## 4 Results and Discussion
### Tests Conducted against Aligned Beacons Aligned  
As a result of the tests involving the simulation of a single crossing and 100 SV crossings, two sets of data were obtained, i.e., a set of five 300 positions and four 300 distance series and a set of five hundred 300 positions and four hundred 300 distance series. Based on these, Figures 6-14 and Table 1 were prepared to analyse the first group of test results. Figure 6 shows an overview of the location of the beacons in the line and the estimated positions relative to them during single and 100 SV crossings. Figure 6 shows a significantly increasing scatter of positions estimated by the GLSA method, before and after the SV passes the extreme beacons. It also shows a uniform and significant increase in the scatter as a function of time of the positions, estimated by the DR method. It also shows that the positions estimated by the EKF and GRA methods are the most concentrated around the reference road positions. The graphs shown in Figures 7 and 8 confirm the conclusions of the analysis of Figure 6. They clearly show the largest distances at the beginning and end of the test in the case of the GLSA method and a uniform increase in the distance throughout the test in the case of the DR method. However, by focusing attention on the shape of the graphs in Figure 8, one can see a spike in the distance at the time points in which the gross errors of the measurements were simulated (every tenth moment \(k\)). This is when the largest distances to the reference positions from the positions estimated by the GLSA method occur. Figures 9 and 10 plot the moving average distance to the reference position from the estimated position during a single and 100 SV crossings as a function of time. The graphs evaluated against each other in Figures 9 and 10 clearly show that neither method provides an estimate of the position with the smallest moving mean distances to the reference position throughout the test. In the time interval \(\langle 0\,\mathrm{s};\ \ 90\,\mathrm{s}\rangle\), the DR method proved to be the best in this respect. In the time interval of \(\langle 90\,\mathrm{s};\ \ 900\,\mathrm{s}\rangle\), it gives way to the GRA method. In the time interval \(\langle 900\,\mathrm{s};\ \ 1000\,\mathrm{s}\rangle\), it gives way to the EKF method. Figures 11 and 12 show plots of moving RMS to the reference position from the estimated position during a single and 100 SV crossings as a function of time. Figure 8: Arithmetic mean distance to reference position from the estimated position during 100 SV crossings as a function of time. Figure 10: Moving average distance to reference position from the estimated position during 100 SV crossings as a function of time (logarithmic scale—log base = 10). Figure 9: Moving average distance to reference position from the estimated position during a single SV crossing as a function of time (logarithmic scale—log base = 10). Figures 11 and 12 clearly show that, throughout the test, the moving RMS value of the distance remains similar for the EKF method (equal scatter around the moving average distance) and increases monotonically uniformly for the DR method. The graphs relating to the GLSA and GRA methods are parallel to each other and, in terms of the variability of the waveform, are decreasing in the time interval \(\langle 0\,\mathrm{s};500\,\mathrm{s}\rangle\) and increasing in the time interval \(\langle 500\,\mathrm{s};1000\,\mathrm{s}\rangle\). It should be noted here that the GRA method obtains the smallest moving RMS distance values (smallest spread around the moving average distance) over the most extensive time interval \(\langle 250\,\mathrm{s};850\,\mathrm{s}\rangle\) (for more than half of the test duration).