# Reliability Analysis of Hybrid Laser INS Under Multi-Mode Failure Conditions
## 3 Reliability Analysis Model for the Hybrid INS Based on Kriging Surrogate Model
### Representation of Random Uncertainty  
According to the design requirements for the mixed inertial navigation system, the structural stiffness of the base, platform, and internal and external frames must be increased to reduce the deviation of the platform coordinate system from the true inertial coordinate system due to elastic deformation under overload conditions such as impacts  
Figure 10: Construction of hybrid inertial navigation kriging proxy model. and vibrations. The internal and external frames, serving as the direct structural components connecting the platform and the shaft-end assembly as well as the base assembly, have high requirements for part stiffness, strength, dimensions, and tolerances. However, significant uncertainties exist in the design and manufacturing processes of these parts. This uncertainty mainly arises from manufacturing process errors related to the material property parameters selected for the components. Taking aluminum alloy as an example, its forming performance is related to the alloy composition ratio, which can result in significant strength variations among different series of aluminum alloys. Additionally, during the structural processing of the components, factors such as manufacturing environment, technical conditions, and installation errors lead to uncertainties in geometric dimensions. This paper selects key dimensional parameters and material performance parameters during the part design process as the subjects of study. Combining engineering experience with existing data, it is known that the elastic moduli of the materials for the outer frame \(E_{Outer\ Frame}\), inner frame \(E_{Inner\ Frame}\), and foundation support \(E_{Founation\ Support}\), as well as the thicknesses \(H_{Outer\ Frame}\), \(W_{Outer\ Frame}\), \(H_{Inner\ Frame}\), and \(W_{Inner\ Frame}\) of the outer and inner frames, follow a normal distribution. The specific distribution information is presented in Table 4. At the same time, since the outer frame assembly is a critical load-bearing component connecting the base and the platform, this paper emphasizes selecting key parameters from the outer frame design process and interface dimensions with the shaft-end and platform components. These parameters include the roundness \(R\) of the outer frame, thickness \(H\) of the outer frame, width \(B\) of the outer frame, mounting interface \(D\) of the platform, motor mounting interface 1 \(d_{1}\), and motor mounting interface 2 \(d_{2}\), as illustrated in Figure 11. The specific distribution information for these parameters is presented in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter Name** & **Mark** & **Unit** & **Mean Value \(\mu_{X_{i}}\)** & **Standard Deviation \(\sigma_{X_{i}}\)** & **Distribution Type** \\ \hline \(E_{Outer\ Frame}\) & \(X_{1}\) & Pa & 7.8 \(\times\) 10\({}^{10}\) & 15.6 \(\times\) 10\({}^{9}\) & Normal distribution \\ \hline \(E_{Inner\ Frame}\) & \(X_{2}\) & Pa & 7.8 \(\times\) 10\({}^{10}\) & 15.6 \(\times\) 10\({}^{9}\) & Normal distribution \\ \hline \(E_{Founation\ Support}\) & \(X_{3}\) & Pa & 5.9 \(\times\) 10\({}^{10}\) & 11.8 \(\times\) 10\({}^{9}\) & Normal distribution \\ \hline \(H_{Outer\ Frame}\) & \(X_{4}\) & mm & 7.00 & 0.14 & Normal distribution \\ \hline \(W_{Outer\ Frame}\) & \(X_{5}\) & mm & 60.00 & 1.20 & Normal distribution \\ \hline \(H_{Inner\ Frame}\) & \(X_{6}\) & mm & 7.90 & 0.16 & Normal distribution \\ \hline \(W_{Inner\ Frame}\) & \(X_{7}\) & mm & 65.00 & 1.30 & Normal distribution \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distribution of mixed inertial navigation structural size parameters and material performance parameters.