ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Review of integral correction methods for orbit calculation
Received date: 2025-07-08
Revised date: 2025-09-01
Accepted date: 2025-09-26
Online published: 2025-10-09
Supported by
National Natural Science Fundation of China(52425212);National Key Research and Development Program of China(2021YFA0717100);Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(CX2025033)
To address the high-efficiency orbit calculation problem of spacecraft, this paper reviews research on integral correction methods. First, a variety of integral correction methods based on different principles are systematically introduced within a unified notation framework, and a classification comparison is conducted. Then, focusing on the advantages of these methods in large-step and parallel computation, the research progress in parameter optimization and parallel acceleration is summarized. Subsequently, the unique superiority of integral correction methods is elaborated by typical applications such as spacecraft orbit design, high-precision orbit propagation, and spacecraft guidance. Finally, combining the methods’ characteristics and the orbit calculation requirements, the development trends and directions worth studying are analyzed and proposed.
Changtao WANG , Honghua DAI , Yichao DONG , Lin SHI , Wenchuan YANG , Xiaokui YUE . Review of integral correction methods for orbit calculation[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2026 , 47(5) : 332528 -332528 . DOI: 10.7527/S1000-6893.2025.32528
| [1] | LUO Y Z, YANG Z. A review of uncertainty propagation in orbital mechanics[J]. Progress in Aerospace Sciences, 2017, 89: 23-39. |
| [2] | 金紫涵, 温昶煊, 乔栋. 卫星解体碎片云对低轨星座的碰撞影响分析[J]. 航空学报, 2024, 45(): 211-220. |
| JIN Z H, WEN C X, QIAO D. Impact analysis of satellite debris cloud on low-orbit constellation[J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(S1): 211-220 (in Chinese). | |
| [3] | 卢哲俊, 胡卫东. 基于随机有限集的空间碎片群运动状态估计[J]. 航空学报, 2017, 38(11): 321200. |
| LU Z J, HU W D. State estimation of space debris group based on random finite set[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(11): 321200 (in Chinese). | |
| [4] | 李罡, 解放, 吴一凡, 等. 深空导航技术发展展望[J]. 测绘学报, 2025, 54(3): 397-409. |
| LI G, XIE F, WU Y F, et al. Prospects for the development of deep space navigation technologies[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(3): 397-409 (in Chinese). | |
| [5] | 李培佳, 黄勇, 樊敏, 等. 嫦娥五号探测器交会对接段定轨精度研究[J]. 中国科学: 物理学 力学 天文学, 2021, 51(11): 66-77. |
| LI P J, HUANG Y, FAN M, et al. Orbit determination for Chang’E-5 mission in rendezvous and docking[J]. Scientia Sinica (Physica, Mechanica & Astronomica), 2021, 51(11): 66-77 (in Chinese). | |
| [6] | 王巍, 邢朝洋, 冯文帅. 自主导航技术发展现状与趋势[J]. 航空学报, 2021, 42(11): 525049. |
| WANG W, XING C Y, FENG W S. State of the art and perspectives of autonomous navigation technology[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(11): 525049 (in Chinese). | |
| [7] | 何昊, 王鹏. 高速变形飞行器制导控制一体化设计方法[J]. 航空学报, 2024, 45(): 299-312. |
| HE H, WANG P. Integrated guidance and control method for high-speed morphing wing aircraft[J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(S1):299-312 (in Chinese). | |
| [8] | 魏昊, 蔡光斌, 凡永华, 等. 高超声速飞行器再入滑翔段在线制导[J]. 北京航空航天大学学报, 2025, 51(1): 183-192. |
| WEI H, CAI G B, FAN Y H, et al. Online guidance for hypersonic vehicles in glide-reentry segment[J]. Journal of Beijing University of Aeronautics and Astronautics, 2025, 51(1): 183-192 (in Chinese). | |
| [9] | 禹春梅. 飞行器在线轨迹规划与制导控制方法研究[D]. 长沙: 国防科技大学, 2018: 1-42. |
| YU C M. Research on online trajectory planning and guidance control for aircraft[D]. Changsha: National University of Defense Technology, 2018: 1-42 (in Chinese). | |
| [10] | 冯浩阳, 汪雪川, 岳晓奎, 等. 航天器轨道递推及Lambert问题计算方法综述[J]. 航空学报, 2023, 44(13): 028027. |
