Wiggling Through the World: The mechanics of slithering locomotion

Wiggling Through the World: The mechanics of slithering locomotion depend on the

surroundings

Author(s): Daniel I. Goldman and David L. Hu

Source:

American Scientist

, July-August 2010, Vol. 98, No. 4 (July-August 2010), pp.

314-323

Published by: Sigma Xi, The Scientific Research Honor Society

Stable URL: https://www.jstor.org/stable/27859538

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Wiggling Through the World

The mechanics of slithering locomotion depend on the surroundings

Daniel I. Goldman and David L. Hu

Movement is critical for the surviv

al of animals. Eels swim, hawks

soar,' moles tunnel and squirrels leap

to perform vital tasks such as foraging,

mating and escaping pr?dation. These

feats of locomotion involve the coordi

nation of complex biophysical process

es that span scales from the tiny (ion

channels in nerve fibers that depolar

ize to send and receive information) to

the intermediate (muscles and tendons

coupling to skeletal elements to gener

ate motion of body parts) to the large

(interaction of body parts such as feet

and hands with their surroundings to

effectively generate traction). A goal of

locomotion science is to uncover gen

eral principles of movement through

the development of models across

sizes. This challenge requires the col

laboration of biologists, physicists, math

ematicians and engineers. Such animal

locomotion studies are also inspiring the

design of vehicles with mobility equal to

or greater than that of animals.

Many animals move effectively with

out the use of limbs. Some legged liz

Daniel I. Goldman is an assistant professor in the

School of Physics at the Georgia Institute of Tech

nology. He received his PhD. in physics from the

University of Texas at Austin in the Center for Non

linear Dynamics, studying fluidization of granular

media. He followed with postdoctoral training at the

University of California at Berkeley in the Depart

ment of Integrative Biology, studying biomechan

ics of climbing and running cockroaches. He has

been awarded a Burroughs Wellcome Fund Career

Award at the Scientific Interface and a Sigma Xi

Young Faculty Award. David L. Hu is an assistant

professor of mechanical engineering and biology at

the Georgia Institute of Technology. He earned his

doctorate in mathematics from the Massachusetts

Institute of Technology, then served as an NSF

postdoctoral fellow and an instructor at the Applied

Math Lab of New York University. Address for

Goldman: School of Physics, 837 State Street NW,

Georgia Institute of Technology, Atlanta, G A 30332

0430. E-mail: [email protected]

ards, for example, are known to forsake

their limbs entirely and wiggle their

bodies to move through dense grasses

or sandy environments. For other crea

tures, such as the thousands of species

of snakes, slugs and worms (and a few

lizards), legs were so superfluous that

they have been completely limbless

for millions of years. Their long, flex

ible bodies enable them* to enter tight

crevices and to traverse long distances

through complex and often tortuous

substrates such as the tops of trees, un

derneath the soil or inside the digestive

tracts of other organisms.

Our recent laboratory work and

models of terrestrial limbless locomo

tion have elucidated the mechanisms

that make undulatory locomotion effec

tive in two distinct environments: above

ground where snakes slither (Hu) and

within flowing substrates such as sand

where sandfish lizards "swim" (Gold

man). We hope to give a glimpse of

how such animals can move at speeds

of several body-lengths per second by

describing the movement of snakes and

sandfish in turn,' and drawing attention

to the specific adaptations these animals

use to enhance their performance in par

ticular habitats. Although the motion of

these snakes and sandfish may appear

similar, these animals propel themselves

using substrate interactions that are dis

tinct to their respective environments, as

we have determined with mathematical

and physical modeling.

Making the Model

The common features of our approaches

to mathematical modeling are derived

from applying what's called the resistive

force technique, developed to model the

locomotion of small organisms in fluids.

Consider a small rod-shaped "slice" of

an undulating reptile. Muscular forces

acting on the segment, combined with

the inertia of the body, generate equal

and opposite reaction forces from the an

imal's environment. If the animal moves

its body in particular ways, the sum of all

the reaction forces on the animal's body

can propel the animal's center of mass

forward. To simplify our model, we will

assume that our animals undulate in a

plane. Thus we can decompose the force

acting on the segment into two pieces,

one tangential and the other perpendicu

lar to the segment's surface. These are

called the axial and normal forces, respec

tively (see Figure 2). For an undulating

animal to move, the summation of the

forward components of the normal forc

es on the animal must exceed the back

ward components of the axial forces. To

determine if that is the case, we first must

calculate the reaction forces from the

animal's environment. Although math

ematically this approach seems straight

forward, it requires input of models of

the environment, and this is where the

new modeling challenge begins.

