# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 2 Tolerance Analysis Considering Form Errors
### Assembly Simulation Considering Form Errors  
The spaceborne phased array antenna will serve in a gravity-free environment. In order to accurately predict the thermal deformation, the antenna is assembled with low stress. Therefore, only the rigid geometric error was considered here and the flexible deformation error was ignored. A simple but accurate assembly simulation method was developed in this paper to calculate the accumulation error.  
The assembly process of the frame and the base plate is similar with the assembly process of the base plate and the reflection plate. They are all bolted and there is no interference between two parts. Taking the assembly of the frame and the base plate as an example, the two parts are connected by 5 pairs of bolts. When there is geometric error, the rigid assembly of the frame and the base plate is completed once 3 pairs of the 10 bolt holes contact each other.  
The process of the assembly simulation method is listed as follow.  
Step 1: Identifying the initial state of the frame and the base plate, the coordinates of the bolts can be calculated by Equations (7)-(9). The five pairs of bolts are illustrated in Figure 6a, the base plate (part B) is located directly above the frame (part A).  
Step 2: Fixing the frame (part A), then move the base plate (part B) vertically downwards until the two parts contact each other. Recording the first contact point as \(C_{1}\), that is the first bolts in contact. Recording the translation matrix as \(T_{1}\).  
\[T_{1}=\left[0,0,\min(Bz-Fz)\right]^{T} \tag{27}\]where \(Bz\) represents the z coordinate of the bolt holes on the base plate, and \(Fz\) represents the z coordinate of the bolt holes on the Frame.  
Step 3: Part B will drop down because of the gravity. Translating the origin of the coordinate system to the point \(C_{1}\) by a matrix \(T_{2}=C_{1}\) first, then denoting the center of gravity of the part B as \(O_{1}\), and the center of gravity of the part A as \(O_{2}\). Finally, the rotation axis is defined as line \(L_{1}\)  
\[L_{1}=\frac{\overrightarrow{O_{1}C_{1}}\times\overrightarrow{O_{2}C_{1}}}{ \left|\overrightarrow{O_{1}C_{1}}\times\overrightarrow{O_{2}C_{1}}\right|} \tag{28}\]  
which must past point \(C_{1}\), as shown in Figure 6b.  
Figure 6: Cont.  
Step 4: Rotate part B around the axis \(L_{1}\) until it collides with part A. Then the second contact point \(C_{2}\) is determined. Recording the first rotation matrix \(R_{1}\). Details can be found in Appendix B.  
Step 5: Setting line \(C_{1}C_{2}\) as the rotation axis, as shown in Figure 6c, rotate part B until it collides with part A, then the third contact point \(C_{3}\) is determined and the rigid assembly is achieved. Recording the second rotation matrix \(R_{2}\).  
Figure 6: (**a**) The initial state of the frame and the base plate. (**b**) The first contact point and the rotation axis. (**c**) The second contact point and the second rotation axis. (**d**) Three contact points and the final state of the two parts.  
Step 6: Substitute the rotation matrix \(R_{1}\), \(R_{2}\) and the translation matrix \(T\) into the NURBS model, then the final state \(nB\) of the surface is determined, as illustrated in Figure 6d.  
\[nB^{T}=R_{2}R_{1}((B-T_{1})-T_{2})^{T}+T_{2}^{T} \tag{29}\]  
The final state of all the elements can be calculated based on Equation (9), and the corresponding performance of the antenna can be calculated by Equation (13).