深空探测小推力轨迹优化的间接法与伪谱法研究
|
摘要
小推力推进是深空探测中的关键技术，这种高效的推进方式得益于它的高比冲。它具有推力幅值小、长时间连续作用的特征，这导致动力学方程不可积，从而使得小推力轨迹设计与脉冲情形完全不同。目前航天工程迫切需要寻找高效可靠的小推力轨迹设计方法。本论文从这一实际需求出发，系统地研究了小推力轨迹优化设计的间接法和伪谱法。
论文研究了含有多次引力辅助的小推力轨迹优化设计，以最优控制理论为基础，系统地给出了这类小推力轨迹的间接解法。不同于文献中的形状曲线逼近和参数优化，本文从最优控制的基本原理出发，将引力辅助作为内点约束，通过详细推导得到了该轨迹优化问题的完整最优必要条件。文中将含有多次引力辅助小推力轨迹优化问题转化为多点边值问题并构造出相应的打靶方程，给出了各个打靶变量的初值设定策略，最后用打靶法得到了最优解。与参数优化方法相比，本文方法在计算效率和优化效果上都反映出良好的性能。
论文系统地研究了伪谱法在小推力轨迹优化设计中的应用。文中构造了小推力轨迹优化设计的伪谱法格式，验证了该方法与间接法的等效性，并建立了相应的乘子等价映射。文中通过伪谱离散将连续轨迹优化问题转化为非线性规划问题并结合非线性规划求解器求得了能量最优轨迹。论文进一步详细分析了伪谱法求解燃料最优bang-bang控制的困难，进而构造了伪谱-同伦混合方法。该方法首先利用同伦算法将燃料最优转化为能量最优问题，在伪谱法求得能量最优解后再使用同伦算法使能量最优解逐步趋近燃料最优解。伪谱-同伦混合方法综合了伪谱法和同伦方法的优势，在求解bang-bang控制方面比普通伪谱法具有更好的精度和效率，同时也化解了同伦方法的初始化困难。
论文构造了含时变内点约束轨迹优化问题的分块伪谱法，建立并证明了相应的乘子等价映射。文中通过分块伪谱离散将轨迹优化问题离散为非线性规划问题并推导了相应的一阶必要条件。文中详细证明了(内点)一阶必要条件与间接法中的(内点)最优必要条件的等价性，从而推广了原有的乘子等价映射。该映射给出了分块伪谱法与多点边值问题之间的联系，它既能估计协态，也能估计与内点约束相关的乘子及内点时刻。因此，这既是对伪谱法理论的补充和完善，也是将伪谱-同伦混合方法拓展至含有时变内点约束轨迹优化问题的理论基础。
Low thrust propulsion is one of the key techniques for deep space exploration, andsuch propulsion systems are really efficient due to their high specific impulse. However,the thrust is very ‘low’ and applied on the spacecraft continuously&lastingly, whichrenders the dynamic equations nonintegrable and thus makes design of low thrusttrajectories totally different from that of the impulsive trajectory. It is now an urgentneed to construct effective and reliable ways to design low thrust trajectory in aerospaceengineering. With such practical demands in mind, this thesis studies indirect andpseudospectral methods systematically for their applications to low thrust trajectoryoptimization.
Based upon optimal control theory, a systematic way for designing low thrusttrajectory via multiple gravity assists (GAs) is presented. Different from the commonshaped-based approach and parameter optimization, gravity assists are taken as interiorpoint constraints here and the first order optimal necessary conditions are derived usingoptimal control principle. The trajectory optimization with multiple GAs is formulatedas a multiple point boundary value problem (MPBVP) and the associated shootingequations are then constructed. The ways to generate reasonable initial guesses for theshooting variables are detailed and the MPBVP is finally solved by the simple shootingmethod. In comparison with parameter optimization, the present method’s efficiencyand optimality is satisfactory.
The thesis systematically studies the pseudospectral method for its applications tolow thrust trajectory optimization. A pseudospectral scheme for low thrust trajectoryoptimization is first formulated. Its equivalence with the indirect method is verified andthe associated covector mapping is established. The continuous trajectory optimizationis transformed into a discrete nonlinear programming problem (NLP) and theenergy-optimal solutions are obtained with a NLP solver. Furthermore, the thesisanalyzes various difficulties confronted by the pseudospectral method when solvingfuel-optimal bang-bang controls and a pseudospectral-homotopic hybrid method is thenproposed to circumvent these difficulties. With the homotopic algorithm, the hybridscheme first transforms the fuel-optimal problem to an energy-optimal one which can be solved efficiently by the pseudospectral method. And the energy-optimal results arethen continued to the desirable fuel-optimal ones by the homotopic algorithm. Thishybrid method combines beneficial features of the pseudospectral method andhomotopic approach, yielding bang-bang controls more accurately and efficiently thanthe former, and also resolving the latter’s initiation difficulty via the former.
A multi-phase pseudospectral method is constructed for low thrust trajectoryoptimization involving time varying interior point constrains (IPCs), and the associatedcovector mapping is formulated and proved. Through introducing multi-phasepseudospectral grids, the continuous trajectory with IPCs is discretized into a NLP,whose first-order necessary conditions (i.e., Karush–Kuhn–Tucker or KKT conditions)are then derived. The (interior) KKT conditions are shown equivalent to the (interior)optimal necessary conditions in the indirect method, thus extending the previouscovector mapping to the cases involving time varying IPCs. This extended covectormapping connects the multi-phase pseudospectral method and the MPBVP. It canestimate the costates, as well as the Lagrange multipliers associated with the IPCs andthe interior time. Thus, it is not only a theoretical improvement of the pseudospectralmethod, but also the theoretical basis for extending the pseudospectral-homotopichybrid method to trajectory optimization involving time varying IPCs.
