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A non-linear dynamic approach to the motion offour-wheel-steering vehicles under various operationconditionsQ Han1 and L Dai2*

1Department of Mechanics, College of Traffic and Communications, South China University of Technology,Guangzhou, Peoples Republic of China2Industrial Systems Engineering, University of Regina, Regina, SK, Canada

The manuscript was received on 24 February 2005 and was accepted after revision for publication on 11 January 2008.

DOI: 10.1243/09544070JAUTO43

Abstract: This research aims to analyse systematically the non-linear dynamic behaviour offour-wheel-steering (4WS) vehicles. A practical non-linear dynamic model of multiple degreesof freedom for 4WS vehicles is established. The non-linear model presented is shown to befeasible for the vehicles under normal and the other operation manoeuvres. Compared withthose of documented investigations, this model may be employed to analyse the motions ofthe vehicle and motions of the vehicle wheels in turning and braking processes together withthe consideration of the effects of air drag and wind. Numerical simulation for the motion ofthe 4WS vehicles under various operation manoeuvres is performed with the modelestablished. Comparison of the behaviours of 4WS and two-wheel-steering vehicles is alsopresented with respect to the inputs on the steering system, the manoeuvrability, the stability,and the relationship between the steering phase and vehicle speed.

Keywords: vehicle dynamics, manoeuvrability, non-linear motion, four-wheel-steeringvehicle, two-wheel-steering vehicle, numerical simulation, non-linear tyres

1 INTRODUCTION

Four-wheel-steering (4WS) vehicles show advan-

tages of smaller turning radius, tight-space man-

oeuvrability, and reduction in driver fatigue. Re-

cently, companies such as General Motors and

Chevrolet have pushed and promoted the use of

4WS techniques, and investigations on the behaviour

of 4WS vehicles have attracted great attention from

the scientists and engineers [14]. Linear manoeuvr-

ing equations were reported for analysing the

motion of 4WS vehicles [5]. Itoh et al. [6, 7]

developed a numerical approach for the steady state

turning of a 4WS tractor. The numerically deter-

mined forces on the tractor tyres were compared

with those obtained in field tests. The stability of a

4WS vehicle is crucial for the operations of positive

and negative phase steering. Lateral motion stabili-

zation for a 4WS vehicle has been reported [8].

Stability and non-linear behaviour of 4WS vehicles

such as Hopf bifurcation were also found among the

recent studies [9, 10]. However, a systematic and

thorough investigation on the response of 4WS

vehicles subjected to the tyre forces generated by

the road surface, the aerodynamics resistance, and

the loadings caused by operation conditions of the

real world is still needed.

The present research proposes an approach for

accurately and effectively analysing the motion of

4WS vehicles. A complete 4WS vehicle model with

multiple degrees of freedom in counting non-linear

effects is presented. The non-linear model estab-

lished is shown to be a feasible 4WS vehicle model

for various manoeuvres (greater lateral accelera-

tions, possibly combined with longitudinal accelera-

tion or braking). The non-linear coupling effects of

sprung and unsprung parts are taken into considera-tion. The model counts the joint effects of operations

*Corresponding author: Industrial Systems Engineering, Uni-

versity of Regina, 3737 Wascana Parkway, Regina, SK, S4S 0A2,Canada. email: [email protected]

535

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and road conditions to the motions, such as the

translation, surge, sway, yaw, pitch, and roll of the

vehicle body under the conditions of various opera-

tion manoeuvres. The four rotational degrees of

freedom of the vehicles four wheels are taken into

account. To obtain a systematic and accurate

analysis of the non-linear dynamics, the high-value

lateral and longitudinal accelerations, the accelerat-

ing and braking processes, the aerodynamic drag of

the vehicle, and the shift of vertical loads due to the

pitch and roll of the vehicle body are investigated in

the present research. The non-linear terms are

naturally introduced into the dynamic system with

considerations of the lateral wheel forces and

geometric relationships relating to the kinematic

motion and slip angles. Based on the analytical

model established, numerical simulations for thedynamic response of the 4WS vehicle are carried out.

The characteristics of 4WS vehicles in comparison

with two-wheel-steering (2WS) vehicles are demon-

strated and analysed.

