# Using Allan Variance Based Semi-Parameter Model to Calibrate Pointing Errors of Alt-az Telescopes
## 3 Online Calibration of Pointing Errors
### Estimation of Nonlinear Errors
\begin{array}{l}{\sigma_{A}}^{2}(\tau)={\sigma_{Q}}^{2}(\tau)+{\sigma_{N}}^{ 2}(\tau)+{\sigma_{B}}^{2}(\tau)+{\sigma_{R}}^{2}(\tau)+{\sigma_{K}}^{2}(\tau) \\ {}=3Q^{2}/\tau^{2}+N^{2}/\tau+B^{2}(2\ln 2/\pi)+K^{2}\tau/3+R^{2}\tau^{2}/2 \end{array}  \tag{9}\]  
As the value of variance is very small in general case, the calculation of its standard deviation can help improve the fitting precision, thus the formula can be approximately expressed as:  
\[{\sigma_{A}}(\tau)=\sum_{n=-2}^{2}A_{n}\tau^{n/2} \tag{10}\]  
When modeling random errors, white noise can be used directly as the driving noise of the system without modeling. The quantization noise can be transformed into the equivalent white noise, thus no further modeling is needed. To compensate for colored noises, corresponding models need to be established. For the remaining three kinds of colored noises in the alt-az telescope, the modeling process is as follows: First, obtain the power spectral density function of each colored noise, then the transfer function of each colored noise can be calculated through spectral decomposition theorem, and, finally, the differential equation can be derived from the transfer function of each colored noise. By adding all differential expressions of colored noises, the final model is set up. As for zero bias stability, the power spectral density function is:  
\[S_{B}(w)=B^{2}/w \tag{11}\]  
According to the spectral decomposition theorem, the power spectral density function and the transfer function of shaping filter satisfy the following relationship. \[S(w)=|G(jw)|^{2} \tag{12}\]  
Thus, the transfer function of irrational form is obtained:  
\[G_{B}(jw)=B/\sqrt{jw} \tag{13}\]  
If the power spectral density function is irrational, then the calculated transfer function will be irrational too. Then, a favorable approximation is needed for the sake of analytical convenience. The zero bias instability can be approximated by first order Gauss-Markov process [11]. \[\dot{G}_{B}(jw)=\beta B/(\beta+jw) \tag{14}\]  
After the Fourier inverse transformation, the differential equation of the zero bias instability is obtained. \[\dot{d}_{B}(t)+\beta d_{B}(t)=\beta Bu_{1}(t) \tag{15}\]  
where \(u_{1}(t)\) is the unit Gauss white noise. As for rate random walk, the power spectral density is:  
\[S_{K}(w)=K^{2}/w^{2} \tag{16}\]  
The transfer function of the shaping filter is:  
\[G_{K}(jw)=K/jw \tag{17}\]  
After the Fourier inverse transformation, the differential equation for random walk is obtained. \[\dot{d}_{K}(t)=Ku_{2}(t) \tag{18}\]  
where \(u_{2}(t)\) is the unit gauss white noise. As for rate ramp, the power spectral density is:  
\[G_{R}(w)=R^{2}/w^{3} \tag{19}\]  
The transfer function of the shaping filter is:  
\[S_{B}(w)=B^{2}/w \tag{11}\]0  
The rate ramp error can be approximated by the second order Gauss-Markov process [11]. The approximate transfer function is:  
\[S_{B}(w)=B^{2}/w \tag{11}\]1  
After the inverse Fourier transform, the differential equation of rate ramp is obtained:  
\[S_{B}(w)=B^{2}/w \tag{11}\]2  
Allan variance method can be employed to identify the output of the mount model. After getting all the Allan variance coefficients of noises, taking five groups of independent Gauss white noise as driven noises, the output of the five equations are added to form the final model. Thus, the Allan variance correction part \(C\) for the remaining nonlinear errors is obtained:  
\[S_{B}(w)=B^{2}/w \tag{11}\]3  
where \(u_{1},u_{2},u_{3},u_{4},u_{5}\) are five independent Gauss white noises; \(s\) is the differential operator; and \(Q,N,B,K,R\) denote quantization noise, angle random walk, zero bias stability, angular rate random walk, rate ramp, respectively, whose values are determined by calculation result of Allan variance.