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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 EngineeringThe Strand, London WC2R 2LS, UK

Email: said.al-milli, k.althoefer, [email protected]

Abstract – Unmanned ground vehicles are widely used in

industries where repetitive tasks or high risk missions arerequired. Such vehicles usually operate on soft deformable

terrains and still require human supervision due to thecomplexity of the interaction between the vehicle and theterrain. The dynamic models currently used in predicting trackforces on such terrains are empirical and out dated. This papercritically investigates the dynamics involved in a steady-stateskid-steering manoeuvre of tracked vehicles and presents anovel technique for predicting track forces on soft terrains. Thesimulation results are in good agreement with previous field

measurements. The results show that the coefficient of lateralresistance varies not only with turning radius but also with

vehicle velocity, an aspect that has not been taken intoconsideration in previous models. Finally, a traversabilitycriterion is presented.

I. INTRODUCTION

Unmanned ground vehicles (UGVs) are widely used inindustries 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. Suchintervention hinders operational speeds as well as potential

applications. To reduce human intervention requires a

thorough understanding of terrain mechanics, vehicledynamics 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 effortare 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 identificationalgorithm is developed for tracked UGVs. The NewtonRaphson 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 vehicmobility and vehicle-terrain interactions [8], W. Steed

(1950) studied the mechanics of skid-steering on firm

ground [9]. J. Y. Wong (1989-2001) has excelled in bot

fields while basing his theory on Bekker and Steeds [1]. Sl

estimation is one of most recent research topics in the fie

of vehicle kinematics and dynamics. A slip estimatio

algorithm has been developed in [10]. The algorithm based on the kinematics model of a tracked vehicle as we

as trajectory measurements.

Traversability prediction is another research topic of prim

importance in UGV automation. Its use is highly required navigation and control of UGVs especially in high rismissions and in planetary exploration. In [11] a rule-baseapproach to extract terrain surface characteristics from a s

of stereo vision cameras is presented. The surfaccharacteristics that are identified are: terrain roughnes

slope, discontinuity and hardness. Due to the lack

analytical modeling, the algorithm has poor identification o

vehicle-terrain interaction dynamics. In [7] an acceleromet

is used to classify the terrain in which the vehicle traverse

on as to overcome traversability and safety issues involve

in uneven, rough, and sloped terrain.

This paper discusses up-to-date dynamic modelling o

tracked vehicles and presents a simulation that analytical predicts the performance of tracked vehicles on so

deformable terrains. A traversability criterion is alsintroduced. Simulation results are verified by comparin

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

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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 ofthe outer and inner tracks may be computed respectively

using the following equations:

B M CF R F oo /r long ++=

, (1) B M CF R F

ii/

r long −+= , (2)

where Ro, Ri is the longitudinal motion resistance force of

outer and inner track respectively, CF long is the longitudinal

component of the centrifugal force, M r 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 followingequation [1]:

β

µ

β

µ

cos

2

cos

'2

2

g

la

gR

lV s

t

y

t

o == , (3)

where l is the track contact length, V is the longitudinal

vehicle velocity, a y is the centrifugal acceleration of thevehicle (V

2/ 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 asfollows [1]:

g

la s

t

yo µ 2

= . (4)

Due to the shifting of centre of turn so, the moment of

turning resistance M r 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 M r is found to be [1]:

−=

2

14 t

yt r

g

aWl M

µ

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]:

r

y

o Bg

hWaW R µ

+=

2, (6)

r

y

i Bg

hWaW R 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 CF long is

found to be [1]:

'2 gR

sWaCF

o y

long = . (8)

Substitution of equations (5-8) into eqs. (1) and (2) give

the thrusts of the outer and inner tracks [1]:

−++⋅

+=

2

14'22 t

yt o y

r

y

o g

aWl

gR

sWa

Bg

hWaW F

µ

µ µ

, (9)

−−+⋅

−=

2

14'22 t

yt o y

r

y

i g

aWl

gR

sWa

Bg

hWaW F

µ

µ µ

. (10)

III. ESTIMATION OF LATERAL COEFFICIENT OFRESISTANCE µT

In the previous section, the coefficient of lateral resistan

is considered constant when predicting the thrusts of thouter and inner tracks. Figure 2 shows plots of the outer an

inner tracks when using equations (9-10), respectively,

conjunction with a constant value of µt and with the vehic parameters given in Table 1, Appendix. It is evident fro

the plots that an increase in turning radius increases th

difference of outer and inner thrusts which is unreasonab

in real situations as experimental results show that th

difference in thrusts is increased with small turning radii anreduced with large turning radii [3].

