Visual Servoing and Pose Estimation with Cameras Obeying the Unified Model - Visual Servoing via Advanced Numerical Methods - page 233

13.5.1 Variation of the Interaction Matrix with respect to Depth

Distribution

Figure 13.3 gives the variations of the interaction matrix entries of I 1 and s I = 1

√ I 1
with respect to translational motion applied along the optical axis to the four random
coplanar points defined in the camera frame as

⎛ ⎞

− 0 . 3258 − 0 . 0811 0 . 1487 0 . 2583

⎝ ⎠ .

− 0 . 0458 0 . 1470 − 0 . 1052 0 . 0039

X o = (13.26)

1 . 0000 1 . 0000 1 . 0000 1 . 0000

The set of points has been chosen to be approximatively centered with respect to the
z − axis ( m 100 ≈ 0 m 010 ≈ 0). For this reason, it can be seen that L x ≈ L x 1 ≈ L y ≈

L y 1 ≈ 0] ). In practice,

the features I 1 and s I also depend mainly on the translational motion with respect
to the object axis of view. From Figures 13.3(a-b), it can be seen that L z 1 is almost
constant and largely invariant to the object depth. On the other hand L z decreases to
0 when the object depth increases.

0( L I 1 =[ L x , L y , L z , 0 , 0 , 0] and L s I =[ L x 1 , L y 1 , L z 1 , 0 , 0 ,

1.5

0.2

L x
L y
L z

0

1

−0.2

L x 1
L y 1
L z 1

−0.4

0.5

−0.6

−0.8

0

−1

−0.5 −1.2

0 1 2 3

0 0.5 1 1.5 2 2.5 3

(a) (b)

Fig. 13.3 Variations of the interaction matrix with respect to depth distribution (in meter):
(a) results for I 1 ; and (b) results for s I =

1
√ I 1

13.5.2 Visual Servoing Results

In these simulations, the set of points is composed of 4 noncoplanar points. For all
the following simulations, the desired position corresponds to the 3D points coordi-
nates defined in the camera frame as

⎛ ⎞

0 − 0 . 200 . 2

⎝ ⎠ .

X d = 0 . 20 − 0 . 20 (13.27)

0 . 91 11 . 2

From the four set of points 4 different triangles can be obtained. For each triangle,
the invariant s I =

1
√ I 1

is computed to control the translational motion.