2 MODELLING OF 4WS VEHICLES

2.1 Model of the vehicle

The 4WS vehicle studied in this research is assumed

to be a symmetric body of mass Mwith four wheels,

as shown in Fig. 1. For such a vehicle system, the

total mass of the vehicle is considered to consist of

Ms and Mu, the masses of vehicle body and the

unsprung part of the vehicle respectively. Also, O9 is

designated as the mass centre of the whole vehicle

Os as the mass centre of the vehicle body, and Ou as

the centre of the unsprung mass, as shown in Fig. 2.

Assume that the vehicle is symmetric about the

longitudinal midplane and all the three mass centres

are located in the symmetric plane.

With the vehicle thus defined, based on Fig. 3, the

velocity relationships can be obtained from Fig. 1 as

U~ _XX~u cosy{vsiny

V~ _YY~u sinyzvcosy1

where u and v are the longitudinal and lateralvelocities of the vehicle in the x and y directionsrespectively, Uand Vare the vehicle velocities in the

X and Y directions respectively, and y is the yawangle. Differentiating both sides of equation (1) gives

ay~YY cosy{XXsiny~ _vvzru

ax~YY sinyzXXcosy~ _uu{rv2

In the above equations, ax and ay are the

projections of the absolute acceleration of point Ou

on the moving coordinate system x, y, z, and r is the yaw angular velocity. The unsprung part is consid-

ered as a rigid body with translational motion and

the sprung part is considered as a mass point, which

is articulated with the unsprung part by a crank. The

crank can rotate with respect to the xaxis and yaxis

to simulate the roll and pitch of the sprung part of

the vehicle.

Figure 1 also shows the relationship between the

roll and pitch of the sprung part of the vehicle. At an

arbitrary moment, the crank is at position AC, which

can be obtained by rotating the crank twice. The

crank is first rotated an angle w with respect to the xaxis, and it is then rotated an angle h with respect to

the y0 axis. Accordingly, the angular velocities of the

three rotations can be expressed as

r~ _yy

p~ _ww

q~ _hh3

Assuming that i, j, and k are the three unit vectors

along the coordinate axes x, y, and z, the position ofthe centre of mass of the sprung part of the vehicleFig. 1 A simplified model of the vehicle body

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can be expressed by a vector rAC. Now

rOC~rOAzrAC~cizrAC 4

where

rAC~h {sinh cosw izsinw j{cosh cosw k 5

The acceleration ac of the centre of mass of the

sprung part is composed of three parts, namely the

acceleration of Ou, the relative acceleration arc, andthe Coriolis acceleration akc, i.e. ac5aec+arc+akc.

From equation (2)

aec~ _uu{rv iz _vvzru j 6

or, in another form

aec~d2rOC

dt2

~{h

2{sin hzw _hhz _ww

2

zcos hzw hhzww

{sin h{w _hh{ _ww

2zcos h{w hh{ww

ii

zh cosw ww{sinw _ww 2 !

j

zh

2cos hzw _hhz _ww

2

zsin hzw hhzww

zcos h{w _hh{ _ww 2

zsin h{w hh{ww i

k 7

Fig. 2 A model of the 4WS vehicle

Fig. 3 Two frames of coordinates

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It should be noted that

drOCdt~{

h

2cos hzw _hhz _ww

zcos h{w _hh{ _ww

h ii

zh cosw _wwjzh2sin hzw _hhz _ww

hzsin h{w _hh{ _ww

ik 8

Therefore, the Coriolis acceleration akc can be

expressed in the form

akc~2r|drOC

dt

~2rk|drOC

dt

~{hr cos hzw _hhz _ww h

zcos h{w _hh{ _ww i

j{2hrcosw _wwi 9

Finally, the acceleration ac, of the centre of mass of

the sprung part can be written as

ac~aeczarczakc~AiizAjjzAkk 10

where

Ai~{h

2{sin hzw _hhz _ww

2

zcos hzw hhzww

{sin h{w _hh{ _ww 2

zcos h{w hh{ww i

{2hrcosw _wwz _uu{rv 11

Aj~h cos w ww{sin w _ww 2 !

{hr cos hzw _hhz _ww hzcos h{w _hh{ _ww

iz _vvzru 12

Ak~h

2cos hzw _hhz _ww

2zsin hzw hhzww

zcos h{w _hh{ _ww 2

zsin h{w hh{ww !

13

The inertia force of the unsprung mass can beexpressed as

Fu~{Mu _uu{rv iz _vvzru j 14

The inertia force of the sprung mass is

Fs~{Ms AiizAj jzAkk

15

The moment about the z axis can be expressed as

Mz~{ Iz_rr{Is

xz_pp

k 16

where Iz is the moment of inertia of the whole

vehicle with respect to the z axis, and Isxz is the

product of the inertia of Ms, and angular velocities r

and p are as defined in equation (3).

