12/20/2013

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A Terradynamics for Legged Locomotion

on Granular Media

Tingnan Zhang*, Chen Li*†, and Daniel I. Goldman* *School of Physics, Georgia Institute of Technology

†University of California at Berkeley

Li, Zhang, Goldman, Science (2013)

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mud

sand debris

Martian soil

Many natural, particulate media can flow under stress

JPL

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JPL

Flowing substrates are challenging to move on

Difficult to gain purchase without slipping for wheeled and

tracked vehicles alike

RHex on dirt/mud

Car on sand Tank on soil Rover on Martian soil

Kod*lab

Kumagai (2004), IEEE Spectrum

Slowed 50

Lizard vs. snake by BBC Ghost crab

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X-ray video, slowed 50×

Complicated morphology + kinematics

Li, Hsieh, and Goldman, J. Exp. Biol. (2012)

Li, Umbanhowar, Komsuoglu, Koditschek,

and Goldman, PNAS (2009)

SandBot (RHex-class)

Zebra-tailed lizard

5 cm

Slowed 10× 10 cm

Challenges: Limb-ground interaction is complex

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Dickinson et al. (2000), Science

Vogel (1996), Life in moving fluids In fluids, Navier-Stokes

equations + moving boundary

conditions

Flying

Swimming

Comprehensive force models are lacking for general particulate media

Challenges: No comprehensive force models

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Is terramechanics applicable?

M. G. Bekker (1960), Off-the-road locomotion, research and

development in terramechanics

J. Y. Wong (2010), Terramechanics and off-road vehicle engineering

Terramechanics for

legged locomotion

Classical terramechanics can accurately and

quickly predict forces and performance for

(large) wheeled and tracked vehicles

?

penetrometer bevameter

Based on penetration resistance, pressure-

sinkage, and shear resistance tests, not

developed for legged locomotion

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elasticity dissipation

friction

Discrete Element Method

Zhang, Qian, Li, Masarati, Hoover, Birkmeyer, Pullin, Fearing, and Goldman, Intl. J. Robotic. Res. (2013)

Maladen, Ding, Umbanhowar, Kamor, and

Goldman, J. Roy. Soc. Interface (2011)

Cons: Slow, impractical for large scales

Pros: Accurate

dynaRoACH (10 cm, 20 g) on 3 mm glass particles

multi-body dynamic simulation coupled to DEM

(One simulation could take a few days)

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Hypothesis: Linear superposition of independent

element forces predicts net forces

Vertical plane

• Inspired by resistive force theory for low Re number swimmers

• Valid in non-inertial regime (negligible particle inertia)

• Works for sand-swimming in horizontal plane Lauga & Powers, Rev. Prog. Phys. (2009)

Maladen, Ding, Li, Goldman, Science (2009)

Continuum model approach?

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Measuring stresses using a plate element

Fully immersed

and far from

bottom

Extraction

Flu

idiz

atio

n

z (cm)

Total force

~ 1 mm

poppy seeds

(above surface)

(below surface)

Stresses are hydrostatic-like

– Video taken at boundary for

illustration

– Force measured in the bulk

– v = 0.01 m/s

– Video played 10 faster

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Stresses per unit depth vs. orientation, movement

direction Vertical Horizontal

Black curves: z,x = 0

Complex dependence

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leg is divided into 30 segments

Fz experiment

Fx experiment

(rad) (rad)

Fz model

Fx model

Net force

Segmental force

(on a larger scale)

– Video taken at boundary for

illustration

– Force measured in the bulk

– v = 0.01 m/s

– Video played 10 faster

~ 1 mm

poppy seeds

Net forces on c-leg: Experiment vs. model

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leg is divided into 30 segments

Fz experiment

Fx experiment

(rad) (rad)

F (N) F (N)

Fz model

Fx model

Net force

Segmental force

(on a larger scale)

– Video taken at boundary for

illustration

– Force measured in the bulk

– v = 0.01 m/s

– Video played 10 faster

~ 1 mm

poppy seeds

Net forces on c-leg: Experiment vs. model

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Stress profiles and model accuracy are generic

loosely packed

closely packed

loosely packed

closely packed

closely packed

Poppy seeds

0.3 mm glass particles

3 mm glass particles

(Photo credit: Sarah Sharpe)

Applicability to granular media of various

particle size, density, friction, and compaction

Generic stress profiles

Single measurement

with an off-the-shelf

penetrometer

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Application on natural sands

Yuma sand

Palm sand

Experimental measurement

Prediction using generic profile

Yuma sand under

microscope

0.06-3mm

z (cm)

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Using resistive force model to predict legged locomotion

– Each body plate and leg is divided into

30 elements

– Total force F and torque are

calculated using resistive force model

– Body movement is calculated by:

Ghiringhelli et al., Nonlinear Dynamics (1999)

– Legs of similar friction to plate element

– Leg speeds < 0.6 m/s (non-inertial regime)

– Motion mostly confined in the vertical plane

Multibody dynamic simulator (MBDyn)

10 cm

Xplorer (150g)

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c-leg

Robot moving on granular media using c-legs

Experiment

Simulation

f = 2.0 Hz, slowed 5

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Terradynamics is accurate and efficient

Much faster than DEM

e.g. 10 seconds vs. 30 days for 1 second of

locomotion on a bed of 5,000,000 poppy seeds

(~106 times speed-up)

Predicts speed Predicts ground reaction forces

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• In collaboration with Dr. Karl

Iagnemma’s group at MIT.

