}}$$

(8)

Given the variables and the parameters of the 11-DOF FCM, new vectors and matrices are correspondingly defined as (9) and (10) [30,34]. Using these definitions, (6) and (8) are correspondingly represented as (11) and (12). With (9), (10) and (11), (1) is represented as (13). By combining (9), (10) with (4), (2) and (3) are represented as (14). By replacing f of (14) with that of (13), (14) is arranged into (15), which is rearranged into a new matrix-vector form as (16). In (16), new matrices, M, K, B, F and E, are defined. With the vectors in (9), new vectors, z and x, are also defined as (17). Using those new vectors and matrices of (17), (16) is rewritten into (18). With the same procedure as above, the EoMs of the SPM and USPMs, (2) and (3), are arranged into (18), which is rearranged into a new matrix-vector form of (19). The state vector x of the FCM is defined as (17). With (17) and (19), the SSE of the 11-DOF FCM is obtained as (20).

$$\begin{array}{c}{z}_s\triangleq {\left[\begin{array}{cccc}{z}_{s1}& \cdots & z_{s7}& z_{s8}\end{array}\right]}^T,\hspace{0.33em}{z}_u\triangleq {\left[\begin{array}{cccc}{z}_{u1}& \cdots & z_{u7}& z_{u8}\end{array}\right]}^T,\hspace{0.33em}{z}_r=w\triangleq {\left[\begin{array}{cccc}{z}_{r1}& \cdots & z_{r7}& z_{r8}\end{array}\right]}^T\\ p\triangleq {\left[\begin{array}{ccc}{z}_c& \varphi & \theta \end{array}\right]}^T,\hspace{0.33em}u\triangleq {\left[\begin{array}{cccc}{u}_1& \cdots & u_7& u_8\end{array}\right]}^T\\ z_f\triangleq {\left[\begin{array}{cccc}{z}_{s1}& z_{s2}& z_{s7}& z_{s8}\end{array}\right]}^T,\hspace{0.33em}v\triangleq {\left[\begin{array}{cccc}{u}_1& u_2& u_7& u_8\end{array}\right]}^T\end{array}$$

(9)

$$\begin{array}{c}{M}_s\triangleq \mathrm{diag}\left(m_s,I_x,I_y\right),\hspace{0.33em}{M}_u\triangleq \mathrm{diag}\left(m_{u1},\cdots ,m_{u7},m_{u8}\right)\\ K_s\triangleq \mathrm{diag}\left(k_{s1},\cdots ,k_{s7},k_{s8}\right),\hspace{0.33em}{K}_t\triangleq \mathrm{diag}\left(k_{t1},\cdots ,k_{t7},k_{t8}\right),\\ B_s\triangleq \mathrm{diag}\left(b_{s1},\cdots ,b_{s7},b_{s8}\right)\end{array}$$

(10)

$$z_s=G^{T}p,\hspace{1em}{\dot{z}}_s=G^T\dot{p}$$

(11)

$$z_f=H^{T}p,\hspace{1em}{\dot{z}}_f=H^T\dot{p}$$

(12)

$$f=-K_s\left(z_s-z_u\right)-B_s\left({\dot{z}}_s-{\dot{z}}_u\right)+u\hspace{0.33em}=-K_s\left(G^{T}p-z_u\right)-B_s\left(G^T\dot{p}-{\dot{z}}_u\right)+u$$

(13)

$$\left\{\begin{array}{l}{M}_s\ddot{p}=Gf\\ M_u{\ddot{z}}_u=-K_t\left(z_u-z_r\right)-f\end{array}\right.$$

(14)

$$\left\{\begin{array}{l}{M}_s\ddot{p}=-G{K}_s\left(G^{T}p-z_u\right)-G{B}_s\left(G^T\dot{p}-{\dot{z}}_u\right)+Gu\\ M_u{\ddot{z}}_u=K_s\left(G^{T}p-z_u\right)+B_s\left(G^T\dot{p}-{\dot{z}}_u\right)+K_t\left(z_u-z_r\right)-u\end{array}\right.$$

