# 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  
In reliability theory, the performance function is a function of basic random variables. Let \(Y=g(x)\) be the performance function of the mixed inertial navigation structure, where \(x=[x_{1},x_{2},\cdots x_{m}]\) represents an \(m\)-dimensional random variable with a joint probability density function \(f(x)\). The failure domain is defined as Equation (10):  
\[F=\{x:g(x)\leq 0\} \tag{10}\]  
The indicator function (Equation (11)) for the failure domain is \(I_{F}(x)\):  
\[I_{F}(x)=\left\{ \begin{array}{c}1\ x\in F\\ 0\ x\
otin F\end{array} \right\} \tag{11}\]  
Thus, the failure probability can be expressed as Equation (12):  
\[P_{f}=\int_{F}f(x)dx=\int_{R^{n}}I_{F}(x)f(x)dx=E(I_{F}) \tag{12}\]  
Using the Monte Carlo method in reliability theory, the failure probability of the mixed inertial navigation system can be computed using the mathematical expectation of the failure domain indicator function (13):  
\[P_{f}=\frac{1}{N}\sum_{j=1}^{N}I_{F}(x_{j})=\frac{N_{f}}{N} \tag{13}\]  
where \(N\) is the total number of samples, and \(N_{f}\) is the number of samples falling into the failure domain. This paper establishes a reliability analysis model for the mixed inertial navigation system based on failure probability, focusing on the following two aspects:  
Figure 11: Structural design parameters of the outer frame. 3.1. Considering the Failure Criterion for the Navigation Function of the Mixed Inertial Navigation System Under Overload Conditions  
As shown in Figure 9, when the coordinate system of the inertial navigation system (\(o-x_{bt}y_{bt}z_{bt}\)) deviates from the virtual body coordinate system (\(o-x_{bt0}y_{bt0}z_{bt0}\)) beyond a certain range during impact overload conditions, it is determined that the navigation function of the mixed inertial navigation system has failed. By examining the \(X\), \(Y\), and \(Z\) axes of the mixed inertial navigation system, the degree of deviation of the coordinate systems due to elastic displacements under large overload impacts is studied. Failure functions for each of the three axes are established. According to the requirements of a certain new type of mixed laser inertial navigation system and performance enhancement task specification, the allowable fluctuation errors \(\theta_{X}^{*}\), \(\theta_{Y}^{*}\), and \(\theta_{Z}^{*}\) in the \(X\), \(Y\), and \(Z\) axes due to elastic displacement are set, as detailed in Table 6. Among them, it is required that the maximum installation error of instruments in three directions installed in a mechanical environment should not exceed 8 inches, the safety factor of the outer frame in a mechanical environment should not be less than 2.0 (safety factor = ultimate stress/allowable stress), the deformation displacement should not exceed 0.052 mm, and the weight should not exceed 0.8 Kg. To ensure the normal operation of the mixed inertial navigation system, it is crucial that the fluctuation range of the three coordinate axes of the inertial navigation system under impact overload conditions meets the requirements for stable navigation functionality. The failure function \(g(x)\) of the mixed inertial navigation system is thus established, as detailed in Equations (14)-(16), with the corresponding functions being \(g_{1}(x),\ g_{2}(x),\ g_{3}(x)\), where \(x=[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}]\). The function corresponding to the failure of the \(X\)-axis inertial navigation function is Equation (14):  
\[g_{1}(x)=\theta_{X}^{*}-|\theta_{Xbt}-\theta_{Xbt0}| \tag{14}\]  
The function corresponding to the failure of the \(Y\)-axis inertial navigation function is Equation (15):  
\[g_{2}(x)=\theta_{Y}^{*}-|\theta_{Ybt}-\theta_{Ybt0}| \tag{15}\]  
The function corresponding to the failure of the \(Z\)-axis inertial navigation function is Equation (16):  
\[g_{3}(x)=\theta_{Z}^{*}-|\theta_{Zbt}-\theta_{Zbt0}| \tag{16}\]  
3.2.