36 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
3.2.2 PREDICTION ROLLOVER WARNING
A new predictive LTR (PLTR) is developed by Chad [37] can provide a time-advanced measure
of rollover propensity and, therefore, offers significant benefits for closed-loop rollover preven-
tion. First, a common rollover index (LTR
e
) based on the LTR is given as:
LTR
e
D
2h
T
w
g
a
y
C g sin
; (3.12)
where a
y
is the measured lateral acceleration of the vehicle, h is the distance from sprung mass
CG to roll center, T
w
is the track width, and is the vehicle roll angle.
With this index, the time between detection of potential rollover characteristics and the
moment rollover occurs may sometimes be too small for a rollover prevention system to stop
the vehicle from rolling over. Hence, a new predictive rollover index, i.e., PLTR, is developed.
is predictive index indicates future vehicle rollover propensity for a wide range of vehicle
maneuvers, based on data collected in the current time frame. e PLTR is defined as follows:
PLTR
t
0
.t / D LTR
.
t
0
/
C L
P
TR
.
t
0
/
t; (3.13)
where t is the preview time, and t
0
is the current time.
Considering the LTR
e
from Equation (3.12), we have
PLTR
t
0
.t / D LTR
.
t
0
/
C
2h
T
w
g
a
y
C g sin
t; (3.14)
or
PLTR
t
0
.t / D LTR
.
t
0
/
C
2h
T
w
g
a
y
C g
t: (3.15)
Equation (3.15) shows the calculation of the PLTR at time t
0
that is predicted for a future
time horizon t . a
y
is typically noisy, and it is difficult to obtain a smooth value of its derivative.
A filtering technique is first used to address this problem, as shown in the following equation:
PLTR
t
0
.t / D
2h
d
a
y
g
C sin
C
2h
T
w
g
s
s C 1
a
y
C
s
s C 1
Pa
y
C g
P
t; (3.16)
where is the time constant.
e lateral acceleration derivative in the second term can be further estimated from the
lateral dynamics. By utilizing a linear approximation and the small angle assumption, the lateral
dynamics equation can be written as:
ma
y
D C
0
ˇ C
1
r
U
C 2C
f
ı; (3.17)
where, C
0
D 2C
f
C 2C
r
and C
1
D 2aC
f
2bC
r
. C
f
and C
r
are the cornering stiffness values
for the front and rear tires, respectively. r is the yaw rate of the vehicle and U is the vehicle
speed.