空间拦截最优轨道设计
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摘要
空间技术对一个国家的政治、经济、军事越来越重要,是综合国力的根本保障。空间拦截是一种保障空间安全和战略优势的关键性空间技术。天基拦截尽管技术复杂,但是可以拦截低、中、高轨道目标,比地基拦截更具有战略价值。天基异面直接上升拦截与共轨拦截和共面拦截相比,作战意图更隐蔽,具有拦截异面目标的能力等优点,此外利用一个搭载多颗拦截器的天基平台拦截既可以彻底破坏由分布在多个轨道面上的多个目标组成的星座的功能,又可以降低成本。异面拦截需要初始轨道面外变轨,必然导致燃料损耗较大,因此燃料约束对转移轨道设计具有决定性作用,考虑燃料约束的最优轨道设计是异面拦截的核心问题。
提出了一种考虑拦截器初始速度的转移角的求解方法,根据Lambert定理建立了圆锥曲线轨道转移时间与半长轴的对应关系,利用几何学分析了椭圆转移轨道的虚焦点位置对椭圆转移轨道形态的影响,以及转移角对转移时间和偏心率的影响。给出了由轨道半长轴确定空间两点间圆锥曲线转移轨道的方法,随后给出了一种固定发射时刻和转移时间的情况下利用半长轴迭代的圆锥曲线转移轨道求解算法。并仿真验证了该算法对椭圆和双曲线转移轨道均适用。
分析了单脉冲多圈飞行拦截的特征速度与转移轨道半长轴之间的关系,指出其最优解实际上为2N+1条满足时间约束的转移轨道中燃料较省的,而不是最省燃料轨道。推导了单脉冲拦截的特征速度与半长轴的导数关系,随后给出了求解最省燃料轨道的计算方法。提出一种脉冲分解多圈飞行拦截,将单脉冲分解成方向相同的两次脉冲,使得拦截器在某特定的椭圆轨道上多圈飞行以消耗掉多余的转移时间,利用剩余转移时间沿最省燃料轨道拦截目标。几何上证明了这种拦截的特征速度与最省燃料轨道的特征速度相同,并且给出了该方法的解的存在性条件。通过两个实例验证了利用该方法设计的转移轨道比单脉冲多圈飞行拦截更节省燃料,同时解的存在性对转移时间的长度要求更低。
给出了一种固定发射点到目标轨道的最小特征速度曲线求解方法,通过对共面拦截和异面拦截的最小特征速度曲线的比较,证明了共面拦截中转移角180°最优解的Hohmann变轨在异面拦截中导致极大的燃料损耗,并且分析了拦截器初始轨道根数对最小特征速度曲线的影响。提出一种基于最小特征速度曲线的异面拦截区求解方法,并且分别给出了一圈内飞行拦截、单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截的防区求解算法。通过仿真研究了单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截可以通过增加飞行圈数将目标纳入防区,可以即时发射拦截目标,而一圈内飞行拦截常常需要改变发射点才能拦截目标。
一圈内飞行拦截特征速度等高线图可以直接利用圆锥曲线Lambert算法求解。但是由于单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截在转移时间不够长的情况下其解并不存在,采用同样的步骤求解特征速度等高线图会产生奇异值。利用一圈内飞行拦截的解替换奇异值,提出一种求解单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截的特征速度等高线图的方法。通过对发射条件的比较,单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截都能充分利用了拦截区,但后者更节省燃料。最后通过实例验证了利用脉冲分解多圈飞行拦截特征速度等高线图设计的拦截轨道可以对分布在异面轨道的星座上的目标进行拦截。
Space technology becomes more and more important to politics, economy and military, as an ultimate guarantee of nation’s power. Space interception is the key of space technology to protect space security and strategic dominance. Although it is complicated in technology, space-based interception, which can intercept a target in any low, middle and high orbit, has more strategic value than ground-based interception. Compare with same orbit interception and coplanar interception, space-based noncoplanar direct ascend interception has some advantages, such as covert purpose, the capability of intercepting targets in noncoplanar orbits. Moreover, a space-based platform with multiple interceptors can not only completely destroy the function of constellation, made up of targets in multiple orbit planes, but also save cost. Since noncoplanar interception will transfer out of initial plane, it consumes lots of fuel consequentially. Because fuel constraint is a crucial fact to design the transfer trajectory, optimal trajectory considering fuel constraint is a primary problem for noncoplanar interception.
