## 3 Results and Implementation of the Optical System to Improve Pointing Accuracy  
By changing the conical scanning amplitude \(\mathrm{r_{s}}\) and frequency \(\omega\) (Equations (11) and (12)), it was possible to evaluate the number of times (N) in which the system reached the requested pointing accuracy (Figure 18). In the case of conical scanning (Figure 18), N = N\({}_{\mathrm{max}}\) when the \(\mathrm{r_{s}}\) value is 0.009 and decreases up to N = 0 when the \(\mathrm{r_{s}}>\) 0.04. A similar approach has been used in the case of [PERSON]'s lemniscate scanning (Figure 19), in which the times in which the system has reached the requested pointing accuracy have been evaluated by varying the half-width \(a\) of the Lemniscate and the frequency \(\omega\) (Equations (13) and (14)). In the case of lemniscate scanning (Figure 19), there is a slightly increasing trend of the function as \(a\) changes with a number \(N\) in the thousands, and N = N\({}_{\mathrm{max}}\) when \(a\) is equal to 0.07. The presented results in Figures 18 and 19 are related to Scenario 01 of ship motions. (Table 2). The results have been investigated for values of \(\mathrm{r_{s}}\) and \(a\) from 0.0007 to 0.07 and values of \(\omega\) from 100 to 5000 Hz, as presented in Figure 19 (or Figure 18). Figure 19: Results for lemniscate scanning. Figure 18: Results for conical scanning. In both cases of adopted scanning strategies, N = N\({}_{\text{max}}\) when the scanning frequency \(\omega\) is equal to 5000 Hz (higher values of \(\omega\) were not considered for system feasibility). In general, the analysis of the other three scenarios concerning ship motions did not show any substantial differences in the results (Table 6), with N = N\({}_{\text{max}}\) for \(\omega\) equal to 5000 Hz, r\({}_{\text{s}}\) = 0.009 for conical scanning and \(a\) = 0.07 for lemniscate scanning. In all scenarios, an increasing number of transitions was determined as the scanning frequency increased. Considering that four different scenarios of several sea conditions were analyzed, and a similar trends were determined, it can be stated that the pointing, when a scanning strategy is adopted, is not so particularly influenced by navigation conditions as it is by errors in the acquisition of the satellite coordinates and the ship's motions. This substantial difference between conical and lemniscate scanning in terms of N\({}_{\text{max}}\) lies in the shape of the chosen scanning methodology, with lemniscate scanning being more effective in ensuring the transition from the pointing to the tracking phase of the system. The improvement in pointing accuracy, considering Scenario 02 of ship motions (Table 2) and scanning methodologies, analyzing the results for N = N\({}_{\text{max}}\), leads to the satellite's first targeting time (deviation from the initial position to the targeting position) of 0.05 s for conical scanning and 0.013 s for lemniscate scanning. The greater probability of targeting the satellite led to the adoption of lemniscate scanning methodology. The shifting of the line of sight arising from the adopted scanning methodology in order to improve pointing accuracy can be carried out through the swaying of the mirror inside the optical system reflecting the laser beam (Figure 20). In order to ensure the feasibility of the system, a lemniscate scanning frequency \(\omega\) of 3500 Hz and a value \(a\) equal to 0.007 (Equations (13) and (14)) of the lemniscate half-width was chosen (the choice of \(\omega\) and r\({}_{\text{s}}\) parameters was made considering actuator specifications from the catalogue [31] using an iterative method).