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m
,
sat
u
0
(
u
(
i
)
)=
m
-order identity matrix.
Co
{.}
denotes a convex hull. For
u
∈
ℜ
|
u
(
i
)
|,
u
0(
i
)
) with
u
0(
i
)
>
,...,
sign
(
u
(
i
)
) min(
0,
i
= 1
m
.
15.2
Problem Statement
The problem of positioning a camera with respect to a mobile visual target is ad-
dressed. The system under consideration consists of a camera which is supported
by a robotic system. The camera is free to execute any horizontal translations and
rotations about the vertical axis.
The objective of the chapter is to design a controller to stabilize the camera with
respect to the target despite the following conditions:
•
(C1)
the depth of the target points with respect to the camera frame, is bounded
but unknown;
•
(C2)
the visual signal errors, in the image, must remain bounded during the sta-
bilization process to ensure visibility;
•
(C3)
the velocity and the acceleration of the camera remain bounded to satisfy
the limits on the actuators dynamics; and
•
(C4)
the velocity vector of the target is supposed to be square integrable but
unknown.
15.2.1
System Description
Let us now describe more precisely the system under consideration. The target is
made of three points
E
i
,
i
= 1
3, equispaced on a horizontal line, and located at
the same height as the camera optical center
C
.Let
R
be a frame attached to the scene
and
R
c
a frame attached to the camera, having its origin at
C
and its
z
-axis directed
along the optical axis. Let
T
,
2
,
3
denote the reduced kinematic screw of the camera
which expresses the translational and rotational velocities of
R
C
with respect to
R
.
The target is assumed to move as if it was fixed to a unicycle: it can rotate about
its central point
E
2
whereas its linear velocity
v
E
is always perpendicular to the line
(
E
1
∈
ℜ
,
E
denotes the target rotational velocity.
Furthermore, we make the hypothesis that the camera intrinsic parameters are
known and consider the metric pinhole model with focal length
f
= 1.
We denote by
l
E
3
) as indicated in Figure 15.1.
ω
>
0 the distance between the target points,
α
the angle between
the target line and the optical axis, and
the angle between the optical axis and
the line (
CE
2
) (see Figure 15.1). Without lost of generality we will assume that the
camera is initially located in the left-half-plane delimited by the line (
E
1
,
η
E
3
) and
that the distance
d
=
CE
2
is bounded as
d
∈
[
d
min
,
d
max
]
.
(15.1)
Furthermore, to prevent from projection singularities, it is considered that
α
∈
[
−
π
+
α
min
,−
α
min
]
(15.2)
where
α
min
>
0 is a small angle.
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