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can be studied with respect to system (15.44) by using nested dead-zone nonlinear-
ities as in [20].
Remark 15.2.
The disturbance free case (
i.e.
,
= 0) corresponds to the case of a
fix target. In this case, the description of the closed-loop system is simpler. Further-
more, the condition C4 does not exist any more. Thus, the relations of Proposition
15.1 can be simplified, by removing roughly speaking all the lines and columns due
to the presence of
ω
ω
, and therefore are equivalent to those of Proposition 4 in [22].
15.4.2
Optimization Issues
It is important to note that relations (15.36), (15.37), (15.38) and (15.39) of Proposi-
tion 15.1 are LMIs. Depending on the energy bound on the disturbance,
1
,isgiven
by the designer or not, the following optimization problems can be considered:
δ
•
given
δ
1
, we want to optimize the size of the sets
E
0
and
E
1
. This case can be
addressed by the following convex optimization problem:
min
,
ε
,
ζ
,
δ
,
σ
ζ
+
δ
+
σ
W
,
R
1
,
Y
,
Z
,
S
subject to
(15.45)
relations (15
.
36)
,
(15
.
37)
,
(15
.
38)
,
(15
.
39)
,
σ
δ
I
6
Y
3
I
6
≥
0
,
≥
0
.
I
6
W
The last two constraints are added to guarantee a satisfactory conditioning num-
ber for matrices
and
W
;and
•
δ
1
being a decision variable, we want to minimize it. This problem comes to
find the largest disturbance tolerance. In this case, we can add
K
δ
1
in the previous
criteria; the other decision variables may be kept in order to satisfy a certain
trade-off between the size of the sets
E
0
,
E
1
and
δ
1
.
15.5
Application
This section presents two applications of the proposed visual servo control approach
for driving a wheeled robot equipped with a camera mounted on a pan-platform
(Figure 15.2).
With respect to the world frame
R
(
O
Z
),
x
and
y
are the coordinates
of the robot reference point
M
located at mid-distance between the wheels. Let
R
M
(
M
,
X
,
Y
,
Z
M
) be a frame attached to the robot, with
X
M
directed along the
robot main axis, and
R
P
(
P
,
X
M
,
Y
M
,
Z
P
) a frame attached to the pan-platform, its ori-
gin
P
being located at its center of rotation
P
.Let
,
X
P
,
Y
P
,
θ
denote the angle between
X
M
and the
X
-axis, and
θ
p
the angle between
X
P
and the
X
M
. The camera is rigidly
fixed to the pan-platform. The transformation between
R
P
and the camera frame
R
C
consists of an horizontal translation of vector
ab
0
and a rotation of angle
2
about the
Y
P
-axis.
D
x
is the distance between
M
and
P
. The velocity of the robot,
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