| FENG H Y, WANG X C, YUE X K, et al. A survey of computational methods for spacecraft orbit propagation and Lambert problems[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(13): 028027 (in Chinese). | |
| [11] | HAIRER E N. Solving ordinary differential equations Ⅰ : Nonstiff problems[M]. Cham: Springer, 2009: 3-528. |
| [12] | FILIPPI S, GR?F J. New Runge-Kutta-Nystr?m formula-pairs of order 8(7), 9(8), 10(9) and 11(10) for differential equations of the form y″=f(x, y)[J]. Journal of Computational and Applied Mathematics, 1986, 14(3): 361-370. |
| [13] | FEHLBERG E, MARSHALL G C. Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems[R]. Washington D.C.: NASA, 1969. |
| [14] | BERRY M M, HEALY L M. Implementation of Gauss-Jackson integration for orbit propagation[J]. The Journal of the Astronautical Sciences, 2004, 52(3): 331-357. |
| [15] | 罗志才, 周浩, 钟波, 等. Gauss-Jackson积分器算法分析与验证[J]. 武汉大学学报(信息科学版), 2013, 38(11): 1364-1368. |
| LUO Z C, ZHOU H, ZHONG B, et al. Analysis and validation of Gauss-Jackson integral algorithm[J]. Geomatics and Information Science of Wuhan University, 2013, 38(11): 1364-1368 (in Chinese). | |
| [16] | CLENSHAW C W, NORTON H J. The solution of nonlinear ordinary differential equations in Chebyshev series[J]. The Computer Journal, 1963, 6(1): 88-92. |
| [17] | FUKUSHIMA T. Vector integration of dynamical motions by the Picard-Chebyshev method[J]. The Astronomical Journal, 1997, 113: 2325. |
| [18] | BAI X L. Modified Chebyshev-Picard iteration methods for solution of initial value and boundary value problems [D]. College Station: Texas A&M University, 2010. |
| [19] | BAI X L, JUNKINS J L. Modified Chebyshev-Picard iteration methods for orbit propagation[J]. The Journal of the Astronautical Sciences, 2011, 58(4): 583-613. |
| [20] | BAI X L, JUNKINS J L. Modified Chebyshev-Picard iteration methods for solution of boundary value problems[J]. The Journal of the Astronautical Sciences, 2011, 58(4): 615-642. |
| [21] | WANG X C, YUE X K, DAI H H, et al. Feedback-accelerated Picard iteration for orbit propagation and lambert’s problem[J]. Journal of Guidance, Control, and Dynamics, 2017, 40(10): 2442-2451. |
| [22] | WOOLLANDS R, JUNKINS J L. Nonlinear differential equation solvers via adaptive Picard-Chebyshev iteration: applications in astrodynamics[J]. Journal of Guidance, Control, and Dynamics, 2019, 42(5): 1007-1022. |
| [23] | WANG Y K, NI G Y, LIU Y C. Multistep Newton-Picard method for nonlinear differential equations[J]. Journal of Guidance, Control, and Dynamics, 2020, 43(11): 2148-2155. |
| [24] | ATALLAH A M, WOOLLANDS R M, ELGOHARY T A, et al. Accuracy and efficiency comparison of six numerical integrators for propagating perturbed orbits[J]. The Journal of the Astronautical Sciences, 2020, 67(2): 511-538. |
| [25] | YAN Z P, DAI H H, WANG Q S, et al. Harmonic balance methods: A review and recent developments[J]. Computer Modeling in Engineering & Sciences, 2023, 137(2): 1419-1459. |
| [26] | WANG C T, DAI H H, YANG W C, et al. High-efficiency unscented Kalman filter for multi-target tra-jectory estimation[J]. Aerospace Science and Technol-ogy, 2025, 159: 109962. |
| [27] | MASAT A, COLOMBO C, BOUTONNET A. Surfing chaotic perturbations in interplanetary multi-flyby trajectories: Augmented Picard-Chebyshev integration for parallel and GPU computing architectures[C]∥AIAA Scitech 2022 Forum. Reston: AIAA, 2022. |
| [28] | WANG X C, HE W, FENG H Y, et al. Fast and accurate predictor-corrector methods using feedback-accelerated Picard iteration for strongly nonlinear problems[J]. Computer Modeling in Engineering and Sciences, 2024, 139(2): 1263-1294. |
| [29] | FENG H Y, YUE X K, WANG X C. A class of linearization-based collocation methods for initial value and boundary value engineering problems[J]. Computer Physics Communications, 2023, 283: 108601. |
