Steering Forces/Flexible Wheels - UMD

Steering Forces/Flexible Wheels ENAE 788X - Planetary Surface Robotics

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

Steering Forces/Flexible Wheels• Final project grading rubric• Wheel-soil interactions in steering• Flexible wheels

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© 2020 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu

http://spacecraft.ssl.umd.edu

Case Study: Final Design Project ENAE 788X - Planetary Surface Robotics


Final Project Grading Rubric (1)• Final design of rover (20 pts.)

– Solid models of design– Design evolution throughout as the analysis progressed– “Baseball card” with details of mass, power, etc.

• Trade studies (NOT an exhaustive list!) (20 pts.)– Number, size, configuration of wheels– Diameter and width of wheels– Size and number of grousers– Suspension design– Steering design– Alternate design approaches (e.g., tracks, legs, hybrid)

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Final Project Grading Rubric (2)• Vehicle stability (10 pts.)

– Slope (up, down, cross)– Acceleration/deceleration– Turning– Combinations of above

• Terrain ability (“trafficability”) (10 pts.)– Weight transfer over obstacles– Climbing/descending vertical or inclined planes– Hang-up limit (e.g., high-centering, wheel capture)

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Final Project Grading Rubric (3)• Design Details (20 pts.)

– Suspension dynamics– Development of drive actuator requirements– Detailed wheel-motor design– Development of steering actuator requirements– Detailed steering mechanism design– Mass budget (with margin)– Power budget (with margin)

• “Above and beyond” - e.g., attention to detail, particularly good presentation slides, name/logo,innovative design, high feasibility (20 pts.)

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Steering Forces – Coordinate System

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from Ishigami et.al., “Terramechanics-based Model for Steering Maneuver of Planetary Exploration Rovers on Loose Soil” J. Field Robotics 24 (3), 233-250

β = tan−1Vy

Vx



Forces Acting on Steered Wheel

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s = {(rω − Vx)/rω (rω > Vx : driving )(rω − Vx)/Vx (rω < Vx : braking )

from Yoshida and Ishigami, “Steering Characteristics of a Rigid Wheel for Exploration on Loose Soil” 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems

Fx = rb∫θf

θr

{τx(θ)cos θ − σ(θ)sin θ} dθ

β = tan−1Vy

Vx



σ(θ) = σmax (cos θ − cos θf

cos θm − cos θf )n

Normal Stress under Wheel

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( for θm < θ < θf)

σ(θ) = σmax

cos {θf −θ − θr

θm − θr (θf − θm)} − cos θf

cos θm − cos θf

n

( for θr < θ < θm)

σmax = rn ( kc

b+ kϕ) (cos θm − cos θf)

n




Calculating Shear Stresses

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θm = (a0 + a1s) θf

a0 and a1 depend on wheel-soil interactions, but generally

a0 ≈ 0.4 and 0 ≤ a1 ≤ 0.3

τx(θ) = (c + σ(θ)tan ϕ)[1 − e−jx(θ)/kx]τy(θ) = (c + σ(θ)tan ϕ)[1 − e−jy(θ)/ky]

jx(θ) = r [θf − θ − (1 − s)(sin θf − sin θ)]jy(θ) = r(1 − s)(θf − θ) tan β

kx and ky are shear displacements in those directions



Lateral Wheel Force Fy

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Fy = Fu + Fs

Fu = Force from shear stress under the wheelFu = Force from bulldozing with the side of the wheel

Soil deformation jy = ∫t

0Vydt = ∫

θf

θV sin β

1ω

dθ

=V sin β

ω (θf − θ)= r(1 − s)(θf − θ) tan β

Fu = rb∫θf

θr

τy(θ)dθ



h ⋅ c +12

ρh2 (cot Xc − tan α′ ) + (cot Xc − tan α′ )2

tan α, + cot ϕ

Bulldozing Force with Side of Wheel

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Rb =cot Xc + tan (Xc + ϕ)

1 − tan α ⋅ tan (Xc + ϕ)×

From Hegedus' bulldozing resistance formula,

Angle of approach a′ should be =0 for side of wheel

h(θ) = r (cos θ − cos θf) + c0s

Xc =π4

−ϕ2



Wheel Sideways Bulldozing Force

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Rb = {cot Xc + tan (Xc + ϕ)} h ⋅ c +12

ρh2 (cot Xc +cot2 Xc

cot ϕ )

