航天器燃耗最优轨道直接/间接混合法延拓求解

doi:10.7527/S1000-6893.2016.0183

Abstract

Abstract:

A continuation method is proposed for fixed-time minimum-fuel spacecraft trajectory optimization given initial and terminal states. The continuation method, based on the direct/indirect hybrid method, starts with a transfer solution with two impulses, followed by calculating firstly the full-propelling transfers and then the fuel-optimal transfers (including continuous and impulsive thrust ones) with continuation on thrust amplitude. All fuel-optimal solutions solved satisfy the necessary optimality conditions derived from the primer vector theory. A thrust switching presetting method based on the variational trends of switching function curves and a simple step-length adaptation rule based on the previous calculation results are proposed to enable the continuation automatic. The necessary optimality conditions for the impulsive trajectory expressed by the modified equinoctial orbit elements are given and verified, and the state transition matrix of costates of modified equinoctial orbit elements for coast arcs is derived. Three numerical examples are given to represent various transfer scenarios. Continuation can be regarded as improvement and extension of the direct/indirect hybrid trajectory optimization method. Continuation procedure and results can help to improve our understanding of the relations of the optimal control trajectories to system parameters.

Key words: modified equinoctial orbit elements, fuel-optimal trajectory, direct/indirect hybrid method, continuation, thrust switching sequence presetting, step-length adaptation

CLC Number:

V412.4⁺1

MENG Yazhe. Minimum-fuel spacecraft transfer trajectories solved by direct/indirect hybrid continuation method[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(1): 320168-320168.

