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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]
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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. 18 Yaw angular velocity versus time
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. 22 Yaw angular velocity versus time
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).
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