基于同伦方法的地月系L_2点小推力转移轨道优化
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摘要
针对地月系下航天器从GEO轨道到L2点的时间最优小推力转移轨道问题,基于庞德里亚金极值原理,推导了限制性三体问题模型下的小推力转移轨道优化问题的最优性一阶必要条件,即推力保持最大值,且方向始终沿主矢量反方向,并将优化问题转换为两点边值问题。通过与同伦方法相结合,解决了间接法求解过程中收敛域小的困难。首先构造了针对推力幅值进行同伦的同伦函数,以大推力幅值的轨道转移问题作为同伦初始问题,然后选取连续同伦中的伪弧长法为同伦曲线跟踪方法,通过迭代求解了不同同伦参数值下的子问题,最终得到原问题下的小推力转移轨道。最后,在数值仿真中得到了不同推力值下的转移轨道,验证了该同伦方法在求解小推力转移轨道中的有效性。
The minimum-time low-thrust transfer trajectory optimization problem from a GEO to L2 Libration point was studied. The optimal conditions of the minimum-time problem for the circular-restricted three-body problem were deduced based on the Pontryagin maximum principle,and the optimization problem was transformed into a two-point boundary-value problem. The homotopy method was applied to overcome the difficulties such as the narrow convergence domain in the indirect method. The homotopy function for thrust magnitude was constructed and the high-thrust problem was considered as the homotopy initial problem. Then the continuous pseudo arclength continuation method was selected as the homotopy path tracking method. By solving the sub-problems with different homotopy parameter iteratively,the low-thrust transfer trajectory of the original problem was finally obtained. In the end,the transfer trajectories with different thrust magnitude were obtained in the numerical simulation,and the effectiveness of the homotopy method was verified.
The minimum-time low-thrust transfer trajectory optimization problem from a GEO to L2 Libration point was studied. The optimal conditions of the minimum-time problem for the circular-restricted three-body problem were deduced based on the Pontryagin maximum principle,and the optimization problem was transformed into a two-point boundary-value problem. The homotopy method was applied to overcome the difficulties such as the narrow convergence domain in the indirect method. The homotopy function for thrust magnitude was constructed and the high-thrust problem was considered as the homotopy initial problem. Then the continuous pseudo arclength continuation method was selected as the homotopy path tracking method. By solving the sub-problems with different homotopy parameter iteratively,the low-thrust transfer trajectory of the original problem was finally obtained. In the end,the transfer trajectories with different thrust magnitude were obtained in the numerical simulation,and the effectiveness of the homotopy method was verified.
引文
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[3]薛锐,崔雁,王晓磊.嫦娥五号飞行试验器服务舱环月期间的角动量管理[J].空间控制技术与应用,2016,42(4):53-56.Xue R,Cui Y,Wang X. Angular momentum management ofCE-5T service module during surrounding the moon[J]. Aero-space Control and Application,2016,42(04):53-56.(inChinese)
[4]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,Jiang F. Survey of low-thrust trajectory optimizationmethods for deep space exploration[J]. Mechanics in Engi-neering,2011,33(3):1-6.(in Chinese)
[5] Betts J T. Very low-thrust trajectory optimization using a directSQP method[J]. Journal of Computational&Applied Mathe-matics,2000,120(1-2):27-40.
[6] Topputo F,Zhang C. Survey of direct transcription for low-thrust space trajectory optimization with applications[J]. Ab-stract and Applied Analysis,2014,2014(2):1-15.
[7] Jiang F,Baoyin H,Li J. Practical techniques for low-thrusttrajectory optimization with homotopic approach[J]. Journal ofGuidance,Control,and Dynamics,2012,35(1):245-258.
[8] Zhang C,Topputo F,Bernellizazzera F,et al. Low-thrustminimum-fuel optimization in the circular restricted three-bodyproblem[J]. Journal of Guidance Control Dynamics,2015,38(8):1-9.
[9] Howell K C,Ozimek M T. Low-thrust transfers in the Earth-Moon system, including applications tolibrationpoint orbits[J]. Journal of Guidance Control Dynamics,2008,33(2):533-549.
[10] Haberkorn T,Martinon P,Gergaud J. Low thrust minimum-fuel orbital transfer:ahomotopic approach[J]. Journal ofGuidance Control&Dynamics,2004,27(6):1046-1060.
[11] Gergaud J,Haberkorn T. Homotopy method for minimum con-sumption orbit transfer problem[J]. Esaim ControlOptimisation&Calculus of Variations,2006,12(2):294-310.
[12] Caillau J B,Farrés A. On local optima in minimum time con-trol of the restricted three-body problem[M]//Recent Ad-vances in Celestial and Space Mechanics. Springer Interna-tional Publishing,2016,209-302.
[13] Pan B,Lu P,Pan X,et al. Double-homotopymethod for sol-ving optimal control problems[J]. Journal of Guidance Con-trol&Dynamics,2016,39(8):1-15.
[14] Seydel R,Hlavacek V. Role of continuation in engineering a-nalysis[J]. Chemical Engineering Science,1987,42(6):1281-1295.
[15] Allgower E L,Georg K. Introduction to Numerical Continua-tion Methods[M]. Colorado State University,1990:61-74.