2.5. MULTI-FREEDOM MODEL 15
A
3
D
2
6
6
6
6
6
6
6
6
6
6
6
4
k
b
C k
f
m
sf
gh
f
k
b
k
f
0
k
b
k
b
C k
r
m
sr
gh
r
0 k
r
k
f
0
k
uf
C m
uf
g
.h
uf
h
cf
/ k
f
0
0 k
r
0
k
ur
C m
ur
g
.h
ur
h
cr
/ k
r
3
7
7
7
7
7
7
7
7
7
7
7
5
I
A
4
D
2
6
6
6
4
l
f
0
0 l
r
l
f
0
0 l
r
3
7
7
7
5
I A
5
D
"
1 0
0 1
#
:
Because the vehicle moves when cornering, the lateral velocity and yaw rate do not vanish.
Hence, the dynamics of vehicle rollover can be described by Equation (2.34) in the partial un-
known state variables v, r, and vector U
s
, V . Setting the state vector as x D
Pv Pr
P
U
s
P
V
T
.
en Equation (2.35) can be rewritten into (2.36):
Px D Ax C Bı; (2.36)
where
A D M
1
q
A
q
; B D M
1
q
B
q
:
e distance between the first axle and the virtual rear axle is the equivalent wheelbase.
According to some papers, the method to determine the equivalent wheelbase is obtained. First,
the linear 2-DOF model of the vehicle can be set up based on Equations (2.25) and (2.26),
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
"
P
ˇ
Pr
#
D
2
6
6
4
k
f
k
m
k
r
mU
ak
f
C b
1
k
m
C c
1
k
r
mU
2
1
ak
f
C b
1
k
m
C c
1
k
r
I
Z
a
2
k
f
b
2
1
k
m
c
2
1
k
r
I
Z
U
3
7
7
5
"
ˇ
r
#
C
2
6
4
k
f
mU
ak
f
I
Z
3
7
5
Œ
ı
ˇ D
v
U
:
(2.37)
e yaw rate is fixed at steady state, now
P
ˇ D 0 and Pr D 0, so
ˇ
r
D
2
6
6
4
k
f
k
m
K
r
mU
ak
f
C b
1
k
m
C c
1
k
r
mU
2
1
ak
f
C b
1
k
m
C c
1
k
r
I
Z
a
2
k
f
b
2
1
k
m
c
2
1
k
r
I
Z
U
3
7
7
5
1
2
6
4
k
f
mU
ak
f
I
Z
3
7
5
Œ
ı
: (2.38)
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