Journal of Nonlinear Science

, Volume 20, Issue 1, pp 47–79 | Cite as

Poincaré–Cosserat Equations for the Lighthill Three-dimensional Large Amplitude Elongated Body Theory: Application to Robotics

  • Frederic BoyerEmail author
  • Mathieu Porez
  • Alban Leroyer


In this article, we describe a dynamic model of the three-dimensional eel swimming. This model is analytical and suited to the online control of eel-like robots. The proposed solution is based on the Large Amplitude Elongated Body Theory of Lighthill and a framework recently presented in Boyer et al. (IEEE Trans. Robot. 22:763–775, 2006) for the dynamic modeling of hyper-redundant robots. This framework was named “macro-continuous” since, at this macroscopic scale, the robot (or the animal) is considered as a Cosserat beam internally (and continuously) actuated. This article introduces new results in two directions. Firstly, it extends the Lighthill theory to the case of a self-propelled body swimming in three dimensions, while including a model of the internal control torque. Secondly, this generalization of the Lighthill model is achieved due to a new set of equations, which are also derived in this article. These equations generalize the Poincaré equations of a Cosserat beam to an open system containing a fluid stratified around the slender beam.


Swimming dynamics Eel-like robots Hyper-redundant locomotion Lie groups Lagrangian reduction Poincaré–Cosserat equations 

Mathematics Subject Classification (2000)



