Origin and nature of lunar soil • Soil mechanics · 2012. 9. 6. · Effects of Soil Mechanics...

Terramechanics ENAE 788X - Planetary Surface Robotics

U N I V E R S I T Y O FMARYLAND

Terramechanics

• Origin and nature of lunar soil• Soil mechanics• Rigid wheel mechanics

1

© 2012 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu



Notes about Revised Course Schedule

• No class next week (9/11 and 9/13)• Makeup lectures to be announced

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Lunar Regolith• Broken down from larger pieces over time• Major constituents

– Rock fragments– Mineral fragments– Glassy particles

• Local environment– 10-12 torr (= 1.22x10 -10 Pa = 1.93x10 -14 psi)– Meteorites at velocities >105 m/sec– Galactic cosmic rays, solar particles– Temperature range +250°F – -250°F

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Regolith Creation Process

• Only “weathering” phenomenon on the moon is meteoritic impact!

• Weathering processes– Comminution: breaking rocks and minerals into smaller

particles– Agglutination: welding fragments together with molten

glass formed by impact energy– Solar wind spallation and implantation (miniscule)– Fire fountaining (dormant)

4



JSC-1 Simulant

• Ash vented from Merriam Crater in San Francisco volcano field near Flagstaff, AZ

• K-Ar dated at 150,000 years old ± 30,000• Major constituents SiO2, TiO2, Al2O3, Fe2O3, FeO,

MgO, CaO, Na2O, other <1%• Represents low-Ti regolith from lunar mare• MLS-1 simulant (U.Minn.) preferred for simulation

of highland material

5



Wheel-Soil Interaction

Wheel rolling over soil does work– Compression– “Bulldozing”

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from Gibbesch and Schafer, “Advanced and Simulation Methods of Planetary Rover Mobility on Soft Terrain” 8th ESA Workshop on Advanced Space Technologies for Robotics and Automation, Noordwijk, The Netherlands, November, 2004



Soil Testing Apparatus

Bevameter (force vs. displacement)

Internal friction angle ϕ

Shear deformation modulus K

7



Soil Characterization – Direct Shear

8

Soil

cross-section A

T

WShear Stress τ =

T

A

Normal Stress σ =W

A

c = Cohesion

φ = angle of internal friction

Normal Stress σ

Shear Stress τ



Modeling Soil Reaction to a Wheel

Assume soil reaction is like a (nonlinear) spring

P = kzn

9

P = applied pressure

z = compression depth

k, n = heuristic parametersP

z



Effects of Soil Mechanics

10

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10

n=0n=0.5n=1.0n=1.5

Scal

edP

ress

ure

P kSoil Penetration Depth z



Wheel-Soil Interactions

11

zo

R

W

wheel width b

area of compressed soil A = bd

distance rolled d

Displacement EnergyE

A=

�F

Adz =

�Pdz

E

A=

� zo

0Pdz =

� zo

0kzndz = k

zn+1o

n + 1



Rolling Resistance

12

Total EnergyE

AA =

E

Abd = k

zn+1o

n+ 1bd

Given a force resisting rolling ≡ R,

the energy required to roll a distance d is

Eroll = Rd

Eroll = Edisplacement ⇒ Rd =E

Abd



Rolling Resistance

13

For n = 1 : P = kz;E

A= k

z2o2;R =

1

2kbz2o

For n =1

2: P 2 = k2z;

E

A=

2

3kz

32o ;R =

2

3kbz

32o

For n = 0 : P = k;E

A= kzo;R = kbzo

Generic case: P = kzn;E

A= k

zn+1o

n+ 1;R = kb

zn+1o

n+ 1



Soil Displacement Calculations

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zo

R

W

wheel width b

θ

θo

dF

Fx

Fz

R−� θo

0dF sin θ = 0

−W +

� θo

0dF cos θ = 0

dF = Pb dF cos θ = −Pb dx

dF sin θ = Pb dz

R =

� θo

0Pb dz W = −

� θo

0Pb dx

In general, P = kxn

W = −� zo

0bkzndx




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zo

wheel width b

θ

A

Bz

D

2

AB =D

2− (zo − z)

x

x2 =

�D

2

�2

− AB2=

�D

2

�2

−�D

2− (zo − z)

�2

=

�D

2

�2

−�D

2

�2

+ 2D

2(zo − z)− (zo − z)2

x2 = [D − (zo − z)](zo − z)



Soil Compression Calculations

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But D � zo − z

x2 ≈ D(zo − z) ⇒ 2x dx = −D dz

so from W = −� zo

0bkzndx we get W = −

� zo

0bkzn

−D

2xdz

W = −bk

� zo

0zn

�−D

2√D√zo − z

�dz

W = bk

� zo

0zn

� √Ddz

2√zo − z

�dz




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Define zo − z ≡ t2 ⇒ dz = −2tdt

