Information Technology Reference
In-Depth Information
2.2
Kernel-based Visual Servoing
To aid the reader in understanding the basics of KBVS, we describe the details of
the algorithm in a simple two dimensional case [12] and refer the reader to Kallem
et al.
[13] for the details of the SO(3) case, roll about the optical axis, translation
along the optical axis, and combined motion derivations.
The kernel-projected value in KBVS, derived from the kernel-based tracking lit-
erature [4, 9, 8, 7, 17], is a weighted measurement of an image based on the a sam-
pling kernel. A Lyapunov function is formed from the vector of kernel-projected
values and the KBVS control input ensures Lyapunov stability, as shown below.
A key concept in the demonstration of stability is the equivalence of the kernel-
projected value under a change of coordinates: ideally there is no difference between
the kernel-projected value under a transformation of the image or the inverse trans-
formation of the kernel. In the two dimensional case discussed below, this transfor-
mation is simply translation parallel to the image plane of the camera.
Throughout the analytical derivation of KBVS, several assumptions are made
about the scene and the signals that appear in the computation. First, we assume
that the image plane is continuous (rather than discrete) and infinite, namely the
image plane is a copy of
2
. Second, we assume the scene to be planar. Finally, we
require that pixels are constantly illuminated across all image frames.
We assume the robot, as seen in Figure 2.1, is the kinematic plant
R
x
=
u
(2.1)
2
drives the
robot parallel to the image plane of the camera. Let the image be represented as a
y
]
T
,
where the configuration of the robot is described as
x
=[
x
and
u
∈
R
Fig. 2.1
The robot used for the experiments done in this chapter
Search WWH ::
Custom Search