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Comparison of DEM and Traditional Modeling

Methods for Simulating Steady-State Wheel-

Terrain Interaction for Small Vehicles

7th Americas Conference of the ISTVS

Tuesday, November 5

Tampa, Florida

William Smith

Daniel Melanz

Carmine Senatore, Karl Iagnemma

Huei Peng

University of Michigan

University of Wisconsin

Massachusetts Institute of Technology

University of Michigan

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Motivation • Small vehicles

– Military Defense

• IED disposal

• Reconnaissance

– Planetary Exploration

• Mars rovers

– Search and Rescue/Disaster

• Fukushima power plant

• Terramechanics is important for steady-state and dynamic operation

– Surface roughness is proportionally much larger

• Need to evaluate DEM compared to the established ‘Bekker’ method

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Source: JPL

Goal: Evaluate three terramechanics methods for predicting single

wheel performance of small vehicles on granular terrain

B. Trease, et. al., “Dynamic modeling and soil mechanics for path planning

of the Mars exploration rovers,” in IDETC/CIE, Washington, D.C., 2011.

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TERRAMECHANICS METHODS Bekker

Dynamic Bekker

Discrete Element Method

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• Wheel forces are functions of the normal and shear stresses acting

along the wheel-soil interface

• Drawbar

• Normal:

• Torque:

• The term ‘Bekker method’ characterizes the semi-empirical terramechanics models

pioneered by M.G. Bekker, primarily during the 1950s and 1960s.

Modeling Method: “Bekker”

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cos sinf

rnormal

F b r d

f

r

drawbar cos sinF rb d

f

r

2wheelT r b d

• Advantages

– Computationally efficient compared to other techniques

– Many soil coefficients can be determined through simple soil tests

• Limitations:

– Describes steady-state relationships, not dynamic equations, limiting its applicability for

transient operation (e.g. multibody vehicle simulations)

– Modeling more complex interactions require significant modifications to the method

• These modifications often result in an increased number of empirical terms

– Soil dynamics are not considered

– Wheel shape

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Dynamic Bekker Method • The ‘Dynamic’ Bekker method addresses two

limitations of the Bekker method:

– Multibody dynamics

– Complex soil profiles

• The wheel is treated as a free body with inertia

• The soil is discretized so the Bekker stress

equations can be applied to each region

• In this paper:

– Single rigid body representing the wheel

• Bilaterally constrained to move at a specified linear and

angular velocity

– Multiple rigid bodies representing the soil

• Connected to springs, which are constrained in vertical

direction

– Bekker equations are applied in the same manner

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B. Trease, et. al., “Dynamic modeling and soil mechanics for path planning

of the Mars exploration rovers,” in IDETC/CIE, Washington, D.C., 2011.

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Discrete Element Method • Soil is modeled as a granular material made of

many particles

• Each particle is capable of free body motion

• Forces occur upon contact with other particles

or the environment (walls)

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• Advantages

– Discrete nature ideal for granular soils

– Flexible simulation method not limited to wheel-terrain

• Limitations

– Computation resources

– Parameter selection C. J. Coetzee and D. N. J. Els, “Calibration of granular material parameters for

DEM modelling and numercal verification by blade–granular material

interaction,” Journal of Terramechanics, vol. 46, no. 1, pp. 15–26, Feb. 2009.

Normal force

Tangential force

Rolling friction

torque

n n ij n nkF n v

t t t t tkF Δs v

t c n t c nif then F F F F

i jkr r r r, eff n

i j

min , R R

R RT T T F

k k kr, t+ t r, t r

kr r r r r rk

T T ΔT

ΔT Δθ T Δω

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SOIL TESTING Direct Shear

Pressure Sinkage

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Experimental Tests • Direct Shear

– Procedure:

• Pour soil into shear box

• Apply normal pressure to soil

• Move bottom half of box to shear soil

– Settings:

• Shear box 60x60x60mm (WxLxH)

