Points-based Visual Servoing with Central Cameras - Visual Servoing via Advanced Numerical Methods - page 291

16.3.3 Decoupled Visual Servoing

In visual servoing scheme, the control properties are directly linked to the interaction
between the designed features and the camera (or the robot) motion. The behavior of
the camera depends on the coupling between the features and the camera velocities.
For example, the interaction matrix in (16.11) related to the image coordinates of 2D
points is highly nonlinear and coupled. Thereof, large displacements of the camera
became difficult to realize.
Several approaches have been proposed to overcome these problems. Most of
them ensure a good decoupling properties by combining 2D and 3D information
when defining the input of the control law. The related control schemes are called
hybrid visual servoing. In this work, three model free decoupled control schemes
are proposed. Let us first define the observation vector as
s = s θ u

.

The vector s is chosen to be variant to the translational motions of the camera and
can be variant or invariant to the rotational motions, whereas the vector u ,rep-

resenting the rotational information between the current and the desired positions
of the camera, is invariant to the translational motions. Consequently, the global
interaction matrix L related to the features vector s is a block-triangular matrix:
L = L s v

θ

L s ω

.

0 3 L ω

Note that when s is invariant to rotational motions, L becomes a block-diagonal
matrix.

16.3.3.1 Interaction Matrix L ω

The rotation matrix between the current and the desired positions of the central
camera, can be obtained from the estimated homography matrix H . Several repre-
sentations of the rotation are possible. The representation

is the rotation
angle and u is a unit vector along the rotation axis) is chosen since it provides the
largest possible domain for the rotation angle. The corresponding interaction matrix
can be obtained from the time derivative of

θ u (where θ

θ u since it can be expressed with respect

to the central camera velocity screw τ :

= 0 3

L ω τ

d ( u )

dt

θ

where L ω is given by [17]
L ω = I 3 − 2 [ u ] × + 1

[ u ] 2

)
sinc 2 ( 2 )

sinc( θ

−

× .