论文研究了含有多次引力辅助的小推力轨迹优化设计，以最优控制理论为基础，系统地给出了这类小推力轨迹的间接解法。不同于文献中的形状曲线逼近和参数优化，本文从最优控制的基本原理出发，将引力辅助作为内点约束，通过详细推导得到了该轨迹优化问题的完整最优必要条件。文中将含有多次引力辅助小推力轨迹优化问题转化为多点边值问题并构造出相应的打靶方程，给出了各个打靶变量的初值设定策略，最后用打靶法得到了最优解。与参数优化方法相比，本文方法在计算效率和优化效果上都反映出良好的性能。
论文系统地研究了伪谱法在小推力轨迹优化设计中的应用。文中构造了小推力轨迹优化设计的伪谱法格式，验证了该方法与间接法的等效性，并建立了相应的乘子等价映射。文中通过伪谱离散将连续轨迹优化问题转化为非线性规划问题并结合非线性规划求解器求得了能量最优轨迹。论文进一步详细分析了伪谱法求解燃料最优bang-bang控制的困难，进而构造了伪谱-同伦混合方法。该方法首先利用同伦算法将燃料最优转化为能量最优问题，在伪谱法求得能量最优解后再使用同伦算法使能量最优解逐步趋近燃料最优解。伪谱-同伦混合方法综合了伪谱法和同伦方法的优势，在求解bang-bang控制方面比普通伪谱法具有更好的精度和效率，同时也化解了同伦方法的初始化困难。
论文构造了含时变内点约束轨迹优化问题的分块伪谱法，建立并证明了相应的乘子等价映射。文中通过分块伪谱离散将轨迹优化问题离散为非线性规划问题并推导了相应的一阶必要条件。文中详细证明了(内点)一阶必要条件与间接法中的(内点)最优必要条件的等价性，从而推广了原有的乘子等价映射。该映射给出了分块伪谱法与多点边值问题之间的联系，它既能估计协态，也能估计与内点约束相关的乘子及内点时刻。因此，这既是对伪谱法理论的补充和完善，也是将伪谱-同伦混合方法拓展至含有时变内点约束轨迹优化问题的理论基础。
Low thrust propulsion is one of the key techniques for deep space exploration, andsuch propulsion systems are really efficient due to their high specific impulse. However,the thrust is very ‘low’ and applied on the spacecraft continuously&lastingly, whichrenders the dynamic equations nonintegrable and thus makes design of low thrusttrajectories totally different from that of the impulsive trajectory. It is now an urgentneed to construct effective and reliable ways to design low thrust trajectory in aerospaceengineering. With such practical demands in mind, this thesis studies indirect andpseudospectral methods systematically for their applications to low thrust trajectoryoptimization.
Based upon optimal control theory, a systematic way for designing low thrusttrajectory via multiple gravity assists (GAs) is presented. Different from the commonshaped-based approach and parameter optimization, gravity assists are taken as interiorpoint constraints here and the first order optimal necessary conditions are derived usingoptimal control principle. The trajectory optimization with multiple GAs is formulatedas a multiple point boundary value problem (MPBVP) and the associated shootingequations are then constructed. The ways to generate reasonable initial guesses for theshooting variables are detailed and the MPBVP is finally solved by the simple shootingmethod. In comparison with parameter optimization, the present method’s efficiencyand optimality is satisfactory.
The thesis systematically studies the pseudospectral method for its applications tolow thrust trajectory optimization. A pseudospectral scheme for low thrust trajectoryoptimization is first formulated. Its equivalence with the indirect method is verified andthe associated covector mapping is established. The continuous trajectory optimizationis transformed into a discrete nonlinear programming problem (NLP) and theenergy-optimal solutions are obtained with a NLP solver. Furthermore, the thesisanalyzes various difficulties confronted by the pseudospectral method when solvingfuel-optimal bang-bang controls and a pseudospectral-homotopic hybrid method is thenproposed to circumvent these difficulties. With the homotopic algorithm, the hybridscheme first transforms the fuel-optimal problem to an energy-optimal one which can be solved efficiently by the pseudospectral method. And the energy-optimal results arethen continued to the desirable fuel-optimal ones by the homotopic algorithm. Thishybrid method combines beneficial features of the pseudospectral method andhomotopic approach, yielding bang-bang controls more accurately and efficiently thanthe former, and also resolving the latter’s initiation difficulty via the former.
A multi-phase pseudospectral method is constructed for low thrust trajectoryoptimization involving time varying interior point constrains (IPCs), and the associatedcovector mapping is formulated and proved. Through introducing multi-phasepseudospectral grids, the continuous trajectory with IPCs is discretized into a NLP,whose first-order necessary conditions (i.e., Karush–Kuhn–Tucker or KKT conditions)are then derived. The (interior) KKT conditions are shown equivalent to the (interior)optimal necessary conditions in the indirect method, thus extending the previouscovector mapping to the cases involving time varying IPCs. This extended covectormapping connects the multi-phase pseudospectral method and the MPBVP. It canestimate the costates, as well as the Lagrange multipliers associated with the IPCs andthe interior time. Thus, it is not only a theoretical improvement of the pseudospectralmethod, but also the theoretical basis for extending the pseudospectral-homotopichybrid method to trajectory optimization involving time varying IPCs.
引文
[1]李俊峰,宝音贺西.深空探测中的动力学与控制.力学与实践,2007,29(4):1-8
[2] Rayman M D, Varghese P, Lehman D H, et al. Results from the Deep Space1technology validation mission. Acta Astronautica,2000,9:475-487
[3] Kuninaka H, Nishivama K, Funakai I, et al. Asteroid rendezvous of HAYABUSAexplorer using microwave discharge ion engines.29th International ElectricPropulsion Conference, Princeton University,2005
[4] Kugelberg J, Bodin P, Persson S, et al. Accommodating electric propulsion onSMART-1. Acta Astronautica,2004,55(2):121–130
[5] Yan C H, McConaghy T T, Chen K. J, et al. Preliminary design of nuclear electricpropulsion missions to the outer planets. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004
[6] Williams S N, Coverstone-Carroll V. Mars missions using solar electric propulsion,Journal of Spacecraft and Rockets,2000,37(5):71–77
[7] Bate R R, Mueller D D, White J. E. Fundamentals of astodynamics. New York: DoverPublications,1971
[8] Howard Curtis. Orbital mechanics for engineering students.2nd ed. Butterworth-Heinemann,2009
[9] Richard H Battin. An introduction to the mathematics and method of astrodynmaics.AIAA,1999
[10]刘敦,赵钧.空间飞行器动力学.哈尔滨:哈尔滨工业大学出版社,2003
[11]章仁为.卫星轨道姿态动力学与控制.北京:北京航空航天大学出版社,1998
[12]刘林,胡松杰,王歆.航天动力学引论.南京:南京大学出版社,2006
[13] Betts J T. Survey of numerical methods for trajectory optimization. Journal ofGuidance, Control, and Dynamics,1998,21(2):193–207
[14]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.力学与实践,2011,33(3):1-6
[15] Hull D G. Conversion of optimal control problems into parameter optimizationproblems. Journal of Guidance, Control, and Dynamics,1997,20(1):57-60
[16] Bruce Conway (editor). Spacecraft trajectory optimization. New York: CambridgeUniversity Press,2010
[17] Betts J T. Practical Methods for optimal control using nonlinear programming.Society for Industrial and Applied Mathematics, Philadelphia, P. A,2001
[18] Bauer T P. Low-thrust perturbation guidance [D]. California Institute of Technology,1982
[19]任远,崔平远,栾恩杰.基于标称轨道的小推力轨道设计方法.吉林大学学报,2006,36(6):999-1002
[20] Arthur E B, Yu-Chi H. Applied optimal control theory. Revised ed. Taylor&Francis,1975
[21] Pontryagin L S. Mathematical theory in optimal control process. CRC Press,1987
[22] Mcintryre J E. The Pontryagin Maximum Principle. National Aeronautics and SpaceAdministration,1968
[23] Lawden D F. Optimal trajectories for space navigation. London: Butterworths,1963
[24] Kirk D E. Optimal control theory: an Introduction. Dover Publications,1970.
