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This result was expected since applying a translational velocity
v
1
to the frame
F
F
2
but taking into
account the change of frame (
v
2
=
1
R
2
v
1
). This variation is thus natural, since the
translational velocities to apply to the camera frame have to depend on its orien-
tation. Finally, this result shows that rotational motions do not change the rank of
the interaction matrix of the features used to control the translational DOF. In other
words, the rotational motions do not introduce singularities on the interaction matrix
and any rank change of the latter depends only on the translational motions.
1
is equivalent to applying a translational velocity to the frame
13.4.2.2
Variation of the Interaction Matrix with respect to Depth
Obtaining constant interaction matrix entries means that the selected features de-
pend linearly of the corresponding DOF. In this way, in [18, 30], it was shown that
for good
z
-axis closed-loop behavior in IBVS, one should choose image features
that scale as
s
Z
(
Z
is the object depth). This means that the variation with respect
to depth is a constant (
i.e.
the system is linear). In the case where the object is de-
fined by an image region, the following feature has been proposed in [30] to control
the motions along the optical axis:
∼
1
√
m
00
s
r
=
where
m
00
is the bidimensional moment of order 0 (that is the object surface in the
image) using the conventional perspective projection model. In the case where the
object is defined by a set of discrete points, the selected optimal feature was
1
(
s
d
=
(13.22)
μ
20
+
μ
02
)
where
ij
are the central moments computed from a set of discrete points (see [30]
for more details). Unfortunately,
s
r
and
s
d
allows only obtaining invariance to rota-
tions around the optical axis and not to all 3D rotations. For this reason,
s
I
=
μ
1
√
I
1
1
will be used instead of
s
r
. To explain the choice of
s
I
=
√
I
1
, let us first determine
how the polynomial invariant
I
1
behaves when
Z
increases by considering each term
of its formula. Let us consider the definition of the projection onto the unit sphere:
⎧
⎨
√
X
2
+
Y
2
+
Z
2
X
x
s
=
√
X
2
+
Y
2
+
Z
2
Y
y
s
=
(13.23)
⎩
Z
√
X
2
+
Y
2
+
Z
2
.
z
s
=
From the definition of the projection onto the unit sphere, it can be obtained that if
the depth
Z
increases (
i.e. X
Z
), the point projection coordinates have
the following behaviors with respect to depth:
x
s
∼
Z
and
Y
1
Z
,
y
s
∼
1
Z
and
z
s
∼
1. It follows
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