# Tie-System Calibration for the Experimental Setup of Large Deployable Reflectors
## 3 Results  
In order to verify the validity of the proposed method, a case study of an asymmetric large deployable reflector, designed by Thales Alenia Space [53], is described. The relevant parameters and geometric data are listed as follows:  
* Focal length: 6 m
* Number of free nodes: 296
* Number of vertices: 14
* Number of total cables: 1044
* Cable section: 4 mm\({}^{2}\)
* Young modulus of cables: \(8.3\times 10^{10}\) N/m\({}^{2}\)
* Initial RMS error: 0.5872 mm
* Design value of the RMS faceting error: 0.21 mm  
The value of the spring constant for tie cables ranges from \(2\times 10^{3}\) N/m to \(68\times 10^{3}\) N/m, with a radial step of \(11\times 10^{3}\) N/m starting from the centre (central node) to the outer ring cables. The initial RMS error on the front net was simulated by introducing an additional value to each node proportionally to the length of the tie connected to it. By imposing the equilibrium in System (1), we can determine the construction lengths described in Section 2. Nevertheless, from Figure 8 it can be noted that the maximum error \(e_{t^{0}}\) obtained was about 1 mm, representing only 1% of the total cables; the largest percentage (86%) showed an error between 0.6 mm and 1 mm,  
Figure 7: Flowchart for the algorithm of the proposed method. RMS, Root Mean Square. In Figure 9 the local faceting error for the chosen initial configuration is shown. The local faceting error is calculated by considering the centroid of the triangular facets into which the surface can be decomposed [64; 65; 66; 67]. As can be observed, the faceting error follows the shape of the asymmetric reflector since the chosen error is proportional to the tie lengths. As a matter of fact, the central zone is the one with the shortest cables and, therefore, with the lowest faceting error. On the contrary, the tie lengths, and consequently the errors, grow; moving from the centre to the outer perimeter of the net. The optimization method described in the previous sections is first applied to the rigid ring truss support. Figure 10 shows the error of each free node of the front net with respect to the ideal surface, coupled with the stretching/shortening value necessary to reach the desired surface accuracy. The reason why the correction values are all positive is that the initial error is simulated by positioning all nodes of the front net above the ideal surface, so there is a need to shorten the tie lengths to satisfy the RMS design value. The bars are grouped by spring constant value. Figure 8: Pie chart of the error (absolute value), grouped by measuring ranges (mm), between the measured construction lengths and those obtained by solving for equilibrium in System (1). Figure 9: Faceted RMS (mm) of the front tension truss in the initial configuration. The corresponding faceting error is shown in Figure 11. As can be observed, the RMS of the faceting error decreases until the required value of 0.2036 mm is furnished, as per specification. Then, the same analysis has been performed for the flexible ring truss support. It can be noted that, in Figure 12, the correction values are lower than the corresponding values for the rigid truss support: This is because the additional tensioning of ties further deforms the shape of the truss support, resulting in a closure of the support itself [68]. As a result, this causes a slight lowering of the front net, thus reducing the shortening action of the tie lengths. Finally, Figure 13 shows the faceting error distribution on the front net. Even in this case, the final RMS faceting error reached 0.2045 mm, representing the design value of the RMS. Comparing Figures 11 and 13, it can be observed that the faceting error distribution is more uniform for the flexible ring truss support. This result can be  
Figure 11: Faceting error (mm) distribution on the front tension truss, obtained for the rigid ring truss support. Figure 10: Error and correction values (rigid case).