The moment about the x axis is

Mx~{ Is

x _pp{Is

xz_rr

i 17

where Is

x is the moment of inertia of the sprung masswith respect to the x axis.

The moment about the y axis is

My~{Is

y_qqj 18

where Isy is the moment of inertia of the sprung mass

with respect to the y axis.

2.2 Model of the tyre force

The additive turning of the rear wheels can be the

turning in either the negative phase or the positivephase. In the negative phase, the rear wheels turn in

the opposite direction to the front wheels whereas,

in the positive phase, the rear wheels turn in the

same direction as the front wheels. To express this

relationship quantitatively, the turning angle ratio Klis introduced. It is defined as the ratio between the

turning angles of the rear wheel and front wheel and

is given by

Kl~dr

df19

where dr is the turning angle of the rear wheel and dfis the turning angle of the front wheel.

In Fig. 4, the velocity of the front right wheel is

expressed as

VfR~Vzr|rOG

~uizvjzrk| ajzd

2j

~ u{d

2r

iz vzar j 20

As such, tan(df+

afR)5

(v+

ar)/(u2

d/2r). Similarly,the lateral slip angles of all four wheels of the vehicle

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can be given by

afR~arctanvzar

u{rd=2

{df

afL~arctanvzar

uzrd=2

{df

arR~arctanv{br

u{rd=2

{dr

arL~arctanv{br

uzrd=2

{dr

21Using the non-linear tyre force model [11]

Ffi~{ C1fafi{C3fa3fi

Fri~{ C1rari{C3ra

3ri

22

where Ffi and Fri (i5R, L, i.e. the subscripts for rightand left respectively) are the tyre forces of the frontand rear wheels respectively. This model is empiricaland the parameters of the model are to bedetermined in the experimental measurements

corresponding to the tyre and road conditions.Variation in the parameters can be easily implemen-ted in the numerical calculations in vehicle beha-viour analysis. With this tyre model, the tyre forcedirections are perpendicular to the centre-lines ofthe tyres. This tyre force model is widely used forvehicle behaviour analysis. In this research, thethree-dimensional non-linear response of 4WS ve-hicles under normal and other operation man-oeuvres of high accelerations is the main focus.Therefore, this model is suitable for the research.However, other tyre force models with considera-

tions of the factors such as dynamic response andslip force [12], tyre saturation [13], pavement effects

[14], and tyre friction [15] can be easily implementedin the vehicle model established in the presentresearch, if so desired.

3 AERODYNAMIC RESISTANCE

A straight-line driving vehicle is influenced by the

side wind and the wind caused by the travelling

velocity of the vehicle. The side wind can be

ignored under normal weather conditions and the

travelling wind can be considered as obeying the

formulae

Fx~1

2CxruAx

Fy~1

2CyrvAy

23

where Cx and Cy are the coefficients of aerodynamic

resistance in the x and y axes respectively, and Axand Ay are the projected areas of the vehicle.

Equation (23) therefore governs the aerodynamic

resistance.

4 MOTION OF THE TYRES AND NORMAL LOADS

4.1 Starting, accelerating, and normal driving

In these cases considered, the governing equations

for the driving wheel (Fig. 5) can be expressed as

Ir _vvrL~T{RFtrL{eFzrL

Ir_vvrR~T{RFtrR{eFzrR 24

Fig. 4 Top view of the vehicle



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and for the driven wheel (Fig. 6)

If _vvfL~RFtfL{eFzfL

If _vvfR~RFtfR{eFzfR25

4.2 Braking

In braking, the equations of dynamics are

If _vvfL~RFtfL{eFzfL{TbrkfL

If _vvfR~RFtfR{eFzfR{TbrkfR

Ir _vvrL~RFtrL{eFzrL{TbrkrL

Ir _vvrR~RFtrR{eFzrR{TbrkrR26

In the above equations, Ii (i5 f, r) are the moments of

inertia of the front wheel and rear wheel, vij(i, j5 f, r,

L, R) are the angular accelerations of the four wheels,the subscripts L and R designating the left and right

wheels respectively. In equation (26), R is the tyre

radius, Trepresents the driving torque, Tbrkij(i, j5 f, r)

are the braking torques, Fzij(i, j5 f, r, L, R) are the

normal loads on the tyres, e is the offset distance of

the normal load, and Ftij(i, j5 f, r, R, L) are the

tangential loads on the tyres. These tangential loads

are directly determined by the driving torque.