References

[1] LAWDEN D F. Optimal trajectories for space navigation[M]. London:Butterworths, 1963:54-69.
[2] HANDELSMAN M, LION P M. Primer vector on fixed-time impulsive trajectories[J]. AIAA Journal, 1967, 6(1):127-132.
[3] JEZEWSKI D J, ROZENDAAL H L. An efficient method for calculating optimal free-space n-impulse trajectories[J]. AIAA Journal, 1968, 6(11):2160-2165.
[4] HULL D G. Conversion of optimal control problems into parameter optimization problems[J]. Journal of Guidance, Control, and Dynamics, 1997, 20(1):57-60.
[5] HARGRAVES C R, PARIS S W. Direct trajectory optimization using nonlinear programming and collocation[J]. Journal of Guidance, Control, and Dynamics, 1987, 10(4):338-342.
[6] FAHROO F, ROSS I M. Direct trajectory optimization by a Chebyshev pseudospectral method[J]. Journal of Guidance, Control, and Dynamics, 2002, 25(1):160-166.
[7] GAO Y, KLUEVER C A. Low-thrust interplanetary orbit transfers using hybrid trajectory optimization method with multiple shooting[C]//Proceedings of AIAA/AAS Astrodynamics Specialist Conference. Reston:AIAA, 2004:726-747.
[8] KLUEVER C A, PIERSON B L. Optimal earth-moon trajectories using nuclear electric propulsion[J]. Journal of Guidance, Control, and Dynamics, 1997, 20(2):239-245.
[9] BETTS J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(2):193-207.
[10] 高扬. 电火箭星际航行:技术进展、轨道设计与综合优化[J]. 力学学报, 2011, 43(6):991-1019. GAO Y. Interplanetary travel with electric propulsion:Technological progress, trajectory design, and comprehensive optimization[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(6):991-1019(in Chinese).
[11] 李俊峰, 蒋方华. 连续小推力航天器的深空探测轨道优化方法综述[J]. 力学与实践, 2011, 33(3):1-6. LI J F, JIANG F H. Survey of low-thrust trajectory optimization methods for deep space exploration[J]. Mechanics in Engineering, 2011, 33(3):1-6(in Chinese).
[12] CONWAY B A. A survey of methods available for the numerical optimization of continuous dynamic systems[J]. Journal of Optimization Theory and Applications, 2012, 152(2):271-306.
[13] DIXON L C W, BARTHOLOMEW-BIGGS M C. Adjoint-control transformations for solving practical optimal control problems[J]. Optimal Control Applications and Methods, 1981, 2:365-381.
[14] KLUEVER C A. Optimal earth-moon trajectories using combined chemical-electric propulsion[J]. Journal of Guidance, Control, and Dynamics, 1997, 20(2):253-258.
[15] RUSSEL R P. Primer vector theory applied to global low-thrust trade studies[J]. Journal of Guidance, Control, and Dynamics, 2007, 30(2):460-472.
[16] 何胜茂. 多段拼接小推力转移轨道优化设计方法[D]. 北京:中国科学院大学, 2012:26-31. HE S M. Design and optimization of low-thrust transfer trajectories with multiple patched orbital segments[D]. Beijing:University of Chinese Academy of Science, 2012:26-31(in Chinese).
[17] FAHROO F, ROSS I M. Costate estimation by a legendre pseudospectral method[J]. Journal of Guidance, Control, and Dynamics, 2001, 24(2):270-277.
[18] GUO T, JIANG F, LI J. Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization[J]. Acta Astronautica, 2012, 71(71):38-50.
[19] SEYWALD H, KUMAR R R. Finite difference scheme for automatic costate calculation[J]. Journal of Guidance, Control, and Dynamics, 1996, 19(1):231-239.
[20] PINES S. Constants of the motion for optimum thrust trajectories in a central force field[J]. AIAA Journal, 1964, 2(11):2010-2014.
[21] RANIERI C L, OCAMPO C A. Indirect optimization of spiral trajectories[J]. Journal of Guidance, Control, and Dynamics, 2006, 29(6):1360-1366.
[22] LEE D, BANG H. Efficient initial costates estimation for optimal spiral orbit transfer trajectories design[J]. Journal of Guidance, Control, and Dynamics, 2009, 32(6):1943-1947.
[23] THORNE J D, HALL C D. Approximate initial lagrange costates for continuous-thrust spacecraft[J]. Journal of Guidance, Control, and Dynamics, 1996, 19(2):283-288.
[24] YAN H, WU H. Initial adjoint-variable guess technique and its application in optimal orbital transfer[J]. Journal of Guidance, Control, and Dynamics, 1999, 22(3):490-492.
[25] MINTER C, FULLER-ROWELL T. A robust algorithm for solving unconstrained two-point boundary value problems (AAS 05-129)[J]. Advances in the Astronautical Sciences, 2005, 120(1):409-444.
[26] 刘滔, 何兆伟, 赵育善. 持续推力时间最优轨道机动问题的改进鲁棒算法[J]. 宇航学报, 2008, 29(4):1216-1221. LIU T, HE Z W, ZHAO Y S. Continuous-thrust orbit maneuver optimization using modified robust algorithm[J]. Journal of Astronautics, 2008, 29(4):1216-1221(in Chinese).
[27] SHEN H X, CASALINO L, LI H Y. Adjoints estimation methods for impulsive moon-to-earth trajectories in the restricted three-body problem[J]. Optimal Control Applications and Methods, 2014, 36(4):463-474.
[28] SHEN H X, CASALINO L, LUO Y Z. Global search capabilities of indirect methods for impulsive transfers[J]. The Journal of the Astronautical Sciences, 2015, 62(3):212-232.
