Skip to main content
SpringerLink
Log in

Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

In this paper we study the locomotion of a shape-changing body swimming in a two-dimensional perfect fluid of infinite extent. The shape changes are prescribed as functions of time satisfying constraints ensuring that they result from the work of internal forces only: conditions necessary for the locomotion to be termed self-propelled. The net rigid motion of the body results from the exchange of momentum between these shape changes and the surrounding fluid.

The aim of this paper is three-fold.

First, it describes a rigorous framework for the study of animal locomotion in fluid. Our model differs from previous ones mostly in that the number of degrees of freedom related to the shape changes is infinite. The Euler–Lagrange equation is obtained by applying the least action principle to the system body fluid. The formalism of Analytic Mechanics provides a simple way to handle the strong coupling between the internal dynamics of the body causing the shape changes and the dynamics of the fluid. The Euler–Lagrange equation takes the form of a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs). The existence and uniqueness of solutions for this system are rigorously proved.

Second, we are interested in making clear the connection between shape changes and internal forces. Although classical, it can be quite surprising to select the shape changes to play the role of control because the internal forces they are due to seem to be a more natural and realistic choice. We prove that, when the number of degrees of freedom relating to the shape changes is finite, both choices are actually equivalent in the sense that there is a one-to-one relation between shape changes and internal forces.

Third, we show how the control problem, consisting in associating with each shape change the resulting trajectory of the swimming body, can be analysed within the framework of geometric control theory. This allows us to take advantage of the powerful tools of differential geometry, such as the notion of Lie brackets or the Orbit Theorem and to obtain the first theoretical result (to our knowledge) of control for a swimming body in an ideal fluid. We derive some interesting and surprising tracking properties: For instance, for any given shape changes producing a net displacement in the fluid (say, moving forward), we prove that other shape changes arbitrarily close to the previous ones exist, which lead to a completely different motion (for instance, moving backward): This phenomenon will be called Moonwalking. Most of our results are illustrated by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004). Control Theory and Optimization, II

    MATH  Google Scholar 

  • Alouges, F., DeSimone, A., Lefebvre, A.: Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18(3), 277–302 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Carling, J., Williams, T., Bowtell, G.: Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier–Stokes equations and Newton’s laws of motion. J. Exp. Biol. 201, 3143–3166 (1998)

    Google Scholar 

  • Cartan, H.: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes. Avec le concours de Reiji Takahashi. Enseignement des Sciences. Hermann, Paris (1961)

  • Chambrion, T., Sigalotti, M.: Tracking control for an ellipsoidal submarine driven by Kirchhoff’s laws. IEEE Trans. Autom. Control 53(1), 339–349 (2008)

    Article  MathSciNet  Google Scholar 

  • Cherman, A., Delgado, J., Duda, F., Ehlers, K., Koiller, J., Montgomery, R.: Low Reynolds number swimming in two dimensions. In: Hamiltonian Systems and Celestial Mechanics, Pátzcuaro, 1998. World Sci. Monogr. Ser. Math., vol. 6, pp. 32–62. World Sci. Publ., River Edge (2000)

    Chapter  Google Scholar 

  • Childress, S.: Mechanics of Swimming and Flying. Cambridge Studies in Mathematical Biology, vol. 2. Cambridge University Press, Cambridge (1981)

    Book  MATH  Google Scholar 

  • Galdi, G.P.: On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148(1), 53–88 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Isidori, A.: Nonlinear Control Systems: An Introduction. Lecture Notes in Control and Information Sciences, vol. 72. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  • Jakubczyk, B.: Introduction to geometric nonlinear control; controllability and Lie bracket. In: Mathematical Control Theory, Part 1, 2 (Trieste, 2001), ICTP Lect. Notes, VIII, pp. 107–168 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002)

  • Jurdjevic, V.: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, vol. 52. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  • Jurdjevic, V., Sussmann, H.J.: Control systems on Lie groups. J. Differ. Equ. 12, 313–329 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  • Kanso, E., Marsden, J.: Optimal motion of an articulated body in a perfect fluid. In: IEEE Conference on Decision and Control, pp. 2511–2516 (2005)

    Google Scholar 

  • Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15(4), 255–289 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Kelly, S.D., Murray, R.M.: Modelling efficient pisciform swimming for control. Int. J. Robust Nonlinear Control 10(4), 217–241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Kozlov, V.V., Onishchenko, D.A.: Motion of a body with undeformable shell and variable mass geometry in an unbounded perfect fluid. J. Dyn. Differ. Equ. 15(2–3), 553–570 (2003a)

