Image Processing Reference

In-Depth Information

of great importance whether for the analysis of observability [TRE 96], or of the con-

vergence of the extended Kalman filter [SON 85]. As a result, the analysis of observ-

ability becomes much easier if we rewrite equations [6.11] and [6.13] as follows:

X
k
+1
=
F

X
k
+
U

0

≡

z
k
=
H
k
X
k

where:

F
=Φ
t
k
+1
,t
k
=
Id

α

t
k

αId

t
k
+1
−

0

Id

[6.30]

X
k
X
t
k

H
k
=
cos
θ
k
,

sin
θ
k
,
0
,
0

−

≡

If we now assume that the observer is not maneuvering,
U

0. We then have:

z
0
=
H
0
X
0

z
1
=
H
1
F

X
0

[6.31]

.

z
k
=
H
k
F
k

X
0

with:

F
k
=
ccId

kαId

0

Id

so that the observability matrix

O

can be written:

⎛

⎞

⎛

⎞

cH
0

H
1
F

.

H
k
F
k

cos
θ
0

−

sin
θ
0

0

0

⎝

⎠

⎝

⎠

cos
θ
1

−

sin
θ
1

α
cos
θ
1

−

α
sin
θ
1

O

=

=

[6.32]

.

cos
θ
k

−

sin
θ
k

kα
cos
θ
k

−

kα
sin
θ
k

Factoring the observability matrix as follows is quite helpful:

⎛

⎝

⎞

⎠

−

r
y
(0)

r
x
(0)

0

0

r
y
(1)

−

r
x
(1)

αr
y
(1)

−

αr
x
(1)

O

=Δ
θ
Δ
r

.

r
y
(
k
)

−

r
x
(
k
)

kαr
y
(
k
)

−

kαr
x
(
k
)

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