A calculated method of transfer angle considering interceptor initial velocity was presented. The relationship between transfer time of conic trajectory and semi-major axis was set up based on Lambert theorem. The effect of vacant focus location to shape of elliptical transfer trajectory and transfer angle to transfer time and eccentricity was analyzed using geometry. A method determining conic transfer trajectory between two points in space by a fixed semi-major axis was given. An arithmetic solving conic transfer trajectory by iterating semi-major axis with a specified launch time and transfer time was given. Some simulation proved that this arithmetic was valid to elliptic and hyperbolic transfer trajectory.
For single impulse multiple-revolution interception, the relationship between characteristic velocity (?v) and semi-major axis of transfer trajectory was considered, It was proposed that the optimal solution actually is the less fuel trajectory among 2N+1 trajectories satisfying time constraint, but not the minimum fuel trajectory (MFT). A derivate formula between ?v of single impulse interception and semi-major axis was derived. As the single impulse was dissembled into two impulses with the same direction, a interceptor could consume the redundant transfer time by costing multiple-revolution on some specified elliptic orbit, and intercept a target on MFT in the rest transfer time. It was proved that ?v of this interception coincide with that of minimum fuel transfer in geometry. The existence of solutions was studied. Some simulations show that this intercept can save fuel and the existence of solutions is more loosely restrictive on the length of transfer time than single impulse multiple-revolution interception.
A calculating method of minimum characteristic velocity curve for a fixed launch point to a target orbit was interpreted. Camparing the two minimum characteristic velocity curves of coplanar and noncoplanar interception study, it was proved that Hohmann transfer, which was the optimal solution of coplanar interception with transfer angle 180°, cost large mount of fuel in noncoplanar interception. The effect of the initial orbital elements of interceptor to minimum characteristic velocity curve is analyzed. A method of computing noncoplanar intercept range based on minimum characteristic velocity curve was proposed. Furthermore, arithmetics of calculating the defence range for lacking-one-revolution interception, single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception were given, respectively. Some simulations prove that single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception can cover the target into the defence range by more revolutions and launch interceptor immediately, but lacking-one-revolution interception often has to change the launch point to intercept the target.
Contour of characteristic velocity for lacking-one-revolution interception can be calculated directly by using conic Lambert arithmetic. On the other hand, using the same process to draw contour of characteristic velocity, there will create singularities for single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception, whose solutions are not existed when transfer time is not long enough. Two methods calculating contour of characteristic velocity for single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception was presented, by using the solutions of lacking-one-revolution interception to replace the singularities. Comparing the launch conditions, although both single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception can utilize intercept range well, the latter saves fuel. Finally, an example proves that the intercept trajectory, designed by contour of characteristic velocity for impulse dissembled multiple-revolution interception, can intercept targets of a constellation distributing on noncoplanar orbits.
提出了一种考虑拦截器初始速度的转移角的求解方法,根据Lambert定理建立了圆锥曲线轨道转移时间与半长轴的对应关系,利用几何学分析了椭圆转移轨道的虚焦点位置对椭圆转移轨道形态的影响,以及转移角对转移时间和偏心率的影响。给出了由轨道半长轴确定空间两点间圆锥曲线转移轨道的方法,随后给出了一种固定发射时刻和转移时间的情况下利用半长轴迭代的圆锥曲线转移轨道求解算法。并仿真验证了该算法对椭圆和双曲线转移轨道均适用。
分析了单脉冲多圈飞行拦截的特征速度与转移轨道半长轴之间的关系,指出其最优解实际上为2N+1条满足时间约束的转移轨道中燃料较省的,而不是最省燃料轨道。推导了单脉冲拦截的特征速度与半长轴的导数关系,随后给出了求解最省燃料轨道的计算方法。提出一种脉冲分解多圈飞行拦截,将单脉冲分解成方向相同的两次脉冲,使得拦截器在某特定的椭圆轨道上多圈飞行以消耗掉多余的转移时间,利用剩余转移时间沿最省燃料轨道拦截目标。几何上证明了这种拦截的特征速度与最省燃料轨道的特征速度相同,并且给出了该方法的解的存在性条件。通过两个实例验证了利用该方法设计的转移轨道比单脉冲多圈飞行拦截更节省燃料,同时解的存在性对转移时间的长度要求更低。
