I.A. Ulchenko1, A.A. Nikolaev2, D.V. Khazheev3, N.A. Volkov4, M.S. Glushkov5, A.A. Belkanov6

1–6 Morinformsystema-Agat Concern (Moscow, Russia)
1ivan_ulchenko@mail.ru, 2A.A.Nikolaev@yandex.ru, 3Hazheev.D@yandex.ru, 4nvolkov32@yandex.ru, 5glushkov.maxim@gmail.com, 6workBelkanov@gmail.com

The modern automatic ship control systems development, training simulators, ship operator assistance systems, as well as various onboard systems and complexes, requires the application of ship motion dynamics mathematical models. A comparative analysis of such models is presented in this study. Simplified models widely described in literature and developments were considered: the Nomoto model, second-order Nomoto model, first-order Nomoto model, Bech model, Norrbin model, and the model from the IEC-62065 standard. The mathematical models parameter identification was performed using optimization methods based on trial records of the tugboat Nikolay Semenchenko. The models accuracy is compared using the root mean square error (RMSE) for the ship’s yaw rate and coordinates.

Objective – ship dynamics mathematical models comparative analysis.

The review of native and foreign literature was conducted. Mathematical models of ship motion dynamics were implemented as software code, and model parameters were identified using optimization methods based on trial records. A comparative analysis of model accuracy revealed that the Nomoto model demonstrated the best results in terms of yaw rate RMSE, while the Bech model performed best in coordinate simulation accuracy. The results of the Bech, Nomoto, second-order Nomoto, and IEC-62065 models were similar (difference in yaw rate RMSE within 4.34%, coordinate RMSE within 9.1%). The IEC-62065 standard model is used for certifying automatic ship control systems. Therefore, the Bech, Nomoto, and second-order Nomoto models, which showed comparable results, can also be applied to developing automatic ship control systems. The first-order Nomoto and Norrbin models performed worse due to their lower dynamic order.

The research results can be applied to developing mathematical models of ship motion dynamics and automatic ship motion control systems.

Ulchenko I.A., Nikolaev A.A., Khazheev D.V., Volkov N.A., Glushkov M.S., Belkanov A.A. Ship dynamics mathematical models comparative analysis. Nonlinear World. 2025. V. 23. № 2. P. 38–49. DOI: https:// doi.org/10.18127/ j20700970-202501-05 (In Russian)