W = bk√D

� √zo

0(zo − t2)ndt

W ≈ bk√Dzo3

zno (3− n)

for n = 1 ⇒ W =2

3bkzo

�Dzo

for n =1

2⇒ W =

5

6bkzo

√D

for n = 0 ⇒ W = bk�Dzo

Taylor Series expansion (zo − t2)n ∼= zno − nzn−1o t2 + · · ·



Rolling Resistance as f(W)

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for n = 0 ⇒ W = bk�

Dzo ⇒ zo =

�W

bk

�2 1

D

R = kbzo ⇒ R =kb

(kb)2W 2

D⇒ R =

W 2

kbD

for n =1

2⇒ W =

5

6bkzo

√D ⇒ zo =

6

5

W

bk√D

R =2

3kbz

32o ⇒ R =

2

3kb

�6

5

W

kb√D

� 32

=2

3

�6

5

� 32 W

32

√kbD

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R = 0.876W

32

√kbD

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Rolling Resistance as f(W)

19

for n = 1 ⇒ W =2

3bkz

32o

√D ⇒ z2o =

�3W

2kb√D

� 43

R =1

2kbz2o ⇒ R =

1

2kb

�3W

2kb√D

� 43

=1

2

�3

2

� 43�

W 4

kbD2

� 13

R = 0.859

�W 4

kbD2

� 13



Rolling Resistance as f(W) (Generic)

20

W =bk√Dzo3

zno (3− n) =bk√D

3zn+ 1

2o (3− n)

zn+ 1

2o =

3

(3− n)

W

bk√D

zn+1o =

�3

3− n

W

bk√D

� n+1

n+12=

�3

3− n

W

bk√D

� 2(n+1)2n+1

R =bk

n+ 1zn+1o =

bk

n+ 1

�3

3− n

W

bk√D

� 2(n+1)2n+1

R =1

n+ 1

�3

3− n

W√D

� 2(n+1)2n+1

�1

bk

� 12n+1



More Detailed Soil Compression Equation

k =kc

b+ kφ

P =�

kc

b+ kφ

�zn

b = wheel width

kc units⇒< N/m(n+1) >

kφ units⇒< N/m(n+2) >

kc = modulus of cohesion of soil deformation

kφ = modulus of friction of soil deformation

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Soil Characteristics

22

soil type n kc � N

mn+1� kφ � N

mn+2�

Dry Sand 1.1 990 1,528,000

Lunar Regolith 1.0 1400 820,000

Sandy Loam 0.7 5270 1,515,000

Sandy Loam (MER-B) 1.0 28,000 7,600,000

Slope Soil(MER-B) 0.8 6800 210,000

Clay (Earth) 0.5 13,190 692,200



Equations for Compression Resistance

Ww = weight on wheel

d = wheel diameter

Rc = compression resistance (per wheel)

Rc =�

bk

n + 1

�zn+1

z =�

3Ww

(3− n)bk√

d

� 22n+1

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Soil Compression – Reece Formulation

kc units⇒< N/m(n+1) >kφ units⇒< N/m(n+2) >

24

Problem is that kc and kφ have variable dimensions,

P =

�kcb

+ kφ

�zn

based on n

Reece Formulation: nondimensionalize by b



Compression Resistance (Lunar Soil)

kc = 0.14 N/cm2

kφ = 0.827 N/cm3

n = 1

Rc =1

n + 1(kc + bkφ)

−12n+1

�3Ww

(3− n)√

d

� 2(n+1)2n+1

Rc =12

(kc + bkφ)−13

�3Ww

2√

d

� 43

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Apollo Lunar Roving Vehicle Example

check units -

z =�

3 ∗ 2532(0.14 + 17.4 ∗ 0.827)

√82

� 23

= 2.03 cm

Rc =12

(0.14 + 17.4 ∗ 0.827)−13

�3 ∗ 2532√

82

� 43

= 29.8 N

�N−1/3

cm−2/3

� �N4/3

cm2/3

�= N

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Rolling and Gravitation Resistance

• Rolling resistance (tires, bearings, etc.)