(15)

$$\underset{M}{\underbrace{\left[\begin{array}{cc}{M}_s& 0_{3\times 8}\\ 0_{8\times 3}& M_u\end{array}\right]}}\left[\begin{array}{c}\ddot{p}\\ {\ddot{z}}_u\end{array}\right]=\underset{K}{\underbrace{\left[\begin{array}{cc}-G{K}_s{G}^T& G{K}_s\\ K_s{G}^T& -K_s-K_t\end{array}\right]}}\left[\begin{array}{c}p\\ z_u\end{array}\right]+\underset{B}{\underbrace{\left[\begin{array}{cc}-G{B}_s{G}^T& G{B}_s\\ B_s{G}^T& -B_s\end{array}\right]}}\left[\begin{array}{c}\dot{p}\\ {\dot{z}}_u\end{array}\right]+\underset{E}{\underbrace{\left[\begin{array}{c}{0}_{3\times 8}\\ K_t\end{array}\right]}}w+\underset{F}{\underbrace{\left[\begin{array}{c}G\\ -I_{8\times 8}\end{array}\right]}}u$$

(16)

$$z\triangleq \left[\begin{array}{c}p\\ z_u\end{array}\right],\hspace{0.33em}x\triangleq \left[\begin{array}{c}z\\ \dot{z}\end{array}\right]$$

(17)

$$M\ddot{z}=Kz+B\dot{z}+Fu+Ew$$

(18)

$$\left[\begin{array}{c}\dot{z}\\ \ddot{z}\end{array}\right]=\underset{A}{\underbrace{\left[\begin{array}{cc}{0}_{7\times 7}& 0_{7\times 7}\\ M^{-1}K& M^{-1}B\end{array}\right]}}\left[\begin{array}{c}z\\ \dot{z}\end{array}\right]+\underset{B_1}{\underbrace{\left[\begin{array}{c}{0}_{7\times 4}\\ M^{-1}E\end{array}\right]}}w+\underset{B_2}{\underbrace{\left[\begin{array}{c}{0}_{7\times 4}\\ M^{-1}F\end{array}\right]}}u$$

(19)

$$\dot{x}=Ax+B_{1}w+B_{2}u$$

(20)

2.2. Design of LQR

When applying LQOC to controller design for 8 × 8 ACVs, an LQCF should be defined for ride comfort, durability and turret stabilization. In this paper, the LQCF, J, is set as (21). In LQCF, each weight ρ_i is set by Bryson’s rule, i.e., ρ_i = 1/ξ_i², where ξ_i is the maximum allowable value (MAV) on the related term [37]. For ride comfort and durability enhancement, a_z should be reduced. For this purpose, ξ₁ should be set lower. For better road holding, the tire deflection should be reduced. For this purpose, ξ₈ and ξ₉ should be set lower. For better turret stabilization, ϕ and θ should be maintained as small as possible. For this purpose, ξ₆ and ξ₇ should be set lower.

The LQCF J can be rewritten into (22) with the weighting matrices Ω, Θ and Φ, which are calculated from A, B₂ and ρ_i in J. With LQCF and Ω, Θ and Φ, LQR is obtained as (23). The gain matrix of LQR, K_LQR, has the dimension of 8 × 22, which is too many to be implemented on real 8 × 8 ACVs. Furthermore, when implementing K_LQR on real 8 × 8 ACVs, 22 state variables should be measured. To overcome these problems, an SOF control is selected as a feedback structure instead of FSF in this paper.

$$J={\displaystyle \underset{0}{\overset{\infty }{\int }}\left\{\begin{array}{c}{\rho }_1{\ddot{z}}_c^2+{\rho }_2{\ddot{\varphi }}^2+{\rho }_3{\ddot{\theta }}^2+{\rho }_4{\dot{\varphi }}^2+{\rho }_5{\dot{\theta }}^2+{\rho }_6{\varphi }^2+{\rho }_7{\theta }^2\\ +{\rho }_8{\displaystyle \sum _{i=1}^8{\left(z_{si}-z_{ui}\right)}^2}+{\rho }_9{\displaystyle \sum _{i=1}^8{z}_{ui}^2}+{\rho }_{10}{\displaystyle \sum _{i=1}^8{u}_i^2}\end{array}\right\}dt}$$