It should be noted that

Ir~IfzjIt, j~0 T~0 12 T=0

&27

where It is the total moment of inertia of all themoving parts connected to the driving wheels.

The total braking torque Tbrk is controlled by the

driver and distributed on the front and rear wheels

according to the rules

TbrkfL~TbrkfR~KbfTbrk

TbrkrL~TbrkrR~ 1{Kbf Tbrk28

The driving torque T on the driven wheels istransmitted from the torque output of the engineby the transmission system. If the torque output ofthe engine is denoted by Me, the transmission ratioof the transmission system is given by ig, thetransmission ratio of the main reducer is designatedbyi0, and the mechanical efficiency of the transmis-

sion system is represented bygT, then

T~1

2igi0gTMe N m 29

where

Me~9549pene

N m 30

In the above equations, pe and ne are the power andthe corresponding rotating speed of the engine.Their values can be found from the corresponding

engine characteristics curve. Taking only the max-imum power pemax and the corresponding crankrotating speed np, the external characteristics curveof the engine of pe versus ne can be given by theexpression

pe~pemax c1nenpzc2

nenp

2{

nenp

3" #kW 31

In equation (31), the units of pe are kilowatts andthe units of ne are revolutions per minute. In the

equation development, only the pure rolling of thetyres is considered. The velocity of the wheel centre

Fig. 6 Free-body diagram of the driven wheel

Fig. 5 Free-body diagram of the driving wheel

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and therefore the velocity of the centre of mass ofthe vehicle can be given by the relations

VfR~RvfR

VfL~RvfL

VrR~RvrR

VrL~RvrL32

and

VfR~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu{

dr

2

2z vzar 2

s

VfL~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuz

dr

2

2z vzar 2

s

VrR~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu{

dr

2

2z v{br 2

s

VrL~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuz

dr

2

2z v{br 2

s

33

The pitch and roll of the vehicle body actually

redistribute the normal loads among the wheels.This redistribution of the normal loads has a greatinfluence on the vehicle performance. The normalload on the tyre exerted by the ground can bedivided into two portions: the static and dynamicloads. These normal loads on the four tyres can bedescribed by the equations

FzfR~Mg

2

b

L{

_uu{rv

g

hcg

L

&

{KRSFhcg

d

_vvzruzMs=Mh _pp

g {Msh

Md sin w !'

zMsAihsg

2L34

FzfL~Mg

2

b

L{

_uu{rv

g

hcg

L

&

zKRSFhcg

d

_vvzruzMs=Mh _pp

g{

Msh

Mdsin w

!'

{

MsAihsg

2L 35

FzrR~Mg

2

a

Lz

_uu{rv

g

hcg

L

&

{ 1{KRSF hcg

d

_vvzruzMs=Mh _pp

g

{Msh

Md

sin w !'z

MsAihsg

2L36

FzrL~Mg

2

a

Lz

_uu{rv

g

hcg

L

&

z 1{KRSF hcg

d

_vvzruzMs=Mh _pp

g{

Msh

Mdsin w

!'

{MsAihsg

2L37

In the above equations, L5a+b, hcg is the

distance from the centre of mass of the vehicle to

the ground, hsg is the distance between the centre of

mass of the sprung part and the ground, and KRSF is a

stiffness coefficient.

5 GOVERNING EQUATIONS

On the basis of the models and related equations

developed, the governing equations of the vehicle

are as follows. The motion in the ydirection is givenby

Mu _vvzru zMsAj{ FfLzFfR cos df

{ FrLzFrR cos dr

z FtfLzFtfR sin df

z FtrLzFtrR sin dr~0 38

The rotation about the z axis is given by

Xi

Mzi~Iz_rr{Is

xz _pp 39

The moment caused by the weight of the sprung

part is

Mmg~h {sin h cos w izsin w j{cos h cosw k

|Msgk~Mshg sin h cosw jzsin w i 40

The rotation about the x axis is given by

Isx _pp{Is

xz_rr~{Msh Aksin wzAjcos h cosw zMshgsinw{Kww 41



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The rotation about the y axis is given by

Isy _qq~Mshgsin h cos w

zMsh {Aksin hzAicos h cos w{Khh 42

where Kh and Kw are the additional resilience

moments caused by the unit roll angle and the pitch

angle respectively.

