## 2 System Modelling
### Modelling of the Hydro-Pneumatic Suspension System  
The vehicle studied in this paper is a rescue vehicle with an adjustable hydro-pneumatic suspension system. The corresponding model has six degrees of freedom which represent the vertical, roll, and lateral dynamics of the vehicle. The 6-DOF model consists of two parts, one is a 4-DOF roll motion model (Figure 1a) and the other is a 2-DOF steer-yaw model (Figure 1b). The 4-DOF roll motion model comprises a lumped spring mass, and two unspring masses are developed. The 4-DOF are composed of the translational motions in the vertical direction of the spring mass and the two unspring masses and the roll motion of the spring mass. The 2-DOF of the steering model consist of the lateral and yaw motions of the vehicle.  
For convenience, we give the linear approximation for the damping and stiffness of the hydro-pneumatic suspension by fitting, and the fitted stiffness and damping of the suspension are shown in Table 1. In addition, two tires are modeled by means of two linear springs. The parameters of the rescue vehicle's suspension system are displayed in Table 1.  
According to Figure 2 and [PERSON]'s law, the dynamic motions of the rescue vehicle are modelled as follows.  
Figure 1: Schematic of a six-degrees-of-freedom (6-DOF) rescue vehicle model: (**a**) the 4-DOF roll motion model; (**b**) the 2-DOF steer-yaw model.  
Rolling dynamic motion:  
\[I_{x}\bar{\varphi}=T[(F_{l}+f_{ul})-(F_{r}+f_{ur})] \tag{1}\]  
Lateral motion:  
\[m_{s}v_{x}(\dot{\beta}+\gamma)=F_{fy}-F_{ry}-m_{s}g\varphi \tag{2}\]  
Yaw motion:  
\[I_{z}\dot{\gamma}=F_{fy}\dot{I}_{f}\dot{-}F_{ry}\dot{I}_{r} \tag{3}\]  
where \(\varphi,\gamma,\beta\) represent the roll angle, yaw rate, and side-slip angle, respectively; \(v_{x}\) is the speed of the vehicle;  
Vertical motion equation of the spring mass:  
\[\begin{array}{l}m_{s}\ddot{z}_{s}=F_{l}+F_{r}+f_{ul}+f_{ur}\\ F_{l}=c_{s}(\dot{z}_{ul}-\dot{z}_{sl})+k_{s}(\dot{z}_{ul}-\dot{z}_{sl})\\ F_{r}=c_{s}(\dot{z}_{ur}-\dot{z}_{sr})+k_{s}(\dot{z}_{ur}-\dot{z}_{sr})\end{array} \tag{4}\]  
where \(z_{s}\) denotes vertical displacement of sprung mass; \(f_{ul}\) and \(f_{ur}\) indicate the adjustable control forces of the suspension.  
Vertical motion equation of the unspring mass:  
\[\begin{array}{l}m_{ul}\ddot{z}_{ul}=k_{t}(z_{ql}-z_{ul})-F_{l}-f_{ul}\\ m_{ur}\ddot{z}_{ur}=k_{t}(z_{qr}-z_{ur})-F_{r}-f_{ur}\end{array} \tag{5}\]  
where \(z_{sl}=z_{s}+Tqz_{sr}=z_{s}-T\varphi\), \(z_{ql},z_{qr}\) indicate the vertical displacements of the wheels excited by the road surface, \(z_{ul},z_{ur}\) is the vertical displacements of unspring mass. The subscript \(l\) and \(r\) of the symbols represent the left and right suspension.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Value** & **Unit** \\ \hline Spring mass & \(m_{s}\) & 1525 & kg \\ Unspring mass & \(m_{u}\) & 50 & kg \\ Tread & \(T\) & 1.56 & \(m\) \\ Suspension spring fitting stiffness & \(k_{s}\) & 35,000 & N\(\cdot\) m/rad \\ Suspension spring fitting damping & C\({}_{s}\) & 980 & N\(\cdot\) m \(\cdot\) s/rad \\ Tire vertical elastic stiffness & \(k_{t}\) & 190,000 & N\(\cdot\) m/rad \\ Yaw moment of inertia & \(I_{r}\) & 2500 & kg\(\cdot\)m\({}^{2}\) \\ Roll moment of inertia & \(I_{x}\) & 460 & kg\(\cdot\)m\({}^{2}\) \\ Distance from the center of gravity to the center of the roll & \(h\) & 0.45 & m \\ Distance from the center of gravity to the ground & \(h_{r}\) & 0.5 & m \\ Distance between the center of gravity and the front axle & \(l_{f}\) & 1.27 & m \\ Distance between the center of gravity and the rear axle & \(l_{r}\) & 1.94 & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the rescue vehicle’s suspension system.  
Figure 2: Steering angle of tires in the [PERSON] tire simulation.