# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 2 Tolerance Analysis Considering Form Errors
### Tolerance Modeling Considering Form Errors  
To model the tolerance of a feature considering form errors, the geometric error of a feature must be represented precisely. A novel tolerance modeling method was proposed based on NURBS surface and the separation method. The main process of the modeling method was illustrated by taking the reflection plate in Figure 1a as example.  
NURBS surface has many excellent properties, such as the affine invariance, strong convex hull property, local modification scheme, and differentiability [22]. Since all these properties perfectly satisfy the needs of flatness representation, the reflection plate with form errors is represented by a NURBS surface \(S(\alpha,\beta)\) defined by Equation (7), where \(N_{m,p}(\alpha)\) is the basis function in u direction and \(N_{n,q}(\beta)\) is the basis function in the v direction, and \(P_{m,n}(x_{mn},y_{mn},z_{mn})\) is the control point.  
\[S(\alpha,\beta)=\frac{\sum\limits_{m=0}^{M}\sum\limits_{n=0}^{N}N_{m,p}(\alpha )N_{n,q}(\beta)\cdot\omega_{m,n}P_{m,n}(x_{mn},y_{mn},z_{mn})}{\sum\limits_{m=0 }^{M}\sum\limits_{n=0}^{N}N_{m,p}(\alpha)N_{n,q}(\beta)\cdot\omega_{m,n}} \tag{7}\]  
Defining a \(5\times 5\) grid on the NURBS surface (representing the reflection plate), each node \(A_{H}(x_{kl},y_{kl},z_{kl})\) was determined through measurement, as shown in Figure 5. Then the control points \(P_{m,n}(x_{mn},y_{mn},z_{nm})\) can be calculated by global interpolation [23].  
\[\textit{global interpolation}:\ \left\{P_{1,1},P_{1,2},\cdots P_{m,n}\right\}= \hbar(A_{1,1},A_{1,2},\cdots A_{kl}) \tag{8}\]  
For each sample of geometric error of the plate, the measurement points are well-determined, then the NURBS surface \(S(\alpha,\beta)\) representing the reflection plate was well-determined by substituting Equation (8) into Equation (7).  
Figure 5: _Cont._  
Since only flatness of the reflection plate was considered, the x-directional and y-directional coordinates are assumed ideal, all the other points on the reflection plate can be then calculated by the point inversion algorithm based on the [PERSON] iteration method [24]. For example, the horn elements \(E_{ij}\) can be determined by Equation (9). Noticing that only the z-directional coordinate \(ez_{ij}\) contains geometric error.  
\[\begin{array}{l}point\text{\ \it inversion}:\ \left(\alpha_{ij},\beta_{ij} \right)=\lambda\left(ex_{ij},ey_{ij}\right)\\ E_{ij}\left(ex_{ij},ey_{ij},ez_{ij}\right)=S\left(\alpha_{ij},\beta_{ij} \right)\end{array} \tag{9}\]  
The z-directional coordinate of the measurement point was assumed to be a normally distributed variable according to the manufacture.  
\[z_{kl}\sim N\left(0,\sigma^{2}\right) \tag{10}\]  
The standard deviation \(\sigma\) is determined by the flatness \(P\) according to the 3-sigma theory.  
\[\sigma=\frac{1}{6}p \tag{11}\]  
To analyze the effect of different value of \(P\) on the performance of the antenna, \(P\) was assumed as a uniformly distributed random variable.  
\[P\sim U(a,b) \tag{12}\]  
where \(a\) and \(b\) were constrained by the manufacture [25].  
Ultimately, the uncertainty of the flatness was substituted equivalently by the uncertainty of measurement point \(z_{kl}\) and the uncertainty of the parameter \(\sigma\) through Equations (7)-(12).