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

Frederic Boyer, Mathieu Porez  and Alban Leroyer

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

 

JournalJournal of Nonlinear Science
PublisherSpringer New York
ISSN0938-8974 (Print) 1432-1467 (Online)
IssueVolume 20, Number 1 / February, 2010
DOI10.1007/s00332-009-9050-5
Pages47-79
Subject CollectionMathematics and Statistics