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basin of attraction (the thickest 1-level set) towards several distinct directions simul-
taneously. Figure 4(b) suggests that PW-BQLFs constitute a promising alternative,
still at a reasonable complexity. Note that in Figures 4(a)-4(b), if
e
s
a
s
b
=(
v
1
v
2
)
,
then
E
s
a
s
b
reads as
⎛
⎞
0000
v
1
v
2
v
1
v
2
0000
⎝
⎠
.
E
s
a
s
b
=
v
1
0
v
2
000
0
v
1
0
v
2
00
00
v
1
v
2
00
10.6
Conclusion
A new framework has been proposed to the analysis of a large class of 3D and
2D eye-in-hand visual servos. Aside from convergence, various criteria (
e.g.
ac-
tuators, 2D or 3D constraints) can be handled. The genuine problem has been
turned into the analysis of a nonlinear rational system subject to rational constraints.
By using DARs and extensions of the S-procedure, LMI conditions could be ob-
tained, enabling the definition of a “multicriteria basin of attraction” as a level
set of a BQLF. Though the consequent optimization programs are convex and of
polynomial complexity, the total computational cost gets fairly important if one
makes the union of basins of attraction computed separately. The extension to PW-
BQLFs was shown to lead to tolerably conservative conclusions still at a reasonable
computational cost.
Short-term incremental developments will concern an evaluation of the method
on more complex case studies, and its comparison with solutions based on other
types of Lyapunov functions,
e.g.
polyquadratic Lyapunov functions, or homo-
geneous polynomial Lyapunov functions in the vein of [5]. Multicriteria synthe-
sis will be considered as well, and actuators saturations will be tolerated rather
than avoided, which is meaningful in a kinematic context. Last, following the
duality acknowledged in [1], a new solution to visual-based localization will be
assessed.
More fundamental mean-term objective are twofold. On a control theory as-
pect, more efficient relaxations will be envisaged so as to get less conservative
LMI conditions. On the robotics side, the aim will be to “bridge the gap” be-
tween the suggested approach and existing developments in visual servoing. For
instance, other visual features will be inspected. One challenge is to get a rational
matrix function
C
.,.
) of the open-loop rational model (10.2) in closed-form, as has
been the case for spotted visual features, which leads to a better conditioning of
the problem. The aim is to limit the conservativeness of the conclusions, so that
an LMI analysis or synthesis strategy can provide an answer to difficult robotics
problems.
(
Acknowledgements.
The first author would like to thank Prof. Alexandre Trofino for early
fruitful discussions, and Prof. Michel Courdesses for his comments on the draft of this work.
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