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N
OMI
N
AL
C
ASE
+
−50
%
UN
C
ERT
A
INT
Y
ON
z
* IN
T
HE
C
ONT
R
OLL
E
R
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
tZ
tZ
(a)
(b)
Fig. 10.3
Multicriteria analysis of a 2 DOF image-based servo [12]: (a) nominal case; and (b)
±
50% uncertainty on
z
∗
. The horizontal and vertical axes correspond to the relative sensor-
target translation and rotation. The point (0
,
0) corresponds to the achievement of the posi-
tioning task. The admissible sensor-target situations are on the right of the left vertical curves,
which portray the actuators and visibility limits. Contrarily to (a), in (b) several curves express
the constraint on
v
z
, which depends on χ
Very often,
e.g.
when the visual servoing task is defined from the image data only,
the depth
z
∗
of the target spots at the reference situation can just be approximated
up to a bounded constant uncertainty
by
z
∗
=
z
∗
(1 +
χ
χ
). Building the control law
(
s
∗
,
z
∗
)] instead of [
(
s
∗
,
z
∗
)] may cause serious convergence problems [18].
with [
J
J
Simulations considering
= 0 are reported on Figure 10.4. To
limit the conservativeness, the considered biquadratic Lyapunov functions depend
on the uncertainty
χ
∈
[
−
0
.
5; +0
.
5] and ˙
χ
χ
through the matrix function
Θ
(
.,.
). Here,
V
(
x
,
χ
) writes as
)=(
t
z
,
N
2
)
P
(
t
z
,
N
2
V
(
x
).
Even if the extents of the elementary basins composing
,
χ
t
z
N
,
,
t
z
,
N
,
χ
t
z
N
,
,
t
z
,
N
,
χ
˜
in Figure 10.3 have
not been intensively optimized, BQLFs seem fairly limited. Figure 4(a) shows that
BQLFs do not provide enough DOF in order to expand the consequent multicriteria
E
1
1
0.8
0.8
0.6
0.6
1
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
1
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
0
1
2
3
4
5
6
7
8
t
z
−1
0
1
2
3
4
5
6
7
8
(a)
(b)
Fig. 10.4
2D servo multicriteria basin of attraction extended towards all the circled points:
(a) obtained from a single BQLF; and (b) obtained from a PW-BQLF
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