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

Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites
ϖ ϖ ∂O
ϖ ∂O
ϖ 1 ∂ O ϖ 2 1 ∂ O ϖ 2
yk
= y′
k + ( ) ′ ∆xk
+ ( ) ′ ∆z
k + ( ) ′ ( ∆x
) + ( ) ′ ( ∆ ) + ...
2 2 k
z
2 2 k
∂x
∂z
∂x
∂z
where „ ´ “ means related terms are computed by approximate ρ xk ′ at epoch tk .
Neglecting the second and higher order terms and supposing
∂f
∂O
�
Fk
= ( ) ′ k, Hk
= ( ) ′ k �
∂x
∂x
�
ϖ ϖ ϖ ϖ ϖ ϖ
∆xk′= xk − xk′ , ∆z
= zk − zk′
�
�
the linear dynamic and observation equations can be expressed as
ϖ
∆&x ϖ ϖ
′= ∆ ′+ �
k Fk xk wk
�
ϖ ϖ ϖ ϖ �
yk = Hk∆xk + Gk∆zk + ε k��
2
72
2
(6-40)
(6-41)
(6-42)
where ρ wk is the dynamic noise and ρ ε is the measurement noise and we can further assume that
ϖ ϖ
Ew ( ) = 0, E(
ε ) = 0.
k k
The satellite dynamic system equation becomes
ϖ ϖ ϖ
∆x = Φ ∆x + Γ w
k k, k−1 k−1 k, k−1 k−1
(6-43)
where Φ kk , −1 is a solution of following differential equation
dΦ
dt
= FΦ
(6-44)
Eq.(6-44) shows that if the dimension of state vector is 6, actually 36 differential equations in Eq.(6-44) must be
solved. In the orbit computation, the dimension of state vector is more than 6, so the solution of Eq.(6-44) is a
time-consuming task for computer. Eq.(6-44) can only be rigorously solved using numerical integration method.
For actually application, some approximation approach may be used. From Kalman filter algorithm, ϖ xk ′ should
be first computed using numerical integration, ϖ xk ′ is also called reference value (or ϖ ϖ Λ ϖ
x0′ , x1′ , , xk′ series called
reference orbit) and should be precisely computed, as the errors of the reference orbit will affect the accuracy of
orbit determination; for computation of ∆ ϖ x k and Pkk-1, Φ kk , −1 may not be so precise as for the computation of
ϖ
xk ′ . Therefore approximation approach can be used to compute Φ kk , −1 for ∆ ϖ x k and Pkk-1. The computation
burden will be significantly reduced and the accuracy of orbit determination will be kept.
The approximation approach of the solution of Eq.(6-44) in the stationary system can be expressed as
Φ( t) e Ft
= (6-45)
Supposing t=h, Φ is a system transition matrix, then according to Tarlor series
Fh
1 1
Φ( h) = e = I + Fh+ Fh* Fh+ Fh* Fh* Fh+
....
2!
3!
where
I an identity matrix.
Referring to Eq.(6-11) to Eq.(6-13), F in Eq.(6-46) may be written as
(6-46)