- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Elastohydrodynamic instabilities of a soft robotic arm in a viscous fluid

Mohamed Warda*

Ronojoy Adhikari

  • Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom

  • *Contact author: mrmaw2@cam.ac.uk
PDF

Phys. Rev. Research 8, 013229 – Published 2 March, 2026

DOI: https://doi.org/10.1103/c914-x8r2

Abstract

The design and control of soft robots operating in fluid environments requires a careful understanding of the interplay between large elastic body deformations and hydrodynamic forces. Here, we show that this interplay leads to elastohydrodynamic instabilities in a clamped soft robotic arm driven terminally by a constant pressure in a viscous fluid. We model the arm as a Cosserat rod that can stretch, shear, and bend. We obtain invariant, geometrically exact, nonlinear equations of motion by using Cartan’s method of moving frames. Stability to small perturbations of a straight rod is governed by a non-Hermitian linear operator. Eigenanalysis shows that stability is lost through a Hopf bifurcation with the increase of pressure above a first threshold. A surprising return to stability is obtained with further increase of pressure beyond a second threshold. Numerical solutions of the nonlinear equations, using a geometrically exact spectral method, confirm stable limit-cycle oscillations between these two pressure thresholds. An asymptotic analysis in the beam limit rationalizes these results analytically. This counterintuitive sequence of bifurcations underscores the subtle nature of the elastohydrodynamic coupling in Cosserat rods and emphasizes their importance for the control of the viscous dynamics of soft robots.

    View figure in article

      Physics Subject Headings (PhySH)

      Article Text

      References (39)