In fluid environments, such as those

typically encountered by swimming

spermatozoa, nematodes or sea snakes,

we can use what are called the Navier

Stokes equations to account for force pro

duced by flows. These equations, named

after physicists Claude-Louis Navier and

George Gabriel Stokes, apply Newton's

second law to fluid motion, and account

for pressure and viscosity, in order to

describe fluid movement. However,

these partial differential equations are

often impossible to solve analytically,

so there are many approximate models

(such as Stokes' Law for viscously dom

inated disturbances) that can be used to

rapidly calculate forces experienced by

swimmers and fliers.

In contrast, environmental models for

terrestrial locomotion can be more com

plex. Although dry friction can describe

interactions with spme surfaces, we lack

validated equations for many materials

such as mud and sand. For surfaces that

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Figure 1. Although their undulations look similar, these reptiles' movements are governed by different mechanics. A corn snake (Elaphe gut

tata) slithers across photoelastic gelatin, which transmits light where the snake applies the greatest force. Rather than pressing itself uniformly

along the ground, the snake lifts sections of its body while slithering to increase speed and efficiency. A high-speed x-ray image (inset) of a 10

centimeter-long sandfish lizard (Scincus scincus) reveals that to "swim" within sand, the animal does not use its limbs but propels itself using

only body undulation. (Images courtesy of the authors and Sarah Steinmetz, Mateo Garcia and Lionel London.)

can be approximated by Coulomb (or

dry) friction, resistance force is indepen

dent of speed, proportional to the ap

plied normal force and opposite to the di

rection of motion. The response of a dry,

granular material such as sand can be

like a solid or a fluid, depending on ap

plied stresses and the compaction of the

material. Moreover, the compaction can

change after a disturbance and thus drag

resistance in granular media depends on

the history of how it was perturbed.

Despite these differences, slithering

locomotion appears to work well in wa

ter, on flat land and, as shown by one

of our recent discoveries, even through

granular material. Undulatory locomo

tion in dry environments, both above

and below ground, has an important fea

ture that makes the mathematics more

tractable: The inertia of both the organ

ism and the surroundings are negligible

compared to frictional forces?thus to

stop moving forward, the animal simply

stops slithering. This is unlike a large

snake swimming in water?if it stops

undulating, it coasts for a distance re

lated to its initial speed.

Above-ground Mechanics

Snakes, and some snakelike lizards with

out legs, are a highly successful class of

terrestrial limbless creatures. They have

evolved to span three orders of magni

tude in length, from the centimeter-scale

threadsnakes to 10-meter-long anacon

das. All possess the same basic body de

sign: a flexible tube of flesh covered in

hardened scales. This form provides them

with tremendous versatility: They can

slither vertically up tree trunks, transition

from slithering on land to swimming in

water without changing gait, travel on

land using a similar amount of energy to

a legged organism of the same weight,

and some, such as the 2-meter-long black

mamba, sprint nearly as fast as a human

can run (5 meters per second).

Working in 2009 with Michael Shelley

of New York University, one of us (Hu)

. focused on the locomotion of juvenile

milk and corn snakes because of their

ability to slither in terrestrial habitats

such as prairies and rocky slopes. Like

all snakes, they are capable of several

gaits, or sequences of placements of their

limbless body on the ground. We investi

gated the most common of their limbless

gaits, slithering. This gait is also known

to biologists as lateral undulation, and its

utility to locomotion in snakes has been

previously described on the basis of so

called push points: Snakes slither by driv

ing their flanks laterally against neigh

boring rocks and branches found along

the ground. Thus, early experiments

involved snakes slithering through a

pegboard. A snake can generate forward

motion on the boards because its com

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Figure 2. In the resistive-force modeling technique, used to predict the speed of undulatory

motion, the body is partitioned into elements (red box) that move through the environment

by sending an undulatory wave pattern down the body from head to tail. Movement of the

elements results in reaction forces, which can be divided into a normal force that acts perpen

dicular to the element, and an axial force that is parallel. The sum of the forward normal forces

must exceed the backwards axial forces in order to propel the entire body forward.

bined muscular forces on the surround

ing pegs exceeds the sliding friction force

on its belly. However, snakes can also

slither easily on relatively featureless ter

rain, such as sand or bare rock, which

do not provide obvious push points. We

sought to understand this physical re

gime in our experiments, in part because

of the relative ease with which frictional

interactions could be incorporated into

our mathematical modeling.

