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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)
Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites � ∂x& ∂x& ∂x& ∂x& ∂x& ∂x& � � ∂x ∂y ∂z ∂x& ∂y& ∂z& � � ∂y& ∂y& ∂y& ∂y& ∂y& ∂y& � � � � ∂x ∂y ∂z ∂x& ∂y& ∂z& � � ∂z& ∂z& ∂z& ∂z& ∂z& ∂z& � � � � ∂x ∂y ∂z ∂x& ∂y& ∂z& � F = 2 2 2 2 2 2 �∂ U ∂ U ∂ U ∂ U ∂ U ∂ U � � 2 ∂x ∂∂ xy ∂∂ x z ∂∂ xx& ∂∂ xy& ∂∂ xz& � � � 2 2 2 2 2 2 �∂ U ∂ U ∂ U ∂ U ∂ U ∂ U � � 2 ∂∂ xy ∂y ∂∂ zy ∂∂ xy & & ∂∂ yy ∂∂ z&y � � � 2 2 2 2 2 2 �∂ U ∂ U ∂ U ∂ U ∂ U ∂ U � � 2 �∂∂ xz ∂∂ yz ∂z ∂∂ xz & ∂∂ yz & ∂∂ z&z� � i.e. � 0 0 0 1 0 0� � 0 0 0 0 1 0� � � � 0 0 0 0 0 1� � 2 2 2 ∂ U ∂ U ∂ U � � 0 0 0 2 � F = � ∂x ∂∂ yx ∂∂ zx � 2 2 2 �∂ U ∂ U ∂ U � � 0 0 0 2 � ∂∂ xy y zy � ∂ ∂∂ � 2 2 2 �∂ U ∂ U ∂ U � � 0 0 0 2 �∂∂ xz ∂∂ yz ∂z � � In Eq.(6-42), 2 2 ∂ U ∂ U = ∂∂ xy ∂∂ yx 2 2 ∂ U ∂ U = ∂∂ x z ∂∂ z x 2 2 ∂ U ∂ U = ∂∂ yz ∂∂ zy 73 (6-47) Considering the various perturbations, the general derivatives in Eq.(6-47) are very difficult to be expressed. In the following, only the geopotential, solar and lunar influences are considered, then U = Ue + Us + Um where subscript e means the geopotential, s means Sun and m Moon. 1) Geopotential Rewritting Eq.(5-1) as follows, GM ae Ue GM r r P C m S m n N n = + �� n+ 1 n= 2 m= 0 where nm (sin ϕ)( nm cos λ+ nm sin λ ) (6-48) G Newtonian gravitational constant M the Earth’s mass (GM=398.600415×10 12 m 3 s -2 ) Pnm(sin ϕ) associated Legendre function λϕ , geographic longitude and latitude of satellite spherical harmonic coefficients. Cnm , Snm Ue is in the earth-fixed coordinate system. Clearly, geopotential Ue is directly related to the spherical components r, ϕλ, , not Cartesian components xyz , , . From relations,
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PRECISE ORBIT DETERMINATION OF GLOB
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Summary GNSS-2 (Global Navigation S
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3.2.2.1 Principle..................
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9.3.3 The Effect of Distribution of
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Chapter 1 Introduction 1.2 Developm
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Chapter 2 Observations of Orbit Det
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Chapter 3 Orbit Tracking System and
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Chapter 3 Orbit Tracking System and
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Page 40 and 41: Chapter 4 Major Error Sources of Sa
Page 52 and 53: Chapter 5 Perturbation Models of IG
Page 74 and 75: Chapter 6 Algorithms of Orbit Deter
Page 94 and 95: Chapter 7 Orbit Determination Using
Page 104 and 105: Chapter 8 Geostationary Orbit Deter
Page 122 and 123: Chapter 9 Software and Simulation R
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Chapter 9 Software and Simulation R
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Conclusion 144
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References Carpino M. (1995): SAEG
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References Klobuchar, J. A.(1996):
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References Traquilla, J. M. and J.
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Acknowledgments 152
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