It is suggested that µt is expressed as a function of turnin

radius [1-3] as well as vehicle velocity. Kar [2] used a

empirical formula, which was developed by Crosheck [4],

estimate µt as a function of the turning radius. The formurequires three constants E 1, E 2, and E 3 that are determine

empirically. This constrains the validity of the formula to th

vehicle specifications and operational conditions that we

used in determining the constants. Slip estimates of th

tracks are also required as inputs to the Crosheck formul

but the equations Kar used to determine track slippage on

relate slip to turning radius. Thus a change in vehic

velocity or soil properties has no effect on the estimation o µt . Figure 3 shows a reproduction of the track thrusts fro

the models used by Kar [2]. Note that the thrust values fo

the outer and inner tracks over lap for velocities 0.0447 m

and 2.235 m/s.

A number of empirical models were developed fo predicting track thrusts, and hence estimating the coefficieof lateral resistance, on firm ground and their simulatio

results were demonstrated as well as compared

experimental test results [3]. Wong derived a Gener

Theory for Skid-Steering on Firm Ground [1] th

analytically predicts track thrusts as well as longitudinal an

lateral moments that are acting on the tracks. Wong’s theor

is an expansion to Steeds pioneering work on skid-steerinof tracked vehicles on firm ground. One of the majo

alterations Wong made was the consideration of th

dependency of shear stress to shear displacement whereSteeds model assumed that the shear stress reaches imaximum value instantly according to Coulomb’s law ofriction. Moreover, Steeds neglected track width in h

model whereas Wong considered the whole contact area o

the track. Wong shows that it is possible to estimate

analytically by using the general theory [1]. Although Won

intended his theory to be applied to tracked vehicles on firm

ground, the principles outlined by Wong to estimate µt w

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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)

T h r u s t o f O u t e r T

r a c k ( k N )

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)

T h r u s t o f I n n e r

T r a c k ( k N )

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)

O u t e r T r a c k T h r u s t ( k

N )

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)

I n n e r T r a c k T h r u s t ( k N )

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, K c = 0.95

kN/mn+2, K φ = 1528.43 kN/ mn+2, µr = 0.0765

The kinematics involved in a steady-state turn of a tracke

vehicle is shown in Figure 4. According to the gener

theory, the shear displacements j Xo and jYo at a point ( x1, yon the outer track in the X and Y directions respectively, wirespect to a fixed frame of reference XY , are expressed bthe following equations [1]:

( )

( )

Ω−−+−

−

Ω−−+

+++′′=

o

z o y

o

z o y

x Xo

r

y scl y

r

y scl

xc

B

R j

ω

ω

1

1

1

1

2/sin

1

2/

cos2 , (11)

( )

( )

Ω−−++

−+−

Ω−−+

+++′′=

o

z o y

o y

o

z o y

xYo

r

y scl y sc

l

r

y scl xc

B R j

ω

ω

1

1

1

1

2/cos

2

2/sin

2 , (12)

where Ω z is the angular speed (yaw velocity) about turnin

centre O, R'' is the lateral distance between O and centre

gravity CG and is equal to 22

o s R −′ , c x and c y are the later

and longitudinal distances between CG and longitudinal an

lateral centre lines of the vehicle hull respectively, r is thsprocket radius and ωo is the sprocket angular velocity of th

outer track. Thus, the resultant shear displacement jo of poi

( x1, y1) on the outer track is:

22

Yo Xoo j j j += . (13)

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

turn. From Wong [1]

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

Wong [1]

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Similarly, the shear displacements j Xi 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

z o y

i

z o y

x Xi

r

y scl y

r

y scl xc

B R j

ω

ω

2

2

2

2

2/sin

12/

cos2

, (14)

( )

( )