6 NUMERICAL ANALYSIS

Implementing the governing equation derived

above, the motion of a 4WS vehicle can be

quantified. The responses of the vehicle to three

types of turning displacement input are numerically

investigated and compared with that of a 2WSvehicle. Any existing numerical methods, such as

the RungeKutta method for solving differential

equations, can be utilized for stimulating the motion

of a 4WS vehicle with implementation of the

equations developed. For good convergence and

high accuracy of the numerical calculations, the

numerical computations are carried out with the

newly developed PT method [16], though other

numerical methods can also be used. Numerical

values of the parameters expressed in the governing

equations and utilized in the numerical computa-

tions are listed in Table 1.

6.1 Dynamic responses of vehicles under a linearangular turning displacement input

The turning angular displacement for this case is

exhibited graphically in Fig. 7. Corresponding to

such a step input on the steering system, the motion

of the vehicle is quantified with the governing

equations derived previously.

Figure 8 shows the paths of the 2WS and 4WS

vehicles for this case. Figure 9 illustrates the varia-

tion of the lateral velocity of the vehicle with respectto time. The angular velocities in the yaw plane are

shown in Fig. 10 for the two types of vehicle.

Figures 11 and 12 exhibit the variations in roll

angular velocity and pitch velocity respectively with

respect to time. The comparisons of the relations

between yaw angular velocities and the correspond-

ing yaw angular displacements are illustrated in

Fig. 13. Figure 14 compares the angular velocities in

roll and pitch planes. Comparison of the angular

velocity and angular acceleration in the pitch plane

is given in Fig. 15. The figures are presented in a way

to help readers to make comparisons conveniently.However, readers may need to note the different

scales, especially the scales of the vertical axes, and

the ranges of the variables involved in the figures.

The results obtained above are for the motions of

2WS and 4WS vehicles in both positive-phase

steering and negative-phase steering. From the

numerical calculations performed, the following

conclusions can be made corresponding to Figs 8

to 15.

1. Taking R* as the turning radius, it can be

observed from Fig. 8 that the magnitudes of

Table 1 Parameters used in the calculation

Symbol Value Units

c 0.026 mh 0.2 m

Mu 670 kg Ms 1160 kg Iz 2500 kg m

2

Isxz 0 kg m2

Izx 750 kg m2

Isy 1600 kg m2

a 0.9 md 1.33 mb 1.7 mC1f 44 400 N/radC3f 44 400 N/rad

3

C1r 43 600 N/radC3r 43 600 N/rad

3

r 1.2258 N s2/m4

Cx 0.32 Cy 0.35

Ax 2.1 m2

Ay 5.7 m2

hsg 0.556 mKw 85 000 N/radKh 76 185 N/radIf 2.1 kg m

2

It 0.136 kg m2

R 0.3 me 0.014 mKbf 0.55 hcg 0.5 mKRSF 0.444

Me 170 N mig 13.7 i0 0.85

gt 0.97 Fig. 7. Turning angle versus time

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Fig. 8 Paths of 4WS and 2WS vehicles

Fig. 9 Velocity v of the vehicle versus time

Fig. 10 Yaw angular velocity versus time

Fig. 11 Roll angular displacement versus time



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Fig. 12 Pitch angular displacement versus time

Fig. 13 Yaw angular velocity versus yaw angular displacement

Fig. 14 Roll angular velocity versus roll angular displacement

Fig. 15 Pitch angular velocity versus pitch angular displacement

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the turning radii satisfy the inequality of

RKl~{0:3vRKl~0vRKl~0:3. Based on the numerical

calculations made for the present research, the

turning radius of the vehicle is also proportional

to the travelling speed of the vehicle. It may thus

be stated that the 4WS vehicle has the smallest

turning radius in negative phase steering; hence

the steering is relatively easy. This implies that the

4WS vehicles have better manoeuvrability than

2WS vehicles do. However, 4WS vehicles have the

largest turning radius in positive-phase steering,

as shown in Fig. 8. As such, the vehicles present

the characteristics of insufficient turning in the

positive-phase steering case. The 4WS system is

therefore suitable for enhancing the manoeuvr-

ability at low driving speeds and for increasing the

stability at intermediate and high speeds. Inknowing this, negative-phase steering should be

employed for the cases in which low speed and

large turning angle are required, whereas posi-

tive-phase steering should be used for improving

the driving stability in the cases of high speeds.

2. The transverse velocities of the vehicles are not

constant, as they should not be corresponding to

the input given. This can be seen from Fig. 9. In

fact, all the motions with the given input are not

constants. The displacements of roll, yaw, and

pitch varied for a while and then stabilized.

