一种无奇异点的高阶次引力位及其梯度计算方法

doi:10.11947/j.AGCS.2025.20250181

测绘学报 ›› 2025, Vol. 54 ›› Issue (9): 1572-1582.doi: 10.11947/j.AGCS.2025.20250181

一种无奇异点的高阶次引力位及其梯度计算方法

李桢¹^,²(), 贺正航¹^,³, 施闯¹^,²^,³

^1.卫星导航与移动通信融合技术工业和信息化部重点实验室，北京　100191
^2.北京航空航天大学空间与地球科学学院，北京　100191
^3.北京航空航天大学电子信息工程学院，北京　100191

收稿日期:2025-05-16 修回日期:2025-07-29 出版日期:2025-10-10 发布日期:2025-10-10
作者简介:李桢（1989—），男，博士，助理研究员，研究方向为航天器轨道动力学、精密定轨、空间碎片等。E-mail：hpulizhen@163.com
基金资助:
国家重点研发计划(2023YFB3906503)

A high-degree gravitational potential and gradient calculation method without singularities

Zhen LI¹^,²(), Zhenghang HE¹^,³, Chuang SHI¹^,²^,³

^1.Laboratory of Navigation and Communication Fusion Technology, Ministry of Industry and Information Technology, Beijing 100191, China
^2.School of Space and Earth Sciences, Beihang University, Beijing 100191, China
^3.School of Electronic Information Engineering, Beihang University, Beijing 100191, China

Received:2025-05-16 Revised:2025-07-29 Online:2025-10-10 Published:2025-10-10
About author:LI Zhen (1989—), male, PhD, assistant researcher, majors in spacecraft orbital dynamics, precise orbit determination, space debris, et al. E-mail: hpulizhen@163.com
Supported by:
The National Key Research and Development Program of China(2023YFB3906503)

摘要/Abstract

摘要：

地球引力场球谐模型在极低轨道卫星精密定轨与高精度惯性导航系统等领域具有重要的应用价值。Cunningham递推算法是一种直角坐标系下的球谐函数递推算法，能够无奇异地计算全球任意阶次的引力位、引力加速度和引力梯度，主要应用于卫星动力学定轨中的引力计算。随着重力场模型阶次的不断提升，该算法递推式因阶乘项导致数值溢出的问题日益凸显。本文引入改进的比例因子，优化递推关系，控制递推函数中阶乘项增长，缓解了数值溢出问题。本文算法在笛卡儿坐标系下，采用双精度浮点数编程运算，可在不产生数值溢出的情况下，计算至1000阶次。与现有主流球谐递推方法相比，本文算法的单次引力位、引力加速度的计算效率分别提升了16.8%、8.0%。

关键词: 缔合勒让德函数, 递推优化, 高阶引力位, 笛卡儿坐标系, 数值稳定性

Abstract:

The spherical harmonic model of the Earth's gravitational field holds significant application value in areas such as precise orbit determination for very low earth orbit (VLEO) satellites and high-precision inertial navigation systems (INS). The Cunningham recurrence algorithm is a Cartesian coordinate-based spherical harmonic recurrence method capable of singularity-free computation of gravitational potential, acceleration, and gradients to any degree globally. It is primarily applied for gravitational calculations in satellite dynamic orbit determination. However, as the degree of gravitational field models continues to increase, the recurrence relations in this algorithm suffer from numerical overflow issues due to factorial terms. This study introduces a novel scaling factor to optimize the recurrence relations, thereby controlling the growth of factorial terms within the recurrence functions and mitigating the numerical overflow problem. The improved algorithm, implemented in Cartesian coordinates using double-precision floating-point arithmetic, enables computation up to degree 1000 without generating numerical overflow. Compared to existing mainstream spherical harmonic recurrence methods, the computational efficiency for single gravitational potential and acceleration calculations is increased by 16.8% and 8.0%, respectively.

Key words: associated Legendre functions, recursion optimization, high-degree gravity potential, Cartesian coordinate system, numerical stability

中图分类号:

P223

李桢, 贺正航, 施闯. 一种无奇异点的高阶次引力位及其梯度计算方法[J]. 测绘学报, 2025, 54(9): 1572-1582.

Zhen LI, Zhenghang HE, Chuang SHI. A high-degree gravitational potential and gradient calculation method without singularities[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1572-1582.

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计算内容	格网数据U	格网数据	轨道数据U	轨道数据
改进CM算法	1.51	2.59	1.47	2.66
跨阶次递推法	1.76	2.88	1.82	2.83
CM算法	2.42	3.71	2.44	3.65
标准列递推法	2.37	3.96	2.43	4.01

一种无奇异点的高阶次引力位及其梯度计算方法

A high-degree gravitational potential and gradient calculation method without singularities

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