Ω−−++

−+−

Ω−−+

++−′′=

i

z o y

o y

i

z o y

xYi

r

y scl y

scl

r

y scl xc

B R j

ω

ω

2

2

2

2

2/cos

2

2/sin

2, (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:

22

Yi Xii j j j += . (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 F yo and F yi as well asthe lateral forces F xo and F xi acting on the outer and inner

tracks are expressed respectively by the following equations[1]:

( )∫ ∫−+

−+− −

−−−=

o y

o y

o scl

scl

b

b

K j

o yo dydxe F 2/

2/

2/

2/111

/sin1 δ µ σ ,(17)

( )∫ ∫−+

−+− −

−−−=

o y

o y

i scl

scl

b

b

K j

i yi dydxe F 2/

2/

2/

2/222

/sin1 δ µ σ , (18)

( )∫ ∫−+

−+− −

−−−=

o y

o y

o scl

scl

b

b

K j

o xo dydxe F 2/

2/

2/

2/111

/cos1 δ µ σ , (19)

( )∫ ∫−+

−+− −

−−−=

o y

o y

i scl

scl

b

b

K j

i xi dydxe F 2/

2/

2/

2/222

/cos1 δ µ σ , (20)

where σ o and σ i are the normal pressures on the outer andinner 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 lateraldirections of the points on the outer and inner tracks

respectively.

The turning moments M Lo and M Li due to the longitudinal

shear forces acting on the outer and inner tracks with respect

to Ov are expressed as follows [1]:

( )∫ ∫−+

−+− −

−−

+−=

o y

o y

o scl

scl

b

b

K j

o Lo dydxe x B

M 2/

2/

2/

2/111

/

1 sin12

δ µ σ , (21)

( )∫ ∫−+

−+− −

−−

+−=

o y

o y

i scl

scl

b

b

K j

i Li dydxe x B

M 2/

2/

2/

2/222

/

1 sin12

δ µ σ , (22)

The moments of turning resistance M ro and M ri due to the

lateral shear forces acting on the outer and inner tracks with

respect to O1 and O2 are expressed as follows [1]:

( )∫ ∫−+

−+− −

−−−=

o y

o y

o scl

scl

b

b

K j

oro dydxe y M 2/

2/

2/

2/111

/

1 cos1 δ µ σ , (23)

( )∫ ∫−+

−+− −

−−−=

o y

o y

i scl

scl

b

b

K j

iri dydxe y M 2/

2/

2/

2/222

/

2 cos1 δ µ σ . (24)

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

equations:

( )

( )[ ] ( )2

1

2

1

11

2/

2/sin

z o z x

o z x

yr xc B R

r xc B R

Ω+−Ω+++′′

−Ω+++′′=

ω δ

, (25)

( )

( )[ ] ( )2

2

2

2

22

2/

2/sin

z i z x

i z x

yr xc B R

r xc B R

Ω+−Ω+++′′

−Ω+++′′=

ω

ω δ

, (26)

( )[ ] ( )2

1

2

1

11

2/cos

z o z x

z

yr xc B R

y

Ω+−Ω+++′′

Ω−=

ω δ

, (27)

( )[ ] ( )2

2

2

2

22

2/cos

z i z x

z

yr xc B R

y

Ω+−Ω+++′′

Ω−=

ω δ

. (28)

Wong suggests that the coefficient of lateral resistance

in equation (5) can be derived by equating the sum of th

moments of turning resistance M ro and M ri computed froequations (23) and (24) to M r in equation (5):

riror M M M += . (29)

IV. TRAVERSABILITY CRITERION

To identify vehicle capabilities on different terrains it

important to compute the maximum tractive effort that track can develop before the track slippage reaches 100slip and consequently the vehicle loses control. The tractiv

effort is expressed by [1]:

( ) ( )

−−+=

− K il e

il

K W Ac F

/11tanφ , (30)

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

Thus, the maximum tractive effort is computed when th

slip i is equal to 100%, i.e.:

( ) ( )

−−+=

− K l e

l

K W Ac F

/

max 11tanφ . (31)

As can be seen from previous equations the slip i and th

sprocket angular velocities ωo and ωi of the outer and inntracks, respectively, are required in the analysis. Thu

further kinematic equations need to be defined to comple

the system of equations. According to Wong [1] the angul

velocity ratio K s for a given turning radius R' and vehic

velocity V is expressed as follows:

( )( )

( )( )o

i

si B R

i B R K

−−

−+=

1'2

1'2 . (41)

Thus

( ) ( )[ ]io s

iii K r

V

−+−=

11

2ω and

i so K ω ω = , (42)

where io and ii are the track slips of the outer and inn

tracks respectively.

