# Using Allan Variance Based Semi-Parameter Model to Calibrate Pointing Errors of Alt-az Telescopes
## 2 Error Analysis and Modeling of Pointing Errors  
A typical alt-az telescope consists of optical imaging system (OIS) and target tracking system (TTS). In OIS, optical imaging sensors, such as CCD camera, infrared camera and visible spectrum camera, collect image information about targets of interest. In TTS, the two degrees-of-freedom servo system, rotating in azimuth axis and elevation axis, ultimately determines the LOS pointing performances of the instrument in three aspects: tracking to keep targets of interest within the field of view of CCD camera, stabilizing the LOS to obtain high-quality images, and determining the exact location according to position sensors. Some auxiliary equipment is also required in most alt-az telescopes, including gyroscopes, motors and encoders. The Denavit-Hartenberg (D-H) transformation [5] is adopted to demonstrate the mount model (MM). The objective of telescope rack modeling is to identify the geometric errors at specific positions and apply them to remove errors in practice.  
In an ideal condition, the azimuth axis, the elevation axis and the optical axis are three perpendicular axes in zero elevation position. However, during the assembly and operation processes, perpendicularity errors, rotation errors due to variable relative motions, measurement errors owing to the offset and imprecision of encoders occur and influence each other. Most servo-control systems of telescopes employ the structure of inner stabilization loop and outer tracking loop. The former eliminates turbulences and movements of the rack and the latter guarantees that the LOS maintains pointing to targets of interest. As a result of drift, coulomb friction, backlash and offsets of auxiliary equipment, servo-control errors including stability and tracking errors, the final LOS pointing direction of the telescope is shaped by modeling the errors according to the order they occur. Besides the above-mentioned errors, nonlinear errors remain because of the influence of environmental factors such as wind, vibration, gravity, temperature and atmosphere [6,7], which cannot be demonstrated by a physical mount model but a statistical method.  
The static pointing precision of telescope LOS pointing can be illustrated according to the inconsistency between the actual and ideal pointing direction. It is calculated by the value of encoders and miss distances in CCD camera in practice. As shown in Figure 1[7], the error vector of LOS pointing is \(\Delta=[\Delta A,\Delta E]^{T}\). The pointing errors can be expressed by the following formula:  
\[\left\{\begin{array}{l}\Delta A=\hat{A}-A=f(A,E)+\varepsilon\\ \Delta E=\hat{E}-E=g(A,E)+\eta\end{array}\right. \tag{1}\]  
where \(\Delta A\) and \(\Delta E\) are pointing errors in azimuth axis and in elevation axis, \(\hat{A}\) and \(\hat{E}\) are measurements of encoders in azimuth axis and in elevation axis, \(A\) and \(E\) are the actual angular positions, \(f(A,E)\) and \(g(A,E)\) are expressions of the mount model in azimuth axis and in elevation axis associated with both azimuth and elevation angles, and \(\varepsilon\) and \(\eta\) are independent errors related to measurements in azimuth axis and in elevation axis.