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reduce the domain of stability of the system. However, in most parts of applications
of image-based control the depth of the target points is unknown, and the interac-
tion matrix is computed at the final position [4]. An adaptive 2D-vision based servo
controller robust with respect to depth measurement was proposed in [24]. Ensur-
ing the visibility during the motion is also an important issue. Different tracking
methods that combine geometric information with local image measurements have
been proposed to this end in [15]. The idea of mixing feature-based control with
path-planning in the image was proposed in [16]. Robust control is particularly in-
teresting when dealing with moving targets for which the unknown velocity can
be bounded in some sense. In [9], generalized predictive control (GPC) and
H
∞
fast visual servoing were proposed for tracking moving target points with a medi-
cal manipulator. Advanced control techniques allow to consider different kinds of
constraints simultaneously at the control synthesis level. A general framework for
image-based positioning control design, based on robust quadratic methods and dif-
ferential inclusions, was proposed in [21]. The main drawback of this approach was
that the conditions were given in the form of bilinear matrix inequalities. A uni-
fied approach for position-based and image-based control, through rational systems
representation, was also proposed in [1]. Other recent methods were published, as
for instance switching strategies for ensuring visibility constraint [6], generation
of circular trajectories for minimizing the trajectory length [7], use of catadioptric
camera [13].
In this chapter, techniques allowing to design a multicriteria image-based con-
troller in order to track moving targets with square integrable velocity are proposed.
The considered approach is based on both polytopic and norm-bounded represen-
tations of uncertainties and uses an original sector condition for the description of
saturation terms. The proposed controller allows to stabilize the camera despite un-
known values of the target points depth, bounds on admissible visual features errors
to ensure visibility, and limits on the camera velocity and acceleration. Lyapunov
analysis and LMI-based optimization procedures are used to characterize a com-
pact neighborhood of the origin inside which the system trajectories remain during
the tracking. When the target is at rest, the asymptotic stability of the closed-loop
system is proved and a maximal stability domain is determined.
The chapter is organized as follows. Section 15.2 precisely describes the system
under consideration and formally states the control design problem. In Section 15.3,
some preliminary results exhibiting useful properties of the closed-loop system are
presented. Section 15.4 is devoted to control design results. Section 15.5 addresses
simulation results regarding the case of a camera mounted on a pan-platform sup-
ported by a wheeled robot. Finally, some concluding remarks together with forth-
coming issues end the chapter.
Notation.
For
x
,
y
n
,
x
∈
ℜ
y
means that
x
(
i
)
−
y
(
i
)
≥
∀
,...,
0,
i
= 1
n
. The elements
m
×
n
are denoted by
A
(
i
,
l
)
,
i
= 1
∈
ℜ
,...,
,...,
of
A
m
,
l
= 1
n
.
A
(
i
)
denotes the
i
th row
>
−
of
A
. For two symmetric matrices,
A
and
B
,
A
B
is positive
definite.
A
denotes the transpose of
A
.
D
(
x
) denotes a diagonal matrix obtained
from vector
x
.1
m
denotes the m-order vector 1
m
=[1
B
means that
A
1]
∈
ℜ
m
.
I
m
denotes the
...
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