Points-based Visual Servoing with Central Cameras - Visual Servoing via Advanced Numerical Methods - page 293

16.3.3.3 Norm-ratio-based Visual Servoing

ρ is invariant to rotational
motion. In the sequel, this property will be exploited in a new control scheme allow-
ing us to decouple translational motions from the rotational ones. At this end, let us
now define s as

As it can be seen in (16.12), the ratio between ρ and

ρ 3 ) .
The interaction matrix J s corresponding to s is obtained by stacking the interaction
matrices given by (16.12) for each point. In this case, the global interaction matrix
L is a block-diagonal matrix:

s = log(

ρ 1 ) log( ρ 2 ) log(

L = L s v

0 3

.

0 3 L ω

As above-mentioned, the translational and rotational controls are fully decoupled.
If the system is correctly calibrated and the measurements are noiseless, the system
is stable since ρ i

is positive.

ρ i

16.3.3.4 Scaled 3D Point-based Visual Servoing

Visual servoing scheme based on 3D points benefits of nice decoupling properties
[19] [2]. Recently, Tatsambon et al. show in [25] that similar decoupling properties
than the ones obtained with 3D points can be obtained using visual features related
to the spherical projection of a sphere: the 3D coordinates of the center of the sphere
computed up to a scale (the inverse of the sphere radius). However, even if such an
approach is theoretically attractive, it is limited by a major practical issue since
spherical object has to be observed.
Consider a 3D point with coordinates X =[ XYZ ] with respect to the frame

F m . The corresponding point onto the unit sphere is X s and X =

X

ρ X s .

Let us now choose s as

1
ρ

s = σ X s = X , (16.13)

F of the camera.
The feature vector s is thus defined as a vector containing the 3D point coordinates
up to a constant scale factor. Its corresponding interaction matrix can be obtained
directly from (16.10):

ρ is the 2-norm of where X with respect to the desired position

L X =

1
ρ

1
ρ

L s = − I 3 [ s ] .

×

ρ which
appears as a gain on the translational velocities. A nonzero positive value attributed
to

As it is shown in the expression of L s , the only unknown parameter is

ρ will thus ensure the global asymptotic stability of the control law. The ratio
between the real value of

ρ and the estimated one ρ will act as an over-gain in the

translational velocities.