Information Technology Reference
In-Depth Information
10.2.1
State Space Formulation
This section first introduces the notations underlying the mathematical modeling.
Then, an open-loop and a closed-loop state space model are drawn up, enabling the
statement of the problem.
10.2.1.1
Frames
,
x
O
,
y
O
,
z
O
) is
The forthcoming state space model hinges on three frames.
F
O
=(
O
,
x
S
,
y
S
,
z
S
) is rigidly linked to the camera, with
S
the optical center and
z
S
on the optical axis. The third frame
F
T
=(
T
associated to the world.
F
S
=(
S
,
x
T
,
y
T
,
z
T
),at-
tached to the target, is defined as the situation to be reached by
F
S
. The target is fitted
with
M
spots
T
1
,
T
2
,...
T
M
, arranged so that to any configuration of their perspec-
tive projections
S
1
,
S
M
onto the camera image plane corresponds a unique
sensor-target relative situation. Let (
−
ST
i
)
(
F
S
)
=(
x
i
y
i
z
i
)
and (
−
TT
i
)
(
F
T
)
=(
a
i
b
i
c
i
)
be the coordinates of
T
i
,
i
= 1
S
2
,...
,...,
M
, in frames
F
S
and
F
T
. The metric coordi-
nates (
−
SS
i
)
(
F
S
)
=(
x
i
y
i
f
)
of
S
i
,
i
= 1
M
, in frame
F
S
thus satisfy
x
i
=
f
x
i
z
i
,...,
and
y
i
=
f
y
i
z
i
, with
f
the camera focal length. In addition, the reference values of
x
i
and
y
i
write as
x
∗
i
=
f
a
i
and
y
∗
i
=
f
b
i
c
i
.
c
i
10.2.1.2
Open-loop State Space Model
The way the 6 degrees of freedom (DOF) camera is caused to move and its in-
teraction with the environment are described by an open-loop state space model.
In the considered kinematic context, its control input is the velocity screw
u
(
v
x
v
y
v
z
ω
x
ω
y
ω
z
)
, with (
v
x
v
y
v
z
)
and (
ω
x
ω
y
ω
z
)
the entries in
F
S
of the translational
and rotational velocities of
F
S
with respect to
F
O
.
As every variable of the system in open-loop is a memoryless function of the
sensor-target relative situation, the state vector can be set to
x
=(
t
r
)
with
t
(resp.
r
) a parametrization of the relative translation (resp. relative attitude) between
F
S
and
F
T
. The subvectors
t
and
r
can respectively be made up with the entries
(
−
ST
)
(
F
S
)
=(
t
x
t
y
t
z
)
in frame
F
S
of the vector joining
S
and
T
, and with the Bryant
angles
1
(
λμν
)
turning (
x
S
,
y
S
,
z
S
) into (
x
T
,
y
T
,
z
T
). The output vector
y
is defined
as the input to the controller. For a position-based scheme,
y
is equal to
x
.When
considering an image-based servo, one sets
y
=
s
x
M
y
M
)
s
∗
,where
s
−
(
x
1
y
1
...
depicts the projection of the visual features, and
s
∗
is its reference value.
The state equation, which explains the effect of the velocity screw onto the rela-
tive sensor-target situation, is obtained from rigid body kinematics. For 2D servos,
the output equation accounts for the interaction between the sensor-target relative
situation and the coordinates of the features' perspective projections. The state space
model, detailed in the companion report of [11], has the general form
r
is single-valued (μ
=
±
2
,
z
S
⊥
z
T
) as soon as a visibility constraint is added to the
problem.
1
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