Terramechanics MARYLAND. Overview of wheel-soil interactions Sources of rolling resistance on soil Rover propulsion analysis U N I V E R S I T Y O F

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1 Overview of wheel-soil interactions Sources of rolling resistance on soil Rover propulsion analysis 2007 David L. Akin - All rights reserved

2 Soil Mechanics Wheel rolling over soil does work Compression Bulldozing Assume soil reaction is like a (nonlinear) spring P = kz n P = applied pressure z = compression depth k, n = heuristic parameters

3 Soil Testing Apparatus Internal friction angle ϕ Bevameter (force vs. displacement) Shear deformation modulus K

4 More Detailed Soil Compression Equation k = k c b + k φ k c = modulus of cohesion of soil deformation P = b = wheel width ( ) kc b + k φ z n k c units < N/m (n+1) > k φ = modulus of friction of soil deformation k φ units < N/m (n+2) >

5 Derivations Displacement Energy E A = F A dz = If we sink to a depth z o, E A = zo 0 P dz = zo 0 kz n dz = k zn+1 o n + 1 Total Energy E A A = E zn+1 o bd = k A n + 1 bd Given a force resisting rolling R, the energy required to roll a distance d is E roll = Rd E roll = E displacement Rd = E A bd P dz wheel width b distance rolled d z o W area of compressed soil A = bd For n = 1 : P = kz; E A = k z2 o 2 ; R = 1 2 kbz2 o n = 1 2 : P 2 = k 2 z; E A = 2 3 kz 3 2 o ; R = 2 3 kbz 3 2 o R Generic case: n = 0 : P = k; E A = kz o; R = kbz o P = kz n ; E A = k zn+1 o n + 1 zn+1 o ; R = kb n + 1

6 Equations for Compression Resistance z = ( ) 2 3W w (3 n)bk 2n+1 d W w = weight on wheel R c = d = wheel diameter ( ) bk z n+1 n + 1 R c = compression resistance (per wheel)

7 Compression Resistance (Lunar Soil) R c = 1 n + 1 (k c + bk φ ) 1 2n+1 ( 3Ww (3 n) d n = 1 k c = 0.14 N/cm 2 k φ = N/cm 3 R c = 1 ( 2 (k c + bk φ ) 1 3Ww 3 2 d ) 4 3 ) 2(n+1) 2n+1

8 Apollo Lunar Roving Vehicle Example z = ( R c = 1 2 check units - ( N 1/ ( ) ( ) cm 2/3 ) ( N 4/3 cm 2/3 ) 2 3 = 2.03 cm ( ) 2 82 = N ) 4 3 = 29.8 N

9 Rolling and Gravitation Resistance Rolling resistance (tires, bearings, etc.) R r = W v c f W v = weight of vehicle c f = coefficient of friction (typ. 0.05) Gravitational resistance R g = W v sin θ slope LRV examples (15 slope) R r = 51 N R g = 262 N

10 Tractive Force per Wheel (No Grousers) H = [AC b + W w tan φ b ] A = area of contact C b = coefficient of soil/wheel cohesion s = wheel slip ratio [ 1 K l φ b = wheel/soil friction angle K = coefficient of soil slip l = length of contact patch (1 e sl K ) ]

11 Tractive Force per Wheel (With Grousers) H = [blc b ( 1 + 2h b ) ( N g + W tan φ b h b arctan b )] [ 1 K h l ( ) ] 1 e sl K A = area of contact = bl C b = soil/wheel cohesion = N/cm 2 φ b = wheel/soil friction angle = 35 s = wheel slip ratio (typ ) K = coefficient of soil slip = 1.8 cm l = length of contact patch = D ( 2 cos 1 1 2z h = height of grouser All values typical for lunar soil D )

12 Effect of Soil Thrust Fraction Soil Thrust Fraction [ 1 K l ( 1 e sl K ) ] 1.2 Soil Thrust Fraction K/l= Slip Ratio

13 Basic Equation of Vehicle Propulsion DP = H (R c + R b + R g + R r ) DP: Drawbar pull (residual drive force) H: Maximum tractive force of wheels R c : Compaction resistance R b : Bulldozing resistance R g : Gravitational resistance R r : Rolling resistance (internal)

14 Example: Wheelbarrow (Single) Wheel R = (k c + k φ b) 1 2(n+1) 2n+1 W 2n+1 1 n + 1 ( ) 3 2(n+1) 2n+1 (n+1) D 2n+1 3 n Compaction Drawbar Resistance (N) (N) b=0.1 m D=0.3 m Dry Sand Sandy Loam Clay Lunar MER-B Sandy Loam MER-B Slope Soil Wheel Weight (N)

15 Effects of Wheel Parameters 200 Compaction Drawbar Resistance (N) (N) W=500 N b=0.1 b=0.25 b= Wheel Diameter (m)

16 Effect of Soil Spring Constant on R/W Resistance/Weight n=0 n=0.5 n= Wheel 1 m diameter x 0.2 m width Soil "k" value (N/m)

17 Soil Type and Wheel Load Drawbar Compaction Resistance (N) Wheel 1 m diameter x 0.2 m width Weight on Wheel (N) Clay Dry Sand Snow

18 Soil Type and Specific Resistance 10 Resistance/Weight Clay Dry Sand Snow 0.1 Wheel 1 m diameter x 0.2 m width Weight on Wheel (N)

19 Effect of Wheel Diameter and Width Wheel Load = 1000 N Compaction Resistance Resistance (N) (N) 1000 D=1 m D=1 D=2 m D=2 D=4 m D=4 Dual m Quad Wheel Width (m)

20 Effect of Slope 600 Gravitational Drawbar Resistance (N) (N) Slope (deg)

21 Wheel Test Apparatus Wheel testing done at MIT Field and Space Robotics Laboratory Independent control of motion and wheel velocity provides controllable slip s = 1 V ωr

22 Wheel Torque vs. Time 9 grousers 18 grousers ϕ=0.24

23 Sinkage vs. Slip Ratio

24 Drawbar Pull vs. Slip Ratio

25 Motor Torque vs. Slip Ratio

26 Required Traction for Wall Climbing

27 Bump/Slope Traction Requirements