给出了一种固定发射点到目标轨道的最小特征速度曲线求解方法,通过对共面拦截和异面拦截的最小特征速度曲线的比较,证明了共面拦截中转移角180°最优解的Hohmann变轨在异面拦截中导致极大的燃料损耗,并且分析了拦截器初始轨道根数对最小特征速度曲线的影响。提出一种基于最小特征速度曲线的异面拦截区求解方法,并且分别给出了一圈内飞行拦截、单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截的防区求解算法。通过仿真研究了单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截可以通过增加飞行圈数将目标纳入防区,可以即时发射拦截目标,而一圈内飞行拦截常常需要改变发射点才能拦截目标。
一圈内飞行拦截特征速度等高线图可以直接利用圆锥曲线Lambert算法求解。但是由于单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截在转移时间不够长的情况下其解并不存在,采用同样的步骤求解特征速度等高线图会产生奇异值。利用一圈内飞行拦截的解替换奇异值,提出一种求解单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截的特征速度等高线图的方法。通过对发射条件的比较,单脉冲多圈飞行拦截和脉冲分解多圈飞行拦截都能充分利用了拦截区,但后者更节省燃料。最后通过实例验证了利用脉冲分解多圈飞行拦截特征速度等高线图设计的拦截轨道可以对分布在异面轨道的星座上的目标进行拦截。
Space technology becomes more and more important to politics, economy and military, as an ultimate guarantee of nation’s power. Space interception is the key of space technology to protect space security and strategic dominance. Although it is complicated in technology, space-based interception, which can intercept a target in any low, middle and high orbit, has more strategic value than ground-based interception. Compare with same orbit interception and coplanar interception, space-based noncoplanar direct ascend interception has some advantages, such as covert purpose, the capability of intercepting targets in noncoplanar orbits. Moreover, a space-based platform with multiple interceptors can not only completely destroy the function of constellation, made up of targets in multiple orbit planes, but also save cost. Since noncoplanar interception will transfer out of initial plane, it consumes lots of fuel consequentially. Because fuel constraint is a crucial fact to design the transfer trajectory, optimal trajectory considering fuel constraint is a primary problem for noncoplanar interception.
A calculated method of transfer angle considering interceptor initial velocity was presented. The relationship between transfer time of conic trajectory and semi-major axis was set up based on Lambert theorem. The effect of vacant focus location to shape of elliptical transfer trajectory and transfer angle to transfer time and eccentricity was analyzed using geometry. A method determining conic transfer trajectory between two points in space by a fixed semi-major axis was given. An arithmetic solving conic transfer trajectory by iterating semi-major axis with a specified launch time and transfer time was given. Some simulation proved that this arithmetic was valid to elliptic and hyperbolic transfer trajectory.
For single impulse multiple-revolution interception, the relationship between characteristic velocity (?v) and semi-major axis of transfer trajectory was considered, It was proposed that the optimal solution actually is the less fuel trajectory among 2N+1 trajectories satisfying time constraint, but not the minimum fuel trajectory (MFT). A derivate formula between ?v of single impulse interception and semi-major axis was derived. As the single impulse was dissembled into two impulses with the same direction, a interceptor could consume the redundant transfer time by costing multiple-revolution on some specified elliptic orbit, and intercept a target on MFT in the rest transfer time. It was proved that ?v of this interception coincide with that of minimum fuel transfer in geometry. The existence of solutions was studied. Some simulations show that this intercept can save fuel and the existence of solutions is more loosely restrictive on the length of transfer time than single impulse multiple-revolution interception.
A calculating method of minimum characteristic velocity curve for a fixed launch point to a target orbit was interpreted. Camparing the two minimum characteristic velocity curves of coplanar and noncoplanar interception study, it was proved that Hohmann transfer, which was the optimal solution of coplanar interception with transfer angle 180°, cost large mount of fuel in noncoplanar interception. The effect of the initial orbital elements of interceptor to minimum characteristic velocity curve is analyzed. A method of computing noncoplanar intercept range based on minimum characteristic velocity curve was proposed. Furthermore, arithmetics of calculating the defence range for lacking-one-revolution interception, single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception were given, respectively. Some simulations prove that single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception can cover the target into the defence range by more revolutions and launch interceptor immediately, but lacking-one-revolution interception often has to change the launch point to intercept the target.