[25] Ranieri C. L, Ocampo C A. Indirect optimization of spiral trajectories. Journal ofGuidance, Control, and Dynamics,2006,29(6):1360-1366
[26] Ranieri C. L, Ocampo C A. Indirect optimization of three-Dimensional finite-burninginterplanetary transfers including spiral dynamics. Journal of Guidance, Control, andDynamics,2009,32(2):444–454
[27] Mantia M L, Casalino L. Indirect Optimization of Low-Thrust Capture Trajectories.Journal of Guidance, Control, and Dynamics,2006,29(4):1011-1014
[28] Jiang F H, Li J F, Baoyin H X. Practical techniques for low-thrust trajectoryoptimization with homotopic approach. Journal of Guidance, Control, and Dynamics,2012,35(1):245-258
[29] Martell C A, Lawton J A. Adjoint variable solutions via an auxiliary optimizationproblem. Journal of Guidance, Control, and Dynamics,1995,18(6):1267-1272
[30] Hans S, Renjith R K. Finite difference scheme for automatic costate calculation.Journal of Guidance, Control, and Dynamics,1996,19(1):231-239
[31] Dixon L C W, Biggs M C. The advantages of adjoint-control transformations whendetermining optimal trajectories by Pontryagin’s Maximum Principle. AeronauticalJournal,1976,76(735):169–174
[32] Yan H, Wu H X. Initial adjoint variable guess technique and its application in optimalorbital transfer. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Boston, MA,1998
[33]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.航天控制,2000,3:44-55
[34] Eugene L A, Kurt G. Introduction to numerical continuation Methods. Society forIndustrial Mathematics,1987
[35] Bertrand R, Epenoy R. New smoothing techniques for solving bang–bang optimalcontrol problems-numerical results and statistical interpretation. Optimal ControlApplications and Methods,2002,4:171-197
[36] Haberkorn T, Martinon P, Gergaud J. Low-thrust minimum-fuel orbital transfer: ahomotopic approach. Journal of Guidance, Control, and Dynamics,2004,27(6):1046-1060
[37] Gergaud J, Haberkorn T. Homotopy method for minimum consumption orbit transferproblem. Control, Optimization and Calculus of Variations,2006,12(2):294-310
[38] Bai X L, Turner J D, Junkins J L. Optimal thrust design of a mission to apophis basedon a homotopy method. AAS/AIAA Spaceflight Mechanics Meeting, Savannah,Georgia,2009
[39] Bai X L, Turner J D, Junkins J L. Bang-bang control design by combingpseudospectral method with a novel homotopy algorithm. AIAA Guidance,Navigation, and Control Conference, Chicago, Illinois,2009
[40] Mischa K. Continuous low-thrust trajectory optimization: techniques and applications[D]. Virginia Polytechnic Institute and State University,2005
[41] Bock H G, Plitt K J. A Multiple shooting algorithm for direct solution of optimalcontrol problems. Proceedings of the Ninth Triennial World Congress of IFAC,Pergamon, New York,1998
[42] Rainer C, Peter R. Optimal control of rigid-link manipulators by indirect methods.GAMM-Mitt,2008,31(1):27-58
[43] Kluever C A, Pierson B L. Optimal low-thrust earthmoon transfers with a switchingfunction structure. Journal of the Astronautical Sciences,1994,42(3):269-283
[44] Kluever C A, Pierson B L. Optimal low-thrust, three-dimensional Earth-Moontrajectories. Journal of Guidance, Control, and Dynamics,1995,18(4):830-837
[45] Kluever C A. Optimal low-thrust, Earth-Moon trajectories [D]. Iowa State Universityof Science and Technology,1993
[46] Gao Y. Advances in low-thrust trajectory optimization and flight mechanics [D].University of Missouri,2003
[47] Gao Y, Kluever C A. Low-thrust interplanetary orbit transfers using hybrid trajectoryoptimization method with multiple shooting. AIAA/AAS Astrodynamics SpecialistConference, Providence, Rhode Island,2004
[48]任远,崔平远,栾恩杰.利用混合法进行地球-火星小推力轨道设计.哈尔滨工业大学学报,2007,39(3):359-362
[49] Hargraves C, Paris S. Direct trajectory optimization using nonlinear programming andcollocation. Journal of Guidance, Control, and Dynamics,1987,4:338–342
[50] Enright P J, Conway B A. Discrete approximations to optimal trajectories using directtranscription and nonlinear programming. Journal of Guidance, Control, andDynamics,1992,15(4):994–1002
[51] Betts J T, Sven O E. Optimal low thrust trajectories to the Moon. SIAM JournalApplied Dynamical Systems,2003,2(2):144-170
[52] Betts J T. Optimal interplanetary orbit transfers by direct transcription. Journal of theAstronautical Sciences,1994,42:247–268
[53] Betts J T. Using sparse nonlinear programming to compute low thrust orbit transfers,Journal of the Astronautical Sciences,1993,41:349–371
[54] Betts J T. Very low-thrust trajectory optimization using a direct SQP method. Journalof Computational and Applied Mathematics,2000,120(1-2):27-40
[55] Herman A L, Conway B A. Optimal, low-thrust, Earth-Moon orbit transfer. Journal ofGuidance Control And Dynamics,1998,21(1):141-147
