Model of metameric locomotion in smooth active directional filaments with curvature fluctuations

¹ Wenzhou Institute, University of Chinese Academy of Sciences - Wenzhou, Zhejiang 325001, China
² School of Physical Sciences, University of Chinese Academy of Sciences - Beijing 100049, China
³ School of Physical Sciences and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences Beijing 100049, China
⁴ Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences - Beijing 100190, China
⁵ Oujiang Laboratory - Wenzhou, Zhejiang 325000, China

^(a) duguangle@ucas.ac.cn (corresponding author)
^(b) rudolfpodgornik@ucas.ac.cn

Received: 17 June 2021
Accepted: 17 November 2021

Abstract

Locomotion in segmented animals, such as annelids and myriapods (centipedes and millipedes), is generated by a coordinated movement known as metameric locomotion, which can be also implemented in robots designed to perform specific tasks. We introduce a theoretical model, based on an active directional motion of the head segment and a passive trailing of the rest of the body segments, in order to formalize and study the metameric locomotion. The model is specifically formulated as a steered Ornstein-Uhlenbeck curvature process, preserving the continuity of the curvature along the whole body filament, and thus supersedes the simple active Brownian model, which would be inapplicable in this case. We obtain the probability density by analytically solving the Fokker-Planck equation pertinent to the model. We also calculate explicitly the correlators, such as the mean-square orientational fluctuations, the orientational correlation function and the mean-square separation between the head and tail segments, both analytically either via the Fokker-Planck equation or directly by either solving analytically or implementing it numerically from the Langevin equations. The analytical and numerical results coincide. Our theoretical model can help understand the locomotion of metameric animals and instruct the design of metameric robots.