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p
π
p
K
m
u
π
m
u
Camera
X
z
X
s
y
1
x
F
m
z
m
y
m
C
m
x
m
z
y
ξ
z
s
x
y
s
C
p
F
p
x
s
Convex Mirror/Lens
(a)
(b)
Fig. 13.1
(a) axis convention; and (b) unified image formation
where
L
s
is the interaction matrix related to
s
. The control scheme is usually de-
signed to reach an exponential decoupled decrease of the visual features to their
desired value
s
∗
. If we consider an eye-in-hand system observing a static object, the
control law is defined as
V
c
=
−
λ
L
s
+
(
s
s
∗
)
,
−
(13.4)
where
L
s
is a model or an approximation of
L
s
,
L
s
+
the pseudo-inverse of
L
s
,
a
positive gain tuning the time to convergence, and
V
c
the camera velocity sent to the
low-level robot controller. The nonlinearities in system (13.4) explain the difference
of behaviors in image space and in 3D space, and the inadequate robot trajectory that
occurs sometimes when the displacement to realize is large (of course, for small
displacements such that the variations of
L
s
are negligible, a correct behavior is
obtained). An important issue is thus to determine visual features allowing to reduce
the nonlinearities in (13.4). Furthermore, using (13.4) local minima can be reached
when the number of features is not minimal. Therefore, one would like to chose a
minimal representation (the number of features is equal to the number of DOF), but
without singularities and robust with respect to image noise.
The problem of pose estimation consists in determining the rigid transforma-
tion
c
M
o
between the object frame
λ
F
c
in unknown posi-
tion using the corresponding object image (see Figure 13.2). It is well known that
the relation between an object point with coordinates
F
o
and the camera frame
X
c
=(
X
c
,
Y
c
,
Z
c
,
1) in
F
c
and
X
,
,
,
o
=(
X
o
Y
o
Z
o
1) in
F
o
can be written as
o
=
c
R
o
c
t
o
c
=
c
M
o
X
X
X
.
(13.5)
o
0
31
1
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