# Tie-System Calibration for the Experimental Setup of Large Deployable Reflectors
## 2 Experimental Settings for Calibration
### Rigid Ring Truss Support: Construction Length Determination  
Tie-cable regulation is usually independent of the actuators used for the LDA deployment [38; 39; 40] and from the control system [41; 42; 43; 44]. Here, the method to regulate tie cable tension in an antenna with rigid ring truss support is first described. Once the deployment has been carried out and all node co-ordinates of the net have been measured, the reflective surface will typically deviate from the design configuration due to mechanical errors. As a consequence of this deviation, the system of equilibrium equation is not satisfied for the current configuration. Then, denoting with \(E_{ij}\), \(A_{ij}\), \(Lc_{ij}\), and \(Lc_{ij}^{0}\) the Young modulus, the cross-section area, the measured length, and the construction length of cable \(ij\), respectively, and with \(k_{ij}\), \(Lt_{ij}\), and \(Lt_{ij}^{0}\) the spring constant, the measured length, and the construction length of tie \(ij\), respectively, the system of nonlinear equations for each free node \(i\), with \(j\) adjacent nodes, is not satisfied:  
\[\left\{ \begin{array}{ll}\sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0 }}{Lc_{ij}^{0}}\frac{x_{i}-x_{j}}{Lc_{ij}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0})\frac{x_ {i}-x_{j}}{Lt_{ij}}]&\
eq 0\\ \sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0}}{Lc_{ij}^{0}}\frac{y_{ i}-y_{j}}{Lc_{ij}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0})\frac{y_{i}-y_{j}}{Lt_{ij}}]&\
eq 0 \\ \sum\limits_{j}[E_{ij}A_{ij}\frac{Lc_{ij}-Lc_{ij}^{0}}{Lc_{ij}^{0}}\frac{z_{ i}-z_{j}}{Lc_{ij}}+k_{ij}(Lt_{ij}-Lt_{ij}^{0})\frac{z_{i}-z_{j}}{Lt_{ij}}]&\
eq 0 \end{array} \right., \tag{1}\]  
where \(x_{l}\), \(y_{l}\), and \(z_{l}\) are the measured Cartesian co-ordinates of the \(l\)-th node. It is noteworthy to remark that the only measured parameters are the node co-ordinates; while the measured length is derived using the Euclidian norm. The remaining parameters of the previous system are design parameters, instead; each affected by different types of error. Here, denoting by \(L_{ij}\) and \(L_{ij}^{0}\) the measured and construction length for both cable and tie, respectively, we choose to gather all these errors inside the construction length \(L_{ij}^{0}\), defined as the distance between the centers of the eyelets \(i\) and \(j\) belonging to the same cable, as shown in Figure 4. This choice is motivated by the fact that the construction length of the cables is affected by two main sources of errors, such as the manufacturing errors generated during the cutting operation of (Computer Numerical Control) CNC machines [45], and the assembly errors coming from the bad placement of the eyelets (which are necessary to connect two or more cables). In order to restore the equilibrium condition in System (1), only the design parameters can be adjusted, while the measured parameters describe the real configuration of equilibrium: The measured configuration is already in equilibrium and, thus, System (1) is to be satisfied in this configuration without changing the node co-ordinates. Then, as each cable works in traction force only, the constraint \(L_{ij}\geq L_{ij}^{0}\) must be imposed. To solve the system of nonlinear equations, Matlab(r) provides the command fsolve, but it does not allow the inclusion of any constraints. To overcome this problem, the nonlinear programming solver fmincon can be used by giving a constant objective function and setting (1) as the nonlinear equality constraint,in addition to the linear inequality constraint \(L_{ij}\geq L_{ij}^{0}\). The resulting constrained optimization problem is described below:  
\[\left\{ \begin{array}{ll}\texttt{find}&L_{ij}^{0},\ \forall\texttt{cables\ and\ ties}\\ \texttt{min}&\texttt{constant\ objective\ function}\\ \texttt{s.t.}&L_{ij}\geq L_{ij}^{0}\end{array} \right. \tag{2}\]  
Hence, a first check is required.