弹道导弹扰动引力快速逼近的算法研究
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摘要
本研究系总参测绘局课题——远程导弹惯性制导系统的重力矢量实时测定系
统——的一部分。全文围绕着远程导弹弹道点扰动引力的快速计算展开，主要讨
论了四个方面的问题：首先是对现有扰动引力计算方法进行分析和比较，据此认
为利用点质量模型和低价位系数模型组合法计算扰动引力是理论严密、计算精密
的方法，并将它作为后面进一步研究的基础。其次，通过对常用数值逼近方法的
考察，选取三次等距B样条函数对用上述方法计算出的某设计弹道的扰动引力进
行了逼近，对结果进行分析后认为其适合于弹道扰动引力的快速计算。文章接着
提出利用有限元内插方法对上述设计弹道的扰动引力进行逼近，重点集中在空间
单元的剖分和单元形状函数的选取上，分析结果显示此方法也适合进行弹道扰动
引力的快速计算。最后，对上面所得的算法进行了集成，编制了软件以满足实用
中便捷和连续弹道计算的要求。
Revolving around the fast computation of the disturbing gravity of ballistic
missile trajectory this dissertation mainly focuses on the following problems: firstly,
analysis and comparison of the existing computing methods of the disturbing gravity
are made, according to which the method of point mass model is considered strict and
exact and thus is selected as the foundation of subsequent research; secondly, after a
full review of the numerical interpolation methods in conunon use, because of its
excellent property' the cubic isometrical B-spline is chosen to interpolate the
disturbing gravity along some pre-designed trajectory and considered fit fOr the fast
computation of disturbing gravity along a IIilssile trajectory after the analysis of
result; thirdly, a finite element interpo1ation method is put forward in this paper, with
emphasis laid on the partition of spatial elements and the selection of shaPe function
of element, and consequent result shows that this method is also fit fOr the fast
comPutation of disturbing gravity of IIilssile trajectory; finally, the above methods are
integrated into software so as to be convenient and successive and automatic fOr
further use.
统——的一部分。全文围绕着远程导弹弹道点扰动引力的快速计算展开，主要讨
论了四个方面的问题：首先是对现有扰动引力计算方法进行分析和比较，据此认
为利用点质量模型和低价位系数模型组合法计算扰动引力是理论严密、计算精密
的方法，并将它作为后面进一步研究的基础。其次，通过对常用数值逼近方法的
考察，选取三次等距B样条函数对用上述方法计算出的某设计弹道的扰动引力进
行了逼近，对结果进行分析后认为其适合于弹道扰动引力的快速计算。文章接着
提出利用有限元内插方法对上述设计弹道的扰动引力进行逼近，重点集中在空间
单元的剖分和单元形状函数的选取上，分析结果显示此方法也适合进行弹道扰动
引力的快速计算。最后，对上面所得的算法进行了集成，编制了软件以满足实用
中便捷和连续弹道计算的要求。
Revolving around the fast computation of the disturbing gravity of ballistic
missile trajectory this dissertation mainly focuses on the following problems: firstly,
analysis and comparison of the existing computing methods of the disturbing gravity
are made, according to which the method of point mass model is considered strict and
exact and thus is selected as the foundation of subsequent research; secondly, after a
full review of the numerical interpolation methods in conunon use, because of its
excellent property' the cubic isometrical B-spline is chosen to interpolate the
disturbing gravity along some pre-designed trajectory and considered fit fOr the fast
computation of disturbing gravity along a IIilssile trajectory after the analysis of
result; thirdly, a finite element interpo1ation method is put forward in this paper, with
emphasis laid on the partition of spatial elements and the selection of shaPe function
of element, and consequent result shows that this method is also fit fOr the fast
comPutation of disturbing gravity of IIilssile trajectory; finally, the above methods are
integrated into software so as to be convenient and successive and automatic fOr
further use.
