Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, Copepod nauplii and Copepod robot

Abstract : This article deals with relation between sub-Riemannian geometry, Hamiltonian dynamics and micro-swimmers. Sub-Riemannian geometry is developed in the framework of geometric optimal control and following the Maximum Principle, the geodesics are solutions of an Hamiltonian differential equation. The optimal problem involves the calculation of the conjugate and cut loci to obtain the sub-Riemannian sphere, combining formal and numerical methods. Adapted software provide a numerical evaluation of these objects and as a consequence of sub-Riemannian geometry, the injectivity radius is zero and homotopy methods allows us to evaluate the sub-Riemannian spheres by performing a continuation on the radius. We present an application of the various techniques to revisit the analysis of micro-swimmers, based on the understanding of the swimming mechanism of an abundant variety of zooplanktons called copepods. Using this non academic example, a combination of geometric and numerical calculations on the geodesics equations determines the optimal solution, i.e. the most efficient stroke using continuation techniques initialized by strokes with small amplitudes. The model of swimming at low Reynolds number where inertia is neglected is validated by observations performed by Takagi's team at Hawaii laboratory, showing the agreement between the predicted and observed displacements. The framework of SR-geometry, where the mechanical energy is the universal criterion which allows to compare the efficiency of different strokes and different swimmers, leads to the challenge of constructing a robotic copepod at macroscopic scale. This is the the final step currently developed in Takagi's laboratory to compare the agreement between control computations, numerical simulations and experiments.
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Pré-publication, Document de travail
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Contributeur : Jérémy Rouot <>
Soumis le : mardi 2 janvier 2018 - 13:33:19
Dernière modification le : jeudi 23 août 2018 - 12:34:02


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  • HAL Id : hal-01653901, version 2


Bernard Bonnard, Monique Chyba, Jérémy Rouot, Daisuke Takagi. Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, Copepod nauplii and Copepod robot. 2018. 〈hal-01653901v2〉



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