Friction is a familiar force that resists

the sliding of one object atop another,

such as the soles of our shoes on the

ground. The force arises from the fact

that no surface is completely flat: Under a

microscope, all surfaces are covered with

tiny peaks and valleys, called asperities,

that snag and deform when two surfaces

are pressed together and slid. This resis

tance to sliding is described by a friction

coefficient, the ratio between the resultant

friction force between two surfaces and

the compressive force applied. For exam

ple, for a snake to slither (or equivalently,

in order to drag a sleeping snake along

the ground), the snake must apply a force

greater than the product of the friction

coefficient and the weight of the snake.

Although snakes' backs are covered

with diamond-shaped scales, their belly

scales are arranged like the overlapping

shingles on the roof of a house, which

snag on the ground when the snake is slid

sideways or tailward. This orientation

gives snakes a handy frictional property:

On certain substrates, their belly scales

have a preferred direction of sliding. By

putting snakes to sleep for a few minutes

(with low doses of anesthetic gas) and

straightening them out in various orien

tations (head down, head up and side

ways) on inclined planes, we measured

the snake's friction coefficient and its de

pendence on the orientation of the snake.

Friction measurements were performed

on cloth fabric whose characteristic length

scale of roughness (0.2 millimeters) was

comparable to the thickness of the snake's

belly scales (0.1 millimeters), enabling the

scales to snag in the cloth. Measurements

of milk snakes on this cloth indicate the

friction coefficient is lowest if the snake

slides forward (0.10), intermediate if slid

ing tailward (0.14) and highest towards

its flanks (0.20). This response is called

frictional anisotrop)) (a physical property

whose value depends upon the ctoection

in which it is measured), without which

the snake would be unable to move for

ward on flat ground.

The necessity of snake scales to loco

motion can be shown by dressing snakes

in an "isotropie jacket;" in our work, this

is a sleeve of fabric snug enough to cling

to the snake without impeding its breath

ing. Friction forces still resist the jacketed

snake's sliding on the ground, generating

forces in both the normal and axial direc

tions of the belly. However, the friction

force magnitude is now equal in every

direction. When the snake generates a

traveling wave, the forces on the snake's

belly sum to zero. Physically, this means

that the snake slithers in place, as if it

were on slippery ice. The same effect can

be achieved by placing snakes on very

sideways friction

0.16

0.12

0.08?

0 50 100 150

angle of snake (degrees)

smooth surface ?rough surface *v theory

Figure 3. Snakes are smooth and abrasion-resistant, enabling them to slide easily through their surroundings (a). However, on their undersides, scales

resemble overlapping shingles ) that can snag on ground protrusions when the snake slides towards the sides or backwards, enabling the snake's

friction to depend on its direction (c). The friction coefficient, which measures the resistance to sliding, is found by placing sleeping snakes on an

inclined plane covered with various materials (d). On smooth surfaces snakes slide easily in any direction, but on rough surfaces, the scales resist mo

tion, making the forward direction the preferred direction of sliding (e). (Unless otherwise noted, all photographs are courtesy of the authors.)

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Figure 4. To test the necessity of snake scales to slithering locomotion, a snake is fitted with an

"isotropie jacket" that prevents its scales from interacting with the ground. When the jacketed

snake tries to slither, it creates frictional propulsive forces whose magnitude is independent of the

direction of motion. Consequently, these forces sum to zero, causing the snake to slither in place.

smooth surfaces such as plastic. On such

substrates, snakes are unable to prog

ress forward unless they lift their bodies

while slithering or transition to another

gait (such as sidewinding, or concertina

mode, in which the snake folds itself like

the pleats of an accordion).

In our mathematical model, we used

the snake's scale properties to determine

the steady speed of the snake's center

of mass. The inputs to the model in

clude the belly friction coefficients and

characteristics of the snake's undulation

kinematics (frequency, wavelength and

amplitude). We assume that along the

snake's length, its weight is uniformly

applied to the ground. Figure 6 shows

our virtual snake juxtaposed by green

arrows denoting the magnitude and di

rection of the friction forces acting on

the snake's belly. The components, of the

arrows pointing in the snake's direction

of motion are responsible for its propul

sion; the remaining components indicate

directions in which the snake's energy is

wasted. Upon adding all the frictional

forces, we were surprised to find that the

speed of our virtual snake was only half

that of the snake we observed in the lab

(8 centimeters per second, or 0.2 body

lengths per second). Some part of snake

behavior clearly was not being captured

in our simplified mathematical model.