| [30] | 陈文斌, 程晋, 吴新明, 等. 微分方程数值解[M]. 上海: 复旦大学出版社, 2014: 55-127. |
| CHEN W B, CHENG J, WU X M, et al. Numerical solution of differential equations[M]. Shanghai: Fudan Press, 2014: 55-127 (in Chinese). | |
| [31] | MA Y Y, PAN B F. Parallel-structured Newton-type guidance by using modified Chebyshev-Picard iteration[J]. Journal of Spacecraft and Rockets, 2020, 58(3): 729-740. |
| [32] | READ J L, YOUNES A B, MACOMBER B, et al. State transition matrix for perturbed orbital motion using modified Chebyshev Picard iteration[J]. The Journal of the Astronautical Sciences, 2015, 62(2): 148-167. |
| [33] | JUNKINS J L, BANI YOUNES A, WOOLLANDS R M, et al. Picard iteration, Chebyshev polynomials and Chebyshev-Picard methods: Application in astrodynamics[J]. The Journal of the Astronautical Sciences, 2013, 60(3): 623-653. |
| [34] | WOOLLANDS R M, BANI YOUNES A, JUNKINS J L. New solutions for the perturbed lambert problem using regularization and Picard iteration[J]. Journal of Guidance, Control, and Dynamics, 2015, 38(9): 1548-1562. |
| [35] | SWENSON T, WOOLLANDS R, JUNKINS J, et al. Application of modified Chebyshev Picard iteration to differential correction for improved robustness and computation time[J]. The Journal of the Astronautical Sciences, 2017, 64(3): 267-284. |
| [36] | WANG X C, XU Q Y, ATLURI S N. Combination of the variational iteration method and numerical algorithms for nonlinear problems[J]. Applied Mathematical Modelling, 2020, 79: 243-259. |
| [37] | FENG H Y, YUE X K, WANG X C. A quasi-linear local variational iteration method for orbit transfer problems[J]. Aerospace Science and Technology, 2021, 119: 107222. |
| [38] | MA Y Y, PAN B F, HAO C C, et al. Improved sequential convex programming using modified Chebyshev-Picard iteration for ascent trajectory optimization[J]. Aerospace Science and Technology, 2022, 120: 107234. |
| [39] | SONG Y S, PAN B F, FAN Q Y, et al. A computationally efficient sequential convex programming using Chebyshev collocation method[J]. Aerospace Science and Technology, 2023, 141: 108584. |
| [40] | ZHOU Q, WANG Y K, LIU Y C. Chebyshev-Picard iteration methods for solving delay differential equations[J]. Mathematics and Computers in Simulation, 2024, 217: 1-20. |
| [41] | ATALLAH A M, WOOLLANDS R M, BANI YOUNES A, et al. Tuning orthogonal polynomial degree and segment interval length to achieve prescribed precision approximation of irregular functions[C]∥ 2018 Space Flight Mechanics Meeting. Reston: AIAA, 2018. |
| [42] | WANG X C, ELGOHARY T A, ATLURI S. An adaptive local variational iteration method for orbit propagation and strongly nonlinear dynamical systems[C]∥ AIAA Scitech 2020 Forum. Reston: AIAA, 2020. |
| [43] | 张哲, 代洪华, 冯浩阳, 等. 初值约束与两点边值约束轨道动力学方程的快速数值计算方法[J]. 力学学报, 2022, 54(2): 503-516. |
| ZHANG Z, DAI H H, FENG H Y, et al. Efficient numerical method for orbit dynamic functions with initial value and two-point boundary-value constraints[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 503-516 (in Chinese). | |
| [44] | DAI H H, ZHANG Z, WANG X C, et al. Fast and accurate adaptive collocation iteration method for orbit dynamic problems[J]. Chinese Journal of Aeronautics, 2023, 36(9): 231-242. |
| [45] | WANG Y K, NI G Y, LIU Y C. Revised Picard-Chebyshev methods for perturbed orbit propagations[J]. Journal of Guidance, Control, and Dynamics, 2022, 46(1): 161-170. |
| [46] | DORMAND J R, PRINCE P J. A family of embedded Runge-Kutta formulae[J]. Journal of Computational and Applied Mathematics, 1980, 6(1): 19-26. |
| [47] | MASAT A, COLOMBO C, BOUTONNET A. GPU-based high-precision orbital propagation of large sets of initial conditions through Picard-Chebyshev augmentation[J]. Acta Astronautica, 2023, 204: 239-252. |