• Normal pressure: 2080, 5330, 17830 Pa

• Loosely-packed soil (1.55-1.6 g/cm3)

• Shear rate 18 μm/s

• Pressure-Sinkage

– Procedure:

• Prepare soil (till, mix, etc)

• Move plate at constant rate into soil

– Settings:

• Plate 5x15cm (WxL)

• Loosely-packed soil (1.55-1.6 g/cm3)

• Penetration rate 10 mm/s

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Bekker Parameter Identification Direct Shear:

• Bekker parameters c, ϕ, and K were determined by numerically minimizing the error given by:

Pressure-Sinkage:

• Bekker parameters k, and n were determined by numerically minimizing the error given by:

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2

, ,

arg min tan 1 expc K

jj c

K

2

plate,

arg min

n

k n

zz kb

Parameter Value

c [Pa] 139.280

ϕ [rad] 0.606

K [m] 5.151x10-4

k [Pa] 2.541x105

n [-] 1.387

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DEM Direct Shear Tests • Same shear box dimensions as experimental

• Normal pressure applied by using a rigid body of

closely-packed particles with necessary density

• Increased shear rate required to limit computation

cost

– Reducing the shear rate further had negligible

impact on simulation results

• Computation Time:

– Simulations were run on a single core of an Intel

Xeon 5160 (3.0 GHz), at a rate of 40 to 20 cpu

minutes per simulation second for time steps 1.5

and 3.8 μs, respectively

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Parameter Value

shear box dimensions [mm] 60 x 60 x 60 (W x L x H)

normal load [Pa] 2080, 5330, 17830

shear rate [mm/s] 0.66

shear displacement [mm] 6.6

number of soil particles ~640

time step [sec] 1.5 - 3.8x10-6

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DEM Pressure-Sinkage Tests • Same size plate dimensions as experimental

• Soil bin was 3x size of plate (recommended by

MIT to limit edge effects)

– Periodic boundaries used to further remove wall

effects

• Same sinkage rate as experimental

• Computation Time:

– Simulations were run on a single core of an Intel

Xeon 5160 (3.0 GHz), at a rate of 6 to 3 hours

cpu per simulation second for time steps 1.5 and

3.8 μs, respectively

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Parameter Value

soil bin dimensions [mm] 150 x 400 x 160

(W x L x H)

plate dimensions [mm] 50 x 130 x 10

(W x L x H)

sinkage rate [mm/s] 10.0

maximum sinkage [mm] 20.0

number of soil particles ~30,000

time step [sec] 1.5 - 3.8x10-6

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• Rolling Resistance Torque

• Spring/Damper Components

• Coefficients

New Rolling Resistance Model • The direct-shear and pressure-sinkage soil

tests have widely different shear/sinkage

rates

– Shear rate 18 μm/s

– Penetration rate 10 mm/s

• The properties of granular soil have been

shown to be rate dependent

– Increasing drag force when velocity increased

from 1 to 50 mm/s [1]

– DEM pile formation simulations found rolling

friction depended on the relative motion

between particles [2]

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2

i j

r n r, eff

i j

r r eff

2.25

2

R Rk k

R R

k I

2

r, eff r tmin , 1.0v

2 2i i j j

eff 2 2i i j j

1.4M R M R

IM R M R

k k kr, t+ t r, t r

r r r

kr r rk

T T ΔT

T Δω

ΔT Δθ

i jkr r r r, eff n

i j

min , R R

R RT T T F

[1] B. Yeomans, C. M. Saaj, and M. Van Winnendael, “Walking

planetary rovers – Experimental analysis and modelling of leg thrust

in loose granular soils,” J. Terramechanics, vol. 50, no. 2, pp. 107–

120, Apr. 2013.

[2] A. P. Grima and P. W. Wypych, “Discrete element simulations of

granular pile formation: Method for calibrating discrete element

models,” Eng. Comput., vol. 28, no. 3, pp. 314–339, 2011.

