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Y
c
Z
Y
c
Y
p
X
p
C
θ
p
Y
b
a
M
D
x
P
y
θ
X
M
M
X
x
O
Fig. 15.2
Camera fixed on a pan-platform supported by a wheeled robot
which constitutes the actual system input is described by the vector
q
=
v
1
v
2
v
3
,
where
v
1
and
v
2
are the linear and the angular velocities of the cart with respect to
R
,
respectively, while
v
3
is the pan-platform angular velocity with respect to the robot
main axis
X
M
. The system kinematics is
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎤
x
y
˙
cos
θ
00
v
1
v
2
v
3
sin
00
010
001
θ
⎣
⎦
.
=
˙
θ
p
The kinematic screw of the camera
T
is linked to the velocity vector
q
by the relation
T
=
J
(
q
)
q
,where
J
(
q
) is the robot Jacobian given by
⎡
⎤
−
sin
θ
p
D
x
cos
θ
p
+
aa
⎣
⎦
.
J
(
q
)=
cos
θ
p
D
x
sin
θ
p
−
b
−
b
(15.46)
0
−
1
−
1
As shown in [18], the rotational degree of freedom of the pan-platform allows to
overcome the nonholonomic constraint of the wheeled base.
15.5.0.1
Application 1
In the first application, the robot has to track a moving target and stabilize its cam-
era in front of it when it stops. The initial robot configuration is
xy
θθ
p
=
4
175
rad
, and the initial value of the state vector is
.
85
m
−
0
.
8
m
0
rad
0
.
ξ
(0)=
0849
m
0
ms
−
1
0
ms
−
1
0
rads
−
1
. At the beginning of the
task, the coordinates of the three target points, with respect to
R
,are:
E
1
10
m
0
−
0
.
12
m
−
0
.
0159
m
0
.
5
m
,
.
E
2
10
m
0
m
and
E
3
10
m
5
m
. According to condition C4, the target veloc-
ity is supposed to be unknown but square integrable, and its bound is defined by
taking
−
0
.
δ
1
= 12
.
1949 in (15.11). The interval of distance between the robot and the
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