(21)

$$J={\displaystyle \underset{0}{\overset{\infty }{\int }}\left\{{\left[\begin{array}{c}x\\ u\end{array}\right]}^T\left[\begin{array}{cc}\Omega & \Theta \\ {\Theta }^T& \Phi \end{array}\right]\left[\begin{array}{c}x\\ u\end{array}\right]\right\}dt}$$

(22)

$$u=-K_{LQR}x=-{\Phi }^{-1}{B}_2^{T}Px$$

(23)

2.3. Lotus Modal Control

Lotus modal control is to independently control three modes, i.e., the VRP modes, of the SPM [38,39,40,41]. For VRP modes, the vertical force, F_zc, and the roll and pitch moments, M_ϕ and M_θ, should be generated to control the corresponding mode of the SPM. These three control inputs are grouped into a single vector, m, as shown in (24). From the relationship of (4), (25) is derived. From (25), (26) is obtained, where (•)+ is the pseudo-inverse of •. By using (26), the three control inputs, F_zc, M_ϕ and M_θ, in m are converted into eight ones in u. This is called Lotus modal control [38,39,40,41]. If m is determined by a control method in Lotus modal control, it can be converted into the control input at each suspension. Similarly, from (8), (27) is obtained. From (27), the four control inputs at S₁, S₂, S₇ and S₈ are determined from m as (28). By combining (26) and (27), the relationship between u and v is obtained as (29). In (29), G⁺H represents the transform from four control inputs u₁, u₂, u₇ and u₈ in S₁, S₂, S₇ and S₈ to eight ones u₁, …, u₇ and u₈ in S₁, S₂, …, S₇ and S₈. In this paper, m is determined by the LQ SOF controller.

$$m={\left[\begin{array}{ccc}{F}_{zc}& M_{\varphi }& M_{\theta }\end{array}\right]}^T$$

(24)

$$m=Gu$$

(25)

$$u=G^{+}m$$

(26)

$$m=Hv$$

(27)

$$v=H^{+}m$$

(28)

$$u=G^{+}Hv$$

(29)

2.4. Design of LQ SOF Controller

For 8 × 8 ACVs, this paper presents the corner damping control (CDC) which adopts SOF control as a feedback structure. Generally, SOF control uses a set of sensor outputs, which are calculated from the state vector, x, with an output matrix. The time-derivatives of the vectors, p and z_f, are obtained as (30) and (31) whose C₁, C₂, C₃ and C₄ are output matrices, respectively.

$$\left\{\begin{array}{c}p=\left[\begin{array}{cccc}{I}_3& 0_{3\times 8}& 0_{3\times 3}& 0_{3\times 8}\end{array}\right]x=C_{1}x\\ \dot{p}=\left[\begin{array}{cccc}{0}_{3\times 3}& 0_{3\times 8}& I_3& 0_{3\times 8}\end{array}\right]x=C_{2}x\end{array}\right.$$

(30)

$$\left\{\begin{array}{l}{z}_s=H^{T}p=H^T{C}_{1}x=C_{3}x\hfill \\ {\dot{z}}_f=H^T\dot{p}=H^T{C}_{2}x=C_{4}x\hfill \end{array}\right.$$

(31)

With the vectors of (30) and (31), four LQ SOF controllers are designed. The first and second SOF controllers are presented by calculating the vector m, as given in (32) and (34), respectively. Using (26), the first and second SOF controllers, u_SOF1 and u_SOF2, are given in (33) and (35), respectively. u_SOF1 needs three signals in $\dot{p}$. For this reason, u_SOF1 has three gain elements. In u_SOF2, the vector m is calculated as (34). Different from u_SOF1, u_SOF2 needs four signals, i.e., corner vertical velocities, in ${\dot{z}}_f$. For this reason, u_SOF2 is the first case of CDC. As shown in (34), u_SOF2 has 5 gain elements in K₂, which use the symmetries between left/right and front/rear corners with respect to the roll and pitch motions, respectively. As shown in (32) and (34), u_SOF1 and u_SOF2 are designed with Lotus modal control [38,39,40,41].

$$m=\left[\begin{array}{c}{F}_{zc}\\ M_{\varphi }\\ M_{\theta }\end{array}\right]=\underset{K_1}{\underbrace{\left[\begin{array}{ccc}{k}_1& 0& 0\\ 0& k_2& 0\\ 0& 0& k_3\end{array}\right]}}\underset{\dot{p}}{\underbrace{\left[\begin{array}{c}{\dot{z}}_c\\ \dot{\varphi }\\ \dot{\theta }\end{array}\right]}}=K_1\dot{p}$$

