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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

76Z10 

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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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