# 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  
Imprecision during the manufacture, assembly, installation and operation processes has complex impacts on LOS pointing precision of alt-az telescopes. Thus, the mount model above-mentioned is unable to take all the cases into consideration, especially the highly coupled errors and environmental factors [9]. Error sources excluded are usually nonlinear and unavoidable, having complex statistical properties, and cannot be expressed exactly through kinematic modeling. Furthermore, small-angle approximations and errors related to measurements have increased the nonlinearity of residual errors. To solve abovementioned problems and achieve higher pointing precision, a hybrid model that can handle with both geometric errors and nonlinear terms is proposed, called Allan variance based Semi-Parametric model. An additional correction term for nonlinear errors is added to the mount model, thus Equation (2) can be rewritten as:  
\[\Delta =R^{T}x+C+\mu \tag{4}\] \[C =\left[ \begin{array}{c}\varepsilon_{c}\\ \eta_{c}\end{array} \right] \tag{5}\]  
where \(\Delta\), \(R\), and \(x\) are analogous to parameters of the mount model and \(\mu\) is assumed to be a white noise. Equation (4) represents the AVSPM, which inherits parametric parts \(Rx\) from the mount model and adds the Allan variance correction item \(C\) for residual nonlinear errors that are not considered in the mount model. Therefore, the AVSPM maintains high interpretability of geometric errors as mount models and retains certain flexibility of the nonlinear errors using a numerical method. AVSPM has been proven to outperform the mount model significantly in handling with complicated situations in practice. Allan variance [10] method is a general method for analyzing frequency stability in time domain. This method is used to detect the contributions of all kinds of error sources in the noise data, and separate out and quantify these error sources. It is applicable to the analysis of the error characteristics of inertial sensors such as the fiber optic gyroscope and any other precision measuring instruments. The Allan variance method is as follows: To sample the residual of mount model at sampling period \(\tau_{0}\), total up to \(N\) points. Divide all the points into \(K\) groups, each group consists of \(M\) sampling values \((M\leq(N-1)/2)\). The sampling time for each group is \(\tau=M\tau_{0}\), namely correlation time. The mean of each group is:  
\[\overline{w}_{k}(M)=\frac{1}{M}\sum_{i=1}^{M}w_{((k-1)M+i)}(k=1,2,\cdots,K) \tag{6}\]  
The Allan variance is defined as:  
\[\sigma_{A}{}^{2}(\tau)=\frac{1}{2(k-1)}\sum_{k=1}^{K-1}(\overline{w}_{k+1}(M )-\overline{w}_{k}(M))^{2} \tag{7}\]  
The Allan variance can be obtained according to different correlation time \(\tau\). The relationship between the Allan variance and the bilateral power spectral density of the noise in the sampled data is as follows:  
\[\sigma_{A}{}^{2}(\tau)=4\int_{0}^{\infty}S_{w}(f)\frac{\sin^{4}(\pi fz)}{(\pi f z )^{2}}df \tag{8}\]Formula above shows that a filter, whose transfer function is \(\sin^{4}(\pi fz)/{(\pi fz)}^{2}\), makes the noise of the output proportional to the Allan variance. Therefore, the Allan variance can separate, identify and quantify all kinds of noise items in the data. Due to the influence of system noises, environmental noises and other factors, random errors occur in the output signal of the LOS pointing, including quantization noise (Q), zero bias stability (B), angle random walk (N), rate ramp (R), angular rate random walk (K), exponential correlation noise (E), sinusoidal noise (S) and so on. If noise terms are independent in statistical properties, the Allan variance can be expressed as the sum of squares of various noise terms. In the actual data test, the first five items have the greatest contribution to the total error, therefore the total Allan variance can be represented by the first five items by approximation. \[