## 2 Material and Methods
### Dynamic Model of Hydroneumatic Suspension
#### 2.1.2 Seven-DOF Model  
The seven-DOF vehicle model system is a strongly coupled multiple-input and multiple-output system. Figure 3 illustrates the structural diagram of a vehicle with seven DOFs. The seven DOFs are as follow: vertical displacement \(z_{s}\), roll angle \(\theta\), pitch angle \(\varphi\), and \(A,\ B,\ C,\ \text{and}\ D\), which are the four positions under the spring quality of vertical displacement for \(z_{ui}\ (i=1,\ 2,\ 3,\ 4)\). In Figure 3, \(m_{s}\) is the body quality; \(J_{x}\) represents the pitching moment of inertia; \(J_{y}\) represents the roll moment of inertia; \(m_{ui}\ (i=1,\ 2,\ 3,\ 4)\) indicates the four unsprung mass locations, denoted as \(A,\ B,\ C,\ \text{and}\ D\); \(k_{si}\ (i=1,\ 2,\ 3,\ 4)\) indicates the four locations (\(A,\ B,\ C,\ \text{and}\ D\))' suspension spring stiffness; \(c_{si}\ (i=1,\ 2,\ 3,\ 4)\) corresponds to the four positions (\(A,\ B,\ C,\ \text{and}\ D\))' suspension shock absorber damping; \(k_{ti}\ (i=1,\ 2,\ 3,\ 4)\) is the stiffness of four tires; \(z_{gi}\ (i=1,\ 2,\ 3,\ 4)\) represents the displacement of four tires; \(f_{i}\ (i=1,\ 2,\ 3,\ 4)\) represents the controllable output force; \(t_{f}\) and \(t_{r}\) represent half of the track width for the front and rear suspensions, respectively; and \(a\) and \(b\) are the distances from the center of mass to front and rear axes, respectively.  
When the car body's vibration is small, \(\sin\theta\approx\theta\) and \(\sin\varphi\approx\varphi\). The vertical displacement of bodies A, B, C, and D \(z_{si}\) is as follows:  
\[\begin{split} z_{s1}&\approx z_{s}-a\theta+t_{f} \varphi\\ z_{s2}&\approx z_{s}-a\theta-t_{f}\varphi\\ z_{s3}&\approx z_{s}+b\theta+t_{r}\varphi\\ z_{s4}&\approx z_{s}+b\theta-t_{r}\varphi\\ \end{split} \tag{1}\]  
Figure 3: Seven-DOF model.  
Figure 2: Active X-type interconnected hydropneumatic suspension structure: 1—hydraulic pump; 2—safety valve; 3—servo valve; and 4—angle sensor.  
If we define \(Z_{m}=[\,z_{s}\,\theta\,\varphi\,]\), \(Z_{s}=[z_{s2}\,z_{s2}\,z_{s3}\,z_{s4}]\). Adopting matrix notation allows for a simplified representation, as follows:  
\[Z_{s}=L^{T}Z_{m} \tag{2}\]  
where \(L\) is the transformation matrix, which can be expressed as  
\[L=\begin{bmatrix}1&1&1&1\\ -a&-a&b&b\\ t_{f}&-t_{f}&t_{r}&-t_{r}\end{bmatrix} \tag{3}\]  
Using [PERSON]'s second law, a differential equation governing the seven-DOF vibrations of the vehicle is formulated. Specifically, the differential equation related to vertical vibrations is as follows:  
\[M_{s}\ddot{z}_{s}-\sum\limits_{i=1}^{4}(F_{ksi}+F_{csi}+f_{i})=0 \tag{4}\]  
The differential equation of pitch motion in the vertical plane can be formulated as follows:  
\[J_{x}\ddot{\theta}+a\sum\limits_{i=1,2}(F_{ksi}+F_{csi}+f_{i})-b\sum\limits_{i =3,4}(F_{ksi}+F_{csi}+f_{i})=0 \tag{5}\]  
The differential equation of the car body's roll motion can be formulated as follows:  
\[\begin{array}{l}J_{y}\ddot{\varphi}+t_{f}\sum\limits_{i=1,2}(-1)^{i}(F_{ksi} +F_{csi}+f_{i})+t_{r}\sum\limits_{i=3,4}(-1)^{i}(F_{ksi}+F_{csi}+f_{i})\\ +\frac{k_{\varphi f}}{t_{f}}\sum\limits_{i=1}^{2}(-1)^{i}z_{\varphi i}+\frac{k _{\varphi f}}{t_{r}}\sum\limits_{i=3}^{4}(-1)^{i}z_{\varphi i}=0\end{array} \tag{6}\]  
The differential equation of the unsprung mass' vibration can be formulated as follows:  
\[\begin{array}{l}m_{u1}\ddot{z}_{u1}-F_{kt1}+F_{ks1}+F_{cs1}+f_{1}+\frac{k_{ \varphi f}}{t_{r}^{2}}(z_{\varphi 1}-z_{\varphi 2})=0\\ m_{u2}\ddot{z}_{u2}-F_{kt2}+F_{ks2}+F_{cs2}+f_{2}-\frac{k_{\varphi f}}{t_{r}^{ 2}}(z_{\varphi 1}-z_{\varphi 2})=0\\ m_{u3}\ddot{z}_{u3}-F_{kt3}+F_{ks3}+F_{cs3}+f_{3}+\frac{k_{\varphi f}}{t_{r}^{ 2}}(z_{\varphi 3}-z_{\varphi 4})=0\\ m_{u4}\ddot{z}_{u4}-F_{kt4}+F_{ks4}+F_{cs4}+f_{4}-\frac{k_{\varphi f}}{t_{r}^ {2}}(z_{\varphi 3}-z_{\varphi 4})=0\end{array} \tag{7}\]  
In Equations (4)-(7), \(F_{ksi}\) (i = 1, 2, 3, 4) is the hydropneumatic spring force, \(F_{csi}\) (i = 1, 2, 3, 4) provides the damping force, and \(F_{kit}\) (i = 1, 2, 3, 4) provides the force of tire stiffness.