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

Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites
If assuming
T
k
−1 −1
k k xk A = H R H + P′
A
A
A
11
12
21
22
T −1
k Rk
Gk
T −1
k Rk
H k
T −1
k Rk
Gk
T −1
ϖ −1
ϖ
k Rk
yk
+ Px′
xk′
= H
= G
= G
b1
= H
b = G R
ϖ
y ,
2
T
k
−1
k
we can get
~ ϖ −1
xk = A11
b1
− A
~ ϖ −1
= −A
A A
z k
where
−1
11
22
k
k
−1
−1
11 A12
A22
b2
−1
−1
21 11 b1
+ ( A22
−1
22
−1
21)
+ A
−1
22
A
−1
21 A11
A12
A
−1
22
) b
2
70
(6-20)
(6-21)
A = ( A11
− A12
A A
(6-22)
If matrices D and C are positive definite and others are arbitrary matrices, following relation holds
−1
−1
−1
−1
−1
−1
( ACB + D)
= D − D A(
BD A + C ) BD
(6-23)
− 1
From Eq.(6-23), A11 becomes
A
−1
T −1
−1
T −1
T −1
−1
T −1
−1
11 = [ H k Rk
H k + Px′
− H k Rk
Gk
( Gk
Rk
Gk
) Gk
Rk
H k ]
k
{ } 1
−1
T −1
−1
T −1
−1
T −1
−
Px′ + H k [ Rk
− Rk
Gk
( Gk
Rk
Gk
) Gk
Rk
] H
k
T −1
−1
T −1
−1
T −1
−1
T −1
P x′
− P { } k x′
H
k k [ Rk
− Rk
Gk
( Gk
Rk
Gk
) Gk
Rk
] + H k Px′
H
k k H k Px′
k
T
T
T −1
−1
T −1
−1
−1
P x′
− P { } k x′
H
k k H k Px′
H
k k + Rk
I k − Gk
( Gk
Rk
Gk
) Gk
Rk
] H k Px′
k
= k
=
=
The gain matrix is defined as
K
[ (6-24)
{ } 1
T
T −1
−1
T −1
−1
−
H P′
H + R [ I − G ( G R G ) G R ]
T
k = Px′
H
k k k xk
k k k k k k k k k
(6-25)
Substituting Eq.(6-22) into Eq.(6-20), taking into account Eq.(6-27) yields
~ ϖ ϖ ϖ
T −1
T −1
−1
T −1
ϖ
x = x′
− K H x′
+ ( I − K H ) P′
H R [ I − G ( G R G ) G R ] y
(6-26)
k
k
k
k
k
k
k
xk
Assuming
−1
[ I − G
T −1
( G R G
−1
T −1
−1
−1
) G R ] = [ R − R G
T −1
( G R G
−1
T −1
) G R ] = W
Rk k k k k k k k k k k k k k k
then
( I − K H
T −1
) P′
H R [ I − G
T −1
( G R G
−1
T −
) G Rk
k
k
xk
= ( I − K H ) P′
H W
k k xk
k
T
k
k
−1
' ' T ' T −1
'
x − Px
H k [ H k Px
H k + W ] H k P
k k
k
xk
= { P
} H W
From Eq.(6-23), the equation above becomes
− 1 T −1
−1
T −1
[ P′
+ H W H ] H
= W
x k
k
k
k
k
k
k
k
k
k
T
k
k
k
−1
1
]
k
k
k
k
k
k
−1
(6-27)
(6-28)
According to the relation (D and C should be positive definite)
−1
−1
−1
−1
CB ( ACB + D)
= ( C + BD A)
BD
(6-29)
then Eq.(6-28) becomes
T
T
= P′
H ( H P′
H + W )
xk k k xk
= P′
H
xk
T
k
k
−1
T
T −1
−1
T −1
−1
−1
{ H k Px′
H k + Rk
I − Gk
( Gk
Rk
Gk
) Gk
Rk
] } = K k
k
[ (6-30)