A novel port-Hamiltonian framework for clustered tensegrity systems

Authors

Mingji Wang, Lingchong Gao, Fei Li, Ningning Song, Gang Wang, Johannes Fottner, Haijun Peng

Abstract

Clustered tensegrity systems (CTSs) are lightweight, energy-efficient, and modular, making them ideal for engineering applications. The port-Hamiltonian (pH) framework is well-suited for their design and analysis due to its effectiveness in modeling dynamic systems. However, when CTSs are modeled within the pH framework, several challenges arise: (1) Current spatial discretization methods for pH systems are complex and inefficient for CTSs, and cannot directly represent strong nonlinear coupling. (2) Current symplectic time discretization methods for pH systems are inefficient and unstable when solving stiff equations. First, a spatial discretization method based on positional finite element method (PFEM) is proposed. It can accurately capture strong nonlinear coupling from large-scale rotations and deformations without relying on rotation matrices. Then, a stiff problem modification based on symplectic time discretization is proposed. Combined with a Quasi-Newton strategy, it significantly improves computational efficiency with few losses of precision. Numerical simulations show that discrete pH systems based on PFEM can efficiently and accurately capture the energy and dynamic behavior of CTSs. The proposed modification effectively improves stiff problem and significantly enhances computational efficiency.

Keywords

Port-Hamiltonian; Clustered tensegrity systems; Positional finite element method; Stiff problems; Quasi-Newton method

Citation

• Journal: Mechanism and Machine Theory
• Year: 2025
• Volume: 215
• Issue:
• Pages: 106177
• Publisher: Elsevier BV
• DOI: 10.1016/j.mechmachtheory.2025.106177

BibTeX

@article{Wang_2025,
  title={{A novel port-Hamiltonian framework for clustered tensegrity systems}},
  volume={215},
  ISSN={0094-114X},
  DOI={10.1016/j.mechmachtheory.2025.106177},
  journal={Mechanism and Machine Theory},
  publisher={Elsevier BV},
  author={Wang, Mingji and Gao, Lingchong and Li, Fei and Song, Ningning and Wang, Gang and Fottner, Johannes and Peng, Haijun},
  year={2025},
  pages={106177}
}

