## 1 Introduction
Among the inducements of CE, the most important factors are CCR geometric errors (which are mainly dihedral angle error and planeness error) and APT elements assembly errors. The assembly error mainly refers to the reflector elements' posture error or the rotation error of the APT axis, which will directly affect the beam-pointing direction. On the other hand, because light diffracts when it pass through any objects, if the beam is reflected by the CCR, the faculae is affected not only by the APT reflector elements but also by the CCR. Under general conditions, however, the faculae \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Abbreviation** & **Explanation** & **Symbol** & **Explanation** \\ \hline APT & Acquisition, pointing, and tracking & GEO & Geosynchronous Earth Orbit \\ CCR & Cube corner retro-reflector & OPD & Optical path difference \\ EDRS & Europe data relay system & \(\delta\) & Dihedral angle error \\ CE & Coaxiality error & \(W\) (\(\jmath\)) & Wavefront \\ PAA & Point-ahead angle & \(\Delta d\) & CCR Geometrical error \\ FSM & Fast-steering mirror & \(ds\) & OPD (dihedral angle error induced) \\ FM & Folding mirror & \(dt\) & OPD (planeness error induced) \\ Az & Azimuth & N1, N2... & Newton’s rings \\ El & Elevation & \(\alpha\) & Angle rotated around the \(x\)-axis \\ PAM & Point-ahead mirror & \(\beta\) & Angle rotated around the \(y\)-axis \\ CCD & Charge coupled device & \(\gamma\) & Angle rotated around the \(z\)-axis \\ LEO & Low Earth Orbit & \(\varepsilon\) & Angle between vectors \\ MEO & Medium Earth Orbit & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables and parameters detail explanation. distortion caused by the reflector elements can be ignored during the calibration process. Thus the faculae centroid error is mainly caused by the dihedral angle error and planneness error from the CCR. The dihedral error is caused by the perpendicular error angle between two faces of the CCR, while the planneness error is caused by the unevenness of each face of the CCR. These two factors can affect the spot energy distribution and reduce coaxiality. Presently, some researchers introduced the influence of the reflective element assembly error on beam-pointing accuracy of the APT system. For example, [PERSON] et al. [11] investigated the static position errors during the assembly of the HY-2 satellite optical communication terminal, while [PERSON] et al. examined the pointing error caused by the axis error angle in a periscope-type APT [12]. The dihedral angle error and planneness error of the CCR and their influence on the laser spot energy distribution have been studied extensively, for example, by [PERSON], who analyzed the planneness error of the CCR, establishing the mathematical model of planneness error [13], and by [PERSON], who analyzed the CCR geometric error and studied the far-field diffractive characteristics of the CCR [14]. However, the CE has not been studied in the presence of assembly error. Moreover, considering the CCR error, in some studies, the CCR is only used to calibrate the signal-receiving-transmission coaxial sub-module of the APT but not the whole APT system [15]. Therefore, some factors have not been considered in previous studies. Hence, this study mainly focuses on the characteristics of the CE change rule, attempting to analyze the influence of coupling factors on the CE and focusing on the challenge of establishing the analysis model under the influence of coupling factors. The solution to this problem can effectively help us obtain the variation characteristics of the CE, improving assembly efficiency and enhancing the performance of the APT systems. This study examines the geometric error of the CCR and the assembly error of the APT system, establishes the CE analysis model, and investigates the change characteristics. The remainder of this paper is organized as follows: The CE measurement method based on CCR is introduced in the Section 2.