Study on steerability of articulated tracked vehicles — Part 1. Theoretical and experimental analysis

Journal ofTerramechanics, Vol. 23, No. 2, pp. 69-83, 1986. Printed in Great Britain.

0022--4898/8653.00+0.00 Pergamon Journals Ltd. © 1986 ISTVS

STUDY ON STEERABILITY OF ARTICULATED TRACKED VEHICLES PART 1. THEORETICAL AND EXPERIMENTAL ANALYSIS -

-

K. WATANABE* and M. KITANO'~

Summary--This paper presents theoretical and experimental analysis of steering performance of articulated tracked vehicleson levelground. A mathematicalmodel for predictingthe steerabilityof articulated units has been developed and computerized for numerical application. The accuracy of the analog has been verified by scale model tests. From the results of the simulation and scale model tests it was found that steerability was significantly improved and required sprocket torques for steering and track slippage were considerably decreased in articulated tracked vehicles when compared with a single and coupled tracked vehicles. INTRODUCTION A TRACKED vehicle exhibits excellent mobility in off-road application because of its low ground pressure. It is normally steered by slipping the tracks with some form of steering device. This means the sprockets require considerable power. Track slippage and sinkage increase power requirements when steering on soft ground. Articulated tracked vehicles with articulated steering systems have been recently developed in order to improve steering performance of skid-steered tracked vehicles. For example, the BV206 [1] is an articulated tracked vehicle with a hydrostatic articulated steering system. The steering dynamics and mechanisms of these vehicles are complicated, so a theoretical analysis of their steering motion has never been presented. In previous papers, we have presented the available mathematical models of single tracked vehicles [2, 3] and coupled (non-articulated) tracked vehicles [4] for predicting the steering performance in stationary and non-stationary turning motion. The equation of plane motion of the coupled tracked units was established and numerically solved on a computer for various steering conditions. The model was experimentally verified on actual vehicles and scale models. It was found that the velocity of the vehicles, their coupled point, steering time lag and mass ratio have an important influence on the steering performance. We also found a coupled tracked vehicle is inferior in steerability when compared to a single tracked vehicle. In this paper, a mathematical model for predicting the plane turning motion of articulated tracked vehicles has been developed and compared with turning characteristicsofsingle and coupled tracked vehicles. In order to validate the actual application of the mathematical

*Research Associate, Department of Mechanical Engineering, The National Defense Academy, 1-10.20 Hashirimizu, Yokosuka, 239 Japan. tProfessor, Department of Mechanical Engineering, The National Defense Academy, 1-10-20 I-Iashirimizu, Yokosuka, 239 Japan. 69

70

K. WATANABEand M. KITANO

model, a scale model was used in turning maneuver tests upon the hard surface ground. The turning radius, sprocket torques, track slippage and sinkage were measured. As a result, it was found that mobility, required sprocket power, track slippage and sinkage in steering on soft ground were significantly smaller in articulated vehicles as compared with single and coupled tracked vehicles, and the steerability of articulated tracked vehicles was excellent at high speed. MATHEMATICAL MODEL OF PLANE MOTION 1. Coordinate system and assumptions Figure 1 shows the coordinate systems and the motion of articulated tracked vehicles on the level ground. The X-Y coordinate system is fixed on the ground and its origin of axes coincides with the center of gravity of the front units at time zero. The origins, 0,0' of bodycentered axis systems x-y, x'-y' are fixed at the center of gravity of each vehicle respectively. The model consists of the front vehicle of mass m and the rear vehicle of mass m'. The length of the track-ground contact and the tread of the tracks are denoted by L,L' and B,B' respectively. The heights of the center of gravity and articulation point measured from ground surface are H J-/' and Hh. The distances from the articulation point to the center of gravity of front and rear vehicles are defined by I a n d / ' . In creating the mathematical model for plane motion of the articulated units, the following assumptions have been made; 1. n road wheels with independent suspensions are fastened on each side of the hull at regular intervals. 2. Frictional coefficient between track and ground is given by # = ~0 (1-e-kg, where the parameters #0, S and K denote the maximum friction coefficient, track slip ratio and a constant, respectively. 3. The vertical loads in the interface between tracks and ground are concentrated under the road wheels. 4. Steering input is given by articulation steering angle. 5. Air resistance and gyroscopic effects due to rotating parts are neglected in the interest of simplicity.

