# Tie-System Calibration for the Experimental Setup of Large Deployable Reflectors
## 2 Experimental Settings for Calibration
### Rigid Ring Truss Support: Tie Calibration  
Once the configuration satisfying the static equilibrium has been found, the algorithm continues with the estimation of the values of stretching or shortening for each tension tie-cable, which ensures that the surface accuracy of the reflector can be met. Figure 5 shows the screw adjustment system for a tie. One fixed part is connected to a node of the front net, while one mobile part--adjustable with a screw--is connected to a node of the rear net. Now, the system of nonlinear equations of each free node \(i\) can be written as follows:  
\[\left\{\begin{array}{ll}\sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0} }{Lc_{ij}^{0}}\frac{x_{i}-x_{j}}{Lc_{ij}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0}+\delta Lt _{ij})\frac{x_{i}-x_{j}}{Lt_{ij}}]&=0\\ \sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0}}{Lc_{ij}^{0}}\frac{y_{i} -y_{j}}{Lc_{ij}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0}+\delta Lt_{ij})\frac{y_{i}-y_{j}}{ Lt_{ij}}]&=0\\ \sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0}}{Lc_{ij}^{0}}\frac{z_{i} -z_{j}}{Lc_{ij}^{0}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0}+\delta Lt_{ij})\frac{z_{i}-z_{ j}}{Lt_{ij}}]&=0\end{array}\right.. \tag{4}\]  
Figure 4: Layout of a cable: The construction length \(L_{0}\) is affected by manufacturing errors generated during the cutting operation and assembly errors coming from a bad placement of the eyelets necessary to connect two or more cables.  
This system, similar to (1), is used to determine the vector \(\mathbf{L^{0}}\), with the difference that the variables to be found are the stretching/shortening values \(\delta Lt_{ij}\) and the co-ordinates of the free nodes of the front and rear net. The values \(\delta Lt_{ij}\) can be positive or negative: Here, we assume positive values for tie-shortening and negative for tie-stretching. Additionally, this system is subject to some constraints: One condition for all net cables is \(Lc\geq Lc^{0}\). For the tension tie-cables, for which the values \(\delta Lt_{ij}\) are to be considered, the constraint condition is that the final force \(F_{ij}\) must be one of traction:  
\[F_{ij}=k_{ij}(Lt_{ij}-Lt_{ij}^{0})+k_{ij}\delta Lt_{ij}\geq 0\Rightarrow Lt_{ ij}+\delta Lt_{ij}\geq Lt_{ij}^{0}. \tag{5}\]  
Finally, the third constraint is related to the surface accuracy, since the RMS error must be lower than the desired design value \(RMS_{target}\). This is a typical constrained optimization problem, largely used in the design optimization of complex systems [46; 47; 48; 49] and mechanisms [50; 51; 52]. The optimization problem can be summarised as follows:  
\[\left\{\begin{array}{ll}\texttt{find}&\mathbf{x_{1}},\mathbf{y_{1}},\mathbf{ z_{1}}\,\texttt{and}\,\delta\mathbf{Lt}\\ \texttt{min}&\texttt{constant objective function}\\ \texttt{s.t.}&Lc_{ij}\geq Lc_{ij}^{0}&\texttt{net cables}\\ &Lt_{ij}+\delta Lt_{ij}\geq Lt_{ij}^{0}&\texttt{tension tie cables}\\ &RMS\leq RMS_{target}&\texttt{reflecting surface requirement}\end{array}\right., \tag{6}\]  
where \(\mathbf{x_{1}}\), \(\mathbf{y_{1}}\), and \(\mathbf{z_{1}}\) are the free node co-ordinates and \(\delta\mathbf{Lt}\) is the array containing all corrections \(\delta Lt_{ij}\). The RMS is calculated by measuring the minimum distance of the free nodes of the front net, compared  
Figure 5: Screw adjustment in the tie system: (**left**) Tie system before adjustment; and (**right**) tie system after the screw regulation.  
to the ideal surface of the paraboloid [53]. The initial condition for the free nodes is represented by their experimental measurement, while the guess value for \(\delta\mathbf{Lt}\) is set equal to zero.