Previous investigators have observed

snakes altering their weight distribution

by lifting the peaks and troughs of their

undulating bodies, concentrating their

weight on the rentiaining points of contact

with the ground. This body lifting is most

clear in sidewinding, in which the snake

travels laterally and leaves a trail in sand

that resembles discrete "footsteps" rather

than a continuous winding path. Our

experiments with snakes on mirrored

surfaces and photoelastic gelatin (which

transmits light when compressed) indi

cate that snakes are also capable of lifting

their bellies while they slither forward.

In our model, we showed theoretically

that such dynamic load balancing leads

to increases in speed of 35 percent and in

efficiency of 50 percent. Why such a large

advantage? Figure 6 shows the directions

of propulsive forces (friction) everywhere

along the snake. The peaks and troughs

of the curves show propulsive-force ar

rows that point normal to the direction

of snake motion, the direction in which

energy is wasted. Because friction is pro

portional to the weight applied, the snake

generates more thrust if it lifts its body in

these regions and increases its weight

elsewhere. Thus, slithering shares certain

features with human walking. When we

walk, we transfer weight from our hind

foot to the leading foot by lifting the hind

foot rather than dragging it. Similarly, a

snake lifts the parts of its body that are

doing the least useful work. This adjust

ment is a simple change to their weight

distribution that snakes can perform. By

working to understand weight distri

butions further, we may one day know

how speedier snakes such as the black

mamba can move so quickly.

Underground Mechanics

Many desert organisms, including

snakes, moles, lizards and scorpions,

disappear into sand to avoid predators

and heat as well as to catch prey. The

sandfish, which one of us (Goldman)

studies, is a 10-centimeter-long desert

lizard with fringed toes on its four limbs,

a shovel-shaped snout and a flattened

belly and flanks?these features are hy

pothesized to aid it in burying itself and

sand-swimming. Like snakes, ite scales

are smooth and abrasion resistant. Un

like snakes, the belly scales on the sand

fish do not overlap. Although there have

been many hypotheses about how such

organisms move within a medium, un

til our work there had been almost no

detailed studies of kinematics, in part

Figure 5. A time-lapse illustration shows the differing results of a snake slithering across a rough

surface (top) versus a smooth surface (bottom). On both surfaces, the snake generates waves of

constant amplitude, wavelength and frequency. However, only on the rough surface do the fric

tional forces generated create forward motion. On the smooth surface, the snake slithers in place.

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2010 July-August 317

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Figure 6. A corn snake is placed on a mirrored surface (a) in order to see when there is a reflection of the belly, indicating that the snake has

lifted its body. Models use the snake's frictional properties and the kinematics of its body wave to calculate the frictional propulsive forces for

snakes with a uniform (b) and a nonuniform (c) weight distribution. The blue curves trace the snake's orientation, the orange dots show its cen

ter of mass, and green arrows indicate the direction and magnitude of the friction force applied by the ground to the snake as it moves from left

to right. Arrows pointing to the left are thrust forces; ones in other directions show energy wasted. Orange lines show sections of the body with

a normal force less than the weight of the section. Lifting increases the amount of forward force at the snake's inflection points (black dots).

because opaque sand makes subsurface

visualization challenging.

The sandfish uses its limbs to move

rapidly on the surface of the sand but

when startled, it points its snout down

and quickly disappears (within half a

second) beneath the sand. Once fully

submerged, the sandfish is quite chal

lenging to locate. The physics that gov

erns the propulsive forces in the granular

world into which the sandfish descends

are quite different from the frictional

forces that are important for slithering on

solid surfaces, mainly because granular

materials such as sand can yield (flow)

and solidify in response to perturbation.

Desert sand is typically dry and com

posed of roughly spherical particles from

0.1 to 0.3 millimeters in diameter that

only interact on contact through energy

dissipating forces such as viscoelastic

ity, plastic deformation and friction. De

pending on the applied stress, granular

media can display a range of physical

behaviors with features characteristic of

gases, fluids and solids. For example,

a pile of grains on a flat board behaves

like a yield-stress fluid: It acts like a solid

if the pile is not tilted too much, but at

sufficiently high angles of inclination

it undergoes a transition to a fluid that

flows downhill. There is not yet a funda

mental comprehension of the mechanics

of these materials at the level known for

fluids such as water and air.