2. Kinematics of vehicles and tracks In Fig. 1, let Vx and Vy be components of total velocity V of the vehicle combination. The yaw angle, side slip angle and directional angle of the course are defined by 0,/3 and ~o, respectively. Then the slip velocities of the track under arbitrary road wheels can be represented schematically by the solid arrows as shown in Fig. 2. A dot over a variable denotes the first derivative of the quantity with respect to time, and 0 is yawing velocity. The velocity V and forward and lateral accelerations ax, ay of the center of gravity of the front vehicle in terms of the moving axis (x,y) are;

(2)

vy- vxo

J

STEERABILITYOF ARTICULATEDTRACKEDVEHICLES / ~lrv 4

71

Front veh,cl.e

~,~h, C'

\ I

e

FIG. 1.

,~

Y'

L'

"~'~/,t

Reor vehicle

e'

Coordinatesystemof aniculat~ tracked vehiclesin turning motion.

X

Fs¢. 2.

The slip motion of tracks.

As mentioned above, steering of an articulated vehicle is accomplished by relative turns in the front-rear vehicle plane. Therefore, from the kinematic diagram in Fig. 3, the relationships between the articulate angle ¢, yaw angle 8 and velocity components of the articulation point can be written as follows,

72

K. WATANABEand M. KITANO

0'-0-~

Vx'= Vxcos~" + ( Vx + ~ t~)sin~"

t ~ ~t

Vy

t 0 -

(3)

Vx sin~" + (Vy + l~O) cos~"

h

t /x,

~.///~

v~cos ~ +( v**le)s,n r~ ".,1/

/"

v~

FIG. 3. The velocitycomponents of articulation point. The side slip angle B and the side slip angular velocity/3 are 13 = tan -1

(VylVx)

] (4) v,

The positions (X, Y) of the front vehicle are obtained by integration of the velocity V with respect to time, t

X = Xo - f v

1

cos ~p dt l

o t

Y= go - f , v hence

¢ = 0 -/3

sin ~0 dt

(5)

] (6)

STEER.ABILITYOF ARTICULATEDTRACKEDVEHICLES

73

where X0 and Y0 denote the initial position of the front vehicle. The transverse axis passing through the center of rotation is defined by the ~ axis. The distance between the origin of the body-centered axis and .~ is the offset D, that is

D = Vy/O

(7)

and the radius R of the trajectory of the front vehicle is

g

(8)

=

The corresponding equations for the rear vehicle can be developed in the same fashion as for the front vehicle, and the symbols referred to the rear vehicle are indicated by the mark 66t95 In order to obtain the friction forces between the track shoe and the ground, the slip velocities and displacements of track have to be determined first of all. Let V,1 and I/,2be the running velocities of the inner and outer tracks, respectively. The x,y components V~x/,Vsytof the slip velocity of the track shoe under the (id) road wheel can be written,

:

rx

+-

TBo

v,/

-

i-1 n-1

1/ 2

(9)

LO (]= 1 ; - , j = 2 ; + )

where subscript i indicates the order of road wheel from the front of vehicle, a n d j indicates the inner track when j= 1 and the outer track when j=2. Assume that eu is the slip angle of the track shoe with regard to the x axis under the (ij) road wheel. The geometrical relationship of track slippage is

~ i/ = c°s-1 (Vsx/ / ~gs2x/ + Vs2vi .

(10)

Also, the slip radii ,4/of the tracks are obtained as that outlined in the previous paper [2--4].

a/ -- vsx//0.

(1 l)

3. Load distribution and frictional forces along the tracks In order to determine the shear stress in interaction between the track and ground the normal stress along the tracks has to be defined. As indicated in Fig. 4, the longitudinal, lateral and vertical forces acting at the center of gravity of each vehicle are m~,x, may and mg, (m'~x, m'~v and m'g for the rear vehicles). The components of reaction forces in the x,y direction acting at the articulation point are Px,Py respectively. The load distribution, P,j acting on the shoes under the (id) road wheel during a turn may be defined as follows [2, 3];

74

K. WATANABE and M. KITANO

Z'

2

Y'~' H

8'q

Z'

m'a',

Z

x'

-

~P,=j.

?

