Port-Hamiltonian modeling of rigid multibody systems

Authors

Thomas Berger, René Hochdahl, Timo Reis, Robert Seifried

Abstract

We employ a port-Hamiltonian approach to model nonlinear rigid multibody systems subject to both position and velocity constraints. Our formulation accommodates Cartesian and redundant coordinates, respectively, and captures kinematic as well as gyroscopic effects. The resulting equations take the form of nonlinear differential-algebraic equations that inherently preserve an energy balance. We show that the proposed class is closed under interconnection, and we provide several examples to illustrate the theory.

Keywords

Port-Hamiltonian systems; Multibody systems; Position and velocity constraints; Dirac structures; Lagrangian submanifolds; Resistive relations; Differential-algebraic equations; 34A09; 37J39; 53D12; 70E55; 93C10

Citation

• Journal: Nonlinear Dynamics
• Year: 2025
• Volume:
• Issue:
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s11071-025-11776-y

BibTeX

@article{Berger_2025,
  title={{Port-Hamiltonian modeling of rigid multibody systems}},
  ISSN={1573-269X},
  DOI={10.1007/s11071-025-11776-y},
  journal={Nonlinear Dynamics},
  publisher={Springer Science and Business Media LLC},
  author={Berger, Thomas and Hochdahl, René and Reis, Timo and Seifried, Robert},
  year={2025}
}

