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where
θ
0
]
q
(
t
)
K
+
q
(
t
)) =
Ke
[
0
=
Kb
0
d
f
n
fT
K
+
,
(7.10)
Γ
(
θ
and
Φ
0
θ
0
]=log(
R
0
)
,
b
0
d
f
=
b
d
f
and
⎧
⎨
with
[
q
(
t
)=
t
if
PC1
,
⎩
2
t
3
+ 3
t
2
q
(
t
)=
−
if
PC2
.
The path given by Proposition 7.1 corresponds to a shortest distance path of the ro-
tation matrix (minimal geodesic) with respect to an adequately chosen Riemannian
metric on
SO
(3) and to a straight line translation.
7.3
The Constrained Problem
As previously, we assume that the current position of the camera with respect to its
desired position is given by the rotation matrix
R
(
t
) and the translation vector
b
(
t
).
Let
u
and
be the axis and the rotation angle obtained from
R
(
t
) and define the
camera state as
x
=[
u
θ
b
d
f
]
. Consider now the dynamical system described by
θ
the state equations
x
(
t
)=
f
(
x
(
t
)
,
U
(
t
)
,
t
)=
U
.
(7.11)
U
denotes the input vector. The state is known at the initial (
t
= 0) and final time
(
t
=
t
f
). The problem is to find a piecewise smooth input vector
U
from a known
initial state
x
0
to a desired
x
f
so that the cost function
J
=
t
f
t
0
U
T
U
dt
(7.12)
is minimized with boundary conditions
G
(
t
0
)
∝
G
0
,
G
(
t
f
)
∝
I
3
×
3
,
and so that a set of
l
image points
p
i
lies on [
u
min
u
max
]
×
[
v
min
v
max
] (where
u
min
,
u
max
,
u
min
,
u
max
are the image limits):
⎧
⎨
u
i
−
u
max
≤
0
,
i
= 1
···
l
,
u
min
−
u
i
≤
0
,
i
= 1
···
l
,
(7.13)
⎩
v
i
−
v
max
≤
0
,
i
= 1
···
l
,
v
min
−
v
i
≤
0
,
i
= 1
···
l
.
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