1. Vagushchenko L.L., Cymbal N.N. Sistemy avtomaticheskogo upravleniya dvizheniem sudna. Odessa: Feniks. 2007. 328 c. (In Russian).
2. Lukomskij Yu.A., Chugunov V.S. Sistemy upravleniya morskimi podvizhnymi ob"ektami. L.: Sudostroenie. 1988. 272 c. (In Russian).
3. Os'kin D.A., Bocharova V.V., Osipov S.V. Matematicheskie modeli dinamiki sudov, osnashchennyh vintorulevymi kolonkami. Vestnik Astrahanskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Upravlenie, vychislitel'naya tekhnika i informatika. 2023. № 3. S. 124–132 (In Russian).
4. Galeev R.E. O vybore matematicheskoj modeli dlya postroeniya traektorii dvizheniya sudna v sisteme podderzhki prinyatiya reshenij sudovoditelem. Nauchnye problemy vodnogo transporta. 2022. № 74. S. 162–173 (In Russian).
5. Ueng S.K., Lin D. Liu C.H. A Ship Motion Simulation System. Virtual reality. 2008. V. 12. P. 65–76.
6. Ambrosovskij V.M., Ambrosovskaya E.B. Matematicheskie modeli dvizheniya perednego hoda MPO dlya otladochnyh stendov. Dep. ruk. VINITI ot. 2018. № 62 (In Russian).
7. Sutulo S., Moreira L., Soares C.G. Mathematical models for ship path prediction in manoeuvring simulation systems. Ocean engineering. 2002. V. 29. № 1. P. 1–19.
8. Das L.C.S., Talole S.E. Evolution of ship’s mathematical model from control point of view. System. 2016. V. 1000. 6 p.
9. Yudin Yu.I., Sotnikov I.I. Matematicheskie modeli ploskoparallel'nogo dvizheniya sudna. Klassifikaciya i kriticheskij analiz. Vestnik Murmanskogo gosudarstvennogo tekhnicheskogo universiteta. 2006. T. 9. № 2. S. 209a (In Russian).
10. Sobolev G.V. Upravlyaemost' korablya i avtomatizaciya sudovozhdeniya. L.: Sudostroenie. 1976. T. 480 (In Russian).
11. Pavlenko V.G. i dr. Hodkost' i upravlyaemost' sudov. M.: Transport. 1991. S. 234–240 (In Russian).
12. Gofman A.D. Dvizhitel'no-rulevoj kompleks i manevrirovanie sudna: Spravochnik. L.: Sudostroenie. 1988 (In Russian).
13. Yudin Yu.I., Poznyakov S.I. Manevrennye harakteristiki sudna kak funkcii parametrov ego matematicheskoj modeli. Vestnik Murmanskogo gosudarstvennogo tekhnicheskogo universiteta. 2006. T. 9. № 2. S. 234–239 (In Russian).
14. Poznyakov S.I., Yudin Yu.I. Sravnenie matematicheskih modelej s tochki zreniya koefficientov vliyaniya. Vestnik Murmanskogo gosudarstvennogo tekhnicheskogo universiteta. 2006. T. 9. № 2. S. 241–245 (In Russian).
15. Vojtkunskij Ya.I., Pershic R.Ya., Titov I.A. Spravochnik po teorii korablya. L.: Sudpromgiz. 1960. 228 s. (In Russian)
16. Blanke M., Knudsen M. A sensitivity approach to identification of ship dynamics from sea trial data. IFAC Proceedings Volumes. 1998. V. 31. № 30. P. 241–249.
17. Sutulo S., Soares C.G. Nomoto-type manoeuvring mathematical models and their applicability to simulation tasks. Ocean Engineering. 2024. V. 304. P. 661–699.
18. Yasukawa H., Yoshimura Y. Introduction of MMG standard method for ship maneuvering predictions. Journal of marine science and technology. 2015. V. 20. P. 37–52.
19. Yoshimura Y. Mathematical Model for Manoeuvring Ship Motion (MMG Model). Workshop on Mathematical Models for Operations involving Ship-Ship Interaction. 2005. P. 1–6.
20. Sotnikov I.I. Matematicheskie modeli, vychislitel'nye skhemy analiza i komp'yuternoe modelirovanie dvizheniya sudna: Dis. … kand. tekhn. nauk. Novgorodskij gosudarstvennyj universitet im. Yaroslava Mudrogo. Nizhnij Novgorod. 2007. 200 c. (In Russian)
21. Afremov A.Sh., Martisov G.G., Nemzer A.I. i dr. Sredstva aktivnogo upravleniya sudami. Izd. 2-e. SPb.: Krylovskij gosudarstvennyj nauchnyj centr. 2016. 182 s. (In Russian)
22. International standart IEC 62065:2014 [Elektronnyj resurs]. Rezhim dostupa: https://webstore.iec.ch/en/publication/6431 (data obrashcheniya: 01.09.2024).
23. Glushkov S.V., Mazhirin I.A., Tyul'kanov A.K. Matematicheskaya model' dvizheniya sudov s nestandartnym dvizhitel'no-rulevym kompleksom. Vestnik Murmanskogo gosudarstvennogo tekhnicheskogo universiteta. 2018. T. 21. № 4. S. 548–557 (In Russian).
24. Ambrosovskaya E.B., Ambrosovskij V.M., Romaev D.V. Matematicheskie modeli v otladochnyh stendah dlya sudovyh sistem upravleniya. Morskie intellektual'nye tekhnologii. 2023. S. 89 (In Russian).
25. Nomoto K. et al. On the steering qualities of ships. International shipbuilding progress. 1957. V. 4. № 35. P. 354–370.
26. Bech M.I., Smitt L. Analogue simulation of ship maneuvers. Hydro-Og and Aerodynamics Laboratory, Hy. 1969. V. 14.
27. Norrbin N. On the design and analysis of the zig-zag test on base of quasi-linear frequency response. 1963. № SSPA B 104-3.
28. Burylin Ya.V. Identifikaciya nelinejnoj modeli dvizheniya sudna i adaptivnoe upravlenie po traektorii: Dis. … kand. tekhn. nauk. Morskoj gosudarstvennyj universitet im. admirala G.I. Nevel'skogo. Novorossijsk. 2018. 133 c. (In Russian)
29. Chinchukova E.P. Sistemy adaptivnogo upravleniya dvizheniem sudna po kursu: Dis. … kand. tekhn. nauk. Morskoj gosudarstvennyj universitet im. admirala G.I. Nevel'skogo. Vladivostok. 2020. 131 c. (In Russian)
30. Bahvalov N.S., Zhidkov N.P., Kobel'kov G.M. Chislennye metody. Izd. 3-e. M.: BINOM. Laboratoriya znanij. 2004. 636 s. (In Russian)