Information Technology Reference
In-Depth Information
9.2.1
Notation and Problem Statement
The notation used in this chapter is as follows:
•
R
: real number set;
•
SO
(3): set of all 3
×
3 rotation matrices;
•
0
n
: null
n
×
1 vector;
•
0
m
×
n
: null
m
×
n
matrix;
•
I
n
:
n
×
n
identity matrix;
•
e
i
:
i
-th column of
I
3
;
•
X
: Euclidean norm;
•
X
∞
: infinity norm;
X
T
: transpose; and
•
•
s.t.: subject to.
In an eye-in-hand visual servo system, such as a robotic manipulator or a mobile
platform with a camera mounted on an end-effector, the goal consists of controlling
the robot so that the end-effector reaches a desired location by exploiting as feed-
back information the image projections of some object features. In particular, the
goal is assumed to be achieved when the object features in the current view match
the corresponding ones in the desired view, which have previously been recorded.
Image noise (for instance due to image quantization, lighting, features extraction,
etc
) unavoidably affects the image measurements,
i.e.
the estimate of the position
of the object features in the image. This means that the matching between object
features in the current view and in the desired view can never be ensured as image
measurement errors are nondeterministic. Moreover, even when this matching is
realized, there is no guarantee that the robot end-effector has reached the desired
location since the available image measurements are corrupted.
This chapter addresses the problem of bounding the worst-case robot positioning
error introduced by image measurement errors, which depends on the level of image
noise, camera parameters, and object features.
9.2.2
Mathematical Formulation of the Problem
Let
F
abs
be an absolute frame in the 3D space. We denote with
F
=(
O
,
c
) the current
camera frame expressed with respect to
F
abs
,where
O
∈
SO
(3) is a rotation matrix
3
defining the orientation, and
c
∈
R
is a vector defining the translation. Similarly,
we denote with
F
∗
=(
O
∗
,
c
∗
) the desired camera frame.
3
be a set of 3D points expressed with respect to
F
abs
.The
i
-th
3D point
q
i
projects onto
F
at the point
p
i
=(
p
i
,
1
,
Let
q
1
,...,
q
N
∈
R
1)
T
3
p
i
,
2
,
∈
R
given by
d
i
p
i
=
AO
T
(
q
i
−
c
)
(9.1)
×
3
is the upper-
triangular matrix containing the camera intrinsic parameters according to
3
where
d
i
is the depth of the point with respect to
F
,and
A
∈
R
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