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

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Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites 6.3.1 Batch Processing Precise satellite orbit determination is usually post-processed in batch mode, i.e. all observation data available in a tracking session is collected and processed together. Using this mode the highest accuracy of orbit determination can be achieved. The basic method can be briefly described as follows. Assuming that the satellite state vector is ϖ x = {, x y, z,&, x y&, z&} , the state transition matrix is Φ( ti, ti−1 ) and the observation is ρ i . If the observation set is available in the following forms, L = { ρ ( t ), ρ ( t ),..., ρ ( t ); ρ ( t ), ρ ( t ),..., ρ ( t );...; ρ ( t ), ρ ( t ),..., ρ ( t )} then L L L 1 2 k 1 1 2 1 n 1 1 2 2 2 n 2 1 k 2 k n k , � ∆ρ1( t1) � � ∆ρ 2( t2) � = � � ϖ ϖ = H1Φ( t1, t0) x0 + ε 1 � Λ � � � �∆ρ n ( t1 ) � �∆ρ 1( t2) � � ∆ρ 2( t2) � = � � ϖ ϖ = H2Φ( t2, t0) x0 + ε 2 � Λ � � � �∆ρ n ( t2 ) � ............... �∆ρ 1( tk ) � � ∆ρ t � k = � 2 ( ) � ϖ ϖ = HkΦ( tk, t0) x0 + ε k � Λ � � � �∆ρ n ( tk ) � 80 (6-85) (6-86) (6-87) where Li the sub-observation matrix at epoch i, ∆ρ i( t j) the difference between computation and observation at tracking station i and Hi epoch tj, the observation coefficient at epoch i, Φ( ti, t0) ϖ x0 the state transition matrix between epoch ti and t0, the initial state vector including initial orbit parameters and initial ambiguities. ϖ x 0 can be solved using least-squares solution �L1 � � H1Φ( t1, t0) � ϖ � L � � H t t � ϖ ϖ L = � 2 � = � 2Φ( 2, 0) �x 0 + ε �... � � ... � � � � � �Lk��HkΦ( tk, t0) � ϖ ϖϖ ϖ ϖ T T H PHx + H PL = 0 0 ϖ ϖ ϖ ϖ ϖ T T x =−( H PH) H PL 0 where ϖ H � H1Φ( t1, t0) � �H t t � = � 2Φ( 2, 0) � � ... � � � �HkΦ( tk, t0) � (6-88)
Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites P the weight matrix of all observations 6.3.2 Sequential Processing According to discussion in section §6.2.1, sequential processing is used for both real-time and post-processing applications. Orbit determination will be updated when the new observations are available. The basic equations are Eq.(6-77), Eq.(6-81), Eq.(6-82) and Eq.(6-83). 6.3.3 Comparison Between Batch and Sequential Processing From the mathematical point of view, the result of batch processing is different from sequential processing, only at the final epoch the results should be the same. This can be explained by the following figures Figure 6-1 Batch Processing Figure 6-2 Sequential Processing In Figure 6-1 and Figure 6-2, the box represents the observation, vertical axis represents the time in which observation has been made, horizontal axis stands for epoch at which observation has been processed by sequential or batch processing mode. From Figure 6-1 and Figure 6-2 it can be seen that batch processing uses all observations during whole observation session to determine the satellite orbit. Sequential processing uses all observations available until the current to determine satellite orbit. From Figure 6-1, at epoch 1 the orbit results are produced by processing all observations. From epoch 2, epoch 3 ... until epoch k, the used observations are same for all epochs by batch processing. From Figure 6-2, procedures are different. At first epoch only observation at first epoch is used; at second epoch only first and second observations are used; at k epoch all observations from first epoch to k-th epoch are used. In other words, sequential processing only uses past and current observations and does not use future observations. From the description above we can see that the accuracy of orbit determination for the whole orbit arc length by batch processing is uniform, but the accuracy of orbit determination by sequential processing is not uniform. For sequential processing the accuracy of orbit determination will gradually become better and better as the new observations come and are processed. The exact proof can be given as follows: starting from simple situation, only two observations at two epochs y1, y2are considered. From batch processing � y1 � Hx 1 1 1 � � � y2 � H2 21x1 2 � � � x � x = � ′ � � � � ′ � � � � ′ � + � ε � � � Φ �ε � � � �ε � 1 t The solution 1 3 T T T T T T [ 1 1 ] [ ] 1 21 2 2 1 1 2 21 1 1 1 1 1 1 1 21 2 2 1 − − − − − − − ( ) ( ) 2 81 (6-88) x = H R H + P′ + Φ H R H Φ H R y + P′ x′ + Φ H R y (6-89) 1 1 1 1 and x = Φ T − ( H R H − + P′ T T − ) + Φ H R H Φ −1 T − ( H R y − + P′ T T − x′ ) + Φ H R y (6-90) [ 1 1 ] [ ] 1 21 2 2 1 2 21 1 1 1 1 1 1 1 21 2 2 1 2 2 21 1 1 1 1 2 3 ... ... ... ... ... ... ... k observation epoch t 1 2 3 ... ... ... ... ... ... ... k observation epoch
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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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Page 38 and 39: Chapter 3 Orbit Tracking System and
Page 40 and 41: Chapter 4 Major Error Sources of Sa
Page 52 and 53: Chapter 5 Perturbation Models of IG
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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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