DYNAMIC ANALYSIS AND TRAVERSABILITY PREDICTION OF TRACKED VEHICLES ON SOFT TERRAIN

Said Al-Milli, Kaspar Althoefer, Lakmal D. Seneviratne

King’s College London, Department of Mechanical Engineering The Strand, London WC2R 2LS, UK

Email: said.al-milli, k.althoefer, lakmal.seneviratne@kcl.ac.uk Abstract – Unmanned ground vehicles are widely used in industries where repetitive tasks or high risk missions are required. Such vehicles usually operate on soft deformable terrains and still require human supervision due to the complexity of the interaction between the vehicle and the terrain. The dynamic models currently used in predicting track forces on such terrains are empirical and out dated. This paper critically investigates the dynamics involved in a steady-state skid-steering manoeuvre of tracked vehicles and presents a novel technique for predicting track forces on soft terrains. The simulation results are in good agreement with previous field measurements. The results show that the coefficient of lateral resistance varies not only with turning radius but also with vehicle velocity, an aspect that has not been taken into consideration in previous models. Finally, a traversability criterion is presented.

I. INTRODUCTION Unmanned ground vehicles (UGVs) are widely used in

industries where repetitive tasks or high risk missions are required such as planetary exploration, military, agriculture, mining and construction. Among numerous types of vehicles, skid-steered tracked vehicles are mostly used as they are capable of traversing over a wide range of terrains. There has been growing interest in improving the autonomy of such vehicles due to the continuous human intervention in controlling the majority of currently used UGVs. Such intervention hinders operational speeds as well as potential applications. To reduce human intervention requires a thorough understanding of terrain mechanics, vehicle dynamics and the dynamics of vehicle-terrain interaction.

Soil properties play an important role in determining the maximum tractive effort that a track can develop, as well as determining the variation of thrust accompanied by slippage. Soil classification has been well established in [1]. The primary properties involved in identifying the tractive effort are cohesion c, angle of internal shearing friction φ, and shear deformation modulus K. Recently, there has been a rising interest in identifying soil properties [KCL Refs]. In [5] an online, or in real time, soil parameter identification algorithm is developed for tracked UGVs. The Newton Raphson numerical method was used to iteratively identify soil properties based on measured sprocket torques with little computational effort for achieving high accuracy. In [6] an online parameter estimation algorithm is developed for wheeled planetary rovers. The identification process is also based on sprocket toque measurements.

Kinematics and dynamics of tracked vehicles is not a new topic. Research on the mobility and the handling characteristics of tracked vehicles dates back to the 1950s.

While M.G. Bekker (1956-1960) focused on vehicle mobility and vehicle-terrain interactions [8], W. Steeds (1950) studied the mechanics of skid-steering on firm ground [9]. J. Y. Wong (1989-2001) has excelled in both fields while basing his theory on Bekker and Steeds [1]. Slip estimation is one of most recent research topics in the field of vehicle kinematics and dynamics. A slip estimation algorithm has been developed in [10]. The algorithm is based on the kinematics model of a tracked vehicle as well as trajectory measurements.

Traversability prediction is another research topic of prime importance in UGV automation. Its use is highly required in navigation and control of UGVs especially in high risk missions and in planetary exploration. In [11] a rule-based approach to extract terrain surface characteristics from a set of stereo vision cameras is presented. The surface characteristics that are identified are: terrain roughness, slope, discontinuity and hardness. Due to the lack of analytical modeling, the algorithm has poor identification of vehicle-terrain interaction dynamics. In [7] an accelerometer is used to classify the terrain in which the vehicle traverses on as to overcome traversability and safety issues involved in uneven, rough, and sloped terrain.

This paper discusses up-to-date dynamic modelling of tracked vehicles and presents a simulation that analytically predicts the performance of tracked vehicles on soft deformable terrains. A traversability criterion is also introduced. Simulation results are verified by comparing with available experimental data.

Figure 1: Forces acting on a tracked vehicle during a turn at high

speeds. (From Theory of Land Locomotion, M.G. Bekker)

Proceedings of the 2007 IEEE International Conference on

Networking, Sensing and Control, London, UK, 15-17 April 2007

MonA01

1-4244-1076-2/07/$25.00 ©2007 IEEE 279

II. DYNAMICS OF SKID-STEERING The forces involved in a tracked vehicle during a skid-

steering manoeuvre are shown in Figure 1. Following the derivation given by Wong [1], the thrusts of

the outer and inner tracks may be computed respectively using the following equations:

BMCFRF oo /rlong ++= , (1) BMCFRF ii /rlong −+= , (2)

where Ro, Ri is the longitudinal motion resistance force of outer and inner track respectively, CFlong is the longitudinal component of the centrifugal force, Mr is the moment of turning resistance and B is the tread of the vehicle.