(32)

$$u_{SOF1}=G^{+}m=G^{+}{K}_1\dot{p}=G^{+}{K}_1{C}_{2}x=K_{SOF1}x$$

(33)

$$m=\left[\begin{array}{c}{F}_z\\ M_{\varphi }\\ M_{\theta }\end{array}\right]=\underset{K_2}{\underbrace{\left[\begin{array}{cccc}{k}_1& k_1& k_1& k_1\\ k_2& -k_2& k_3& -k_3\\ k_4& k_4& -k_5& -k_5\end{array}\right]}}\underset{{\dot{z}}_f}{\underbrace{\left[\begin{array}{c}{\dot{z}}_{s1}\\ {\dot{z}}_{s2}\\ {\dot{z}}_{s7}\\ {\dot{z}}_{s8}\end{array}\right]}}=K_2{\dot{z}}_f$$

(34)

$$u_{SOF2}=G^{+}m=G^{+}{K}_2{\dot{z}}_f=G^{+}{K}_2{C}_{4}x=K_{SOF2}x$$

(35)

The third SOF controller, u_SOF3, is presented as (36). As shown in (36), the control input of u_SOF3 is simply obtained by multiplying the gains by the vertical velocities in ${\dot{z}}_s$. u_SOF3 has two gain elements and needs eight signals, i.e., corner vertical velocities, in ${\dot{z}}_s$. For this reason, u_SOF3 is the second case of CDC. The block diagonal structure of u_SOF3 originated from the symmetries between the front and rear corners of the SPM.

$$\begin{array}{l}{u}_{SOF3}=\underset{K_3}{\underbrace{\mathrm{diag}\left(\left[\begin{array}{llllllll}{k}_1\hfill & k_1\hfill & k_2\hfill & k_2\hfill & k_1\hfill & k_1\hfill & k_2\hfill & k_2\hfill \end{array}\right]\right)}}\underset{{\dot{z}}_s}{\underbrace{\left[\begin{array}{c}{\dot{z}}_{s1}\\ {\dot{z}}_{s2}\\ ⋮\\ {\dot{z}}_{s8}\end{array}\right]}}\hspace{0.33em}\hspace{0.17em}\\ \hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.17em}=K_3{\dot{z}}_s=K_3{G}^T\dot{p}=K_3{G}^T{C}_{2}x=K_{SOF3}x\end{array}$$

(36)

The fourth SOF controller, u_SOF4, uses the transform, (29), which converts four control inputs to eight ones. For this reason, before designing u_SOF4, it is necessary to calculate the vector v with ${\dot{z}}_f$ as given in (37). u_SOF4 uses four signals, i.e., corner vertical velocities, in ${\dot{z}}_f$. For this reason, u_SOF4 is the third case of CDC. With the vector v, u_SOF4 is presented as (38). As shown in (38), the transform, (29), is used to derive eight control inputs from four ones, and u_SOF4 needs only two gain elements.

$$v=\underset{K_4}{\underbrace{\mathrm{diag}\left(\left[\begin{array}{llll}{k}_1\hfill & k_2\hfill & k_1\hfill & k_2\hfill \end{array}\right]\right)}}\underset{{\dot{z}}_f}{\underbrace{\left[\begin{array}{c}{\dot{z}}_{s1}\\ {\dot{z}}_{s2}\\ {\dot{z}}_{s7}\\ {\dot{z}}_{s8}\end{array}\right]}}=K_4{\dot{z}}_f$$

(37)

$$u_{SOF4}=G^{+}Hv=G^{+}H{K}_4{\dot{z}}_f=G^{+}H{K}_4{C}_{4}x=K_{SOF4}x$$

(38)

Three SOF controllers, u_SOF2, u_SOF3 and u_SOF4, use the vertical velocities of each corner of the SPM. For this reason, this can be regarded as a damping or derivative control with corner vertical velocity. Let denote the SOF controllers, u_SOF2, u_SOF3 and u_SOF4, as CDC.

