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

Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites
ϖ
x
ϖ
ϖ
k = Φ k,
k−1
xk
−1
+ Γk,
k −1w
k−1
69
(6-14)
The observation equation is
yk H k xk
Gk
z k ε k
ϖ ϖ ϖ ϖ
= + +
(6-15)
where
ϖ
xk ϖ
yk ϖ
zk n dimensional signal state vector such as satellite orbit and dynamic parameters
m dimensional observation vector
p dimensional systematic parameter vector like tracking station coordinates, etc.
Φ kk , −1 n×n dimensional state transition matrix
ϖ
wk dynamic system noise vector
ϖ
ε k observation noise vector
Gk Γkk , −1
Hk m×p dimensional coefficient matrix of non-random parameters
coefficient matrix of dynamic system noise vector
m×n dimensional observation coefficient matrix
The a priori statistic information of ϖ~ x0 is given by the expectancy Ex { }
ϖ 0
ϖ 0 0
and the covariance Dx { } = P .
Obviously, according to the assumption in Eq.(6-14) and (6-15), ϖ xk is a random parameter vector and ϖ zk is a
non-random parameter vector. In order to solve the problem, Eq.(6-14) and Eq.(6-15) can be rewritten in the
following matrix form:
ϖ
ϖ ϖ
�y
k � �H
k Gk
��x
k � � ε k �
� ϖ � = � ��
ϖ � + � ϖ �
(6-16)
�x
k � � I x 0 ��z
k � ��
ε xk
��
where ϖ x k is assumed to be pseudo-observation, ϖ ε xk is the pseudo-observation noise.
The a priori values ϖ xk ′ and covariance matrix Pxk ′ can be approximated from Eq.(6-14). The results are
ϖ ~ ϖ
x = Φ − x −
k′
k,
k 1 k 1
P′
xk
= Φ
T
T
k,
k−1
Px
Φ k k k k Q
k 1 , −1
+ Γ , −1
k −1Γ
−
k,
k −1
(6-17)
(6-18)
where ϖ~ x k−1 is the estimator of ϖ x k−1 at the epoch of k −1; P is the covariance matrix of xk −1 ϖ~ x k−1 .
T
Qk − 1 cov( wk
−1,
wk
−1)
=
ϖ ϖ
is the system state noise covariance at k −1.
Covariance matrix of the observation is
�Rk
0 �
R = � �
� ′
�
0 Pxk
��
where Rk is the covariance matrix of observation vector ϖ yk ; Pxk ′ represents a priori information of system
parameters.
Using the least squares method, we can obtain
�
~ ϖ
� �
��
T
xk
H k
�~
ϖ � = ��
T
��
z k ��
�� ��
Gk
��
−1
I x R
k k
��
0 ��
��
��H
k
−1
��
P′
��
��
I x x
k k
−1
G ��
� � T
k H k
��
�
0
T
��
�� ��
Gk
��
−1
I x R
k k
��
0 ��
��
0
�
ϖ
0 �y
k �
−1
�
′
� ϖ �
Px
��
′
�
x
k k �
� T −1
−1
H k Rk
H k + Px′
k = � T −1
��
Gk
Rk
H k
−1
T −1
� � T
H k Rk
Gk
H k
T −1
� � T
Gk
Rk
Gk
��
��
Gk
��
−1
I x R
k k
��
0 ��
��
0
�
ϖ
0 �y
k �
−1
�
′
� ϖ �
Px
��
�x
k k �
� T −1
−1
H k Rk
H k + Px′
k
= � T −1
��
Gk
Rk
H k
−1
T −1
� � T −1
ϖ −1
ϖ
H
+ ′ ′ �
k Rk
Gk
H k Rk
yk
Px
x
k k
T −1
� � T −1
ϖ �
Gk
Rk
Gk
��
��
Gk
Rk
yk
��
ϖ ϖ
ϖ ϖ
Here, x′ , z′
are a priori unbiased estimates of x , z respectively.
k k
k k
(6-19)