## 6 Research and Analysis of the Obtained Results
### Test No. 1  
The test consisted of making three and then one hundred passages along the test section. COG, SOG and \(\Delta z\) were measured every one second. \(r\), \(\alpha\), \(\beta\) were measured every 5 s.  
In Figure 8 determined routs using different methods of positioning are presented. Figures 9-12 present graphs depicting simulated measurement errors (\(\Delta_{COG}\), \(\Delta_{SOG}\), \(\Delta_{r}\), \(\Delta_{\alpha}\), \(\Delta_{\beta}\)) during the first passage of the test section (these graphs will not be displayed during the presentation of the results of further testing). Figures 9-12 clearly show that the dispersion of the measurement errors are random and limited. The error value does not exceed \(3\sigma\) fluctuating:  
* \(\Delta_{COG}\) in the range of \(\left\langle-15^{\circ},\ 15^{\circ}\right\rangle\),
* \(\Delta_{SOG}\) in the range of \(\left\langle-0.8\ \mathrm{m/s},\ 0.8\ \mathrm{m/s}\right\rangle\),
* \(\Delta_{\alpha}\), \(\Delta_{\beta}\) in the range of \(\left\langle-9^{\circ},\ 9^{\circ}\right\rangle\),
* \(\Delta_{r}\) in the range of \(\left\langle-0.6\ m,\ 0.6\ \mathrm{m}\right\rangle\). Figure 11: Graph of simulated measurement error \(\Delta_{x}\), \(\Delta_{\beta}\). Figure 8: Determined routes (single passage). Figure 12: Graph of simulated measurement error \(\Delta_{r}\). Figure 10: Graph of simulated measurement error \(\Delta_{SOG}\). Figure 9: Graph of simulated measurement error \(\Delta_{COG}\). Figure 16: Determined route (single passage). Figure 17: Graph of the reference position distance from the position estimated by EKF and DR methods and determined by the USBL system (single passage). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Positioning Method**} & **Minimum Distance to the** & **Maximum Distance to the** & **Average Distance to the** \\  & **Reference Position (m)** & **Reference Position (m)** & **Reference Position (m)** \\ \hline EKF & 0.1 & 6.4 & 2.3 \\ DR & 0.1 & 22.6 & 11.8 \\ USBL & 0.2 & 81.6 & 23.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of statistical positioning errors by EKF, DR, and USBL system (single passage). Figure 15: Graph of the reference position distance from the position estimated by EKF and DR methods and determined by the USBL system (single passage). On the basis of graphs analysis of Figures 14-17, as well as statistical parameters from Tables 2 and 3, it can be stated that the accuracy of estimated position coordinates using the EKF method is higher than those estimated using the DR method and determined by the USBL system. The reference position distance from the estimated one by EKF method does not exceed 7 m. Unfortunately, on the graphs it can be noticed that time intervals have occurred in which the reference position is further from the position estimated by EKF method, in comparison to the estimated position by DR method or determined using the USBL system. One can ask oneself a question here: how often do such time intervals occur? Generalized results of statistical analysis can answer that question. It is based on two additional indicators. The former describes the average \(\xi\) distance to the reference position with estimated calculation on the basis of \(n\) distance \(\xi\) obtained for \(n\) passages along the test section at the same moment of \(k\), in accordance with the formula:  
\[\xi\left(k\right)=  
One Hundred Passages  
Generalizing the results of test No. 1, based on Figure 19, it can be stated that the average value of the reference position distance from the estimated position is constant in the case of EKF, it increases linearly for DR and decreases and then increases linearly in the case of the USBL system. The second indicator to the generalized analysis is based on the histogram showing the frequency of the same distance to the reference position from the estimated position. The histogram is presented in Figure 20.  
It results from the histogram that a two meter distance from the estimated position by EKF occurred most often.