For comparison, another SOF controller is designed [42,43]. This SOF controller, u_SOF5, calculates F_zc, M_ϕ and M_θ, from 6 sensor signals, as given in (39). Then, the transform, (26), converts three control forces and moments to eight ones. u_SOF5 is presented as (40). In (40), the matrix C₅ converts the state vector x to the vector q. As shown in (39), u_SOF5 requires 6 sensor signals and 6 gain elements. Different from the above four SOF controllers, u_SOF5 requires the vertical displacement, z_c, the roll and pitch angles, ϕ and θ. Those variables are much more difficult to measure with a sensor in real 8 × 8 ACVs. For this reason, u_SOF5 is used for comparison. In fact, u_SOF5 has three PD controllers along the directions of VRP motions, as presented in the previous paper [8]. However, the detailed procedure on how to select the gains of PD controllers was not given. In the previous paper, 6 gain elements in K₅ were obtained by applying sliding mode control and LQOC [42,43]. For this reason, this paper adopts LQOC to calculate 6 gain elements in K₅. If K₅ is optimized for LQCF J, u_SOF5 is called the LQ optimal PD controller.

$$m=\underset{K_5}{\underbrace{\left[\begin{array}{llllll}{k}_1\hfill & k_2\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & k_3\hfill & k_4\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & k_5\hfill & k_6\hfill \end{array}\right]}}\underset{q}{\underbrace{\left[\begin{array}{l}{z}_c\hfill \\ {\dot{z}}_c\hfill \\ \varphi \hfill \\ \dot{\varphi }\hfill \\ \theta \hfill \\ \dot{\theta }\hfill \end{array}\right]}}=K_{5}q$$

(39)

$$u_{SOF5}=G^{+}m=G^{+}{K}_5\dot{q}=G^{+}{K}_5{C}_{5}x=K_{SOF5}x$$

(40)

For the four SOF controllers, it is necessary to measure the elements in the vector,${\dot{z}}_s$, or the vector, ${\dot{z}}_f$. In this paper, it is assumed that v_z,$\dot{\varphi }$ and $\dot{\theta }$ are measured with an inertial measurement unit (IMU), attached on the SPM. In the previous works, this can be easily obtained by applying high-pass and low-pass filters (HPF and LPF) sequentially to accelerometer signals measured at each corner [31,32,33,34,35]. In this paper, the vertical velocities in ${\dot{z}}_s$ can be calculated from v_z, $\dot{\varphi }$ and $\dot{\theta }$ by using (41). On the other hand, if the vertical accelerations at the corners, ➊, ➋, ➐ and ➑, are measured, the vertical velocities in ${\dot{z}}_f$ are obtained by applying HPF and LPF to the acceleration signals. In this paper, the vertical velocities in ${\dot{z}}_f$ are calculated from v_z,$\dot{\varphi }$ and $\dot{\theta }$ by using (42).

$${\dot{z}}_s=G^T\dot{p}\hspace{1em}\iff \hspace{1em}\dot{p}={\left(G^T\right)}^{+}{\dot{z}}_s$$

(41)

$${\dot{z}}_f=H^T\dot{p}\hspace{1em}\iff \hspace{1em}\dot{p}={\left(H^T\right)}^{+}{\dot{z}}_f$$

(42)

The SOF controllers presented above should be optimized with LQCF given in (21). In other words, it is necessary to find the gain matrices K₁, K₂, K₃, K₄ and K₅ of the SOF controllers which minimize the LQCF, J. Let denote those SOF controllers optimized with LQCF as LQSOF1, LQSOF2, LQSOF3, LQSOF4 and LQSOF5, respectively. To date, there have been no analytical methods to optimize the gain matrices in terms of J [28,29,30]. Furthermore, the gain matrices, K₁, K₃, K₃, K₄ and K₅ are structured. This makes the problem more difficult to solve.

