## 3 LSDM Assembly Error Model in Microgravity Assembly on the Ground
### Assembly Error Propagation of Single-Loop Closed Chain  
This section builds the assembly error propagation model in a single-loop closed chain and considers three factors: the assembly positioning error, the truss rod deformation, and the hinge clearance. The process of constructing an error model is progressive, which means that, initially, only the positioning error is considered and, then, hinge clearance effects are considered. Finally, all three factors are accounted for. Assembly positioning error is inevitable during the assembly process. Typically, the assembly process instruction provides each hinge's positional tolerance, and the operator uses a laser tracker to measure each hinge's spatial position. Based on the given tolerance, the operator determines whether the hinge's positioning error meets the process requirements. The hinge clearance fit states are shown in Figure 8. The state is stable on the ground, but the balance is broken in space. When considering the positioning error and hinge clearance, the hinge center coordinates in space are calculated as follows:  
\[\left\{ \begin{array}{ll}x_{s}=x+\Delta x,&\text{support}\\ x_{s}=x+\Delta x+\varepsilon_{x},&\text{supported joint}\\ z_{s}=z+\Delta z+\Delta r_{1},&\text{support}\\ z_{s}=z+\Delta z-\Delta r_{2}+\varepsilon_{z},&\text{supported joint}\\ (\Delta x)^{2}+(\Delta y)^{2}\leq\xi^{2}\\ \varepsilon_{x}\in[-\Delta r_{1}-\Delta r_{2},\ \Delta r_{1}+\Delta r_{2}]\\ \varepsilon_{z}\in[0,\ 2\Delta r_{1}+2\Delta r_{2}]\end{array} \right. \tag{5}\]  
where \(x_{s}\), \(z_{s}\) are the hinge joint center coordinates in space, and they are not constant but vary within certain ranges; x, z represent the designed coordinates of the hinge; \(\Delta x\) and \(\Delta z\) represent the hinge's positioning errors in the directions of X and Z, respectively; \(\xi\) is the hinge's positioning tolerance; \(\Delta r_{1}\) is the clearance between shaft and support; \(\Delta r_{2}\) is the clearance between shaft and supported joint; and \(\varepsilon_{x}\) and \(\varepsilon_{z}\) are the components of the center offset between the support and the supported joint in X direction and Z direction, respectively. Equation (5) is the assembly error model considering the positioning error and clearance that has different states on the ground and in space. Then, the truss rod deformations should be considered in the above assembly error model. A detailed analysis of the truss rod deformation is provided in Section 2.3. However, simply inserting the rods' length changes into the assembly error model when there is coupling between rods is not feasible. In this paper, the truss rod deformation of the closed-chain structure is introduced into the assembly error model, as shown in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline
**Rod Types** & **Characteristic** & **Schematic Diagram** \\ \hline \multirow{3}{*}{Independent Rod} & The rod connects hinges at both & \\  & ends, and the length between the & \\  & two hinges is affected by rod & \\  & deformation. & \\ \hline \multirow{3}{*}{Coupling Rod} & In addition to the hinge at both & \\  & ends of the rod, there is also one & \\  & or more hinges at the middle & \\  & position, and the middle hinge is & \\  & fixed to the rod. & \\ \hline \multirow{3}{*}{Equivalent Fixed-Length Rod} & The rod is determined by two & \\  & supports fixed on the antenna & \\ \cline{1-1}  & panel or satellite body, and the rod & \\ \cline{1-1}  & is not affected by deformation. & \\ \hline \end{tabular}
\end{table} 
Table 1: Rod types of the LSDM. A closed-chain structure is composed of three or more truss rods connected by hinges. It contains at least one equivalent fixed-length rod. For the one without equivalent fixed-length rods, the known lengths of the truss rods determined by the previous closed-chain can be used as the equivalent fixed-length rod for the current closed chain.