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using our features and using the point Cartesian coordinates (a conventional per-
spective projection has been considered). The obtained results are given on Figure
13.5. From Figure 13.5(a), it can be seen that a nice decrease of the features errors is
obtained using our features. Furthermore, from Figure 13.5(b), since the considered
translational motion is null, the translational velocity computed using the invariants
to rotations are null (thanks to the invariance to rotations). Further, as for feature
errors, Figure 13.5(c) shows a nice decrease of the rotational velocities. The re-
sults obtained using the point Cartesian coordinates to control the camera position
are given on Figures 13.5(d-f). Figure 13.5(d) shows a nice decrease of the feature
errors. On the other hand, the behavior of the velocities is far from satisfactory. In-
deed, a strong translational motion is observed (see Figure 13.5(e)) and since the
rotational DOF are coupled with the translational one, this introduced also a strong
oscillations of the rotational velocities (see Figure 13.5(f)).
u =
40 deg
θ
6
.
42
,
19
.
26
,
128
.
.
(13.29)
For the third simulation, the motion between the initial and desired camera poses
defined by (13.30) and (13.31) has been considered. The same desired camera pose
as for the first experiment was used. From Figure 13.6(a), it can be noticed that
the feature errors behavior is very satisfactory. The same satisfactory behavior is
obtained for translational and rotational velocities (see Figures 13.6(b-c)). Indeed,
nice decreases of the feature errors as well as for the velocities are obtained. On
the other hand the results obtained using the point Cartesian coordinates show a
strong translational motion generated by the wide rotation and also oscillations of
the whole velocities (see Figures 13.6(e-f))
u =
40 deg
θ
6
.
42
,
19
.
26
,
128
.
(13.30)
t 1 =
1 m
0
., −
0
.
3
,
.
(13.31)
13.5.3
Pose Estimation Results
In this part, our pose estimation method is compared with the linear method pro-
posed by Ansar in [1] and the iterative method proposed by Araujo [2]. The identity
matrix has been used to initialize i M o for our method and for the iterative method
proposed in [2]. The combination of the linear method and iterative method pro-
posed by Araujo is also tested. In other words, the results obtained by the linear
method will be used as initialization to the iterative method. The following setup
has been used:
an object composed of four points forming a square defined as follows has been
considered:
0
.
20
.
2
0
.
20
.
2
;
X o =
0
2
1111
.
2
0
.
20
.
20
.
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