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  1. Alamir, M., El-Rafei, M., Hafidi, G., Marchand, N., Porez, M., Boyer, F.: Feedback design for 3-D movement of an eel-like robot. In: Proceedings of IEEE Inter. Conf. on Robotics and Automation, Rome, Italy, pp. 256–261 (2007) Google Scholar
  2. Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications è l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier XVI, 319–361 (1966) CrossRefGoogle Scholar
  3. Arnold, V.I.: Mathematical Methods in Classical Mechanics. Springer, Berlin (1988) Google Scholar
  4. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967) zbMATHGoogle Scholar
  5. Boyer, F., Primault, D.: Finite element of slender beams in finite transformations—a geometrically exact approach. Int. J. Numer. Methods Eng. 59, 669–702 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  6. Boyer, F., Primault, D.: The Poincaré–Chetayev equations and flexible multibody systems. J. Appl. Math. Mech. 69, 925–942 (2005) MathSciNetCrossRefGoogle Scholar
  7. Boyer, F., Porez, M., Khalil, W.: Macro-continuous computed torque algorithm for the three-dimensional eel-like robot. IEEE Trans. Robot. 22, 763–775 (2006) CrossRefGoogle Scholar
  8. Boyer, F., Porez, M., Leroyer, A., Visonneau, M.: Fast dynamics of an eel-like robot—comparisons with Navier–Stokes simulations. IEEE Trans. Robot. 24, 1274–1288 (2008) CrossRefGoogle Scholar
  9. Breder, C.M.: The locomotion of fishes. Zoologica, N.Y. 4, 159–256 (1926) Google Scholar
  10. Carling, J., Williams, T.L., 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, 3243–3166 (1998) Google Scholar
  11. Cheng, J.-Y., Chahine, G.L.: Computational hydrodynamics of animal swimming: boundary element method and three-dimensional vortex wake structure. Comput. Biochem. Physiol. 131, 51–60 (2001) CrossRefGoogle Scholar
  12. Colgate, J.E., Lynch, K.M.: Mechanics and control of swimming: a review. IEEE J. Ocean. Eng. 29, 660–673 (2004) CrossRefGoogle Scholar
  13. del Valle, G., Campos, I., Jiménez, J.L.: A Hamilton–Jacobi approach to the rocket problem. Eur. J. Phys. 17, 253–257 (1996) CrossRefGoogle Scholar
  14. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  15. Farnell, D.J.J., David, T., Barton, D.C.: Numerical model of self-propulsion in a fluid. J. R. Soc. Interface 2, 79–88 (2005) CrossRefGoogle Scholar
  16. Featherstone, R.: The calculation of robot dynamics using articulated-body inertias. Int. J. Robot. Res. 2, 13–30 (1983) CrossRefGoogle Scholar
  17. Featherstone, R., Orin, D.: Robot dynamics: equation and algorithms. In: Proceedings of IEEE Int. Conf. on Robotics and Automation, San Francisco, USA, vol. 1, pp. 826–834 (2000) Google Scholar
  18. Gray, J.: Studies in animal locomotion, I: the movement of fish with special reference to the eel. J. Exp. Biol. 10, 88–104 (1933) Google Scholar
  19. Gray, J.: Studies in animal locomotion, VI: the propulsive powers of the dolphin. J. Exp. Biol. 13, 192–199 (1936) Google Scholar
  20. Hill, S.J.: Large amplitude fish swimming. PhD thesis, University of Leeds (1998) Google Scholar
  21. Hoerner, S.F.: Fluid Dynamics Drag. Hoerner Fluid Dynamics (1965) Google Scholar
  22. Ijspeert, A.J., Crespi, A., Cabelguen, J.-M.: Simulation and robotics studies of salamander locomotion. Applying neurobiological principles to the control of locomotion in robots. Neuroinformatics 5, 171–196 (2005) CrossRefGoogle Scholar
  23. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.: Locomotion of articulated bodies in a perfect planar fluid. J. Nonlinear Sci. 15(4), 255–289 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  24. Kelly, S.D., Murray, R.M.: Modeling efficient pisciform swimming for control. Int. J. Robust Nonlinear Control 10, 217–241 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  25. Khalil, W., Dombre, E.: Modeling, Identification and Control of Robots. Penton Science, London (2002) Google Scholar
  26. Lamb, H.: Hydrodynamics. Dover, New York (1932) zbMATHGoogle Scholar
  27. Leroyer, A., Visonneau, M.: Numerical methods for RANSE simulations of a self-propelled fish-like body. J. Fluids Struct. 20, 975–991 (2005) CrossRefGoogle Scholar
  28. Lighthill, J.: Note on the swimming of slender fish. J. Fluid Mech. 9, 305–307 (1960) MathSciNetCrossRefGoogle Scholar
  29. Lighthill, J.: Aquatic animal propulsion of high hydro-mechanical efficiency. J. Fluid Mech. 44, 265–301 (1970) zbMATHCrossRefGoogle Scholar
  30. Lighthill, J.: Large-amplitude elongated body theory of fish locomotion. Proc. R. Soc. 179, 125–138 (1971) CrossRefGoogle Scholar
  31. Lighthill, J.: Mathematical Biofluiddynamics. SIAM, Philadelphia (1975) zbMATHCrossRefGoogle Scholar
  32. Liu, Q., Kawachi, K.: A numerical study of undulatory swimming. J. Comput. Phys. 155, 223–247 (1999) zbMATHCrossRefGoogle Scholar
  33. Luh, J.Y.S., Walker, M.W., Paul, R.C.P.: On-line computational scheme for mechanical manipulator. J. Dyn. Syst. Meas. Control 102, 69–76 (1980) MathSciNetCrossRefGoogle Scholar
  34. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, Berlin (1999) zbMATHGoogle Scholar
  35. McIsaac, K.A., Ostrowski, J.P.: A geometric approach to anguilliform locomotion modelling of an underwater eel robot. In: Proceedings of IEEE Int. Conf. on Robotics and Automation, Detroit, USA, vol. 4, pp. 2843–2848 (1999) Google Scholar
  36. McMillen, T., Holmes, P.: An elastic rod model for anguilliform swimming. J. Math. Biol. 53, 843–886 (2006) MathSciNetCrossRefGoogle Scholar
  37. Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970) Google Scholar
  38. Melli, J.B., Rowley, C.W., Rufat, D.S.: Motion planning for an articulated body in a perfect fluid. SIAM J. Appl. Dyn. Syst. 5, 650–669 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  39. Munk, M.M.: The aerodynamic forces on airship hulls. National Advisory Committee for Aeronautics, p. 184 (1924) Google Scholar
  40. Ostrowski, J.P.: Computing reduced equations for robotic systems with constraints and symmetries. IEEE Trans. Robot. Autom. 15, 111–123 (1999) MathSciNetCrossRefGoogle Scholar
  41. Poincaré, H.: Sur une forme nouvelle des équations de la mécanique. C. R. Acad. Sci. Paris 132, 369–371 (1901) zbMATHGoogle Scholar
  42. Reissner, E.: On a one-dimensional large displacement finite-strain theory. Stud. Appl. Math. 52, 87–95 (1973) zbMATHGoogle Scholar
  43. Ringuette, M.J.: Vortex formation and drag on low aspect ratio, normal flat plates. PhD thesis, California Institute of Technology, Pasadena, CA (2004) Google Scholar
  44. Simo, J.C.: A finite strain beam formulation: the three-dimensional dynamic problem, Part I: formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72, 267–304 (1985) MathSciNetCrossRefGoogle Scholar
  45. Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model, part II: computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986) zbMATHCrossRefGoogle Scholar
  46. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions. A geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66, 125–161 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  47. Susbielles, G., Bratu, C.: Vagues et Ouvrages Pétroliers en Mer. Editions Technip, Paris (1981) Google Scholar
  48. Taylor, G.I.: Analysis of the swimming of long narrow animals. Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 214, 158–183 (1952) zbMATHCrossRefGoogle Scholar
  49. Triantafyllou, M.S., Triantafyllou, G.S.: An efficient swimming machine. Sci. Am. 272, 64–70 (1995) CrossRefGoogle Scholar
  50. Wolfgang, M.J.: Hydrodynamics of flexible-body swimming motions. PhD thesis, Massachusetts Institute of Technology (1999) Google Scholar
  51. Wu, T.Y.-T.: Swimming of a waving plate. J. Fluid Mech. 10, 321–355 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  52. Yamada, H., Chigisaki, S., Mori, M., Takita, K., Ogami, K., Hirose, S.: Development of amphibious snake-like robot. ACM-R5. In: Proceedings of 36th Int. Symposium on Robotics, Japan (2005) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.EMNIRCCyNNantes Cedex 3France
  2. 2.EPFL-BIRG, INN 241LausanneSwitzerland
  3. 3.ECNLMFNantes Cedex 3France

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