Acta Aeronautica et Astronautica Sinica ›› 2023, Vol. 44 ›› Issue (13): 28027-028027.doi: 10.7527/S1000-6893.2022.28027
• Reviews • Previous Articles Next Articles
Haoyang FENG1,2, Xuechuan WANG1,2(), Xiaokui YUE1,2, Changtao WANG1,2
Received:
2022-09-20
Revised:
2022-10-25
Accepted:
2022-12-14
Online:
2023-07-15
Published:
2022-12-27
Contact:
Xuechuan WANG
E-mail:xcwang@nwpu.edu.cn
Supported by:
CLC Number:
Haoyang FENG, Xuechuan WANG, Xiaokui YUE, Changtao WANG. A survey of computational methods for spacecraft orbit ropagation and Lambert problems[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(13): 28027-028027.
Table 1
Comparison between Encke method, Picard iteration method and Adomian decomposition method
方法 | 原理 | 优点 | 缺点 |
---|---|---|---|
Encke法[ | 在微小摄动力下,以Kepler轨道为基准轨道,建立实际轨道相对于基准轨道偏差的微分方程 | 允许的积分步长较大,计算效率较高 | 当轨道偏差大到一定程度,需重新选取基准轨道并重新计算,较为繁琐 |
Picard迭代法[ | 将微分方程转化为积分形式的Picard迭代序列,逐次逼近求解 | 原理和形式简单,便于和其他方法结合使用 | 每一步迭代都涉及积分,计算较为不便,收敛域小 |
Adomian分解法[ | 以一组函数的和表示非线性问题的解,在初始估计的基础上,通过迭代不断加入修正量以得到高精度解 | 适用范围广,计算简便,收敛速度快,可以得到近似解析解 | 局部收敛,Adomian多项式的计算较为复杂 |
Table 5
Comparison between shooting-type methods
打靶类方法 | 原理 | 优点 | 缺点 | 收敛域 | 收敛速度 |
---|---|---|---|---|---|
打靶法/牛顿法[ | 利用Jacobian矩阵确定搜索方向,对估计解不断修正 | 收敛快,精度高 | 初值敏感 | 小 | 二次收敛 |
拟牛顿法[ | 用拟牛顿矩阵代替Jacobian矩阵,以修正估计解 | 无需计算Jacobian矩阵,降低了计算量,超线性收敛 | 收敛速度降低,拟牛顿矩阵往往是稠密的,而Jacobian矩阵可能是稀疏的 | 介于一阶收敛和二阶收敛之间 | |
多重打靶法[ | 将待求解区间划分为一系列子区间,分别应用打靶法 | 稳定性更好,收敛域更大 | 需要为多个节点处的多个未知变量构造初始估计 | 大于牛顿法 | |
隐式打靶法[ | 根据初始估计求出隐式约束的偏差,利用变分法则和共轭函数法得到初始估计的修正量,迭代求解 | 可求解包含隐式终端约束的边值问题 | 协态变量的初始估计不易构造 | 快 |
Table 6
Comparison between quasi-linearization method, finite difference method and homotopy method
方法 | 原理 | 优点 | 缺点 | 收敛域 | 收敛速度 |
---|---|---|---|---|---|
拟线性化方法[ | 将非线性微分方程转化为迭代形式的线性微分方程,利用线性微分方程的解逐次逼近原问题的解 | 收敛速度快,精度高;初始估计容易构造 | 需多次求解线性微分方程 | 大于牛顿法 | 二次收敛 |
有限差分法[ | 利用差分原理,将微分方程转化为代数方程以近似求解 | 原理简单,便于应用;求解线性常微分方程无需迭代 | 无法直接求解非线性微分方程,需借助其他方法;计算效率和精度不高 | ||
牛顿-同伦法[ | 构造同伦映射,通过逐步调整同伦参数,将简单问题的解延拓到困难问题的解 | 降低了初始猜测的难度,增强了牛顿法的稳定性 | 初始猜测选取不当会导致求解效率低 | 全局收敛 | 较慢 |
1 | 孟占峰, 高珊, 盛瑞卿. 嫦娥五号月球轨道交会导引策略设计[J]. 航空学报, 2023, 44(5): 326584. |
MENG Z F, GAO S, SHENG R Q. Lunar orbit rendezvous phasing design for Chang’e-5 mission[J]. 航空学报, 2023, 44(5): 326584 (in Chinese). | |
2 | FREY S, COLOMBO C. Transformation of satellite breakup distribution for probabilistic orbital collision hazard analysis[J]. Journal of Guidance, Control, and Dynamics, 2020, 44(1): 88-105. |
3 | ROMANO M, COLOMBO C, PÉREZ J M S. Verification of planetary protection requirements with symplectic methods and Monte Carlo line sampling[C]∥68th International Astronautical Congress. Adelaide: IAF, 2017: 7574-7588. |
4 | 肖业伦. 航天器飞行动力学原理[M]. 北京:宇航出版社,1995:57-59. |
XIAO Y L. Principles of spacecraft flight dynamics[M]. Beijing: China Astronautic Publishing House, 1995: 57-59 (in Chinese). | |
5 | INCE E L. Ordinary differential equations[M]. New York: Dover Publications, 1956: 63-66. |
6 | 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: 161-162. |
7 | PARKER G E, SOCHACKI J S. Implementing the Picard iteration[J]. Neural, Parallel & Scientific Computations, 1996, 4(1): 97–112. |
8 | LARA M. Earth satellite dynamics by Picard iterations[DB/OL]. arXiv preprint: 2205.04310, 2022. |
