Points-based Visual Servoing with Central Cameras - Visual Servoing via Advanced Numerical Methods - page 285

value of the features and the desired one is small, and if the task to realize constrains
all the available DOF, that may be a good choice. However, as soon as the error is
large, problems may appear such as reaching local minimum or task singularities
[3]. Hybrid visual servoing is an alternative to the two previous control schemes. In
this case, the visual features gather 2D and 3D information.
The way to design adequate visual features is directly linked to the modeling of
their interaction with the robot motion, from which all control properties can be
analyzed theoretically. If the interaction is too complex ( i.e. highly nonlinear and
coupled), the analysis becomes impossible and the behavior of the system is gener-
ally not satisfactory in difficult configurations where large displacements (especially
rotational ones) have to be realized. To overcome these problems, it is possible to
combine path-planning and visual servoing, since tracking planned trajectories al-
lows the error to always remain small [20]. A second approach is to use the measures
to build particular visual features that will ensure expected properties of the control
scheme (refer for instance to [21, 14, 5, 13, 12, 4, 24]).
This chapter is concerned with homography-based visual servo control tech-
niques with central catadioptric cameras. This framework, also called 2 1/2 D
visual servoing [15] in the case where the image features are points, exploits a com-
bination of reconstructed Euclidean information and image features in the control
design. The 3D information is extracted from an homography matrix relating two
views of a reference plane. As a consequence, the 2 1/2 D visual servoing scheme
does not require any 3D model of the target. Unfortunately, in such approach when
conventional cameras are used, the image of the target is not guaranteed to remain
in the camera field of view. To overcome this deficiency, 2 1/2 D visual servoing
is first extended to the entire class of central cameras (including pinhole cameras,
central catadioptric cameras and some fisheye cameras [6]). It will be shown that as
when a conventional camera is employed, the resulting interaction matrix is block-
triangular with partial decoupling properties. Then two new control schemes will
be proposed. The basic idea of the first one is to control the translational motions
using a scaled 3D point directly obtained from the image points coordinates and
the homography matrix. Compared to the conventional 2 1/2 D visual servoing, it
allows to obtain better camera trajectory since the translation is controlled in the 3D
space while the interaction matrix remains block-triangular. Then, a hybrid scheme
which allow us to fully decouple rotational motions from translational ones ( i.e. the
resulting interaction matrix is square block-diagonal) will be proposed. For the three
proposed control schemes, it will be also shown that the equilibrium point is glob-
ally stable even in the presence of errors in the norm of 3D points which appears in
the interaction matrices.

16.2 Modeling

In this section, the unified cental projection model using the unitary sphere is briefly
recalled. Then, Euclidean reconstruction from the generic homography matrix is
addressed.