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WHEEL TESTS

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Settings: Bekker Method • Some of the Bekker parameters cannot be determined from soil tests

– Parameters a0 and a1 used to determine the location of maximum shear stress

– These parameters can only be determined experimentally using wheel tests

• Parameter values were chosen assuming coefficients from the literature

– The goal is to predict, not to fit, wheel performance

• Computation time

– ~43 ms to solve for a given slip ratio and normal load (using standard iterative

solving method, implemented in C)

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Parameter Value

a0 [-] 0.18

a1 [-] 0.32

θr [rad] 0

J. Wong and A. Reece, “Prediction of rigid wheel performance based on the analysis of soil-wheel stresses part I.

Performance of driven rigid wheels,” J. Terramechanics, vol. 4, pp. 81–98, 1967.

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Settings: Dynamic Bekker Method • The dynamic Bekker method computes a time

series, rather than a single value, which

requires the selection of a time step

– A convergence analysis was performed to

evaluate the steady state wheel sinkage at

varying time step values

• A time step between 1x10-3 and 1x10-4 was found

to obtain convergence

• Similarly, the number of soil nodes (or soil

spacing) must also be determined

– A convergence analysis was performed to

evaluate the steady state wheel sinkage at

varying node spacing

• Convergence occurred for 300 or more nodes

(corresponding to a node spacing of 3.3mm or

less)

• Computation time

– ~45 cpu seconds per simulation second (single

core 2.2 GHz)

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Settings: DEM

Parameter Value

soil bed dimensions [mm] 600 x 1000 x 160

(W x L x H)

number of soil particles ~300,000

number of wheel particles ~12,000

time step [sec] 2.2x10-6

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• DEM wheel composed of 1cm diameter overlapping particles, grouped to act as a single rigid body

• Experimental wheel/soil bin dimensions were maintained

• Procedure:

– Wheel placed on soil, allowed to rest for 0.5 seconds

– An x-axis force and a y-axis torque were applied to the wheel for 1 second to ramp-up the longitudinal and angular velocities

– Wheel was simulated for a distance of 0.7m, or until steady-state was reached

• Computation time

– ~8.5 cpu hours per simulation second (8 cores 3.0 GHz)

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Steady-State Results

• DEM shows better quantitative and qualitative agreement – Greatest benefit near zero slip

– Bekker has discontinuity around zero slip

• Bekker and dynamic Bekker are almost identical (expected)

– Differences are a result of implementation of normal stress in dynamic Bekker

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Time Series Results

– Dynamic Bekker shows oscillation due to stiff system (no damping)

– Experimental results show low frequency periodicity, which reflects the

periodic failure pattern within the soil

– DEM results have a lower frequency with higher amplitude, likely a

result of the relatively large particle sizes used

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CONCLUSIONS

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Recap

Bekker method

– Extremely computationally efficient: ~43x10-3 cpu seconds

– Poor prediction of wheel performance using soil test tuning

– Some parameters cannot be determined from soil tests

Dynamic Bekker method

– Computationally efficient: ~45 cpu seconds/sim second

– Similar steady-state performance to Bekker method

Discrete element method

– Computationally inefficient: ~24.5x104 cpu seconds/sim second

– Significantly better prediction of wheel performance

– Also provides some time-series information

– All parameters determined from soil test tuning

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DEM-Tuned Bekker Method • Bekker parameters can be tuned to produce similar results as DEM

• When the Bekker method capabilities are sufficient, may be able to tune to DEM

simulations

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Parameter Soil-Tuned Values DEM-Tuned Values

c [Pa] 139.280 96.240

ϕ [rad] 0.606 0.606

K [m] 5.151x10-4 4.534x10-3

k [Pa] 2.541x105 2.305x104

n [-] 1.387 0.418

a0 [-] 0.18* 0.09

a1 [-] 0.32* 0.90

θr [rad] 0 0

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THANK YOU Questions?

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