# Design and Stability Analysis of Six-Degree-of-Freedom Hydro-Pneumatic Spring Wheel-Leg
## 4 Analysis of Horizontal Rotational Stability
### Displacement Load Variation of Hydro-Pneumatic Springs
#### 4.1.2 Displacement Load Variation of Right Hydro-Pneumatic Spring  
At the static equilibrium position, the horizontal distance between the upper and lower ball joints of the right hydro-pneumatic spring is given as  
\[b_{0}=\sqrt{s_{0}^{2}-h_{c}^{2}}. \tag{23}\]  
The angle between the line connecting the lower ball joint and the center of rotation and the line connecting the upper ball joint and the center of rotation at the static equilibrium position is given as  
\[\varphi=\arccos\left(\frac{l_{c}^{2}+r_{c}^{2}-h_{0}^{2}}{2l_{c}r_{c}}\right). \tag{24}\]  
After rotation by an angle \(\theta\), the horizontal projection length of the hydro-pneumatic spring is given as  
\[b_{1}=\sqrt{l_{c}^{2}+r_{c}^{2}-2l_{c}r_{c}cos(\varphi+\theta)}. \tag{25}\]  
Thus, the changed length of the hydro-pneumatic spring is given as  
\[s_{2}=\sqrt{b_{1}^{2}+h_{c}^{2}}. \tag{26}\]The rotational lever arm corresponding to the right hydro-pneumatic spring is given as  
\[\begin{cases}\beta=\arccos\left(\frac{\left(\frac{\beta_{c}^{2}+b_{1}^{2}-\tau_{c}^{ 2}}{2L_{b}b_{1}}\right)}{2L_{b}b_{1}}\right).\\ L_{2}=l_{c}\sin\beta\end{cases} \tag{27}\]  
According to Equation (25), the load on the left hydro-pneumatic spring after rotation by an angle \(\theta\) is expressed as  
\[F_{2}=\frac{P_{A0}\cdot S_{A}}{\left(1-\frac{\left(\beta_{0}-\tau_{c}\right)D ^{2}}{h_{0}d^{2}}\right)^{n}}. \tag{28}\]  
Thus, the unstable rotational torque provided by the left hydro-pneumatic spring is given as  
\[N_{2}=F_{2}L_{2}\frac{b_{1}}{s_{2}}. \tag{29}\]  
Summarizing the above equations, with the variation of the horizontal rotation angle \(\theta\), the unstable rotational torque is given as  
\[\begin{split}&\text{N}_{y}=2(N_{1}+N_{2})=2\Big{(}-F_{1}L_{1} \frac{a_{1}}{s_{1}}+F_{2}L_{2}\frac{b_{1}}{s_{2}}\Big{)}\\ &=\frac{-2L_{1}P_{A0}\cdot S_{A}a_{1}}{s_{1}\left(1-\frac{\left( \sqrt{2}+\tau_{c}^{2}-2l_{c}\cos(\varphi-\theta)+b_{c}^{2}-\tau_{c}}{h_{0}d^{2 }}\right)^{n}}+\frac{2L_{2}P_{A0}\cdot S_{A}b_{1}}{s_{2}\left(1-\frac{\left( \sqrt{2}+\tau_{c}^{2}-2l_{c}\cos(\varphi+\theta)+b_{c}^{2}-\tau_{c}}{h_{0}d^{2 }}\right)\right)^{n}}}{h_{0}d^{2}}\Big{)}^{n}\end{split} \tag{30}\]  
The initial parameters from Table 2 were input into the model, and the resulting curve of hydraulic rod force versus the rotation angle is shown in Figure 9(a). The curve of the hydraulic rod extension versus the rotation angle is shown in Figure 9(b). The curves of the rotational torque, resistance torque, and resultant torque versus the rotation angle are presented in Figure 9(c).  
As indicated in Figure 10, in the absence of the spring's action, the hydro-pneumatic springs of the six-arm structure varied with horizontal rotation. After deviating from the initial position, during counterclockwise rotation, the rotational lever arm of the left hydro-pneumatic spring decreased but the elastic force increased more rapidly. This was due to the nonlinear stiffness characteristics of the hydro-pneumatic springs. Ultimately, within a certain range, there existed a resisting torque during counterclockwise rotation. It indicates that the six-arm hydro-pneumatic spring structure had a strong self-stabilizing property.