Information Technology Reference
In-Depth Information
Note also that theoretically in this case
L
−
1
u
. This nice property can advan-
tageously be exploited to compute the control vector.
In practice, estimated camera parameters are used. The estimated rotation pa-
rameter θ
ω
θ
u
=
θ
u
can be written as a nonlinear function of the real ones
ψ
(
θ
u
).Since
L
−
1
ω
θ
u
= θ
u
, the closed-loop equation of the rotation control is
θ
u
dt
=
−
λ
L
ω
ψ
(
θ
u
)
.
The asymptotic stability of this system has been studied for conventional camera
(
has a simple analytical form [16]. How-
ever, the stability analysis remains an open problem when
ξ
= 0) since in this case the function
ψ
ξ
= 0 since the nonlinear
function
ψ
is much more complex in this case.
16.3.3.2
2 1/2 D Visual Servoing
2 1/2 D visual servoing has been first proposed by Malis and Chaumette in case of
conventional camera (
= 0). In this section, the original scheme is extended to the
entire class of central cameras. In order to control the translational motion, let us
define
s
as
ξ
s
=
s
1
s
2
where
s
1
=[
xy
]
and
s
2
= log(
) are respectively the coordinates of an image point
and the logarithm of the norm of its corresponding 3D point. The error between the
current value log(
ρ
ρ
) can be estimated using (16.8)
ρ
) and the desired value log(
s
2
= log(
since
s
2
−
).
The corresponding interaction matrix
L
s
can be written as
L
s
=
J
s
1
J
s
2
L
X
σ
where the Jacobian matrix
J
s
1
is given by (16.11), and
J
s
2
can be easily computed:
ρ
−
2
X
.
J
s
2
=
L
s
=
L
s
v
L
s
ω
can be obtained by stacking the interaction matrix in (16.11) and
000
1
σρ
2
(
x
2
+
y
2
)
Φ
ξ
−
1
L
s
2
=
J
s
2
L
X
=
−
Φ
−
Φ
,
(16.12)
x
y
1+γξ
Z
+
ξρ
ρ
1+
γξ
ρ
can be estimated only
once during an off-line learning stage. If the system is supposed correctly calibrated
and that measurements are noiseless, then the control law is asymptotically stable
for any positive value ρ
. However, the robustness with respect to calibration and
measurement errors still remains an open problem.
(
x
2
+
y
2
)
.
with
Φ
=
=
Note that the parameter
γ
+
ξ
Search WWH ::
Custom Search