人卫运动方程的数值解法及激光测距技术在精密定轨中的应用研究
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摘要
描述人造卫星运动的微分方程非常复杂，不可能给出解的解析表达式。由天体摄动运动方程给出的小参数幂级数解等分析解满足不了人造卫星应用所需要的定轨精度要求，这就使得解常微分方程的数值方法在卫星动力学方法中占有十分重要的地位。
本文从应用的角度研究了天体运动方程的数值解法。首先，目前在卫星动力学中常用的所有人造卫星运动方程的数值解法有待改进的最重要的问题就是稳定性问题。无论是单步法还是多步法，某一步产生的误差都会传播下去，误差将会积累。当这种误差的积累得到控制时，对应的数值解法才是稳定的。
数值稳定性问题与步长的选择有关。定步长的缺点在于不管是积分曲线的各个点的曲率如何都取一样的步长，这样肯定会在曲率大的一些点带来比较严重的误差，这种误差将在每一步的计算中积累。如果以积分曲线每一步的曲率大小来控制步长大小，可以避免这种累积误差。从这个想法出发，在文中实现了以积分曲线的曲率控制步长大小的作法，在曲率大的点步长小，而在曲率小的点步长大，这是非常合理的选择，也是本文的一个创新点。
其次，众所周知Taylor级数法虽然公式简单且数值稳定性好，但是高阶导数的计算非常麻烦，难以实用。在本论文中，给出了以右函数f 在一些任意点的值表示x 在 点的高阶导数的方法，从而把人造卫星运动方程的数值解法转化为非线性方程组的迭代求解法。这种积分器精度高，数值稳定性也好。
本文作者同时作了激光测距技术在人造卫星精密定轨中的应用方面的一些研究工作，利用全球激光测距资料对Lageos-1卫星进行了定轨，精度达到1cm，并归算了长春SLR站的站坐标。分析了长春站的观测资料的处理结果，得出一些结论。进而设计了长春SLR站实现白天观测的实施方案。
The differential equations that describe the artificial satellite motion are very complicated, and it is impossible to get the analytic express of solution. Its analytic solution such as small parameter power series solution cannot meet the orbit determination accuracy that artificial satellite application needs. So this makes numerical method in ordinary differential equations for the satellite dynamics occupy very important position.
This paper studied the numerical methods in celestial motion equations from the applied aspect. First, now in all common numerical methods for the satellite dynamics equations the most important problem that needs to be improved is stability problem. Whether are a single-step methods or multi-step methods, the error in calculating every step will spread and accumulate. When the error accumulation can be controlled, the relative
method is just stable.
Numerical stability is relative with the choice of step-size. The weakness of fixed step-size is that step-size is not variable regardless the curvature of each point of the integral curve change or not, which will affirmatively bring serious error in these points that the value of curvature is bigger. And, this kind of error will accumulate in calculating process. If the change of step-size is controlled by the value of curvature of integral curve, the result can avoid such an error accumulation. From this viewpoint, we fulfilled to control the step-size with the curvature of integral curve. The step-size is smaller in calculating these points that the value of the curvature is bigger and the step-size is bigger in calculating these points that the value of the curvature is smaller, which is rather reasonable choice and is also a creative point in this paper.
The next in order, it is well known that the formula of Taylor series method is simple and its stability is good, but its high level derivative is not easy to be gotten and so it is difficult in practice. This paper points out that the values of the right function at arbitrarily points can express the high level derivatives of at , and gives out the formulas. So we translated the numerical methods for artificial satellite motion equation into the iterative method for a set of nonlinear equations. This kind of integrator is high accurate and great stable.
Author also did some research work for the application of Satellite Laser Ranging(SLR) technique to precise orbit determination. We calculated the
orbit elements of Lageos-1 with using the worldwide SLR data and the orbit accuracy attain 1 cm, meanwhile calculated the coordinates of Changchun SLR Station. Some conclusions were gotten by analyzing the data processing results of Changchun SLR Station. Further more, author designed an implementary project for daytime observation at Changchun SLR Station.
本文从应用的角度研究了天体运动方程的数值解法。首先，目前在卫星动力学中常用的所有人造卫星运动方程的数值解法有待改进的最重要的问题就是稳定性问题。无论是单步法还是多步法，某一步产生的误差都会传播下去，误差将会积累。当这种误差的积累得到控制时，对应的数值解法才是稳定的。
数值稳定性问题与步长的选择有关。定步长的缺点在于不管是积分曲线的各个点的曲率如何都取一样的步长，这样肯定会在曲率大的一些点带来比较严重的误差，这种误差将在每一步的计算中积累。如果以积分曲线每一步的曲率大小来控制步长大小，可以避免这种累积误差。从这个想法出发，在文中实现了以积分曲线的曲率控制步长大小的作法，在曲率大的点步长小，而在曲率小的点步长大，这是非常合理的选择，也是本文的一个创新点。
其次，众所周知Taylor级数法虽然公式简单且数值稳定性好，但是高阶导数的计算非常麻烦，难以实用。在本论文中，给出了以右函数f 在一些任意点的值表示x 在 点的高阶导数的方法，从而把人造卫星运动方程的数值解法转化为非线性方程组的迭代求解法。这种积分器精度高，数值稳定性也好。
本文作者同时作了激光测距技术在人造卫星精密定轨中的应用方面的一些研究工作，利用全球激光测距资料对Lageos-1卫星进行了定轨，精度达到1cm，并归算了长春SLR站的站坐标。分析了长春站的观测资料的处理结果，得出一些结论。进而设计了长春SLR站实现白天观测的实施方案。
The differential equations that describe the artificial satellite motion are very complicated, and it is impossible to get the analytic express of solution. Its analytic solution such as small parameter power series solution cannot meet the orbit determination accuracy that artificial satellite application needs. So this makes numerical method in ordinary differential equations for the satellite dynamics occupy very important position.
This paper studied the numerical methods in celestial motion equations from the applied aspect. First, now in all common numerical methods for the satellite dynamics equations the most important problem that needs to be improved is stability problem. Whether are a single-step methods or multi-step methods, the error in calculating every step will spread and accumulate. When the error accumulation can be controlled, the relative
method is just stable.
Numerical stability is relative with the choice of step-size. The weakness of fixed step-size is that step-size is not variable regardless the curvature of each point of the integral curve change or not, which will affirmatively bring serious error in these points that the value of curvature is bigger. And, this kind of error will accumulate in calculating process. If the change of step-size is controlled by the value of curvature of integral curve, the result can avoid such an error accumulation. From this viewpoint, we fulfilled to control the step-size with the curvature of integral curve. The step-size is smaller in calculating these points that the value of the curvature is bigger and the step-size is bigger in calculating these points that the value of the curvature is smaller, which is rather reasonable choice and is also a creative point in this paper.
The next in order, it is well known that the formula of Taylor series method is simple and its stability is good, but its high level derivative is not easy to be gotten and so it is difficult in practice. This paper points out that the values of the right function at arbitrarily points can express the high level derivatives of at , and gives out the formulas. So we translated the numerical methods for artificial satellite motion equation into the iterative method for a set of nonlinear equations. This kind of integrator is high accurate and great stable.
Author also did some research work for the application of Satellite Laser Ranging(SLR) technique to precise orbit determination. We calculated the
orbit elements of Lageos-1 with using the worldwide SLR data and the orbit accuracy attain 1 cm, meanwhile calculated the coordinates of Changchun SLR Station. Some conclusions were gotten by analyzing the data processing results of Changchun SLR Station. Further more, author designed an implementary project for daytime observation at Changchun SLR Station.
引文
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[3] Bettis D G. A Runge-Kutta Nystrom Algorithm. Celest Mech,1973,8: 229─233
[4] Black W. Practical Considerations in the Numerical Solution of the N-body Problem by Taylor Series Methods. Celset. Mech.. 1973,8:357─370
[5] Broucke R. Solution of the N-body Problem with Recurrent Power Series. Celset. Mech.. 1971,4:110─115
[6] Brouwer D, Clemence G N. Methods of celestial mechanics. Academic Press, New York, 1961, Chapter Ⅺ,ⅩⅦ
[7] Burlirsch,R., Stoer,J. Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods. Num.Math., 1966,8:1-13.
[8] Butcher,J.C. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John-Wiley & Sons. 1987.
[9] Channell,P.J. Symplectic Integration Algorithm. Los Alamos National Laboratory internal report. 1983, AT-6,ATN-83-9.
[10] Cowell P H, Crommelin A C D. Appendix to Greenwich Observations for 1909. Edinburgh. 1909,84
[11] Degnan J J. ed. Satellite Laser Ranging in the 1990s: Report of the 1994 Belmont Workshop. NASA Conference Publication 3283. 1994, ⅲ-ⅳ,71-76
[12] Degnan J J. Millimeter accuracy satellite laser ranging: A review. Contributions of Space Geodesy to Geodynamics: Technology. Geodynamics Series, Vol 25, AGU, 1993. 133~16
[13] Dormand J R, Ince, P J, Celest.Mech., 1978,18,223-232.
[14] Dou-xing Cui, Garfinkel B. A Variant of the Hori-Lie Series Method. Celest Mech, 1985,35: 89─94
[15] Dou-xing Cui, Tian-yi Huang. A Numerical Method for Differential Equations of Second Order. Celest Mech, 1994,55: 405─409
[16] Dou-xing Cui. A Method for Computation of Perturbance. Celest Mech, 1984,32: 1─13
[17] Dou-xing Cui. Gravitational Theory Based on Relativistic Mechanics. International Journal of Theoretical Physics, 1996,35: 1473─1480
[18] Emslie A K, Walker I W. Studies in the Application of Recurrence Relations to Special Perturbation Methods Ⅱ. Celest Mech. 1973,19:147─162
[19] Everhart E. Implicit Single-Sequence Methods for Integrating Orbits. Celest Mech. 1974,10:35─55
[20] Feagin T. Multistep Methods of Numerical Integration Using Back-Corrections. Celest Mech. 1976,13: 111─120
[21] Fehlberg E. Classical Fifth-,Sixth-,Seventh-,and Eight-order Runge-Kutta Formulas with Stepsize Control. NASA 1968,TR R-287
[22] Fehlberg E. Classical Fifth-,Sixth-,Seventh-,and Eight-order Runge-Kutta Formulas with Stepsize Control for General Second-order Differential Equations. NASA 1974,TR R-432
[23] Fehlberg E. NASA TR, 1972,R-381
[24] Fehlberg E. Z Angew Math Mech,1964,44
[25] Feng K., Qin.M-Z. The Symplectic Methods for the computation of Hamiltonian Equations. In: Zhu and Guo ed. Numerical Methods for Partial Differential Equations. 1987,
[26] Fox K. Numerical Integration of the Equations of Motion of Celestial Mechanics. Celest.Mech., 1984,33,127-142.
[27] Garfinkel B. On the perturbation matrics of celestial mechanics. Astronom. J. 1944,51(2) 44-48
[28] Goldstein H. Classical Mechanics. Addison-Wesley, Reading. Massachusetts, 1950. Chapter 2,7,8,9
[29] Gragg,W,1965, SINUM,2,384-403.
[30] Grobner W. Die Lie-Reihen und ihre Anwendungen. VEB Deutscher Verlar der Wissenschaften, Berlin, 1967
[31] Hanslmeier A, Dvorak R. Numerical Integration with Lie-Series. Astron.Astrophys. 1984,132:203-207
[32] Huang C. et al. Relativistic effect for near-earth satellite orbit determination. Celest. Mech. Dyn. Astr. 1990,48
[33] Huang T Y, Innanen K A. The Accuracy Check in Numerical Integration of Dynamical Systems. Astron.I., 1983,88,870-876.
[34] Huang T Y, Valtonen M J. Two multiderivative Multistep (MDMS) Integrator. Celest. Mech.,1988
[35] Hull T E. et al. Technical Report No.29, Dept. of Computer Science, University of Toronto. 1972
[36] Hyung J R. TOPEX orbit determination using GPS tracking system., University of Texas at Austin. 1992, CSR-92-3
[37] Jacchia L G. Revised static models of the thermosphere and exosphere with empirical temperature profiles. SP332,SAO, Cambridge, 1971
[38] Jacchia L G. Thermosphere temperature, density and composition. New models, SP375,SAO, Cambridge, 1977
[39] King R W, Bock Y. Documentation for the MIT GPS analysis software: GAMIT, version 9.94. Mass. Inst. Of Technol., Cambridge, 1999
[40] Kinsoshita H. Third-Order Solution of an Artificial Satellite Theory. In: Szebehely D ed al. Dynamics of Planets and Satellite and Theories of their Motion. Reidel Pub. Co.,Dordrecht-Holland, 1976,241─257
[41] Kinsoshita H, Nakai H. Numerical integration methods in dynamical astronomy. Celest. Mech. 45, 231-242
[42] Kirchner G, Koidl F, Prochazka I, et al. SPAD time walk compensation and return energy dependent ranging. In: Schluter W, Schreiber U, Dassing R, eds. Proceedings of 11th International Workshop on Laser Ranging Instrumentation. Frankfurt: Verlag Des Bundesamtes fur Kartographie und Geodasie, 1998，521~525
[43] Knocke P, Ries J C. Earth radiation pressure effects on satellites. University of Texas at Austin. 1987, CSR-8701
[44] Lambert J C. Computational Methods in Ordinery Differential Equations. John Wiley & Sons Ltd. 1973, 55
[45] Lanczos. Variational Principles of mechanics. Unit of Toronto Press. Toronto, 1949
[46] Liu Chengzhi，Zhao You etc。Performance and observation summary of Changchun Satellite Laser Ranging Station. Chinese Science Bulletin. 2002,47(13):1070─1072
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