# Reliability Analysis of Hybrid Laser INS Under Multi-Mode Failure Conditions
## 3 Reliability Analysis Model for the Hybrid INS Based on Kriging Surrogate Model
### Reliability Analysis Model for Mixed Inertial Navigation Based on Failure
Consideration of Structural Strength Failure Criteria as the Failure Criterion for the Outer Frame Assembly Under Overload Conditions  
This paper considers the strength failure criterion as the failure criterion for the outer frame assembly structure under overload conditions, utilizing both the maximum stress criterion and the maximum strain criterion to determine whether the outer frame structure has failed. The maximum stress criterion states that material failure occurs when the maximum stress under the applied load reaches the material's strength limit. The maximum \begin{table}
\begin{tabular}{c c} \hline \hline
**Failure Indicators** & **Failure Boundary** \\ \hline \(\theta_{X}^{*}\) & \(\leq\)12\({}^{*}\) \\ \(\theta_{Y}^{*}\) & \(\leq\)12\({}^{*}\) \\ \(\theta_{Z}^{*}\) & \(\leq\)12\({}^{*}\) \\ \(n_{need}\) & \(\geq\)2.0 \\ \(\delta_{need}\) & \(\geq\)0.052 mm \\ \(W_{need}\) & \(\geq\)0.8 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Related data for mixed inertial navigation function failure. strain criterion indicates that material failure occurs when the maximum strain under the applied load reaches the material's strength limit. In this paper, the maximum stress criterion is defined using the ratio of maximum stress to allowable stress as the safety factor (Failure Indicator 1). The maximum strain criterion uses the maximum strain of the structure under overload as Failure Indicator 2. Additionally, with the development and design of inertial measurement devices trending towards lightweight and miniaturization, mass is selected as a critical metric (Failure Indicator 3). The corresponding performance functions are denoted as \(g_{4}(x),\ g_{5}(x),\ g_{6}(x)\), where \(x=[x_{8},x_{9},x_{10},x_{11},x_{12},x_{13}]\). The function corresponding to failure under the maximum stress criterion is Equation (17):  
\[g_{4}(x)=n_{F_{stress,\max}}-n_{need} \tag{17}\]  
The function corresponding to failure under the maximum strain criterion is Equation (18):  
\[g_{5}(x)=\delta_{deformation\_\max}-\delta_{need} \tag{18}\]  
The function corresponding to mass failure is Equation (19):  
\[g_{6}(x)=W_{quality\_\max}-W_{need} \tag{19}\]  
Addressing the uncertainty in the parameters of the outer frame structure, this paper first separately considers the two failure modes: failure under the maximum stress criterion and failure under the maximum strain criterion. A comparative analysis of the performance of the outer frame structure using aluminum alloy and SiCp/Al composite materials is conducted, with the outer frame roundness \(R\), thickness \(H\), width \(B\), platform mounting interface \(D\), motor mounting interface 1 \(d_{1}\), and motor mounting interface 2 \(d_{2}\) as inputs. As shown in Figure 3, the outputs are the safety factor corresponding to the maximum stress and the maximum strain under mechanical overload conditions. A Kriging surrogate model is established to accurately predict these output indicators. Based on the Kriging surrogate model, the Sobol sequence sampling method is employed to simulate the random uncertainties of each design parameter and study their impact on failure probability under different failure modes. Moreover, to comprehensively assess the performance of aluminum alloy and SiCp/Al materials, it is important to note that the occurrence of any one failure mode signifies the failure of the outer frame structure. Therefore, the three failure modes are treated as a series relationship, and the overall performance function can be expressed as Equation (20):  
\[Z(x)=\min(g_{1}(x),g_{2}(x),g_{3}(x)) \tag{20}\]  
If \(g(x)\leq 0\), the mixed inertial navigation system experiences functional failure. The failure domain \(F\) for the mixed inertial navigation function can be expressed as Equation (21):  
\[F=\{x:g(x)\leq 0\} \tag{21}\]  
The probability of failure \(P_{f}\) representing the system output response falling within the failure probability (i.e., the failure domain \(F\)) is provided by Equation (22):  
\[P_{f}=P(x:x\in F) \tag{22}\]