In order to satisfy the equilibrium condition in the lateral direction, the centre of turn is to lie at a distance so ahead of the traverse centreline of the track ground contact area AC, as shown in Figure 1. so is computed using the following equation [1]:

βµ

βµ

cos2

cos'2

2

gla

gRlVs

t

y

to == , (3)

where l is the track contact length, V is the longitudinal vehicle velocity, ay is the centrifugal acceleration of the vehicle (V2/R'), R' is the turning radius with slip taken into consideration and µt is the lateral coefficient of resistance.

It should be noted that since R' is normally large when compared with l, which makes β small and so cos β may be assumed to be equal to 1. Equation (3) is then reduced as follows [1]:

g

last

yo µ2= . (4)

Due to the shifting of centre of turn so, the moment of turning resistance Mr will have two components: the first is the moment of lateral resistance about 0', and the second is the moment of the centrifugal force about 0'. The moment of turning resistance Mr is found to be [1]:

−=

2

14 t

ytr g

aWlMµ

µ , (5)

where W is the vehicle weight. The centrifugal force also causes lateral load transfer.

Thus, the longitudinal motion resistances of the outer and inner track Ro and Ri will not be identical [1]:

ry

o BghWaWR µ

+=

2, (6)

ry

i BghWaWR µ

−=

2, (7)

where h is the height of centre of gravity above the track-ground contact area and µr is the coefficient of motion resistance in the longitudinal direction.

The centrifugal force in the longitudinal direction CFlong is found to be [1]:

'2gRsWa

CF oylong = . (8)

Substitution of equations (5-8) into eqs. (1) and (2) gives the thrusts of the outer and inner tracks [1]:

−++⋅

+=

2

14'22 t

ytoyr

yo g

aWlgR

sWaBg

hWaWFµ

µµ , (9)

−−+⋅

−=

2

14'22 t

ytoyr

yi g

aWlgR

sWaBg

hWaWFµ

µµ . (10)

III. ESTIMATION OF LATERAL COEFFICIENT OF RESISTANCE µT

In the previous section, the coefficient of lateral resistance is considered constant when predicting the thrusts of the outer and inner tracks. Figure 2 shows plots of the outer and inner tracks when using equations (9-10), respectively, in conjunction with a constant value of µt and with the vehicle parameters given in Table 1, Appendix. It is evident from the plots that an increase in turning radius increases the difference of outer and inner thrusts which is unreasonable in real situations as experimental results show that the difference in thrusts is increased with small turning radii and reduced with large turning radii [3].

It is suggested that µt is expressed as a function of turning radius [1-3] as well as vehicle velocity. Kar [2] used an empirical formula, which was developed by Crosheck [4], to estimate µt as a function of the turning radius. The formula requires three constants E1, E2, and E3 that are determined empirically. This constrains the validity of the formula to the vehicle specifications and operational conditions that were used in determining the constants. Slip estimates of the tracks are also required as inputs to the Crosheck formula, but the equations Kar used to determine track slippage only relate slip to turning radius. Thus a change in vehicle velocity or soil properties has no effect on the estimation of µt. Figure 3 shows a reproduction of the track thrusts from the models used by Kar [2]. Note that the thrust values for the outer and inner tracks over lap for velocities 0.0447 m/s and 2.235 m/s.

A number of empirical models were developed for predicting track thrusts, and hence estimating the coefficient of lateral resistance, on firm ground and their simulation results were demonstrated as well as compared to experimental test results [3]. Wong derived a General Theory for Skid-Steering on Firm Ground [1] that analytically predicts track thrusts as well as longitudinal and lateral moments that are acting on the tracks. Wong’s theory is an expansion to Steeds pioneering work on skid-steering of tracked vehicles on firm ground. One of the major alterations Wong made was the consideration of the dependency of shear stress to shear displacement whereas Steeds model assumed that the shear stress reaches its maximum value instantly according to Coulomb’s law of friction. Moreover, Steeds neglected track width in his model whereas Wong considered the whole contact area of the track. Wong shows that it is possible to estimate µt analytically by using the general theory [1]. Although Wong intended his theory to be applied to tracked vehicles on firm ground, the principles outlined by Wong to estimate µt will

280

be considered for soft terrain due to its attractiveness as an analytical generalised solution.

