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0.2
1
0.15
0.1
0.5
0.05
0
0
−0.05
−0.1
−0.5
−0.15
−0.2
−0.25
−1
2
4
6
8
10
2
4
6
8
10
Iterations
Iterations
(a)
(b)
1
1.4
0.5
1
0
0.6
0.2
−0.5
0
−0.2
−1
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Iterations
Iterations
(c)
(d)
1
Fig. 13.4 Results using I 1 : (a) features errors; (b) velocities (in m/s). Using s I =
I 1 :(c)
features errors; and (d) velocities (in m/s)
In a first simulation, we show the advantage of using s I = 1
I 1 instead of using
directly I 1 . For this purpose, the translational motion given by (13.28) has been con-
sidered between the desired and the initial camera poses. Further, in the control law
(13.4), the scalar
that tunes the velocity has been set to 1 and the interaction ma-
trix computed at each iteration is used ( i.e. L s = L s ). If the system were completely
linear, the convergence would be obtained in only one iteration. The nonlinearity of
the system has as effect to damp or to magnify the camera velocities. In our case ( i.e.
λ
λ
= 1), the nonlinearity can slow the convergence (damping the velocity) or it can
produce oscillations (magnifying the velocity). The results obtained using I t = 1
I 1
and using I 1 are given on Figure 13.4. From Figures 13.4(a-b), oscillations can be
observed for the features errors as well as for the velocities obtained using I 1 before
converging (after 9 iterations). On the other hand, a fast convergence is obtained
using I t = 1
I 1 without oscillations (after only two iterations the system has almost
converged). This shows that using I t =
1
I 1 , the system behaved almost as a linear
system.
t 0 = 0
6 m
.
2
,
0
.
3
,
0
.
.
(13.28)
In a second simulation, the rotational motion defined by the rotation vector
(13.29) has been considered. The rotation matrix is obtained from the rotation vec-
tor
θ
u using the well known Rodrigues formula. We compare the system behavior
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