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y
=
x
(3D servo)
x
=
F
(
x
)
u
;
or
y
=(
x
1
−
x
∗
1
y
1
−
y
∗
1
(10.1)
...
x
M
−
x
∗
M
y
M
−
y
∗
M
)
=
H
(
x
)
.
(2D servo)
10.2.1.3
Closed-loop Model and Problem Statement
Let a visual feedback with state vector
x
c
and no external input be connected
to (10.1). Denote by
x
(
x
x
c
)
∈
R
n
the state vector of the consequent autonomous
closed-loop system. The camera converges to the reference situation whenever the
closed-loop equilibrium
x
∗
=
0
is asymptotically stable. This state space formula-
tion prevents “local minima” [3],
i.e.
convergence to poses such that
u
=
0
while
s
=
s
∗
. Instead, the convergence of
s
to
s
∗
is a consequence of the attraction of
x
∗
=
0
.
Moreover, as all the variables depicting the closed-loop system come as mem-
oryless functions of
x
, the fulfillment of the other criteria can be turned into the
boundedness of some functions
ζ
j
,
ζ
j
]. For instance, the
visual features' projections can be restricted to the limits of the camera image plane
by defining
ζ
j
(
x
) by suitable intervals [
s
∗
j
, including for 3D servos. Actuators saturations can be dealt
with as well,
e.g.
by defining some
ζ
j
=
s
j
−
ζ
j
's as entries of the velocity screw
u
or norms of
subvectors extracted from
u
. 3D constraints, such as constraining the camera motion
inside a corridor, can be handled even for 2D servos by bounding some distances
ζ
j
=
d
3D
j
. Last, imposing bounds on the control signal
u
or the differences
s
j
−
s
∗
j
enable the avoidance of differential singularities in the loop transfers,
e.g.
when
using some 2D “inverse Jacobian” control schemes.
Without loss of generality, each so-called
additional variable
ζ
j
(
.
),
j
∈
Ξ
J
,is
defined so that
ζ
j
(
0
)=
0
.
10.2.2
The Rational Systems Framework
In the following, the multicriteria visual servoing problem is recast into the rational
systems framework, and the induced potentialities are analyzed.
10.2.2.1
Rewriting the Multicriteria Analysis/Control Problem
Definition 10.1 (Rational System [13]).
Asystemissaid
rational
if it is defined by
x
y
=
A
x
u
(
x
,
χ
)
B
(
x
,
χ
)
,
(10.2)
C
(
x
,
χ
)
D
(
x
,
χ
)
n
x
n
where
x
∈X ⊂
R
is the state vector and
χ
∈X
χ
⊂
R
χ
is the vector of uncertain
parameters;
X
and
X
χ
are given polytopic sets containing the origin
0
; and
A
(
.,.
)
,
B
(
.,.
)
,
C
(
.,.
)
,
D
(
.,.
)
are rational matrix functions of
(
x
,
χ
)
well-defined on
X×X
χ
,
i.e.
with no singular entries for all
(
x
∈X×X
χ
.
The open-loop system (10.1) involves all the attitude coordinates in trigonomet-
ric functions. So, it can be easily turned into a rational form,
e.g.
by redefining
,
χ
)
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