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⎧
⎨
L
3
x
= 2
n
V
(
n
k
)
+
k
n
n
V
i
×
×
L
3
y
= 2
n
V
(
n
k
)
+
k
n
n
V
j
(5.22)
×
×
⎩
L
3
z
= 2
k
n
n
V
k
×
.
However, the interaction matrix is very often computed at the desired position [4].
Indeed, this way to proceed avoid to compute on-line 3D information like the depths
for example. We also here consider this case. More precisely, we consider that, at
the desired position the depth of all the points where the luminance is measured are
equal to a constant value
Z
∗
. That means that we consider that the object is planar
and that the camera and the object planes are parallel at this position. This case is
depicted on the Figure 5.2. Here, since we suppose that
J
n
=
0
and
n
=
−
k
,itis
straightforward to show that
L
2
=
0
. Besides, since
n
=
−
k
and
L
=
−
k
,wehave
k
.Wealsohave
J
R
=
0
. Consequently, from (5.21),
L
1
becomes
R
=
−
L
1
=
k
J
V
L
x
+
L
3
−
(5.23)
while
L
3
writes
000
2
V
i
0
. Finally, using explicitly
V
,
J
V
2
V
j
−
−
and
L
x
,
we simply obtain
x
Z
x
0
1
x
2
+
y
2
Z
y
Z
−
L
1
=
(5.24)
y
−
x
where
Z
=
Z
∗
2
.
As it can be seen, even if the computation of the vectors
L
1
and
L
2
to derive the
interaction matrix is not straightforward, their final expression is very simple and
easy to compute on-line.
x
5.3
Visual Servoing Control Law
The interaction matrix associated to the luminance being known, the control law
can be derived. Usually it is based on a desired behavior for the error signal
e
.More
often, an exponential decoupled decrease of this signal is required, that is
e
=
−
λ
e
where
λ
is a positive scalar. Therefore, expressing the temporal derivative of
e
,we
have
e
=
L
s
v
=
−
λ
e
(5.25)
leading to the classical control law given in (5.2) when considering that only an
approximation or an estimation of the interaction matrix is available.
However, we think that presenting the design of a control law from an optimiza-
tion problem, as proposed in [16], can lead to more powerful control laws.
5.3.1
Visual Servoing as an Optimization Problem
In that case, the cost function that we have to minimize with respect to the camera
current pose writes as follows
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