L

I_'

Reor vehicle

Front vehicle

i

FIG. 4. Load distribution analog on track shoes. for the front vehicle,

-

Pi/

mg +_ 1 nB ( m a y H + g ' H h ) 2n

3 ( n + 1 -2i)

n(n + 1) L

(max H + Px Hh)

(12)

(/=I + , / = 2 . - ) for the rear vehicle,

P 'i/

-

m'g 2 n'

1 (m'a'y H' +e;,Hh) +- n ' B '

3 (n' + 1 - 2i) n' (n' + 1) L' (m'a'xH" - - PxH'h)

(13) f/=l.+,/=2 . - ) where, m and m' are the masses of the vehicles, and g is acceleration due to gravity. The shear strength o f earth materials varies greatly for different types o f soil and is dependent on the confining pressure and time rate o f loading (shearing). The friction coefficient o f the tracks, however, is generally approximated by a pull-slip equation as the following f o r m [6, 7] /~=At o (1 - e - K S )

(14)

where /% and K are positive constants determined from pull-slip tests. ~t0 is a m a x i m u m friction coefficient at slip ratio S = I , and K is a constant determined by the cohesive and frictional properties o f soil.

STEERABILITYOF ARTICULATEDTRACKEDVEHICLES

75

Then the x and y components of the frictional forces of the tracks under the (ij) road wheel are expressed respectively as follows; for the front vehicle;

Q x i / = P i~. u

cos ($i/ + ") ] (15)

Qyij = Pi/" u ~in ($ij + '0 for the rear vehicle;

,

,

Qxu =P'//'u cos($ u +~r)

) (16)

f

Qyi/

t

=

l

P i~ " I~ sin (~ii

+

lr)

4. Balance of forces of differential system Figure 5 shows the forces being exerted on the articulated tracked vehicles in turning maneuvers. As the driving power is transmitted through differential systems to the driving sprockets, it may be considered that output torques of both sprockets are equal. However internal friction forces are developed in the differential gears and sprocket-track interface in compliance with track tension. If F, and Fn denote internal friction forces of the two driving sprockets, the circumferential forces of each sprocket El, F2 can be written as follows; FI =1;'[1 + ~, Qxi l i=1 (17) n

F2 =Ff~ + Y, Qxi2 i= 1 £ Or `

X

.~2" #,.~"."-x ,,.~".?-./~Yo' ,," ,4,,,,~N - ,,.~/. ,/'~-.,

X-"

0~,~, / FIG. 5.

~-- :-,'~

Rear ~ehlcfe Forces and moments being exerted on articulated vehicles.

76

K. WATANABE and M. KITANO

hence II

ll

F{, + Z Qxil =F¢.= + Z Oxi2 i=1

(18)

i=1

It is generally considered that the internal friction forces Fa, Ea for both sprockets have the same value. Therefore, the total friction forces o f both tracks under the road wheels can be written respectively as follows;

i=1

Qxil = ~' Qxi2 i=1

(19) II

t

H

t

~-" Qxil = ~" Qxi2

i=1

i=1

where, Qx,j,Q'xo are x,x' c o m p o n e n t s of friction forces o f the track under the (id) road wheel.

5. Equations of motion F r o m Fig. 5, the forces Px,P, acting on the articulation point and the steering torque M a r e defined as follows;

Px =Px cos~" + P y sin~" Pv = P x sin~" - P'v cos ~"

(20)

' t

M=M '

The equations of motion of articulated units can be expressed by the followingequations from the dynamicbalance betweenall forcesand momentsabout the z,z' axes acting on two articulated units, for the front vehicle; n

2

m((rx+VyO) = Z Z Q x i / - P x - ( R l i=l /=1

n

+R2)

2

m ( 1 ) y - VxO) = Z E Q,,i!-ev i=1

/=1

"

"

(2l) B

[z "0"=-

n

n

2

~ (Qxi2-Qxil)- ~ ~ Qyi] (

2 i--1

i=1/=1

_py~ + B

S (R,-R~)+M

1

i-

2

-n --1) L

1

STEERABILITYOF ARTICULATEDTRACKED VEHICLES

77

for the rear vehicle; n'

t

m' (It x + V y 0') =

2

z P.~j+e:,-(R'l +n':) P

I

i=l]=l

n 2 m ( ~ - v ; b'l = Z z Q'>,~/ - e ) i=Ii=I I

*

t

P

)

>L,j

(22)

I zt 0., =

F

t n' 2 1 -- i=11=1 Y-" y" Qyi/ ( - ~ -

t

2 i--I ( Q x i 2 - Q x n )

+ e y, £, + TB

( R'I - R ' ~ ) -

i-I n'l

M'

By substituting equations (1)-(20) into equations (21)-(22), the equations of motion of an articulated tracked vehicle may be represented directly in terms of the unknown quantities