• Gravitational resistance

• LRV examples (15° slope)

Rr = Wvcf

Wv = weight of vehicle

Rg = Wv sin θslope

Rg = 262 N

cf = coefficient of friction (typ. 0.05)

Rr = 51 N

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Bulldozing Resistance

28

Rb =b sin (α + φ)2 sinα cos φ

�2zcKc + γz2Kγ

�+

π�3oγ (90− φ)540

+cπ�2o180

tan�

45 +φ

2

�

α = angle of attack of wheel in soil ≡ cos−1

�1− 2z

D

�

γ = density of soil�

kgm3

�

�o = length of soil rupture ≡ z tan2

�45− φ

2

�



Bulldozing Resistance

• “Bulldozing” is the process of pushing soil up ahead of the wheel

• Ranges from a small factor to a huge one, depending on soil and wheel factors

• Will be covered in detail in a later lecture

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Tractive Force per Wheel (No Grousers)

A = area of contact

Cb = coefficient of soil/wheel cohesion

� = length of contact patch

K = coefficient of soil slips = wheel slip ratio

φb = wheel/soil friction angle

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H = [ACb + Ww tanφb]�1− K

�

�1− e

−s�K

��



Tractive Force per Wheel (With Grousers)

h = height of grouser

s = wheel slip ratio (typ. 0.02-0.05)K = coefficient of soil slip = 1.8 cm

Cb = soil/wheel cohesion = 0.017 N/cm2

φb = wheel/soil friction angle = 35◦

� = length of contact patch =D

2cos−1

�1− 2z

D

�

All values typical for lunar soil

A = area of contact ∼= b�

H =�b�Cb

�1 +

2h

b

�Ng + W tanφb

�1 + 0.64

h

barctan

b

h

�� 1− K

�

�1− e

− s�K

��

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Effect of Soil Thrust Fraction

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Slip Ratio

So

il T

hru

st F

ract

ion

K/l=0.10.250.512410

Soil Thrust Fraction�1− K

�

�1− e−

s�K

��

32



Basic Equation of Vehicle Propulsion

• DP: Drawbar pull (residual drive force)• H: Maximum tractive force of wheels• Rc: Compaction resistance• Rb: Bulldozing resistance• Rg: Gravitational resistance• Rr: Rolling resistance (internal)

DP = H − (Rc + Rb + Rg + Rr)

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Example: Wheelbarrow (Single) Wheel

0

100

200

300

400

500

600

0 200 400 600 800 1000

Wheel Weight (N)

Dra

wb

ar

Resi

stan

ce (

N)

Dry SandSandy LoamClayLunarMER-B Sandy LoamMER-B Slope Soil

b=0.1 mD=0.3 m

R = (kc + kφb)−1

2n+1 W2(n+1)2n+1

1n + 1

�3

3− n

� 2(n+1)2n+1

D−(n+1)2n+1

Com

pact

ion

Res

ista

nce

(N)

34



Effects of Wheel Parameters

0

20

40

60

80

100

120

140

160

180

200

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Wheel Diameter (m)

Dra

wb

ar

Resi

stan

ce (

N)

b=0.1b=0.25b=0.5

W=500 N

Com

pact

ion

Res

ista

nce

(N)

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Effect of Soil “Spring Constant” on R/W

0.1

1

10

100

1000

10000

1 10 100 1000 10000

Soil "k" value (N/m)

Resi

stan

ce/

Weig

ht

n=0n=0.5n=1

Wheel 1 m diameter x 0.2 m width

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Soil Type and Wheel Load

1

10

100

1000

10000

100000

1000000

10 100 1000 10000

Weight on Wheel (N)

Dra

wb

ar

Resi

stan

ce (

N)

ClayDry SandSnow


Com

pact

ion

Res

ista

nce

(N)

37



Soil Type and Specific Resistance

0.1

1

10

10 100 1000

Weight on Wheel (N)

Resi

stan

ce/

Weig

ht

ClayDry SandSnow


38



Effect of Wheel Diameter and Width

100

1000

10000

0.01 0.1 1

Wheel Width (m)

Resi

stan

ce (

N)

D=1 mD=2 mD=4 m

100

1000

10000

0.01 0.1 1

Wheel Width (m)

Resi

stan

ce (

N)

D=1 mD=2 mD=4 mDualQuad

Wheel Load = 1000 N

Com

pact

ion

Res

ista

nce

(N)

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Effect of Slope

-600

-400

-200

0

200

400

600

-100 -50 0 50 100

Slope (deg)

Dra

wb

ar

Resi

stan

ce (

N)

Gra

vita

tiona

l Res

ista

nce

(N)

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Wheel Test Apparatus• Wheel testing done at MIT Field and Space Robotics

Laboratory• Independent control of motion and wheel velocity provides

controllable slips = 1− V

ωr

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Wheel Torque vs. Time

ϕ=0.24

9 grousers 18 grousers

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Sinkage vs. Slip Ratio

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Drawbar Pull vs. Slip Ratio

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Motor Torque vs. Slip Ratio

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Origin and nature of lunar soil • Soil mechanics · 2012. 9. 6. · Effects of Soil Mechanics...

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Transcript of Origin and nature of lunar soil • Soil mechanics · 2012. 9. 6. · Effects of Soil Mechanics...