## 3 Research  
In the simulation studies, the SV was assumed to sail along a straight section at a constant speed in the vicinity of three beacons, in the time interval of \(\tau\). The onboard navigation equipment determines \(COG(k)\) and \(SOG(k)\). A measuring system based on an omnidirectional camera determines \(RB_{1}(k)\), \(RB_{2}(k)\), \(RB_{3}(k)\) and \(D_{1}(k)\), \(D_{2}(k)\), \(D_{3}(k)\) to the beacons with known positions, at any moment in time, \(k(k\in\tau)\). All determined values are subject to a measurement error represented by a randomly varying correction \(\xi\) with normal distribution \(\text{N}[0,1]\), generated by the computer [31]. The limits of this correction correspond to a multiple of the mean error value, characterising the accuracy of the measurement performed with a given device. In every tenth moment, the \(k=\{10,20,30,\dots,10\cdot\left\lvert k_{last}/10\right\rvert\}\) value \(\xi\in\left\langle-10;-5\right\rangle\ \vee\ \left\langle 5;\ 10\right\rangle\) simulates the gross error of measurement, while in other moments, the \(k\) value \(\xi\in\left\langle-3;3\right\rangle\) simulates the mean error of measurement with \(P\approx 0.99\) (with \(k_{last}\) being the last moment \(k\in\tau\)). Calibration measurements \(RB_{1}(k)\), \(RB_{2}(k)\), \(RB_{3}(k)\), and \(D_{1}(k),D_{2}(k),D_{3}(k)\) to the beacons are calculated from the reference position SV, determined by the DR method, and based on the initial invariant values of \(COG(k=0)\) and \(SOG(k=0)\). Figure 5: Position accuracy areas are determined by the GLSA method based on relative bearings and distances taken relative to three navigation marks aligned: (**a**) in a line, (**b**) in a triangle. (\(\sigma_{D}=20\ m\)—arbitrarily assumed, \(\sigma_{RG}=0.25^{\circ}\)—assumed based on [30]; the distance between beacons is approximately 2600 m). Based on the measurement results obtained in this way, at each successive moment of \((k+1)\) time interval \(\tau\), the coordinates of the SV position are determined using DR, EKF, GLSA, and GRA methods in parallel. Whereby, the approximate geodetic adjustment position is the one determined at the previous time moment \(k\) (by the same geodetic method) after being shifted by a vector calculated according to Formula (1). That is, the same vector of the position determined by the EKF method in the prediction stage is shifted. A unique software application prepared in the integrated development environment C++ Builder [32], with installed template library package for linear algebra, Eigen [33], was used to conduct the simulation studies as specified. During its operation, it performs four primary tasks in parallel:  
1. Simulating the next position of the reference trajectory of SV movement, using the DR method with a constant time step \(\Delta t(k+1)\) based on initial, invariant (reference) values of \(COG(k=0)\) and \(SOG(k=0)\);
2. Simulating variables amendments: 1. \(\xi_{COG}(k+1)\)--added to the model \(COG(k=0)\), 2. \(\xi_{SOG}(k+1)\)--added to the model \(SOG(k=0)\), 3. \(\xi_{RB_{1}}(k+1)\), \(\xi_{RB_{2}}(k+1)\), \(\xi_{RB_{3}}(k+1)\)--added to the reference relative bearings \(RBr_{1}(k+1)\), \(RBr_{2}(k+1)\), \(RBr_{3}(k+1)\) calculated from the next position of the reference SV trajectory, up to three beacons, and taking into account the reference value of the \(COG(k=0)\); 4. \(\xi_{D_{1}}(k+1)\), \(\xi_{D_{2}}(k+1)\), \(\xi_{D_{3}}(k+1)\)--added to the reference distances \(Dr_{1}(k+1)\), \(Dr_{2}(k+1)\), \(Dr_{3}(k+1)\), calculated from successive positions of the simulated SV reference trajectory, up to three beacons;
3. DR-based estimation of SV position coordinates: 1. \(COG(k+1)=COG(k=0)+\xi_{COG}(k+1)\), 2. \(SOG(k+1)=SOG(k=0)+\xi_{SOG}(k+1)\);
4. Estimation of SV position coordinates with EKF, GLSA, and GRA methods based on the following: 1. \(COG(k+1)=COG(k=0)+\xi_{COG}(k+1)\), 2. \(SOG(k+1)=SOG(k=0)+\xi_{SOG}(k+1)\), 3. \(RB_{1}(k+1)=RBr_{1}(k+1)+\xi_{RB_{1}}(k+1)\), 4. \(RB_{2}(k+1)=RBr_{2}(k+1)+\xi_{RB_{2}}(k+1)\), 5.