# Tie-System Calibration for the Experimental Setup of Large Deployable Reflectors
## 1 Introduction  
Satellite communications have become increasingly wide-spread, thanks to the possibility of having high transmitting capacities at a relatively low cost, compared to the establishment of a terrestrial broadcasting network. Deployable reflectors represent the most widely-used type of structure, due to their features, such as large scale, high packaging efficiency, high accuracy, and low weight. Typically, their architecture consists of a deployable ring truss support, two cable nets (facing each other and linked by a series of tension ties), and a RF mesh attached to the backside of the front net. The most representative kind of this type of reflector is the AstroMesh [1], and its components are shown in Figure 1.  
The electromagnetic performance of these antennae is closely related to the shape of the reflector surface. In turn, this depends on the position of the nodes located on the front net. Therefore, it is clear that the measurement process of 3D node co-ordinates needs to be made with extreme accuracy, so as to avoid invalidating the real root mean square (RMS) value. Several measurement systems have been developed, depending on the application area. Generally, they can be divided into three categories:  
* Photogrammetry,
* Laser tracker, and
* Laser radar. Photogrammetry is a measurement technique that uses two-dimensional images of an object to obtain its dimensions. As depicted in Figure 2, photogrammetry is based exclusively on angle measurements: Three-dimensional co-ordinates are calculated by an optical triangulation (or intersection) of two or more images taken from different positions. The object to be measured is identified by targets mounted on it, usually made of reflective material to produce a high contrast between the target and the background [2]. Typically, a calibrated scale bar is integrated into the object, in order to reproduce it in true scale. At the end of the measurements, a dedicated software calculates the 3D co-ordinates in the chosen Cartesian co-ordinate system (\(x,y,z\)). Laser trackers and laser radars, alternatively, are measurement systems based on the estimation of two angles and one length, as shown in Figure 3. Two high-resolution encoders measure the azimuth (\(\theta\)) and elevation (\(\varphi\)) angles, whereas the radial co-ordinate (\(r\)) relative to the center of the target is measured by means of optical interference [3]. Several types of target can be mounted, but the most widely-used is the spherically mounted retro-reflector (SMR). Figure 1: AstroMesh. Figure 2: Photogrammetry. Differing from the laser tracker, a laser radar does not require a retro-reflector. As a matter of fact, it is capable of measuring the surface of an object with just 1% of the reflected signal [4]. Each of these measurement systems, however, has advantages and disadvantages, which need to be assessed on the basis of various factors. [PERSON] et al. [4] identified influencing factors to be taken into account before making the measurement: Task requirements, part restrictions, and environmental restrictions. Regarding task requirements, the main element to be considered when measuring the position of the nodes is accuracy, since the permissible RMS error on the reflecting surface is less than 1 mm. Whilst the laser tracker has errors on the order of 1 \(\upmu\)m, it is too costly due to the high price of each SMR and given the number of nodes (at least a hundred). The photogrammetry turned out to be the best choice [5; 6], considering the affordable cost and accuracy of the measure. Furthermore, the possibility of obtaining multiple images from different angles makes it possible to overcome the lower accuracy of the angular encoders of laser trackers. The acquisition of node co-ordinates is essential for the next calibration step, in order to meet the RMS design requirements of the reflecting surface. Generally, the main strategies followed by LDR developers and designers is to use the tie system, connecting front and rear nets, in order to locally move single nodes.