Effects of grousers • Example solutions for rolling resistance

Terramechanics 2 ENAE 788X - Planetary Surface Robotics

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

Terramechanics 2

• Effects of grousers • Example solutions for rolling resistance • Discussion of first steps in term project

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© 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu

http://spacecraft.ssl.umd.edu



Grousers (on Tracked Vehicle)

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Grousers (MSL Wheels)

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Mars Rover Wheels

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Tractive Force per Wheel (No Grousers)

A = area of contact

Cb = coe�cient of soil/wheel cohesion

� = length of contact patch

K = coe�cient of soil slips = wheel slip ratio

�b = wheel/soil friction angle

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H = [ACb + Ww tan�b]⇤1� K

⌥

�1� e

�s⇥K

⇥⌅

s = 1� V

�r



Tractive Force per Wheel (With Grousers)

h = height of grouser

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

Cb = soil/wheel cohesion = 0.017 N/cm2

�b = 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 tan�b

⇤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

⇥⌅

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Basic Equation of Vehicle Propulsion

DP = H � (Rc + Rb + Rg + Rr)

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• 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)



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

�r

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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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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 + k�b)�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)

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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 10000Weight on Wheel (N)

Dra

wb

ar

Resi

stan

ce (

N)

ClayDry SandSnow


Com

pact

ion

Res

ista

nce

(N)

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Soil Type and Specific Resistance

0.1

1

10

10 100 1000

Weight on Wheel (N)

Resi

stan

ce/

Weig

ht

ClayDry SandSnow


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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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Bulldozing Resistance (Reminder)

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Rb =b sin (↵+ �)

2 sin↵ cos�

�2zcKc + �z2K�

�

General case:

For tracked vehicles, only the first term applies:

Rb =b sin (↵+ �)

2 sin↵ cos�


�

ò

= z tan2✓⇡

4� �

2

◆+`3o

�

3

⇣⇡2� �

⌘+ c`2

o

1 + tan

✓⇡

4+

�

2

◆�

All angles in radians!

ò

⌘ soil disruption depth, and is not the same as contact length `



Bulldozing Resistance (Rb)

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Symbol Description Value

Angle of internal friction 30-40 degrees for Lunar Regolith

γ Soil density 0.002595 N/cm3 for Lunar Regolith

Co Cohesive strength of soil 0.017 N/cm2 for Lunar RegolithLo

(degrees)Distance of rupture

Kc (degrees)

Modulus of cohesion of soil deformation

Kγ (degrees)

Modulus of density of soil deformation

Nc (radians)

Coefficient of passive earth pressure

α(degrees)

Angle of approach of the wheel



Bulldozing Example (1)

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� = 33o = 0.576 rad

↵ = cos

�1

✓1� 2z

d

◆= cos

�1

✓1� 2(1.812)

81.2

◆= 17.18o = 0.2999 rad

Nq =

e(1.5⇡��) tan�

2 cos

2⇣

⇡4 +

�2

⌘= 32.23

ò

= z1 tan2

✓⇡

2� �

4

◆= 0.5341

Nc =Nq � 1

tan�= 48.09




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N� =2(Nq + 1) tan�

1 + 0.4 sin 4�= 33.27

Kc = (Nc � tan�) cos2 � = 33.37

K� =

✓2N�

tan�+ 1

◆cos

2 � = 72.77

� = 0.002595N

cm3; c

o

= 0.017N

cm2

` =d

2

cos

�1

✓1� 2z

d

◆= 12.18 cm



Terzhagi Parameters

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Red lines represent values calculated for this example




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Rb =b sin (↵+ �)

2 sin↵ cos�


�

+`3o

�

3

⇣⇡2� �

⌘+ c`2

o

1 + tan

✓⇡

4+

�

2

◆�

hRbi = cm

✓cm

N

cm2+

N

cm3cm2

◆+ cm3 N

cm3+

N

cm2cm2

Rb = 94.98 + 0.000131 + 0.014 = 95.00 N per leading wheel

Rb,total

= 190.0 N

Course Overview ENAE 788X - Planetary Surface Robotics


AAR Design Project Statement• Perform a detailed design of a small astronaut

assistance rover, emphasizing mobility systems– Chassis systems (e.g., wheels, steering, suspension…)– Support systems (e.g., energy storage)– Navigation and guidance system (e.g., sensors,

algorithms...)

• Design for Moon, then assess feasibility of systems for Mars, and conversion to Earth analogue rover

• This is not a hardware project - focus is on detailed design (but may be built later!)

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Objectives for Design Project (1)• UMd SSL is proposing to do month-long

simulations of lunar/Mars science exploration missions to examine impact of robotics

• Primary missions would be held at HI-SEAS in Hawaii

• Rover design should facilitate shipping to/from Hawaii– Minimal size and mass– Modular construction for packing (ideally each less than

50 lbs with packaging)

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Aerial View of HI-SEAS Site

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Sample HI-SEAS Terrain

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24° slope




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First Steps in Rover Design• Assume a vehicle gross weight• Examine rolling resistance for design parameters

– Number of wheels– Diameter of wheels– Width of wheels

• Calculate torque transfer to ground– Required torque for climbing maximum slope– Design of grousers for required torque

• Next step: chassis design

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Documents

Effects of grousers • Example solutions for rolling resistance