15
C H A P T E R 3
Tilting Vehicle Dynamics
is chapter develops mathematical models to capture the dynamical behavior of NTVs. Based
on a classical modeling process for conventional automobiles, an integrated handling and roll
dynamics for NTV are derived. Topics closely related to NTVs such as derivation of the rollover
index, re-configurable actuators, wheel configurations, and suspension kinematics on stability
are discussed based on the derived model. e vehicle model and rollover index will be utilized
in the controller development in Chapter 4.
3.1 TIRE FORCES AND LATERAL DYNAMICS
For modeling the lateral dynamics of a NTV, one major challenge comes from its flexible wheel
configurations. Unlike four-wheeled conventional cars, NTVs can have both three-wheeled and
four-wheeled configurations. Among those three-wheelers, there is an extra design flexibility to
place the centered wheel either at the front axle (a.k.a. delta configuration), or at the rear axle
(a.k.a. tadpole configuration).
To address this in the vehicle modeling process, a general wheel-configuration model in
the lateral plane is considered, as shown in Figure 3.1. Six-wheel modules are considered, which
are denoted as f l; fc; f r; rl; rc; rr, respectively. By enabling lateral and longitudinal tire forces
on corners that match the target vehicle configuration while disabling the rest, all configurations
of NTVs can be covered with the proposed generic model. It should also be noted that for any
centered wheels, its track width T
wj
is considered zero.
e lateral and yaw dynamics, considering the net lateral forces F
yCG
and yaw moment
M
zCG
, can be written as
Pv D
1
m
F
yCG
ur
Pr D
1
I
z
M
CG
;
(3.1)
where m and I
z
are the mass and yaw inertia of the vehicle; and u, v, and r denote vehicle
longitudinal speed, lateral speed, and yaw rate, respectively.
Generalized forces F
yCG
and M
zCG
on the right-hand-side of Eq. (3.1) are dependent on
the tire forces produced at each corner as well as actual wheel configurations. Indexing axles with
i 2 ff; rg (for front and rear axles), and wheels on each axle using j 2 fl; r; cg (for left, right,
and centered wheels), generalized forces and moments applied at the center-of-gravity (CoG)
16 3. TILTING VEHICLE DYNAMICS
F
xrl
F
xrc
F
xrr
F
xfl
F
xfc
F
xfr
F
yfl
F
yfc
F
yfr
F
yrl
F
yrc
F
yrr
δ
fr
δ
rr
α
fr
v
r
u
b
T
w
a
Figure 3.1: General double-track vehicle handling model for an NTV.
can be determined from individual tire force components as
F
yCG
D
X
i;j
F
xij
sin
ı
ij
C F
yij
cos
ı
ij
M
zCG
D
X
i;j
M
zij
D
X
i;j
0
@
T
wj
2
F
xij
cos
ı
ij
F
yij
sin
ı
ij
Ca
i
F
xij
sin
ı
ij
C F
yij
cos
ı
ij
1
A
;
(3.2)
where F
xij
and F
yij
are longitudinal and lateral forces generated at wheel corner ij; ı
ij
denotes
the steering angle applied on each wheel; and T
wj
and a
i
are generalized track width and axle
to CoG distance, respectively. Geometrical parameters a; b; T
w
are illustrated in Figure 3.1.
T
wj
D
8
<
:
CT
w
j D r
0 j D c
T
w
j D l
; a
i
D
a i D f
b i D r:
e force produced by pneumatic tires F
xij
; F
yij
at corner ij has been studied both theoret-
ically and experimentally [35–37]. In general, longitudinal and lateral tire forces can be written
3.1. TIRE FORCES AND LATERAL DYNAMICS 17
as
F
xij
D f
x
˛
ij
; S
ij
; F
zij
;
ij
F
yij
D f
y
˛
ij
; S
ij
; F
zij
;
ij
;
(3.3)
where ˛
ij
; S
ij
; F
zij
, and
ij
represent the side slip angle, the slip ratio, the normal force, and
the camber angle of each wheel ij, respectively. It should be mentioned that both conventional
automotive tires as well as motorcycles tires could be fitted with the model shown in Eq. (3.3)
using different parameter sets.
Motorcycle tires generate lateral force mainly through camber angles
ij
known as camber
thrust [38], while F
yij
from conventional automotive tires are dominated by side-slips angles ˛
ij
.
Differences in contact patch locations [11, 22] between these two types of tires are ignored in
this research. e side slip angles ˛
ij
and camber angles
ij
at each wheel ij are calculated as
˛
ij
D ı
ij
v C a
i
r
u
ij
D K
ij
;
(3.4)
where denotes vehicle roll angle and K
ij
is the camber-by-roll coefficient determined by sus-
pension and tilting mechanism designs.
For tractability in real-time control applications, an alternative to tire force expressed in
Eq. (3.3) is its linearized form. More specifically, for small slip ratios, the longitudinal force
contributed by tire ij is proportional to the driving or braking torque Q
ij
as
F
xij
D
Q
ij
R
wij
; (3.5)
where R
wij
denotes the effective rolling radius of wheel ij.
e nonlinear lateral force F
yij
in Eq. (3.3) could also be approximated by an affine-linear
tire model, which linearizes lateral forces at the current operation points of side-slip and camber
angles with the zeroth and first-order terms of the Taylor expansion [39, 40]. Denote the lateral
tire force, cornering, and camber stiffness of tire ij at the operating point as
N
F
yij
,
Q
C
˛i
, and
Q
C
i
,
respectively; the affine-linear lateral tire force model can be expressed as
F
yij
D
N
F
yij
C
Q
C
˛ij
˛
ij
N˛
ij
C
Q
C
ij
ij
N
ij
: (3.6)
To capture tire saturation properties, the maximum longitudinal and lateral tire forces
should be dependent on the normal force and the road friction conditions. For combined slip
scenarios, the tire force constraints can be formulated using the friction ellipse as
F
xij
F
xij max
2
C
F
yij
F
yij max
2
1; (3.7)
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