Contour of characteristic velocity for lacking-one-revolution interception can be calculated directly by using conic Lambert arithmetic. On the other hand, using the same process to draw contour of characteristic velocity, there will create singularities for single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception, whose solutions are not existed when transfer time is not long enough. Two methods calculating contour of characteristic velocity for single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception was presented, by using the solutions of lacking-one-revolution interception to replace the singularities. Comparing the launch conditions, although both single impulse multiple-revolution interception and impulse dissembled multiple-revolution interception can utilize intercept range well, the latter saves fuel. Finally, an example proves that the intercept trajectory, designed by contour of characteristic velocity for impulse dissembled multiple-revolution interception, can intercept targets of a constellation distributing on noncoplanar orbits.
引文
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66王石,文援兰,戴金海.用进化-模拟退火算法求解Lambert方程.系统仿真学报. 2007, 19(2): 450-452
67 Dawei Zhang, Shenmin Song, Guangren Duan. Fuel and Time Optimal Transfer of Spacecrafers Rendezvous Using Lambert’s Theorem and Improved Genetic Algorithm. IEEE Xplore. 2008:1-5
68 Luo Yazhong, Lei Yongjun, Tang guojin. Remarks on a Benchmark Nonlinear Constrained Optimization Problem. Journal of System Science and Eletronics. 2006, 17(3):551-553
69 Luo Yazhong, Li Haiyang, Tang Guojin. Hybrid Approach to Optimize a Rendezvous Phasing Strategy. Journal of Guidance, Control and Dynamics. 2007, 30(1):185-191
70 Luo Yazhong, Tang Guojin, Li Haiyang. Optimization of Multi-Impulse Minimum-Time Rendezvous Using a Hybrid Genetic Algorithms. Aerospace Science and Technoloty. 2006, 10(6): 534-540
71 Luo Yazhong, Tang Guojin, Wang Zhonggui, etc. Optimization of Perturbedand Constrained Fuel-Optimal Impulse Rendezvous Using a Hybrid Approach. Engineering Optimization. 2006, 38(8): 959-973
72 Luo Yazhong, Tang Guojin, Zhou Lini. Simulated Annealing for Solving Near-Optimal Low-Thrust Orbit Transfer. Engineering Optimization. 2005, 37(2): 201-216
73 Luo Yazhong, Tang Aerospace Science and Technoloty. Spacecraft Optimal Rendezvous Controller Design Using Simulated Annealing. Aerospace Scinence and Technology. 2005, 9(8): 632-637
74 Luo Yazhong, Tang Guojin. Guojin. Parallel Simulated Annealing Using Simplex Mehtod. AIAA Journal. 2006, 44(12): 3143-3146
75 Luo Yazhong, Tang Guojin. Rendezvous Phasing Special-Point Maneuver Mixed Constinuous-Discrete Optimization Using Simulated Annealing. Aerospace Science and Technology. 2006, 10(7): 652-658
76 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimal Multi-Objective Linearzed Impulsive Rendezvsou. J. of Guidance Control and Dynamics. 2007, 30(2): 383-389
77 Ya-Zhong Luo, Guo-Jin Tang, Geoff Parks. Multi-Objective Optimization of Perturbed Impulsive Rendezvous Trajectories Using Physical Programming. J. of Guidance Control and Dynamics. 2008, 31(6): 1829-1832
78 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimal Multi-Objective Nonlinear Impulsive Rendezvous. J of Guidance Control and Dynamics. 2007, 30(4): 994-1002
79 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimization of Multiple-Impulse Multiple-Renvolution Rendezvsou-Phasing Maneuvers. J. of Guidance Control and Dynamics. 2007, 30(4): 946-952
80李晨光,肖业伦.多脉冲C-W交会的优化方法.宇航学报. 2006, 27(2): 172-177
81卢山,陈统,徐世杰.基于自适应模拟退火遗传算法的最优Lambert转移.北京航空航天大学学报. 2007, 33(10): 1191-1195
82陈统,徐世杰.基于遗传算法的最优Lambert双脉冲转移.北京航空航天大学学报. 2007, 33(3): 273-277
83范淑梅.基于自适应遗传算法的航天器快速轨道机动研究.硕士学位论文.山东大学. 2008
84秦帅.遗传算法在航天器轨道机动中的应用研究.硕士学位论文.哈尔滨工业大学. 2007
85张洪波,郑伟,汤国建.混合遗传算法在远程交会轨道设计中的应用.航天控制. 2006, 24(2): 34-37