9 | ADOMIAN G. Solving frontier problems of physics: The decomposition method[M]. Dordrecht: Kluwer Academic Publishers, 1994:6-17. |
10 | 刘俊英. 几个非线性方程的Adomian近似解析解[D]. 呼和浩特: 内蒙古师范大学, 2010: 3-8. |
LIU J Y. Adomian approximate analytic solutions of several nonlinear equations[D]. Hohhot: Inner Mongolia Normal University, 2010: 3-8 (in Chinese). | |
11 | GABET L. The theoretical foundation of the Adomian method[J]. Computers & Mathematics With Applications, 1994, 27(12): 41-52. |
12 | HE C Y, YANG Y, CARTER B, et al. Nonlinear uncertainty propagation of orbital mechanics subject to stochastic error in atmospheric mass density models[J], Trans. JSASS Aerospace Tech. Japan, 2017, 14(31): 1-8. |
13 | FEHLBERG E. Low-order classical Runge-Kutta fomulas with stepsize control and their application to some heat transfer problems: NASA-TR-R-315[R]. Washington, D.C.: NASA, 1969. |
14 | FILIPPI S, GRÄF J. New Runge–Kutta–Nyström 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. |
15 | FOX K. Numerical integration of the equations of motion of celestial mechanics[J]. Celestial Mechanics, 1984, 33(2): 127-142. |
16 | 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. |
17 | 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). | |
18 | 余彪, 范东明, 游为, 等. GRACE重力反演中的轨道数值积分方法分析[J]. 宇航学报, 2017, 38(3): 253-261. |
YU B, FAN D M, YOU W, et al. Analysis of orbit numerical integration methods in Earth’s gravitational field recovery by GRACE[J]. Journal of Astronautics, 2017, 38(3): 253-261 (in Chinese). | |
19 | SÜLI E, MAYERS D F. An introduction to numerical analysis[M]. Cambridge: Cambridge University Press, 2003: 329-331. |
20 | MONTENBRUCK O, GILL E, LUTZE F H. Satellite orbits: models, methods, and applications[J]. Applied Mechanics Reviews, 2002, 55(2): B27-B28. |
21 | SHAMPINE L F, REICHELT M W. The MATLAB ODE suite[J]. SIAM Journal on Scientific Computing, 1997, 18(1): 1-22. |
22 | 黄国强, 陆宇平, 南英. 飞行器轨迹优化数值算法综述[J]. 中国科学: 技术科学, 2012, 42(9): 1016-1036. |
HUANG G Q, LU Y P, NAN Y. Overview of numerical algorithms for aircraft trajectory optimization[J]. Scientia Sinica (Technologica), 2012, 42(9): 1016-1036 (in Chinese). | |
23 | 崔乃刚, 郭冬子, 李坤原, 等. 飞行器轨迹优化数值解法综述[J]. 战术导弹技术, 2020(5): 37-51, 75. |
CUI N G, GUO D Z, LI K Y, et al. A survey of numerical methods for aircraft trajectory optimization[J]. Tactical Missile Technology, 2020(5): 37-51, 75 (in Chinese). | |
24 | EL-BAGHDADY G I, EL-AZAB M S. Chebyshev-Gauss-Lobatto Pseudo-spectral method for one-dimensional advection-diffusion equation with variable coefficients[J]. Sohag Journal of Mathematics, 2016, 3(1): 7-14. |
25 | WU B L, WANG D W, POH E K, et al. Nonlinear optimization of low-thrust trajectory for satellite formation: Legendre pseudospectral approach[J]. Journal of Guidance, Control, and Dynamics, 2009, 32(4): 1371-1381. |
26 | 雍恩米, 唐国金, 陈磊. 基于Gauss伪谱方法的高超声速飞行器再入轨迹快速优化[J]. 宇航学报, 2008, 29(6): 1766-1772. |
YONG E M, TANG G J, CHEN L. Rapid trajectory optimization for hypersonic reentry vehicle via Gauss pseudospectral method[J]. Journal of Astronautics, 2008, 29(6): 1766-1772 (in Chinese). | |
27 | EIDE J D, HAGER W W, RAO A V. Modified Legendre-Gauss-Radau collocation method for optimal control problems with nonsmooth solutions[J]. Journal of Optimization Theory and Applications, 2021, 191(2): 600-633. |
