# Assembly Tolerance Design Based on Skin Model Shapes Considering Processing Feature Degradation
## 3 Constrained 3D Assembly Simulation and Tolerance Synthesis  
### Constrained Assembly Simulation  
Then, the tolerance models generated by the methods described in Section 2 are used in assembly simulations. Generally, the process of assembling two parts involves the transformation of one part from its initial condition to its assembling condition, while the other part is relatively static. In this paper, the former part is called assembly part \(Q\), and the latter part is called reference part \(P\). The assembling transformation \(\xi\) is in the form of a twist because of its splendid performance in representing the orientation, and its independence from the coordinate system transformation. Parts are considered under the geometric basis of Skin Model Shapes in this paper, which means that \(Q\) and \(P\) are in the form of meshes or point clouds. So, the assembly part \(Q\) is transformed through \(\xi\) from its initial condition \(Q\) to assembling condition \(Q^{\prime}\), as shown in Equation (13):  
\[Q_{\prime}=\xi\cdot Q \tag{13}\]  
Twist \(\xi\) is essentially an affine transformation. It can always be decomposed to a motion screw \(\xi_{i}=(s_{i},v_{i})\) in a three-dimensional (3D) velocity vector field, containing a rotation around its axis and a translation along the axis, as shown in Equation (14):  
\[\xi_{i}=s_{i}+p_{i}\times v_{i} \tag{14}\]  
where \(p_{i}\) is the vector from the spatial origin to point \(i\). \(v_{i}\) is the axial velocity vector at point \(i\). \(s_{i}\) is the tangential velocity vector at point \(i\). Assembly twists of all the points of the assembly part consist the assembly transformation: \(\xi=\{\xi_{1},\xi_{2},\cdots,\xi_{i},\cdots\}\). Mathematically, \(Q^{\prime}\) is the global extremum of an object function containing \(Q\), \(P\), and twist \(\xi\) under assembly constraints, as shown in Equation (15):  
\[\xi=\text{argmin}(f(P,Q^{\prime})) \tag{15}\]  
Choose the root mean square Euclidean distance function in Equation (16) of the closest point pairs in point clouds \(P\) and \(Q\):  
\[f=RMSED(P,Q^{\prime})=\sqrt{\frac{1}{n}\sum_{p_{i},q_{i}\in CPP(i)}\|p_{i}-q_{ i}\|^{2}} \tag{16}\]  
Establish the iteration object function based on Equations (13)-(16). The end condition of the iteration is the appearance of three pairs of contact points in the assembly. A pair of contact points contains a point in \(P\) and a point in \(Q\). Furthermore, the Euclidean distance between the two points is smaller than a threshold value. Usually, using the iterative closest points (ICP) is a common method under the circumstances to find the target. However, assembly entity constraints and assembly force constraints have restricted the direct use of ICP. Assembly entity constraints restrict the intersection of part entity in assembly simulation. As shown in Figure 2a, the direct use of ICP causes an unreasonable intersection of part materials. However, if the rule of assembly entity constraints is considered, ICP would give a result that is much more rational (in Figure 2b). A signed distance field is introduced to supervise the compliance of the assembly entity constraints. The positive direction of the signed distance field is always against the assembling direction. The signed distance vector \(d_{i}^{stdp}(i=1,2,\cdots,n)\) is from the points in \(P\) to the corresponding closest point in \(Q\). When the angle between \(d_{i}^{stdv}\) and the positive direction is under 90 degrees, \(d_{i}^{stdv}\) is labeled positive. Otherwise, it is labeled negative. As shown in Figure 2c, all the signed distance vectors \(d_{i}^{stdv}(i=1,2,\cdots,n)\) are positive during iteration before assembling. However, negative signeddistance vectors appear when the rule of assembly entity constraints is violated (in Figure 2d). So, the signals of \(d_{i}^{sky}(i=1,2,\cdots,n)\) are repeatedly checked during the iteration. Assembly force constraints ensure the stability of assembling conditions through restricting the location of contact points according to the assembly force.