[29] HABERKORN T, MARTINON P, GERGAUD J. Low thrust minimum-fuel orbital transfer:An homotopic approach[J]. Journal of Guidance, Control, and Dynamics, 2004, 27(6):1046-1060.
[30] FOWLER W T, O'NEILL P M. Relationship between coast arc length and switching function value during optimization[J]. Journal of Spacecraft and Rockets, 1976, 13(7):445-446.
[31] ENRIGHT P J, CONWAY B A. Discrete approximations to optimal trajectories using direct transcription and nonlinear programming[J]. Journal of Guidance, Control, and Dynamics, 1992, 15(4):994-1002.
[32] GAO Y. Near-optimal very low-thrust earth-orbit transfers and guidance schemes[J]. Journal of Guidance, Control, and Dynamics, 2007, 30(2):529-539.
[33] ZUIANI F, VASILE M. Extended analytical formulas for the perturbed keplerian motion under a constant control acceleration[J]. Celestial Mechanics and Dynamical Astronomy, 2015, 121(3):275-300.
[34] BERTRAND R, EPENOY R. New smoothing techniques for solving bang-bang optimal control problems-Numerical results and statistical interpretation[J]. Optimal Control Applications and Methods, 2002, 23(4):171-197.
[35] GERGAUD J, HABERKORN T. Homotopy method for minimum consumption orbit transfer problem[J]. ESAIM:Control, Optimisation and Calculus of Variations, 2006, 12(2):294-310.
[36] JIANG F, BAOYIN H, LI J. Practical techniques for low-thrust trajectory optimization with homotopic approach[J]. Journal of Guidance, Control, and Dynamics, 2012, 35(1):245-258.
[37] ZHANG C, TOPPUTO F, BERNELLI-ZAZZERA F, et al. Low-thrust minimum-fuel optimization in the circular restricted three-body problem[J]. Journal of Guidance, Control, and Dynamics, 2015, 38(8):1501-1510.
[38] PETUKHOV V. Optimization of interplanetary trajectories for spacecraft with ideally regulated engines using the continuation method[J]. Cosmic Research, 2008, 46(3):219-232.
[39] PETUKHOV V. Method of continuation for optimization of interplanetary low-thrust trajectories[J]. Cosmic Research, 2012, 50(3):249-261.
[40] LI J, XI X N. Fuel-optimal low-thrust reconfiguration of formation-flying satellites via homotopic approach[J]. Journal of Guidance, Control, and Dynamics, 2012, 35(6):1709-1717.
[41] MITANI S, YAMAKAWA H. Continuous-thrust transfer with control magnitude and direction constraints using smoothing techniques[J]. Journal of Guidance, Control, and Dynamics, 2012, 36(1):163-174.
[42] CHUANG C H, TROY G, JOHN H. Fuel-optimal, low- and medium-thrust orbit transfers in large numbers of burns:AIAA-1994-3650[R]. Reston:AIAA, 1994.
[43] 朱小龙, 刘迎春, 高扬. 航天器最优受控绕飞轨迹推力幅值延拓设计方法[J]. 力学学报, 2014, 46(5):756-769. ZHU X L, LIU Y C, GAO Y. Thrust-amplitude continuation design approach for solving spacecraft optimal controlled fly-around trajectory[J]. Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(5):756-769(in Chinese).
[44] GLANDORF D R. Lagrange multipliers and the state transition matrix for coasting arcs[J]. AIAA Journal, 1969, 7(2):363-365.
[45] FERNANDES S D S. Universal closed-form of lagrangian multipliers for coast-arcs of optimum space trajectories[J]. Journal of the Brazilian Society of Mechanical Sciences & Engineering, 2003, 25(4):336-340.
[46] XU Y. Enhancement in optimal multiple-burn trajectory computation by switching function analysis[J]. Journal of Spacecraft and Rockets, 2006, 44(1):264-272.
[47] JAMISON B R, COVERSTONE V. Analytical study of the primer vector and orbit transfer switching function[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(1):235-245.
[48] WALKER M J H, IRELAND B, OWENS J. A set modified equinoctial orbit elements[J]. Celestial Mechanics, 1985, 36(4):409-419.
[49] LIU H, TONGUE B H. Indirect spacecraft trajectory optimization using modified equinoctial elements[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(2):619-623.
[50] VOLK O. Johann heinrich lambert and the determination of orbits for planets and comets[J]. Celestial Mechanics, 1980, 21(2):237-250.
[51] GAUSS C F. Theoriy of the motion of the heavenly bodies moving about the sun in conic sections:A translation of carl frdr. Gauss "theoria motus":With an appendix. By ch. H. Davis[M]. Boston:Little, Brown and Comp., 1857:161-233.
[52] BATTIN R H. An introduction to the mathematics and methods of astrodynamics[M]. Reston:AIAA, 1999:141-236.
[53] VALLADO D A. Fundamentals of astrodynamics and applications[M]. New York:Springer Science & Business Media, 2007:419-495.
[54] 佘明生. 轨道转移[M]//杨嘉墀, 范剑峰. 航天器轨道动力学与控制. 北京:中国宇航出版社, 2009:333-335. SHE M S. Orbital tranfer[M]//YANG J X, FAN J F. Spacecraft orbital dynamics and control. Beijing:China Astronautic Publishing House, 2009:333-335(in Chinese).
[55] IZZO D. Revisiting lambert's problem[J]. Celestial Mechanics and Dynamical Astronomy, 2015, 121(1):1-15.

Minimum-fuel spacecraft transfer trajectories solved by direct/indirect hybrid continuation method

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