28 | GARG D, PATTERSON M A, FRANCOLIN C, et al. Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method[J]. Computational Optimization and Applications, 2011, 49(2): 335-358. |
29 | BOYD J P. Chebyshev and Fourier spectral methods[M]. 2nd ed. New York: Dover Publications, 2000:1-126. |
30 | CHEN Q F, ZHANG Y D, LIAO S Y, et al. Newton–kantorovich/pseudospectral solution to perturbed astrodynamic two-point boundary-value problems[J]. Journal of Guidance, Control, and Dynamics, 2013, 36(2): 485-498. |
31 | GARG D, PATTERSON M, HAGER W W, et al. A unified framework for the numerical solution of optimal control problems using pseudospectral methods[J]. Automatica, 2010, 46(11): 1843-1851. |
32 | FAHROO F, ROSS I M. Advances in pseudospectral methods for optimal control: AIAA-2008-7309[R]. Reston: AIAA, 2008. |
33 | 曹喜滨, 张相宇, 王峰. 采用Gauss伪谱法的小推力日-火Halo轨道转移优化设计[J]. 宇航学报, 2013, 34(8): 1047-1054. |
CAO X B, ZHANG X Y, WANG F. Optimization of low-thrust transfer trajectory for the Sun-Mars halo orbit based on Gauss pseudospectral method[J]. Journal of Astronautics, 2013, 34(8): 1047-1054 (in Chinese). | |
34 | 董凯凯, 罗建军, 马卫华, 等. 非合作目标交会的双层MPC全局轨迹规划控制[J]. 航空学报, 2021, 42(11): 524903. |
DONG K K, LUO J J, MA W H, et al. Global trajectory planning and control of rendezvous of non-cooperative targets based on double-layer MPC[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(11): 524903 (in Chinese). | |
35 | 徐少兵, 李升波, 成波. 最优控制问题的Legendre伪谱法求解及其应用[J]. 控制与决策, 2014, 29(12): 2113-2120. |
XU S B, LI S B, CHENG B. Theory and application of Legendre pseudo-spectral method for solving optimal control problem[J]. Control and Decision, 2014, 29(12): 2113-2120 (in Chinese). | |
36 | FAHROO F, ROSS I M. On discrete-time optimality conditions for pseudospectral methods: AIAA-2006-6304[R]. Reston: AIAA, 2006. |
37 | HUNTINGTON G T. Advancement and analysis of a Gauss pseudospectral transcription for optimal control problems[D]. Cambridge: Massachusetts Institute of Technology, 2007: 115-146. |
38 | 唐国金, 罗亚中, 雍恩米. 航天器轨迹优化理论、方法及应用[M]. 北京: 科学出版社, 2012: 144-190. |
TANG G J, LUO Y Z, YONG E M. Theory, method and application of spacecraft trajectory optimization[M]. Beijing: Science Press, 2012: 144-190 (in Chinese). | |
39 | 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. |
40 | 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. |
41 | WANG X C. Combination of the variational iteration method and numerical algorithms for nonlinear problems[J]. Applied Mathematical Modelling, 2020, 79: 243-259. |
42 | 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. |
43 | HA S N. A nonlinear shooting method for two-point boundary value problems[J]. Computers & Mathematics With Applications, 2001, 42(10-11): 1411-1420. |
44 | KELLER H B. Numerical solution of two point boundary value problems[M]. Philadelphia: Society for Industrial and Applied Mathematics, 1976. |
45 | 伍佩钰. 非线性方程组的非精确Broyden方法[D]. 长沙: 长沙理工大学, 2017: 1-10. |
WU P Y. Inexact broyden methods for solving nonlinear equations[D]. Changsha: Changsha University of Science & Technology, 2017: 1-10 (in Chinese). | |
46 | 安邦, 郭艺丹, 赵磊. 基于Broyden秩1算法的磁浮列车再生制动分析[J]. 中国安全科学学报, 2018, 28(S1): 17-21. |
