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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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Notes about Revised Course Schedule
• No class next week (9/11 and 9/13)• Makeup lectures to be announced
2
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
3
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Wheel-Soil Interaction
Wheel rolling over soil does work– Compression– “Bulldozing”
6
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Testing Apparatus
Bevameter (force vs. displacement)
Internal friction angle ϕ
Shear deformation modulus K
7
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Displacement Calculations
14
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Displacement Calculations
15
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)
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Compression Calculations
16
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Displacement Calculations
17
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 + · · ·
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Rolling Resistance as f(W)
18
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
34
R = 0.876W
32
√kbD
34
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
21
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
23
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
25
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
26
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
27
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
�
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
29
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
30
H = [ACb + Ww tanφb]�1− K
�
�1− e
−s�K
��
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
��
31
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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)
33
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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)
35
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
36
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
Wheel 1 m diameter x 0.2 m width
Com
pact
ion
Res
ista
nce
(N)
37
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Soil Type and Specific Resistance
0.1
1
10
10 100 1000
Weight on Wheel (N)
Resi
stan
ce/
Weig
ht
ClayDry SandSnow
Wheel 1 m diameter x 0.2 m width
38
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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)
39
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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)
40
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
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
41
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Wheel Torque vs. Time
ϕ=0.24
9 grousers 18 grousers
42
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Sinkage vs. Slip Ratio
43
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Drawbar Pull vs. Slip Ratio
44
Terramechanics ENAE 788X - Planetary Surface Robotics
U N I V E R S I T Y O FMARYLAND
Motor Torque vs. Slip Ratio
45