0 10 20 30 40 500

10

20

30

40

50

Turning Radius (m)

Thr

ust o

f Out

er T

rack

(kN

)

0.0447 m/s 2.235 m/s 4.47 m/s 6.25 m/s

0 5 10 15 20 25-40

-20

0

20

40

60

Turning Radius (m)

Thr

ust o

f Inn

er T

rack

(kN

)

0.0447 m/s 2.235 m/s 4.47 m/s 6.25 m/s

Figure 2 Outer & inner track thrusts vs turning radius at different

velocities. ( µt = 0.45 and µr = 0.076)

0 10 20 30 400

10

20

30

40

50

60

70

Turning Radius (m)

Out

er T

rack

Thr

ust

(kN

)

0.0447m/s 2.235m/s 4.47m/s 6.25m/s

0 10 20 30 40-60

-40

-20

0

20

40

Turning Radius (m)

Inn

er T

rack

Thr

ust

(kN

)

0.0447m/s 2.235m/s 4.47m/s 6.25m/s

Figure 3 Outer & inner track thrusts based on Kar [2]. Vehicle

Specifications are shown in Table 1. Soil parameters: n = 1.1, Kc = 0.95 kN/mn+2, Kφ = 1528.43 kN/ mn+2, µr = 0.0765

The kinematics involved in a steady-state turn of a tracked vehicle is shown in Figure 4. According to the general theory, the shear displacements jXo and jYo at a point (x1, y1) on the outer track in the X and Y directions respectively, with respect to a fixed frame of reference XY, are expressed by the following equations [1]:

( )

( )

Ω−−+−

−

Ω−−+

+++′′=

o

zoy

o

zoyxXo

ryscl

y

ryscl

xcBRj

ω

ω

11

11

2/sin

12/

cos2 , (11)

( )

( )

Ω−−++

−+−

Ω−−+

+++′′=

o

zoyoy

o

zoyxYo

ryscl

yscl

ryscl

xcBRj

ω

ω

11

11

2/cos

2

2/sin

2 , (12)

where Ωz is the angular speed (yaw velocity) about turning centre O, R'' is the lateral distance between O and centre of gravity CG and is equal to 22

osR −′ , cx and cy are the lateral and longitudinal distances between CG and longitudinal and lateral centre lines of the vehicle hull respectively, r is the sprocket radius and ωo is the sprocket angular velocity of the outer track. Thus, the resultant shear displacement jo of point (x1, y1) on the outer track is:

22YoXoo jjj += . (13)

Figure 4 Kinematics of the outer and inner tracks during a steady-state

turn. From Wong [1]

Figure 5 Kinetics of a tracked vehicle during a steady-state turn. From

Wong [1]

281

Similarly, the shear displacements jXi and jYi at a point (x2, y2) on the inner track in the X and Y directions respectively are expressed as follows:

( )

( )

Ω−−+−

−

Ω−−+

++−′′=

i

zoy

i

zoyxXi

ryscl

y

ryscl

xcBRj

ω

ω

22

22

2/sin

12/

cos2 , (14)

( )

( )

Ω−−++

−+−

Ω−−+

++−′′=

i

zoy

oyi

zoyxYi

ryscl

y

sclr

ysclxcBRj

ω

ω

22

22

2/cos

22/

sin2

, (15)

where ωi is the sprocket angular velocity of the inner track. Thus, the resultant shear displacement ji of the point (x2, y2) on the inner track is:

22YiXii jjj += . (16)

Figure 5 shows the forces and moments that are acting at points (x1, y1) and (x2, y2) on the outer and inner tracks respectively. The longitudinal forces Fyo and Fyi as well as the lateral forces Fxo and Fxi acting on the outer and inner tracks are expressed respectively by the following equations [1]:

( )∫ ∫−+

−+− −

−−−= oy

oy

oscl

scl

b

b

Kjoyo dydxeF

2/

2/

2/

2/ 111/ sin1 δµσ ,(17)

( )∫ ∫−+

−+− −

−−−= oy

oy

iscl

scl

b

b

Kjiyi dydxeF

2/

2/

2/

2/ 222/ sin1 δµσ , (18)

