## 4 Results and Discussion
### Selection of the Best Method for Estimating Position Coordinates  
Sections 4.1 and 4.2 analyse the test results. They showed that the GRA and EKF methods are interchangeably the best in terms of the accuracy of SV position coordinate estimation. On the one hand, they confirm the effectiveness of the algorithms of both methods (see Figures 1 and 3) in suppressing the results of measurements with gross error. On the other hand, they show the interchangeability of the two methods between each other in obtaining the highest accuracy of estimated position coordinates (see Figures 10 and 19). This mainly results from the way beacons are positioned and the location of the SV in relation to them (see Figures 6 and 15).  
Given the above, it made sense to find an objective indicator to select the most accurate positioning method depending on the SV's location in relation to the beacons and the measurement errors of the navigation equipment.  
The authors of the article propose to use, for this purpose, the mean error of the \(M_{xy}\) of the coordinates of the positions estimated by the GRA method. However, the value \(M_{xy}\) is determined on the assumption that the mean error values of the measurements have been determined 'ideally', i.e., the coefficient of variation (VC) has a value equal to one (see relation (13)). Determination of the limit value \(M_{xy}\), beyond which the EKF method should be changed to GRA, should be carried out as a result of simulation tests performed for a specific navigation system.  
Figures 24 and 25 show graphs of the mean error of the \(M_{xy}\) of the GRA-estimated position coordinates over the course of 100 SV crossings relative to the aligned and triangular marks.  
Figure 24: Graph of mean error \(M_{xy}\) of GRA-estimated position coordinates over the course of 100 SV crossings, relative to aligned marks as a function of time.  
Figure 23: Arithmetic mean of the percentages of the distance to the reference point from the estimated point during 100 SV crossings.  
Figure 22: Percentage of distance to the reference point from the estimated point during a single SV crossing.  
The graphs in Figures 24 and 25 show that the value of the \(M_{xy}\) depends not only on the iteratively determined values of the equivalent weight matrix \(\widetilde{\mathbf{P}}_{\mathrm{x}}\) (see Figure 3), but also, and mainly, on the spacing of the beacons relative to each other and the position of the SV relative to them (becoming then, so to speak, the so-called dilution of precision factor [36]).  
However, by comparing Figures 24 and 25 with Figures 10, 12, 19, and 21, it can be tentatively concluded that the GRA method should be used when the value of the \(M_{xy}\) will be less than 1.6 m.  
Nevertheless, it is essential to realise that the limit value \(M_{xy}\) was estimated based on simulation tests carried out under straightforward assumptions, e.g., SV only flows along a line.  
Therefore, the authors of the article recommend setting a limit value \(M_{xy}\) depending on the number of marks and the type of measurements performed (e.g., direction only or distance only, direction and distance) and simulating the SV of a move-the-lawn path along the area around the beacons within the achievable measurement range.  
These generalised conclusions give rise to the hypothesis that the proposed methodology has great potential in supporting SV positioning in coastal and harbour waters. Along with the continuous development of measurement systems (vision, acoustic, or radar) and the high computing power, it will undoubtedly be used on a larger scale in the future. Its first implementation is planned as part of the research project entitled \"Hydroacoustic positioning system for unmanned autonomous underwater vehicles\" (no. DOB-SZAFIR/01/B/003/04/2021), co-financed by the Polish National Center for Research and Development (NCBR).