Research article Special Issues

Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle

  • Received: 07 December 2023 Revised: 12 February 2024 Accepted: 05 March 2024 Published: 14 March 2024
  • MSC : 70E60, 93C85

  • Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.

    Citation: Simone Fiori. Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle[J]. AIMS Mathematics, 2024, 9(4): 10157-10184. doi: 10.3934/math.2024497

    Related Papers:

  • Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.



    加载中


    [1] A. F. Molland, Chapter 10 - Underwater vehicles, In: The Maritime Engineering Reference Book – A Guide to Ship Design, Construction and Operation, 2008, 728–783.
    [2] B. Allotta, R. Costanzi, L. Pugi, A. Ridolfi, Identification of the main hydrodynamic parameters of Typhoon AUV from a reduced experimental dataset, Ocean Eng., 147 (2018), 77–88. https://doi.org/10.1016/j.oceaneng.2017.10.032 doi: 10.1016/j.oceaneng.2017.10.032
    [3] L. D. L. Barker, M. V. Jakuba, A. D. Bowen, C. R. German, T. Maksym, L. Mayer, et al., Scientific challenges and present capabilities in underwater robotic vehicle design and navigation for oceanographic exploration under-ice, Remote Sens., 12 (2020), 2588. https://doi.org/10.3390/rs12162588 doi: 10.3390/rs12162588
    [4] Y. Bestaoui, A Lagrangian approach to modeling of an airship with wind and varying mass effects, In: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA, 2010. https://doi.org/10.2514/6.2010-40
    [5] L. Bevilacqua, W. Kleczka, E. Kreuzer, On the mathematical modeling of ROVs, IFAC Proc. Volumes, 24 (1991), 51–54. https://doi.org/10.1016/S1474-6670(17)51031-9 doi: 10.1016/S1474-6670(17)51031-9
    [6] E. Borhaug, A. Pavlov, K. Y. Pettersen, Integral LOS control for path following of underactuated marine surface vessels in the presence of constant ocean currents, In: 2008 47th IEEE Conference on Decision and Control, 2008, 4984–4991. https://doi.org/10.1109/CDC.2008.4739352
    [7] E. Celledoni, E. Çokaj, A. Leone, D. Murari, B. Owren, Lie group integrators for mechanical systems, Int. J. Comput. Math., 99 (2022), 58–88. https://doi.org/10.1080/00207160.2021.1966772 doi: 10.1080/00207160.2021.1966772
    [8] C. W. Chen, N. M. Yan, Prediction of added mass for an autonomous underwater vehicle moving near sea bottom using panel method, In: 2017 4th International Conference on Information Science and Control Engineering (ICISCE), 2017, 1094–1098. https://doi.org/10.1109/ICISCE.2017.228
    [9] M. Chyba, T. Haberkorn, R. N. Smith, G.Wilkens, A geometric analysis of trajectory design for underwater vehicles, Discrete Cont. Dynam. Sys. B, 11 (2009), 233–262. https://doi.org/10.3934/dcdsb.2009.11.233 doi: 10.3934/dcdsb.2009.11.233
    [10] S. Fiori, Model formulation over Lie groups and numerical methods to simulate the motion of gyrostats and quadrotors, Mathematics, 7 (2019), 935. https://doi.org/10.3390/math7100935 doi: 10.3390/math7100935
    [11] S. Fiori, Manifold calculus in system theory and control - Fundamentals and first-order systems, Symmetry, 13 (2021), 2092. https://doi.org/10.3390/sym13112092 doi: 10.3390/sym13112092
    [12] T. I. Fossen, O. E. Fjellstad, Nonlinear modelling of marine vehicles in 6 degrees of freedom, Math. Model. Syst., 1 (1995), 17–27. https://doi.org/10.1080/13873959508837004 doi: 10.1080/13873959508837004
    [13] T. I. Fossen, K. Y. Pettersen, Modeling of Underwater Vehicles, Berlin: Springer Berlin Heidelberg, 2018. https://doi.org/10.1007/978-3-642-41610-1_12-1
    [14] T. I. Fossen, A nonlinear unified state-space model for ship maneuvering and control in a seaway, Int. J. Bifurc. Chaos, 15 (2005), 2717–2746. https://doi.org/10.1142/S0218127405013691 doi: 10.1142/S0218127405013691
    [15] J. González-García, A. Gómez-Espinosa, E. Cuan-Urquizo, L. G. García-Valdovinos, T. Salgado-Jiménez, J. A. E. Cabello, Autonomous underwater vehicles: Localization, navigation, and communication for collaborative missions, Appl. Sci., 10 (2020), 1256. https://doi.org/10.3390/app10041256 doi: 10.3390/app10041256
    [16] E. Y. Hong, H. G. Soon, M. Chitre, Depth control of an autonomous underwater vehicle, starfish, In: Oceans'10 IEEE Sydney, 2010. https://doi.org/10.1109/OCEANSSYD.2010.5603566
    [17] P. Jagtap, P. Raut, P. Kumar, A. Gupta, N. Singh, F. Kazi, Control of autonomous underwater vehicle using reduced order model predictive control in three dimensional space, IFAC Papers OnLine, 49 (2016), 772–777. https://doi.org/10.1016/j.ifacol.2016.03.150 doi: 10.1016/j.ifacol.2016.03.150