( )∫ ∫−+

−+− −

−−−= oy

oy

oscl

scl

b

b

Kjoxo dydxeF

2/

2/

2/

2/ 111/ cos1 δµσ , (19)

( )∫ ∫−+

−+− −

−−−= oy

oy

iscl

scl

b

b

Kjixi dydxeF

2/

2/

2/

2/ 222/ cos1 δµσ , (20)

where σo and σi are the normal pressures on the outer and inner tracks respectively, µ is the coefficient of friction between the track and the ground, and δ1 and δ2 are the angles between the resultant sliding velocities and the lateral directions of the points on the outer and inner tracks respectively.

The turning moments MLo and MLi due to the longitudinal shear forces acting on the outer and inner tracks with respect to Ov are expressed as follows [1]:

( )∫ ∫−+

−+− −

−−

+−= oy

oy

oscl

scl

b

b

KjoLo dydxexBM

2/

2/

2/

2/ 111/

1 sin12

δµσ , (21)

( )∫ ∫−+

−+− −

−−

+−= oy

oy

iscl

scl

b

b

KjiLi dydxexBM

2/

2/

2/

2/ 222/

1 sin12

δµσ , (22)

The moments of turning resistance Mro and Mri due to the lateral shear forces acting on the outer and inner tracks with respect to O1 and O2 are expressed as follows [1]:

( )∫ ∫−+

−+− −

−−−= oy

oy

oscl

scl

b

b

Kjoro dydxeyM

2/

2/

2/

2/ 111/

1 cos1 δµσ , (23)

( )∫ ∫−+

−+− −

−−−= oy

oy

iscl

scl

b

b

Kjiri dydxeyM

2/

2/

2/

2/ 222/

2 cos1 δµσ . (24)

The angles δ1 and δ2 are defined by the following set of equations:

( )( )[ ] ( )2

12

1

11

2/

2/sinzozx

ozx

yrxcBR

rxcBR

Ω+−Ω+++′′−Ω+++′′

=ω

ωδ , (25)

( )( )[ ] ( )2

22

2

22

2/

2/sinzizx

izx

yrxcBR

rxcBR

Ω+−Ω+++′′−Ω+++′′

=ω

ωδ, (26)

( )[ ] ( )21

21

11

2/cos

zozx

z

yrxcBR

y

Ω+−Ω+++′′Ω−=

ωδ , (27)

( )[ ] ( )22

22

22

2/cos

zizx

z

yrxcBR

y

Ω+−Ω+++′′Ω−=

ωδ . (28)

Wong suggests that the coefficient of lateral resistance µt in equation (5) can be derived by equating the sum of the moments of turning resistance Mro and Mri computed from equations (23) and (24) to Mr in equation (5):

riror MMM += . (29)

IV. TRAVERSABILITY CRITERION To identify vehicle capabilities on different terrains it is

important to compute the maximum tractive effort that a track can develop before the track slippage reaches 100% slip and consequently the vehicle loses control. The tractive effort is expressed by [1]:

( ) ( )

−−+= − Kile

ilKWAcF /11tanφ , (30)

where A is the track contact area ( = bl) and i is the track slip.

Thus, the maximum tractive effort is computed when the slip i is equal to 100%, i.e.:

( ) ( )

−−+= − Kle

lKWAcF /

max 11tanφ . (31)

As can be seen from previous equations the slip i and the sprocket angular velocities ωo and ωi of the outer and inner tracks, respectively, are required in the analysis. Thus further kinematic equations need to be defined to complete the system of equations. According to Wong [1] the angular velocity ratio Ks for a given turning radius R' and vehicle velocity V is expressed as follows:

( )( )( )( )o

is iBR

iBRK

−−−+

=1'21'2 . (41)

Thus

( ) ( )[ ]iosi iiKr

V−+−

=11

2ω and iso K ωω = , (42)

where io and ii are the track slips of the outer and inner tracks respectively.

The slips io and ii of the outer and inner tracks can be numerically computed using equation (30) since the thrusts can be found from equations (9) and (10) and consequently the sprocket angular velocities are computed from equations (41) and (42) for a given vehicle velocity V and turning radius R'. If the given trajectory specifications (V and R') for a tracked vehicle, with known specifications, that is traversing on a soil, with identified properties (K, c, φ), causes any of the tracks to reach 100% slip or over, then it is not possible for that vehicle to conduct that turning manoeuvre.

