## 1 Introduction
### Related Works  
Computing the position and orientation between assembled parts is a key issue of tolerance analysis [7] and has received much research attention. Various research of tolerance analysis has been carried out assuming that manufactured surfaces only have size and position errors, and some approaches to compute PODs were proposed. These approaches are generally based on establishing and solving models of two- or three-dimensional assembly constraints of motion and rotation [8; 9; 10]. For instance, two-dimensional assembly constraint models can be established according to the moving distances between discrete points or vertices if the manufactured surfaces are simplified as curves or lines deviating from the ideal circumstances [11; 12]. In addition, three-dimensional constraint assembly models can be acquired by analyzing the  
Figure 1: Assembly cases of single and multiple planes. Figure 2: Position and orientation deviations generated after multiple planes of two parts are assembled. DOF, degree of freedom. **(a)** Ideal circumstances, **(b)** Actual circumstances, **(c)** Positions of unconstrained DOFs under varying circumstances, **(d)** Number of the contact points on each mating plane under varying circumstances. constrained DOFs if the following two conditions are met. First, the manufactured surfaces should be simplified as the surfaces of ideal geometric features that deviate from the ideal circumstances. Second, the manufactured surfaces should be described by tolerance models of the deviation domain, Jacobian-torsor, or small displacement torsor [13; 14; 15]. Because the non-ideal morphologies of the manufactured surfaces are ignored by the approaches mentioned above, the acquired PODs differ from the actual circumstances even though the size and position errors are simultaneously considered, which directly affects the accuracy of tolerance analysis. To avoid the influence of ignoring form errors on tolerance analysis, the problems of mating between two non-ideal planes have been extensively studied. For instance, inspired by the idea that the distances between the corresponding points on two mating planes will converge and the interference will never occur if two non-ideal planes are mated [16], the approach of constrained registration (CR) has been proposed [17]. Constrained registration can determine PODs by continuously adjusting the assembly position of the mating plane until the signed projected distance, signed normal distance, or convex hull volume is minimized, and the constraint condition of a non-interference is simultaneously ensured. Consequently, CR is often employed to determine the \"best-fit\" assembly position between two parts [18; 19]. Several approaches based on determining contact points have been proposed under the condition that the assembly of two non-ideal planes is judged according to the six-point positioning principle. For instance, [PERSON] has proposed an approach of difference surface (DS), wherein three contact points of a mating plane are determined successively via searching for the points with limit coordinates on the DS [20]. Difference surface is well suited for computing or analyzing the position and the orientation between two parts of 3-2-1 positioning [21; 22; 23]. Moreover, [PERSON] [24] and [PERSON] [25] have also proposed approaches that determine PODs of a plane by searching for three contact points successively. The main principle is that the position and the orientation between two assembled parts result from the continuous adjustment of the assembled part position from the ideal circumstances until the six-point positioning principle is met. The adjustment magnitude of the position and orientation is the key in determining contact points. This magnitude can be acquired by computing distances between the corresponding points on the mating planes based on the methods of projecting [26], raying [27], or gap surface [28].