## 7 Numerical Examples  
The structural parameters of an improved Delta parallel mechanism are \(r_{i1}=200\) mm, \(r_{i2}=200\) mm, \(a_{i}=200\) mm, \(c_{i}=400\) mm, \(\alpha_{1}=0^{\circ}\), \(\alpha_{2}=120^{\circ}\), \(\alpha_{3}=240^{\circ}\) (\(i=1,2,3\)). The initial posture is: \(\theta_{i1}=40.54^{\circ}\), \(\theta_{i2}=112.33^{\circ}\), \(\theta_{i3}=0^{\circ}\) (\(i=1,2,3\)). The trajectory of point \(P\) is: \(x=-30\sin(120^{\circ}t)\), \(y=-500-20t\), \(z=-30\cos(120^{\circ}t)+30\), and the time is 3 s. Assuming that all the original errors are independent of each other and have a normal distribution with mean and standard deviations shown in Table 2.  
Figure 4 shows the change of the rotation angles of the three driving arms during the movement. When \(\varepsilon_{1}=-0.1\) and \(\varepsilon_{2}=0.1\), the results of the kinematic reliability in the \(x\), \(y\) and \(z\) directions are as shown in Figure 5. The kinematic reliability of the mechanism in the \(y\)-direction is observed to be larger than that in the \(x\) and \(z\) directions corresponding to the given coordinate system and motion trajectory. For the 12 random variables in Table 2, each random variable takes three level values, and 193 samples are obtained based on the same central point applying Box-Behnken sampling as shown in Appendix B (the same central point data are calculated as one). Meanwhile, according to each group of samples, the 193 corresponding output position errors of the mechanism at \(t=0.5\) s are calculated using a virtual prototype simulation and are listed in the last column of Appendix B. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Original Input Errors (\(i=1,2,3\))** & **Mean \(\mu\)** & **Standard Deviation \(\sigma\)** \\ \hline Radius error of moving platform \(\Delta r_{i2}/\)mm & 0.02 & 0.01 \\ Length error of driving arm \(\Delta a_{i}/\)mm & 0.06 & 0.0067 \\ Clearance error of revolute pair \(\Delta\varphi_{i}/\)mm & 0.10 & 0.01 \\ Driving error at any time \(\Delta\theta_{i1}/\)rad & 0.03 & 0.1173 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean and standard deviation of original input errors. According to the data in Appendix B and Equations (4) and (5), the response surface function \(Y\) of the output position error of the improved Delta parallel mechanism in the combined direction is obtained. Assuming that the maximum allowable limit of the mechanism is \(U=1\) mm, the limit state function \(g(x)\) is derived according to Equation (26). \[g(x)=U-Y. \tag{38}\]  
The limit state function \(g(x)\) is shown in Appendix Equation (A1). As shown in Figure 6, the frequency histogram of \(g(x)\) is obtained using the Monte Carlo method based on Equation (38). The \(g(x)\) distribution diagram is shown in Figure 7. The distribution is noted as linear indicating that \(g(x)\) has a normal distribution. Figure 4: Change of the rotation angle \(\Theta_{\mathrm{i1}}\) of three driving arms. Figure 5: Calculation results of kinematic reliability. Figure 6: Frequency distribution histogram of \(g(x)\). In addition, the data \(\mu_{g}\) and \(\sigma_{g}\) are obtained through the Monte Carlo method based on Equation (38). \[\mu_{g}=128.6927\,\sigma_{g}=40.0166\]  
Then we can calculate the reliability index \(\beta\) and reliability \(R\) from Equations (33) and (34). \[\beta=3.216,\ R=0.9994\]  
From Equations (27), (28) and (31), the mean \(\mu_{g}\) of the limit state function is obtained, with its expression shown in Appendix Equation (A2). From Equations (29), (30) and (32), the variance \(Dg\) is obtained, and the function expression is shown in Appendix Equation (A3). According to Appendix Equation (A2) and Appendix Equation (A3), the values \(\mu_{g}=128.0974\) and the \(\sigma_{g}=39.8660\) are calculated, which are nearly equal to the data calculated by the Monte Carlo method.