## 3 An Integrated Tolerance Theory for the Kinematic Accuracy Analysis of Landing
### Robust Analysis  
The overall mathematical model is \(\Delta=\sqrt{\delta_{1}^{2}+\delta_{3}^{2}}\). \(\tau_{eA}\), \(\tau_{eB}\), \(\tau_{eC}\), \(\tau_{eD}\), \(\tau_{eK}\), and \(\tau_{eL}\) can be seen as the control factors, their mid-values are corresponding levels as they are shown in Table 5. Letting \(\theta\)\(133.4^{\circ}\) to \(233.4^{\circ}\), calculations are performed every \(10^{\circ}\). The Monte Carlo method is used in the calculation, \(L_{27}(3^{5})\) is selected as the orthogonal array and shown in Table 1.  
Using Matlab and Minitab, the numerical computation was carried out. For ANVOA analysis, the null hypothesis was: there is no significant effect of a certain joint clearance to the final output results, while the alternative hypothesis was: there is a significant effect of a certain joint clearance to the final output results. The contributions of joint clearances are depicted in Figures 6-9; and SN ratios are depicted in Figures 10-12. \begin{table}
\begin{tabular}{c c} \hline \hline
**Components** & **Tolerances (mm)** \\ \hline \multirow{6}{*}{\(\tau_{eA}\)} & Level 1: \(\tau_{eA}\sim(0.027,0.00283^{2})\) (rough) \\  & Level 2: \(\tau_{eA}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eA}\sim(0.047,0.00283^{2})\) (precise) \\  & Level 1: \(\tau_{eB}\sim(0.027,0.00283^{2})\) (rough) \\  & Level 2: \(\tau_{eB}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eB}\sim(0.047,0.00283^{2})\) (precise) \\  & Level 1: \(\tau_{eC}\sim(0.027,0.00283^{2})\) (rough) \\  & Level 2: \(\tau_{eC}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eC}\sim(0.047,0.00283^{2})\) (precise) \\  & Level 1: \(\tau_{eD}\sim(0.027,0.00283^{2})\) (rough) \\  & Level 2: \(\tau_{eD}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eD}\sim(0.047,0.00283^{2})\) (precise) \\  & Level 1: \(\tau_{eK}\sim(0.027,0.00283^{2})\) (rough) \\  & Level 2: \(\tau_{eK}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eK}\sim(0.047,0.00283^{2})\) (precise) \\  & Level 1: \(\tau_{eL}\sim(0.027,0.00283^{2})\) (rough) \\ \(\tau_{eL}\) & Level 2: \(\tau_{eL}\sim(0.037,0.00283^{2})\) (medium) \\  & Level 3: \(\tau_{eL}\sim(0.047,0.00283^{2})\) (precise) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Tolerances and experiment levels. Figure 8: (**a**) The contributions of \(\mathbf{r}_{eK}\); (**b**) The contributions of \(\mathbf{T}_{eL}\). Figure 6: (**a**) The contributions of \(\mathbf{r}_{eA}\); (**b**) The contributions of \(\mathbf{T}_{eB}\). Figures 6-9 display the contributions of \(r_{eA},r_{eB},r_{eC},r_{eD},r_{eK}\), and \(r_{eL}\), and they change as the input variable \(\theta\) changes. In Figure 6a, when the input variable \(\theta\) is within the range of 133\({}^{\circ}\) 178\({}^{\circ}\), the contribution of \(r_{eA}\) increases from 17% to 42%; when \(\theta\) is 183\({}^{\circ}\), the contribution drops to 4.2%, and then gradually increases again until it reaches 17%. In Figure 6b, the highest contribution of \(r_{eB}\) is 28%, and the lowest contribution is 2%; they happen when \(\theta\) are 163\({}^{\circ}\) and 228\({}^{\circ}\) respectively. As shown in Figure 7, the contributions of \(r_{eC}\) and \(r_{eD}\), change in the range of 12%\(\sim\)17%, and 0%\(\sim\)20% respectively. They have one feature in common; that is, during the period 168\({}^{\circ}\)\(\sim\)178\({}^{\circ}\), they both first increase, then decrease. In Figure 8, the tendencies of the two curves are similar. They both decrease first; then during the period of 178\({}^{\circ}\)\(\sim\)183\({}^{\circ}\), suddenly increase; and then decrease again. The ranges of them are within 1% \(\sim\)28% and 0%\(\sim\)34% respectively. In Figure 9, it can be seen that the contributions of \(r_{eA},r_{eK}\), and \(r_{eL}\) are comparatively larger than the other three contributions. The confidence probabilities of \(r_{eA}\) and \(r_{eC}\) are much less than 5%, therefore, the null hypothesis is denied. They have significant effects on the final results.