AN B, GUO Y D, ZHAO L. Analysis of regenerative braking of maglev train based on Broyden rank 1 algorithm[J]. China Safety Science Journal, 2018, 28(S1): 17-21 (in Chinese). | |
47 | 范文亮, 刘丞, 李正良. 基于HLRF法与修正对称秩1方法的改进可靠度方法[J]. 工程力学, 2022, 39(9): 1-9. |
FAN W L, LIU C, LI Z L. Improved reliability method based on HLRF and modified symmetric rank 1 method[J]. Engineering Mechanics, 2022, 39(9): 1-9 (in Chinese). | |
48 | LI D H, FUKUSHIMA M. A globally and superlinearly convergent Gauss: Newton-based BFGS method for symmetric nonlinear equations[J]. SIAM Journal on Numerical Analysis, 1999, 37(1): 152–172. |
49 | DENNIS J E, MORÉ J J. A characterization of superlinear convergence and its application to Quasi-Newton methods[J]. Mathematics of Computation, 1974, 28(126): 549-560. |
50 | 梁雨婷, 谭毅, 程楠, 等. 一种混合算法求解多体问题下行星际转移轨道[J]. 计算机与数字工程, 2010, 38(4): 15-17, 29. |
LIANG Y T, TAN Y, CHENG N, et al. An hybrid method for interplanetary transfer trajectory design in multibody problem[J]. Computer & Digital Engineering, 2010, 38(4): 15-17, 29 (in Chinese). | |
51 | 张洪波, 郑伟, 汤国建. 采用打靶法设计考虑地球扁率的机动轨道[J]. 宇航学报, 2008, 29(4): 1177-1181. |
ZHANG H B, ZHENG W, TANG G J. Maneuver trajectory design with J2 correction based on shooting procedure[J]. Journal of Astronautics, 2008, 29(4): 1177-1181 (in Chinese). | |
52 | NA T Y. Computational methods in engineering boundary value problems[M]. New York: Academic Press, 1979: 76-85. |
53 | MORRISON D, RILEY J, ZANCANARO J F. Multiple shooting method for two-point boundary value problems[J]. Communications of the ACM, 1962, 5: 613-614. |
54 | CHARTIER P, PHILIPPE B. A parallel shooting technique for solving dissipative ODE's[J]. Computing, 1993, 51(3): 209-236. |
55 | 胡朝江, 陈士橹. 改进直接多重打靶算法及其应用[J]. 飞行力学, 2004, 22(1): 14-17. |
HU C J, CHEN S L. An improved multiple shooting algorithm for direction solution of optimal control problem and its application[J]. Flight Dynamics, 2004, 22(1): 14-17 (in Chinese). | |
56 | 崔文豪. J2摄动下的卫星编队队形重构与队形保持方法研究[D]. 哈尔滨: 哈尔滨工程大学, 2019: 27-34. |
CUI W H. Research on the satellite formation reconfiguration and keeping under J2 perturbation[D]. Harbin: Harbin Engineering University, 2019: 27-34 (in Chinese). | |
57 | 张光澄. 最优控制计算方法[M]. 成都: 成都科技大学出版社, 1991: 113-117. |
ZHANG G C. Optimal control calculation method[M]. Chengdu: Chengdu University of Science and Technolo-gy Press, 1991: 113-117 (in Chinese). | |
58 | 王大轶, 李铁寿, 马兴瑞. 月球最优软着陆两点边值问题的数值解法[J]. 航天控制, 2000, 18(3): 44-49, 55. |
WANG D Y, LI T S, MA X R. Numerical solution of TPBVP in optimal lunar soft landing[J]. Aerospace Control, 2000, 18(3): 44-49, 55 (in Chinese). | |
59 | 张雷, 曾蓉, 陈聆. 非线性最优控制计算方法及其应用[M]. 北京: 科学出版社, 2015: 116-128. |
ZHANG L, ZENG R, CHEN L. Calculation method of nonlinear optimal control and its application[M]. Beijing: Science Press, 2015: 116-128 (in Chinese). | |
60 | 彭坤, 彭睿, 黄震, 等. 求解最优月球软着陆轨道的隐式打靶法[J]. 航空学报, 2019, 40(7): 322641. |
PENG K, PENG R, HUANG Z, et al. Implicit shooting method to solve optimal Lunar soft landing trajectory[J]. Acta Aeronautica et Astronautica Sinica, 2019, 40(7): 322641 (in Chinese). | |
61 | LEE E S. Quasilinearization and invariant imbedding, with applications to chemical engineering and adaptive control[M]. New York: Academic Press, 1968 |