Fs = ∫θf

θr

Rbdl = ∫θf

θr

Rb(r − h(θ)cos θ)dθ

Fs = {cot Xc + tan (Xc + ϕ)} ×

∫θf

θr

h(θ) +12

ρh2(θ)(cot Xc +cot2 Xc

cot ϕ ) (r − h(θ)cos θ)dθ



Resolution into Rover Axes

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FC = Fx sin β + Fy cos β

FB = Fx cos β − Fy sin β



Procedure for Solution of Side Force• Input the vertical load , the longitudinal slip

ratio , and the steering angle

• Determine the angle of contact and the angle of departure from the model of wheel sinkage

• Determine the vertical stress and the shear stresses , under the wheel

• Determine the longitudinal wheel force

• Determine the lateral wheel force

Fzs β

θfθr

σ(θ)τx(θ) τy(θ)

Fx

Fy

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Simulation Parameters and Values

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Side Force as a Function of Slip

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Relationship between and Fx Fy

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Experimental Set-Up

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Experimental Results

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Four-Wheel Test Parameters

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Four-Wheeled Test Vehicle

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Drawbar Pull vs. Slip Ratio

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Side Force vs. Slip Ratio

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Typical Flexible Wheels

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Sharma et. al., “Systematic design and development of a flexible wheel for low mass lunar rover ” J. Terramechanics 76 (2018)



Flexible (“Elastic”) Wheel Analysis

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Ground Pressure: Pgr =W

ℓcb

b = wheel width

ℓc = length of contact patch

W = weight on wheel

Pgr = [ kc

b+ kϕ]

12n + 1

[ 3W

(3 − n)b D ]2n

2n + 1



Bekker Substitution Circle

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The performance of an elastic wheel can be modeled by that of a larger rigid wheel

R*R0

= 1 +f0h

+f0h

R * = equivalent radius of rigid wheelR0 = actual radius of flexible wheelh = sinkage of flexible wheel in soil ( = z)f0 = deflection of flexible wheel under load

Value of is usually determined by finite element analysisf0

G. Sharma et al., Journal of Terramechanics 76 (2018)



Flexible Wheel Analysis

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H = (cA + W tan φ)[1 −ksl (1 − e

−slk )]

Drawbar Pull

θf = 2 sin−1 ( l2r )

Contact Angle

ng min =360∘

θf

ng max =2πrlqp

lqp =hb

tan ( π4 − ϕ

2 )

hb = height of grousers

lqp = length of stress zone due to grousers



Grouser Traction

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Nϕ = tan2 ( π4

+ϕ2 )

Fp = b ( 12

γsh2b Nϕ +

Wbl

hbNϕ + 2chb Nϕ)Grouser traction



Experimental Setup

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R0 = 90 mm

b = 50 mm


R * = 325 mm



Flexible Wheel Tested Characteristics

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Smooth Wheel Performance vs. Slip

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Drawbar Pull vs. Wheel Loading

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(Number of grousers)G. Sharma et al., Journal of Terramechanics 76 (2018)



A Slightly More Rigorous Approach

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Y. Favaedi et al., Journal of Terramechanics 48 (2011)



Sinkage and Normal Pressure

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Z = L cos θ − L1 cos θ1

σ1(θ) = − ( kc

Bs+ kϕ) (L cos θ − L1 cos θ1)n

Along arc CD:

Along flat section BC:

Z0 = Ldf cos θdf − L1 cos θ1

σdf = − K (Ldf cos θdf − L1 cos θ1)n

σ2(θ) = − K (Ldf cos θdf − L2 cos θ2)n

Z2 = Ldf cos θdf − L2 cos θ2

Along arc AB:



Fast Forwarding…

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• Calculate slip velocity under the wheel (not a constant!)

• Calculate shear displacement and soil adhesion• Calculate soil thrust and drawbar pull by sections

(CD, BC, AB)• Add in effects of grousers



Rigid vs. Flexible Wheel Performance

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Experimental vs. Theoretical Sinkage

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Experimental Theoretical




Experimental vs. Theoretical Torque

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Experimental vs. Theoretical DP

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Closing Comments• Full analysis of flexible wheels gets really complex

– the current SOA is for finite element modeling of both the wheel and the soil

• Soil is a lot more complex than is typically modeled

• If experimental results are within the general vicinity of the predicted values, declare victory and submit the paper

• Terramechanics is hard!

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Steering Forces/Flexible Wheels - UMD