# Using Allan Variance Based Semi-Parameter Model to Calibrate Pointing Errors of Alt-az Telescopes
## 4 Experiment and Results  
The experiment was conducted to compare performances between the proposed AVSPM and the classical mount model. The experiment was carried out through a 1 m alt-az telescope, as shown in Figure 2, which was designed with an angular movement ranging from 0\({}^{\circ}\) to 360\({}^{\circ}\) in azimuth and 0\({}^{\circ}\) to 90\({}^{\circ}\) in elevation. Actual LOS directions and corresponding pointing errors of stars were obtained from the alt-az telescope. The 176 stars in use were randomly distributed in the whole sky and uniformly selected from the FK5 Catalogue, as demonstrated in Figure 3.  
The quantization noise, angle random walk, zero bias stability, rate ramp and angular rate random walk can be obtained by fitting, and the estimated values are:  
\[ \begin{array}{l}Q=\ A_{-2}/3600\sqrt{3}\\ N=\ A_{-1}/60\\ B=\ A_{0}/0.6643\\ R=\ 60\sqrt{3}A_{1}\\ K=\ 3600\sqrt{2}A_{2}\end{array}  \tag{24}\]  
As for zero bias stability, the relative amplitude error of the approximate transfer function and the real transfer function is:  
\[\varepsilon_{B}=10*\log\bigl{|}\bigl{(}\bigl{|}\hat{G}_{B}(jw)\bigr{|}-\bigl{|} G_{B}(jw)\bigr{|}\bigr{)}/\bigl{|}G_{B}(jw)\bigr{|}\bigr{|}=10*\log\bigl{|}\sqrt{w}/ \sqrt{1+w^{2}}-1\bigr{|} \tag{25}\]  
Considering low-frequency characteristics of the zero bias instability, the domain is restricted from 0.05 Hz to 0.1 Hz; within this range, if \(\beta\) equals 4 Hz and the amplitude error of the approximate  
Figure 3: Star map distribution of the experiment. Figure 2: The alt-az telescope. transfer function \(\hat{G}_{B}(jw)\) is less than \(-3\) dB. Similarly, for rate ramp, the relative amplitude error of the approximate transfer function and the real transfer function is:  
\[\varepsilon_{R}=10*\log\bigl{|}\bigl{(}\bigl{|}\hat{G}_{R}(jw)\bigr{|}-\bigl{|}G _{R}(jw)\bigr{|}\bigr{)}/\bigl{|}G_{R}(jw)\bigr{|}\bigr{|}=10*\log\bigl{|}\sqrt{ w^{3}}/\sqrt{w^{4}+w_{0}{}^{4}}-1\bigr{|} \tag{26}\]  
Considering the low-frequency characteristics of the rate ramp, the domain is restricted from 0.005 Hz to 0.01 Hz; within this range, if \(w_{0}\) equals 0.05 rad/s, the amplitude error of the approximate transfer function \(\hat{G}_{R}(jw)\) is less than \(-3\) dB. The CCD camera must be calibrated before the experiment begins, aiming at determining the equivalent weight between miss distances and angles in azimuth and elevation axis. After that, the specific procedure to compare performances of the AVSPM and the classical mount model is as follows:  
1. Preparations for calibration process were made, namely blind-pointing measurements, including measuring LOS pointing errors at positions covering the whole sky and determining parameters of the mount model which most meet with the observed data. 2. Active sampling such as blind-pointing measurements was carried out to acquire the correction coefficients of mount model. For the calibration portion, the residual errors after compensation of the mount model is necessary. The proposed AVSPM and conventional mount model were employed to calibrate LOS pointing errors at every pointing angle, making corrections available for the servo system and actuators. A numerical model called Spherical Harmonic Function Model (SHFM) was introduced for contrast. In SHFM, the spherical function is expanded into Fourier series, and then the zonal harmonic terms of series to fourth order are taken to shape the final model. 3. While the LOS was turned to different directions to cover the whole sky within the movement range, all of the observed data were recorded synchronously, including pointing angles, corresponding errors and time. As the baseline of these evaluation models, measurements of the original LOS pointing angles and corresponding errors are shown in Figure 4. Owing to the impacts of geometric and nonlinear errors, the LOS pointing was of poor pointing precision, 18.1477\" in azimuth and 26.3599\" in elevation, which is mainly caused by the inaccuracy during manufacture and assembly processes.