The behavior of granular media is

sensitive to their conditions and prepa

ration: One of us (Goldman) has recently

investigated how a parameter called the

packing fraction?the ratio of the material

volume of grains to the occupied vol

ume?controls the material's response

to sustained perturbation (such as the

movement of an object through the

sand). Although the range of naturally

occurring disordered packings occupied

by approximately spherical dry grains

is small (58 percent at the loosest and 63

percent at the most tightly packed), these

different states behave quite differently:

A loosely packed collection of grains be

haves in a more fluidlike manner, flow

ing smoothly in response to disturbance,

whereas in a tightly packed collection of

grains the drag force nearly doubles and

the grains flow in a halting, abrupt man

ner when an object is dragged through

them. This is a consequence of the ability

of loose packings to flow by particles

pushing into free volume, whereas tight

packings flow as groups of particles cre

ate new volume by expansion (called di

lation). Unlike in fluids such as water, in

which force on a moving object increases

with rising velocities, in granular me

dia for low enough velocities, forces on

objects are approximately independent

of speed, because velocity-independent

frictional interactions dominate the par

ticle interactions in this regime.

Modeling such behavior is also a chal

lenge. Although there has been much

progress in describing the gaslike state

of shaken granular media, drag laws are

not available for flows in which fluid

and solid states coexist. A successful ap

proach to modeling these materials is

to instead apply a more "brute force"

approach?let a computer follow the

motion and interaction of millions of in

dividual grains subject to collision rules

and gravity. This procedure is called mo

lecular dynamics. Once validated, such

models can give insight into particle

level flows, Auctioning as a virtual mi

croscope into the medium.

A Quick Burial

It is fascinating to contemplate how a

sandfish moves within sand. Does the

animal use its limbs to paddle or does

it undulate like an eel, or both? Once

its head breaks up the material, does

it remain a fluid or does it solidify fast

enough that subsequent portions of the

body have to refluidize it? Do chang

es in material compaction (loosely to

tightly packed sand) change the behav

ior and movement pattern of the sand

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Figure 7. It is difficult to see the pattern of a snake adjusting its weight as it slithers, because the snake's body is often in contact with the ground. One

way to visualize regions of higher applied force is to film a snake slithering on gelatin, illuminated from below, between cross-polarizing filters (left).

The orientation of the filters initially blocks the polarized light from reaching the imager, but when the snake pushes on the gelatin, the light is ro

tated and can be detected by the camera, as shown by the luminous regions (middle and right). Although the adhesiveness of the gelatin can interfere

with the snake's natural locomotion, this technique suggests that less of the snake may be in contact with the ground than was previously believed.

fish? To investigate questions such as

these, one of our (Goldman's) doctoral

students, Ryan Maladen, uses a combi

nation of x-ray imaging techniques to

visualize subsurface motion.

To control the properties of the sand

encountered by the animal, we use a

custom-made fluidized bed, a device in

which a collection of grains placed into a

container with a porous bottom is driven

upward by a flow of air. Below a critical

flow rate, the grains remain in a solid

state, but above this threshold, the grains

take on the properties of a fluid. Once the

airflow is stopped, the grains settle into

a loosely packed state; subsequent per

turbations by either pulses of air or con

trolled vibrations to the bed can create

repeatable states of different compaction.

EHiring experiments, the sandfish is

placed in a holding pen connected to

the bed, then a gate on the pen is lifted.

Once the animal realizes it has access to

the sand, it immediately runs out of the

pen toward the material, with its back

straight and using its limbs for propul

sion. High-speed video reveals that to

bury itself the animal uses a combina

tion of its limbs and its body to push

itself into the sand. The burial time is

rapid and does not depend significantly

on the packing fraction of the sand.

Once below the surface of the sand,

direct visualization with high-speed

cameras become impossible. To image

movement within the material, we rely

on high-speed x-ray video. The sand

(and the lizard) are placed between an

x-ray source and a scintillating material

(called an image intensifier), which con

verts the x-ray photons into electrons,

Figure 8. The sandfish lizard, native to the deserts of north Africa, uses its shovel-shaped

snout, smooth skin and powerful body to undulate through sand at speeds of up to two body

lengths per second. The 10-centimeter-long reptile can bury itself in less than a second. On

the surface it uses its limbs for locomotion, but once buried, it holds its limbs at its sides and

moves solely by propagating a sinusoidal wave along its body.

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time (seconds)

Figure 9. Drag force on a small rod as a function of time demonstrates the difference in resistance

between loosely (blue) and tightly (red) packed granular media. The force in loosely packed par

ticles is constant after an initial transient increase when material flows around the rod. Tightly

packed material displays large oscillations in force as material in front of the rod periodically

yields and a block of solidified material is forced to the surface. The rod (inset) is 2 centimeters

long and is dragged at 1 centimeter per second; the particles are 0.3-millimeter glass beads.

which strike another surface that emits

visible light, which is then captured by a

conventional high-speed camera.

