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⎡
⎤
⎡
⎤
100
0
v
r
v
r
sin
−
⎣
⎦
⎣
⎦
e
1
(
q
,
t
)=
−
γ
0
q
(
t
)+
θ
e
,
(15.48)
L
Y
2
J
0
H
B
where
L
Y
2
=
−
z
2
1 +
Y
2
is the interaction matrix associated with
Y
2
,and
J
is the Jacobian matrix given by (15.46). For this first task a kinematic controller
is defined by imposing an exponential decay of the task function, as it is classically
done for sensor-based control in robotics:
e
1
=
1
/
z
2
Y
2
/
−
λ
e
1
. The following expression of
q
is deduced:
H
−
1
(
e
1
+
B
)
q
=
−
λ
.
(15.49)
Ta s k 2
. The second task is the visual servo control task described in Section 15.2.2
but with a fixed target (see also Proposition 4 in [22]). The multicriteria controller
T
=
sat
u
0
(
) is deduced from Proposition 15.1. The expression of the robot ve-
locity vector
q
that constitutes the actual control input of the robot is then deduced
from the value of
T
and the robot Jacobian (15.46). In order to smooth the transi-
tion between the two tasks, the approach proposed in [19] is followed. Each task
e
i
,
i
= 1
K
ξ
2, is valid within a region
W
i
:
W
1
is a neighborhood of the wall inside
which the proximetric data can be measured, and
W
2
is the angular sector centered
at the visual target position, which is defined by (15.1) and (15.2). Once the robot
has reached the region
W
1
∩
,
t
) and its time derivative are
evaluated, while the robot still executes the first task. At the switching time
t
s
the
vision-based control loop is activated by considering the initial conditions
e
2
(
q
W
2
the task function
e
2
(
q
,
,
t
s
)
and
e
2
(
q
t
s
).
Simulation results
. At the beginning of task 1 the robot's configuration is given
by
x
= 0m,
y
=
,
24 rad. The reference wall is defined
by the line
y
= 1 m with respect to frame
R
. As we impose the security distance
of 1 m the actual reference path is the line
y
= 0. The parameters of the second
−
2m,
θ
0
= 0rad,and
θ
p
= 0
.
task are as follows: The coordinates of the target point are
E
1
14
033 m
,
.
42 m 2
.
E
2
14
533 m
and
E
3
14
033 m
with respect to
R
.Wefixed
.
42 m 1
.
.
42 m 1
.
d
min
= 2
454 m, and
d
max
= 10 m. The reference values for the projected target
points are:
Y
1
= 0
.
2m,
Y
2
= 0m,and
Y
3
=
.
−
0
.
2 m. To ensure target visibility
we consider
β
= 0
.
4. The bounds on the kinematic screw and the acceleration of
the camera are:
u
1
=
1m/s1m/s10rad/s
,and
u
0
=
1m/s
2
1m/s
2
5rad/s
2
.
The switching time
t
s
was fixed to 55 s. By applying the proposed control scheme
with the MATLAB
LMI Control Toolbox we obtained the following values of
the control gain
K
:
⎡
⎤
−
9
.
4931 27
.
1505
−
9
.
4931
−
4
.
7075
0
−
4
.
3744
⎣
⎦
.
K
=
−
19
.
6803
0
−
19
.
6803
0
−
6
.
2776
0
−
29
.
4251 55
.
1338
−
29
.
4251
−
4
.
4403
0
−
21
.
6589
Figure 15.6 represents the robot trajectory and the convergence of the visual data
Y
i
to their reference value
Y
i
. Figure 15.7 represents the camera's kinematic screw
along the motion and a zoom on the vision-based controller during the beginning
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