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then
d
f
)
T
P
q
f
(
x
)
(
x
T
d
P
q
d
t
T
P
q
f
=
(
J
(
x
)
f
(
x
))
(
x
)
+
f
(
x
)
f
(
x
)
d
t
T
P
q
J
+
f
(
x
)
(
x
)
f
(
x
)
(4.35)
Given that the matrices
P
q
are independent of time and of the state vector, namely:
d
P
q
d
t
=
0
(4.36)
and that the transpose of a product of matrices is the product of the transposed in
permuted order:
T
B
T
A
T
(
AB
)
=
(4.37)
the Eq. (
4.35
) can be rewritten as:
d
f
)
0
T
P
q
f
(
x
)
(
x
d
P
q
d
t
T
J
T
P
q
f
T
T
P
q
J
=
f
(
x
)
(
x
)
(
x
)
+
f
(
x
)
f
(
x
)
+
f
(
x
)
(
x
)
f
(
x
)
d
t
T
J
)
f
T
P
q
+
(4.38)
=
f
(
x
)
(
x
)
P
q
J
(
x
(
x
)
Furthermore,
ϕ
q
(
)
ϕ
q
(
)
ϕ
q
(
)
d
x
d
x
d
x
=
x
˙
=
f
(
x
)
(4.39)
d
t
d
x
d
x
Substituting the expressions (
4.38
) and (
4.39
)in(
4.32
):
ϕ
q
(
d
f
T
J
f
s
ϕ
q
(
x
)
V
T
P
q
+
T
P
q
f
(
x
)
=
x
)
f
(
x
)
(
x
)
P
q
J
(
x
)
(
x
)
+
f
(
x
)
(
x
)
(
x
)
d
x
q
=
1
(4.40)
T
,so:
As
ϕ
q
(
x
)
are scalar functions, they can permute their position with
f
(
x
)
f
d
f
J
f
s
ϕ
q
(
x
)
V
T
T
P
q
+
T
P
q
f
(
x
)
=
(
x
)
ϕ
q
(
x
)
(
x
)
P
q
J
(
x
)
(
x
)
+
f
(
x
)
(
x
)
(
x
)
d
x
q
=
1
(4.41)
d
ϕ
q
(
x
)
The same applies to the product
f
(
x
)
, since
d
x
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