Subsurface Swimming

When we first observed the subsurface

movement of the sandfish, we were

amazed to see that it was moving for

ward at nearly two body lengths a sec

ond using a large-amplitude undulation

of its body, and without using its limbs?

traveling faster than milk snakes move

on flat ground. The sandfish "swam" to

a depth of about 4 centimeters and then

stopped undulating, presumably feeling

safe. The sandfish moved forward by

propagating a traveling sinusoidal wave

backward from head to tail at a particu

lar frequency?the greater the frequency,

the faster it went. In tightly packed me

dia, the animal used frequencies up to

four undulations per second.

There were several interesting features

of the sandfish locomotion: We expected

that, as we increased the packing fraction

to generate more resistive media, the ani

mal should slow down. Measurements

we made on small rods in granular media

showed that drag force in tightly packed

sand is nearly double that in loosely

packed sand. Surprisingly, we found that

volume fraction had no effect on speed

for a given wave frequency. Further, the

animal's movements were the same in

both loosely and tightly packed media; it

was impossible to determine simply by

looking at the x-ray images if the animal

was swimming in more or less resistive

materials. Stranger yet, we found that on

average, the sandfish swam faster in the

tightly packed material than in the loosely

packed material. It did this by increasing

undulation frequency to a higher range

(its maximum frequency was nearly dou

bled in tight versus loose packings).

In experiments and models, we use

a number called the wave efficiency to

characterize the locomotion of the sand

fish?it is commonly used to character

ize the locomotion of other swimmers

(such as millimeter-long nematode

worms) in deformable media. Wave ef

ficiency is defined as the average swim

rrting speed divided by the wave speed

(which is a product of the frequency and

the wavelength). If there is no move

ment of the material and no slip of the

animal, the wave efficiency is one and

the animal effectively moves in a tube. If

the animal were in a vacuum with noth

ing to push against, the wave efficiency

would be zero. In granular media, we

found that the sandfish swam with a

wave efficiency of 0.5, nearly twice that

of nematodes in fluids (about 0.2) and

greater than snakes on frictional surfac

es (about 0.3). Remarkably, the wave ef

ficiency of the sandfish did not depend

on the compaction of the sand.

Modeling the Sandfish

To explain some of these phenom

ena, a doctoral student in one of our

(Goldman's) groups, Yang Ding, first

Figure 10. The physics that governs interactions with granular media can be quite complex. A container of grains (6-millimeter plastic spheres)

remains solid until an aluminum ball, 5 centimeters in diameter (left), impacts the surface, at which point the grains become a fluid (middle).

Once the ball comes to rest, the grains can resolidify. A simulation technique called molecular dynamics visualizes the forces and velocities

on the grains (right). At this instant in time, dark blue particles are moving slowly and behaving as a near-solid, whereas redder particles sur

rounding the impactor are moving more rapidly. (Two images at left courtesy of Andrei Savu.)

320 American Scientist, Volume 98

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5 10

position (centimeters)

Figure 11. In this laboratory apparatus (left) used to visualize the movement of a burying and swimming sandfish, the animal is first held in

a holding pen as a flow of air from below the bed fluidizes the sand. Repeated pulses of air generate larger packing fractions, then the air is

turned off. A high-speed x-ray camera takes video of the sandfish (inset, top right) once the hatch to the holding pen is lifted. The video data

can be used to graph (right) the tracked position of the animal's back as it is running on the surface (white region of the graph), burying itself

(gray region) and swimming below the surface (tan region). The tracks show the sinusoidal traveling wave that propels the lizard forward. The

pattern of the tracks reveals a wave efficiency (the ratio of forward swimming speed to wave speed) of 0.5.

applied the resistive-force technique

(RFT) described above for snakes. We

hypothesized that this approach would

be applicable in sand because the x-ray

images indicated that the material near

the animal was flowing. Also, dissipa

tion in granular media is large and dis

turbances from different regions of the

sand do not influence each other so that

forces from separate elements (slices) of

the body can be linearly combined.

Unlike the fractional interaction that de

scribes snake locomotion, the appropriate

force laws for granular media were un

known. Ding and Chen Li, another doc

toral student in the group, developed em

pirical force laws by measuring the drag

components on a stainless-steel cylinder

(that approximated the skin friction of the

sandfish) as it was pulled through sand

at different angles relative to the displace

ment direction. The axial law resembled

that of a true fluid, but the normal force

law was "enhanced" relative to what we

would have predicted from a true fluid or

from pure Coulomb friction. However, as

we expected, both forces were indepen

dent of the magnitude of drag speed.