V. SIMULATION RESULTS A MATLAB based simulation was constructed according

to the kinematic and dynamic equations, as previously

282

described, and the general theory by Wong [1]. To ensure that the simulation gives valid results, its results are compared with experimental data and simulation results of previous developments such as the empirical models used in [2]. As mentioned previously, Crosheck’s empirical formula to estimate µt as a function of turning radius was used by Kar [2].

100

101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Turning Radius (m)

Mu t

KAR [2] 0.0447 m/s 2.235 m/s 4.47 m/s 6.25 m/s

Figure 6 Plots of lateral coefficient of resistance vs turning radius at

different vehicle velocities according to the same specifications as used by Kar [2]. Additional soil parameters are: K = 2.5cm, c = 1.04 kPa, φ

= 28º and µ = 0.4.

Figure 6 shows the variation of the estimates of the coefficient of lateral resistance µt, with turning radius, deduced from Kar as well as results of µt estimates from our simulation. The vehicle specifications and soil properties used by Kar as well as in our simulation can be found in Table 1 and Table 3 respectively in the Appendix. As shown in the figure, it is evident that vehicle velocity has significant influence on the value of µt. Nevertheless, the simulation results form enveloping curves, along the velocity range, to the estimates made by Kar.

100

101

102

-60

-40

-20

0

20

40

60

Turning Radius (m)

Thr

ust

(kN

)

0.0447 2.235 4.47 6.25V = m/s

Fo

Fi

Fmax

Figure 7 Plots of outer and inner track thrusts vs turning radius at different vehicle velocities according to the same specifications as in

Figure 6.

Figure 7 shows the resultant outer and inner track thrusts based on the estimates of µt as shown in Figure 6. The maximum tractive effort Fmax, which the vehicle can achieve on the soil before the tracks reach 100% slip, is also shown in Figure 7. This is given as a guide to vehicle capabilities as well as for predicting whether a particular maneuver is possible to achieve or not. Previous simulation results such as the ones shown in Figure 3 demonstrated that the difference in thrusts is reduced as the vehicle velocity increases for the same turning radius. This is only true if the estimates of µt are insensitive to vehicle velocity, but in reality they are not. One of the limitations of the simulation developed is that the convergence of the µt estimates is less successful at radii where the thrusts are close to the maximum tractive effort or at high vehicle velocities.

Figure 8 shows the variation of outer and inner sprocket torques with turning radius of the experimental field measurements on a sandy terrain taken from Ehlert et al. [3]. Predictions from our simulation are also shown in the figure. The vehicle specifications and soil properties used by Ehlert as well as in the simulation can be found in Table 2 and Table 4, Appendix. The additional soil properties, K and µ, that are required in the simulation were not provided in the reference, thus the values used in our simulation were derived by fitting the simulated results to the measured data given by Ehlert. The figure shows that the simulation results are in good agreement with the experimental values provided. Due to the shortage of experimental data at various vehicle velocities, it is not possible to validate the developed simulation along the range of velocities that the vehicle can achieve. Ehlert claimed that that the sprocket-torque-characteristic of tracked vehicles traversing on soft terrain is in principle similar to traversing on firm ground. His claim is true with regards to the shape of the curves but may not be true with regards to torque values at different velocity ranges.

100

101

102

-15

-10

-5

0

5

10

15

20

25

Turning Radius (m)

Tor

que

(kN

m)

EHLERT et al. [3] Simulation

Mi

MoVelocity (V = 14.0 m/s)

Figure 8 Plots of outer and inner sprocket torques vs turning radius

according to the same specifications as used by Ehlert et al. [3]. Additional soil parameters are: K = 2.5cm and µ = 0.33.

283

VI. CONCLUSIONS AND FUTURE WORK The kinematics and dynamics involved in a steady-state

skid-steering manoeuvre were investigated and discussed thoroughly. Wong’s general theory has been utilised to estimate the coefficient of lateral resistance µt for tracked vehicles on soft deformable terrain. A simulation to predict track thrusts during a steady-state turning manoeuvre is presented. The simulation takes into account soil properties, vehicle dynamic behaviour and track slips and is based purely on analytical modelling. The results of the simulation are compared with previously developed empirical models as well as experimental field measurements. The results show good agreement with experimental data as well as superior estimates to previously developed estimation models. Future work will focus on extending the simulation so as to predict, in real time, the traversability of tracked vehicles traveling on unknown terrain. This will require interfacing the simulation with available online soil parameter estimation and online slip estimation algorithms. The effects of ground sloping and non-steady (dynamic) turning manoeuvres will also be studied. The objective is to develop a trajectory tracking strategy based on the analytical models discussed and developed in this paper.

