DOI: 10.5061/dryad.bk3j9kdp2
title: Code from: Competition between elasticity and adhesion in caterpillar locomotion

description: In recent years, there has been a growing interest in understanding
 animals locomotion mechanisms for developing bio-inspired micro- or
 nano-robots capable of overcoming obstacles and navigating in confined
 environments. Among non-pedal crawlers, caterpillars exhibit one of the
 most stable and efficient gait strategies, utilizing muscle contractions
 and substrate grip. Although several approaches have been proposed to
 model their locomotion, little is known about the competition between body
 elasticity and adhesion, which we demonstrate playing a central role in
 crawling gait. Preliminarily, experimental observations and measurements
 were performed on Pieris brassicae larvae, gaining insights into
 fundamental features characterizing caterpillar locomotion and estimating
 key geometrical and mechanical parameters. A minimal but effective
 one-dimensional discrete model was thus conceived to capture all the
 relevant aspects of the movement. Inter-mass springs model the deformable
 body units, Winkler-like constraints with an adhesion threshold reproduce
 elastic interactions and attaching/detaching events at prolegs-substrate
 interface, and a triggering muscle contraction initiates the larva's
 crawling cycle, generating the observed travelling wave. After
 demonstrating theoretically that caterpillars move obeying quasi-static
 laws, we proved robustness of the proposed approach by showing very good
 agreement between theoretical outcomes and experimental evidence, so
 paving the way for new optimization strategies in soft robotics.
description: # Code from: Competition between elasticity and adhesion in caterpillar
 locomotion The provided code is divided into four distinct parts, each
 addressing a specific aspect of the modeling process for tracing back the
 one-dimensional caterpillar locomotion. For the users' utility, two
 files are given: the Mathematica notebook and its pdf print. **Part 1:
 Pre-processing Phase** In this initial phase, data is collected and stored
 to trigger the locomotion model. This includes the necessary parameters
 and conditions required to simulate the movement. Following the notation
 adopted in the paper, the same symbols are considered in the Mathematica
 notebook for representing the subunits mass, the inter-masses springs and
 Winkler-like constraints stiffness, the resting length of the springs, the
 displacement threshold, the damping coefficient, the damping ratio, and
 the inelastic trigger. **Part 2: Calculation of Potential Terms** This
 section computes the elastic and kinetic potential terms that are
 fundamental for obtaining the system's equations. These terms are
 derived based on the mechanical properties and movement characteristics of
 the caterpillar. **Part 3: Derivation of Equations via Euler-Lagrange** In
 this step, the Euler-Lagrange equations are applied to derive the motion
 equations, using the potential terms calculated in Part 2. This allows for
 a detailed understanding of the system’s dynamics in terms of generalized
 coordinates. By turning off the inertial and dissipative terms, the
 elasto-static problem can be easily derived. **Part 4: Elasto-static and
 Dynamic Response Analysis** The final phase calculates both the
 elasto-static and dynamic responses of the system. The results are then
 compared in a post-processing step, where a plot is generated to visualize
 and analyze the differences between the two response types. For more
 details, please see the workflow explicitly reported in the published
 paper.