If one of K_SOF1, K_SOF2, K_SOF3, K_SOF4 and K_SOF5 is selected, let this be K_s, as given in (43). With K_s, A and B₂, the closed-loop system matrix, A_c, is calculated as (43). For the purpose of finding the optimum of K_s, the optimization problem is formulated as (44) [30,31,32,39,42]. To solve this problem, the covariance matrix adaptation–evolutionary strategy (CMA-ES) is applied as the meta-heuristic algorithm [36].

$$A_c=A+B_2{K}_S,\hspace{1em}{K}_S\in \left[K_{SOF1},K_{SOF2},K_{SOF3},K_{SOF4},K_{SOF5}\right]$$

(43)

$$\begin{array}{l}\underset{K_S}{\min } \mathrm{trace}\left(P_s\right)\hspace{1em}\hfill \\ s.t.\hspace{1em}\left\{\begin{array}{l}{P}_s=P_s^T>0\\ \max \left(\mathrm{Re}\left[A_c\right]\right)<0\\ A_c^T{P}_s+P_s{A}_c+Q+K_S^T{N}^T+N{K}_S+K_S^{T}R{K}_S=0\end{array}\right.\hfill \end{array}$$

(44)

3. Simulation and Discussion

In this section, frequency response analysis is carried out and simulation is conducted to assess the performance of five SOF controllers, i.e., LQSOF1, LQSOF2, LQSOF3, LQSOF4 and LQSOF5. With simulation results, those controllers are compared to one another in view of a_z, ϕ and θ.

3.1. Simulation Environment

In the simulation, the target vehicle is 8 × 8 armored combat vehicle, a built-in model provided in TruckSim [44]. The parameters depicted in Figure 1 were referred from the target vehicle in TruckSim. In the LQCF, J, the MAVs are given in

Table 1. The maximum allowable values (MAVs) in LQCF.

MAV	Value	MAV	Value	MAV	Value
ξ1	0.5 m/s2	ξ2	20.0 deg/s2	ξ3	20.0 deg/s2
ξ4	20.0 deg/s	ξ5	20.0 deg/s	ξ6	0.5 deg
ξ7	0.5 deg	ξ8	0.1 m	ξ9	0.1 m
ξ10	20,000.0 N

. As shown in Table 1, the MAVs, ξ₁, ξ₆ and ξ₇, were set low for the purpose of reducing a_z, ϕ and θ.

The five LQ SOF controllers were implemented by MATLAB/Simulink and simulated on the environment with MATLAB/Simulink and TruckSim. In this paper, the bandwidth of the active actuator is set to 10 Hz, and there were no limits or saturations on the maximum force generated by the actuator.

For the simulation on TruckSim, four road profiles were selected. Figure 4 shows these road profiles, provided in TruckSim. The first is Left/Right Bumps (LRBs). The second is Chassis Twisted Sine Wave Road (CTSWR) where the wavelength and the maximum height are correspondingly 14 m and 0.15 m. The third and fourth are Bounce Sine Sweep Road (BSSR) and Roll Cross Slope Sine Sweep Road (RCSSSR), respectively. The first two profiles are used to check the time responses, and the last two ones are used to analyze the frequency responses. The vehicle passes those road profiles at speeds of 25, 15, 40 and 20 km/h, respectively. On those road profiles, the initial vehicle speed is maintained over the simulation period by a built-in speed controller in TruckSim.

Table 2. Summary of the optimization results for LQ SOF controllers.

Controller	J	Number of Outputs	Number of Gains
LQR	12,378	22	96
LQSOF1	38,136	3	3
LQSOF2	38,343	4	5
LQSOF3	42,005	8	2
LQSOF4	45,995	4	2
LQSOF5	37,589	6	6

shows the optimization results of LQ SOF controllers. As shown in Table 2, LQSOF1, LQSOF2 and LQSOF5 are nearly identical to each other. This is caused by the fact that these two controllers use the Lotus modal control. Among them, LQSOF5 shows the best performance because it has 6 gain elements and uses z_c, ϕ and θ, which are nearly impossible to measure in real 8 × 8 ACVs. In other words, the differences between LQSOF1, LQSOF2 and LQSOF5 were caused by the number and type of sensor signals. LQSOF3 and LQSOF4 are worse than LQSOF1 and LQSOF2 because they use a smaller number of controller gains, as shown in Table 2. In view of measurement and implementation, LQSOF1 is preferred to the others because it requires three signals that can be easily measured with IMU on the SPM and has only three gains. In view of CDC consisting of LQSOF2, LQSOF3 and LQSOF4, LQSOF2 shows the minimum J.

