Odd elasticity in a three-link microswimmer: feedback equivalence, global controllability, and the cost of non-reciprocity
Abstract
We study a Purcell three-link microswimmer whose joints are \emph{odd-elastic}: the torsional stiffness is non-Hermitian, its antisymmetric part ko injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost --- odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar detK=k2+ko2. From this we deduce that the abnormal extremals of the energy problem are unchanged by ko, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan (2,3,5) sub-Riemannian structure, deformed only by the metric scaling gχ=(1+χ2)g0. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of ko. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.
Related
BibTeX
@misc{attanasi2026elasticity,
title = {{Odd elasticity in a three-link microswimmer: feedback equivalence, global controllability, and the cost of non-reciprocity}},
author = {Rossella Attanasi and Gaetano Napoli and Marta Zoppello},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02777},
eprint = {2608.02777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02777},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Odd elasticity in a three-link microswimmer: feedback equivalence, global controllability, and the cost of non-reciprocity
AU - Rossella Attanasi
AU - Gaetano Napoli
AU - Marta Zoppello
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2608.02777
UR - https://arxiv.org/abs/2608.02777
AB - We study a Purcell three-link microswimmer whose joints are \emph{odd-elastic}: the torsional stiffness is non-Hermitian, its antisymmetric part k_o injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost --- odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar detK=k^2+k_o^2. From this we deduce that the abnormal extremals of the energy problem are unchanged by k_o, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan (2,3,5) sub-Riemannian structure, deformed only by the metric scaling g_χ=(1+χ^2)g_0. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of k_o. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.
ER -