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Passivity-preserving splitting methods for rigid body systems

  • Elena Celledoni
  • Eirik Hoel Høiseth
  • Nataliya Ramzina
Article

Abstract

A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input–output passive. We propose passivity-preserving splitting methods for the numerical solution of a class of problems which includes this system as a special case. We prove the passivity-preservation property for the splitting methods, and we investigate stability and energy behaviour in numerical experiments. Implementation is discussed in detail for a special case where the splitting gives rise to the subsequent integration of two completely integrable flows. The equations for the attitude are reformulated on \(\mathit{SO}(3)\) using rotation matrices rather than local parameterisations with Euler angles.

Keywords

Passivity Structure preservation Differential equations Time integration Multibody dynamics 

Notes

Acknowledgements

This work has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691070, and from The Research Council of Norway. We are grateful to T.I. Fossen for useful discussions. We are also grateful to Sergio Blanes and Fernando Casas for useful discussions regarding splitting methods, and for providing highly accurate coefficients for the splitting methods of order 4 and 6 used in the numerical experiments. Part of this work was done while visiting Massey University, Palmerston North, New Zealand, and La Trobe University, Melbourne, Australia.

References

  1. 1.
    Baker, A.: Matrix Groups: An Introduction to Lie Group Theory. Springer Undergraduate Mathematics Series. Springer, London (2002) CrossRefMATHGoogle Scholar
  2. 2.
    Blanes, S., Casas, F., Murua, A.: Splitting methods in the numerical integration of non-autonomous dynamical systems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 106(1), 49–66 (2012).  https://doi.org/10.1007/s13398-011-0024-8 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blanes, S., Casas, F., Murua, A.: Splitting methods with complex coefficients. Bol. Soc. Esp. Mat. Apl. 50, 47–61 (2010) MathSciNetMATHGoogle Scholar
  4. 4.
    Blanes, S., Casas, F., Ros, J.: High order optimised geometric integrators for linear differential equations. BIT Numer. Math. 42(2), 262–284 (2002).  https://doi.org/10.1023/A:1021942823832 CrossRefMATHGoogle Scholar
  5. 5.
    Bou Rabee, N., Marsden, J.E.: Hamilton–Pontryagin integrators on Lie groups Part I: Introduction and structure-preserving properties. Found. Comput. Math. 9, 197–219 (2009).  https://doi.org/10.1007/s10208-008-9030-4 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.W.R., Wright, W.: Energy-preserving Runge–Kutta methods. Modél. Math. Anal. Numér. 43(4), 645–649 (2009) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Celledoni, E., Säfström, N.: Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions. J. Phys. A 39(19), 5463–5478 (2006).  https://doi.org/10.1088/0305-4470/39/19/S08 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Celledoni, E., Fassò, F., Säfström, N., Zanna, A.: The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput. 30(4), 2084–2112 (2008).  https://doi.org/10.1137/070704393 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Egeland, O., Gravdahl, J.T.: Modeling and Simulation for Automatic Control. Marine Cybernetics, Trondheim (2002). Corrected second printing (2003) Google Scholar
  10. 10.
    Fossen, T.I.: Marine Control Systems. Marine Cybernetics, Trondheim (2002) Google Scholar
  11. 11.
    Jensen, G.A., Säfström, N., Nguyen, T.D., Fossen, T.I.: Modeling and control of offshore pipelay operations based on a finite strain pipe model. In: Proceedings of American Control Conference, St. Louis, MO, USA, pp. 4717–4722 (2009).  https://doi.org/10.1109/ACC.2009.5160110 Google Scholar
  12. 12.
    Jensen, G.A., Säfström, N., Nguyen, T.D., Fossen, T.I.: A nonlinear PDE formulation for offshore vessel pipeline installation. Ocean Eng. 37(4), 365–377 (2010).  https://doi.org/10.1016/j.oceaneng.2009.12.009 CrossRefGoogle Scholar
  13. 13.
    Gustafsson, E.: Accurate discretizations of torqued rigid body dynamics. Master thesis, Norwegian University of Science and Technology, Trondheim (2010) Google Scholar
  14. 14.
    Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010) MathSciNetMATHGoogle Scholar
  15. 15.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006) MATHGoogle Scholar
  16. 16.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996) CrossRefMATHGoogle Scholar
  17. 17.
    Holm, D.D.: Geometric Mechanics, Part II: Rotating, Translating and Rolling. Imperial College Press, London (2008) CrossRefMATHGoogle Scholar
  18. 18.
    Lamb, H.: Hydrodynamics, 4th edn. Cambridge University Press, Cambridge (1916) MATHGoogle Scholar
  19. 19.
    Kirchhoff, G.R.: Ueber die Bewegung eines Rotationskörpers i einer Flüssigkeit. Crelle t. LXXI (1869) [Ges. Abh. p. 376], Mechanik, c. XIX Google Scholar
  20. 20.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn. Texts in Applied Mathematics, vol. 17. Springer, New York (1999) CrossRefMATHGoogle Scholar
  21. 21.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 357, 1021–1045 (1999) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    MSS. Marine Systems Simulator (2010). http://www.marinecontrol.org. Viewed 2014-04-04
  24. 24.
    Müller, A.: Screw and Lie group theory in rigid body dynamics. Multibody Syst. Dyn. 42, 219–248 (2018).  https://doi.org/10.1007/s11044-017-9583-6 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jahnke, T., Lubich, C.: Error bounds for exponential operator splitting. BIT Numer. Math. 40, 735–744 (2000) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Perez, T., Fossen, T.I.: Kinematic models for manoeuvring and seakeeping of marine vessels. Model. Identif. Control 28(1), 19–30 (2007).  https://doi.org/10.4173/mic.2007.1.3 CrossRefGoogle Scholar
  27. 27.
    Säfström, N.: Modeling and simulation of rigid body and rod dynamics via geometric methods. PhD thesis, Norwegian University of Science and Technology, Trondheim (2009) Google Scholar
  28. 28.
    Van der Schaft, A.: Port-Hamiltonian Systems: An Introductory Survey. European Mathematical Society Publishing House (EMS Ph), Madrid (2006) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNTNUTrondheimNorway

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