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ˆ
ˆ
c
H
w
(
t
) (at least for the subspace of in-
terest), the objective of the estimation algorithm, often dubbed
localization
or
pose
estimation
algorithm, is to reconstruct the relative position between
ξ
(
t
) can be derived by the knowledge of
C
c
and
W
.
ˆ
c
H
w
(
t
) is composed of a rotation
R
and a translation
T
, (18.3)
Notice that if
T
]
w
P
. The fundamental matrix
F
associated with this
projective map is defined as
F
=
K
c
S
(
T
)
RK
−
c
,where
S
(
T
) is the skew-symmetric
matrix associated with the translation
T
. Trivially, if
I
p
d
is the feature position in the
desired camera position, we have
I
is rewritten as
λ
p
=
K
c
[
R
|
0]
w
P
, with the fundamental matrix
F
satisfying the condition
I
p
d
F
I
p
= 0. Therefore, assuming a calibrated camera
(
K
c
is known), the fundamental matrix can be robustly estimated knowing image
features correspondences
I
p
I
λ
p
d
=
K
c
[
I
|
I
p
d
([18]). Therefore, two main design requirements
↔
are derived:
1. full camera calibration (as implicitly assumed in the rest of the chapter);
2. a priori knowledge of
w
P
for each feature.
The latter condition is not strictly needed if the algorithm that performs the servo-
ing works in agreement with a mapping algorithm. In this case, the visual servoing
scheme can be applied to previously unknown portion of the environment, thus in-
creasing the autonomous capability of the mobile vehicle. The VSLAM for servoing
is an example of such an architecture. In fact, since the PBVS approach abstracts
sensor information to a higher level of representation, it allows the integration of
different sensorial sources, thus make it suitable to cooperate with SLAM based ar-
chitectures. In our example of a camera mounted on a mobile robot, for instance,
the synergistic use of odometry and visual feedback is viable if these information
are described in the same coordinate frame, where they can be fused coherently.
Summarizing, the PBVS approach estimates the error between the current and
desired robot position through an approximation of
ˆ
c
H
w
(
t
). For this reason, we can
conclude that the PBVS approach verifies the Definition 18.1, which implies the
satisfaction of Definition 18.2 in accordance with the accuracy of the estimation
algorithm.
18.2.2
Image-based Visual Servoing
IBVS and other sensor-level control schemes have several advantages in relation to
PBVS, such as robustness (or even insensitivity) to modelling errors, and hence suit-
ability to unstructured scenes and environments ([24]). Although IBVS is demon-
strated to be quite effective for manipulators ([28]), its control design turns out to
be more challenging for nonholonomic mobile vehicles. Indeed, the
image Jacobian
(18.4), which relates image features and robot's motion, cannot be used to solve the
general stabilization problem directly, a fact reported in famous results on nonholo-
nomic systems stabilization ([3]). However, as presented here, the classical visual
servoing scheme can be used together with path planners on image maps ([4]). Such
maps can be either given a priori or constructed on-line, according to a VSLAM
approach.
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