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of the pose estimates required by such schemes has a decisive effect on their stabil-
ity and accuracy, their robustness to calibration or measurement errors may happen
to be poor. In addition, ensuring the visibility of the target at the controller design
stage is a nontrivial issue. On the other hand, image-based (or 2D) servos [14] state
the control problem in terms of some visual features extracted from the image,
e.g.
points, lines, moments,
etc
, which intrinsically makes them more robust to calibra-
tion errors and image noise. Yet, they present two drawbacks, all the more significant
as the displacement to perform is large: a satisfactory apparent motion in the image
may correspond to a contorted 3D trajectory, and the camera may converge to a
“local minimum” distinct from the desired goal.
In view of the above, many approaches have been developed towards the analysis
or synthesis of visual servos in regard to criteria embracing convergence, visibility
constraints, actuators saturations, singularity avoidance and 3D effective motion. To
cite few, stability analyses are proposed in [3, 17, 22] for specific image-based and
hybrid strategies. Visibility constraints can be ensured through path-planning [20],
navigation functions [10], as well as partitioned or switching strategies [7, 6].
Joint position limits are handled through constrained quadratic optimization in [15].
Within the task function framework, they are considered together with singularities
avoidance in [19] and with visibility constraints in [23] thanks to LMI optimization.
The work developed hereafter proposes a generic approach to the multicriteria
analysis of a large class of position-based and image-based servos. It is organized
as follows. Section 10.2 introduces the nonlinear rational systems framework and
argues its suitability to the analysis and synthesis problems. Section 10.3 outlines
the foundations of the method, namely elements of the Lyapunov theory and LMIs.
After detailing the multicriteria analysis method in Section 10.4, a case study is
considered in Section 10.5. A conclusion ends the chapter.
Notation.
Scalars, vectors, and matrices are respectively denoted as in
x
,
x
,
X
.The
zero vector, the identity and zero matrices are respectively termed
0
,
,and
may be subscripted by their dimensions. The transpose operator is represented by
.
The notation
I
and
O
is symmetric and pos-
itive definite (resp. semidefinite.) For block matrices, the symbol
M
>
0 (resp.
M
≥
0) means that the matrix
M
stands for sym-
metric blocks outside the main diagonal. Given a polytope
X
,
V
(
X
) denotes the
set of all its vertices. For any positive integer
N
,
Ξ
N
stands for the set
{
1
,...,
N
}
.
10.2
The Rational Systems Framework to Visual Servos
Multicriteria Analysis and Synthesis
Visual servos which aim to drive a perspective camera to a unique relative situation
with respect to a still target in a static environment are considered. Dynamic effects
in the camera motion are neglected, and dedicated spots serve as visual features. A
state space model uniting 3D and 2D schemes is first set up. Therein, the actuators,
the sensor and the image processing system are supposed perfect and instantaneous,
so they do not appear. Nonlinear rational systems are then shown to constitute a
sound and versatile framework to the multicriteria analysis.
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