Information Technology Reference
In-Depth Information
(
t
x
t
y
t
z
tan(
2
) tan(
2
) tan(
4
)
)
. Connecting the obtained model (10.2) (with zero
matrix functions
x
A
.,.
D
.,.
(
) and
(
) therein) with a rational controller
x
c
u
=
K
c
(
x
x
c
y
,
x
c
)
K
cy
(
x
,
x
c
)
(10.3)
K
u
(
x
,
x
c
)
K
uy
(
x
,
x
c
)
n
c
, with
of state vector
x
c
∈X
c
⊂
R
X
c
a given polytope enclosing
0
, leads to the
autonomous closed-loop rational system
x
=
A
x
=(
x
x
c
)
∈
˜
˜
n
(
x
,
χ
)
x
,
X
X×X
c
⊂
R
,
n
=
n
x
+
n
c
,
0
∈
X×X
χ
.
(10.4)
A
˜
X×X
χ
. As outlined above, the asymptotic sta-
bility of the equilibrium
x
∗
=
0
of (10.4) whatever the uncertainty
(
.,.
) is assumed well-defined on
is sufficient to
the convergence of the camera from initial poses next to the goal. This condition is
in essence local, and must be complemented by regional stability considerations so
as to cope with farther initial sensor-target situations.
The relationships on the additional variables related to the other criteria can be
turned into
χ
∈
Ξ
J
, ∀
χ
∈X
χ
,
ζ
j
=
Z
j
(
x
∀
j
,
χ
)
x
∈
[
ζ
j
,
ζ
j
]
,
(10.5)
˜
with each
Z
j
(
.,.
) a rational column vector function well-defined on
X×X
χ
.
10.2.2.2
Potentialities for Visual Servoing
The general equation (10.3) depicts a dynamic gain-scheduled nonlinear controller,
whose parameters are set on-the-fly to rational functions of the relative sensor-
target situation parametrization
x
and/or the controller state vector
x
c
. Noticeably,
(10.3) can specialize into simpler visual feedbacks, such as dynamic linear 3D or
2D schemes, and encompasses most “classical” strategies. For instance, the classi-
cal “inverse 3D-Jacobian” schemes
u
=
−
λ
B
−
1
(0)
x
and
u
=
−
λ
B
−
1
(
x
)
x
, with
B
(
.
)
such that
x
=
B
(
x
)
u
, respectively correspond to a linear static state feedback
u
=
K
x
and to a nonlinear rational static state feedback
u
=
K
(
x
)
x
. Similarly, the “inverse
(
s
∗
,
z
∗
)]
+
(
s
s
∗
) and
u
=
z
)]
+
(
s
s
∗
),
2D-Jacobian” controllers
u
=
−
λ
[
J
−
−
λ
[
J
(
s
,
−
with
J
(
.,.
) the interaction matrix defined from
s
=
J
(
s
,
z
)
u
, can also be dealt with.
Indeed,
) is a rational function of
s
and of the vector
z
made up with the depths
z
i
=
−
ST
i
.
z
S
, so that these 2D servos respectively correspond to a linear static output
feedback
u
=
J
(
.,.
(
x
)
y
.
Last, the problem statement can be enriched. Dynamic effects can be taken into
account in the open-loop model by building
u
with the forces and torques which
cause the camera motion, and by augmenting
x
with velocities. A finer modeling
of the camera can be handled as well. Uncertainties affecting rationally the camera
parameters, the measurements, or the target model coefficients
a
i
K
y
and to a nonlinear rational static output feedback
u
=
K
c
i
, can be also
inserted. The suggested framework can constitute a sound basis to the definition of a
“standard problem” [25] of visual servoing, through the introduction of a penalized
output
z
(playing in some way the role of a task function [21]) and of a criterion on
z
or on the transfer from a relevant input signal
w
to
z
.
,
b
i
,
Search WWH ::
Custom Search