## 4 Optimization Design  
### Robust Bi-Objective Optimization Problem Formulation  
A bi-objective robust optimization design was performed to further improve the structural safety and environmental adaptability of the ISU system. The optimization simultaneously addressed two critical performance aspects: maximizing acceleration robustness and reducing the peak side force (\(F_{cs}\)). These objectives are essential for maintaining ride comfort and preventing mechanical failures in varying operational environments. According to previous studies, the operational temperature of hydropneumatic suspension systems can range from as low as \(T=-20\)\({}^{\circ}\)C to as high as \(T=200\)\({}^{\circ}\)C under extreme conditions. However, the actual working range generally lies between \(T=20\)\({}^{\circ}\)C and \(T=120\)\({}^{\circ}\)C under typical operating scenarios [28]. Based on this, three representative  
Figure 12: Vertical and RMS acceleration under sinusoidal excitation (amplitude = 0.05 m, frequency = 3 Hz). temperatures--\(T=20\)\({}^{\circ}\)C, 70\({}^{\circ}\)C, and 120\({}^{\circ}\)C--were selected in this study to assess realistic thermal environments while avoiding extreme edge cases. The dynamic response of the suspension system was significantly affected by temperature and excitation frequency. The typical travel speed range for vehicles, such as tanks that use ISU suspensions, lies between 10 and 50 km/h. Sinusoidal excitation with frequencies between 1 and 3 Hz can effectively cover most off-road driving scenarios, considering that the typical road impact is between 4 and 6 m. Nine operating conditions were rendered by combining selected temperatures with excitation frequencies of 1, 2, and 3 Hz to evaluate its robustness fully. The acceleration robustness metric considers both the average and standard deviation of the RMS acceleration under various temperature and frequency conditions, where the mean value (\(\mu_{\text{RMS}}\)) represents the overall performance level, and the standard deviation (\(\sigma_{RMS}\)) indicates how much the performance fluctuates with environmental changes. A smaller standard deviation implies higher stability and a more robust system under uncertain conditions. Excessive side force (\(F_{cs}\)) presents serious reliability concerns for the ISU system. Accordingly, minimizing the value of \(F_{cs}\) is essential to improve durability and operational stability. Five key design parameters were selected as optimization variables, including crank-arm length (\(L_{1}\)), connecting-rod length (\(L_{3}\)), orifice diameter (\(D_{z}\)), check-valve diameter (\(D_{d}\)), and number of damping orifices (\(nd\)). The RMS acceleration must be lower than 25 m/s\({}^{2}\) to ensure passenger comfort, while \(L_{2}\) must be maintained between 0.13 and 0.19 m to meet installation dimensions. The bi-objective robust optimization problem proposed in this study can mathematically be expressed as follows:  
\[\text{\emph{Design Variables}}:\ \mathbf{v}=\Big{[}L_{1},L_{3},D_{z},D_{d},nd \Big{]}^{T}\]  
\[\text{\emph{Objective Functions}}:\ \left\{ \begin{array}{l}\underset{\mathbf{v}}{min}\ f_{1}( \mathbf{v})=\mu_{\text{RMS}}(\mathbf{v})+\lambda\sigma_{\text{RMS}}(\mathbf{ v}),\\ \underset{\mathbf{v}}{min}\ f_{2}(\mathbf{v})=maxF_{\text{side}}(\mathbf{v}). \end{array} \right.\]  
\[\text{\emph{Constraints}}:\ \left\{ \begin{array}{l}a_{\text{RMS}}(\mathbf{v},T,f) \leq 25\ \text{m/s}^{2}\\ 0.13\ \text{m}\leq L_{2}(\mathbf{v})\leq 0.19\ \text{m}\\ \mathbf{v}_{\text{min}}\leq\mathbf{v}\leq\mathbf{v}_{\text{max}}\end{array} \right.\]  
where the robustness weighting factor \(\lambda\) was set equal to 3. Selecting an appropriate robust weighting factor is crucial to balance the mean performance and its variability in robust optimization. Based on previous research and practical experience, the weighting factor \(\lambda\) usually ranges between 2 and 5.