**5. Research Results**

The aim of the computer simulation research of the FNAC and MGAC algorithms to determine the optimal safe-ship trajectory in collision situations was to evaluate methods to solve the problem formulated in this work by using fuzzy-set theory as a multistage and matrix game theory as a multistep decision-making process.

The computer simulation of the FNAC and MGAC algorithms was carried out in MATLAB software on an example of the real navigational situation of passing J = 3 encountered ships in the Skagerrak Strait in good visibility Ds = 0.2 nm and the restricted visibility Ds = 2.0 nm (nautical miles) (Table 1).


**Table 1.** Data of own-ship and met ships: 1, 2 and 3.

The situation was registered on board r/v HORYZONT II, a research and training vessel of the Gdynia Maritime University, on the radar screen of the ARPA anticollision system Raytheon. The sample results of the performed computer simulations for the navigational situation when the own-ship passed three met ships in a good and restricted visibility at sea are presented in Figures 9 and 10, respectively.

The trajectories of the own-ship, shown in Figures 9 and 10, are optimal, but in different ways.

The FNAC trajectory ensures minimum risk of collision of own-ship, taking into account the uncertainty of the control process, described by the fuzzy set membership functions of state and control constraints, and collision risk, not including maneuvers of encountered ships.

On the other hand, the trajectory of MGAC ensures a minimum collision-risk of the own-ship taking into account the maneuvering of encountered ships in a co-operative or non-co-operative way.

Designated safe trajectories FNAC and MGAC of the own-ship are the reference trajectories for automatic ship's control systems using the autopilot and main engine.

**Figure 9.** Comparison of safe and optimal trajectories of own-ship in the situation of passing three ships, in conditions of good visibility at sea at Ds = 0.2 nm, determined using the following algorithms: FNAC decision chosen trajectory = 0.5766; MGAC final payment = 1.08 nm.

**Figure 10.** Comparison of safe and optimal trajectories of own-ship in the situation of passing three ships, in conditions of restricted visibility at sea at Ds = 2.0 nm, determined using the following algorithms: FNAC decision chosen trajectory = 0.6452; MGAC final payment = 3.83 nm.

#### **6. Conclusions**

This work showed that the proposed concept of applying the fuzzy neural anticollision and matrix game anticollision algorithms is a promising way to solve the considered problem and design a novel anticollision system, allowing to increase the safety of maritime transport.

The FNAC algorithm achieves a short computation time of about one second, while the MGAC algorithm scope of the calculation time is from about two to five seconds.

The obtained solutions obtained using both algorithms meet the requirements of the COLREGs.

The MGAC computer program, designed in MATLAB software, takes into consideration the following: degree of co-operation with the own-ship and encountered ships, COLREGs Rules, advance time for a maneuver calculated with regard to the own-ship dynamic features, and the assessment of the final deviation between the real and reference trajectories.

In summary of these results, the created algorithms can be used as a tool to assist navigator in making maneuver decision, in complex collision situations, when passing through more ships, especially in restricted visibility at sea.

In future works, the sensitivity analysis of safe ship control should be performed to change the parameters of the process model and the inaccuracy of information from the ARPA radar system; moreover, the design of the ARPA system may be considered, extended with the function of a computer-aided maneuvering navigator decision, using the FNAC and MGAC algorithms.

**Author Contributions:** Conceptualization, J.L.; Methodology, M.M.-S.; Software, M.M.-S.; Validation, J.L. and M.M.-S.; Formal analysis, J.L. and M.M.-S.; Investigation, J.L. and M.M.-S.; Resources, J.L. and M.M.-S.; Data curation, M.M.-S.; Writing—original draft preparation, M.M.-S.; Writing—review and editing, J.L.; Visualization, J.L.; Supervision, J.L.; Project administration, J.L.; Funding acquisition, M.M.-S.

**Funding:** This research was funded by a statute research project of Gdynia Maritime University in Poland, No. 446/DS/2018: "Design and simulation tests of marine automation systems in MATLAB/Simulink and LabVIEW software".

**Conflicts of Interest:** The authors declare no conflict of interest regarding the publication of this paper. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **Nomenclature**



### **References**


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