Precise Orbit Determination of Global Navigation Satellite System of ...

Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites
In Eq.(6-100), because
T
( H1 R1 H P )
1 − −1 −1
1 + 1′ and Φ21 2 2 Φ
1
T T
H R H2
21
therefore
T 1 1 T T T T T
( H R H + P′ ) Φ H R H Φ = Φ H R H Φ ( H R H + P′
)
− are positive and symmetrical matrices,
1 1 1
1 1 21 2 2 1
2 21 21 2 2 1
2 21 1 1 1
− − − − − − −1 −1
1 1
then Eq.(6-100) becomes
x = Φ
T −
( H R H
−
+ P′ T T −
) + Φ H R H Φ
−1
T −
( H R y
−
+ P′ T T −
x′ ) + Φ H R y
(6-101)
[ 1 1 ] [ ]
1
21 2 2 1
2 21 1 1 1 1 1 1 1 21 2 2 1 2
2 21 1 1 1
The sequential results are expressed by Eq.(6-99) and Eq.(6-101). Clearly Eq.(6-101) is the same as Eq.(6-90),
i.e. both batch and sequential processing should have the same results at the final epoch. Eq.(6-99) is different
from Eq.(6-89), which means except final epoch, the results of sequential processing at other epochs are different
from batch processing. Similarly, for observations y1, y2, Λ , ynthe results of batch and sequential processing at
the epoch n can be expressed as
n
−1
�
� �
n
�
T −1−1T T −1
T −1−1 T T −1
xn = Φn1�( H1 R1 H1 + P1′ ) + �Φi1HiRi HiΦi1��( H1 R1 y1 + P1′ x1′ ) + �Φi1HiRi
yi�
(6-102)
��
i=
2
��
��
i=
2
��
But actually, due to accuracy limit, after some steps of process the sequential processing has almost the same
results as batch processing, referred to Figure 6-3 and Figure 6-4.
Kalman Results
Results of Batch Process
102
101.5
101
100.5
100
99.5
99
98.5
98
102
101.5
101
100.5
100
99.5
99
98.5
98
S tart of K alm an F ilter
0 24 48 72 96 120
Tim e (hour)
144 168 192 216 240
Figure 6-3 Sequential Processing
83
E nd of K alm an F ilter
0 24 48 72 96 120
Time (hour)
144 168 192 216 240
Figure 6-4 Batch Processing