## 4 Case Study and Discussion
### Position and Orientation Deviation Computation of Mechanical Product
#### 4.2.1 Sliding Workbench and Computation Parameters  
The sliding workbench in Figure 12 is taken as an example to verify both the effectiveness and usefulness of the proposed approach.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Assembly** \\ **States** \\ \end{tabular} } & \multicolumn{6}{c}{**PODs**} \\ \cline{2-7}  & \begin{tabular}{c} **Deviaitons** \\ **along X-Axis** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Deviaitons** \\ **along Y-Axis** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Deviaitons** \\ **along Z-Axis** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Nutation** \\ **angle** \\ **(\({}^{\circ}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Precession** \\ **Angle** \\ **(\({}^{\circ}\))** \\ \end{tabular} &
\begin{tabular}{c} **Spin** \\ **Angle** \\ **(\({}^{\circ}\))** \\ \end{tabular} \\ \hline
1 & \(5.39\times 10^{-7}\) & 0.009 & 0.011 & 0.521 & 90.456 & 90.458 \\
2 & \(0.011\) & \(-1.98\times 10^{-7}\) & 0.014 & 0.621 & 169.859 & 179.864 \\
3 & \(0.007\) & \(5.28\times 10^{-7}\) & 0.012 & 0.383 & 178.996 & 179.994 \\
4 & \(0.009\) & 0.012 & 0.008 & 0.865 & 131.475 & 130.474 \\
5 & \(0.018\) & 0.008 & 0.012 & 1.112 & 136.098 & 156.098 \\
6 & \(0.013\) & 0.011 & 0.008 & 0.964 & 119.716 & 139.716 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Acquired position and orientation deviations (PODs) of the assembly with three mating planes.  
Figure 11: Determined contact points of the six assembly states. The three contact points on the main mating plane are red, the two contacts point on the first auxiliary mating plane are green, and the contact point on the second auxiliary mating plane is yellow.  
According to the six-point positioning principle, five DOFs of PPC are constrained via five contact points. There are two possible cases in which the five contact points are distributed on the support and guide planes, which consequently results in two types of assembly states.  
Assembly state A: Three of the five contact points are distributed on the support planes of PPA and PPB, and the other two are distributed on the guide plane of PPA.  
Assembly state B: Three of the five contact points are distributed on the support planes of PPA and PPB, and the other two are distributed on the guide plane of PPB.  
The surface deviations are supposed to be modeled according to the deviation functions in Table 2, and are required to be limited within the ranges of \(\pm\)0.015 mm and \(\pm\)0.03 mm, respectively. Thus, the surface deviations of each mating plane can be generated after the coefficients of each deviation function are determined according to the required variation ranges. Each mating plane is also discretized with the size of 2 mm. Thus, the points on the support and guide planes of PPC-A1/A2/B1/B2 are 22 \(\times\) 41 = 902 and 7 \(\times\) 41 = 287, and the points on the support and guide planes of PPA and PPB are 21 \(\times\) 501 = 10521 and 6 \(\times\) 501 = 3006.  
The thresholds employed to judge whether the fourth and fifth contact points come into contact are given as 1 \(\times\) 10\({}^{-3}\) mm and 1 \(\times\) 10\({}^{-3\circ}\).