引文
[1] 管泽霖、宁津生，地球形状及外部重力场，测绘出版社，1981．
[2] 方俊，重力测量与地球形状学，科学出版社，1975．
[3] 陆仲连、吴晓平等，弹道导弹重力学，八一出版社，1992．
[4] 陆仲连，地球重力场理论与方法，北京：解放军出版社，1996．
[5] 朱长青，计算方法及其在测绘中的应用，北京：测绘出版社，1988．
[6] 边少锋，大地测量边值问题数值解法与地球重力场逼近，博士论文，1992．
[7] 饶寿期，有限元法和边界元法基础，北京航空航天大学出版社，1990．
[8] 茹家欣，导弹射击准备给地球形状和地球引力场研究提出的任务，军事测绘专辑，Vol．1，1979．
[9] 茹家欣，地面重力异常资料用于远程弹道导弹弹道计算的几个问题，军事测绘专辑，Vol．1，1979．
[10] 张国瑞译，民兵导弹的重力模型，国外导弹技术，No．6，1978．
[11] 吴晓平，局部重力场的点质量模型，测绘学报，Vol．13，No．4，1984．
[12] 吴晓平，在推求地球外部扰动重力场中数据的应用，解放军测绘学院学报，No．4，1992．
[13] 吴晓平，扰动引力场点质量表达法研究，研究报告，1994．
[14] 黄漠涛，潜地战略导弹弹道扰动引力计算与研究，硕士论文，1994．
[15] 吴晓平，远程导弹惯性制导系统的重力矢量实时测定系统，技术报告，1999，3．
[16] 胡健伟、汤怀民，微分方程数值方法，科学出版社，1999．
[17] 张传定，卫星重力测量—基础、模型化方法与数据处理算法，博士论文，2000．
[18] Bjerhammar, A. A new Theory of Geodetic Gravity, Division of Geodesy, The Royal Institute of Technology, No.243, 1964.
[19] Bjerhammar, A., A Review of Discrete Methods in Physical Geodesy, in Approximation Methods in Physical Geodesy, H.Moritz and H.Sunkel, Herbert Wichmann Verlag, Karisruhe, 1978.
[20] Cruz, J.Y. and P. Laskowski, Upward Continuation of Surface Gravity Anomalies,Report No.360,Dept. of Geodetic Science and Surveying, the Ohio State University,December 1984.
[21] Cruz, J.Y. Disturbance Vector in Space from Surface Gravity Anomalies Using Complementary Models, Report No.366,Dept. of Geodetic Science and Surveying,The Ohio State University, August, 1985.
[22] Forsberg, R.A Study of Terrain Reductions, Density Anomalies, and Geophysical Inversion Methods in Gravity Field Modelling,Report No.355, Dept of Geodetic
Science and Surveying , The Ohio State University, April 1984．
[23] Heiskanen, W.A. and H.Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco, 1967．
[24] Jekeli, C. The Downward Continuation to the Earth's Surface of Truncated Spherical and Ellipsoidal Harmonic Series of the Gravity and Height Anomalies, Report No.323, Dept. of Geodetic Science and Surveying , The Ohio State University , December 1981．
[25] Katsambalos, K.E. Simulation Studies on the Computation of the Gravity Vector in Space from Surface Data Considering the Topography of the Earth, Report No.323, Dept. of Geodetic Science and Surveying, The Ohio State University ,December 1981．
[26] Moritz, H. Advanced Physical Geodesy, Herbert Wichmann Verlag, Karisruhe, 1980．
[27] Needham, P.E. The Formation and Evaluation of Detailed Geopotential Models Based on point masses, Report No.l49,Dept. of Geodetic Science and Surveying , The Ohio State University, December 1970．
[28] Paul, M.K. Recurrence Relations for the Truncation Error Coefficients for the Extended Stokes Function, Bulletin Geodesique, Vol, 57 No.2,1983．
[29] Rapp, R.H., Ellipsoidal Corrections for Geoid Undulation Computations, Report No.308, Dept. of Geodetic Science and Surveying ,The Ohio State University, March 1981．
[30] Sjoberg, L., On the Discrete Boundary Value Problem of Physical Geodesy With Harmonic Reduction to an Internal Sphere, Division of Geodesy, The Royal Institute of Technology, Stockholm, 1975．
[31] Sunkel, H. , Point Mass Models and the Anomalous Gravitational Field of the Earth, Report No.328, Dept. of Geodetic Science and Surveying, The Ohio State University, December 1981．
[32] Sunkel, H. , The Generation of a Mass point Model from Surface Gravity Data, Report No.353, Dept. of Geodetic Science and Surveying, The Ohio State University, December 1983．
[33] Tscherining, C.C. and R.H. Rapp, Closed Covariance Expressions for Gravity Anomalies , Geoid Undulations, and Deflection of the Vertical. implied by Anomaly Degree Variance Models, Report No.208, Dept. of Geodetic Science and Surveying, The Ohio State University, May 1974．
