Robosnail 1 Snail-Inspired Fluid Locomotion Brian Chan, M.S Theresa Guo, undergraduate researcher Advisors: Anette Hosoi, Julio Guerrero (SLB) Hatsopolous.

Robosnail 1 Snail-Inspired Fluid Locomotion

Brian Chan, M.STheresa Guo, undergraduate researcherAdvisors: Anette Hosoi, Julio Guerrero (SLB)

Hatsopolous Microfluids LaboratoryDepartment of Mechanical EngineeringMassachusetts Institute of Technology

Schlumberger - Doll Research, SDR

Contents:

Snail locomotion Type 1 Robosnails

Theory Simulations

Robosnail 1A Design Experiment

Robosnail 1B Design Experiment

Robosnail 1C In progress

Conclusions

Robosnail 1A

Robosnail 1B

MotivationTo evaluate the feasibility of using

snail-like locomotion, and to optimize the performance of mechanical Robosnails.

Advantages of snail locomotion: can be configured as a versatile

flexible robot A sealed Robosnail mechanism

can be robust in muddy conditions

Effective locomotion for environments with little or no traction

Snail Locomotion basics

- All snails are separated from the substrate by a fluid layer (mucus)- Locomotive forces must be transferred through this layer to the

substrate- Snails, equipped with a single flexible foot must find a way to generate

fluid forces parallel to the substrate.

Classifying Snail locomotion:

Direct waves/Retrograde waves (Denny 1989)

Direct waves: waves of compression (used by most land snails)

Retrograde waves: waves of expansion (used by most aquatic snails) – also possible flapping motion …

Limax maximus (moving with direct waves)

Robosnail 1: Design using Retrograde Waves(Joint SLB/MIT U.S. Patent Pending)

Driving a flexible waving membrane with a motor

Governing physics:Analysis of thin fluid layers: Lubrication Theory

Assumptions: Height scale much smaller

than length scale pressure varies only in the x

direction Inertia effects negligible

The lubrication equation:

(slight modifications for non-Newtonian fluids)

Conservation of momentum

Robosnail 1: Theory - Physical mechanism

Using lubrication pressures for propulsion

Pressure under a sinusoidal flapping membrane (1 wavelength):

Immediately before the wave trough, fluid is being compressed (high pressure),

Behind the wave trough, fluid is being pulled apart (low pressure)

Pressure acting on sloped surface creates a propulsive force.

x – velocity profile

Volume flux Q:

Pressure p:

From conservation of momentum and mass we find velocity

Robosnail 1: 2D Theory

ws vh

zvhzz

dx

dpu

1

2

1 2

h

ws

h

zvh

zzv

hzz

dx

dpdzuQ

0

223

0 2232

1

hv

vh

dx

dpw

s

212

1 3

03322 *

112

*

1

2

12pdx

ha

QLdx

hv

v

a

Lxp w

s

ws vh

zvhzz

dx

dpu

1

2

1 2

Robosnail 1: 2D Theory

Steady- state horizontal force balance: horizontal component of pressure force and shear stress at membrane balances tractoring force

sxpxx FFF ,,

dxdy

dudx

dx

dhpF

h

LL

x 00

Robosnail 1: 2D Theory

We derive a simple linear tractoring force – velocity function, which resembles a motor torque-speed curve.

AI

F

A

AVs 3434

16

1

where

31

22

II

IA

1

0

1dx

hI

jj

Robosnail 1: Full 3D TheoryFor real Robosnails, we always experience side leakage, hence losses. By

analyzing a differential control volume, we can derive a 3D lubrication equation.

hph 123

Where p is pressure, h is height, and η is viscosity.

Robosnail 1: Full 3D TheoryDeriving a force-speed relationship

[A]: stalled robosnail [B] pure shearing force

In 3D, the force-velocity relationship is still linear

dxh

vFL

sB 0

1dxdz

dudx

dx

dhpF

L

hz

L

A

00

BAt FFF

Robosnail 1: 3D SimulationsNumerical solutions for the pressure show the losses due to leakage; we can

integrate pressure to solve for the tractoring force (maximum, stalled)

Robosnail 1: 3D SimulationsSinusoidal foot profile:

Comparing the tractoring force of 3D Robosnails to the ideal 2D case:

As we expect, the wider the foot, the closer it behaves like the 2D snail.

Robosnail 1A: Experiment

Robosnail 1A: Results: Free velocity

Fluid: glycerol

b/l = 0.6

Robosnail 1A: Results: Stall Force

Fluid: Silicone oil

b/l = 0.6

Strain gauge

Robosnail 1A: Results: Force-velocity

Fluid: Silicone oil

b/l = 0.6

By varying the payload m and the waving velocity vw, we measure different values of vs

Robosnail 1B: Apparatus

- Periodic foot design to eliminate entrance/exit anomalies.

- Replaceable tracks define the height profile.

Robosnail 1B: Apparatus

Core mechanism:

Various replaceable tracks:

Robosnail 1B: Experiment

Robosnail 1B: Results: Free Velocity (Sinusoidal foot)

RS-1B performs better than the 3d solution (due to partial sealing effect of the tank walls) but understandably still not as well as the 2D solution.

Robosnail 1C: Faster-than-wave locomotion

Snail speed is a function of wave shape. For some non-sinusoidal wave shapes we can predict Vs/Vw >1.

That is, a Robosnail that moves faster than it ‘steps’!

We replace the sinusoidal foot with a foot composed of two parabolas, varying the size ratios:

Conclusions

- Lubrication theory predicts a linear force and velocity relationship for both 2D and 3D Robosnails.

- Analytic solutions exist for the 2D case for any given wave height function.

- Numerical simulations give a similar linear force-velocity relation for 3D snails, but with losses dependent on the ratio of snail width to length.

- We have experimental data for sinusoidal wave Robosnails that confirms the numerical results.

- In theory, certain wave shapes exhibit regimes where the snail speed is faster than the wave speed; future experiments will test this theory.

Appendix: 2D theory (detail)

To more easily analyze the fluid flow in the lubrication layer we switch to a reference frame following the waves.

In the new reference frame, Q = constant.

h

ws

h

zvh

zzv

hzz

dx

dpdzuQ

0

223

0 2232

1

hvv

hdx

dpw

s

212

1 3

03322 *

112

*

1

2

12pdx

ha

QLdx

hv

v

a

Lxp w

s

ws vh

zvhzz

dx

dpu

1

2

1 2

Appendix: Dimensionless variables

Experimental constants:L wavelengthμ viscosityh0 average fluid thicknessvw waving velocity

Dimensional variables:x* x-positionb* half-width of footp* pressureh* heighta* foot amplitudevs* snail velocityFx* tractoring force

Dimensionless variablesx = x*/Lb = b*/L

p = *

*

Lbv

ph

w

h = h*/h0

a = a*/h0

vs= vs*/ vw

Fx=w

x

v

hF

**

Acknowledgements

National Science Foundation

Schlumberger Limited

References

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