18 3. TILTING VEHICLE DYNAMICS
where
F
xij max
D
x
F
zij
F
yij max
D
y
F
zij
and
x
;
y
are the friction coefficients in longitudinal and lateral directions, respectively.
3.2 ROLL DYNAMICS AND ROLLOVER INDEX
Roll dynamics is crucial to the operational safety of any NTVs. is section derives a vehicle
roll model considering active tilting actuators. Apart from that, as a measure of the rollover
tendency, a rollover index based on lateral load transfer (LTR) is also developed.
A generic vehicle roll model is adopted and illustrated in Figure 3.2. e suspension system
is approximated using an equivalent rotational spring .K
/ and damper .C
/ connecting the
sprung .m
s
/ and un-sprung mass .m
u
/ at vehicle roll center (RC). e lateral acceleration .a
y
/
is treated as a disturbance to the roll model, and the tilting actuator generates an active control
moment .T
x
/ to regulate vehicle roll motions. e net forces at left- and right-side wheels are
written as .F
zl
; F
zr
/ and .F
yl
; F
yr
/, respectively, for their vertical and lateral components.
Geometrical parameters like the distance between sprung mass CoG to roll center .h
s
/, CoG
height of the un-sprung mass .h
u
/, roll center height .h
rc
/, and track width .T
w
/ are illustrated
in the figure.
e roll dynamics equation can then be derived as,
R
D
1
I
x
T
x
m
s
h
s
cos
.
/
mI
x
a
y
K
m
s
gh
s
sin./
I
x
C
I
x
P
; (3.8)
where I
x
denotes the roll inertia of the vehicle.
By defining the roll rate and roll angle as the states
ŒX
r
D
P
T
, tilting mo-
ment as the control input .U
r
D
Œ
T
x
/, and maneuver-induced lateral acceleration as distur-
bances .W
r
D
a
y
/, roll dynamics in Eq. (3.8) under the small angle assumption can be written
in the state-space form as
P
X
r
D A
r
X
r
C B
r
U
r
C E
r
W
r
; (3.9)
where
A
r
D
"
C
I
x
m
s
gh
s
K
I
x
1 0
#
; B
r
D
"
1
I
x
0
#
; E
r
D
"
m
s
h
s
I
x
0
#
:
For the rollover mitigation control, LTR index [41, 42] is widely adopted as the evaluation
metric for vehicle rollover tendency. Compared with the SSF index discussed in Section 1.2
which considers only geometrical aspects, the LTR is a more proper index to quantify the vehicle