## 6 Reliability Analysis for Transposition System  
### Reliability Evaluation  
According to the transposition system in Figure 2, coordinates \((X_{1},Y_{1})\), \((X_{2},Y_{2})\), \((X_{3},Y_{3})\), and \((X_{4},Y_{4})\), are the vital parameters for the gears' center positions. Parameters \(X_{1}\), \(Y_{1}\), \(X_{2}\), \(Y_{2}\), \(X_{3}\), \(Y_{3}\), \(X_{4}\), and \(Y_{4}\) can be regarded as eight independent random variables. The mean value \(\mathbf{\mu}_{X1}\) (\(i=1,2,\ldots,8\)) is listed in Table 1. Based on the mean values, the center distances of each gear are \(a_{1}\)= 53.59 mm, \(a_{2}\) = 56.87 mm, and \(a_{3}\) = 95.50 mm, respectively. Considering the manufacturing precision, the limit deviation of each gear center is set as 0.21 mm, and the standard deviation \((\mathbf{\mu}_{X2i})^{0.5}\) of each coordinate can be set as 0.035 mm. Also, other statistical moments of the transposition system are listed in Table 2. Based on Tables 1 and 2, the first four moments can be used for the description of the distribution range of each gear. Considering the maximum allowed angle error \(\xi\) = 0.04\({}^{\circ}\) of \(Z\) = \(g_{X}(\mathbf{X})\) = \(\xi\) - \(E\), Equations (16) and (17) are utilized for the system reliability of the transposition mechanism in SFW2512. The reliability index can be calculated, \(\beta\) = 2.1393 and \(R\) = 0.983795, which is compared with the Monte Carlo simulation [39; 40] with 10\({}^{7}\) samples. Comparing them with the FOSM (First-Order Second-Moment) and AFOSM (Advanced First-Order Second-Moment) methods, the results of the system reliability evaluation are listed in Table 3. The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(\mathbf{X_{1}}\) & \(\mathbf{Y_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{X_{4}}\) & \(\mathbf{Y_{4}}\) \\ \hline
3 rd Moment \(\times\) 10\({}^{-3}\) & 1.5 & 1.5 & 2.0 & 2.0 & 3.0 & 3.0 & 3.5 & 3.5 \\
4 th Moment \(\times\) 10\({}^{-5}\) & 6.0 & 6.0 & 6.0 & 6.0 & 8.0 & 8.0 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Moments of random variables in transposition system. results calculated with the proposed method for the transposition system are in agreement with the results from the Monte Carlo simulation with \(10^{7}\) samples. The failure probability is an important factor in evaluating the calculation error. Based on Table 3, the probability values of the proposed method and the Monte Carlo simulation are \(P_{f1}\) = 0.0162 and \(P_{f2}\) = 0.0169. The relative error of failure probability is 4.32%. Regardless of the higher statistical characteristics, system reliability is underestimated. Based on the Monte Carlo simulation, the values of the reliability index and the failure probability are shown in Figure 4. The value achieved by the Monte Carlo simulation approaches 0.9831 with the increasing of the sample size. However, a large number of samples is needed for the improvement of calculation accuracy. Results calculated using the proposed method are in agreement with results calculated through the Monte Carlo simulation with \(10^{7}\) and \(10^{8}\) samples. The calculated result is stable, based on a huge number of samples. The Monte Carlo simulation with \(10^{9}\) samples can also maintain the computational accuracy of the reliability index and the failure probability, costing extensive computational resources. Also, the reliability-based sensitivity indices of mean value and covariance are computed as shown in Table 3 and Equation (23), respectively. Compared with the Monte Carlo simulation, the indices of the reliability-based sensitivity are presented in Figures 5 and 6, respectively. \[