## 4 Case Study and Discussion
### Discussion on the Approach Efficiency  
The proposed approach is a type of search algorithm, and its effectiveness is significantly affected by the computational efficiency. Thus, the approach efficiency should be discussed. According to the case study, it is found that the approach efficiency is mainly affected by three factors: the principle of searching for contact points via progressively approaching position and orientation, the principle of computing the distances between the corresponding points on the mating planes, and the size of the point clouds. * **Effects of the principle of searching for contact points via progressively approaching position and orientation** The principle stipulates that each adjustment iteration continues from the previous assembly position rather than restarts from the initial position. Furthermore, the minimum  
Figure 14: Three types of variation curves. (**a**) Variation curves of the distance deviation along the X-axis, (**b**) Variation curves of the distance deviation along the Z-axis, (**c**) Variation curves of the angle deviation around the Y-axis. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Assembly States**} & \multicolumn{5}{c}{**PODs**} \\ \cline{2-6}  & **Devaiitons** & **Deviations** & **Nutation** & **Precision** & **Spin** \\  & **along X-Axis** & **along Z-Axis** & **Angle** & **Angle** & **Angle** \\  & **(mm)** & **(mm)** & **(c’)** & **(c’)** & **(c’)** \\ \hline A & 0.014 & 0.015 & 0.811 & 0.015 & 1.987 \\ B & 0.009 & 0.014 & 0.525 & 0.791 & 1.631 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Acquired position and orientation deviations (PODs) of the rectangular sliding guide. distance or angle between two mating planes is taken as the value for each adjustment iteration, which can lead to adjustment value convergence in a few iterations. For instance, the position and the orientation are only moved twice, rotated once, and moved once when searching for contact points CP4, CP5, and CP6, as shown in Section 4.1. Therefore, each contact point can be determined efficiently. Moreover, the efficiency is directly affected by the threshold as well. The smaller the threshold is, the more approaches are required to determine a contact point, which inevitably increases the computing cost. In contrast, a lower computing cost will be expended. However, the accuracy of computation will be affected if loose thresholds are given, because the actual contact state of the points are inaccurately described. Therefore, to acquire a reasonable threshold, the requirements of accuracy and efficiency should be balanced. Stringent thresholds should be given if high accuracy is required, even though the computing cost will be sacrificed to some extent. Otherwise, the threshold can be enlarged appropriately, such that the efficiency can be improved. * **Effects of the principle of computing the distances between the corresponding points on the mating planes. ** In addition to the total number of iterations of adjusting position and orientation, the efficiency of each adjustment iteration also plays a vital role in approach efficiency. The distance computation between the corresponding points significantly affects the efficiency of each adjustment step. To compute the distances between the corresponding points, the corresponding relations between the discrete points on the mating planes should be determined first because they are uncertain during the process of progressively approaching position and orientation. Thus, the method of projecting [26] is employed in this study, and its principle is displayed in Figure 15. The target patch of the current point should be determined. Next, a new point should be inserted on the target patch, which means that the point corresponding to the current point is determined. The main factor affecting the efficiency of computing the distances between the corresponding points is the strategy used to determine the target patch of the current point.