[56] Herman A L, Conway B A. Direct optimization using collocation based on high-orderGauss-Lobatto quadrature rules. Journal of Guidance, Control, and Dynamics,1996,19(3):592–599
[57] Coverstone-Carroll, Victoria Williams, Steven N. Optimal low thrust trajectoriesusing differential inclusion concepts.1994,42(4):379-393
[58] Hull, D. G, Speyer J L. Optimal reentry and plane-change trajectories. Journal of theAstronautical Sciences,1982,(2):117-130
[59]尚海滨,崔平远,栾恩杰.地球-火星的燃料最省小推力转移轨道的设计与优化.宇航学报,2006,27(6):1168-1173
[60] Bock H, Plitt K. A multiple shooting algorithm for direct solution of optimal controlproblems. In9th IFAC World Congress,242–247,1984
[61] Leineweber D, Bauer I, Bock H, et al. An efficient multiple shooting based reducedSQP strategy for large-scale dynamic process optimization, Part I: theoretical aspects.Computers and Chemical Engineering,2003,27:157-166
[62] Grimm W, Markl A. Adjoint estimation from a direct multiple shooting method.Journal of Optimization Theory and Applications,1997,92(2),263–283
[63] Gill P. E, Murray W, Saunders M A. User’s guide for SNOPT version7: software forlarge-scale nonlinear programming.2008, http://ccom.ucsd.edu/~peg/
[64] Gill P E, Murray W, Saunders M A. SNOPT: an SQP algorithm for large-scaleconstrained optimization. SIAM Journal on Optimization,2002,12(4):979–1006
[65] Li Y H, Baoyin H X, Gong S P.1st ACT global trajectory optimization competition:Tsinghua University results. Acta Astronautica,2007,61(9):735-741
[66] Zhu K J, Li J F, Baoyin H. X. Multi-swingby optimization of mission to Saturn usingglobal optimization algorithms. Acta Mechanica Sinica,2009,25(6):839-845
[67]王劼,李俊峰,崔乃刚,等.登月飞行器软着陆轨道的遗传算法优化.清华大学学报:自然科学版,2003,43(8):1056-1059
[68] Pontani M, Conway B A. Particle swarm optimization applied to space trajectories.Journal of Guidance, Control, and Dynamics,2010,33(5):1429-1441
[69] Wall B, Conway B A. Near-optimal low-thrust Earth-Mars trajectories via a GeneticAlgorithm. Journal of Guidance, Control, and Dynamics,2005,28(5):1027–1031
[70] Petropoulos A E, Lee S. Optimization of low-thrust orbit transfers using the Q-Lawfor the initial guess. Advances in the Astronautical Sciences,2006,123:2229-2248
[71] Lee S, von Allmen P, Fink W, et al. Design and optimization of low-thrust orbittransfers using the Q-law and evolutionary algorithms. IEEE Aerospace ConferenceProceedings, Big Sky, Montana,2005
[72] Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral methods in fluid dynamics.New York: Springer–Verlag,1988
[73] Trefethen L N. Spectral methods in MATLAB. Society for Industrial and AppliedMathematics, Philadelphia,2000
[74] Jan S H, Sigal G, David G. Spectral methods for time-dependent problems. New York:Cambridge University Press,2007
[75] Gong Q, Kang W, Bedrossian N, et al. Pseudospectral optimal control for military andindustrial applications.46th IEEE Conference on Decision and Control, New Orleans,LA,2007
[76] Kang W, Bedrossian N. Pseudospectral optimal control theory makes debut flight,saves NASA$1M in under three hours. SIAM News,40,2007
[77] Elnagar J, Kazemi M A, Razzaghi M. The pseudospectral Legendre method fordiscretizing optimal control problems. IEEE Transactions on Automatic Control,1995,40(10):1793–1796
[78] Fahroo F, Ross I M. Costate estimation by a Legendre pseudospectral method. Journalof Guidance, Control, and Dynamics,2001,24(2):270–277
[79] Ross I M, Fahroo F. Legendre pseudospectral approximations of optimal controlproblems. Lecture Notes in Control and Information Sciences, New York:Springer–Verlag,2003
[80] Ross I M. A historical introduction to the covector mapping principle. Advances inthe Astronautical Sciences, Univelt, San Diego, CA,2006
[81] Garg D, Patterson M. A, Darby C L, et al. Direct trajectory optimization and costateestimation of finite-horizon and infinite-horizon optimal control problems using aRadau pseudospectral method. Computational Optimization and Applications,2009,49(2):335-358
[82] Kameswaran S, Biegler L T. Convergence rates for the direct transcription of optimalcontrol problems using collocation at Radau points. Computational Optimization andApplications,2008,41(1):81–126
[83] Benson D A, Huntington G T, Thorvaldsen T P, et al. Direct trajectory optimizationand costate estimation via an orthogonal collocation method. Journal of Guidance,Control, and Dynamics,2006,29(6):1435–1440
[84] Benson D A. Gauss pseudospectral transcription for optimal control [D].Massachusetts Institute of Technology,2004
[85] Huntington G T. Advancement and analysis of Gauss pseudospectral transcription foroptimal control problems [D]. Massachusetts Institute of Technology,2007
[86] Vlassenbroeck J, Van D R. A Chebyshev technique for solving nonlinear optimalcontrol problems. IEEE Transactions on Automatic Control,1988,33(4):333–340.
