## 3 Research
\(RB_{3}(k+1)=RBr_{3}(k+1)+\xi_{RB_{3}}(k+1)\), 6. \(D_{1}(k+1)=Dr_{1}(k+1)+\xi_{D_{1}}(k+1)\), 7. \(D_{2}(k+1)=Dr_{2}(k+1)+\xi_{D_{2}}(k+1)\), 8. \(D_{3}(k+1)=Dr_{3}(k+1)+\xi_{D_{3}}(k+1)\);
5. Calculation of statistical parameters and generation of graphs and overview maps for analysing the test results obtained, i.e., 1. Maximum value, arithmetic mean, and standard deviation of the distance to the reference position from the position estimated by DR, EKF, GLSA, and GRA methods; 2. Distance graphs to the reference position from the estimated position, moving average, and moving RMS, to the reference position from the estimated position, and percentage of distance to the reference point from the estimated position, 3. An overview map with the beacon alignment adopted for the study and the resulting SV positions estimated by DR, EKF, GLSA, and GRA methods relative to them. In carrying out the tests with the developed application, it was assumed that SV flows with \(COG(k=0)=90^{\circ}\) and \(SOG(k=0)=5\) m/s. It is also assumed that measurements \(COG(k+1)\), \(SOG(k+1)\), \(RB_{1}(k+1)\), \(RB_{2}(k+1)\), \(RB_{3}(k+1)\), \(D_{1}(k+1)\), \(D_{2}(k+1)\), and \(D_{3}(k+1)\) are performed at the exact moment in time with a constant frequency of 1 Hz, \(\Delta t(k+1)=1\) s. On the other hand, to make the measurements simulated by the application more realistic, they are performed:  
* Onboard navigational equipment: it was assumed that \(\sigma_{\text{COG}}=2^{\circ}\) and \(\sigma_{\text{SOG}}=0.05\) m/s (1% of the measured speed). It was also assumed that the measurement \(COG(k+1)\) was performed with the INS 'NAV-100' and the measurement \(SOG(k+1)\) was performed with the DVL 'NavQuest' [28; 29];
* Measurement system based on an omnidirectional camera: it was arbitrarily assumed that \(\sigma_{RG}=\sigma_{\text{COG}}+0.5^{\circ}\)[34] and \(\sigma_{D}=0.5\) m [35]. It is taken into account that the measurements \(RB_{1}(k+1),RB_{2}(k+1),RB_{3}(k+1)\) are relative to \(COG(k+1)\). The operation of the application thread responsible for the iterative calculations performed by the GLSA and GRA methods was considered complete when the absolute differences of the correction vectors \(\left|\mathbf{V}_{\text{x}}(k+1)-\mathbf{V}_{\text{x}}(k+1)^{-}\right|\) reached values smaller than the value of the vector  
\(\Delta\mathbf{V}_{\text{max}}=[0.01\text{ m }0.001^{\circ}0.01\text{ m }0.001^{\circ}0.01\text{ m }0.001^{\circ}]^{\text{T}}\). In the case of the GRA method, it was arbitrarily assumed that the values of the control parameters of the Danish damping function (relation no. 5): \(l=0.001\), \(m=2.5\), \(g=1.2\). This means that the value of the standardised correction of the undamped single measurement result \(\left[\overline{\mathbf{V}}_{\text{x}}(k+1)\right]_{i,1}\)_for_\(i=1,\ 2,\ldots,2n\) should belong to the interval \(\left\langle-m;m\right\rangle\), with probability \(P=0.988\). The tests were conducted against beacons aligned and triangulated. They consisted of simulating a single crossing (execution of a single test trial) and 100 crossings (implementation of 100 test trials) of the SV along a straight stretch of road that was 1500 m long. Each test trial started at time \(k=0\) from a position with coordinates \((x(k=0)=-2500\text{ m},\)\(y(k=0)=2500\text{ m})\) and ended when the SV navigated to the reference position with coordinates \(\text{x}(k+1)>750\text{ m}\). The beacons were aligned at positions with coordinates: \((-500,\ 0)\), \((0,\ 0)\), \((500,\ 0)\), while in a triangle, by changing the coordinates of the position of the centre beacon to \((0,\ -500)\).