## 4 Results and Discussion
### Influence of Initial Tracking Position of an SDC System under Azimuth Axis Tilt Error  
In practical engineering applications, the initial tracking position of an SDC system is generally due east (i.e., the focal axis points to due east). Currently, total stations, ranging sensors, and laser scanners are mainly used to ensure the initial tracking position of an SDC  
Figure 5: Variation of the effective tracking factor \(\sigma\) of an SDC system with the radial and tangential angles of azimuth axis. system. These measuring instruments have certain errors, which leads to the initial tracking position of the SDC system not being due east. Therefore, this section takes a typical day (the summer solstice) in a typical area (Dunhuang) as an example to study the influence of the initial tracking position of an SDC system on its tracking performance under the azimuth axis tilt error. Figure 6 shows the schematic diagram of the initial tracking position of an SDC system. In order to describe the initial tracking position of an SDC system, two offset angles, \(\gamma_{1}\) and \(\gamma_{2}\), are introduced. Then, any initial tracking position can be considered as the focal axis vector rotated by \(\gamma_{1}\) around the z-axis and then rotated by \(\gamma_{2}\) around the y-axis. The focal axis vector \(\mathrm{N^{\prime}_{f}}\) of the initial tracking position can be calculated by Equation (19), where \(\mathrm{Rot(y,\gamma_{2})}\) and \(\mathrm{Rot(z,\gamma_{1})}\), respectively, represent the rotation matrix around the y- and z-axes, which are shown by Equations (3) and (20), respectively. When \(\gamma_{1}\
eq 0\) and \(\gamma_{2}=0\), the initial tracking position varies in the horizontal plane. When \(\gamma_{1}=0\) and \(\gamma_{2}\
eq 0\), the initial tracking position varies in the vertical plane. When \(\gamma_{1}=\gamma_{2}=0\), it means that the initial tracking position of the SDC system is due east. The value range of \(\gamma_{1}\) and \(\gamma_{2}\) is \([-0.5^{\circ},0.5^{\circ}]\). \[\mathrm{N^{\prime}_{f}}=\mathrm{N_{f}}\cdot\mathrm{Rot(z,\gamma_{1})}\cdot \mathrm{Rot(y,\gamma_{2})} \tag{19}\]  
\[\mathrm{Rot(y,\gamma_{2})}=\left[ \begin{array}{ccc}\cos\gamma_{2}&0&-\mathrm{ sin\gamma_{2}}\\ 0&1&0\\ \mathrm{sin\gamma_{2}}&0&\cos\gamma_{2}\end{array} \right] \tag{20}\]  
Simplify Equation (19) to get \(\mathrm{N^{\prime}_{f}}=[-\cos\gamma_{1}\cos\gamma_{2},\;-\mathrm{sin\gamma_{1 }},\;\mathrm{cos\gamma_{1}}\mathrm{sin\gamma_{2}}]\). According to the results, different initial tracking position offsets have the same trend on the effective tracking factor of an SDC system, only with a numerical value difference. In order to display the data more clearly, this section only lists the variation rules of the initial tracking position offset of \(0.2^{\circ}\) and \(0.4^{\circ}\). Figure 7 shows the variation of the effective tracking factor of an SDC system with different initial tracking positions under the azimuth  
Figure 6: Schematic diagram of initial tracking position of an SDC system. axis tilt error. It can be clearly seen that whether the initial tracking position varies in the horizontal plane or the vertical plane, the larger the deviation angle of the initial tracking position (i.e., the larger \(\gamma_{1}\) or \(\gamma_{2}\)), the worse the tracking performance of the SDC system. Regarding the symmetric initial tracking position offset angle in the due east direction (i.e., \(\gamma_{1}\) = 0.2\({}^{\circ}\) and \(\gamma_{1}\) = \(-\)0.2\({}^{\circ}\)), the tracking performance change trends are exactly opposite. From Figure 7a, it can be seen that when \(\gamma_{1}\) > 0\({}^{\circ}\), the azimuth axis tilt error has the least influence on the tracking performance at the azimuth axis tangential angle \(\phi_{1}\) = 345\({}^{\circ}\) (i.e., 15\({}^{\circ}\) south by east) or \(\phi_{1}\) = 165\({}^{\circ}\) (i.e., 15\({}^{\circ}\) north by west).