[87] Fahroo F, Ross I M. Direct trajectory optimization by a Chebyshev pseudospectralmethod. Journal of Guidance, Control, and Dynamics,2002,25(1):160–166
[88] Gong Q, Ross I M, Fahroo F. Costate computation by a Chebyshev pseudospectralmethod. Journal of Guidance, Control, and Dynamics,2010,33(2):623-628
[89] Welfert B D. Generation of pseudospectral differentiation matrices I. SIAM J.Numer.Anal.1997,34(4):1640–1657
[90] Weideman J A C, Reddy S C. A MATLAB differentiation matrix suite.2001
[91] Rao A V, Benson D A, Darby, et al. Algorithm902: GPOPS, A Matlab Software forSolving Multiple-Phase Optimal Control Problems Using the Gauss PseudospectralMethod. ACM Transactions on Mathematical Software,2010,37(2):22:1-22:39
[92] Ross I M, Gong Q, Sekhavat P. Low-thrust, high-accuracy trajectory optimization.Journal of Guidance Control and Dynamics,2007,30(4):921-933
[93] Ross I M, Fahroo F. Pseudospectral knotting methods for solving optimal controlproblems. Journal of Guidance, Control, and Dynamics,2004,27b(3):397–405
[94] Gong, Q, Ross I M. Autonomous pseudospectral knotting methods for space missionoptimization,16th AAS/AIAA Space Flight Mechanics Meeting, AmericanAstronautical Society Paper06-151,2006
[95] Gong Q, Fahroo F, Ross I M. Spectral algorithm for pseudospectral methods inoptimal control. Journal of Guidance, Control, and Dynamics,2008,31(3):460-471
[96] Gong Q, Kang W, Ross I M. A pseudospectral method for the optimal control ofconstrained feedback linearizable systems. IEEE Transactions on Automatic Control,2006,51(7):1115–1129
[97] Kang W, Gong Q, Ross I M, et al. On the convergence of nonlinear optimal controlusing pseudospectral methods for feedback linearizable control systems. InternationalJournal of Robust and Nonlinear Control,2007,17(14):1251–1277
[98] Gong Q, Ross I M, Kang W, et al. Connections between the covector mappingtheorem and convergence of pseudospectral methods for optimal control.Computational Optimization and Applications,2007,41(3):307-335
[99] Fahroo F, Ross I M. Convergence of the costates does not imply convergence of thecontrol. Journal of Guidance, Control, and Dynamics,2008,31(5):1492-1497
[100] Fahroo F, Ross I M. Pseudospectral methods for infinite-horizon nonlinear optimalcontrol problems. Journal of Guidance, Control, and Dynamics,2008,31(4):927–936
[101] Garg D A, Hager W W, Rao A V. Pseudospectral methods for solving infinite-horizonoptimal control problems. Automatica,2011,47(4):829-837
[102] Ross I M, Fahroo F. Discrete verification of necessary conditions for switchednonlinear optimal control systems. Proceedings of American Control Conference,2:1610-1615,2004
[103] Ross I M, D'Souza C N. Hybrid optimal control framework for mission planning.Journal of Guidance, Control, and Dynamics,2005,28(4):686-697
[104] Huntington G T, Rao A V. Comparison of global and local collocation methods foroptimal control. Journal of Guidance, Control, and Dynamics,2008,31(2):432-436
[105] Bedrossian N S, Bhatt S, Kang W, et al. Zero propellant maneuver guidance. IEEEControl Systems Magazine,2009,29:53-73
[106] Ross I M, Fahroo F. User's Manual for DIDO: A MATLAB package for dynamicoptimization. Naval Postgraduate School Technical Report,2002
[107] Williams P, Blanksby C, Trivailo P. Receding horizon control of tether system usingquasilinearization and Chebyshev pseudospectral approximations. AAS/AIAAAstrodynamics Specialist Conference, Big Sky, MT,2003
[108] Williams P. Jacobi Pseudospectral method for solving optimal control problems.Journal of Guidance, Control, and Dynamics,2004,27(2):293–297
[109] Williams P. Application of pseudospectral methods for receding horizon control.Journal of Guidance, Control, and Dynamics,2004,27(2):310–314
[110]文浩.绳系卫星释放和回收的动力学控制[博士学位论文].南京:南京航空航天大学,2008
[111] Yan H, Fahroo F, Ross I M. Real-Time Computation of neighboring optimal controllaws. AIAA-Guidance, Navigation, and Control Conference AIAA-2002-4657,Monterey, California,2002
[112] Yan H, Alfriend K T. Three-axis magnetic attitude control using pseudospectralcontrol law in eccentric orbits. AAS Spaceflight Mechanics Meeting, Tampa, FL,2006
[113]罗建军,王明光,袁建平.基于伪光谱方法的月球软着陆轨道快速优化.宇航学报,2007,28(5):1119-1122
[114] Cui P Y, Qiao D, Cui H T, et al． Target selection and transfer trajectories design forexploring asteroid mission. Science China Technological Sciences,2010,53(4):1150-1158,
[115] McConaghy T T, Debban T J, Petropoulos A E, et al． Design and optimization oflow-thrust trajectories with gravity assists. Journal of Spacecraft and Rockets,2003,3:380-387
[116] Casalino L, Colasurdo G, Pastrone D. Optimal low-thrust escape trajectories usinggravity assist. Journal of Guidance, Control, and Dynamics,1999,5:637-642
[117] Sims J A. Delta-V gravity-assist trajectory design: theory and practic [D]. PurdueUniversity,1997
[118] Petropoulos A E, Longuski J M. Shape-based analytic representations of low-thrusttrajectories for gravity-assist applications, The AAS/AIAA AstrodynamicsConference, Girdwood, USA,1999
[119] Petropoulos A E, Longuski J M. Shape-based algorithm for automated design oflow-thrust，gravity-assist trajectories. Journal of Spacecraft and Rockets,2004,41(5):87-796
[120] Pascale P D, Vasile M. Preliminary design of low-thrust multiple gravity-assisttrajectories. Journal of Spacecraft and Rockets,2006,5:1065-1076
[121] Wall B J, Conway B A． Shape-based approach to low-thrust rendezvous trajectorydesign. Journal of Guidance, Control, and Dynamics,2009,32(1):95-101
[122]尚海滨,崔平远,徐瑞,等.结合行星借力飞行技术的小推力转移轨道初始设计.宇航学报,2011,32(1):29-38
[123] Howell K. C, Kakoi M. Transfers between the earth-moon and sun-earth systemsusing manifolds and transit orbits. Acta Astronautica,2006,59(1-5):367-380
[124]乔栋.深空探测转移轨道设计方法研究及在小天体探测中的应用[博士学位论文].哈尔滨:哈尔滨工业大学,2007
[125]任远.星际探测中的小推力转移轨道设计与优化方法研究[博士学位论文].哈尔滨:哈尔滨工业大学,2007
[126]徐明,徐世杰.小推力航天器的地月低能转移轨道.航空学报,2008,29(4):781-787
[127] McConaghy T T. GALLOP Version4.5. User's Guide. School of Aeronautics andAstronautics, Purdue University. West Lafayette, Indiana.,2005
[128]关治，陆金甫.数值分析基础.北京：高等教育出版社，2010
[129] Luenberger D. G, Yinyu Y. Linear and nonlinear programming. Springer,2008
[130] Zorich V A, Cooke R. Mathematical analysis. Springer,2004
[131]吴麒.自动控制原理.北京：清华大学出版社，1992
[2] Rayman M D, Varghese P, Lehman D H, et al. Results from the Deep Space1technology validation mission. Acta Astronautica,2000,9:475-487
[3] Kuninaka H, Nishivama K, Funakai I, et al. Asteroid rendezvous of HAYABUSAexplorer using microwave discharge ion engines.29th International ElectricPropulsion Conference, Princeton University,2005
[4] Kugelberg J, Bodin P, Persson S, et al. Accommodating electric propulsion onSMART-1. Acta Astronautica,2004,55(2):121–130
[5] Yan C H, McConaghy T T, Chen K. J, et al. Preliminary design of nuclear electricpropulsion missions to the outer planets. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004
[6] Williams S N, Coverstone-Carroll V. Mars missions using solar electric propulsion,Journal of Spacecraft and Rockets,2000,37(5):71–77
[7] Bate R R, Mueller D D, White J. E. Fundamentals of astodynamics. New York: DoverPublications,1971
[8] Howard Curtis. Orbital mechanics for engineering students.2nd ed. Butterworth-Heinemann,2009
[9] Richard H Battin. An introduction to the mathematics and method of astrodynmaics.AIAA,1999
[10]刘敦,赵钧.空间飞行器动力学.哈尔滨:哈尔滨工业大学出版社,2003
[11]章仁为.卫星轨道姿态动力学与控制.北京:北京航空航天大学出版社,1998
[12]刘林,胡松杰,王歆.航天动力学引论.南京:南京大学出版社,2006
[13] Betts J T. Survey of numerical methods for trajectory optimization. Journal ofGuidance, Control, and Dynamics,1998,21(2):193–207
[14]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.力学与实践,2011,33(3):1-6
[15] Hull D G. Conversion of optimal control problems into parameter optimizationproblems. Journal of Guidance, Control, and Dynamics,1997,20(1):57-60
[16] Bruce Conway (editor). Spacecraft trajectory optimization. New York: CambridgeUniversity Press,2010
[17] Betts J T. Practical Methods for optimal control using nonlinear programming.Society for Industrial and Applied Mathematics, Philadelphia, P. A,2001
[18] Bauer T P. Low-thrust perturbation guidance [D]. California Institute of Technology,1982
[19]任远,崔平远,栾恩杰.基于标称轨道的小推力轨道设计方法.吉林大学学报,2006,36(6):999-1002
[20] Arthur E B, Yu-Chi H. Applied optimal control theory. Revised ed. Taylor&Francis,1975
[21] Pontryagin L S. Mathematical theory in optimal control process. CRC Press,1987
[22] Mcintryre J E. The Pontryagin Maximum Principle. National Aeronautics and SpaceAdministration,1968
[23] Lawden D F. Optimal trajectories for space navigation. London: Butterworths,1963
[24] Kirk D E. Optimal control theory: an Introduction. Dover Publications,1970.
