## 4 Parameter-Identification-Based Configuration Design
### Deployment Implementation and Theoretical RSE  
We have mentioned in Section 2.4 that most PTDAs are deployed with the rotation about fixed oblique axes in space, and this also applies to the antenna proposed in this study. Here, we utilize [PERSON]'s rotation theorem [56; 57] to determine the position of the rotation axis. Firstly, a fixed point \(P\) is determined, about which the panel will rotate. Then, according to [PERSON]'s rotation theorem, there exists such an axis passing through the point that the rotation can be realized by a single rotation around the axis, as illustrated in Figure 7. However, the situation differs due to the adopted panel pose, which is the same as that in Figure 6c for high stowing efficiency, as shown in Figure 7a. Apart from this, radius \(r_{\mathrm{c}}\) is taken as the same magnitude as folded radius \(\frac{D}{2}\) to improve the VSE, as shown in Figure 7a. This manipulation also contributes to the improvement in RSE because a larger \(r_{\mathrm{c}}\) implies not only a larger \(\alpha_{\mathrm{m}}\) in accordance with Equation (1) but eventually a higher RSE according to the process of increasing RSE in Figure 6. These differences from Ref. [45] bring additional and more complex constraints, which are elaborated as follows. As shown in Figure 7b, taking panel 1 as an example, the deployed panel is transformed into the folded one in Figure 7a by rotating about the fixed point \(P\), and the pose of the folded panel is the same as that of the green panel in Figure 6c. Specifically, the following four geometric constraints (GCs) can be summarized. 1. Point \(A_{1}\) is on the envelope (GC 1), and it is conceivable that this constraint is easy to fulfill since line segment \(PA_{1}\) always intersects with the envelope at point \(A_{1}\) after an appropriate rotation about \(P\). 2. Under the first constraint, locating point \(D_{1}\) precisely on the envelope (GC 2) is difficult to realize because the two constraints are coupled. 3. Point \(C_{1}\) has a prespecified upward displacement \(h\) (GC 3) to avoid interference between the panel and central surface. This constraint should be fulfilled under the first two constraints. Figure 7: Deployment implementation based on [PERSON]’s rotation theorem. (**a**) Front view of folded state. (**b**) Top view of deployed state. (1) The folded radius \(\frac{D_{1}}{2}\) is taken the same as radius \(r_{\mathrm{c}}\) of the central aperture for high VSE; (2) the panel is deployed or folded by the rotation about the fixed point \(P\), whose position is elaborately formulated to fulfill the constraints. According to Figure 6c, the focal axis, i.e., coordinate axis \(OZ\), and plane \(A_{1}B_{1}D_{1}\) are coplanar (GC 4). This constraint should be satisfied under the former three GCs. Thus, we lower the requirement first and simply guarantee that axis \(OZ\) is coplanar with points \(A_{1}\) and \(B_{1}\) (weak GC 4). When points \(A_{1}\) and \(B_{1}\) are projected onto plane \(OXY\), their polar angles are the same, denoted as \(\beta\), which is a parameter to be determined, as shown in the top-left subfigure of Figure 7a. In addition, to avoid interference caused by the panel thickness, it is assumed that the distance between \(B_{1}\) and \(B_{2}\) is a prespecified parameter \(s\), named an interference parameter. In summary, the first constraint can be directly satisfied, but the rest of the constraints cause a problem in identifying a proper fixed point \(P\) to fulfill themselves. In response to this problem, an ingenious scheme will be formulated and expanded on in Section 4.2. We now return to Figure 7 and assume that all of the four GCs are satisfied. Based on its immediate geometric relation, we arrive at the following expression:  
\[l_{A_{1}B_{1}}\cos\theta_{\mathrm{r}}+l_{OB_{1}}=\frac{D_{\mathrm{f}}}{2}, \tag{5}\]  
where \(l_{A_{1}B_{1}}=D_{\mathrm{a}}\sin\frac{\pi}{n}\) and \(l_{OB_{1}}=\frac{s}{2\sin\frac{\pi}{n}}\).