## 4 Parameter-Identification-Based Configuration Design
### Parameter Identification of Configuration  
It can be observed from Figures 7 and 8 that the position vector of the fixed point \(P\) can be expressed as the following form:  
\[\mathbf{r}_{P}=\bigg{(}(1-k_{1})r_{\mathrm{c}}\cos\biggl{(}k_{2}\frac{2\pi}{n}+ \theta_{D_{1}}\biggr{)},(1-k_{1})r_{\mathrm{c}}\sin\biggl{(}k_{2}\frac{2\pi}{ n}+\theta_{D_{1}}\biggr{)},\frac{r_{\mathrm{c}}^{2}}{4f},1\bigg{)}^{\mathrm{T}}, \tag{7}\]  
where \(f\) is the focal length and equal to \(\eta D_{\mathrm{a}}\) (\(\eta\) is the focus/diameter ratio). It is commonly known that three non-collinear points in space determine a rigid body and its pose naturally. According to the discussion above, the deployed pose can be represented by point set \(\{A_{1},B_{1},P\}\), and their (including point \(C_{1}\)) corresponding position vectors (or coordinates) are  
Figure 8: Functions of \(k_{1}\) and \(k_{2}\). (**a**) Without parameters. (**b**) Function of parameter \(k_{1}\). (**c**) Function of parameter \(k_{2}\). (1) Parameter \(k_{1}\) can compensate for the downward motion of \(C_{1}\) in subfigure (**a**); (2) parameter \(k_{2}\) can compensate for the inward motion of \(D_{1}\) in subfigure (**b**); (3) if the values of \(k_{1}\) and \(k_{2}\) are appropriately prescribed, both constraints can be satisfied. \[ \begin{cases}\mathbf{r}_{A_{1}}=\left(\frac{D_{1}}{2}\cos\frac{-\pi}{n},\frac{D_{2}}{2} \sin\frac{-\pi}{n},\frac{p_{2}^{2}}{4\bar{q}},1\right)^{\mathsf{T}},\\ \mathbf{r}_{B_{1}}=\left(\frac{D_{2}}{2}\cos\frac{\pi}{n},\frac{D_{2}}{2}\sin\frac {\pi}{n},\frac{p_{2}^{2}}{4\bar{q}},1\right)^{\mathsf{T}},\\ \mathbf{r}_{P}=\left((1-k_{1})r_{\mathrm{c}}\cos\left(k_{2}\frac{2\pi}{n}+\theta_ {D_{1}}\right),(1-k_{1})r_{\mathrm{c}}\sin\left(k_{2}\frac{2\pi}{n}+\theta_{D_ {1}}\right),\frac{r_{\mathrm{c}}^{2}}{4\bar{q}},1\right)^{\mathsf{T}},\\ \mathbf{r}_{C_{1}}=\left(r_{\mathrm{c}}\cos\left(\frac{2\pi}{n}+\theta_{D_{1}} \right),r_{\mathrm{c}}\sin\left(\frac{2\pi}{n}+\theta_{D_{1}}\right),\frac{r _{\mathrm{c}}^{2}}{4\bar{q}},1\right)^{\mathsf{T}},\end{cases}  \tag{8}\]  
respectively. On the other hand, the folded pose in Figure 7a can be represented by the corresponding point set, denoted as \(\{A_{1},B_{1},P\}^{\prime}\). In light of Figures 7 and 8, the coordinates of \(\{A_{1},B_{1},P\}^{\prime}\) can be expressed as  
\[ \begin{cases}\mathbf{r}_{A_{1}}^{\prime}=\left(\frac{D_{1}}{2}\cos\beta,\frac{D_{ 1}}{2}\sin\beta,z_{A_{1}},1\right)^{\mathsf{T}},\\ \mathbf{r}_{B_{1}}^{\prime}=\left(r_{B_{1}}\cos\beta,r_{B_{1}}\sin\beta,z_{B_{1}}, 1\right)^{\mathsf{T}},\\ \mathbf{r}_{P}^{\prime}=\left((1-k_{1})r_{\mathrm{c}}\cos\left(k_{2}\frac{2\pi}{n} +\theta_{D_{1}}\right),(1-k_{1})r_{\mathrm{c}}\sin\left(k_{2}\frac{2\pi}{n}+ \theta_{D_{1}}\right),\frac{r_{\mathrm{c}}^{2}}{4\bar{q}},1\right)^{\mathsf{T }},\end{cases}  \tag{9}\]  
where \(z_{A_{1}},r_{B_{1}}\) and \(z_{B_{1}}\) remain unknown and shall be identified. Given that the panel is a rigid body, we have  
\[ \begin{cases}\|\mathbf{r}_{A_{1}}-\mathbf{r}_{B_{1}}\|_{2}=\|\mathbf{r}_{A_{1}}^{\prime}- \mathbf{r}_{B_{1}}^{\prime}\|_{2},\\ \|\mathbf{r}_{B_{1}}-\mathbf{r}_{P}\|_{2}=\|\mathbf{r}_{B_{1}}^{\prime}-\mathbf{r}_{P}^{\prime} \|_{2},\\ \|\mathbf{r}_{P}-\mathbf{r}_{A_{1}}\|_{2}=\|\mathbf{r}_{P}-\mathbf{r}_{A_{1}}^{\prime}\|_{2}. \end{cases}  \tag{10}\]  
Consequently, if values of \(k_{1}\), \(k_{2}\) and \(\beta\) are given, all equations in Equation (10) can be solved simultaneously for unknowns \(\{z_{A_{1}},r_{B_{1}},z_{B_{1}}\}\). In this way, the deployed and folded poses of the panel can be totally determined, which can be represented by point sets \(\{A_{1},B_{1},P\}\) and \(\{A_{1},B_{1},P\}^{\prime}\), respectively. This also suggests that a scenario is formed, which is suitable for using rigid-body registration algorithms [62; 63] to obtain the deploying-to-folding transformation matrix denoted as \(\mathbf{T}\). Evidently, if \(\beta\) is given, the transformation matrix \(\mathbf{T}\) is a function of \(k_{1}\) and \(k_{2}\), and can be denoted as \(g(k_{1},k_{2})\).