-

m + m' 0 - r a ' £ 'sin~

_

0

- m'~ sin~"

-

m + m'

m(~ + ~'cos~')

m' £

I z + m' ~ (~ + £' cos ~')

- F (l)q =

m'(~ + ~' cos ~') + I ' z + m ~ ( ~ ' + ~ c o s ~ ) _

0 B

F(1), F(2) and F(3) are as follows; F ( 1 ) ---/(1) - f ( 4 )

cos ~"- / ' ( 5 ) sin ~ - m' £' sin ~ .'(

F(2)=f(2)-f(4)

sin~" + f ( 5 ) e o s ~ ' + m ' £ ' cos~'.~"

F ( 3 ) = f ( 3 ) - £ f ( 4 ) sin ~"+ ~f(5) cos ~'+f(6) + ~'f(5) +(m'~'

cos~'+/'z + m ' £ ' 2 ) ~ ""

where, y(1)-(f(6) are written as follows;

n

2 7., Qxo - m V y O i=l ]=l

f(l) = Z

n 2 f ( 2 ) = Z Y~ Qyi]+ m v x o i=l /=l

- (R, + n 2 )

m

F

(2) 1

F

(3) 1

(23)

78

K. WATANABE and M. KITANO n 2 1 ~_, Qvi] i Z=1/:1 " ( -~ -

f ( 3 ) = S - B i=1 (Qxi2 - Q x i l ) B

+ ~

f(4)=m'(

i-I n-I

)L

(Rt - R~)

I (V v + ~ t ~ ) c o s ~ ' -

n' 2 , ~ sin~" I * i Z = l i Z = l Q y , / -

, O, m Vv

+ (R't + R ~ )

f(5)=m'~l(Vy

+ ~t~) sin ~'+ V~ c ° s ~ ' l + •

,

f(6)=-~-

,

i= 1 ( a x i 2 - Q x i l )

i=1/=1

,

-

n' 2 r ~ Z Q y i / + m' V'x O'

1

i=1,,=1 Q y i / ( ~

i-1

L'

-ffr~_1 )

B'

+ ~-- ( R I ' - R 2 ' ) . NUMERICAL ANALYSIS In order to c o m p a r e steerability of articulated vehicles with coupled (non-articulated) vehicles, a numerical analysis has been carried out in regard to the vehicle's characteristics as listed in Table 1. Figure 6 shows an example of the numerically c o m p u t e d results of trajectories for the front vehicles of the articulated and coupled vehicles. Solid lines represent the trajectories of articulated units and dotted lines are for coupled units. TABLE I.

PARAMETERS OF THE ACTUAL VEItlCLE

Ground contact length Vehicle width Mass Height of C.G Numbers of wheel Distances from articulation point to C.G.

L=L' B=B' m=m' H=H' n=n ' I=1'

3.12 m 1.68 m 5450 kg 0.88 m 5 3.12 m

Steering input to articulated units is given by

~" = 8t

(24)

where 6 is angular velocity of articulation and t is time. Numerical analyses have been performed for various steering rates of 6=2,4,6 °/s. It is clearly seen from Fig. 6 that the trajectories depend upon steering rate 6 in the case of articulated units. Yawing velocity increases with respect to steering rate.

STEERABILITYOF ARTICULATEDTRACKED VEHICLES

79

Articulatedvehicle 2

0

s ~ l~ Cvehceil~o u . p l e d

ratio e=118

=40"/

9" I /

//t t ,;!,',' /

• \

/

rs'3~ilt 'l \ - ,,) ,'~, ~ I,t~

/ tlus

~t

1

]

L_

%.

lOs~

/

/

/

~

O

I

/

IX', \

I6 -I lu II

/

I._>- ,'1

L"/

s

I0

20

N

0

X oxis (m)

FIG. 6. Theoreticalturning paths of articulated vehicleand coupled vehicle. In the case of coupled vehicles, the steering input is given by varying the track velocity of the front vehicle as follows;

Vti=Vo +

V, sin ( 7n tits )