    [18] A. Krishnan, J. Kadiyam, S. Mohan, Robust motion control of fully/over-actuated underwater vehicle using sliding surfaces, J. Intell. Robot. Syst., 108 (2023), 60. https://doi.org/10.1007/s10846-023-01918-y doi: 10.1007/s10846-023-01918-y
    [19] S. K. Lee, T. H. Joung, S. J. Cheo, T. S. Jang, J. H. Lee, Evaluation of the added mass for a spheroid-type unmanned underwater vehicle by vertical planar motion mechanism test, Int. J. Naval Archit. Ocean Eng., 3 (2011), 174–180. https://doi.org/10.2478/IJNAOE-2013-0060 doi: 10.2478/IJNAOE-2013-0060
    [20] N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, 33 (1997), 331–346. https://doi.org/10.1016/S0005-1098(96)00176-8 doi: 10.1016/S0005-1098(96)00176-8
    [21] C. C. Liang, T. L. Teng, W. H. Lai, A study of diving depth on deep-diving submersible vehicles, Int. J. Pres. Ves. Pip., 75 (1998), 447–457. https://doi.org/10.1016/S0308-0161(98)00041-6 doi: 10.1016/S0308-0161(98)00041-6
    [22] X. Liang, Y. Pang, L. Wan, B. Wang, Dynamic Modelling and Motion Control for Underwater Vehicles with Fins, Rijeka: Intech Open Access Publisher, 2009. https://doi.org/10.5772/6720
    [23] N. Nordkvist, A. K. Sanyal, A Lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics, In: 49th IEEE Conference on Decision and Control (CDC), 2010, 5414–5419. https://doi.org/10.1109/CDC.2010.5717622
    [24] J. P. Panda, A. Mitra, H. V. Warrior, A review on the hydrodynamic characteristics of autonomous underwater vehicles, Proc. Institut. Mech. Eng. M J. Eng. Marit. Environ., 235 (2021), 15–29. https://doi.org/10.1177/1475090220936896 doi: 10.1177/1475090220936896
    [25] M. Saghafi, R. Lavimi, Optimal design of nose and tail of an autonomous underwater vehicle hull to reduce drag force using numerical simulation, Proc. Institut. Mech. Eng. M J. Eng. Marit. Environ., 234 (2020), 76–88. https://doi.org/10.1177/1475090219863191 doi: 10.1177/1475090219863191
    [26] C. Shen, Y. Shi, B. Buckham, Trajectory tracking control of an autonomous underwater vehicle using Lyapunov-based model predictive control, IEEE T. Industrial Elect., 65 (2018), 5796–5805. https://doi.org/10.1109/TIE.2017.2779442 doi: 10.1109/TIE.2017.2779442
    [27] Ø. N. Smogeli, T. Pérez, T. I. Fossen, A. J. Sørensen, The marine systems simulator statespace model representation for dynamically positioned surface vessels, In: International Maritime Association of the Mediterranean IMAM Conference, Lisbon, 2005.
    [28] SNAME, Nomenclature for treating the motion of a submerged body through a fluid, Soc. Naval Archit. Marine Eng. Tech. Res. Bull., 1950 (1950), 1–5.
    [29] Y. Sun, X. Ran, J. Cao, Y. Li, Deep submergence rescue vehicle docking based on parameter adaptive control with acoustic and visual guidance, Int. J. Adv. Robot. Syst., 17 (2020). https://doi.org/10.1177/1729881420919955
    [30] N. Syahroni, Y. B. Seo, J. W. Choi, Depth control of autonomous underwater vehicle based on open control platform, IFAC Proc. Volumes, 41 (2008), 3707–3712. https://doi.org/10.3182/20080706-5-KR-1001.00626 doi: 10.3182/20080706-5-KR-1001.00626
    [31] A. S. Tijjani, A. Chemori, V. Creuze, Robust adaptive tracking control of underwater vehicles: Design, stability analysis, and experiments, IEEE/ASME Transact. Mechat., 26 (2021), 897–907. https://doi.org/10.1109/TMECH.2020.3012502 doi: 10.1109/TMECH.2020.3012502
    [32] F. Udwadia, R. Kalaba, On the foundations of analytical dynamics, Int. J. Non-Linear Mech., 37 (2002), 1079–1090. https://doi.org/10.1016/S0020-7462(01)00033-6 doi: 10.1016/S0020-7462(01)00033-6
    [33] C. Woolsey, N. Leonard, Stabilizing underwater vehicle motion using internal rotors, Automatica, 38 (2002), 2053–2062. https://doi.org/10.1016/S0005-1098(02)00136-X doi: 10.1016/S0005-1098(02)00136-X
    [34] Y. Zhang, J. Che, Y. Hu, J. Cui, J. Cui, Real-time ocean current compensation for AUV trajectory tracking control using a meta-learning and self-adaptation hybrid approach, Sensors, 23 (2023), 6417. https://doi.org/10.3390/s23146417 doi: 10.3390/s23146417
    [35] J. Zhou, Y. Si, Y. Chen, A review of subsea AUV technology, J. Mar. Sci. Eng., 11 (2023), 1119. https://doi.org/10.3390/jmse11061119 doi: 10.3390/jmse11061119
    [36] K. Zhu, L. Gu, A MIMO nonlinear robust controller for work-class ROVs positioning and trajectory tracking control, In: 2011 Chinese Control and Decision Conference (CCDC), 2011, 2565–2570. https://doi.org/10.1109/CCDC.2011.5968643
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(494) PDF downloads(43) Cited by(0)

Article outline

Figures and Tables

Figures(4)  /  Tables(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog