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 � k1= f( tn, yn) � � 4 4h k2 = f( tn + h, yn + k1) � 27 27 � 2 h � k3 = f[ tn + h, yn + ( k1 + 3k2)] � 9 18 � 1 h � k4 = f[ tn + h, yn + ( k1 + 3k3)] � 3 12 � 1 h � k5 = f[ tn + h, yn + ( k1 + 3k4)] � 2 8 � 2 h � k6 = f[ tn + h, yn + ( 13k1 − 27k3 + 42k4 + 8k5)] � 3 54 � 1 h � k7 = f[ tn + h, yn + ( 389k1 − 54k3 + 966k4 − 824k5+ 243k6)] 6 4320 � � h k8 = f[ tn + h, yn + ( − 231k1 + 81k3 − 1164k4 + 656k5− 122k6 + 800k7)] � 20 � � 5 h k9 = f[ tn + h, yn + ( − 127k1 + 18k3 − 678k4 + 456k5 − 9k6 + 576k7 + 4k8)] � 6 288 � h � k10 = f[ tn + h, yn + ( 1481k1− 81k3+ 7104k4− 3376k5+ 72k6 −5040k7 − 60k8 + 720k9)] � 820 � h � yn+ 1 = yn + ( 41k1 + 27k4 + 272k5+ 27k6 + 216k7 + 216k9 + 41k10) � 840 � The right function f in Eq.(6-5) must be repeatedly computed several times for each integration step. The satellite movement equation is solved step by step, i.e. from known initial state y 0 , y 1 can be computed; from y 1 , y 2 can be computed, the rest is done similarly until y n is obtained. Hence Runge-Kutta algorithm is also called one step method. Because Runge-Kutta algorithm has such a low computation efficiency, normally, it is used during the starting phase of orbit integration. Afterwards, the so-called multi-step algorithm will be used for subsequent integration of satellite movement equation. 6.1.2 Adams-Cowell Algorithm Adams-Cowell’s algorithm is a multi-step numerical integration, which can solve the following form of differential equation = � � 0 = 0 � ) ( y& ( t) f ( t, y) (6-6) y t y Adams’ integration algorithm can be divided into two types, Adams-Moulton (calibration) and Adams-Bashforth (prediction). Using Admas-Moulton algorithm to integrate the differential equation, the solution is given by (Cappellari et al 1976 and Xu 1989) q � * yn+ 1 = yn + h� β k f n−k + 1 � k= 0 � q � * k * �m � � β k = � ( −1) γ m � � � � � � k m= k � � � (6-7) * 1 * 1 * 1 * �1 m = 0� γ m + γ m−1 + γ m−2 + ... + γ m−m = � � 2 3 m + 1 �0 m ≠ 0� �m � k( k −1)( k − 2)...[ k − ( m −1)] � = � � � � � � k � m! �� When yn+1 is computed from yn , the right function is computed only once, the other right functions fn-1, fn-2, ..., fn- 9 , have been already computed before this step, therefore Adams’ algorithm can integrate much faster than 66 (6-5)
Chapter 6 Algorithms of Orbit Determination of IGSO, GEO and MEO Satellites Runge-Kutta algorithm does, but it can not integrate the differential equation directly from initial condition y0, it needs y0, y1, y2, ..., yq that can be provided by Runge-Kutta algorithm. 6.1.3 Cowell Algorithm Cowell integration algorithm can solve following form of differential equations: &y ( t) = f ( t, y) � � y& ( t0) = y& 0 � y( t = � 0) y0 � There are also two kinds of algorithms, prediction and calibration. The Cowell prediction algorithm is: (Cappellari et al 1976 and Xu 1989) q � 2 y = 2y − y + h β f � β σ c n+ 1 k m j 0 = = 1− = � m= k j q � i= 1 σ = 1 n k �m � ( −1) � � � �σ m � k � m−1 � j= 0 1 i n−1 2 c jσ j + 2 � k = 0 m− j k n−k � � � � � � � � � � � � � � � �� and Cowell calibration algorithm can be written as (Cappellari et al 1976 and Xu 1989) q � 2 * yn+ 1 = 2y n − yn−1 + h � β k f n−k + 1 � k= 0 � q � * �m � * � β k = � � � � �σ m � = � k m k � � m−1 � * 2 * � σ m = −� c jσ m− j � j + 2 j= 0 � j � 1 � c j = � i � i= 1 � σ = 1 � 0 � �� 67 (6-8) (6-9) (6-10) In practice, the prediction and calibration are combined to integrate the differential equation, first using prediction to compute the approximate yn+1 and also compute the right function with a certain accuracy and then using calibration to solve more precise yn+1. 6.1.4 Numerical Integration of Satellite Dynamic Movement Equation The satellite movement equation Eq.(6-2) can be equally written as follows
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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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Page 24 and 25: Chapter 2 Observations of Orbit Det
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Page 52 and 53: Chapter 5 Perturbation Models of IG
Page 76 and 77: Chapter 6 Algorithms of Orbit Deter
Page 94 and 95: Chapter 7 Orbit Determination Using
Page 104 and 105: Chapter 8 Geostationary Orbit Deter
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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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