Vo

(25)

and where the final steering ratio of the front vehicle becomes ~= 1.18 at three various steering periods T~=1,2,3 s. It is found from Fig. 6 that the steering performance of coupled vehicles is influenced by the final steering ratio ~ and that is independent of steering times, and it takes a longer time to reach a stationary turning circle. Consequently, it is considered that the steering response is inferior as compared with the articulated vehicle. Figures 7 and 8 show the non-dimensional tractive forces and track slip velocities of the articulated vehicle in comparison with those of the coupled unit, when the steering maneuvers of each vehicle have been set so that each vehicle would be on the same path. The parameters G and G' are the weights of the front and rear units of the articulated vehicles. As mentioned above, in the turning motion of tracked vehicles, tractive forces are closely connected with track slip velocities. The tractive forces and track slip velocities of the articulated unit are very small, especially in the early stage of steering, as compared with those of the coupled unit. EXPERIMENTAL PROCEDURE So as to evaluate the steerability and mobility of the articulated tracked vehicle, the scale model was used in maneuver tests. The overall view of the scale model is shown in Fig. 9. Table 2 shows the parameters of the scale model. The center of gravity is located on the body

80

K. WATANABE and M. KITANO

03

// L~~ u

/

02

OI

5

V=2m/s L/(L*~')=05

\J

GYG : I

I---- F2/G

-/

. . . . . .F ,./ G ,

L -----~b

:2:

:-~ J

[2-_

F2/G

-~ ~ /

I

-0 r --

xi

/"

/

Steering compteted

/

F,/G

I

xD

-02

-03

--

ArticuLated

Steering completed i 2

0

-----

,

CoupLed

i

E

4

6

! i

f

I

8

I0

12

T i m e t (s)

F1G. 7.

Tractive forces of articulated units and coupled units.

A

--

004

---

. . . .

o o

ArticuLated

V =2 m/s

CoupLed

~/(~+~'):05

G'IG =I

002

/"

-4

,' t

"'-~"

0

.t

g

,z"

E 15

\

i

SteerJng completed

I Steeling ~',, I I c o m p l e t e d

l

z

P

I

I

2

4

i

i

l

6

8

I0

Time f (s) F I G . 8.

Slip velocity of articulated units and coupled units.

FIG. 9.

The overall view of the scale model.

12

STEERABILITY O F A R T I C U L A T E D T R A C K E D VEHICLES

TABLE 2.

PARAMETERS

$1

OF T H E S C A L E M O D E L

G r o u n d contact length Vehicle width Mass Height of C.G. Numbers of wheel Distances from articulation po.int to C.G.

0.43 0.29 55.5 0.11

L=L '

B=-B' m=m'

H=/'/'

m m kg m

n=n '

5

1=/'

0.54 m

center. The five road wheels on each side of each unit have independent spring suspensions. The revolutions of both engines and transmissions of front and rear units are synchronized by a propcUcr shaft. The power is transmitted through the differentialgear to sprocket axes and both side sprocket torques are balanced. The connecting plate in which articulationangle can bc selected freelyis used to conduct stationary turning motion tests. In the non-stationary maneuver motion tests, the vehicle is handled by the controlled articulation joint with a stepping motor. The steering input pattern is a linear function such as ~=~t. Turning tests were carded out on hard level ground by using the scale model. The surface of the hard ground was formed by a wooden plate covered with a synthetic resin film about 1 mm thick. The vehicle tracks were covered with rubber pads about 2 mm thick in order to generate the efficient friction performance and ground pressure of tracks. Stationary turning motion and non-stationary maneuvering tests were conducted when the articulation point was equidistant from the center of gravity of each vehicle. In order to measure the turning radius and the trajectories of the centers of gravity, marker lamps were attached over the center of gravity of each unit. The required sprocket torques for steering were measured by using sprocket axes with strain gages. EXPERIMENTALRESULTS Figure 10 shows the comparison between the experimental results and numerical prediction of tractive forces of articulated units, coupled units and a single unit during 03

~--G "--

__

F2/G z~

02

Ar~cuLote~ F~I G

OI u .~

Stogie

i 0 .o-"

.o

G'/G=I ,cL , '1,o5

-01

"8

__ _ , x . . . .

o Z -0.2.

z~--

SingLe ,5,/G i

0

I I

1

I

2

3

4

Non dimensional turning radius (RIL)

F~o. I0.

Relationships between tractive forces and turning radius.

82

K. WATANABEand M. KITANO

stationary turning motion. It is clearly seen that for a single and coupled unit, large driving forces and braking forces are developed in the outer and inner tracks because of the skidsteering. For the articulated tracked vehicles, driving forces of both tracks were balanced by differential gears and the required sprocket torques for steering were considerably smaller than those of single and coupled units. Figure 11 shows an example for numerically computed results of the path curve of the center of gravity of the units compared with the experimental trajectory when the vehicle is steered at an articulation steering angular velocity of c5=3.6 °/s and steering time t= 15 s. The marker lamp is attached on the right side of the center of gravity as a clue to elucidate the steering behavior. The white dotted marks on the experimental trajectory trace show the starting and completion positions in steering maneuver. The good agreement between the experimental and theoretical results, as shown in Fig. 11, suggests that the mathematical model is capable of analyzing the steering motion of articulated units.

