## 3 An Integrated Tolerance Theory for the Kinematic Accuracy Analysis of Landing
### Primary Clearance Design and Tolerance Analysis of the Mechanism  
DC, CB, and AV constitute the main connecting mechanism, while DC, CK, and KL make up the lock mechanism. The simplified diagram is in Figure 4, and the basic parameters are listed in Table 3.  
Figure 3: The landing gear retraction mechanism: (**a**) the retracted position; (**b**) the extended position. The joint clearances of the main connecting and lock mechanism are critical for the kinematic accuracy, and therefore, the tolerance design and robust analysis will be performed next. Based on the experience and engineering handbook, the tolerances of the components are allocated according to the medium level and they are listed in Table 4.  
The kinematic tolerance changes of the landing gear retraction mechanism are along \(x\) and \(z\) direction, and therefore, \(\delta_{1}\) and \(\delta_{3}\) are the final errors. According to the theory described above, the overall mathematical model is deduced as  
Figure 4: The main connecting and lock mechanism. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Components** & **Length(mm)** & **Angle** & **Scope\({}^{\text{o}}\)** \\ \hline AB & 451.77 & \(\theta_{1}\) & 133.14\(\sim\)234.30 (input variable) \\ BC & 315.8 & \(\theta_{2}\) & 86.257\(\sim\)62.389 \\ CD & 465 & \(\theta_{3}\) & 43.481\(\sim\)\(-\)62.389 \\ AD & 704.89 & \(\theta_{4}\) & 152.544 \\ AV & 1334.437 & \(\theta_{5}\) & 24.203 \\ KL & 305 & \(\theta_{6}\) & 122.004\(\sim\)\(-\)141.98 \\ CK & 195 & \(\theta_{7}\) & 122.004\(\sim\)\(-\)141.98 \\ LD & 615.359 & \(\theta_{8}\) & 9.453 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Slider-crank parameters. \(\delta_{1}=\mathbf{T}_{eA}\cdot\cos\theta_{1}+\Delta_{AB}\cdot\cos\theta_{1}+\mathbf{ \tau}_{eB}\cdot\cos\theta_{2}+\Delta_{BC}\cdot\cos\theta_{2}+\mathbf{\tau}_{eC} \cdot\cos(\pi-\theta_{3})+\Delta_{CD}\cdot\cos\theta_{3}+\)  
\(\mathbf{\tau}_{eD}\cdot\cos\theta_{3}+\mathbf{\tau}_{eK}\cdot\cos\theta_{5}+\Delta_{CK }\cdot\cos\theta_{5}+\mathbf{\tau}_{eL}\cdot\cos\theta_{6}+\Delta_{KL}\cdot\cos \theta_{6}\)  
\(\delta_{3}=\mathbf{\tau}_{eA}\cdot\sin\theta_{1}+\Delta_{AB}\cdot\sin\theta_{1}+ \mathbf{\tau}_{eB}\cdot\sin\theta_{2}+\Delta_{BC}\cdot\sin\theta_{2}+\mathbf{\tau}_{eC }\cdot\sin(\pi-\theta_{3})+\Delta_{CD}\cdot\sin\theta_{3}+\)  
\(\mathbf{\tau}_{eD}\cdot\sin\theta_{3}+\mathbf{\tau}_{eK}\cdot\sin\theta_{5}+\Delta_{ CK}\cdot\sin\theta_{5}+\mathbf{\tau}_{eL}\cdot\sin\theta_{6}+\Delta_{KL}\cdot\sin \theta_{6}\)  
\(\Delta=\sqrt{\delta_{1}^{2}+\delta_{3}^{2}}\)  
Letting \(\theta\) change from \(133.4^{\circ}\) to \(233.4^{\circ}\), calculations are performed every \(2^{\circ}\). The Monte Carlo method is used in the calculation, the tolerances of the components and joint clearances obey norm distribution as they are shown in Table 3, and the test number is chosen as 100,000. Mean errors for all angles are depicted in Figure 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Components** & **Tolerances (mm)** \\ \hline \(\Delta_{AB}\) & \(\Delta_{AB}\sim(0,0.133^{2})\) \\ \(\Delta_{BC}\) & \(\Delta_{BC}\sim(0,0.083^{2})\) \\ \(\Delta_{CD}\) & \(\Delta_{CD}\sim(0,0.133^{2})\) \\ \(\Delta_{CK}\) & \(\Delta_{CK}\sim(0,0.083^{2})\) \\ \(\Delta_{KL}\) & \(\Delta_{KL}\sim(0,0.083^{2})\) \\ \(\mathbf{\tau}_{eA}\) & \(\mathbf{\tau}_{eA}\sim(0.047,0.00283^{2})\) \\ \(\mathbf{\tau}_{eB}\) & \(\mathbf{\tau}_{eB}\sim(0,0385,0.00242^{2})\) \\ \(\mathbf{\tau}_{eC}\) & \(\mathbf{\tau}_{eC}\sim(0.0385,0.00242^{2})\) \\ \(\mathbf{\tau}_{eD}\) & \(\mathbf{\tau}_{eD}\sim(0.047,0.00283^{2})\) \\ \(\mathbf{\tau}_{eK}\) & \(\mathbf{\tau}_{eK}\sim(0.0385,0.00242^{2})\) \\ \(\mathbf{\tau}_{eL}\) & \(\mathbf{\tau}_{eL}\sim(0.0385,0.00242^{2})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Primary tolerance design parameters. Figure 5: Original deviations for the primary tolerance design. Looking at Figure 4, When the landing gear mechanism retracts or extends, it is apparent that the fluctuation of the errors is comparatively large. The influences of joint clearances are undeniable.