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z
1
z
2
y
c
α
z
3
E
1
l
ω
E
η
d
z
2
v
E
c
E
2
C
l
d
min
2
E
3
d
2
max
Fig. 15.1
Description of the vision-based task
3, let us denote respectively by
Y
i
and
Y
i
the ordinates of the current
and desired target points in the image. Following the formalism introduced in [10],
a simple choice for the positioning task function is
For
i
= 1
,
2
,
⎡
⎤
e
1
e
2
e
3
⎣
⎦
∈
ℜ
3
e
=
with
Y
i
.
The desired camera position corresponds to
Y
2
= 0and
Y
3
=
e
i
=
Y
i
−
Y
1
. In addition, to
guarantee the visibility of the target, the condition C2 is specified by imposing the
following bound on task components:
−
|
e
i
|≤
β
,
i
= 1
,
2
,
3 ; with
β
>
0
.
(15.3)
Consequently, the angle
η
is bounded by
|
η
|≤
η
max
= arctan (
β
)
<
π
/
2
,
(15.4)
and therefore the depth
z
2
=
d
cos(
η
) of the central point
E
2
is bounded as
∈
,
.
z
2
[
d
min
cos(
η
max
)
d
max
]
(15.5)
The relation between the time-derivative of the task function and the kinematic
screw is given by the optical flow equations:
e
)
T
+
∂
e
e
=
L
(
z
,
(15.6)
∂
t
3
×
3
is the image Jacobian
where
L
(
z
,
e
)
∈
ℜ
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