| [48] | WANG Y K. Parallel numerical Picard iteration methods[J]. Journal of Scientific Computing, 2023, 95(1): 27. |
| [49] | JUNKINS J L, YOUNES A B, BAI X L. Orthogonal polynomial approximation in higher dimensions: Applications in astrodynamics[J]. Advances in the Astronautical Sciences, 2013, 147: 531-594. |
| [50] | YOUNES A B. Orthogonal polynomial approximation in higher dimensions: Applications in astrodynamics [D]. College Station: Texas A&M University, 2013. |
| [51] | MONTENBRUCK O, GILL E. Satellite orbits: Models, methods and applications[M]. Heidelberg: Springer, 2000: 3-347. |
| [52] | ATALLAH A, YOUNES A B. Parallel Chebyshev Picard method[C]∥AIAA Scitech 2020 Forum. Reston: AIAA, 2020. |
| [53] | ATALLAH A M, YOUNES A B. Parallel evaluation of Chebyshev approximations: Applications in astrodynamics[J]. The Journal of the Astronautical Sciences, 2022, 69(3): 692-717. |
| [54] | CHANDRA R, MENON R, DAGUM L, et al. Parallel Programming in OpenMP[M]. San Francisco: Academic Press, 2001: 6-230. |
| [55] | 王昌涛, 代洪华, 张哲, 等. 并行加速的局部变分迭代法及其轨道计算应用[J]. 力学学报, 2023, 55(4): 991-1003. |
| WANG C T, DAI H H, ZHANG Z, et al. Parallel accelerated local variational iteration method and its application in orbit computation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(4): 991-1003 (in Chinese). | |
| [56] | Cook S. CUDA并行程序设计GPU编程指南[M]. 北京: 机械工业出版社, 2014: 10-105. |
| Cook S. CUDA programming[M]. Beijing: China Machine Press, 2014: 10-105 (in Chinese). | |
| [57] | 朱新忠. 星载嵌入式计算机技术与应用[M]. 上海: 上海科学技术出版社, 2023: 1-39. |
| ZHU X Z. Spacecraft embedded computer technologies and applications[M]. Shanghai: Shanghai Scientific & Technical Publishers, 2023: 1-39 (in Chinese). | |
| [58] | 虞志刚, 冯旭, 陆洲, 等. 宇航级处理器发展现状与趋势[J]. 天地一体化信息网络, 2023, 4(1): 50-58. |
| YU Z G, FENG X, LU Z, et al. Development status and trends of space processor[J]. Space-Integrated-Ground Information Networks, 2023, 4(1): 50-58 (in Chinese). | |
| [59] | 陆士强, 梁赫光, 刘东洋. 国产化星载计算机技术现状和发展思考[J]. 移动信息, 2018(6): 126-129. |
| LU S Q, LIANG H G, LIU D Y. Thoughts on the status quo and development of localized onboard computer technology[J]. Mobile Information, 2018(6): 126-129 (in Chinese). | |
| [60] | 李兴伟, 白博, 周军. 基于FPGA的立方星可重构星载处理系统研究[J]. 计算机测量与控制, 2018, 26(8): 172-176. |
| LI X W, BAI B, ZHOU J. Research on reconfigurable board processing system of cube sat based on FPGA[J]. Computer Measurement & Control, 2018, 26(8): 172-176 (in Chinese). | |
| [61] | 徐国栋, 赵丹, 向文豪, 等. 可重构的卫星/运载复用电子系统设计[J]. 航空学报, 2009, 30(7): 1298-1304. |
| XU G D, ZHAO D, XIANG W H, et al. Design of reconfigurable and reusable electronic systems for satellite/launch vehicles[J]. Acta Aeronautica et Astronautica Sinica, 2009, 30(7): 1298-1304 (in Chinese). | |
| [62] | CHOI Y K. Performance debugging frameworks for FPGA high-level synthesis[D]. Los Angeles: University of California, Los Angeles, 2019. |
| [63] | ZHANG J Y, YU H C, DAI H H. Overview of Earth-Moon transfer trajectory modeling and design[J]. Computer Modeling in Engineering and Sciences, 2022, 135(1): 5-43. |
| [64] | 雷汉伦. 平动点、不变流形及低能轨道[D]. 南京: 南京大学, 2015. |
| LEI H L. Equilibrium point, invariant manifold and low-energy trajectory[D]. Nanjing: Nanjing University, 2015 (in Chinese). | |
| [65] | 刘林, 侯锡云. 深空探测器轨道力学[M]. 北京: 电子工业出版社, 2012: 15-117. |
| LIU L, HOU X Y. Deep space exploration orbital mechanics[M]. Beijing: Publishing House of Electronics Industry, 2012: 15-117 (in Chinese). | |
| [66] | 泮斌峰. 航天器制导理论与方法[M]. 北京: 科学出版社, 2023: 1-12. |
| PAN B F. Spacecraft guidance theory and methods [M]. Beijing: Science Press, 2023: 1-12 (in Chinese). | |
| [67] | DI MAURO G, LAWN M, BEVILACQUA R. Survey on guidance navigation and control requirements for spacecraft formation-flying missions[J]. Journal of Guidance, Control, and Dynamics, 2017, 41(3): 581-602. |
/
| 〈 |
|
〉 |