## 3 Results
Since the routes were comprised of several straight sections, each profile was presented as a linear function expressed using the following formula:  
\[x_{i,j}=a_{i,j}\cdot y_{i,j}+b_{i,j}, \tag{5}\]  
where \(i\) is the route number, \(j\) is the section number for the \(i\)-th route, \(x_{i,j}\) and \(y_{i,j}\) are the flat rectangular coordinates PL-2000 of the point \(j\) recorded by a particular receiver on the \(i\)-th route, \(a_{i,j}\) is a slope of the straight line \(j\) for the \(i\)-th route, defined as follows:  
\[a_{i,j}=\frac{x_{i,j+1}-x_{i,j}}{y_{i,j+1}-y_{i,j}}, \tag{6}\]  
\(b_{i,j}\) is a y-intercept of the straight line \(j\) for the \(i\)-th route, defined as follows:  
\[b_{i,j}=x_{i,j}-a_{i}\cdot y_{i,j}. \tag{7}\]  
The distances were then calculated between the recorded coordinates of the geodetic GNSS receiver (converted into two-dimensional x, y coordinates) and the designed route sections. During the calculations, two variants needed to be considered. The first variant assumed that for the measured points, straight lines perpendicular to the designed hydrographic profiles could be drawn (Figure 9a), while in the second variant, it was not possible to determine the perpendicular line (Figure 9b). In the first case, the XTE value was calculated using the following formula:  
\[XTE_{i,k}=\frac{\left|-a_{i,j}\cdot y_{p_{i,k}}+x_{p_{i,k}}-b_{i,j}\right|}{ \sqrt{\left(-a_{i,j}\right)^{2}+1}}, \tag{8}\]  
where \(k\) is the point number recorded by a particular receiver, \(x_{p_{i,k}}\) and \(y_{p_{i,k}}\) are the flat rectangular coordinates PL-2000 of the point \(k\) recorded by a particular receiver on the \(i\)-th route. On the other hand, in the second variant the distance to the nearest route turning point was determined using the following formula:  
\[XTE_{i,k}=\sqrt{\left(x_{p_{i,k}}-x_{i,j}\right)^{2}+\left(y_{p_{i,k}}-y_{i,j }\right)^{2}}. \tag{9}\]After calculating the XTE value, this variable was statistically analysed. Two accuracy measures were adopted as the assessment criteria: XTE (\(p\) = 0.68) and XTE (\(p\) = 0.95) which corresponded to the probabilities of 68% and 95%. Table 2 shows the values of these measures. Based on the results presented in Table 2, it can be concluded that the statistics for the accuracy of steering an USV in the automatic mode, measured by the XTE variable for four representative routes, were very similar to each other and amounted to 0.92-1.4 m for \(p\) = 0.68 and 2.01-2.39 m for \(p\) = 0.95. The measurements demonstrated that the accuracy of maintaining the vessel along the profile was not affected by the route shape (parallel sounding lines and the \"narrowing squares\") and the mutual distance between the profiles (5 and 10 m) (Figure 7). As part of the study, an analysis of the XTE random variable distribution was conducted as well. To this end, Easy Fit software, which verifies the conformity of the measurements with typical statistical distributions using conformity tests (Kolmogorov-Smirnov, Anderson-Darling, and chi-square) was applied. The conducted analyses show that the empirical statistical distribution of the XTE values was most similar to four-parameter generalised gamma distribution with the probability density function [37]:  
\[f(XTE;\chi,\varepsilon,\phi)=\frac{\varepsilon}{\chi\cdot\Gamma(\phi)}\cdot \left(\frac{XTE}{\chi}\right)^{\varepsilon\phi-1}\cdot e^{-(\frac{XTE}{\chi}) ^{\varepsilon}}, \tag{10}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Accuracy Measure** & **Route a (10 m)** & **Route b (5 m)** & **Route c (10 m)** & **Route d (5 m)** \\ \hline Route type & & & & \\ Number of measurements & 458 & 848 & 506 & 869 \\ XTE (\(p\) = 0.68) & 0.92 m & 1.15 m & 1.40 m & 1.27 m \\ XTE (\(p\) = 0.95) & 2.01 m & 2.38 m & 2.20 m & 2.39 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical accuracy measures (XTE) of steering an USV for particular routes.