3.2. Frequency Response Analysis

To assess the performance of the LQ SOF controllers, a frequency response analysis was carried out. The frequency response was drawn with the SSE of the FCM. In the frequency responses, the input is z_r1 and the outputs are a_z, ϕ and θ. LQR is used as a baseline, and LQSOF5 is used for comparison.

Figure 5 shows the frequency response plots of the LQ SOF controllers from z_r1 to a_z, ϕ and θ. As shown in Figure 5 and Table 2, LQ SOF1 and LQSOF2 are nearly identical to each other, and LQSOF3 and LQSOF4 are inferior and superior to LQSOF1 and LQSOF2 in terms of ϕ and θ, respectively. Among the LQ SOF controllers, LQSOF5 is the best in terms of a_z and ϕ, and LQSOF4 is the worst in terms of a_z and ϕ, and the best in terms of θ. On the contrary, LQSOF1 is the best in terms of a_z and ϕ, and the best in terms of θ. As shown in Figure 5, LQSOF5 is nearly identical to LQSOF1. From those plots, it can be known that CDC, i.e., LQSOF2, LQSOF3 and LQSOF4, can provide better performance in terms of θ.

3.3. Simulation on Left/Right Bumps

The simulation was carried out on LRBs given in TruckSim. The initial vehicle speed was set to 25 km/h. Figure 6 shows the simulation results of the LQ SOF controllers on LRBs.

Table 3. Summary of the simulation results of LQ SOF controllers on LRBs in TruckSim.

Controller	J	Max |az| (m/s2)	Max |ϕ| (deg/s)	Max |θ| (deg/s)	Max |Force| (N)
No Control		5.9	3.4	0.9
LQSOF1	38,136	2.9	2.2	0.5	60,493
LQSOF2	38,343	2.8	2.2	0.7	56,021
LQSOF3	42,005	2.6	2.5	0.3	54,317
LQSOF4	45,995	4.2	2.6	0.4	51,870
LQSOF5	37,589	3.1	1.8	0.9	75,936

summarizes the results from Figure 6. As shown in Figure 6 and Table 3, compared to the uncontrolled case, the LQ SOF controllers reduced a_z, ϕ and θ by 47%, 29% and 47% on average, respectively. Among them, LQSOF3 shows the best performance in terms of a_z and θ, contrary to what is expected in Figure 5. LQSOF5, introduced for comparison, shows the best performance in controlling ϕ and has little effect on θ while requiring the largest control input, as shown in Table 3.

3.4. Simulation on Chassis Twisted Sine Sweep Road

Simulation was carried out with the LQ SOF controllers on CTSSR at the speed of 15 km/h. Figure 7 shows the simulation results of the LQ SOF controllers.

Table 4. Summary of the simulation results of LQ SOF controllers on CTSWR in TruckSim.

Controller	J	Max |az| (m/s2)	Max |ϕ| (deg/s)	Max |θ| (deg/s)	Max |Force| (N)
No Control		6.6	7.8	3.5
LQSOF1	38,136	5.5	5.5	1.9	175,713
LQSOF2	38,343	3.7	5.6	2.1	178,898
LQSOF3	42,005	4.8	4.8	1.1	99,518
LQSOF4	45,995	6.1	6.1	1.2	109,376
LQSOF5	37,589	3.5	5.0	1.4	169,789

summarizes the simulation results from Figure 7.

As shown in Figure 7 and Table 4, compared to the uncontrolled case, the LQ SOF controllers reduced a_z, ϕ and θ by 24%, 16% and 54% on average, respectively. LQSOF5, introduced for comparison, shows the best performance in controlling a_z, as expected in Figure 5. Among them, the roll angles were slightly reduced compared to a_z and θ. This is caused by the fact that the wheels were lifted due to high road profiles. In other words, the vertical tire force became zero when the left or right wheels were lifted by upward bumps. Due to this phenomenon, the active actuators have little effect on controlling the roll motion of the SPM.

As shown in Table 2, the LQ SOF controllers proposed in this paper have a smaller number of gains than LQR. As shown in Figure 6 and Figure 7, those LQ SOF controllers show good performance for suspension control in terms of ride comfort, durability and turret stabilization in spite of a smaller number of gains. This is the key contribution of this paper.

