## 2 Problem Formulation
* Mean ellipse error \[a =\sqrt{\frac{1}{2}\cdot\left(\sigma_{x}{}^{2}+\sigma_{y}{}^{2}+\sqrt{p} \right)},\] (8) \[b =\sqrt{\frac{1}{2}\cdot\left(\sigma_{x}{}^{2}+\sigma_{y}{}^{2}-\sqrt{p }\right)},\] (9) \[\varphi =\frac{1}{2}\operatorname{arc}\operatorname{tg}\frac{2\cdot \sigma_{xy}}{\sigma_{x}{}^{2}-\sigma_{y}{}^{2}}\;.\] (10) where: \(p=\left(\sigma_{x}{}^{2}-\sigma_{y}{}^{2}\right)^{2}+4\cdot\sigma_{xy}{}^{2}\), \(q\)--directional angle of the ellipse; \(a\)--length of the large half-axis of the ellipse; \(b\)--length of the small half-axis of the ellipse. Knowing the values of the measurement errors (\(\sigma_{COG}=2^{\circ}\) and \(\sigma_{SOG}=1\%\)) of the measured speed given by the manufacturers of INS 'NS-100' and DVL 'NavQuest 600' in the technical specifications [28; 29] and assuming that the SV flows with uniform rectilinear  
Figure 3: Algorithm for state vector estimation \(\widehat{\mathbf{x}}(k+1)\) using geodetic adjustment with Danish gross error suppression function. motion \(COG(k+1)=90^{\circ}\) and \(SOG(k+1)=6\) m/s, the positions of circles of mean error (having radius length equal to \(M_{xy}\)) and mean error ellipses (having semi-axis lengths equal to \(a\) and \(b\), and a directional angle equal to \(\varphi\)) of the positions determined by the DR method, e.g., after 200 s and after 500 s (which are illustrated in Figure 4), can be calculated. Figure 4 shows that the mean error ellipse extends in the direction perpendicular to the \(COG(k+1)\). Its half-axis length \(b\) is almost twice as short as the half-axis \(a\). Thus, it is the measurement error \(\sigma_{COG}\) which determines the mean error \(M_{xy}\), whose value reaches 108.9 m after the SV has travelled a distance of only 3000 m.  
Let us also assume that the SV can, at any subsequent time \((k+1)\), determine the coordinates of the occupied positions. This is also done using the GLSA method (Figure 2), solely based on course over ground measurements \(COG(k)\) performed by, e.g., GNSS Satellite Compass ComNav [30], and relative bearings \(RB_{1}(k+1),RB_{2}(k+1),RB_{3}(k+1)\) and distances \(D_{1}(k+1),D_{2}(k+1),D_{3}(k+1)\) taken, e.g., with an omnidirectional camera relative to the three navigation marks. The mean error values of the positions determined in this way can then be calculated from the trace of the covariance matrix according to the formula:  
\[M_{xy}=\sqrt{Tr(\mathbf{C_{X}}(k+1))}, \tag{11}\]  
where:  
\[\mathbf{C_{X}}(k+1)=(\mathbf{A^{T}}(k+1)\mathbf{P_{x}}(k+1)\mathbf{A}(k+1))^{- 1}, \tag{12}\]  
assuming that \(\delta_{0}^{2}(k+1)=1\). Figure 5 shows the areas of the accuracy of GLSA-determined positions relative to three beacons set in two characteristic variants. Moreover, Figure 5 shows that the two variants of beacon alignment (a) and (b) allow the GLSA method to determine position with a mean error of less than the 30 m level in many of the drawn areas of accuracy. This means that the GLSA method can outperform the DR method in terms of the accuracy of the SV position determination. This situation may occur in the case of continuous position determination by the DR method after exceeding the contractual time limit, depending on the values of \(\sigma_{COG}\) and \(\sigma_{SOG}\). Taking into account the results of the calculation of the positioning accuracy with the DR method presented in Figure 4, it can be stated, with certainty, that the contractual limit was already exceeded after 120 s. At this point, the mean error \(M_{xy}\) reached a value of 43.6 m.  
Figure 4: Change in mean error and mean error ellipse after SV travels 1200 m and 3000 m.  
Mainly for these reasons, the augmented Kalman filter method is now often used to determine SV positions. In its operation, it can make use of both measurements of motion parameters \(COG(k)\), \(SOG(k)\), as well as additional measurements of relative bearings \(RB_{1}(k+1)\), \(RB_{2}(k+1)\), \(RB_{3}(k+1)\). However, it is essential to realise that the EKF is a good solution when the measurement error values are set 'perfectly'.