## 6 Simulation Results and Discussion
### Discussion
This model supports various functions, including testing different PSD regeneration techniques, validating the accuracy of regenerated PSD against the original model, and generating vibration time-domain signals for further analysis. In this study, time series data were used in the maximum likelihood estimator through a sliding window technique to accurately estimate the standard deviation \(\sigma_{\text{vib}}\) of the satellite platform vibrations (see Figures 13 and 14). Figure 13a shows the original PSD model, presenting a clear baseline for comparison with the estimated results in Figure 13b,c. These estimates, derived using an algorithm, effectively capture the frequency distribution characteristics, validating the algorithm's accuracy in regenerating PSD signals. In Figure 13b,c, the estimated PSD curves closely align with the original PSD, indicating a high level of precision in the estimation process. The sliding window technique, illustrated in Figure 14, plays a critical role in ensuring robust and dynamic \(\sigma_{\text{vib}}\) estimation. By using samples of the vibration signal magnitudes from both the \(x\) and \(y\) directions and focusing on the most recent data, this approach enables decision-making based on current conditions. This method enhances the stability of the maximum likelihood estimator, contributing to a smoother and more accurate \(\sigma_{\text{vib}}\) estimation. These results collectively demonstrate that the PSD model and the proposed estimation methodology provide a reliable basis for simulating satellite and spacecraft vibration characteristics. The alignment between the estimated PSD and the original model suggests that the sliding window technique, combined with maximum likelihood estimation, offers an effective approach to \(\sigma_{\text{vib}}\) estimation in inter-satellite laser link applications. During the acquisition phase of establishing an inter-satellite laser link, precision is hindered by satellite platform vibrations. To minimize the impact of such vibrations on successful acquisition and to avoid multiple scans of the same uncertainty area, several scan parameters can be optimized, including overlap factor, beam divergence, and the size of the uncertainty area, as reported in [40]. In the presence of vibrations, changing the uncertainty area alone, while keeping the beam divergence and overlap factor constant, did not reduce scan gaps. Therefore, optimizing both the overlap factor and beam divergence is essential in the presence of vibrations. The optimization of these scan parameters depends on an accurate estimation of \(\sigma_{\text{vib}}\), whose performance is displayed in Figure 14. In addition to the estimation accuracy, the overlap factor and beam divergence have inherent limitations, as varying these parameters impacts scanning time. Thus, optimal performance represents an interplay between these factors. Figure 18 illustrates the performance of the spiral scan method under satellite platform vibrations, comparing fixed and optimized scan parameters. Figure 18a,b depict the spiral scan path and its perturbation due to vibrations with \(\sigma_{\text{vib}}=5\,\upmu\text{rad}\), where Monte Carlo simulations were conducted using an overlap factor \(a=0.2\) and beam divergence \(\theta_{b}=25\,\upmu\text{rad}\). The results in Figure 18c,d show that the fixed parameters resulted in an average of 1.23 scans with a variance of 0.78, and a mean scan time of 9.38 s with a variance of 30.51 s.  
Figure 17: Probabilistic positions of the satellite within the uncertainty area. (**a**) Spiral scan path. (**b**) Raster scan path. (**c**) Square spiral scan path. (**d**) Hexagon scan path. In contrast, Figure 18e,f highlight the improvements achieved by optimizing \(a\) and \(\theta_{b}\) based on the estimated satellite vibrations. The range of \(a\) was \(0\leq a\leq 0.9\), and \(\theta_{b}\) ranged from 25 \(\upmu\)rad to 35 \(\upmu\)rad.