3. When a 4WS vehicle is in negative-phase steering,the lateral forces on the front and rear tyres

generate the moments in the same rotating

direction with respect to the vehicles centre of

mass. Therefore, the vehicle has relatively large

yaw angular velocities for 2WS and 4WS negative-

phase steering as time increases whereas the yaw

angular velocity is smaller in positive-phase

steering, as can be seen from Figs 10 and 13. It

should be noted, however, that the lateral force

acting at the vehicles centre of mass is relatively

large in the case of positive-phase steering.

Additionally, in comparison with 2WS vehicles,

the increase in the yaw angular velocity of 4WS

vehicles is slower in positive-phase steering, andthe vehicle response time is longer with a

decrease in the yaw velocity. These are the

preferred turning characteristics for vehicles

operating at high speeds.

4. For 4WS vehicles in positive- and negative-phase

steering, the pitch and roll motions of the sprung

portion of the vehicle are very smooth and the

variation in the displacement of the 4WS vehicle

are small, especially for the negative-phase steer-

ing as shown in Figs 14 and 15. This is beneficial

for improving ride comfort.

6.2 Dynamic responses of vehicles under a saw ora half-saw angular turning displacementinput

Sawteeth angular turning is common in vehicle

operations, such as in turning and lane changes.

The dynamic responses of vehicles with a saw

angular turning displacement input are shown in

Figs 16 to 19.

Fig. 16 Turning angle versus time

Fig. 17 Moving paths of the vehicles



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The dynamic responses of vehicles to a given half-

saw angular displacement input are shown in Figs 20

to 23.

Again, in reviewing the figures, readers may need

to pay attention to the different scales of the vertical

axes in the figures.

With the analysis of the results plotted in Figs 16

to 23 for the two types of saw angular turning

displacement input, together with the stability

analyses performed by the present authors for the

4WS and 2WS vehicles [10], the following conclu-

sions can be drawn.

1. When a straight-line driving vehicle passes frontal

obstacles or turns, it can be observed from the


Fig. 19 Lateral acceleration versus time

Fig. 20 Turning angle versus time

Fig. 21 Moving paths of the vehicles

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motion trajectories that 4WS vehicles are more

manoeuvrable than front-wheel turning vehicles.From Figs 17 and 18, 4WS vehicles require tighter

space manoeuvrability. The better manoeuvrabil-

ity and stability of 4WS vehicles over that of 2WS

vehicles may also be seen from Figs 18 and 22. In

examining the peak values of the yaw angular

velocities in the figures for the 4WS and 2WS

vehicles, the following can be concluded as the

positive contributions to the manoeuvrability and

stability of the 4WS vehicles.

(a) The absolute peak values of the yaw angular

velocity of the 4WS vehicle in positive-phasesteering are the smallest for the two types of

input.

(b) The yaw angular velocity of the 4WS vehicle

in positive-phase steering varies in a smaller

range in comparison with that of the 2WS

vehicle, for the two input cases.

2. The better manoeuvrability and stability of 4WS

vehicles over that of 2WS vehicles may also be

observed from Figs 19 and 23. Although the saw

and half-saw angular turning displacement inputs

may generate a slightly higher transient accelera-

tion, however, the lateral accelerations of the

vehicle are stabilized in a short transient time as

shown in Figs 19 and 23. It may also be interest-ing to evaluate the peak values of the lateral

accelerations shown in Figs 19 and 23. The

following can be found from the figures.

(a) The peak values of the lateral acceleration of

the 4WS vehicle in positive-phase steering are

much more symmetrically and smoothly

distributed during the transient period of

time in comparison with that of the 2WS

vehicle, as shown in Figs 19 and 23. This also

contributes to the stability, safety, and man-

oeuvrability of 4WS vehicles, although the

transient period is short.

(b) The total variation range of the peak values of

the lateral acceleration for the 4WS vehicle in

positive-phase steering is smaller than that of

the 2WS vehicle corresponding to the saw

input, during the transient period. However,

the lateral acceleration of the 4WS vehicle

varies in a larger range than that of the 2WS

vehicle for the half-saw case owing to the

larger lateral force acting on the 4WS vehicle

in positive-phase steering, although the range

is much smaller than that in the saw inputcase. It should be noted, however, that the


Fig. 23 Lateral acceleration versus time



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distribution of peak values of the lateral

acceleration of the 4WS vehicle is much

smoother and is concentrated in two loca-

lized areas in comparison with that of the

2WS vehicle, which has an acceleration that

varies sharply, as shown in Fig. 23. All these

have positive influences on the stability,

safety, and manoeuvrability of 4WS vehicles

during the transient period.

