## 3 Method for Improving the Pointing Error
### Pre-compensation for Ship Platform Attitude  
During the initial acquisition stage, the target coordinate value (A, E) in the Northeast celestial coordinate system is converted to \((\mathrm{A_{c}},\mathrm{E_{c}})\) in the deck coordinate system. \((\mathrm{A_{c}},\mathrm{E_{c}})\) is used to guide the ATP to open-loop pointing. As shown in Figure 2a, the Northeast celestial coordinate system is defined as O-XYZ, where OX is due North, OY is due east, and OZ is due sky. The origin, O is the intersection of the azimuth axis and the pitching axis of the tracking equipment. The polar coordinates of the observation point, M are (A, E), while the projection of M on the OXY plane is N. A represents the azimuth angle, that is, the included angle of XON, which is positive when overlooking clockwise. E represents the elevation  
Figure 1: Schematic of acquisition and tracking. (**a**) The shipborne acquisition, tracking, and pointing (ATP) system initiates the acquisition process. The shipborne ATP and satellite activate their beacon lasers and point at each other using the predicted ephemeris. (**b**) The satellite detects the uplink beacon laser and initiates precise tracking. (**c**) The shipborne ATP detects the downlink beacon laser and initiates precise tracking. (**d**) Bidirectional tracking and locking between the satellite and shipborne ATP is established until the laser communication is terminated. angle, which is the angle MON, where up is positive. L represents the distance from the tracking device to the observation point. As shown in Figure 2b, the deck coordinate system \(\rm O-X_{c}Y_{c}Z_{c}\) is defined; the axis \(\rm X_{c}\) is the fore-and-aft line, while the bow direction is forward. The axis \(\rm OZ_{c}\) is positive clock wise; the axis \(\rm Y_{c}\) is perpendicular to the fore-and-aft line in the deck plane, whereas the axis \(\rm OZ_{c}\) is perpendicular to the deck plane, upward is positive. The origin \(\rm O\) is consistent with the Northeast celestial coordinate system. The projection point of \(\rm M\) on plane \(\rm OX_{c}Y_{c}\) is \(\rm N_{c}\). The polar coordinates of \(\rm M\) in the deck coordinate system are defined as \(\rm(A_{c^{\prime}},E_{c})\). \(\rm A_{c}\) represents the azimuth angle of the deck, where clockwise is positive. \(\rm E_{c}\) represents the elevation angle of the deck; upward is positive. As shown in Figure 2c, the roll angle, \(\rm R\) is the angle obtained by rotating the fore-and-aft line of the deck platform, and the port side rising is defined as a positive angle. The pitch angle, \(\rm P\) is the angle included between the fore-and-aft line and the horizontal plane, and \(\rm P\) is the positive angle when the bow is raised. The yaw angle, \(\rm H\) is the rotation angle of the fore-and-aft line about the z-axis, and the clockwise angle is the positive angle. When the shipborne ATP tracks a target, it is necessary to convert (\(\rm A\), \(\rm E\)) of the target in the Northeast celestial coordinate system to \(\rm(A_{c},E_{c})\) of the deck coordinate system for guidance and tracking. In the Northeast celestial coordinate system, the transformation relationship exists between the polar coordinate system and rectangular coordinate system is given by Equation (1):  
\[\left\{ \begin{array}{l}\rm x=Lcos\,EcosA\\ \rm y=Lcos\,EsinA\\ \rm z=Lsin\,E\end{array} \right. \tag{1}\]  
In the deck coordinate system, the transformation relationship between the polar coordinate system and rectangular coordinate system is given by Equation (2):  
\[\left\{ \begin{array}{l}\rm x_{c}=Lcos\,E_{c}cos\,A_{c}\\ \rm y_{c}=Lcos\,E_{c}sin\,A_{c}\\ \rm z_{c}=Lsin\,E_{c}\end{array} \right. \tag{2}\]  
When the ship is affected by yaw, pitch, and roll at the same time, it shall be convertedin the order of yaw, pitch, and roll. The attitude rotation matrix, \(\xi t\) is given by Equation (3):  
\[\xi t=\left[ \begin{array}{cccc}1&0&0\\ 0&\cos\,\rm R&\sin\,\rm R\\ 0&-\sin\,\rm R&\cos\,\rm R\end{array} \right]\left[