    Article  MathSciNet  MATH  Google Scholar 

  • Kozlov, V.V., Onishchenko, D.A.: On the motion in an ideal fluid of a body containing a moving concentrated mass. Prikl. Mat. Mekh. 67(4), 620–633 (2003b)

    MathSciNet  MATH  Google Scholar 

  • Lamb, H.: Hydrodynamics, 6th edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  • Lighthill, J.: Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics, Philadelphia (1975)

    MATH  Google Scholar 

  • Liu, H., Kawachi, K.: A numerical study of undulatory swimming. J. Comput. Phys. 155(2), 223–247 (1999)

    Article  MATH  Google Scholar 

  • Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol. 17. Springer, New York (1999)

    MATH  Google Scholar 

  • Mason, R.: Fluid Locomotion and Trajectory Planning for Shape-Changing Robots. Ph.D. thesis, California Institute of Technology, Department of Mechanical Engineering (June 2003)

  • Mason, R., Burdick, J.: Propulsion and control of deformable bodies in an ideal fluid. In: Proceedings of the 1999 IEEE International Conference on Robotics and Automation. Springer, Berlin (1999)

    Google Scholar 

  • Melli, J.B., Rowley, C.W., Rufat, D.S.: Motion planning for an articulated body in a perfect planar fluid. SIAM J. Appl. Dyn. Syst. 5(4), 650–669 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  • Miloh, T., Galper, A.: Self-propulsion of general deformable shapes in a perfect fluid. Proc. Math. Phys. Sci. 442(1915), 273–299 (1993)

    Article  MATH  Google Scholar 

  • Munnier, A.: On the self-displacement of deformable bodies in a potential fluid flow. Math. Models Methods Appl. Sci. 18(11), 1945–1981 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Munnier, A.: Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalism. J. Nonlinear Sci. 19(6), 665–715 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Munnier, A., Pinçon, B.: Locomotion of articulated bodies in an ideal fluid: 2d model with buoyancy, circulation and collisions. Math. Models Methods Appl. Sci. 20(10), 1899–1940 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  • San Martin, J., Scheid, J.F., Takahashi, T., Tucsnak, M.: An initial and boundary problem modeling fish-like swimming. Arch. Ration. Mech. Anal. 188(3), 429–455 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • San Martín, J., Takahashi, T., Tucsnak, M.: A control theoretic approach to the swimming of microscopic organisms. Q. Appl. Math. 65(3), 405–424 (2007)

    MATH  Google Scholar 

  • Shapere, A., Wilczek, F.: Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 557–585 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Sparenberg, J.A.: Survey of the mathematical theory of fish locomotion. J. Eng. Math. 44(4), 395–448 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 180, 171–188 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor, G.: Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond., Ser. A 209, 447–461 (1951)

    Article  MATH  Google Scholar 

  • Taylor, G.: Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond., Ser. A 214, 158–183 (1952)

    Article  MATH  Google Scholar 

  • Triantafyllou, M.S., Triantafyllou, G.S., Yue, D.K.P.: Hydrodynamics of fishlike swimming. In: Annual Review of Fluid Mechanics, vol. 32, pp. 33–53. Annual Reviews, Palo Alto (2000)

    Google Scholar 

  • Wu, T.Y.: Mathematical biofluiddynamics and mechanophysiology of fish locomotion. Math. Methods Appl. Sci. 24(17–18), 1541–1564 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu, Q., Wolfgang, M.J., Yue, D.K.P., Triantafyllou, M.S.: Three-dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech. 468, 1–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Élie Cartan UMR 7502, Nancy-Université, CNRS, INRIA, B.P. 239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Thomas Chambrion & Alexandre Munnier

  2. INRIA Nancy Grand Est, Projet CORIDA, Nancy, France

    Thomas Chambrion & Alexandre Munnier

Authors
  1. Thomas Chambrion
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandre Munnier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Chambrion.

Additional information

Communicated by E. Kanso.

Authors both supported by CPER MISN AOC. First author supported by ANR GCM, ERC Boscain and BQR Lorraine, and second author by ANR CISIFS and ANR GAOS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chambrion, T., Munnier, A. Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid. J Nonlinear Sci 21, 325–385 (2011). https://doi.org/10.1007/s00332-010-9084-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-010-9084-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation