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A Comparative Study of Equivalent Modeling for
Multi-axle Vehicle
Yubiao Zhang, Yanjun Huang*, Hong Wang, Amir Khajepour
Department of Mechanical and Mechatronics Engineering, University of Waterloo, ON, N2L3G1, Canada
Abstract: In this paper, equivalent modeling methods of a multi-axle vehicle are presented and compared. Firstly,
for the sake of comparison, a single-track model of a three-axle and a two-axle vehicle is developed, and then
existing equivalent modeling derivations are presented and discussed. Next, the proposed model based dynamic
equivalence of force/moment at the centre of gravity (CG) is introduced and optimized. It represents the
approximately equivalent steady state and transient response of the yaw rate and side slip angle, which allows
different cornering stiffness on the central and rear axle. Finally, to demonstrate how the proposed method is
advantageous to the other equivalent models available in the literature, different simulation cases are compared in
the dimension of time-domain, eigenvalues characteristics and frequency-domain. Furthermore, the proposed
method is extended to any multi-axle vehicle configurations and a general expression is formulated.
Keywords: multi-axle vehicle, vehicle modeling and dynamics, optimization
1. Introduction
In order to transport a larger volume of freights via a relatively fixed infrastructure, one way to increase
transportation efficiency is to employ longer vehicles with more axles [1]. These vehicles are often highly
specialized and customized to perform particular tasks like logistic, military, long-distance coaches, and public
transportation. With different handling properties, but driving on existing infrastructure, the safety and stability
problems of multi-axle vehicles continue to be an active research area among researchers and automobile
manufacturers [2-4]. A single-track model is well known and widely used in traditional two-axle vehicles with the
yaw rate and lateral velocity (or sideslip angle at CG) as the degrees of freedom to predict the handling performance
and stability both in time and frequency domain analyses[2,4,5]. An equivalent two-axle model of multi-axle
vehicles can be used to easily study such vehicles’ handling and stability characteristics. Also, compared to the
nonlinear model or even original multi-axle model, it is more computational efficient to design the active model-
based control systems[2,6-8], for instance differential braking, torque vectoring and active steering. As a result,
several equivalent models have been developed in the literature.
There are three major attempts existing in the literature to develop the equivalent model of three-axle vehicles.
William [9,10] presents the basic notions of the equivalent wheelbase and the understeer coefficient of three-axle
vehicles when compared to the two-axle model. It is obtained directly from the steady-state response of the state
space model. Furthermore, Williams [11] extends the concept to vehicles with any arbitrary number of axles. He
also develops a unified way to study handling of multi-axel vehicles by the equivalent wheelbase and understeer
coefficient. The general expression of the characteristic equation with steady-state vehicle dynamics is also extended
to the n-axle vehicle. Ding and Guo [12] apply William’s work to develop the 2M (twice-multiple) equivalent
approach to analyze both the effect of vehicle body roll and n-axle handling on vehicle dynamics. However, the
sideslip angle is not considered from the above work.
Winkler and Gillespie [13-15] generate an equivalent three-axle vehicle wheelbase from a virtual front axle and rear
axle, which is consistent with the handling characteristics of the dynamics equations. The method is only applicable
to the vehicle with one steering front-axle and two driving rear-axles. The basic concept behind their approach is to
move the equivalent front axle forward and move the equivalent rear axle rearward, subjecting to assumption that
the two equivalent axles is free of slipping. The equivalent parameters such as distance and cornering therefore, are
found. In Johannes’s thesis[16], it explains the equivalent modeling approach in detail. In addition, Winkler [14]
further presents a comprehensive study on the steady-state turning of multi-axle vehicles by using Pacejka’s [17]
handling diagram methodology. A generalized expression for an arbitrary number of non-steered axles is achieved.
However, this method is limited to the situation of a steady-state low-speed turn.
Given that the normal load is equally shared between the tandem axles in the rear and all tires have the same
cornering stiffness, Ellis [18] proposes a method of equating the action of a tandem axle bogie (central and rear axle)
to a single equivalent axle acting in the midway of the central axle and rear axle by adding an augmented moment to
produce a compensation for the equivalent two-axle model. Based on a similar steady-state development, the
equivalent wheelbase using Ellis’s method is shown to yield a compatible connection to the general three-axle
development with simplifying assumptions consistent with Williams and Winkler and Gillespie’s derivation.
However, this method is limited by the same cornering stiffness and an equally shared load. The accuracy of the
equivalence is questionable when these conditions change.
In this paper, a new model is derived from dynamic equalization, which means the forces and moment at CG are
always equivalent between the two models. An over-constrained optimization problem is formulated and the
stiffness of the equivalent rear wheels and equivalent distance from CG to equivalent rear axle are found via solving
the optimization problem. As a statement, the method generates an approximately equivalent model not completely
equivalent since it minimizes the errors between the proposed model and the original model. It promises not only an
approximately equivalent steady state response but also an approximately equivalent transient response. In addition,
a high accuracy of sideslip angle response is guaranteed. The proposed method’s efficiency is verified through
simulations in several scenarios to show both transient and steady state behavior in time and frequency domains.
The proposed approach can be used to more conveniently study three-axle vehicles dynamics and stability. In this
way, it is feasible to build a unified model and a holistic controller for vehicles with different numbers of axles.
In Sec. 2, a three-axle vehicle model and its equivalent model are developed. In Sec. 3, the existing equivalent
models in the literature are presented and discussed. Sec. 4 leads to the proposed method of equivalent modeling and
explains how to find the equivalent parameters using optimization. In Sec. 5, simulation results are presented and
compared to demonstrate the advantages of the proposed method. Sec. 6 formulates a general expression for any
multi-axle vehicle and Sec. 7 concludes the paper.
2. Vehicle Single-track Model
A fixed third axle is added to the classic two-axle single-track model as described in Figure 1. Fancher [19] found
that the yaw resisting moment from the distance between dual tires was far less significant when compared to the
moment that from the tandem spread. Hence, the effect of dual tires used on the tandem rear axles is neglected and
thus the single-track model extended to multi-axle vehicle is used.
Figure 1. Derivation of single track model of three-axle vehicle and its equivalent model
The single-track model is widely used in the literature and is not explained here in detail. The equations for lateral
dynamics and yaw motion are directly given by:
()
yf yc yr
m v u F F F
(1)
z yf f yc c yr r
I F l F l F l
(2)
Assuming the slip angles are small, tire lateral forces have a linear relationship to the tire slip angle. The lateral
force of a displaced axle
i
can be expressed as
( ) ( )
( ) ( , ; 0)
i
ii
yi i
i i c r
vl
C i f
u
Fvl
C i c r
u
(3)
In addition, to obtain the sideslip angle on CG, one can easily derive it from
1
tan v
u
(4)
In Equation (1), (2), (3) and (4), we have:
, z
m I
- vehicle mass and the moment of inertia about z axis,
, , ,uv
- vehicle longitudinal speed, lateral speed, yaw rate and body slip angle at vehicle’s CG,
,,
yf yc yr
F F F
- lateral force of front, central and rear tires,
///
f c r
l l l l
-distance from CG to front/central/rear axle, the distance between front axle to rear axle,
//
f c r
C C C
- front/central/rear wheels cornering stiffness,
//
f c r
- front/central/rear wheels steering angle.
By substituting (3) to (1) and (2), the yaw dynamics can be represented by the following state space model:
2
2 2 2
1
f c r f f c c r r f
f
ff
f f c c r r f f c c r r
x
xz
zz
B
A
C C C l C l C l C C
mu mu mu
lC
l C l C l C l C l C l C
I
I I u
(5)
Figure 1 also shows an equivalent model of a two-axle vehicle. It is assumed a single equivalent rear axle can be
used to represent two original axles. To describe the classic two-axle single-track model (i.e. the bicycle model), one
can easily formulate it in the state-space form,
2
22
1
eq
eq
f eq f f req eq f
f
ff
f f req eq f f req eq
x
xz
zz
B
A
C C l C l C C
mu mu mu
lC
l C l C l C l C
I
I I u
(6)
,
req eq
lC
- distance from CG to the equivalent rear axle and the equivalent cornering stiffness representing tires of
the central and rear axle.
3. Existing Equivalent Modeling
There are three equivalent two-axle models in the literature that have been proposed for multi-axle vehicles. These
models are reviewed below.
3.1 Daniel E.Williams Derivation
In this two-axle single-track model, the steady-state vehicle yaw rate response is fully determined by two parameters:
understeer coefficient and wheelbase. The vehicle dynamic response is assumed only dependent on these two
parameters. Williams [10] derives an equivalent rear axle directly from the steady state response of the linear model
when compared with the two-axle model. The understeer coefficient and wheelbase are characterized analogous
with that of the two-axle vehicle.
In the steady state, the time derivatives are zero and the steady-state can be solved in terms of the steering angle. The
equivalent understeering coefficient and equivalent wheelbase for the three-axle model are defined.
Steady yaw rate w.r.t steering angle (two-axle model)
2
()
, req eq f f
ss eq
f eq eq eq f eq
m l C l C
uK
l K u l C C
, (7)
where
eq
K
is known as the understeering coefficient of the two-axle vehicle. The sign of its value depends on the
cornering stiffness of the tires and their position with respect to the CG.
Steady yaw rate w.r.t steering angle (three-axle model)
2 2 2 2
[( ) ]
( ) ( ) ( )
f c f c f r
ss
f f c f c f r r c r c c c r r f f
u l l C C lC C
l l C C l C C l l C C mu l C l C l C
(8)
Rewrite the equation (8) as the form
2 2 2 2
( ) ( ) ( )
( ) ( )
eq eq
ss
f c f c f r r c r c c c r r f f
f
f c f c f r f c f c f r
lK
u
C C l l C C l C C l l m l C l C l C u
l l C C lC C l l C C lC C
(9)
Therefore, in Williams’s method, it is made to be equivalent that
2 2 2
( ) ( ) ,
()
f c f c f r r c r c
eq req eq f
f c f c f r
C C l l C C l C C l l
l l l l
l l C C lC C
(10)
( ) ( )
()
req eq f f c c r r f f
eq eq f eq f c f c f r
m l C l C m l C l C l C
Kl C C l l C C lC C
(11)
The equivalent concerning stiffness is calculated by
()
ff
eq eq f eq f eq
ml C
Cm l l K C l
(12)
3.2 Winkler–Gillespie Derivation
Consider the three-axle vehicle and an equivalent model illustrated in Figure 2, in Winkler–Gillespie’s derivation
[13-15], the vehicle is assumed to be in a steady-state turn at a very low speed such that the lateral acceleration can
be treated as zero. The idea behind is to find equivalent positions of the virtual front and rear axles so that the
equivalent model operate with zero slip angles and generate zero lateral forces at very low speeds. It is defined that
is the distance from central and rear axle to the mid-point between them and the geometric wheelbase
g
l
is the
distance between the tandem center and real front axle.
Figure 2. Derivation of the equivalent wheelbase of a three-axle vehicle in a steady-state low-speed turn
In a steady-state turn at very low speed, the requirements of static equilibrium of lateral forces and yaw moment lead
to
0
y yf yc yr
F F F F
(13)
( ) 2 0
c yf g yr
M F l F
(14)
Tire lateral forces have a linear relationship to the tire slip angle. The lateral force of a displaced tire
i
can be
expressed as
( ); ( ); ( )
yf f yc c yr r
a b b
F C F C F C
R R R
(15)
Explained by Winkler–Gillespie [14] in detail,
R
is the low-speed turn radius,
a
is the distance ahead of the front
tire at which an imaginary tire, steered to the same angle as the real front tire, would have to be located to
experience zero side slip.
b
is the distance aft of the center of the rear suspension where an imaginary, nonsteering
tire would have to be located to experience zero slip.
With substitution and simplification, the virtual distance
a
,
b
, the equivalent wheelbase and cornering stiffness can
be achieved by
2
2 ( )( )
2 ( ) ;
( ) 2 ( )( )
r c r g
r
f g r c r g
C C C l
Cb
ab
C l C C C l
(16)
;
eq g eq c r
l a l b C C C
(17)
Furthermore, if
cr
CC
, it can be simplified to
22
;
eq
eq g eq c r
g f g
C
l l C C C
l C l
(18)
3.3 J.R. Ellis Derivation
Instead of deriving two virtual equivalent axles, Ellis replaces the tandem axle bogie with an equivalent axle in the
midway of the central and rear axles and an augmented moment is added to produce a compensation for the
equivalent two-axle model [18]. However, it is only appropriate in the special cases where the tires are equally
loaded on to the central and rear axles. Ellis shows that when both rear axles have equal cornering stiffness, the yaw
augmented moment
resist
M
produced is
2
2
resist r
C
Mu
(19)
Extending Ellis’ development, modifying the state-space form in Equation (6), one can achieve the corresponding
equivalent wheelbase and cornering stiffness. Fortunately, similar to the development of the Winkler–Gillespie
method, same results are achieved in the three-axle steady-state model [10,14].
4. The Proposed Method
4.1 Equalization based on CG force/moment
To summarize the previous work, it can be seen that Williams’s derivation is based on an equivalent steady-state
yaw rate and the understeering coefficient is also equal to that of the two-axle model. However, the side slip angle of
CG, as another important control variable in stability control system is neglected from this equalization. Winkler–
Gillespie’s derivation is based on the assumption of a steady-state low speed and the lateral acceleration is
approximately equal to zero. The limitation is that the equalization is only admissible in the low-speed situation and
the stability control design is quite restricted using this model. Similar drawbacks arise in J.R. Ellis’s derivation.
The proposed method here is based on dynamic equalization, which means the forces and moment at CG are
identical between two models. The method promises both approximately equivalent steady state response and transient
response. Two equivalent parameters
req
l
and
eq
C
are introduced when deriving the model. From the model of
three-axle and two-axle, the equations for lateral dynamics and yaw motion can be expressed by:
The lateral motion of three-axle model and two-axle model
( ) ( ) + ( ) ( )
fcr
f f c r
vl vl vl
m v u C C C
u u u
(20)
( ) ( ) ( )
f req
f f eq
v l v l
m v u C C
uu
(21)
The yaw motion of three-axle model and two-axle model
( ) ( ) ( )
fcr
z f f f c c r r
vl vl vl
I l C l C l C
u u u
(22)
( ) ( )
f req
z f f f req eq
v l v l
I l C l C
uu
(23)
where the right hand sides of Equation (20) and Equation(21) actually represent the summative lateral force at CG,
and the right hand sides of Equation (22) and Equation(23) represent the summative yaw moment at CG.
From the fundamental vehicle dynamics theory, the forces and moment at CG determines the vehicle motions. In
order to make motion of the three-axle model and the two-axle model be identical, we attempt to equalize the lateral
forces and yaw moment at CG of two models. This means operate the right hand side of Equation (20) and Equation
(21) to be equal, so does to Equation (22) and Equation (23). Assuming that the front wheels stiffness and the
distance from CG to front axle remain unchanged, so the equivalent rear wheels stiffness
eq
C
and equivalent distance
from CG to equivalent rear axle
req
l
are the only two unknown parameters need to be solved. By eliminating term
with
f
l
and
f
C
, we obtain
2 2 2
( ) ( ) ( )
( ) ( ) ( )
req creq c r
eq c r
req eq c c r r
req crreq eq c c r r
req eq c c r r
vl vl vl C C C
C C C
u u u l C l C l C
vl vl vl l C l C l C
l C l C l C
u u u
(24)
Continuing eliminating the terms with vehicle state variables of
v
,
and
u
, three independent equations are
deduced shown as Equation (24). But we only need to solve two unknowns
req
l
and
eq
C
, and this becomes an over-
constrained problem. Therefore, we pursue to find an optimal solution that minimizing the error between three-axle
full model and the proposed model. Defining that
req
l
represents as x-axis and
eq
C
represents as the y-axis and plot
the functions of
eq
C
=
()
req
fl
derived from Equation (24). A generic three-axle vehicle is used and plotting results
are shown in Figure 3. Enlarging the intersections from the left figure, an optimization area is obtained in the right
figure.
Figure 3. Equivalent parameters of
req
l
and
eq
C
4.2 Optimization of equivalent parameters
As shown in Figure 4, to find the optimal
req
l
and
eq
C
, the above over-constrained problem is formulated as a
simple convex optimization problem as follows,
Figure 4. Problem formulation
Minimize :
( , )f x y
where,
3
2 2 2 22
1 2 3 1
( , ) ( ) ( )
ii
i
f x y pp pp pp x x y y
1 1 1
( , ) ( , )
c c r r cr
cr
l C l C CC
C
px C
y
;
22
22
2
( , ) ( , )
c c r r cr
cr
l C l C CC
C
px C
y
2 2
3 3 3
2
22
)
( , ) ( , )
(
c c r r c c r r
c c r r c c r r
l C l
px C l C l C
l C l C l C C
yl
Constraint:
0; 0.xy
Note: The objective function
( , )f x y
is an Euclidean norm, or
2
l
-norm, which means a convex function. The
constraint is also a convex set [20]. The problem is then solved by making the gradient of
( , )f x y
1 2 3 1 2 3
1 2 3 1 2 3
2(3 ) ( )/3
0*
( , )
2(3 ) ( )/ 3
0*
x x x x x x x
x
f x y y y y y y y y
y
The optimal point of
req
l
and
eq
C
is found as
1 2 3 1 2 3
* ( , )
33
x x x y y y
p
.
5. Comparison and Discussion
5.1 Step-steer Manoeuver
To compare the propose model with the three existing equivalent models in the literature, the responses of the
models to a step-steer (i.e. front wheels steering angle = 2°) input to a three-axle vehicle at different speeds (20km/h,
60km/h and 90km/h) are presented. In the simulations, it is assumed that the cornering stiffness of central and rear
axles are identical.
1.32 1.34 1.36 1.38 1.4 1.42
2.24
2.26
2.28
2.3
2.32
2.34
2.36
2.38
2.4
x 105
x
y
p2(x2,y2)
p1(x1,y1)
p(x,y)
p3(x3,y3)
The simulation results of the 3-axle vehicle and the equivalent models are shown in Figure 5. As seen, the results
show the proposed method has over-performed all the previous methods and especially it has a better response of
slip angle. At a low speed (20km/h), all methods have a good response tracking to the original model. This is
because the lateral acceleration is very small and W-G/ J.R. Ellis’s derivation is based on a steady turn at a very low
speed. However, when the speed is increased to 60km/h and 90km/h, the error in sideslip angle increases in other
equivalent models, while the proposed one holds a very good tracking on both yaw rate and side slip angle. One may
note that the Williams’s derivation results in a good tracking only in yaw rate response, and it precisely shows how
the method is developed from equivalent steady state yaw rate.
(a) (b) (c)
(a) 20km/h at step steer
(b) 60km/h at step steer
(c) 90km/h at step steer
Figure 5. Yaw rate and side slip angle response w.r.t different speeds (
1.4 5 N/rad
cr
C C e
)
5.2 Different Combinations of Cornering Stiffness
In the following simulations, the equivalent models are compared in step-steer and lane change manoeuvers with
different cornering stiffness. In this case, the speed of the vehicle is assumed to be 90 km/h and 70 km/h for the step-
steer and lane change maneuvers, respectively.
The results are shown in Figure 6. As seen, the proposed model can accurately predict the yaw rate and side slip
angle of the original three-axle model. The additional manoeuver of single lane change presents a similar
phenomenon.
0 2 4 6 8
0
2
4
6
8
t(sec)
yaw rate(deg/s)
0 2 4 6 8
0
0.5
1
1.5
2
t(sec)
side slip angle(deg)
0 2 4 6 8
0
5
10
15
t(sec)
yaw rate(deg/s)
0 2 4 6 8
0
0.1
0.2
0.3
0.4
t(sec)
side slip angle(deg)
0 2 4 6 8
0
5
10
15
20
t(sec)
0 2 4 6 8
-1
-0.5
0
0.5
t(sec)
0 0.5 1
0
0.5
1
t(sec)
yaw rate(deg/s)
0 2 4 6 8
0
0.5
1
t(sec)
side slip angle(deg)
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
(a) (b) (c)
(a) 90km/h at step steer(
8 4 N/rad; 1.4 5 N/rad
cr
C e C e
)
(b) 90km/h at step steer(
1.4 5 N/rad; 8 4 N/rad
cr
C e C e
)
(c) 70km/h at single lane change(
8 4 N/rad; 1.4 5 N/rad
cr
C e C e
)
Figure 6. Yaw rate and side slip angle response
5.3 Discussion on Changing Tires Stiffness
In practice, it is known that the cornering stiffness depends on the normal tire load, which particularly happens at
turnings. In this section, it is discussed that how the proposed model is influenced by the changes in cornering
stiffness. The obtained three equations of two equivalent parameters are repeated here for convenience.
2 2 2
eq c r
req eq c c r r
req eq c c r r
C C C
l C l C l C
l C l C l C
(25)
Consider the central and rear wheels cornering stiffness changed from a specific range from minimum to maximum
min max
[ , ]
r r r
C C C
. For simplicity, if we assume that the central wheels cornering stiffness has a linear
relationship to rear wheels cornering stiffness,
cr
CC
(26)
and by substituting Equation (26) to Equation(25), and then elimination by substitution, we have,
1c
rq r
el
ll
,
22
21
cr
req ll
l
(27)
From Equation (27) it can be seen that the equivalent parameter
req
l
is independent to the tires stiffness, which
makes sense that
req
l
is a coefficient of distance parameters of
c
l
and
r
l
. Figure 7 shows
req
l
in Equation (27) as a
function of
. As it can be seen in the figure, the changes in equivalent parameter
req
l
is small with the range of
from 0 to 4. Generally, the commercial vehicle are equipped with the same tires, some multi-axle vehicles use dual
tires. The ratio of the central wheels cornering stiffness to rear wheels cornering stiffness is therefore assumed to be
0 2 4 6 8
0
5
10
15
20
t(sec)
yaw rate(deg/s)
0 2 4 6 8
-1.5
-1
-0.5
0
0.5
t(sec)
side slip angle(deg)
0 2 4 6 8
0
10
20
30
t(sec)
yaw rate(deg/s)
0 2 4 6 8
-2
-1
0
1
t(sec)
side slip angle(deg)
0 2 4 6 8
-40
-20
0
20
t(sec)
yaw rate(deg/s)
0 2 4 6 8
-1
-0.5
0
0.5
1
t(sec)
side slip angle(deg)
0 0.5 1
0
0.5
1
t(sec)
yaw rate(deg/s)
0 2 4 6 8
0
0.5
1
t(sec)
side slip angle(deg)
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
1.2 1.22 1.24 1.26 1.28 1.3
1.6
2.1
2.6
3.1
3.6
x 105
lreq (=2)
Ceq
Cer=(1+)Cr
ler*Cer=(lc*+lr)*Cr
ler2*Cer=(lc2*+lr2)*Cr
around 1, 0.5 or 2. But the vehicle loads condition/distribution affects the cornering stiffness on each axle. This
ratio
is strictly correlated with the ratio of the vertical loads on the wheels when all other conditions are the same
(same tires, same inflation pressure, etc.). With comprehensive consideration, plotting
from 0 to 4 in Figure 7
provides enough evidence showing that ratio
has little impacts on deriving equivalent parameter
req
l
.
Figure 7. Equivalent
req
l
w.r.t changing ratio of
c
C
and
r
C
(
cr
CC
)
Since the ratio of
c
C
and
r
C
is fixed for any given vehicle, we study the variation of
eq
C
as a function of
req
l
when
is chosen to be 1, 0.5 and 2 in Figure 8. The results in Figure 8 demonstrate how the equivalent model is
influenced by the changes in cornering stiffness. Results show that when
min max
[ , ]
r r r
C C C
, changes from
60000 to 120000, the equivalent
req
l
stayed at the same range and it could always find the optimal value from this
small range of
req
l
. In addition, the equivalent
eq
C
reasonably change with respect to the changing of
r
C
, which
makes sense according to the vehicle’s dynamic characteristics.
Figure 8. Equivalent parameters of
req
l
and
eq
C
with changing cornering stiffness
0 0.5 1 1.5 2 2.5 3 3.5 4
1.1
1.2
1.3
1.4
1.5
1.6
1.7
lreq
lreq=(*lc+lr)/(1+)
lreq=(*lc2+lr2)/(*lc+lr)
1.32 1.34 1.36 1.38 1.4
1
1.5
2
2.5 x 105
lreq (=1)
Ceq
1.44 1.46 1.48 1.5
0.8
1.2
1.6
x 105
lreq (=0.5)
Ceq
1.2 1.22 1.24 1.26 1.28 1.3
1.6
2.1
2.6
3.1
3.6
x 105
lreq (=2)
Ceq
120000
r
C
90000
r
C
60000
r
C
120000
r
C
120000
r
C
90000
r
C
90000
r
C
60000
r
C
60000
r
C
5.4 Root Locus and Frequency Responses
In order to investigate the driving stability during straight line with the possible disturbance such as wind or uneven
surface, we consider the steering angle to be equal to zero in Equation (5) and (6). Under the assumption of constant
velocity, one arrives at the linear homogenous state space equations
x Ax
- original three-axle model (28)
eq
x A x
- equivalent two-axle model (29)
The system eigenvalues (poles) are the eigenvalues of matrix
A
/
eq
A
, solved by the characteristic equation
det( ) 0sI A
(30)
The linear system modelled with such an equation is known to be asymptotically stable if all the eigenvalues of
matrix
A
/
eq
A
have negative real parts. We define the matrix
A
/
eq
A
eigenvalues in the form of:
1,2 d
s d i
(31)
where
,,d
id
are the imaginary unit, damping coefficient and damped natural frequency, respectively. Furthermore,
the damping ratio
and un-damped natural frequency
n
can be expressed as
22
22
, nd
d
dd
d
(32)
It indicates an unstable system if the damping ratio
is negative. The eigenvalues distribution (root locus) and
damping characteristics are evaluated for the vehicle parameters of velocity and rear cornering stiffness. We used
the other parameters of the vehicle in Section 5.1. Considering the vehicle velocity increases from 20km/h to
120km/h, Figure 9(a) depicts the root locus (system eigenvalues) with respect to a series of speed points. The arrow
shows the increasing direction of the speed. The un-damped natural frequency and damping ratio are plotted as
functions of speed in Figure 9(b). Similarly, Figure 10 plots root locus and damping characteristics w.r.t the
increasing cornering stiffness of rear wheels, and the central wheels cornering stiffness is equal to
r
C
. Results
suggest the vehicle is mathematically stable of all derived models. It is worth noting that the proposed method has a
preferable coincidence in root locus and damping characteristics as well as in time-domain responses.
(a) Root locus (b) Natural frequency and damping ratio
Figure 10. Root locus and damping characteristics w.r.t rear wheels cornering stiffness
After steady-state response analysis, stability analysis, we now turn to frequency domain analysis and this is
necessary to analyze and compare the transient response in vehicle dynamics. Figure 11 shows the frequency
responses of yaw rate and sideslip angle at the speed of 60km/h. Again, in frequency-domain, the results indicate
that the responses from proposed method almost completely overlap to where are from the original three-axle model.
But the other two conventional derivations somehow fail to match it in frequency-domain.
-9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5
-8
-6
-4
-2
0
2
4
6
8
Real Axis
Imag Axis
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
x 105
0
5
10
Natural frequency(rad/s)
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
x 105
0.7
0.8
0.9
1
Rear cornering stiffness(N/rad)
Damping ratio
(a) Root locus (b) Natural frequency and damping ratio
Figure 9. Root locus and damping characteristics w.r.t velocity
-70 -60 -50 -40 -30 -20 -10 0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
Imag Axis
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
20 40 60 80 100 120
5
10
15
20
25
Natural frequency(rad/s)
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
20 40 60 80 100 120
0.7
0.8
0.9
1
Vehicle velocity(km/h)
Damping ratio
(a) Yaw rate (b) Side slip angle
Figure 11. Frequency responses of yaw rate and sideslip angle
6. General Formulation for Any Multi-axle Vehicle
Following the obvious similarities in the development of the two-axle and three-axle equalization, the concept now
is extended to account for a multi-axle vehicle of one front axle and any number of rear axles. Assuming that the
vehicle has one front axle and rear
( 1)n
axles,
( 1,2, , 1)
i
l i n
is distance from CG to
i
th rear axle,
( 1,2, , 1)
i
C i n
is the total wheels cornering stiffness of
i
th rear axle. For convenience, the lateral and yaw
motion of such a multi-axle vehicle is written by
1
1
( ) ( ) + ( )
n
fi
f f i
i
vl vl
m v u C C
uu
(33)
1
1
( ) ( )
n
fi
z f f f i i
i
vl vl
I l C l C
uu
(34)
Next, we make the total force/moment of the right hand side of Equation (21) and Equation (23) equal to the
force/moment items in Equation (33) and Equation (34). With elimination, the following expression is obtained.
1
11
1
1
11
1
22
1
1
( ) ( )
( ) ( )
n
eq i
ni
req i
eq i n
ireq eq i i
ni
req i
req eq i i n
ireq eq i i
i
CC
vl vl
CC
uu
l C l C
vl vl
l C l C
uu
l C l C
(35)
The problem is transfer to find an optimal point of
req
l
and
eq
C
expressed by the known parameters
i
l
and
i
C
.
Referring to the same procedures in Section 4.2, the optimization problem is formulated as
Minimize :
( , )f x y
-30
-20
-10
0
10
20
Magnitude (dB)
10-1 100101102
-90
-45
0
Phase (deg)
Frequency (Hz)
Original three-axle model
D. E.Williams derivation
W–G/J.R. Ellis derivation
The proposed method
-50
-40
-30
-20
-10
Magnitude (dB)
10-2 10-1 100101102
-90
-45
0
45
Phase (deg)
Frequency (Hz)
where,
3
2 2 2 22
1 2 3 1
( , ) ( ) ( )
ii
i
f x y pp pp pp x x y y
1
1
1
1 1 1 11
1
( , ) ( , )
n
ii
n
ii
ni
i
i
lC
p x y C
C
;
121
1
2 2 2 11
1
( , ) ( , )
n
ii
n
ii
ni
i
i
lC
p x y C
C
11
22
11
3 3 3 11
2
11
( )
( , ) ( , )
nn
i i i i
ii
nn
i i i i
ii
l C lC
p x y l C l C
.
The optimal point of
req
l
and
eq
C
is readily found as
1 2 3 1 2 3
* ( , )
33
x x x y y y
p
.
Therefore, the general expression of
req
l
and
eq
C
for any axle-vehicle with of one front axle and
1n
rear axles is
formulated as:
1 2 3 1 2 3
,
33
req eq
x x x y y y
lC
(36)
Substituting
req
l
and
eq
C
into Equation (6), the general formulation of equivalent two-axle model is achieved.
7. Conclusion and Future Work
Different equivalent modeling methods that describe the fundamental lateral dynamics of a three-axle vehicle were
comparatively studied. Major previous work, including D. E.Williams, Winkler-Gillespie, and J.R. Ellis, were
discussed and an approximately equivalent model based on CG force\moment was developed and optimized.
Through different simulation cases, the model presented in this paper performed more accurately compared to the
other methods, and particularly for slip angle response. This is due to the fact that the proposed model is based on a
dynamics equivalence of CG forces and moment. Moreover, the impact of changing cornering stiffness at the rear
axle on the equivalent parameters
req
l
and
eq
C
was investigated and vehicle stability of eigenvalues and frequency-
domain characteristics were discussed and compared.
Using concept of the proposed method, it is possible to extend the work to multi-axle vehicles with any arbitrary
number of front and rear axles. A general formulation is therefore derived for the multi-axle vehicle with one front
axle and any number of rear axles. To point it out, the resulting approximately equivalent model can be expediently
used to study the dynamics and stability characteristics of any multi-axle vehicle without formulation for every case.
Moreover, this simplification provides a great benefit on model-based controller design just as readily as the
common two-axle single-track model.
Reference
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Appendix
TABLE I Parameters for a generic three-axle vehicle
Parameter
Description
Unit
Value
Vehicle mass
kg
2200
Distance of front axle from CG
m
1.6
Distance of rear axle from CG
m
1.65
Distance of central axle from CG
m
1
Distance of front axle from rear axle
m
3.25
Yaw moment of inertia
kg/m2
3000
Front axle cornering stiffness
N/rad
100000
Central axle cornering stiffness
N/rad
depends on the case
Rear axle cornering stiffness
N/rad
depends on the case
A Comparative Study of Equivalent Modeling for
Multi-axle Vehicle
Yubiao Zhang, Yanjun Huang*, Hong Wang, Amir Khajepour
Department of Mechanical and Mechatronics Engineering, University of Waterloo, ON, N2L3G1, Canada
Manuscript ID: NVSD-2017-0129
Dear editors and reviewers,
On behalf of all authors, I greatly appreciate your quick reply on my paper. Please find the only-one-point rebuttal to
your comments/concerns. It is in blue after the point.
Best regards,
Response to Reviewer 3
Reviewer #3: In my opinion, some significant corrections must still be made in Eqs. (20-23).
Considering Eq. (3), which is now correct, I think that the signs in front of all terms with the stiffness coefficients
,
cr
CC
and
eq
C
should be changed in the equilibrium equations (20-23). Please, check them carefully.
Since this is clearly an oversight, I think that there is no need of replying to the reviewer after the correction, unless
the authors believe that I am wrong.
R: Thanks for your careful comments again. I added negative signs in brackets of the equilibrium equations (20-23).
I believe they are correct now.
