## 4 Experimental Examples  
In order to verify the accuracy and effectiveness of the online reconstructed displacement error calibration method presented in this paper, a scaled-down model for a radio telescope test antenna model was constructed, as shown in Figure 7. Under the elevation motion, the support structure's deformation measurements and calibration experiment were carried out. The support structure is made of aluminum alloy material. Its Young's modulus is 7300 MPa, Poisson's ratio is \(v=0.3\), density \(\rho=2557\) kg/m\({}^{3}\), and the entire structure weighs about 43 kg. The support beam is 2000 mm long and has a radius of 20 mm. The total weight of the sub-reflector is 200 kg. Figure 7: Scaled-down model for the radio telescope test antenna. In order to realize the deformation reconstruction for the support legs of the sub-reflector, six fiber grating strain sensors are installed on the surface of each support leg, and three sensors are placed on the same section to capture the surface strain. In the experiments, the strain data is measured by the FBG strain sensor (Fiber Bragg Grating \(\mid\) os1100, Micron Optics, Atlanta, GA, USA) and the FBG demodulator (Optical Sensing Instrument \(\mid\) Si 155, Micron Optics, Atlanta, GA, USA) Strain measurement system. The positions of the six strain sensors are shown in Table 1. Here, \(x_{k}\) represents the relative position, and \((\theta,\beta)\) are the circumferential angle \(\theta\) with angle \(\beta\) between x-axis and the frame, both measured in degrees. The properties of the FBG sensor include 1530-1565 wavelength operating range, 0.23 nm bandwidth, 15 db side-mode rejection ratio, 10 mm grid length, and 1 pm demodulation accuracy. A high-precision single-point laser tracker (LTS, API Tracker 3, Automated Precision Inc., Rockwell, MD, USA) was used to measure the actual displacement at the end of the support leg and at different elevation angles. The resolution of the LTS is 1 um, and the precision is related to the distance between the LTS and the measured object (the ratio is 5 um/m). During the experiment, the distance between the instrument and the test antenna model was 6 m, and the measurement precision of the laser tracker was 0.03 mm. The entire measurement system is shown in Figure 8.  
In the experiments, the global coordinate system was established on the main reflector. In each local coordinate system, the x-axis was established along the neutral axis of each support beam, the y-axis and z-axis were orthogonal in the beam section, and the four support beams had four different local coordinate systems (Figure 9). \begin{table}
\begin{tabular}{c c c} \hline
**Axial Position \(x_{k}\)** & **0.2L** & **0.8L** \\ \hline \((\theta_{1},\beta_{1})\) & (0,0) & (0,45) \\ \((\theta_{2},\beta_{2})\) & (120,0) & (\(-\)120,0) \\ \((\theta_{3},\beta_{3})\) & (\(-\)120,0) & (120,0) \\ \hline \end{tabular}
\end{table} 
Table 1: Strain sensor locations. Figure 8: Deformation measurement system. In order to better validate the effectiveness of the calibration scheme proposed in this paper, the strain and real displacement data at the end of the support leg at each elevation angle were collected at midnight to ensure that the model was in a stable environmental temperature. Assuming the initial position and altitude of 0 degree between the main reflector of the antenna and the horizontal plane, the strain data collected by the fiber demodulator was recorded and returned to zero. The platform was then rotated and stopped every 5 degrees increment to measure the strains and true displacements of the support leg (see Figure 10). From 5 to 60 degrees, the strains \(\varepsilon^{*}\) and displacements at the end of the support legs \(\left\{u_{x}^{act},\ u_{y}^{act},\ u_{z}^{act}\right\}\) were collected. As shown in the second section, the reconstruction value \(\left\{u_{x}^{FEM},\ u_{y}^{FEM},\ u_{z}^{FEM}\right\}\) of the deformation at the end of the support legs can be calculated by \(\varepsilon(x_{i},\theta_{i},\beta_{i})\).