3.5. Simulation on Bounce Sine Sweep and Roll Cross Slope Sine Sweep Roads

For frequency response analysis on simulation results, simulation was carried out with the LQ SOF controllers on BSSR and RCSSSR at speeds of 40 km/h and 20 km/h, respectively. The former is used to assess the performance along the vertical and pitch motions of the SPM, and the latter is used to assess the performance along the vertical and roll motions of the SPM.

Figure 8 shows the frequency responses of a_z and θ obtained from simulation results of the LQ SOF controllers on BSSR. As shown in Figure 8, a_z and θ were reduced by the LQ SOF controllers below 7 Hz. However, under high-frequency excitation over 10 Hz, the performance of the LQ SOF controllers severely deteriorated in terms of a_z. On the contrary, the performance of those controllers on θ was maintained over 10 Hz except for the frequency range within 5 ~ 8 Hz. In particular, LQSOF4 and LQSOF5 show good performance in terms of θ over 10 Hz.

Figure 9 shows the frequency responses of a_z and ϕ obtained from simulation results of the LQ SOF controllers on RCSSSR. As shown in Figure 9, a_z was reduced by the LQ SOF controllers below 6 Hz. On the contrary, the performance of those controllers on ϕ was not significant.

3.6. Discussions with Frequency Response Analysis and Simulation Results

As mentioned earlier, this paper proposes CDC with LQSOF controllers with active suspension for 8 × 8 ACVs. The LQ SOF controllers proposed in this paper require much smaller numbers of sensor signals and gain elements, compared to LQR. This is the key contribution of this paper. From frequency response analysis, the LQ SOF controllers show good performance, close to LQR, for ride comfort, durability and turret stabilization. From simulation results, those controllers show good performance in terms of a_z and θ.

In this paper, the reason why the active suspension is selected as an actuator is that the LQ SOF controllers with the active suspension can give a guideline to design a controller with another actuator such as a semi-active one or MR damper in terms of the bandwidth and the maximum force. Compared to the maximum forces given in Table 3, the LQ SOF controllers on CTSSR show very large control inputs, as shown in Table 4. This is caused by larger disturbance, compared to LRBs. In particular, the maximum forces of LQSOF1 and LQSOF2 were too large to be realized in real 8 × 8 ACV.

In future research, CDC with LQ SOF controllers proposed in this paper will be applied to recently developed heavy 8 × 8 ACVs. As described in the simulation results, the maximum actuator force used for control was quite large, which is not easy to implement in real 8 × 8 ACVs. To overcome the problem, an MR damper can be selected as an actuator. In addition, when applying CDC with LQ SOF controllers can be obtained by using several controller design methodologies such as H_∞ control and sliding mode control. In future research, another controller design methodology will be applied to calculate those control inputs. For this purpose, the methods presented in this paper can be used as a baseline when designing a controller with MR damper for real 8 × 8 ACVs.

4. Conclusions

This paper presented a method to design the active suspension controller for 8 × 8 armored combat vehicles. With the state-space model derived from the 11-DOF full-car model and LQ cost function for ride comfort, LQR is designed. To avoid the full-state feedback of LQR, the corner damping control (CDC) with SOF control structure was proposed. Four SOF controllers from the CDC framework were proposed based on the state-space model. As given in Table 2, CDC with SOF controllers has a much smaller number of gains, compared to LQR. To find the gains of those controllers that minimize the LQ cost function, the heuristic optimization method was applied. This is the first approach to apply LQOC methodology to the design of active suspension controllers for 8 × 8 ACVs. To verify the performance of the LQ SOF controllers, frequency response analysis was carried out with the state-space equation and simulation was conducted with the vehicle simulation package, TruckSim, on four types of road profiles. From the frequency response analysis and the simulation results, it was drawn that the LQ SOF controllers designed with the proposed CDC and SOF structures are effective in terms of ride comfort, durability and turret stabilization.

In future research, CDC with LQ SOF controllers will be applied with a semi-active suspension to recently developed heavy 8 × 8 ACVs. Recently developed MR damper can be used as an actuator for real 8 × 8 ACVs. As a controller design methodology, H_∞ control and sliding mode control will be adopted in future research. For this purpose, the method presented in this paper can be used as a baseline when designing a controller with MR damper for real 8 × 8 ACVs.