Download the bib file

References

• Wu, G. & Niu, B. Hexad robot: A 6-dof parallel PnP robot to accommodate antagonistic rotational capability and structural complexity. Mechanism and Machine Theory 195, 105612 (2024) – 10.1016/j.mechmachtheory.2024.105612
• Liu, Y., Luo, K., Tian, Q. & Hu, H. Force-guided heuristic kinematics control of a continuum robot with variable curvatures. Mechanism and Machine Theory 209, 106007 (2025) – 10.1016/j.mechmachtheory.2025.106007
• Song, N., Peng, H. & Guo, X. Sym-ML: A symplectic machine learning framework for stable dynamic prediction of mechanical system. Mechanism and Machine Theory 206, 105934 (2025) – 10.1016/j.mechmachtheory.2025.105934
• Song, N., Wang, C., Peng, H. & Zhao, J. A study of mechanism-data hybrid-driven method for multibody system via physics-informed neural network. Acta Mech. Sin. 41, (2024) – 10.1007/s10409-024-24159-x
• Zhan, Y. et al. Non-anthropomorphic passive load-bearing lower-limb exoskeleton with a reconfigurable mechanism based on mechanical intelligence. Mechanism and Machine Theory 201, 105753 (2024) – 10.1016/j.mechmachtheory.2024.105753
• Li, M., Feng, H. & Dai, J. S. Topology-manifold-based parametric design of dual-spherical-4R chiral origami mechanisms. Mechanism and Machine Theory 209, 105983 (2025) – 10.1016/j.mechmachtheory.2025.105983
• Cammarata, A. et al. Elastostatic analysis of a module-based shape morphing snake-like robot. Mechanism and Machine Theory 194, 105580 (2024) – 10.1016/j.mechmachtheory.2024.105580
• Chen, Analysis of clustered cable-actuation strategies of V-Expander tensegrity structures. Eng Struct (2024)
• Yang, Delocalized Deformation Enhanced Reusable Energy Absorption Metamaterials Based on Bistable Tensegrity. Adv Funct Mater (2024)
• Liu, A review on tensegrity structures-based robots. Mech. Mach. Theory. (2022)
• Kang, Y. et al. Design and analysis of a flexible struts V-expander tensegrity robot for navigating pipes. Mechanism and Machine Theory 202, 105757 (2024) – 10.1016/j.mechmachtheory.2024.105757
• Moored, K. W. & Bart-Smith, H. Investigation of clustered actuation in tensegrity structures. International Journal of Solids and Structures 46, 3272–3281 (2009) – 10.1016/j.ijsolstr.2009.04.026
• Kan, Z., Peng, H., Chen, B. & Zhong, W. Nonlinear dynamic and deployment analysis of clustered tensegrity structures using a positional formulation FEM. Composite Structures 187, 241–258 (2018) – 10.1016/j.compstruct.2017.12.050
• Bel Hadj Ali, N., Rhode-Barbarigos, L. & Smith, I. F. C. Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm. International Journal of Solids and Structures 48, 637–647 (2011) – 10.1016/j.ijsolstr.2010.10.029
• Lv, Q., Tang, Y., Wang, X. & Li, T. A force-density framework for flexible multi-body dynamic analysis of clustered tensegrity structures. International Journal of Solids and Structures 305, 113098 (2024) – 10.1016/j.ijsolstr.2024.113098
• Peng, H., Wang, M., Yang, H., Li, F. & Kan, Z. Rigid-flexible-soft coupling dynamic modeling and analysis of clustered tensegrity. Nonlinear Dyn 112, 10959–10993 (2024) – 10.1007/s11071-024-09475-1
• Duindam, (2009)
• 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
• Chen, G. & Zhu, G. Symplectic Algorithms for Stable Manifolds in Control Theory. IEEE Trans. Automat. Contr. 67, 3105–3111 (2022) – 10.1109/tac.2021.3093594
• Mehrmann, V. & Unger, B. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32, 395–515 (2023) – 10.1017/s0962492922000083
• Warsewa, A., Böhm, M., Sawodny, O. & Tarín, C. A port-Hamiltonian approach to modeling the structural dynamics of complex systems. Applied Mathematical Modelling 89, 1528–1546 (2021) – 10.1016/j.apm.2020.07.038
• Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics 62, 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• Baaiu, A., Couenne, F., Lefevre, L., Le Gorrec, Y. & Tayakout, M. Structure-preserving infinite dimensional model reduction: Application to adsorption processes. Journal of Process Control 19, 394–404 (2009) – 10.1016/j.jprocont.2008.07.002
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information 38, 493–533 (2020) – 10.1093/imamci/dnaa038
• Zhang, L., Gao, Q. & Zhang, H. W. An efficient algorithm for mechanical analysis of bimodular truss and tensegrity structures. International Journal of Mechanical Sciences 70, 57–68 (2013) – 10.1016/j.ijmecsci.2013.02.002
• Zhang, L., Lu, M. K., Zhang, H. W. & Yan, B. Geometrically nonlinear elasto-plastic analysis of clustered tensegrity based on the co-rotational approach. International Journal of Mechanical Sciences 93, 154–165 (2015) – 10.1016/j.ijmecsci.2015.01.015
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian flexible multibody dynamics. Multibody Syst Dyn 51, 343–375 (2020) – 10.1007/s11044-020-09758-6
• De Veubeke, B. F. The dynamics of flexible bodies. International Journal of Engineering Science 14, 895–913 (1976) – 10.1016/0020-7225(76)90102-6
• Sun, J. & Hu, H. Dynamic topology optimization of flexible multibody systems. Nonlinear Dyn 112, 11711–11743 (2024) – 10.1007/s11071-024-09619-3
• Aoues, S., Di Loreto, M., Eberard, D. & Marquis-Favre, W. Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability. Systems & Control Letters 110, 9–14 (2017) – 10.1016/j.sysconle.2017.10.003