-4

V : 02 mls : 3 . 6 O/s 6"'/O : I

m

2

v

o

4

3

2

I

o

,k" axis (m) (O) Theoretical

FIG. I I.

result

(b)

E x p e r i m e n t a l result

Trajectories of non-stationary turning motion (experimental and theoretical results).

Figure 12 shows the comparison between the tractive forces of the front and rear units during non-stationary turning motion when the angular velocities of the articulation joint are 6=1.8 °/s and 3.6 °/s respectively. In contrast with smaller tractive forces of the rear units, the tractive forces of the front unit increase with time after the steering maneuver has been initiated. The largest cause for the difference between the tractive forces of the two units is the off-tracking of each trajectory. In other words, the front unit pulls the rear one, causing it to turn in a larger radius. This means that slippage and sinkage of tracks of the front unit increases on soft ground. Consequently, by designing the differential gear to equalize tractive forces of the two units, the mobility and steerability of an articulated vehicle might be improved.

STEERABILITY OF ARTICULATED TRACKED VEHICLES

CoL. "~

Front

."-

Reor

0 12

~ o.o8-

Exper

-----

•

•

o

z~

/ ' ~

V-O 2 7 m / s l/(~+~')°05 G'lG

83

/

I

• I

/, .

/

F~o~......~

~ O04r-

/

C

01--

0

"-...

3

6

Z"

9

.~2r"°

12

15

Time t ( s )

FIG. 12. Tractive forces of both sprocket during a non-stationary turn. -

CONCLUSION

In this p a p e r , a m a t h e m a t i c a l m o d e l o f plane m o t i o n for p r e d i c t i n g the steerability a n d m o b i l i t y o f a r t i c u l a t e d vehicles h a s been d e v e l o p e d a n d c o m p u t e r i z e d f o r n u m e r i c a l applications. The e x p e r i m e n t a l results o b t a i n e d b y utilizing the scale m o d e l agreed with the theoretical results, a n d the validity o f the m o d e l h a s been e x p e r i m e n t a l l y verified in t u r n i n g m a n e u v e r tests. The following results are o b t a i n e d ; I. In c o m p a r i s o n with a single frame vehicle a n d t o w - p i n c o u p l e d units, the r e q u i r e d driving forces o f s p r o c k e t s a n d t r a c k slippage a n d sinkage o f a r t i c u l a t e d vehicles for steering are significantly smaller. C o n s e q u e n t l y a r t i c u l a t e d steering systems p r o d u c e a significant gain in m o b i l i t y a n d steerability on soft terrain which w o u l d be i m p a s s a b l e for a single unit. 2. The m a t h e m a t i c a l m o d e l will enable us to o b t a i n n u m e r o u s d a t a indispensable in designing a r t i c u l a t e d vehicles.

REFERENCES [ I] J. LJUNGGR~N,BV206 A new Swedish all terrain vehicle, ProceedingsofISTVS 7th International Conference, Vol. II, 677-689 (1981). [2] M. KIT^NO and H. JVOZAKI, A theoretical analysis of steerability of tracked vehicles, Journal of Terramechanics 13(4), 241-258 (1976). [3] M. Kn'ANO and M. KUMA, An analysis of horizontal plane motion of tracked vehicles, Journal of Terramechanics |4(4), I 1-24 (1978). [4] M. KIT^NO, K. WATANABE,K. SAWAGASmaAand A. KINou, An analysis of plane motion of articulated tracked vehicles. Proceedings of ISTVS 7th Imernalional Conlbrence, Vol. Ill, 1413-1447 ( 1981). [5] K. WATAN^BEandM. KITANO,Studyonstcerabilityofarticulatedtrackedvehicles, ProceedingsoflSTVS8th International Conference, Vol. II, 901-915 ( 1984). [6] J.E. CaOSH~CK,Skid-steering of crawlers. SAE paper, No. 750552, pp. 1-15. [7] R.D. WlSM~Rand H. J. LUTH,Off-road traction prediction for wheeled vehicles. ASAE Paper, No. 72-619, pp. 1-16.