62 | 冯浩阳, 岳晓奎, 汪雪川. 大范围收敛的摄动Lambert问题新型解法:拟线性化-局部变分迭代法[J]. 航空学报, 2021, 42(11): 524699. |
FENG H Y, YUE X K, WANG X C. A novel quasi linearization-local variational iteration method with large convergence domain for solving perturbed Lambert’s problem[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(11): 524699 (in Chinese). | |
63 | SPALDING D B. A novel finite difference formulation for differential expressions involving both first and second derivatives[J]. International Journal for Numerical Methods in Engineering, 1972, 4(4): 551-559. |
64 | REINHARDT H J. Analysis of approximation methods for differential and integral equations[M]. Berlin: Springer, 1985. |
65 | WAWRZYNIAK G G, HOWELL K C. Generating solar sail trajectories in the Earth-Moon system using augmented finite-difference methods[J]. International Journal of Aerospace Engineering, 2011, 2011: 1-13. |
66 | 闫海江, 唐硕, 泮斌峰. 亚轨道飞行器上升段轨迹快速生成方法研究[J]. 科学技术与工程, 2011, 11(10): 2266-2270, 2293. |
YAN H J, TANG S, PAN B F. Rapid ascent trajectory generation of suborbital launch vehicle[J]. Science Technology and Engineering, 2011, 11(10): 2266-2270, 2293 (in Chinese). | |
67 | LIU C S, YEIH W, KUO C L, et al. A scalar homotopy method for solving an over/under-determined system of non-linear algebraic equations[J]. Computer Modeling in Engineering & Sciences, 2009, 53(1):47-71. |
68 | PAN X, PAN B F. Practical homotopy methods for finding the best minimum-fuel transfer in the circular restricted three-body problem[J]. IEEE Access, 2020, 8: 47845-47862. |
69 | 郭铁丁. 深空探测小推力轨迹优化的间接法与伪谱法研究[D]. 北京: 清华大学, 2012: 20-21. |
GUO T D. Study of indirect and pseudospectral methods for low thrust trajectory optimization in deep space exploration[D]. Beijing: Tsinghua University, 2012: 20-21 (in Chinese). | |
70 | 潘迅, 泮斌峰. 基于同伦方法三体问题小推力推进转移轨道设计[J]. 深空探测学报, 2017, 4(3): 270-275. |
PAN X, PAN B F. Optimization of low-thrust transfers using homotopic method in the restricted three-body problem[J]. Journal of Deep Space Exploration, 2017, 4(3): 270-275 (in Chinese). | |
71 | BAI X L, TURNER J, JUNKINS J. Bang-Bang control design by combing pseudospectral method with a novel homotopy algorithm: AIAA-2009-5955[R]. Reston: AIAA, 2009. |
72 | WATSON L T. Globally convergent homotopy algorithms for nonlinear systems of equations[J]. Nonlinear Dynamics, 1990, 1(2): 143-191. |
73 | SHAFTO M. NASA technology roadmaps TA 11: Modeling, simulation, information technology, and processing: TA11[R]. Washington, D.C: NASA, 2015. |
74 | 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. |
75 | 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. |
76 | WANG X C, ELGOHARY T A, ZHANG Z, et al. An adaptive local variational iteration method for orbit propagation in astrodynamics problems[J]. The Journal of the Astronautical Sciences, 2023, 70(2): 1-34. |
77 | 洪蓓, 辛万青. 基于hp自适应伪谱法的滑翔弹道快速优化设计[C]∥中国自动化学会控制理论专业委员会B卷. 上海:上海系统科学出版社, 2011: 1956-1961. |
HONG B, XIN W Q. Rapid gliding trajectory optimization via hp-adaptive pseudospectral method[C]∥Proceedings of the 30th Chinese control Conference. Shanghai: Shanghai Systems Science Press Limited, 2011:1956-1961 (in Chinese). | |
