# Limbless undulatory propulsion on land

Original Entry by Michelle Borkin, AP225 Fall 2009

## Contents

## Overview

Limbless undulatory propulsion on land.

Z. Guo and L. Mahadevan, Proceedings of the National Academy of Sciences (USA), 105, 3179, 2008.

## Keywords

undulatory locomotion, Viscoelastic, Voigt model

## Summary

This paper presents analytical and numerical models of lateral undulatory motion commonly employed by limbless creatures such as snakes to "swim" on land. The critter propagates undulatory waves along its body via muscle contractions. To model this system and determine the important factors effecting this motion, a mathematical model is presented: a nonlinear boundary value problem which takes into account the viscous and elastic features in the muscle tissue and the frictional forces exerted by the environment. In summary, they find that the normalized shape of the animal is determined by its interaction with its environment, and the speed at which it moves is determined by the muscle contraction generated wave's frequency and amplitude. In addition to be of interest to biologists, this active area of research is of key interest to robotics where designing robots to deal with extreme terrains is important. (Dealing with movement across sand or other fluid-like surfaces is important for things ranging from terrestrial robots on beaches or desserts to robots on Mars.)

## Soft Matter

The lateral undulatory motion works by propagating a wave down the creature's body. As show in **Figure 1**, the lateral (side-ways) forces brace the body (the motion in each lateral direction cancels itself out) against the substrate thus allowing the in-plane forces to propel it in a forward (or backward) direction. In part **(a)**, the thickness of the black line illustrates the muscle contractions and black dots the points of inflection. This paper approximates this kind of movement on a solid substrate and takes into account frictional forces.

The key to understanding this movement is in the tissue's response and resistance to deformation. They model this by the linear Voigt model for viscoelasticity:

<math>\sigma = E \epsilon + \eta \dot{\epsilon}</math>

where <math>\sigma</math> is the uniaxial stress in the bulk tissue, <math>E</math> is Young's modulus, <math>\eta</math> is viscosity of the tissue, <math>\epsilon</math> is the strain and <math>\dot{\epsilon}</math> dot is the strain rate. (This is schematically drawn in part **(c)** of the figure.) Thus the passive moment of the tissue is the sum of the elastic and viscous moments of the tissue (<math>M_{p} = M_{e} + M_{\nu}</math>). (This is reflected in the alternating muscle contractions shown in part **(a)**.) Finally, the force per unit length <math>p</math> can be described by the situation where the snake is pushing against an array of pegs (see schematic), is proportional to the active moment, or determined by the local shape of the snake.

Based on the above discussion, the non-dimensionalized set of equations describing the system are:

where <math>T</math> is the tension, <math>N</math> is the transverse shear force, <math>s</math> is the coordinate in the traveling wave frame, <math>\kappa</math> is the curvature of the centerline, <math>\nu_{w}</math> and <math>\nu_{p}</math> are the longitudinal and lateral coefficients of friction, Mo is the dimensionless amplitude of active moment, Pr is the dimensionless lateral resistive force, Be is the dimensionless passive elastic bending stiffness of the organism, and Vi is the dimensionless passive viscous bending stiffness of the organism. These factors characterize the exogenous (<math>\nu_{w}</math>, <math>\nu_{p}</math>, Pr) and the endogenous (Mo, Be, Vi) dynamics in the system. Based on further mathematical analysis of these equations, they find that: the normalized shape for steady motion depends only on exogenous factors (frictional and resistive forces from substrate) and the wave amplitude is small for non-deformable surface and large for deformable surfaces, increasing the viscous bending stiffness of the animal reduces the mechanical efficiency, and that there is an optimal substrate to achieve maximal velocities using this lateral undulatory motion.