VII. ACKNOWLEDGEMENT The author wishes to acknowledge the support by N.

Kurdi, Z. Song and S. Hutangkabodee.

VIII. REFERENCES [1] J. Y. Wong, Theory of Ground Vehicles. John Wiley & Sons, Third

Edition 2001 [2] M.K. Kar, “Prediction of Track Forces in Skid-Steering of Military

Tracked Vehicles,” Journal of Terramechanics, vol.24, no. 1, 1987 [3] W. Ehlert, B.Hug, and I.C. Schmid, “Field Measurements and

Analytical Models as a Basis of Test Stand Simulation of the Turning Resistance of Tracked Vehicles,” Journal of Terramechanics, vol.29, no. 1, 1992

[4] C.F. Chiang, “Handling Characteristics of Tracked Vehicles on Non-Deformable Surfaces,” Master Thesis, Ottawa-Carleton Institute for Mechanical and Arospace Engineering, Carleton University, 1999

[5] Z. Song, S. Hutangkabodee, Y. H. Zweiri, L. D. Seneviratne and K. Althoefer “Identification of Soil Parameters for Unmanned Ground Vehicles Track-Terrain Interaction Dynamics,” SICE Annual conference in Sapporo, 2004

[6] K. Iagnemma H. Shibly S. Dubowsky “On-Line Terrain Parameter Estimation for Planetary Rovers,” Proceedings of the 2002 IEEE international Conference on Robotics & Automation, Washington, DC, 2002

[7] C. Brooks, K. Iagnemma, and S. Dubowsky “Vibration-based Terrain Analysis for Mobile Robots,” Proceedings of the 2005 IEEE International Conference on Robotics and Automation in Barcelona, Spain, 2005

[8] M. G. Bekker, Theoly of Land Locomotion, University of Michigan Press, 1956

[9] Steeds, W., 1950, Tracked Vehicles (in Three Parts), Automobile Engineer, pp 143-148, pp 187- 190 and pp.2 19-222.

[10] Z.B. Song, Yahya H Zweiri, Lakmal D Seneviratne and Kaspar Althoefer “Driver Support System Based on a Non-Linear Slip Observer for Off Road Vehicles,” IFAC, 2005

[11] A. Howard, H. Seraji and E. Tunstel “A Rule-Based Fuzzy Traversability Index for Mobile Robot Navigation,” IEEE Intl. Conf. Robotics and Automation. Seoul, Korea, 3067-3071, 2001

IX. APPENDIX

IX.1 Vehicle Parameters Total Weight (W) 178 kN Cone Index (CI) 828 kPa Height of CG (h) 1.270 m Tread Width (B) 2.3495 m Track Length (l) 3.4544 m Track Width (b) 0.4445 m Sprocket Radius (r) 0.254 m

Table 1 Vehicle specifications taken from Kar [2]

Total Weight (m) 25500 kg Cone Index (CI) 814 kPa Height of CG (h) 1.3 m Tread Width (B) 2.54 m Track Length (l) 3.8 m Track Width (b) 0.45 m Sprocket Radius (r) 0.32 m

Table 2 Vehicle specifications taken from C.F. Chiang [4]

IX.2 Soil Parameters Cohesion1 (c) 1040 Pa Angle of Internal Shear1 (φ) 28º Shear Deformation Modulus1 (K) 0.025 m Coefficient of motion resistance in the longitudinal direction2 (µr) 0.0765

Coefficient of friction (µ) 0.4 Table 3 Soil properties for dry sand according to Kar [2]

Cohesion (c) 0 Pa Angle of Internal Shear (φ) 25º Shear Deformation Modulus1 (K) 0.025 m Coefficient of motion resistance in the longitudinal direction2 (µr) 0.0854

Coefficient of friction (µ) 0.33 Table 4 Soil properties for dry sandy loam according to Ehlert et al. [3]

1 Dry sand soil properties were deduced from [1] 2 µr is estimated using Wismer & Luth as mentioned in [2]

284