[2] 方俊，重力测量与地球形状学，科学出版社，1975．
[3] 陆仲连、吴晓平等，弹道导弹重力学，八一出版社，1992．
[4] 陆仲连，地球重力场理论与方法，北京：解放军出版社，1996．
[5] 朱长青，计算方法及其在测绘中的应用，北京：测绘出版社，1988．
[6] 边少锋，大地测量边值问题数值解法与地球重力场逼近，博士论文，1992．
[7] 饶寿期，有限元法和边界元法基础，北京航空航天大学出版社，1990．
[8] 茹家欣，导弹射击准备给地球形状和地球引力场研究提出的任务，军事测绘专辑，Vol．1，1979．
[9] 茹家欣，地面重力异常资料用于远程弹道导弹弹道计算的几个问题，军事测绘专辑，Vol．1，1979．
[10] 张国瑞译，民兵导弹的重力模型，国外导弹技术，No．6，1978．
[11] 吴晓平，局部重力场的点质量模型，测绘学报，Vol．13，No．4，1984．
[12] 吴晓平，在推求地球外部扰动重力场中数据的应用，解放军测绘学院学报，No．4，1992．
[13] 吴晓平，扰动引力场点质量表达法研究，研究报告，1994．
[14] 黄漠涛，潜地战略导弹弹道扰动引力计算与研究，硕士论文，1994．
[15] 吴晓平，远程导弹惯性制导系统的重力矢量实时测定系统，技术报告，1999，3．
[16] 胡健伟、汤怀民，微分方程数值方法，科学出版社，1999．
[17] 张传定，卫星重力测量—基础、模型化方法与数据处理算法，博士论文，2000．
[18] Bjerhammar, A. A new Theory of Geodetic Gravity, Division of Geodesy, The Royal Institute of Technology, No.243, 1964.
[19] Bjerhammar, A., A Review of Discrete Methods in Physical Geodesy, in Approximation Methods in Physical Geodesy, H.Moritz and H.Sunkel, Herbert Wichmann Verlag, Karisruhe, 1978.
[20] Cruz, J.Y. and P. Laskowski, Upward Continuation of Surface Gravity Anomalies,Report No.360,Dept. of Geodetic Science and Surveying, the Ohio State University,December 1984.
[21] Cruz, J.Y. Disturbance Vector in Space from Surface Gravity Anomalies Using Complementary Models, Report No.366,Dept. of Geodetic Science and Surveying,The Ohio State University, August, 1985.
[22] Forsberg, R.A Study of Terrain Reductions, Density Anomalies, and Geophysical Inversion Methods in Gravity Field Modelling,Report No.355, Dept of Geodetic
Science and Surveying , The Ohio State University, April 1984．
[23] Heiskanen, W.A. and H.Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco, 1967．
[24] Jekeli, C. The Downward Continuation to the Earth's Surface of Truncated Spherical and Ellipsoidal Harmonic Series of the Gravity and Height Anomalies, Report No.323, Dept. of Geodetic Science and Surveying , The Ohio State University , December 1981．
[25] Katsambalos, K.E. Simulation Studies on the Computation of the Gravity Vector in Space from Surface Data Considering the Topography of the Earth, Report No.323, Dept. of Geodetic Science and Surveying, The Ohio State University ,December 1981．
[26] Moritz, H. Advanced Physical Geodesy, Herbert Wichmann Verlag, Karisruhe, 1980．
[27] Needham, P.E. The Formation and Evaluation of Detailed Geopotential Models Based on point masses, Report No.l49,Dept. of Geodetic Science and Surveying , The Ohio State University, December 1970．
[28] Paul, M.K. Recurrence Relations for the Truncation Error Coefficients for the Extended Stokes Function, Bulletin Geodesique, Vol, 57 No.2,1983．
[29] Rapp, R.H., Ellipsoidal Corrections for Geoid Undulation Computations, Report No.308, Dept. of Geodetic Science and Surveying ,The Ohio State University, March 1981．
[30] Sjoberg, L., On the Discrete Boundary Value Problem of Physical Geodesy With Harmonic Reduction to an Internal Sphere, Division of Geodesy, The Royal Institute of Technology, Stockholm, 1975．
[31] Sunkel, H. , Point Mass Models and the Anomalous Gravitational Field of the Earth, Report No.328, Dept. of Geodetic Science and Surveying, The Ohio State University, December 1981．
[32] Sunkel, H. , The Generation of a Mass point Model from Surface Gravity Data, Report No.353, Dept. of Geodetic Science and Surveying, The Ohio State University, December 1983．
[33] Tscherining, C.C. and R.H. Rapp, Closed Covariance Expressions for Gravity Anomalies , Geoid Undulations, and Deflection of the Vertical. implied by Anomaly Degree Variance Models, Report No.208, Dept. of Geodetic Science and Surveying, The Ohio State University, May 1974．
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