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

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Chapter 7 Orbit Determination Using Carrier Phase Observation The observation equation Eq.(7-2) can be used with Kalman filter or batch processing to determine the satellite orbit. The advantage is that there are no initial ambiguities in Doppler measurements. Due to slow changes of the line-of-sight between tracking station and GEO satellite, it is difficult to determine a GEO satellite orbit using Doppler observations. This problem will be further shown in §7.2.2. 7.1.3 Effects of Initial Ambiguities on Orbit Determination Initial ambiguities will affect the accuracy of satellite orbit determination, which can be evaluated below. Rewriting Eq.(7-1): 2 2 2 i i i i i ρ = ( x − x) + ( y − y) + ( z − z) + λN Differentiating Eq.(7-3), 1 ∆ρi = [( xi − x) ∆x+ ( yi − y) ∆y+ ( zi − z) ∆z] + λ∆Ni (7-4) ρ′ where, 2 2 2 i i i ρ ′= ( x − x) + ( y − y) + ( z −z) Assuming that the accuracy of ∆x, ∆y, ∆z are equal and ∆ρ = 0 . The vectors ϖ ϖ ϖ ϖ r = xi + yj + zk � ϖ ϖ ϖ ϖ �� ri = xii + yij + zik � ϖ ϖ ϖ� ∆r = ∆xi + ∆yj + ∆zk�� Then Eq.(7-4) becomes λN i ϖ ϖ ϖ ϖ ϖ ϖ ( r − r r r r r θ i ) ∆ − i ∆ cos = ϖ ϖ = ϖ ϖ r − r r − r i letting θ=0, Eq.(7-6) can be written as λN r i =∆ϖ i 86 (7-3) ϖ ϖ r, ri and ∆ ϖ r can be expressed as Eq.(7-7) shows the influence of initial ambiguity on the accuracy of orbit determination. Using GPS L1 frequency as an example, one cycle error in the initial ambiguity will cause 0.19 meter error in satellite orbit. 7.1.4 Kalman Filter for Orbit Determination using Carrier Phase Observation Using the Kalman filter and carrier-phase observations to determine IGSO, GEO and MEO satellite orbits, the initial ambiguities should be included as system parameters in the satellite dynamical model or as non-random parameters in observation equations. The float solutions are obtained as Kalman filter processes the incoming observations. In order to fix the float ambiguities to integers, it is necessary to take special strategies such as bias optimizing (Blewitt, 1989), LAMBDA (Teunissen 1994 and Jonge et al, 1996), TCAR (Harris, 1996 and Forssell et al 1997), etc.. Assuming that the state variance is ϖ x = {, x y, z,&, x y&, z&, N1, N2, N3 ,..., Nn} in which n is the number of tracking stations, the satellite movement equation has a following form, � f1 � ϖ � dx f � ϖ = f x t t = � 2 [ ( ), ] � dt �... � � � � fn � (7-5) (7-6) (7-7) (7-8)
Chapter 7 Orbit Determination Using Carrier Phase Observation the state transition matrix Φ can be written as follows dΦ = FΦ (7-9) dt where, � ∂f1 ∂f1 ∂f1 ∂f1 ∂f1 ∂f1 ∂f1 ∂f1 � � ... ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N ∂N � � 1 n � �∂f 2 ∂f2 ∂f2 ∂f2 ∂f2 ∂f2 ∂f2 ∂f2 ... � � ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N ∂N � 1 n �∂f ∂f ∂f ∂f ∂f ∂f ∂f ∂f � � 3 3 3 3 3 3 3 3 ... � � ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N1 ∂Nn � �∂f 4 ∂f4 ∂f4 ∂f4 ∂f4 ∂f4 ∂f4 ∂f � 4 F = � ... � � ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N1 ∂Nn � � ... � � ... � � � �∂f k ∂fk ∂fk ∂fk ∂fk ∂fk ∂fk ∂fk ... � � ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N1 ∂N � n � ... ... ... � � ∂fn ∂fn ∂fn ∂fn ∂fn ∂fn ∂fn ∂f � � n ... � �� ∂x ∂y ∂z ∂x& ∂y& ∂z& ∂N1 ∂Nn �� 87 (7-10) The transition matrix Φ with initial ambiguities as parameters is more complex than without initial ambiguities. The Kalman filter algorithms discussed in Chapter 6 can be used with carrier phase observation for satellite orbit determination. According to Eq.(7-1), the carrier phase range equations can be written as 2 2 2 i i i i i ρ = ( x − x) + ( y − y) + ( z − z) + λN linearizing Eq.(7-10), (7-11) 1 ∆ρi = [( xi − x) ∆x+ ( yi − y) ∆y+ ( zi − z) ∆z] + λ∆Ni (7-12) ρ′ Then, ϖ ϖ ϖ y = H∆x + ε (7-13) where, � ∂ρ1 ∂ρ1 ∂ρ1 ∂ρ1 ∂ρ1 ∂ρ1 ∂ρ1 ∂ρ1 � � ... ∂x ∂y ∂z ∂x& ∂y& ∂z&∂N ∂N � � ∆ρ1 � � 1 n � � ∆ρ � � ∂ρ 2 ∂ρ 2 ∂ρ 2 ∂ρ 2 ∂ρ 2 ∂ρ 2 ∂ρ 2 ∂ρ 2 ... � ϖ � 2 � � ∂x ∂y ∂z ∂x& ∂y& ∂z&∂N ∂N � 1 n y = � ∆ρ 3 � , H = � ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ � � � � 3 3 3 3 3 3 3 3 ... � � ... � � ∂x ∂y ∂z ∂x& ∂y& ∂z&∂N1∂Nn� � �∆ρ m � � � ... ... ... � �∂ρ m ∂ρ m ∂ρ m ∂ρ m ∂ρ m ∂ρ m ∂ρ m ∂ρ � m � ... � �� ∂x ∂y ∂z ∂x& ∂y& ∂z&∂N1∂Nn�� ϖ ∆x = { ∆x, ∆y, ∆z, ∆x&, ∆y&, ∆z&, ∆N1,..., ∆Nn} In developing the equations above it is assumed that all observations are continuous. If the cycle slips occur, there are two methods to solve it: first using cycle slip fixing strategies to repair the cycle slips at the observation preprocessing session, therefore the above equations will never be changed; secondly the new initial ambiguities
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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 4 Major Error Sources of Sa
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Chapter 4 Major Error Sources of Sa
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Page 52 and 53: Chapter 5 Perturbation Models of IG
Page 74 and 75: Chapter 6 Algorithms of Orbit Deter
Page 96 and 97: 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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