3. Figure 21 shows how easy the turning operation

graphically described in Fig. 20 is for a 4WS

vehicle. 4WS vehicles can therefore be easily

controlled to respond to the road surface condi-

tions and the curvatures of the road. This results

in higher manoeuvrability for the driver of a 4WS

vehicle in operating the vehicle.

4. 4WS vehicles are generally more stable in opera-tions (larger stable regions) in comparison with

2WS vehicles. The conditions for the stability of

4WS and 2WS vehicles were given in reference

[10].

5. Under the condition of low vehicle speeds, the

lateral velocity and angular velocity are all in the

controllable ranges for 2WS vehicles and 4WS

vehicles in both positive- and negative-phase

steering. In the cases of intermediate or high

vehicle speeds, however, 4WS vehicles in positive-

phase steering provide the highest manoeuvr-

ability and therefore higher safety.

On the basis of the analysis above, it may be

observed that the model established in the present

research can be used to analyse the three-dimen-

sional non-linear behaviour of a vehicle under

various operating and environmental conditions. In

fact, such a non-linear dynamic model with multiple

degrees of freedom for the 4WS vehicles including

the effects of the non-linear tyre force, the air drag

and the wind, together with motions of the vehicle

and vehicle wheels in turning and braking man-

oeuvres has not been found in the current literature.

In the current literature, there are limited research

results available for dynamic responses of 4WS

vehicles; very few archived documents show the

characteristics of the 4WS vehicles with a compar-

ison with those of 2WS vehicles. The numerical

results obtained from the model in the present

research show a good coherence with the results

reported in the limited investigations on the

dynamic behaviours of 4WS and 2WS vehicles found

from the archived documents. A study on the

behaviours of 4WS and 2WS vehicles was reportedby Itoh et al. [6], with a validation on the basis of a

detailed experimental analysis. The 4WS and 2WS

vehicles analysed in the research by Itoh et al. are the

vehicles of identical bodies. This is also the case for

the research in this paper. The research by Itoh et al.

is therefore suitable for qualitative comparison with

the present research results, which are generated

including considerations of the vehicle motion and

vehicle wheels in turning and braking manoeuvres,

together with other factors and the joint effects of

the non-linearities as described in the context of this

paper. Although the research by Itoh et al. is an

investigation on a tractor, it is still appropriate for a

qualitative comparison with the results of the

present research as the research is on turnability

and the difference between the turnabilities of 4WS

and 2WS vehicles, and the turnability of 4WS and

2WS vehicles is one of the main research topics of

both the research studies.

Itoh et al. found from their investigations that the

yaw angular velocity in 4WS vehicles decreased

because of the steering of the rear tyres in the same

direction as the front tyres and that the yaw angular

velocity increased by means of steering the rear tyres

in the opposite direction to that of the front tyres.

This agrees exactly with the conclusions made on

the basis of the present model, with the numerical

results exhibit in Figs 10, 13, 18, and 27. The

conclusions of Itoh et al. on the higher manoeuvr-

ability of 4WS vehicles in positive-phase steering

over that of negative-phase steering and the sharp

turning characteristics of 4WS vehicles in negative-

phase turning also agree with the discussions given

on the numerical results that are generated by the

present vehicle model.

Additionally, the higher manoeuvrability and

stability of 4WS vehicles over that of 2WS vehicles

are recognized by Furukawa et al. [4]. This also

matches the conclusions and results made on the

basis of the model and numerical simulations of the

present research.

7 CONCLUDING REMARKS

This research analyses the non-linear behaviour of a

4WS vehicle via an accurate and effective approach.

With the development of the approach, the kine-

matics of the sprung vehicle body and the unsprung

part of the vehicle can be conveniently studied. A

non-linear vehicle model with multiple degrees of

freedom coupling the sprung and unsprung compo-

nents of the vehicle is developed such that the

effects of non-linear tyre forces, aerodynamic resis-tance, negative- and positive-phase steering, turning

548 Q Han and L Dai


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torque output of the vehicle engine, and mechanical

efficiency of the transmission system can be inte-

grally taken into consideration in vehicle motion

analysis. Such an approach for non-linear vehicle

behaviour analysis has not been found in the current

literature. The non-linear model shows the feasibility

and efficiency for analysing the behaviour of the

vehicle under normal and other operation condi-

tions with high accelerations. The numerical results

generated by the model established show good

agreement with those of the existing investigations

available in the literature. With a specified operation

manoeuvre, the model established allows calcula-

tions of the vehicle path, and the force and moments

acting on the vehicle in three-dimensional aspects.

