## 1 Introduction
### Searching for Contact Points by Progressively Approaching Position and Orientation  
When determining the \(n\)th (the value of \(n\) can be 4, 5, and 6) contact point on the auxiliary mating plane, each procedure of the position and orientation adjustment will break the contact states of the previously acquired \(n-1\) points because of the surface deviations. Consequently, the previously acquired \(n-1\) contact points should be re-determined after each adjustment by the obtained adjustment value. However, not only a new position and orientation but also a new adjustment value will be acquired if the previously acquired \(n-1\) contact points are re-determined, which means that the \(n^{\text{th}}\) contact point on the auxiliary mating plane cannot be determined through one adjustment. Therefore, a strategy for progressively approaching position and orientation is proposed.  
The \(n\)th contact point on the auxiliary plane is found by performing several iterations of the approaching process of the position and orientation, as shown in Figure 5. The approaching process of the position and orientation starts by computing the adjustment value \(\Delta\Omega_{i}^{n}\) of the current position and orientation \(\Delta\Omega_{i}^{n}\). Subsequently, the position and the orientation are adjusted by \(\Delta\Omega_{i}^{n}\) to acquire the new position and orientation \(\Omega_{i+1}^{n}\). Because the contact states of the previously acquired points are broken, the previous \(n-1\) contact points corresponding to the position and orientation \(\Omega_{i+1}^{n}\) should be re-determined through several adjustment procedures. Subsequently, the approaching process of the position and orientation is completed, and the new position and orientation \(\Omega_{i+1}^{update\_n-1}\) are acquired. The next approaching process will be carried out on the basis of the new position and orientation \(\Omega_{i+1}^{update\_n-1}\) because each adjustment starts from the position and orientation acquired at the previous step.  
Figure 4: Possible cases of the main and auxiliary mating planes of the assembly with two mating planes. DOF, degree of freedom. (**a**) Case 1 of the main and auxiliary ones among all the mating planes, (**b**) Case 2 of the main and auxiliary ones among all the mating planes.  
Determining the adjustment value is a key problem when searching for contact points on the auxiliary mating planes. The moving adjustment magnitude for CP4 refers to the minimum distance between the corresponding points on the first auxiliary mating plane along the direction of the constrained moving DOF. The rotating adjustment magnitude for CP5 refers to the minimum angle acquired by rotating the first auxiliary mating plane in the direction of the constrained rotating DOF around the vertical axis through contact point CP4. The moving adjustment magnitude for CP6 refers to the minimum distance between the corresponding points on the second auxiliary mating plane along the direction of the constrained moving DOF.  
Surface deviations have a longer period of geometric fluctuation than surface roughness, which means that oscillations of the position and orientation will never emerge during the adjustment process. In addition, each adjustment procedure continues the previous one rather than restarts from the initial position and orientation. Therefore, the adjustment value converges rapidly during the iterative process. When the adjustment value converges to a certain value, it can be judged that the DOF corresponding to this type of adjustment value is constrained, and the expected contact point is determined. Consequently, the threshold of the adjustment value can be set to judge whether the contact point is determined, and the condition of contact judgment is summarized considering that each type of adjustment value is a six-dimensional vector: If \(|\Delta\Omega_{i}^{n}|\leq|\Delta\Omega_{R}^{n}|\), the contact point is determined. Otherwise, the contact point remains undetermined.