[25] Ranieri C. L, Ocampo C A. Indirect optimization of spiral trajectories. Journal ofGuidance, Control, and Dynamics,2006,29(6):1360-1366
[26] Ranieri C. L, Ocampo C A. Indirect optimization of three-Dimensional finite-burninginterplanetary transfers including spiral dynamics. Journal of Guidance, Control, andDynamics,2009,32(2):444–454
[27] Mantia M L, Casalino L. Indirect Optimization of Low-Thrust Capture Trajectories.Journal of Guidance, Control, and Dynamics,2006,29(4):1011-1014
[28] Jiang F H, Li J F, Baoyin H X. Practical techniques for low-thrust trajectoryoptimization with homotopic approach. Journal of Guidance, Control, and Dynamics,2012,35(1):245-258
[29] Martell C A, Lawton J A. Adjoint variable solutions via an auxiliary optimizationproblem. Journal of Guidance, Control, and Dynamics,1995,18(6):1267-1272
[30] Hans S, Renjith R K. Finite difference scheme for automatic costate calculation.Journal of Guidance, Control, and Dynamics,1996,19(1):231-239
[31] Dixon L C W, Biggs M C. The advantages of adjoint-control transformations whendetermining optimal trajectories by Pontryagin’s Maximum Principle. AeronauticalJournal,1976,76(735):169–174
[32] Yan H, Wu H X. Initial adjoint variable guess technique and its application in optimalorbital transfer. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Boston, MA,1998
[33]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.航天控制,2000,3:44-55
[34] Eugene L A, Kurt G. Introduction to numerical continuation Methods. Society forIndustrial Mathematics,1987
[35] Bertrand R, Epenoy R. New smoothing techniques for solving bang–bang optimalcontrol problems-numerical results and statistical interpretation. Optimal ControlApplications and Methods,2002,4:171-197
[36] Haberkorn T, Martinon P, Gergaud J. Low-thrust minimum-fuel orbital transfer: ahomotopic approach. Journal of Guidance, Control, and Dynamics,2004,27(6):1046-1060
[37] Gergaud J, Haberkorn T. Homotopy method for minimum consumption orbit transferproblem. Control, Optimization and Calculus of Variations,2006,12(2):294-310
[38] Bai X L, Turner J D, Junkins J L. Optimal thrust design of a mission to apophis basedon a homotopy method. AAS/AIAA Spaceflight Mechanics Meeting, Savannah,Georgia,2009
[39] Bai X L, Turner J D, Junkins J L. Bang-bang control design by combingpseudospectral method with a novel homotopy algorithm. AIAA Guidance,Navigation, and Control Conference, Chicago, Illinois,2009
[40] Mischa K. Continuous low-thrust trajectory optimization: techniques and applications[D]. Virginia Polytechnic Institute and State University,2005
[41] Bock H G, Plitt K J. A Multiple shooting algorithm for direct solution of optimalcontrol problems. Proceedings of the Ninth Triennial World Congress of IFAC,Pergamon, New York,1998
[42] Rainer C, Peter R. Optimal control of rigid-link manipulators by indirect methods.GAMM-Mitt,2008,31(1):27-58
[43] Kluever C A, Pierson B L. Optimal low-thrust earthmoon transfers with a switchingfunction structure. Journal of the Astronautical Sciences,1994,42(3):269-283
[44] Kluever C A, Pierson B L. Optimal low-thrust, three-dimensional Earth-Moontrajectories. Journal of Guidance, Control, and Dynamics,1995,18(4):830-837
[45] Kluever C A. Optimal low-thrust, Earth-Moon trajectories [D]. Iowa State Universityof Science and Technology,1993
[46] Gao Y. Advances in low-thrust trajectory optimization and flight mechanics [D].University of Missouri,2003
[47] Gao Y, Kluever C A. Low-thrust interplanetary orbit transfers using hybrid trajectoryoptimization method with multiple shooting. AIAA/AAS Astrodynamics SpecialistConference, Providence, Rhode Island,2004
[48]任远,崔平远,栾恩杰.利用混合法进行地球-火星小推力轨道设计.哈尔滨工业大学学报,2007,39(3):359-362
[49] Hargraves C, Paris S. Direct trajectory optimization using nonlinear programming andcollocation. Journal of Guidance, Control, and Dynamics,1987,4:338–342
[50] Enright P J, Conway B A. Discrete approximations to optimal trajectories using directtranscription and nonlinear programming. Journal of Guidance, Control, andDynamics,1992,15(4):994–1002
[51] Betts J T, Sven O E. Optimal low thrust trajectories to the Moon. SIAM JournalApplied Dynamical Systems,2003,2(2):144-170
[52] Betts J T. Optimal interplanetary orbit transfers by direct transcription. Journal of theAstronautical Sciences,1994,42:247–268
[53] Betts J T. Using sparse nonlinear programming to compute low thrust orbit transfers,Journal of the Astronautical Sciences,1993,41:349–371
[54] Betts J T. Very low-thrust trajectory optimization using a direct SQP method. Journalof Computational and Applied Mathematics,2000,120(1-2):27-40
[55] Herman A L, Conway B A. Optimal, low-thrust, Earth-Moon orbit transfer. Journal ofGuidance Control And Dynamics,1998,21(1):141-147