      1. R. M. Murray, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994).
      2. J. M. Selig, Geometric Fundamentals of Robotics (Springer, New York, 2004).
      3. G. S. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods (Birkhäuser, Boston, 2009).
      4. G. S. Chirikjian and J. W. Burdick, A hyper-redundant manipulator, IEEE Rob. Autom. Mag. 1, 22 (1994).
      5. T. Ranzani, M. Cianchetti, G. Gerboni, I. D. Falco, and A. Menciassi, A soft modular manipulator for minimally invasive surgery: Design and characterization of a single module, IEEE Trans. Rob. 32, 187 (2016).
      6. F. Campisano, A. A. Remirez, S. Caló, J. H. Chandler, K. L. Obstein, R. J. Webster, and P. Valdastri, Online disturbance estimation for improving kinematic accuracy in continuum manipulators, IEEE Rob. Autom. Lett. 5, 2642 (2020).
      7. F. Campisano, S. Caló, A. A. Remirez, J. H. Chandler, K. L. Obstein, R. J. Webster III, and P. Valdastri, Closed-loop control of soft continuum manipulators under tip follower actuation, Int. J. Rob. Res. 40, 923 (2021).
      8. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge University Press, Cambridge, 1892).
      9. J. L. Ericksen and C. Truesdell, Exact theory of stress and strain in rods and shells, Arch. Ration. Mech. Anal. 1, 295 (1957).
      10. S. S. Antman, Nonlinear Problems of Elasticity (Springer, New York, 2005).
      11. M. Gazzola, L. H. Dudte, A. G. McCormick, and L. Mahadevan, Forward and inverse problems in the mechanics of soft filaments, R. Soc. Open Sci. 5, 171628 (2018).
      12. F. Boyer, V. Lebastard, F. Candelier, and F. Renda, Dynamics of continuum and soft robots: A strain parameterization based approach, IEEE Trans. Rob. 37, 847 (2021).
      13. N. Naughton, J. Sun, A. Tekinalp, T. Parthasarathy, G. Chowdhary, and M. Gazzola, Elastica: A compliant mechanics environment for soft robotic control, IEEE Rob. Autom. Lett. 6, 3389 (2021).
      14. Z. Chen, Z. Liu, and X. Han, Model analysis of robotic soft arms including external force effects, Micromachines 13, 1486 (2022).
      15. A. Tekinalp, Y. Bhosale, S. Cui, F. K. Chan, and M. Gazzola, Self-propelling, soft, and slender structures in fluids: Cosserat rods immersed in the velocity–vorticity formulation of the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Eng. 440, 117910 (2025).
      16. A. Laskar and R. Adhikari, Filament actuation by an active colloid at low Reynolds number, New J. Phys. 19, 033021 (2017).
      17. G. De Canio, E. Lauga, and R. E. Goldstein, Spontaneous oscillations of elastic filaments induced by molecular motors, J. R. Soc. Interface 14, 20170491 (2017).
      18. F. Ling, H. Guo, and E. Kanso, Instability-driven oscillations of elastic microfilaments, J. R. Soc. Interface 15, 20180594 (2018).
      19. Y. Man and E. Kanso, Morphological transitions of axially-driven microfilaments, Soft Matter 15, 5163 (2019).
      20. Y. Fily, P. Subramanian, T. M. Schneider, R. Chelakkot, and A. Gopinath, Buckling instabilities and spatio-temporal dynamics of active elastic filaments, J. R. Soc. Interface 17, 20190794 (2020).
      21. B. Clarke, Y. Hwang, and E. E. Keaveny, Bifurcations and nonlinear dynamics of the follower force model for active filaments, Phys. Rev. Fluids 9, 073101 (2024).
      22. K. G. Link, R. D. Guy, B. Thomases, and P. E. Arratia, Effect of fluid elasticity on the emergence of oscillations in an active elastic filament, J. R. Soc. Interface 21, 20240046 (2024).
      23. O. Schnitzer, Onset of spontaneous beating and whirling in the follower force model of an active filament, J. Fluid Mech. 1007, A65 (2025).
      24. M. Beck, Die knicklast des einseitig eingespannten, tangential gedrückten stabes, Z. Angew. Math. Phys. 3, 225 (1952).
      25. V. V. Bolotin, in Nonconservative Problems of the Theory of Elastic Stability, edited by G. Herrmann (Pergamon Press, London, 1963).
      26. H. Ziegler, Principles of Structural Stability (Blaisdell Publishing Company, Waltham, MA, 1968).
      27. J. Carr and M. Z. M. Malhardeen, Beck’s problem, SIAM J. Appl. Math. 37, 261 (1979).
      28. M. Chen, Hopf bifurcation in Beck’s problem, Nonlinear Anal.: Theory Methods Appl. 11, 1061 (1987).
      29. H. Koch and S. S. Antman, Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations, SIAM J. Math. Anal. 32, 360 (2000).
      30. Q. Wang, A comprehensive stability analysis of a cracked beam subjected to follower compression, Int. J. Solids Struct. 41, 4875 (2004).
      31. M. Yan, M. Warda, B. Németh, L. Kikuchi, and R. Adhikari, Geometric field theory for elastohydrodynamics of Cosserat rods, arXiv:2510.18097.
      32. M. Garg and A. Kumar, A slender body theory for the motion of special Cosserat filaments in Stokes flow, Math. Mech. Solids 28, 692 (2023).
      33. T. Kikuchi, Rare events and dynamics in non-equilibrium systems, Ph.D. thesis, Apollo—University of Cambridge Repository, 2023.
      34. L. Kikuchi and R. Adhikari, Cartan media: Geometric continuum mechanics in homogeneous spaces, arXiv:2310.01388.
      35. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
      36. Z. D. Jastrzebski, Nature and Properties of Engineering Materials, Wiley International Edition (Wiley, 1959).
      37. mathematica code and implementation details are available from the author upon request.
      38. P. S. Krishnaprasad and D. P. Tsakiris, G-snakes: Nonholonomic kinematic chains on Lie groups, in Proceedings of the 33rd IEEE Conference on Decision and Control (IEEE, Lake Buena Vista, FL, 1994), pp. 2955–2960.
      39. D. Holm, R. Ivanov, and J. R. Percival, G-strands, J. Nonlinear Sci. 22, 517 (2012).

      Outline

      Information

      Phys. Rev. Research 8, 013229– Published 2 March, 2026
      • Received 29 October 2025
      • Accepted 28 January 2026
      Crossmark - Check for updates button
      Creative Commons Logo

      Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

      Published by the American Physical Society

      Sign In to Your Journals Account

      Filter

      Filter

      Article Lookup

      Enter a citation