Considering the sensitivity of the safety and stability

of 4WS vehicles to the operation conditions, e.g.

negative-phase steering at high speeds would be

dangerous because of the very high yaw rates

produced, it is significant to quantify the kinetic

and dynamic parameters for the proper operation of

a 4WS vehicle. The advantages of 4WS vehicles over

2WS vehicles in terms of the ease of turning,

manoeuvrability, stability, smaller turning radius in

negative-phase turning, tight-space manoeuvrabil-

ity, ride comfort, and safety are evident based on the

quantitative analysis and numerical simulations

presented in this research.

ACKNOWLEDGEMENTS

The authors wish to acknowledge, with thanks, thefinancial support from the Natural Sciences andEngineering Research Council of Canada, the Can-ada Foundation for Innovation, the National NaturalScience Foundation of China (10272046), and theNational Natural Science Foundation of GuangdongProvince (020858).

REFERENCES

1 Cho, Y. H. and Kim, J. Design of optimal four-wheel steering system. Veh. System Dynamics, 1995,24, 661682.

2 Akita, T. and Satoh, K. Development of 4WScontrol algorithms for an SUV. JSAE Rev., 2003, 4,441448.

3 Nagai, M., Ueda, E., and Moran, A. Nonlineardesign approach to four-wheel-steering systemusing neural networks. Veh. System Dynamics,1995, 24, 329342.

4 Furukawa, Y., Yuhara, N., Sano, S., Takeda, H.,

and Matsushita, Y. A review of four-wheel steeringstudies from the viewpoint of vehicle dynamics andcontrol. Veh. System Dynamics, 1989, 18, 151186.

5 You, S. S. and Chai, Y. H. Multi-object controlsynthesis: an application to 4WS passenger vehi-cles. Mechatronics, 1999, 9, 363390.

6 Itoh, H., Oida, A., and Yamazaki, M. Numericalsimulation of a 4WD4WS tractor turning in a ricefield. J. Terramechanics, 1999, 36, 91115.

7 Itoh, H., Oida, A., and Yamazaki, M. Measurementof forces acting on 4WD4WS tractor tires duringsteady-state circular turning in a rice field. J.Terramechanics, 1995, 32, 263283.

8 Abe, M., Kano, Y., Suzuki, K., Shibahata, Y., andFurukawa, Y. Side-slip control to stabilized vehiclelateral motion by direct yaw moment. JSAE Rev.,2001, 22, 413419.

9 Liu, Z., Payre, G., and Bourassa, P. Nonlinearoscillations and chaotic motions in a road vehiclesystem with driver steering control. NonlinearDynamics, 1996, 9, 281304.

10 Dai, L. and Han, Q. Stability and Hopf bifurcationof a nonlinear model for a four-wheel-steeringvehicle system. Commun. Nonlinear Sci. Numer.Simulation, 2004, 9, 331341.

11 Bakker, E., Pacejka, H. B., and Lidener, L. A newtire model with an application in vehicle dynamics

studies. In Proceedings of the Fourth Autotechnol-ogies Conference, Monte Carlo, 1989, SAE paper890087, pp. 8395.

12 Svendenius, J. and Gafvert, M. A. Semi-empiricaldynamic tire model for combined-slip forces. Veh.System Dynamics, 2006, 44, 189208.

13 Raper, R. L., Bailey, A. C., and Burt, E. C. Inflationpressure and dynamic load effects on soil deforma-tion and soiltire interface stresses. Trans. ASAE,1995, 38, 685690.

14 Allen, R. W., Chrstos, J. P., and Rosenthal, T. J. Atire model for use with vehicle dynamics simula-tions on pavement and off-road surfaces. Veh.

System Dynamics, 1997, 27, 318322.15 Deur, J., Asgari, J., and Hrovat, D. A 3D brush-type

dynamic tire friction model. Veh. System Dynamics,2004, 42, 133173.

16 Dai, L. and Singh, M. C. A new approach withpiecewise-constant arguments to approximate andnumerical solutions of oscillatory problems. J.Sound Vibr., 2003, 263, 535548.



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