[56] Herman A L, Conway B A. Direct optimization using collocation based on high-orderGauss-Lobatto quadrature rules. Journal of Guidance, Control, and Dynamics,1996,19(3):592–599
[57] Coverstone-Carroll, Victoria Williams, Steven N. Optimal low thrust trajectoriesusing differential inclusion concepts.1994,42(4):379-393
[58] Hull, D. G, Speyer J L. Optimal reentry and plane-change trajectories. Journal of theAstronautical Sciences,1982,(2):117-130
[59]尚海滨,崔平远,栾恩杰.地球-火星的燃料最省小推力转移轨道的设计与优化.宇航学报,2006,27(6):1168-1173
[60] Bock H, Plitt K. A multiple shooting algorithm for direct solution of optimal controlproblems. In9th IFAC World Congress,242–247,1984
[61] Leineweber D, Bauer I, Bock H, et al. An efficient multiple shooting based reducedSQP strategy for large-scale dynamic process optimization, Part I: theoretical aspects.Computers and Chemical Engineering,2003,27:157-166
[62] Grimm W, Markl A. Adjoint estimation from a direct multiple shooting method.Journal of Optimization Theory and Applications,1997,92(2),263–283
[63] Gill P. E, Murray W, Saunders M A. User’s guide for SNOPT version7: software forlarge-scale nonlinear programming.2008, http://ccom.ucsd.edu/~peg/
[64] Gill P E, Murray W, Saunders M A. SNOPT: an SQP algorithm for large-scaleconstrained optimization. SIAM Journal on Optimization,2002,12(4):979–1006
[65] Li Y H, Baoyin H X, Gong S P.1st ACT global trajectory optimization competition:Tsinghua University results. Acta Astronautica,2007,61(9):735-741
[66] Zhu K J, Li J F, Baoyin H. X. Multi-swingby optimization of mission to Saturn usingglobal optimization algorithms. Acta Mechanica Sinica,2009,25(6):839-845
[67]王劼,李俊峰,崔乃刚,等.登月飞行器软着陆轨道的遗传算法优化.清华大学学报:自然科学版,2003,43(8):1056-1059
[68] Pontani M, Conway B A. Particle swarm optimization applied to space trajectories.Journal of Guidance, Control, and Dynamics,2010,33(5):1429-1441
[69] Wall B, Conway B A. Near-optimal low-thrust Earth-Mars trajectories via a GeneticAlgorithm. Journal of Guidance, Control, and Dynamics,2005,28(5):1027–1031
[70] Petropoulos A E, Lee S. Optimization of low-thrust orbit transfers using the Q-Lawfor the initial guess. Advances in the Astronautical Sciences,2006,123:2229-2248
[71] Lee S, von Allmen P, Fink W, et al. Design and optimization of low-thrust orbittransfers using the Q-law and evolutionary algorithms. IEEE Aerospace ConferenceProceedings, Big Sky, Montana,2005
[72] Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral methods in fluid dynamics.New York: Springer–Verlag,1988
[73] Trefethen L N. Spectral methods in MATLAB. Society for Industrial and AppliedMathematics, Philadelphia,2000
[74] Jan S H, Sigal G, David G. Spectral methods for time-dependent problems. New York:Cambridge University Press,2007
[75] Gong Q, Kang W, Bedrossian N, et al. Pseudospectral optimal control for military andindustrial applications.46th IEEE Conference on Decision and Control, New Orleans,LA,2007
[76] Kang W, Bedrossian N. Pseudospectral optimal control theory makes debut flight,saves NASA$1M in under three hours. SIAM News,40,2007
[77] Elnagar J, Kazemi M A, Razzaghi M. The pseudospectral Legendre method fordiscretizing optimal control problems. IEEE Transactions on Automatic Control,1995,40(10):1793–1796
[78] Fahroo F, Ross I M. Costate estimation by a Legendre pseudospectral method. Journalof Guidance, Control, and Dynamics,2001,24(2):270–277
[79] Ross I M, Fahroo F. Legendre pseudospectral approximations of optimal controlproblems. Lecture Notes in Control and Information Sciences, New York:Springer–Verlag,2003
[80] Ross I M. A historical introduction to the covector mapping principle. Advances inthe Astronautical Sciences, Univelt, San Diego, CA,2006
[81] Garg D, Patterson M. A, Darby C L, et al. Direct trajectory optimization and costateestimation of finite-horizon and infinite-horizon optimal control problems using aRadau pseudospectral method. Computational Optimization and Applications,2009,49(2):335-358
[82] Kameswaran S, Biegler L T. Convergence rates for the direct transcription of optimalcontrol problems using collocation at Radau points. Computational Optimization andApplications,2008,41(1):81–126
[83] Benson D A, Huntington G T, Thorvaldsen T P, et al. Direct trajectory optimizationand costate estimation via an orthogonal collocation method. Journal of Guidance,Control, and Dynamics,2006,29(6):1435–1440
[84] Benson D A. Gauss pseudospectral transcription for optimal control [D].Massachusetts Institute of Technology,2004
[85] Huntington G T. Advancement and analysis of Gauss pseudospectral transcription foroptimal control problems [D]. Massachusetts Institute of Technology,2007
[86] Vlassenbroeck J, Van D R. A Chebyshev technique for solving nonlinear optimalcontrol problems. IEEE Transactions on Automatic Control,1988,33(4):333–340.