We hypothesized that the enhancement

of the normal force in granular media

was responsible for the increased wave

efficiency of the sandfish compared to

animals such as nematodes that use vis

0 1.2

time (seconds)

-f- ?n?

10 15

position (centimeters)

Figure 12. Computer simulation of a sandfish can capture the locomotion pattern of biological sand-swimming. In simulation, a container is

filled with about 150,000 spheres with a diameter of 3 millimeters, whose interaction properties are adjusted to match the 3-millimeter glass

beads used in experiments (top left). The tracks of the undulating simulated sandfish resemble those of an actual one (bottom left). The simu

lation estimates the forces along the body that propel the animal forward (top right). Green arrows indicate reaction forces measured from the

simulation. Also, the simulation allows visualization of grain mobility around the lizard (bottom right) and shows that the bulk of the material

is in a solid state (dark blue) whereas a small region around the sandfish is fluidlike (redder particles).

www.americanscientist.org

2010 July-August 321

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Figure 13. A robotic model of the sandfish swims within a model granular medium, 6-millimeter plastic beads (bottom left). The robot is composed of

seven segments, each with a servo-motor programmed to oscillate at a rate that will generate the appropriate sinusoidal traveling wave {top left). The

robot is covered with a fabric "skin." X-ray imaging shows the robot moving under the surface (middle top). A molecular-dynamics simulation that

incorporates similar geometry (middle bottom) swims like both the robot and the sandfish, and can increase speed with increasing wave frequency

(right). There is close agreement in subsurface-swimming speed versus frequency between the simulation and the robot experiment

cous-fluid forces or snakes that use fric

tion forces. Indeed, when the drag laws

were inserted into the RFT and integrated

using the animal's measured movements,

the model correctly predicted that speed

increased linearly with frequency. More

over, the predicted wave efficiency was

in a range of 0.4 to 0.7, independent of

volume fraction. This result bounded the

measured sandfish wave efficiency, and

its range was a consequence of not accu

rately knowing the drag on the sandfish's

shovel-shaped snout (we made predic

tions for minimal wave efficiency for a

flat-headed model and maximal efficien

cy for a model without a head).

Thus, the RFT demonstrated that the

animal could indeed "swim" within

sand without use of limbs. The medium

in which it was swimming demonstrates

fluidlike properties, although the flow

is better described as a frictional fluid,

as opposed to a viscous fluid. The RFT

model gave a plausible explanation for

the independence of wave efficiency

on packing fraction: As resistance force

increased in close-packed material, so

did the thrust forces that could be gener

ated. The enhanced normal force within

the granular media was responsible for

the high wave efficiency of the sandfish,

which was even greater than that of a

snake moving on a flat surface (without

lifting). That the sandfish wave efficiency

is higher than that of snakes is striking

given that the sandfish must burrow its

entire body through sand.

The RFT made an interesting predic

tion for optimal undulatory locomotion

within a granular medium: The sand

fish can increase its speed by increasing

its wave amplitude while mamtaining

a wave of approximately a fixed single

period (the number of undulations along

the body from head to tail). But since the

animal has a finite length, a larger am

plitude leads to less forward progress at

each cycle?the head moves closer to the

tail so that at the limit of a large ampli

tude the animal is basically moving per

pendicular to the direction of motion. The

RFT predicted a maximum speed at am

plitudes of about 0.2 of the wavelength.

Remarkably, we found that the sandfish

data clustered at this peak, mcUcating that

our animals were determined to flee as

rapidly as possible through the sand.

Virtual and Robot Sandfish

Although the agreement between the

biological data and the RFT model was

encouraging and the prediction of opti

mal swimming was tantalizing, the RFT

approach suffers from some drawbacks.

For one, it is challenging to change pa

rameters?for example, if we wish to

perform the analysis in beads of differ

ent size or with altered surface features

(such as friction), we must remeasure the

empirical force laws, a time-consuming

task. In addition, it was not clear that

certain assumptions in the RFT were val

id during sand-swimming (such as the

assumption that different segments did

not influence each other and the use of a

steady-state, constant-velocity drag force

for oscillating segments) so perhaps the

agreement was only fortuitous.