The slips io and ii of the outer and inner tracks can b

numerically computed using equation (30) since the thrus

can be found from equations (9) and (10) and consequent

the sprocket angular velocities are computed from equation

(41) and (42) for a given vehicle velocity V and turninradius R'. If the given trajectory specifications (V and R') f

a tracked vehicle, with known specifications, that

traversing on a soil, with identified properties ( K , c, φ

causes any of the tracks to reach 100% slip or over, then it

not possible for that vehicle to conduct that turninmanoeuvre.

V. SIMULATION RESULTS

A MATLAB based simulation was constructed accordin

to the kinematic and dynamic equations, as previous

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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 formulato 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)

M u 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 thevelocity range, to the estimates made by Kar.

100

101

102

-60

-40

-20

0

20

40

60

Turning Radius (m)

T h r u s t ( k N )

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 thrus

based on the estimates of µt as shown in Figure 6. Th

maximum tractive effort F max, which the vehicle can achievon the soil before the tracks reach 100% slip, is also showin Figure 7. This is given as a guide to vehicle capabilities well as for predicting whether a particular maneuver

possible to achieve or not. Previous simulation results suc

as the ones shown in Figure 3 demonstrated that thdifference in thrusts is reduced as the vehicle velocit

increases for the same turning radius. This is only true if th

estimates of µt are insensitive to vehicle velocity, but

reality they are not. One of the limitations of the simulatio

developed is that the convergence of the µt estimates is le

successful at radii where the thrusts are close to th

maximum tractive effort or at high vehicle velocities.Figure 8 shows the variation of outer and inner sprock

torques with turning radius of the experimental fiemeasurements on a sandy terrain taken from Ehlert et al. [3

Predictions from our simulation are also shown in the figur

The vehicle specifications and soil properties used by Ehle

as well as in the simulation can be found in Table 2 an

Table 4, Appendix. The additional soil properties, K and

that are required in the simulation were not provided in th

reference, thus the values used in our simulation we

derived by fitting the simulated results to the measured da

given by Ehlert. The figure shows that the simulation resul

are in good agreement with the experimental valu provided. Due to the shortage of experimental data

various vehicle velocities, it is not possible to validate thdeveloped simulation along the range of velocities that th

vehicle can achieve. Ehlert claimed that that the sprocke

torque-characteristic of tracked vehicles traversing on so

terrain is in principle similar to traversing on firm groun

His claim is true with regards to the shape of the curves b

may not be true with regards to torque values at differevelocity ranges.

100

101

10

-15

-10

-5

0

5

10

15

20

25

Turning Radius (m)

T o r q u e ( k N m )

EHLERT et al. [3] Simulation

Mi

Mo

Velocity (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.

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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 trackedvehicles 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 simulationare 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 vehiclestraveling 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) turningmanoeuvres will also be studied. The objective is to developa 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, ThirdEdition 2001

[2] M.K. Kar, “Prediction of Track Forces in Skid-Steering of MilitaryTracked 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 TurningResistance 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 GroundVehicles 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 IEEEinternational 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 MichiganPress, 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 SlipObserver for Off Road Vehicles,” IFAC , 2005

[11] A. Howard, H. Seraji and E. Tunstel “A Rule-Based Fuz

Traversability Index for Mobile Robot Navigation,” IEEE Intl. Co Robotics and Automation. Seoul, Korea, 3067-3071, 2001

IX. APPENDIX

IX.1 Vehicle Parameters

Total Weight (W ) 178 kNCone 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 mTable 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 Shear 1 (φ) 28º

Shear Deformation Modulus1 ( K ) 0.025 m

Coefficient of motion resistancein the longitudinal direction2 ( µr )

0.0765

Coefficient of friction ( µ) 0.4Table 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 resistancein 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 . [

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

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