# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 1 Introduction
The purple dash line represents the linear array when only translation and rotation errors are considered. **(b)** The relation between the straightness and the maximum power density. For each sample of realistic geometric error of the linear array, the maximum power density was calculated based on Equation (A1), totally 10,000 samples were generated. (2) Tolerance analysis considering form errors. These methods can achieve precise tolerance analysis based on the samples by applying precise tolerance model considering form errors. For example, [PERSON] [11] proposed the hybrid dimension-based modeling method to consider all the form errors of a face feature, such as the flatness, waviness, and roughness. Based on the hybrid dimension model, sufficient samples were generated and the tolerance was analyzed. A methodology based on Bezier curves is proposed by [PERSON] [12] to represent the side lines of the surface feature with flexible structural deformations, and then the assembly variation analysis was finished. A metric modal decomposition (MMD) method is developed by [PERSON] [13] to model the form errors of various parts in a mechanism, the basic modes were defined first and the non-ideal surface was defined by the weighted sum of basic modes, then the tolerance analysis was achieved by kernel density estimation [14]. Skin model shapes were used by [PERSON] [15] to represent all the geometric error of the physical interface between a workpiece and its environment. He [16] proposed an assembly tolerance analysis and design method based on skin model shape. This method is highly accurate [15; 16; 17] but computationally expensive [18]. When the tolerance analysis of the linear antenna in Figure 2a is performed by the above methods, the basic process is illustrated in the left part of Figure 3. First, all the feasible values \(\{T_{1},\cdots T_{k},\cdots T_{k}\}\) of the straightness are determined. Then the tolerance model was established and the samples \(G^{k}=\left\{G_{1}^{k},\cdots G_{n}^{k},\cdots G_{N}^{k}\right\}\) of the geometric feature were generated for the specific value \(T_{k}\) (First loop, N times). The performance of the antenna was calculated based on the tolerance propagation model for all feasible values (Second loop, K times). Finally, the optimal tolerance was determined. The double loop process needs a large number of samples and is time consuming. When dealing with the complex antenna as shown in Figure 1a, the double loop process is difficult to apply. A novel tolerance analysis method with single loop process was developed in this paper. First, the tolerance model considering form errors was developed, which can represent the geometric error of a surface feature precisely. Then the tolerance propagation was achieved efficiently based on the novel tolerance model. By introducing the variance separation method, the tolerance analysis was simplified to a single loop process. Finally, the tolerance analysis was implemented, as shown in the right part of Figure 3.  
The rest of this paper is organized as follows. Section 2 discusses the novel tolerance analysis method. Section 3 discusses the numerical simulation results. Finally, Section 4 provides the concluding remarks.