To rigorously test our ideas about op

timal sand-swimming and to improve

our modeling efforts, we took a second

approach using the molecular-dynam

ics techniques described previously. The

model for the interaction of the particles

incorporates contact elasticity, viscous

force during contact to model colli

sional energy loss (called the coefficient

of restitution) and tangential interac

tion (assumed to be Coulomb friction),

and can be calibrated by comparing the

simulation to measurements of forces in

experiments. Once these three param

eters are determined, we find that the

molecular dynamics model has good

predictive ability over a range of experi

mental conditions. For example, inter

action laws of grains measured at one

drag velocity or angle predict the forces

under all other conditions. In these ex

periments and simulations we used 3

millimeter glass beads instead of the

0.3-millimeter beads used in the previ

ous sandfish experiments. This switch

made simulation possible, because we

could reduce the number of particles in

our box by a factor of 1,000?simulations

ran in a matter of days on our desktop

computers instead of years. Experimen

tally, the sandfish swam in the larger

beads using the same wave efficiency as

in the 0.3-millimeter particles.

Once particle properties were known,

we created a virtual sandfish with move

ment patterns taken from our biologi

cal experiments and forces calculated

by molecular-dynamics simulation.

We found that the molecular-dynamics

322 American Scientist, Volume 98

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modeling was in good agreement with

the wave efficiency predicted by the RFT

calculations (as we? as measured in the

biological data). Unlike the empirical

RFT models, molecular dynamics allows

access to particle-level information?we

could visualize the highly damped flow

of particles around the sandfish and esti

mate accurately the force on different ele

ments. Additionally confirming our RFT

model, th? simulated sandfish moved

fastest when it used the kinematics pre

dicted by the RFT and taken from obser

vations of the animal. Further work with

the molecular^dynamics model will al

low us to carefully investigate the phys

ics that creates the scaling of thrust and

drag, which leads to the independence

of wave efficiency on packing fraction.

We wanted to give our models one

more test, so we decided to move back

into the physical world: We would use

the molecular-dynamics simulation to

design a physical model of the sand

fish, a robot, that would swim within

granular material and test the prin

ciples we had learned. For our tests,

we chose segments made of servo

motors popular with hobbyists, so

we were therefore constrained in size

by commercially available actuators.

We also decided to scale the size of the

granular particles to 6 millimeters, so

mat we would not have to simulate bil

lions of grains and small particles would

not get into the motors. Ryan Maladen,

in collaboration with mechanical engi

neer Paul Umbanhowar at Northwest

ern University, built a device that could

undulate with the same wave pattern

as that of the sandfish. The agreement

between robot experiment and robot

simulation was within five percent. We

found, just as predicted by the models,

that the robot swam fastest in the real

world (and in simulation) using the op

timal sandfish kinematics. The optimal

wave efficiency of the robot sandfish

was 0.3 in both experiment and simula

tion. We attributed this result to the finite

number of segments of the robot?in

the simulation, when we increased the

number of segments so that the body

was nearly smooth, the optimal wave

efficiency approached 0.5.

Two Worlds Without Legs

We have studied two very different envi

ronments in which undulatory locomo

tion is effective. In above-ground snakes,

anisotropie belly friction generates

thrust to overe?me drag, whereas un

derground, an animal's sides exploit the

frictional fluidlike properties of granular

media to generate thrust. Although the

drag laws differ between these regimes,

we found that resistive-force modeling

techniques, which originated in hydro

dynamics, can be successfully applied

to organisms on and within dry land.

Our experiments and modeling consid

ered mostly planar motion of the body,

but our findings of the advantages of

dynamic body-lifting in snakes suggests

that motion in the third dimension may

be used to increase performance. In the

sandfish, we must begin to explore 3D

effects as well: Our rays reveal that the

animal does not simply swim in a fixed

horizontal plane, but actually dives into

the material at a shallow angle.

Modeling the interaction between the

organism and its environment enables us

as physical scientists to work with biolo

gists to uncover new behaviors and the

relevant neuromechanics associated with

effectively using long, slender bodies to

move. The modeling approach does

not constrain us to sandfish or snake

morphology: Using our models of the

animals, we can vary body shape and

waveform to understand benefits and

tradeoffs of different locomotor modes in

diverse environments. Of particular inter

est is the importance of gait among limb

less animals. For example, does speed

or efficiency motivate limbless animals

to shift from slithering to sidewinding?

To that end, we plan to develop models

of the internal mechanics of me animals,

which will be useful in determining their

inherent metabolic and muscular limits.

To gain this broad understanding,

we must develop force laws for both

above and below ground that can ac

commodate a wider range of substrates.

For which materials is our approxima

tion of Coulomb friction a good'one?

What happens when an animal buries

into wetted material? Can we use simi

lar empirical forces laws, or if not, how

are they modified? With improved mod

els of environments and organisms, and

their interactions, our approach can helf>

find designs for future limbless robotic

devices that can move through complex

terrain faster and more efficiently than

their natural counterparts.

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http://wv^.arnericanscientist.orq/

igsye$/ia85/pa$ta$px

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