Graphics Programs Reference
In-Depth Information
1
2β
3
β
m
T
2β
m
T
---------
C
22
=
[
4
e
3
e
+
2β
m
T
]
(9.146)
1
2β
2
2β
m
T
β
m
T
---------
C
23
=
C
32
=
[
e
+
12
e
]
(9.147)
1
2β
m
2β
m
T
C
33
=
---------
[
1
e
]
(9.148)
Two limiting cases are of interest:
The short sampling interval case (
T
«
τ
m
),
1.
T
5
T
4
T
3
⁄
20
⁄
8
⁄
6
2σ
2
τ
m
----------
lim
C
=
T
4
T
3
T
2
(9.149)
⁄
8
⁄
3
⁄
2
β
m
T
→
0
T
3
T
2
⁄
6
⁄
2
T
and the state transition matrix is computed from Eq. (9.141) as
TT
2
1
⁄
2
lim
Φ
=
(9.150)
01
T
β
m
T
→
0
0
0
1
which is the same as the case for the
αβγ
filter (constant acceleration).
The long sampling interval ( ). This condition represents the case
when acceleration is a white noise process. The corresponding covariance
and transition matrices are, respectively, given by
T
»
τ
m
2.
2
T
3
τ
m
3
T
2
τ
m
τ
2
---------------
σ
2
lim
C
=
(9.151)
T
2
τ
m
β
m
T
→
∞
2
T
τ
m
τ
m
τ
2
τ
m
1
1
TT
τ
m
01 τ
m
00 0
lim
Φ
=
(9.152)
β
m
T
→
∞
Note that under the condition that , the cross correlation terms and
become very small. It follows that estimates of acceleration are no longer
T
»
τ
m
C
13
C
23
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