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References

• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
• van der Schaft, A. Characterization and partial synthesis of the behavior of resistive circuits at their terminals. Systems & Control Letters 59, 423–428 (2010) – 10.1016/j.sysconle.2010.05.005
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Math. Control Signals Syst. 30, (2018) – 10.1007/s00498-018-0223-3
• Mehl, C., Mehrmann, V. & Wojtylak, M. Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems. SIAM J. Matrix Anal. & Appl. 39, 1489–1519 (2018) – 10.1137/18m1164275
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters 121, 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• B Maschke, Vietnam J. Math. (2020)
• Venkatraman, A. & van der Schaft, A. Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs. European Journal of Control 16, 665–677 (2010) – 10.3166/ejc.16.665-677
• Gernandt, H., Haller, F. E. & Reis, T. A Linear Relation Approach to Port-Hamiltonian Differential-Algebraic Equations. SIAM J. Matrix Anal. Appl. 42, 1011–1044 (2021) – 10.1137/20m1371166
• van der Schaft, A. & Mehrmann, V. Linear port-Hamiltonian DAE systems revisited. Systems & Control Letters 177, 105564 (2023) – 10.1016/j.sysconle.2023.105564
• Mehrmann, V. & van der Schaft, A. Differential–algebraic systems with dissipative Hamiltonian structure. Math. Control Signals Syst. 35, 541–584 (2023) – 10.1007/s00498-023-00349-2
• Gernandt, H., Haller, F. E., Reis, T. & Schaft, A. J. van der. Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics 159, 103959 (2021) – 10.1016/j.geomphys.2020.103959
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43, 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Kurula, M., Zwart, H., van der Schaft, A. & Behrndt, J. Dirac structures and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications 372, 402–422 (2010) – 10.1016/j.jmaa.2010.07.004
• Jäschke, J., Skrepek, N. & Ehrhardt, M. Mixed-dimensional geometric coupling of port-Hamiltonian systems. Applied Mathematics Letters 137, 108508 (2023) – 10.1016/j.aml.2022.108508
• Arnold, V. I. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics (Springer New York, 1989). doi:10.1007/978-1-4757-2063-1 – 10.1007/978-1-4757-2063-1
• Schiehlen, W. & Eberhard, P. Applied Dynamics. (Springer International Publishing, 2014). doi:10.1007/978-3-319-07335-4 – 10.1007/978-3-319-07335-4
• Woernle, C. Multibody Systems. (Springer Berlin Heidelberg, 2024). doi:10.1007/978-3-662-67262-4 – 10.1007/978-3-662-67262-4
• Shabana, A. A. Dynamics of Multibody Systems. (2020) doi:10.1017/9781108757553 – 10.1017/9781108757553
• Kotyczka, P. & Thoma, T. Symplectic discrete-time energy-based control for nonlinear mechanical systems. Automatica 133, 109842 (2021) – 10.1016/j.automatica.2021.109842
• Mehrmann, V. & Morandin, R. Structure-preserving discretization for port-Hamiltonian descriptor systems. 2019 IEEE 58th Conference on Decision and Control (CDC) 6863–6868 (2019) doi:10.1109/cdc40024.2019.9030180 – 10.1109/cdc40024.2019.9030180
• Egger, H., Habrich, O. & Shashkov, V. On the Energy Stable Approximation of Hamiltonian and Gradient Systems. Computational Methods in Applied Mathematics 21, 335–349 (2020) – 10.1515/cmam-2020-0025
• Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters 133, 104530 (2019) – 10.1016/j.sysconle.2019.104530
• Giesselmann, J., Karsai, A. & Tscherpel, T. Energy-consistent Petrov–Galerkin time discretization of port-Hamiltonian systems. The SMAI Journal of computational mathematics 11, 335–367 (2025) – 10.5802/smai-jcm.127
• Reis, T. Some notes on port-Hamiltonian systems on Banach spaces. IFAC-PapersOnLine 54, 223–229 (2021) – 10.1016/j.ifacol.2021.11.082
• Courant, T. J. Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Lang, S. Fundamentals of Differential Geometry. Graduate Texts in Mathematics (Springer New York, 1999). doi:10.1007/978-1-4612-0541-8 – 10.1007/978-1-4612-0541-8
• Lee, J. M. Introduction to Smooth Manifolds. Graduate Texts in Mathematics (Springer New York, 2012). doi:10.1007/978-1-4419-9982-5 – 10.1007/978-1-4419-9982-5
• Armstrong-Hélouvry, B., Dupont, P. & De Wit, C. C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30, 1083–1138 (1994) – 10.1016/0005-1098(94)90209-7
• RI Leine, Dynamics and Bifurcations of Non-smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics (2004)
• Fujimoto, K., Takeuchi, T. & Matsumoto, Y. On port-Hamiltonian modeling and control of quaternion systems. IFAC-PapersOnLine 48, 39–44 (2015) – 10.1016/j.ifacol.2015.10.211
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Smith, M. C. Synthesis of mechanical networks: the inerter. IEEE Trans. Automat. Contr. 47, 1648–1662 (2002) – 10.1109/tac.2002.803532
• Jacobs, H. O. & Yoshimura, H. Interconnection and composition of Dirac structures for Lagrange-Dirac systems. IEEE Conference on Decision and Control and European Control Conference 928–933 (2011) doi:10.1109/cdc.2011.6160480 – 10.1109/cdc.2011.6160480
• Jacobs, H. et al. Interconnection of Lagrange-Dirac Dynamical Systems for Electric Circuits. AIP Conference Proceedings 566–569 (2010) doi:10.1063/1.3498539 – 10.1063/1.3498539
• RN Arnold, Gyrodynamics and Its Engineering Applications (1961)
• Krhač, K., Maschke, B. & van der Schaft, A. Port-Hamiltonian systems with energy and power ports. IFAC-PapersOnLine 58, 280–285 (2024) – 10.1016/j.ifacol.2024.08.294
• Barbero Liñán, M. et al. Morse families and Dirac systems. Journal of Geometric Mechanics 11, 487–510 (2019) – 10.3934/jgm.2019024
• Castrillón López, M. & Ratiu, T. S. Morse Families and Lagrangian Submanifolds. Springer Proceedings in Mathematics & Statistics 65–78 (2016) doi:10.1007/978-3-319-32085-4_6 – 10.1007/978-3-319-32085-4_6