Information Technology Reference
In-Depth Information
•
(
Region IV
) region between the spiral
T
2
P
(bound of
Region III'
) and the horizon-
tal line passing through the feature and the desired position
P
(bound of
Region
V
). The
Region V
is symmetric to
Region IV
. Shortest path:
T
2
∗
T
1
P
;and
•
(
TRegion
)regiondefinedby
T
1
P
and
T
2
P
.
However, in [30] it has been shown that the previous taxonomy is only locally
valid,
i.e.
near the desired configuration. As a consequence, Regions II, II', IV, V
have to be further subdivided. Indeed, there exists a subspace of Regions IV and V,
for which the shortest paths are
SL
−
T
2
∗
T
1
P
or
SL
−
∗
T
2
P
(see Regions VI
and VII in Figure 18.8(b)), and a subspace of of Regions II, II' where the shortest
paths are of kind
T
1
T
1
∗
T
2
−
SL
or
T
2
∗
T
1
−
SL
.
18.4.1
IBVS and Optimal Paths
As the optimal paths are described and a taxonomy of the motion plane is derived,
a visual-based controller able to track the optimal trajectories is needed in order to
close the loop. Even though PBVS solutions can be adopted ([20]), the proposed
IBVS solution avoids localization algorithms and then inherits intrinsic higher ro-
bustness with respect to PBVS approaches ([5]).
The IBVS scheme here proposed is feasible if: 1) optimal paths can be computed
using only visual information and 2) once the vehicle optimal 3D words are “trans-
lated” to the image space. For the former point, it can be shown that only the orien-
tation
Ω
between the initial position
Q
and the desired one
P
is needed ([30]). Since
is defined as
0
ω
Ω
along the optimal path, it can be estimated using
epipolar geometry and the fundamental matrix
F
if at least eight feature points (in a
non singular configuration) are given ([18]). Estimation robustness can be achieved
if even more features are used.
(
t
)
dt
=
Ω
18.4.1.1
Trajectories on the Image Plane
From the definition of the optimal paths in Section 18.4, it follows that only three
specific kind of robot maneuvers needs to be translated: pure rotations,
i.e.
,
ν
= 0,
pure translations,
i.e.
,
ω
= 0, and logarithm spirals. In what follows we assume that
I
p
i
=[
I
x
i
,
I
y
i
]
T
,
I
p
c
=[
I
x
c
,
I
y
c
]
T
and
I
p
d
=[
I
x
d
,
I
y
d
]
T
are the feature initial, current
and desired positions respectively.
Pure rotation
: by plugging into the image Jacobian (18.4), the condition
ν
= 0
= constant and solving for the
I
y
c
feature coordinate, yields to
and
ω
= ¯
ω
I
y
i
cos
arctan
I
x
i
α
x
cos
arctan
I
x
c
I
y
c
=
α
x
,
(18.9)
that is the equation of a conic,
i.e.
, the intersection between the image plane and
the cone with vertex in the camera center (optical center) and base circumference
passing through the 3D feature position. On the other hand, solving for
I
x
c
, one
gets
Search WWH ::
Custom Search