4.3. ENVELOPE-BASED ROLL STABILITY CONTROL 35
k
a
D
2mh
0
T
w
mg
C
2m
s
h
s
T
w
m
m
s
h
s
K
m
s
gh
s
I
k
t
D
2m
s
h
s
T
w
m
1
K
m
s
gh
s
:
e LTR index in the form of Eq. (4.1) could be used to determine whether to apply the
control moment according to a
y
measurement and the LTR
lim
threshold. For a proactive con-
trol action in feed-forward designs, lateral acceleration a
y
could be estimated based on driver’s
steering wheel input ı leads to
LTR
ss
D k
a
u
2
l
0
C K
us
u
2
ı C k
t
T
x
; (4.2)
where K
us
is the vehicle understeer gradient coefficient.
For a given steady-state roll envelope specified by the threshold LTR
lim
, Eq. (4.2) can be
used to calculate the steady-state torque request (T
x
), based on driver’s steering maneuver (ı)
and vehicle operational speed (u) as
T
x
D
8
<
:
0 k
a
u
2
l
0
CK
us
u
2
ı LTR
lim
1
k
t
LTR
lim
k
a
u
2
l
0
CK
us
u
2
ı
k
a
u
2
l
0
CK
us
u
2
ı > LTR
lim
:
(4.3)
e piece-wise function in Eq. (4.3) demonstrates the idea of proposed envelope control:
by comparing the estimated roll stability measure with a pre-designed threshold, the activa-
tion condition of roll mitigation control can be tuned by proper envelope selections in control
schemes.
4.3.3 STEADY-STATE ROLL ANGLE ENVELOPE
Previous section derives a steady-state roll envelope from a torque perspective for feed-forward
tilting controllers. However, information collected from motion sensors on tilting vehicles can
greatly enhance the control performance. For this reason, the steady-state roll envelope is also
derived from a motion perspective in this section. e resultant steady-state roll angle based on
the envelope approach can be used for a more precise control of vehicle roll motions.
Under steady-state conditions, the transient terms in the LTR expression are ignored.
Denoting the threshold for LTR index as LTR
lim
, the desired tilting angle derived from the roll
envelope can be written as
des
D
(
mT
w
2m
s
h
s
sign
a
y
LTR
lim
mh
0
m
s
h
s
g
a
y
ˇ
ˇ
a
y
ˇ
ˇ
a
y
passive
otherwise;
(4.4)
where the lateral acceleration threshold a
y
to activate the tilting control is solved as
a
y
D
mT
w
2m
s
h
s
m
s
h
s
K
m
s
gh
s
C
mh
0
m
s
gh
s
LTR
lim
: (4.5)
36 4. TILTING VEHICLE CONTROL
It should be mentioned that by setting a zero track-width (i.e., T
w
D 0), or a zero rollover
index (i.e., LTR
lim
D 0), the desired tilting angle derived from the envelope approach can be
reduced to its the special form adopted by conventional tilting controls .
des
a
y
=g/. e
general expression given in Eq. (4.4) also considers the sprung and un-sprung mass distribution,
which could be very different for full tilting and partial tilting vehicles, as shown in Table 1.1.
It is also revealed by Eq. (4.5) that smaller track-width (T
w
), as well as a lower rollover index
threshold (LTR
lim
), could activate the tilting control at an earlier stage [70].
A major benefit of the proposed envelope approach for tilting control is the energy saving.
To demonstrate this, the energy consumption on tilting actuators .J
tilt
/ to achieve the desired
roll stability can be calculated using the inverse roll dynamics from Eq. (3.9), as detailed in [70].
Figure 4.2 illustrates the desired tilting angle and energy consumption under a quasi-static pro-
cess for various LTR index thresholds. It can be seen that the energy consumption increases
quadratically as the increase of the lateral acceleration. e energy saving with a higher LTR
threshold is majorly due to the “delayed” activation, which applies control efforts only under
more severe disturbances. e figure also shows that, maintaining a zero rollover index could be
challenging for large a
y
disturbances. e suspension mechanism, as well as the actuator, needs
to be properly designed to allow the excessive roll motion [70].
e energy consumption as a result of the suggested roll envelope approach is very de-
pendent on the driving style. Adopting the steady-state lateral acceleration in the feed-forward
form, as shown in Eq. (4.2), the desired tilting angle in Eq. (4.4) can be represented as a map-
Passive Roll
Active Tilt
φ
des
(deg)
50 12,000
8,000
4,000
0
–50
–5 50
0
(A
y
*
,φ
*
)
A
y
(m/s
)
–5 50
A
y
(m/s
)
J
tilt
(J)
LTR
lim
= 0.00
LTR
lim
= 0.25
LTR
lim
= 0.50
LTR
lim
= 0.75
Figure 4.2: Desired tilting angle and energy consumption.
4.3. ENVELOPE-BASED ROLL STABILITY CONTROL 37
ping dependent solely on vehicle longitudinal speed, steering angle, and threshold of the LTR
index. e contour of the desired roll angle with different LTR
lim
is visualized in Figure 4.3.
e lighter color in the figure denotes smaller tilting angles, while the white area represents
the scenarios where the rollover index is below the threshold, and no active tilting control is
required. As the increase of the LTR threshold, the area represents scenarios that require no
u (km/h)
50
40
30
20
10
0
u (km/h)
u (km/h)
u (km/h)
50
40
30
20
10
0
50
40
30
20
10
0
0 2 4 6 8 10
0
δ (deg)
0 2 4 6 8 10
δ (deg)
0 2 4 6 8 10
δ (deg)
0 2 4 6 8 10
δ (deg)
50
40
30
20
10
0
(a) LTR
lim
= 0.00 (b) LTR
lim
= 0.25
(c) LTR
lim
= 0.50 (d) LTR
lim
= 0.75
Figure 4.3: Contour plot of desired tilting angles with various LTR thresholds (lighter color
denotes smaller angles).
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