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r
denotes the relative pose between
r
and
r
∗
. Therefore, by plugging (5.34)
into (5.26), we obtain an approximation of the cost function in a neighborhood of
r
∗
where
Δ
(
r
)=
1
C
r
H
∗
Δ
2
Δ
r
.
(5.35)
Of course, the graal would be that the eigenvalues of
H
∗
are all equal since in
that case the cost function would be an hypersphere. Indeed, only a global minimum
would exist and a simple steepest descent method would ensure to reach this mini-
mum. Unfortunately, when using the luminance as visual feature, the eigenvalues are
very different
3
. On the other hand, the eigenvectors of
H
∗
point out some directions
where the cost function decreases slowly when its associated eigenvalue is low or
decreases quickly when its associated eigenvalue is high. That means that the cost
function (5.26) presents very narrow valleys. More precisely, an eigenvector associ-
ated to a small eigenvalue corresponds to a valley where the cost varies slowly. In
contrast, the cost function varies strongly along an orthogonal direction. It can be
shown that is in a direction near
(
r
) [6]. These preferential directions where the
variation of the cost function is low are easy explained by the fact that it is very dif-
ficult to distinguish in an image an
x
axis translational motion (respectively
y
) from
a
y
axis rotational motion (respectively
x
). The
z
axis being the camera optical axis.
∇
C
5.3.3
Control Law
As shown in Section 5.3.1, several control laws can be used to minimize (5.26).
We first used the classical control laws based on the GN approach and the ESM
approach [16, 25]. Unfortunately, they may fail, either because they diverged or
because they led to unsuitable 3D motion. It is well-known in optimization theory
that minimizing a cost function that presents narrow valleys is a complex problem.
Therefore, a new control law has to be derived.
We propose the following algorithm to reach its minimum. The camera is first
moved to reach the valleys and next along the axes of the valleys towards the desired
pose. It can be easily done by using a control law formally equal to the one used in
the Levenberg-Marquardt approach (see Section 5.3.1). However, the way to tune
the parameter
is different. We denote this method in the remainder of the chapter
as modified Levenberg-Marquardt (MLM). As stated in the Section 5.3.2, the first
step can be easily done by using a gradient approach, that is by choosing a high
value for
μ
= 1). Once the bottom of valleys has been reached (see [6]
for more details), the parameters
μ
(typically
μ
is forced to decrease to turn the behavior of the
algorithm to a GN approach. The resulting control law is then given by
μ
diag(
H
))
−
1
L
I
e
v
=
−
λ
(
H
+
μ
(5.36)
where
μ
is not a constant value.
3
Note that this phenomenon also holds for most of the geometrical visual features usually
used in visual servoing since a term related to the depth always occurs in the translational
part of the interaction matrix (see (5.7)).
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