## 3 Process of Computing POD of Multi-Mating Planes  
Using the principle of successively determining contact points, the computation of the PODs can be conducted according to the flow in Figure 7.  
Figure 6: Principle of the recursion of contact points. At the beginning of the flow, the view coordinate system and the part definition coordinate systems \(\Theta^{\text{A}}\) and \(\Theta^{\text{B}}\) of the assembled and reference parts are established, and the initial assembly position is acquired by setting \(\Theta^{\text{A}}\) and \(\Theta^{\text{B}}\) to coincide with each other according to the ideal circumstances. Furthermore, \(\Theta^{\text{A}}\) should be adjusted according to the given positions corresponding to the unconstrained DOFs if the assembly is incompletely positioned. According to the principle of modeling from the perspective of manufacturing errors, the surface deviations (\(\mathbf{S_{A}}\), \(\mathbf{S_{B}}\)) of the two parts are generated via the deviation function corresponding to each manufacturing error, as shown in Equations (1) and (2):  
\[dev_{i}=f_{i}(\mathbf{K}_{i},q_{1},q_{2}) \tag{1}\]  
\[F(\mathbf{C},q_{1},q_{2})=\sum_{i=1}^{n}f_{i}(\mathbf{K}_{i},q_{1},q_{2}) \tag{2}\]  
where \(dev_{i}\) refers to the deviation function corresponding to manufacturing error \(i\);\(f_{i}\) refers to the form of \(dev_{i}\); and \(\mathbf{K}_{i}\) refers to the set of all coefficients corresponding to \(dev_{i}\), \(\mathbf{K}_{i}=\{k_{ij}|j=1,2,\cdots,N_{i}\}\). Each coefficient is denoted as \(k_{ij}\), and the number of the coefficients is \(N_{i}\). \(q_{1}\) and \(q_{2}\) represent the coordinates of the points on the ideal plane. \(F\) refers to the total-deviation function, and \(n\) refers to the number of manufacturing errors. \(\mathbf{C}\) represents the coefficient sets of \(F\), \(\mathbf{C}=\bigcup\limits_{i=1}^{n}\mathbf{K}_{i}\). According to \(F\), the coordinates of an arbitrary  
Figure 7: Flowchart of the approach based on successively determining contact points. VCS, view coordinate system. PDCS, part definition coordinate system. point \(P\) on the actual plane can be represented as \((q_{1}^{P},q_{2}^{P},dev^{P})\). However, they should be transformed to the coordinates represented within the PDCS \((x^{P},y^{P},dev^{P})\), wherein the geometry of a part is described. Consequently, the generated surface deviations \(S\) are point clouds, and \(\mathbf{S}=\{P_{m,n}|m=1,2,\cdots,N_{m};n=1,2,\cdots,N_{n}\}\), where \(P_{m,n}\) represents the coordinates within the PDCS of the point on the actual plane. \(N_{m}\) and \(N_{n}\) are the row and column numbers of \(P_{m,n}\). During the computation process, contact points CP1, CP2, and CP3 on the main mating plane are determined first because the position and orientation \(\Omega^{3}\) are the basis for searching for contact point CP4. Next, according to the principle of searching for contact points via progressively approaching position and orientation, contact point CP4 and the position and orientation \(\Omega^{4}\) (i.e., \(\Omega^{update\_3}\)) are determined through several iterations in four steps. First, the minimum distance of the first auxiliary mating plane is computed. Second, it is checked whether the adjustment value meets the distance threshold. Third, the position and the orientation are adjusted. Fourth, the previously acquired three contact points are re-determined. The redetermination of the previously acquired three contact points is a recursive process of contact point determination, which means that the approach returns to the step of determining the three contact points on the main mating plane after the position and the orientation are adjusted each iteration. Similarly, contact point CP5 and the position and orientation \(\Omega^{5}\) (i.e., \(\Omega^{update\_4}\)) are also determined through several iterations in four steps. First, the minimum angle of the first auxiliary assembly plane is computed. Second, it is checked whether the adjustment value meets the angle threshold.