78 | PENG H J, WANG X W, LI M W, et al. An hp symplectic pseudospectral method for nonlinear optimal control[J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 42: 623-644. |
79 | DARBY C L, HAGER W W, RAO A V. An hp-adaptive pseudospectral method for solving optimal control problems[J]. Optimal Control Applications and Methods, 2011, 32(4): 476-502. |
80 | 刘德贵, 宋晓秋. 常微分方程初值问题的并行算法[J]. 数值计算与计算机应用, 1995, 16(3): 214-223. |
LIU D G, SONG X Q. Parallel algorithms for initial value problems for ordinary differential equations[J]. Journal of Unmerical Methods and Computer Applications, 1995, 16(3): 214-223 (in Chinese). | |
81 | SOLODUSHKIN S, IUMANOVA I. Parallel numerical methods for ordinary differential equations: A survey[C]∥CEUR Workshop Proceedings. CEUR-WS, 2016, 1729: 1-10. |
82 | LUO Y Z, YANG Z. A review of uncertainty propagation in orbital mechanics[J]. Progress in Aerospace Sciences, 2017, 89: 23-39. |
83 | ARISTOFF J M, HORWOOD J T, POORE A B. Orbit and uncertainty propagation: A comparison of Gauss-Legendre-, Dormand-Prince-, and Chebyshev-Picard-based approaches[J]. Celestial Mechanics and Dynamical Astronomy, 2014, 118(1): 13-28. |
84 | BALDUCCI M, JONES B, DOOSTAN A. Orbit uncertainty propagation and sensitivity analysis with separated representations[J]. Celestial Mechanics and Dynamical Astronomy, 2017, 129(1): 105-136. |
85 | JULIER S J, UHLMANN J K. Unscented filtering and nonlinear estimation[C]∥Proceedings of the IEEE. Piscataway: IEEE Press, 2004: 401-422. |
86 | LIOU J C, HALL D T, KRISKO P H, et al. LEGEND-a three-dimensional LEO–to–GEO debris evolutionary model[J]. Advances in Space Research, 2004, 34(5): 981-986. |
87 | KUI L, BIN J, CHEN G S, et al. A real-time orbit SATellites Uncertainty propagation and visualization system using graphics computing unit and multi-threading processing: 8A2[R]. Piscataway: IEEE Press, 2015. |
88 | 赵党军, 梁步阁, 杨德贵, 等. 基于序列凸优化的高超声速滑翔式再入轨迹快速优化[J]. 宇航总体技术, 2017, 1(1): 34-40. |
ZHAO D J, LIANG B G, YANG D G, et al. Rapid planning of reentry trajectory via sequential convex optimization[J]. Astronautical Systems Engineering Technology, 2017, 1(1): 34-40 (in Chinese). | |
89 | 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. |
90 | 张帅. 变推力航天器轨道转移问题序列凸优化求解[D]. 哈尔滨: 哈尔滨工业大学, 2019: 1-6. |
ZHANG S. Optimal orbit transfer for continuous thrust spacecraft using sequential convex programming[D]. Harbin: Harbin Institute of Technology, 2019: 1-6 (in Chinese). | |
91 | CHENG X M, LI H F, ZHANG R. Autonomous trajectory planning for space vehicles with a Newton-Kantorovich/convex programming approach[J]. Nonlinear Dynamics, 2017, 89(4): 2795-2814. |
92 | 郭杰, 相岩, 王肖, 等. 基于hp伪谱同伦凸优化的火箭垂直回收在线轨迹规划方法[J]. 宇航学报, 2022, 43(5): 603-614. |
GUO J, XIANG Y, WANG X, et al. Online trajectory planning method for rocket vertical recovery based on hp pseudospectral homotopic convex optimization[J]. Journal of Astronautics, 2022, 43(5): 603-614 (in Chinese). | |
93 | 邓雁鹏, 穆荣军, 彭娜, 等. 月面着陆动力下降段最优轨迹序列凸优化方法[J]. 宇航学报, 2022, 43(8): 1029-1039. |
DENG Y P, MU R J, PENG N, et al. Sequential convex optimization method for lunar landing during powered descent phase[J]. Journal of Astronautics, 2022, 43(8): 1029-1039 (in Chinese). |
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