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control task is to stabilize the robot towards the desired position controlling the
camera position ([4, 5, 23]). More precisely:
Definition 18.1.
Given the desired and the current robot positions, which corre-
spond the desired
C
d
=
{
O
c
d
,
X
c
d
,
Y
c
d
,
Z
c
d
}
and the current
C
c
=
{
O
c
c
,
X
c
c
,
Y
c
c
,
Z
cc
}
reference frames respectively, the stabilization in the desired position is ac-
complished if
C
c
≡
C
d
at the end of the control task.
Remark 18.1.
Since the mobile robot is moving constantly on a plane, the visual
servoing control approach is used only to stabilize at most a 3D subspace of the
state space
n
.
ξ
∈
R
As it is customary in the visual servoing literature, by a suitable robot state variables
change of coordinates,
d
. Hence, the visual servoing control problem
turns into a
point-to-point stabilization
problem,
i.e.
, we require that
W
≡
C
ξ
(
t
)
→
0as
t
→
≡
d
corresponds
+
∞
. In the particular framework proposed in this chapter,
W
C
to choose
X
w
=
Z
c
d
,
Y
w
=
Y
c
d
and
Z
w
=
X
c
d
.
A visual servoing scheme for robot control relays on the straightforward reformu-
lation of Definition 18.1 from the use of image feature positions (albeit the equiva-
lence between the two problems holds only if singularity configurations are avoided,
see [4, 26]).
−
Definition 18.2.
Given n current F
c
=[
I
x
c
1
,
I
y
c
n
]
T
image feature
po-
sitions, the servoing task is fulfilled if at the end of the controlled trajectory, F
c
matches the desired image feature positions F
d
=[
I
x
d
1
,
I
y
c
1
,
I
x
c
2
,...,
I
y
d
1
,
I
x
d
2
,...,
I
y
d
n
]
T
, i.e.,
I
x
d
i
=
I
x
c
i
and
I
y
d
i
=
I
y
c
i
,
∀
i
= 1
,...,
n.
In the presented chapter, we consider the visual servoing with an explicit feasi-
bility constraint: the image features must be always within the FOV of the cam-
era along the robot stabilizing trajectories (henceforth referred to as the
FOV
constraint
), which ensures that a visual feedback can be always performed. Prob-
ably, the problem of keeping the features in view during the robot manoeuvres is
one the most relevant problem to address for effective robot control. Multiple solu-
tions have been proposed in literature, ranging from omnidirectional cameras ([1]),
image path planning ([9]), or switching visual servoing schemes ([7]). In the pre-
sented dissertation, the FOV constraint will be addressed in the controller design,
which simplifies the mechanical set-up and lowers the overall cost. Moreover, we
will not focus on image processing backgrounds, giving for granted the feature ex-
traction, tracking and association among the image features. In particular, we adopt
the well-established scale invariant feature transform (SIFT) proposed in [21].
18.2.1
Position-based Visual Servoing
The main attractive feature of the PBVS approach is probably the relative simplicity
of the control design. Indeed, the control law can be synthesized in the usual work-
ing coordinates for the robot ([6]). Unfortunately, a position estimation algorithm
has to be necessarily provided. More precisely, since the estimated robot posture
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