## 4 Case Study and Discussion
### Position and Orientation Deviation Computation of an Assembly with Three Mating Planes  
The assembly with three mating planes in Figure 8 is a typical case study as presented by [PERSON] (1977) and [PERSON] (2007), which is composed of a cube PA and a block PB. The mating plane sizes of PA and PB are 80 \(\times\) 80 (mm) and 100 \(\times\) 100 (mm), respectively. According to the six-point positioning principle, six DOFs of PA are constrained via six contact points. There are six possible cases in which the six contact points are distributed  
Figure 8: Assembly with three mating planes. on the XOY, XOZ, and YOZ planes, which consequently results in six types of assembly states, as shown in Table 1.  
Considering that the principles of determining the contact points of these assembly states are consistent, the process of the successive determination of contact points is illustrated by taking assembly state 1 as an example in this section. The initial assembly position is set according to the ideal circumstances. Consequently, the defined coordinate systems of PA and PB are coincident, as shown in Figure 9. In addition, the definition coordinate system of PB should always coincide with the view coordinate system, such that adjusting the assembly position can be conveniently realized by adjusting PA.  
To generate surface deviations of each mating plane through the modeling method from the perspective of manufacturing error, the mating plane morphologies of PA and PB are supposed to be machined through end milling and are mainly affected by the inclination error of the spindle axis, straightness error in the vertical plane of the guide, and wear error of the milling cutter. The established deviation functions corresponding to these manufacturing errors are shown in Table 2.  
The surface deviations, described by point clouds, can be acquired after the coefficients of each deviation function are given. In this study, each mating plane is discretized with the size of 2 mm in the length and width directions. Thus, the points on the mating planes of PA and PB are 41 \(\times\) 41 and 51 \(\times\) 51, respectively. The thresholds for judging whether the fourth and sixth contact points come into contact are given as \(1\times 10^{-3}\) mm, and the threshold for the fifth contact points is given as \(1\times 10^{-3}\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Assembly State** & **XOY Plane** & **XOZ Plane** & **YOZ Plane** \\ \hline
4 & 2 contact points & 3 contact points & 1 contact point \\
5 & 1 contact point & 2 contact points & 3 contact points \\
6 & 1 contact point & 3 contact points & 2 contact points \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Assembly states of the assembly with three mating planes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Manufacturing Errors** & **Deviation Functions** \\ \hline Inclination error of the spindle axis & \(dev_{1}(q_{1},q_{2})=R\times(1-cos(q_{2}))\times tan(k_{11})\) \\ Straightness error in the vertical plane of the guide & \(dev_{2}(q_{1},q_{2})=k_{21}\times sin(k_{22}\times q_{1})+k_{23}\) \\ Wear error of the milling cutter & \(dev_{3}(q_{1},q_{2})=k_{31}\times q_{1}+k_{32}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deviation functions corresponding to the three main manufacturing errors of end milling. Figure 9: Established part definition coordinate systems (PDCSs) and view coordinate system. Over the process of determining contact points, the three contact points on the main mating plane are determined through the one-step-adjusting-based strategy. Next, the contact points on each auxiliary mating plane are determined through the strategies of progressively approaching the position and orientation and recursion of the contact points.