[87] Fahroo F, Ross I M. Direct trajectory optimization by a Chebyshev pseudospectralmethod. Journal of Guidance, Control, and Dynamics,2002,25(1):160–166
[88] Gong Q, Ross I M, Fahroo F. Costate computation by a Chebyshev pseudospectralmethod. Journal of Guidance, Control, and Dynamics,2010,33(2):623-628
[89] Welfert B D. Generation of pseudospectral differentiation matrices I. SIAM J.Numer.Anal.1997,34(4):1640–1657
[90] Weideman J A C, Reddy S C. A MATLAB differentiation matrix suite.2001
[91] Rao A V, Benson D A, Darby, et al. Algorithm902: GPOPS, A Matlab Software forSolving Multiple-Phase Optimal Control Problems Using the Gauss PseudospectralMethod. ACM Transactions on Mathematical Software,2010,37(2):22:1-22:39
[92] Ross I M, Gong Q, Sekhavat P. Low-thrust, high-accuracy trajectory optimization.Journal of Guidance Control and Dynamics,2007,30(4):921-933
[93] Ross I M, Fahroo F. Pseudospectral knotting methods for solving optimal controlproblems. Journal of Guidance, Control, and Dynamics,2004,27b(3):397–405
[94] Gong, Q, Ross I M. Autonomous pseudospectral knotting methods for space missionoptimization,16th AAS/AIAA Space Flight Mechanics Meeting, AmericanAstronautical Society Paper06-151,2006
[95] Gong Q, Fahroo F, Ross I M. Spectral algorithm for pseudospectral methods inoptimal control. Journal of Guidance, Control, and Dynamics,2008,31(3):460-471
[96] Gong Q, Kang W, Ross I M. A pseudospectral method for the optimal control ofconstrained feedback linearizable systems. IEEE Transactions on Automatic Control,2006,51(7):1115–1129
[97] Kang W, Gong Q, Ross I M, et al. On the convergence of nonlinear optimal controlusing pseudospectral methods for feedback linearizable control systems. InternationalJournal of Robust and Nonlinear Control,2007,17(14):1251–1277
[98] Gong Q, Ross I M, Kang W, et al. Connections between the covector mappingtheorem and convergence of pseudospectral methods for optimal control.Computational Optimization and Applications,2007,41(3):307-335
[99] Fahroo F, Ross I M. Convergence of the costates does not imply convergence of thecontrol. Journal of Guidance, Control, and Dynamics,2008,31(5):1492-1497
[100] Fahroo F, Ross I M. Pseudospectral methods for infinite-horizon nonlinear optimalcontrol problems. Journal of Guidance, Control, and Dynamics,2008,31(4):927–936
[101] Garg D A, Hager W W, Rao A V. Pseudospectral methods for solving infinite-horizonoptimal control problems. Automatica,2011,47(4):829-837
[102] Ross I M, Fahroo F. Discrete verification of necessary conditions for switchednonlinear optimal control systems. Proceedings of American Control Conference,2:1610-1615,2004
[103] Ross I M, D'Souza C N. Hybrid optimal control framework for mission planning.Journal of Guidance, Control, and Dynamics,2005,28(4):686-697
[104] Huntington G T, Rao A V. Comparison of global and local collocation methods foroptimal control. Journal of Guidance, Control, and Dynamics,2008,31(2):432-436
[105] Bedrossian N S, Bhatt S, Kang W, et al. Zero propellant maneuver guidance. IEEEControl Systems Magazine,2009,29:53-73
[106] Ross I M, Fahroo F. User's Manual for DIDO: A MATLAB package for dynamicoptimization. Naval Postgraduate School Technical Report,2002
[107] Williams P, Blanksby C, Trivailo P. Receding horizon control of tether system usingquasilinearization and Chebyshev pseudospectral approximations. AAS/AIAAAstrodynamics Specialist Conference, Big Sky, MT,2003
[108] Williams P. Jacobi Pseudospectral method for solving optimal control problems.Journal of Guidance, Control, and Dynamics,2004,27(2):293–297
[109] Williams P. Application of pseudospectral methods for receding horizon control.Journal of Guidance, Control, and Dynamics,2004,27(2):310–314
[110]文浩.绳系卫星释放和回收的动力学控制[博士学位论文].南京:南京航空航天大学,2008
[111] Yan H, Fahroo F, Ross I M. Real-Time Computation of neighboring optimal controllaws. AIAA-Guidance, Navigation, and Control Conference AIAA-2002-4657,Monterey, California,2002
[112] Yan H, Alfriend K T. Three-axis magnetic attitude control using pseudospectralcontrol law in eccentric orbits. AAS Spaceflight Mechanics Meeting, Tampa, FL,2006
[113]罗建军,王明光,袁建平.基于伪光谱方法的月球软着陆轨道快速优化.宇航学报,2007,28(5):1119-1122
[114] Cui P Y, Qiao D, Cui H T, et al． Target selection and transfer trajectories design forexploring asteroid mission. Science China Technological Sciences,2010,53(4):1150-1158,
[115] McConaghy T T, Debban T J, Petropoulos A E, et al． Design and optimization oflow-thrust trajectories with gravity assists. Journal of Spacecraft and Rockets,2003,3:380-387
[116] Casalino L, Colasurdo G, Pastrone D. Optimal low-thrust escape trajectories usinggravity assist. Journal of Guidance, Control, and Dynamics,1999,5:637-642
[117] Sims J A. Delta-V gravity-assist trajectory design: theory and practic [D]. PurdueUniversity,1997
[118] Petropoulos A E, Longuski J M. Shape-based analytic representations of low-thrusttrajectories for gravity-assist applications, The AAS/AIAA AstrodynamicsConference, Girdwood, USA,1999
[119] Petropoulos A E, Longuski J M. Shape-based algorithm for automated design oflow-thrust，gravity-assist trajectories. Journal of Spacecraft and Rockets,2004,41(5):87-796
[120] Pascale P D, Vasile M. Preliminary design of low-thrust multiple gravity-assisttrajectories. Journal of Spacecraft and Rockets,2006,5:1065-1076
[121] Wall B J, Conway B A． Shape-based approach to low-thrust rendezvous trajectorydesign. Journal of Guidance, Control, and Dynamics,2009,32(1):95-101
[122]尚海滨,崔平远,徐瑞,等.结合行星借力飞行技术的小推力转移轨道初始设计.宇航学报,2011,32(1):29-38
[123] Howell K. C, Kakoi M. Transfers between the earth-moon and sun-earth systemsusing manifolds and transit orbits. Acta Astronautica,2006,59(1-5):367-380
[124]乔栋.深空探测转移轨道设计方法研究及在小天体探测中的应用[博士学位论文].哈尔滨:哈尔滨工业大学,2007
[125]任远.星际探测中的小推力转移轨道设计与优化方法研究[博士学位论文].哈尔滨:哈尔滨工业大学,2007
[126]徐明,徐世杰.小推力航天器的地月低能转移轨道.航空学报,2008,29(4):781-787
[127] McConaghy T T. GALLOP Version4.5. User's Guide. School of Aeronautics andAstronautics, Purdue University. West Lafayette, Indiana.,2005
[128]关治，陆金甫.数值分析基础.北京：高等教育出版社，2010
[129] Luenberger D. G, Yinyu Y. Linear and nonlinear programming. Springer,2008
[130] Zorich V A, Cooke R. Mathematical analysis. Springer,2004
[131]吴麒.自动控制原理.北京：清华大学出版社，1992