Visual Servoing and Pose Estimation with Cameras Obeying the Unified Model - Visual Servoing via Advanced Numerical Methods - page 229

13.4 Features Choice

In this section, the features choice is detailed. We will first explain how to obtain
features to control the translational DOF with interaction matrices almost constant
with respect to depth distributions. Then, a vector of features to control the whole 6
DOF will be proposed.

13.4.1 Invariants to Rotational Motion

The shape of an object does not change under rotational motions. After a rotational
motion of the camera frame, it can easily be shown that the projected shape on the
sphere also undergoes the same rotational motion. This means that the invariants to
rotation in 3D space are also invariant if the considered points are projected onto the
unit sphere. The decoupled control we propose is based on this invariance property.
This important property will be used to select features invariant to rotations in order
to control the 3 translational DOF. In this way, the following invariant polynomials
to rotations have been proposed in [31] to control the translational DOF:

m 110 − m 101 − m 011 , I 1 = m 200 m 020 + m 200 m 002 + m 020 m 002 − (13.12)

m 300 m 102 + m 210 − m 210 m 012 + m 201 − I 2 = − m 300 m 120 − m 210 m 030 − m 201 m 021

m 201 m 003 + m 120 − m 120 m 102 + 3 m 111 + m 102 − m 030 m 012 + m 021 − − m 021 m 003

+ m 012 ,

(13.13)
I 3 = m 300 + 3 m 300 m 120 + 3 m 300 m 102 + 3 m 210 m 030 + 3 m 210 m 012 + 3 m 201 m 021
+ 3 m 201 m 003 + 3 m 120 m 102 −

3 m 111 + m 030 + 3 m 030 m 012 + 3 m 021 m 003 + m 003 .
(13.14)
The invariants (13.13) and (13.14) are of higher orders than (13.12). They are thus
more sensitive to noise [23, 32]. For this reason, I 1 will be used in this chapter to
control the translational DOF. Therefore, the set of points has to be separated in at
least three subsets to get three independent values of I 1 , which allows controlling
the 3 translational DOF. Furthermore, in order to decrease the variations of the in-
teraction with respect to depth distribution, it is s I =

1
√ I 1

that will be used instead of

I 1 . This will be explained in the following.

13.4.2 Variation of the Interaction Matrix with respect to the

Camera Pose

As it was mentioned above, one of the goals of this work is to decrease the non-
linearity by selecting adequate features. In this way, the invariance property allows
us to setup some interaction matrix entries to 0. These entries will thus be constant
during the servoing task. However, the other entries depend on the camera pose as it