• Zheng, X. et al. Dynamic modeling and experimental verification of a cable-driven continuum manipulator with cable-constrained synchronous rotating mechanisms. Nonlinear Dyn 107, 153–172 (2021) – 10.1007/s11071-021-06989-w
• Mohammadi, N., Rouvinen, A., Korkealaakso, P. & Escalona, J. L. Real-time explicit co-simulation of wire-rope systems for industrial mobile harbor cranes. Nonlinear Dyn 112, 13095–13114 (2024) – 10.1007/s11071-024-09752-z
• Kotyczka, P. & Thoma, T. Symplectic discrete-time energy-based control for nonlinear mechanical systems. Automatica 133, 109842 (2021) – 10.1016/j.automatica.2021.109842
• Zheng, X. et al. An efficient dynamic modeling and simulation method of a cable-constrained synchronous rotating mechanism for continuum space manipulator. Aerospace Science and Technology 119, 107156 (2021) – 10.1016/j.ast.2021.107156
• Talasila, The wave equation as a Port-Hamiltonian system, and a finite-dimensional approximation. (2002)
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics 231, 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics 361, 442–476 (2018) – 10.1016/j.jcp.2018.02.006
• Cardoso-Ribeiro, F. L., Haine, G., Le Gorrec, Y., Matignon, D. & Ramirez, H. Port-Hamiltonian formulations for the modeling, simulation and control of fluids. Computers & Fluids 283, 106407 (2024) – 10.1016/j.compfluid.2024.106407
• Tornabene, F., Fantuzzi, N. & Bacciocchi, M. Strong and weak formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells: Convergence and accuracy. Engineering Analysis with Boundary Elements 92, 3–37 (2018) – 10.1016/j.enganabound.2017.08.020
• Toledo-Zucco, J.-P., Matignon, D., Poussot-Vassal, C. & Le Gorrec, Y. Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems. Systems & Control Letters 194, 105947 (2024) – 10.1016/j.sysconle.2024.105947
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling 75, 940–960 (2019) – 10.1016/j.apm.2019.04.035
• Valvo, Symmetric stiffness matrices for isoparametric finite elements in nonlinear elasticity. Comput Mech (2024)
• Li, F., Yang, H., Gu, G., Wang, Y. & Peng, H. Position and Orientation Tracking Control of a Cable-Driven Tensegrity Continuum Robot. IEEE Trans. Robot. 41, 1791–1811 (2025) – 10.1109/tro.2025.3543292
• Sun, A simple way constructing symplectic Runge-Kutta methods. J. Comput. Math. (2000)
• Verlet, L. Computer ‘Experiments’ on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Phys. Rev. 159, 98–103 (1967) – 10.1103/physrev.159.98
• Bi, H., Wang, B., Ouyang, H., Deng, Z. & Zhang, B. Nonlinear dynamic instability of wrinkled film-substrate structure under axial load. Nonlinear Dyn 106, 2807–2827 (2021) – 10.1007/s11071-021-06976-1
• Li, Q., Wang, B., Deng, Z., Ouyang, H. & Wei, Y. A simple orbit-attitude coupled modelling method for large solar power satellites. Acta Astronautica 145, 83–92 (2018) – 10.1016/j.actaastro.2017.12.037
• Gu, An implicit asynchronous variational integrator for flexible multibody dynamics. Comput. Method. Appl. M. (2022)
• Azzara, R., Filippi, M. & Carrera, E. Geometrically nonlinear transient analyses of rotating structures through high-fidelity models. Composite Structures 343, 118265 (2024) – 10.1016/j.compstruct.2024.118265
• Zhang, Numerical investigation of the wheel-rail interaction in a stacked multibody system with multipoint frictional contact dynamics under strong earthquakes. Mech. Mach. Theory. (2024)
• Zhang, X., Qi, Z., Wang, G. & Guo, S. Model smoothing method of contact-impact dynamics in flexible multibody systems. Mechanism and Machine Theory 138, 124–148 (2019) – 10.1016/j.mechmachtheory.2019.03.039
• Liu, S., Yang, Q., Lv, J. & Fang, H. Modeling of a Six-Bar Tensegrity Robot Using the Port-Hamiltonian Framework and Experimental Validation. IEEE Robot. Autom. Lett. 9, 4439–4446 (2024) – 10.1109/lra.2024.3381819
• Billups, S. C. & Murty, K. G. Complementarity problems. Journal of Computational and Applied Mathematics 124, 303–318 (2000) – 10.1016/s0377-0427(00)00432-5
• Fischer, A. A special newton-type optimization method. Optimization 24, 269–284 (1992) – 10.1080/02331939208843795
• Ali, A dynamic-relaxation formulation for analysis of cable structures with sliding-induced friction. Int. J. Solids Struct. (2017)
• Chen, Analysis of clustered cable-actuation strategies of V-Expander tensegrity structures. Eng Struct. (2023)
• Zhang, J. et al. A Preprogrammable Continuum Robot Inspired by Elephant Trunk for Dexterous Manipulation. Soft Robotics 10, 636–646 (2023) – 10.1089/soro.2022.0048
• Sanz-Serna, J. M. Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica 1, 243–286 (1992) – 10.1017/s0962492900002282
• Sherman, J. & Morrison, W. J. Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix. Ann. Math. Statist. 21, 124–127 (1950) – 10.1214/aoms/1177729893
• Broyden, C. G. A class of methods for solving nonlinear simultaneous equations. Math. Comp. 19, 577–593 (1965) – 10.1090/s0025-5718-1965-0198670-6
• Flanders, Differential Forms with Applications to the Physical Sciences. Courier Corporation (1963)
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3