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problem. The nonlinear global model has a large validity domain and thus it can be
used for large displacements. Nevertheless, the prediction over the prediction hori-
zon can be time consuming. Moreover, this model requires 3D data that are the pose
of the target in the initial camera frame, as well as the target model. To reduce the 3D
knowledge, a solution can be the linearization of the model based on the interaction
matrix.
20.3.2
Local Model Based on the Interaction Matrix
For a point-like feature
s
expressed in normalized coordinates such that
u
=
X
/
Z
and
v
=
Y
/
Z
, the interaction matrix related to
s
is given by [5]
L
s
=
−
1
Z
u
Z
(1 +
u
2
)
v
−
0
uv
.
(20.18)
1
Z
v
Z
1 +
v
2
0
−
−
uv
−
u
The value
Z
is the depth of the 3D point expressed in the camera frame. The rela-
tionship between the camera velocity
τ
and the time variation of the visual features
s
is given by
s
(
t
)=
L
s
(
t
)
τ
(
t
)
.
(20.19)
In [11], this dynamic equation is solved to reconstruct the image data in case of
occlusion. Here, with a simple first order approximation, we obtain
s
(
k
+ 1)=
s
(
k
)+
T
e
L
s
(
k
)
τ
(
k
)
.
(20.20)
To avoid the estimation of the depth parameter at each iteration, its value
Z
∗
given or
measured at the reference position can be used. Consequently, the interaction matrix
(20.18) becomes
L
s
and depends only on the current measure of the visual features.
By considering here the visual features
s
as the state
x
, we obtain the set of equations
describing the process dynamics and outputs (20.7):
x
(
k
+ 1)=
x
(
k
)+
T
e
L
s
(
k
)
τ
(
k
)=
f
(
x
(
k
)
,
τ
(
k
))
(20.21)
s
m
(
k
)=
x
(
k
)=
h
(
x
(
k
))
.
This approximated local model does not require 3D data but only the approximate
value of
Z
∗
. 2D constraints can be taken into account since the model states and
outputs are the visual features. On the other hand, no information is available on the
camera pose and so 3D constraints can not be directly handled. For doing that, as for
the nonlinear model, it would be necessary to reconstruct the initial camera pose by
using the knowledge of the 3D model target. That is of course easily possible but has
not been considered in this chapter. Finally, for large displacements, a problem can
be mentioned as we will see on simulations: the linear and depth approximations
may be too coarse and can lead to control law failures.
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