## 2 Methodology
### Pointing Accuracy Evaluation  
It is necessary that the actual pointing phase take place regardless of the sea conditions during navigation and regardless of the entity responsible for the errors that can occur during the phase of acquiring satellite coordinates and ship motions.  
The accuracy is evaluated in terms of azimuth and elevation error through the Error _fcn_ presented in Figure 7 and described in Section 2.5.  
The vector of pointing errors is given by \(\Delta=[\Delta\)Az, \(\Delta\)El\(]^{\text{T}}\), and pointing errors are defined by Equation (10):  
\[\left\{\begin{aligned} \Delta Az=&\left|Az-\hat{A}z \right|\\ \Delta El=&\left|El-\hat{El}\right|\end{aligned}\right. \tag{10}\]  
where \(\Delta\)Az is the pointing error of the azimuth axis, and \(\Delta\)El is the pointing error of the elevation axis. Az and El are the real azimuth and elevation values of the satellite, while \(\hat{A}z\) and \(\hat{El}\) are the azimuth and elevation value respect to the pointed target (Figure 11). The pointed target differs from the satellite position due to the presence of ship motions (and relative errors) and the errors in satellite coordinates.  
Following a transitional phase during which the pointing system shifts from its initial position to the pointing one, the error is expected to converge towards zero if the satellite is precisely pointed. Figure 12 depicts the accuracy error in pointing in terms of \(\Delta\)Az and \(\Delta\)El, i.e., the difference, in terms of azimuth and elevation, between the real position of the satellite and the position pointed by the system.  
Figure 12a depicts the errors in both transition and pointing phases, while Figure 12b provides a more focused representation of the error during the pointing phase. The system requires an error less than 100 \(\upmu\)rad to enable quantum communication; in Figure 12c, the red circle identifies the upper limit of the \(\Delta\)Az and \(\Delta\)El errors that can be admitted to achieve the satellite's pointing.  
Figure 10: Satellite’s trajectory (red) and ship position (black).  
If the system is able to always point the satellite, the error of the pointing phase is contained within this circle area (locking area, Figure 12c).  
In the example shown, it can be seen that the system achieves the required pointing accuracy only once in the 10 s time interval considered for the simulation. The result obtained may be different in other scenarios, and the satellite may never be pointed; therefore, it will be necessary to incorporate methodologies to achieve a higher level of pointing accuracy. It is also considered necessary to increase the pointing speed in terms of the time in which the system can point the satellite in order to reduce the information lag. Once the satellite has been pointed, the tracking phase can begin [10], but it is still considered necessary for the system to be able to point the satellite with greater accuracy so that even in the case of target loss, the satellite can be re-targeted quickly.  
Figure 11: Pointing errors.  
Figure 12: Pointing accuracy. (**a**) Pointing accuracy graph presenting both transition and pointing phase. (**b**) Focus on pointing error occurring in pointing phase. (**c**) Red circle represents the maximum admitted error by the system to enable the satellite pointing. Here the ATP system reach the requested pointing accuracy once (blue line crossing the red circle one time).