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Considering the computation and assumption from above, to practically imple-
ment a KBVS controller we must relax several of the assumptions made about the
scene and signals. First, the image plane is not continuous. Our kernel-projected
value then becomes a discretized summation, rather than a continuous integral:
=
≈
w
K
[
w
]
I
[
w
(
t
)]
ξ
K
(
w
)
I
(
w
(
t
))
d
w
.
(2.13)
I
Additionally, the image plane in not infinite. In (2.13), the domain of the integration
would be infinite whereas the domain of the summation extends to the boundaries
of the image. We work around this problem by choosing kernels with finite support.
That is, the kernels are selected such that the weighting is zero at the boundaries of
the image. This allows us to truncate the integration, as the integral over the domain
outside the image boundaries will essentially be zero.
Another limitation is the assumption of a planar scene. This is necessary to avoid
issues of parallax between near and far objects in the scene. Scene parallax could
significantly affect the size of the domain of attraction. In our experimental con-
figurations we have constructed a planar scene. In real world implementations, one
can ensure that the kernels and their finite support are wholly contained within an
approximately planar object.
Because we are making a weighted measurement on the intensities of each pixel
individually, we need to assume that each point in space is providing the same mea-
surement as it moves across the pixels of the camera. This is also known as the
brightness constancy constraint. To avoid problems of varying lighting, we normal-
ize the image at each time step to the maximum pixel value.
The first aim of the following sections is to verify the KBVS method empirically,
taking into consideration the fact that we are violating the assumptions of the an-
alytical solution. During experimentation, we are using the workarounds for each
of the nonideal issues as described above. Additionally, the analysis above only de-
termined that the KBVS control input produces an asymptotically stable system in
a neighborhood of the goal configuration, but gave no insight into the size of the
neighborhood. The second aim of the empirical validation is to characterize the do-
main of attraction for each of the degrees of freedom discussed above.
2.3
Empirical Validation
Empirical validation of KBVS is a two step process. The first is to characterize the
domain of attraction for a kernel and scene combination. As discussed during the
analytical derivation, there are scenes and/or kernel combinations that could result
in either a poor domain of attraction or instability. After finding a good set of kernels
for the scene, per the Jacobian rank condition, and having ascertained the expected
domain of attraction, the second step is to use the expected domain of attraction for
initial conditions to a set of experiments.
Subsection 2.3.1 discusses the domain of attraction for several of the combina-
tions of degrees of freedom for which we have an analytical solution as indicated in
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