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Substituting the above expressions in (
A.18
),
∂ω
∂ω
j
j
c
kj
r
j
r
j
N
j
N
j
N
j
N
j
r
=
1
r
=
1
r
=
1
r
=
1
c
kj
r
r
ω
−
ω
∂
x
q
∂
x
q
∂
c
kj
∂
x
q
=
N
j
2
r
=
1
ω
r
j
(A.21)
∂ω
∂ω
r
j
r
j
N
j
N
j
M
i
M
i
r
=
1
s
=
1
r
=
1
s
=
1
c
kj
ω
s
j
c
kj
ω
s
j
−
∂
x
q
∂
x
q
=
,
N
j
2
r
=
1
ω
r
j
which can be written as
∂ω
r
j
N
j
s
c
kj
−
c
kj
)
x
q
ω
j
(
∂
∂
c
kj
r
,
s
=
1
x
q
=
.
(A.22)
N
j
2
∂
r
=
1
ω
r
j
Substituting (
A.17
) and (
A.22
)in(
A.16
):
⎛
⎝
⎞
⎠
w
i
M
i
∂
x
q
w
p
b
p
b
l
ji
−
i
(
ji
)
∂
l
,
p
=
1
m
c
kj
+
M
i
2
j
=
1
l
=
1
∂
b
ji
c
kj
w
i
n
n
m
∂ω
x
k
=
˜
x
k
.
˜
r
j
N
j
∂
x
q
s
c
kj
−
c
kj
)
k
=
0
k
=
0
j
=
1
x
q
ω
j
(
∂
r
,
s
=
1
+
b
ji
N
j
2
r
=
1
ω
r
j
(A.23)
Therefore, the second term in the Jacobian matrix is:
n
j
=
1
b
ji
c
kj
x
k
∂
k
=
0
m
∂
m
=
b
ji
c
qj
x
q
j
=
1
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