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3.2. ROLLOVER WARNING 35
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
Steering Angle (rad)
Test 1
Test 2
0 1 2 3 4 5 6 7 8
Time (s)
Figure 3.13: e used steering angles in test 1 and test 2.
10
9
8
7
6
5
4
3
2
1
0
TTR (s)
Test 1
Test 2
0 1 2 3 4 5 6 7 8
Temps (s)
Figure 3.14: Comparison of TTR in test 1 and test 2.
in the rollover detection. is advantage is very interesting since the rollover must be avoided in
a matter of seconds.
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.
3.2. ROLLOVER WARNING 37
e derivative of Equation (3.17) can be written as
Pa
y
D
C
0
a
y
rU
C
1
Pr
mU
C
2C
f
m
1
sw
s C 1
1
SR
P
ı
d
; (3.18)
where .ı=ı
d
/ D .1=SR/ .1=
sw
s C 1/, ı
d
is the drivers steering-wheel angle,
sw
is the steering
ﬁrst-order time constant, and SR is the steering ratio.
By using this model-based ﬁlter, the noise from the diﬀerentiation of the steering-wheel
angle can be ﬁltered out using a low-pass ﬁlter. Moreover, the driver’s steering input information
plays an important role in predicting the rollover index due to the inherent time delay between
the steering input and its inﬂuence on vehicle roll.
e new PLTR is displayed as follows:
PLTR
t
0
.t / D
2h
T
w
a
y
.
t
0
/
g
C sin
C
2h
T
w
g
s
s C 1
a
y
.
t
0
/
C
s
s C 1
C
0
a
y
ru
C
1
Pr
mU
C
2C
f
m
1
sw
s C 1
1
SR
P
ı
d
!
C g
P
#
t:
(3.19)
Filter s
2
=..s C 1/.
sw
s C 1// is used on the drivers steering angle. Prediction time t
needs to be selected to be long enough to cover the rollover prevention system response time.
e term sin./ is approximately proportional to lateral acceleration. Hence, sin./ can be
replaced by ka
y
. e value of constant k depends on the CG height and suspension parameters
and will have to be accordingly tuned for each vehicle. For small roll angles, the term can be
entirely ignored. Finally, the ﬁnal form of the new PLTR is given as follows:
PLTR
t
0
.t / D
2h
T
w
g
.1 C kg/a
y
.
t
0
/
C
2h
T
w
g
s
s C 1
a
y
.t
0
/
C
s
s C 1
C
0
a
y
ru
C
1
Pr
mU
C
2C
f
m
1
sw
s C 1
1
SR
P
ı
d
!
C g
P
#
t:
(3.20)
e simulation was performed using a stock Humvee vehicle model in Carsim to illustrate
the eﬀectiveness of the PLTR with 0.3 s predictive time [37]. Figure 3.15 shows how the PLTR
matches the actual LTR proﬁle and the predictive quality of the PLTR. It is shown that the
PLTR shows a time advance (of the order of 100 ms) compared with the LTR. Otherwise, the
PLTR roughly matches the shape of the LTR trajectory.
Figure 3.16 shows the calculated LTR and PLTR from experimental vehicle test data for
the “Sine with Dwell” maneuver (open loop). e plots show a good correlation between the
simulation study and the actual implementation in the vehicle.
Figures 3.17 and 3.18 further present the calculation of the LTR and the PLTR from ex-
perimental vehicle testing data for the North Atlantic Treaty Organization double-lane change
38 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
LTR
PLTR
0 1 2 3 4
Time (s)
Figure 3.15: “Sine with Dwell” at a steering amplitude of 113.8
ı
(simulation).
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
LTR
PLTR
0 1 2 3 54
Time (s)
Figure 3.16: “Sine with Dwell” at a steering amplitude of 113.8
ı
(experimental measurements).
3.2. ROLLOVER WARNING 39
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
LTR
PLTR
7654 8 109
Time (s)
Figure 3.17: Double-lane change (experimental measurements).
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
LTR
PLTR
0 1 2 3 54 6 7 8 109
Time (s)
Figure 3.18: Fishhook maneuver (experimental measurements).
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