## 2 Configuration Design of Truss-Link Mechanism
### Configuration Design  
The configuration is simplified to a two-dimensional domain to obtain the truss-link's dimensions in the stowed configuration (Figure 2). By simplifying to a two-dimensional geometry, it is possible to reduce the _DOF_ and variables used to obtain dimensions and coordinates of the body compared to traditional three-dimensional (3D) methods. The dimensions and coordinates were defined in the formula in the reference [23], which was derived using geometric construction methods and bar-groups methods. Upon verification of the formula's results, it was found that there were some errors, which were fixed accordingly in this study as presented in Equations (2)-(10). With the panel folded, the length and angle of each truss-link are derived from input variables in Tables 2 and 3. Input variables including \(\varphi_{1}\), \(\varphi_{4}\), \(\varphi_{6}\), \(L_{1}\), \(L_{4}\), \(L_{6}\), \(X_{A}\), \(X_{C}\), \(X_{D}\), \(X_{F}\), and \(X_{G}\) are defined by the mechanical designer considering the size of the panel according to the design requirements. Through Equations (2)-(10), output variables, including \(\varphi_{2}\), \(\varphi_{3}\), \(\varphi_{5}\), \(\varphi_{7}\), \(L_{2}\), \(L_{3}\), \(L_{5}\), \(L_{7}\), \(X_{B}\), \(X_{E}\), and \(X_{H}\), were determined. Joint O corresponds to the origin of absolute coordinates as the rotation center of the hinge installed between the bus and the inner panel; Joint I is the rotation center between the inner panel and outer panel; Joint F is the intersection point of all truss-links. Furthermore, to keep the deployment state, three latching points, B, E, and H, were considered, as shown in Figure 2b. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Type 1** & **Type 2** \\  & **(RADARSAT-1)** & **(RADARSAT-2)** \\ \hline Body (\(L\)) & Bus (1) + Links (11) = 12 & Bus (1) + Links (9) = 10 \\ \(J_{1},J_{2}\) & \(J_{1}=12\), \(J_{2}=0\) & \(J_{1}=10\), \(J_{2}=0\) \\ Degrees of freedom & \(3\times(12-1)-2\times(12)=9\) & \(3\times(10-1)-2\times(10)=7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation of total DOF. Figure 1: Configuration of 2-D truss-link mechanism: (a) Type 1 (RADARSAT-1) (b) Type 2 (RADARSAT-2). \[L_{3}=\sqrt[2]{\left(x_{F}-x_{C}\right)^{2}+\left(z_{F}-z_{C}\right)^{2}} \tag{2}\]  
\[\left(x_{H},z_{H}\right)=\left(x_{G}+L_{6}\cos\varphi_{6},z_{G}-L_{6}\sin\varphi _{6}\right) \tag{3}\]  
\[L_{7}=\sqrt[2]{\left(x_{F}-\left(x_{G}+L_{6}\cos\varphi_{6}\right)\right)^{2}+ \left(z_{F}-\left(z_{G}-L_{6}\sin\varphi_{6}\right)\right)^{2}} \tag{4}\]  
\[\left(x_{B},z_{B}\right)=\left(x_{A}-L_{1}\cos\varphi_{1},z_{A}+L_{1}\sin \varphi_{1}\right) \tag{5}\]  
\[L_{2}=\sqrt[2]{\left(x_{A}-L_{1}\cos\varphi_{1}\right)-x_{F})^{2}+\left(z_{A}+ L_{1}\sin\varphi_{1}\right)-z_{F})^{2}} \tag{6}\]  
\[\varphi_{2}=\cos^{-1}(\frac{x_{B}-x_{F}}{L_{2}}) \tag{7}\]  
\[\left(x_{E},z_{E}\right)=\left(x_{D}-L_{4}\cos\varphi_{4},z_{D}+L_{4}\sin \varphi_{4}\right) \tag{8}\]  
\[L_{5}=\sqrt[2]{\left(x_{D}-L_{4}\cos\varphi_{4}\right)-x_{F})^{2}+\left(z_{D} +L_{4}\sin\varphi_{4}\right)-z_{F})^{2}} \tag{9}\]  
Figure 1:0 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Input** & **Output** \\ \hline Angle variables & \(\varphi_{1}\), \(\varphi_{4}\), \(\varphi_{6}\) & \(\varphi_{2}\), \(\varphi_{3}\), \(\varphi_{5}\), \(\varphi_{7}\) \\ Length variables & \(L_{1}\), \(L_{4}\), \(L_{6}\) & \(L_{2}\), \(L_{3}\), \(L_{5}\), \(L_{7}\) \\ Absolute coordinate variables & A, C, D, F, G & B, E, H \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design variables of the truss-link mechanism.