• Experiments performed by Carmine

Senatore from MIT and Mark

Kingsbury from Crab lab.

RFT wheel test (in collaboration with MIT)

Photo by carmine

Vertical bearing

Horizontal bearing

Force spring

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MER wheel on fluidized bed

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Wheels and testing conditions McMaster

Small

McMaster

Large

3D Printed MIT Smooth

Diameter [mm] 152.4 203.2

145 (to lug tips) 260

Width [mm] 44.5 50.8 76.2 160

Fz Tested [N] 7 20 10, 18 60, 120

Terrain State Tested

(Poppy seeds)

Loose Loose and

Compact

Loose and compact

(only for 18 N)

Loose and

Compact

McMaster

Small

McMaster

Large

3D Printed MIT Smooth (approx. to scale)

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-0.5 0 0.5

-5

0

5

10

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

Drawbar vs. slip ratio in experiment and model

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3D Printed

Fz = 18 N

Compact

-0.5 0 0.5

-5

0

5

10

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.2

0

0.2

0.4

0.6

0.8

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

15

20

25

Slip

Sin

ka

ge

[m

m]

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6-10

-8

-6

-4

-2

0

2

4

6

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.2

0

0.2

0.4

0.6

0.8

1

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.65

10

15

20

25

30

35

Slip

Sin

ka

ge

[m

m]

McMaster

Large

Fz = 20 N

Compact

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1. Developed a resistive force model in the vertical plane for legged

locomotion on granular media (for slow intrusions)

2. Resistive force model predicts forces (without any fitting

parameters) on intruders of complex morphology and kinematics

3. Resistive force model + multi-body simulation predicts legged robot

performance

4. RFT is able to predict wheel performance under a wide range of

conditions.

Summary

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Acknowledgements:

Yang Ding, Nick Gravish, Paul Umbanhowar, Gareth Meirion-Griffith, and Hal Komsuoglu for

discussion. Jeff Shen for robot modification. Pierangelo Masarati for MBDyn support. Sarah

Sharpe for taking the photos of granular materials. Paul Umbanhowar and Hamid Marvi for

assistance with natural sand collection.

Funded by: Burrough’s Wellcome Fund, ARL MAST CTA, ARO, NSF PoLS and Miller Research

Fellowship (C.L.).

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A convenient model flowing substrate for locomotion studies:

representative, relevant, relatively simple, controllable

~ 1 mm

poppy seeds

Air flow

Jackson (2000),

The Dynamics of

Fluidized Particles

Air flow

A fluidized bed

prepares repeatable

packing states

Starting point: level, uniform, dry granular media

Granular media (e.g., sand and gravel):

collections of discrete particles that interact

through dissipative, repulsive contact forces

1 cm

Nedderman (1992), Statics and Kinematics of Granular Materials

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McMaster

Small

Fz = 7 N

Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-4

-3

-2

-1

0

1

2

3

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

15

20

25

30

Slip

Sin

ka

ge

[m

m]

-0.5 0 0.5

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Slip

To

rqu

e [N

m]

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-10

-5

0

5

Slip

Dra

wb

ar

[N]

Experiment

WR

RFTMcMaster

Large

Fz = 20 N

Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

20

30

40

50

Slip

Sin

ka

ge

[m

m]

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6-10

-8

-6

-4

-2

0

2

4

6

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.2

0

0.2

0.4

0.6

0.8

1

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.65

10

15

20

25

30

35

Slip

Sin

ka

ge

[m

m]

McMaster

Large

Fz = 20 N

Compact

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3D Printed

Fz = 10 N

Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-5

-4

-3

-2

-1

0

1

2

3

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.1

0

0.1

0.2

0.3

0.4

Slip

To

rqu

e [N

m]

-0.5 0 0.5

10

15

20

25

Slip

Sin

ka

ge

[m

m]

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3D Printed

Fz = 18 N

Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-10

-5

0

5

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.2

0

0.2

0.4

0.6

Slip

To

rqu

e [N

m]

-0.5 0 0.5

10

15

20

25

30

35

Slip

Sin

ka

ge

[m

m]

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3D Printed

Fz = 18 N

Compact

-0.5 0 0.5

-5

0

5

10

Slip

Dra

wb

ar

[N]

Experiment

WR

RFT

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.2

0

0.2

0.4

0.6

0.8

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

15

20

25

Slip

Sin

ka

ge

[m

m]

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-0.5 0 0.5

-1

0

1

2

3

4

Slip

To

rqu

e [N

m]

MIT Wheel

Fz = 60 N

Compact/Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-30

-20

-10

0

10

20

30

Slip

Dra

wb

ar

[N]

Experiment L

Experiment C

WR L

WR C

RFT L

RFT C

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

20

30

40

50

Slip

Sin

ka

ge

[m

m]

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MIT Wheel

Fz = 120 N

Compact/Loose

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

10

20

30

40

50

60

Slip

Sin

ka

ge

[m

m]

-0.5 0 0.5

-2

0

2

4

6

8

Slip

To

rqu

e [N

m]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-60

-40

-20

0

20

40

Slip

Dra

wb

ar

[N]

Experiment L

Experiment C

WR L

WR C

RFT L

RFT C