86齐映红,曹喜滨.三脉冲最优交会问题的解法.吉林大学学报. 2006, 36(4): 608-612
87张秋华,韩琦,孙毅.基于模糊推力的基向量曲线求解方法.宇航学报. 2007, 28(4): 1002-1006
88 D. F. Lawden. Optimal Trajectories for Space Navigation . London Butter Worths. 1963
89 P. M. Lion, M. Handelsman. Primer Vector on Fixed-Time Impulsive Trajectories . AIAA Journal. 1968, 6(1): 127-132
90 David R. Glandorf. Lagrange Multipliers and the State Transition Matrix for Coasting Arcs . AIAA Journal. 1969, 7(2): 363-365
91 David R. Glandorf. Primer Vector Theory for Matched-Conic Trajectories. AIAA Journal. 1970, 8(1):155-156
92 J. E. Prussing. Optimal Impulsive Linear Systems: Sufficient Conditions and Maximum Number of Impulse . Journal of the Astronautical Science. 1995, 43(2): 195-206
93 Thomas E. Carter. Necessary and Sufficient Conditions for Optimal Impulsive Rendezvous with Linear Equations of Motion. Dynamics and Control. 2000, 10: 219-227
94 D.J. Jezewski and H. L. Rozenkdal. An Efficient Method for Calculating Optimal Free-Space N-Impulse Trajecteries. AIAA Journal. 1968, 6(11): 2160-2165
95 Larry R. Gross, John E. Prussing. Optimal Multiple-Impulse Direct Ascent Fixed-Time Rendezvous. AIAA Journal. 1974,12(7): 885 - 890
96 John E. Prussing, Jeng-Hua Chiu. Optimal Multiple-Impulse Time-Fixed Rendezvous Between Circular Orbits . J.Guidance. 1986, 9(1): 17-25
97 Der-Ren Taur, Victoria coverstone Carroll, John E. Prussing. Optimal Impulsive Time-Fixed Orbital Rendezvous and Interception with Path Constraints. Journal of Guidance, Control, and Dynamics. 1995, 18(1): 54–60
98 Der-Ren Taur. Optimal Impulsive Time-Fixed Orbital Rendezvous and Interception with Path Constraints. Ph. D. Thesis, The University of Illinois at Urbana-Champaign, 1989
99 William Gary Hechathorn. Optimal Impulse Direct Ascent Time-Fixed Orbital Interception .Ph. D. Thesis, The University of Illinois at Urbana-Champaign. 1985
100 John E. Prussing, Linda J. Wellnitz, Willian G. Hechathorn. Optimal impulsive time-fixed direct-ascent interception . J. Guidance. 1989, 12(4): 487–494
101 Suzannah Sandrik. Primer-Optimized Results and Trends for Circular Phasing and Other Circle-to-Circle Impulsive Coplanar Rendezvous. Doctoral Thesis of University of Illinois at Urbana-Champaign. 2006
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104 N. X. Vinh. Integration of the Primer Vector in a Central Force Field. Journal of Optimization Theory and Applications. 1972,9(1): 51-58
105 Jan-Won Jo, John E. Prussing. Procedure for Applying Second-Order Conditions. Journal of Guidance, Control, and Dynamics. 2000, 23(2): 241-250
106 John E. Prussing, Suzannah L. Sandrik. Second-Order Necessary and Sufficient Conditions Applied to Continuous-Thrust Trajectories . Journal of Guidance, Control, and Dynamics. 2005, 28(4): 812-816
107 John E. Prussing, Suzannah L. Sandrik. Second-Order Necessary and Sufficient Conditions Applied to Low-Thrust Trajectories: AIAA/AAS Astrodynamics Specialist Conference and Exhibit. 5-8 August 2002, Monterey, California, AIAA-2002-4726: 1-15
108 John E. Prussing. Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions. Journal of Spacecraft and Rockets. 2003, 40(6): 952-959.
109 Haijun Shen. Optimal Scheduling for Satellite Refuelling in Circular Orbits. Ph. D. Thesis, Georgia Institute of Technology. 2003
110 Haijun Shen, Panagiotis Tsiotras. Optimization of very-low-thrust, many-revolution spacecraft trajectories. AIAA Guidance, Navigation, and Control Conference and Exhibit, 5-8 August, Monterey, California, 2002: 1-11
111 Haijun Shen, Panagiotis Tsiotras. Using Battin’s Method to Obtain Multiple-Revolution Lambert’s Solutions. AAS 03-568 : 1-18
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114韩潮,谢华伟.空间交会中多圈Lambert变轨算法研究.中国空间科学技术.2004,5(9):9-14
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27 John E. Prussing. Geometrical Interpretation of the Anglesαandβin Lambert's Problem . J. Guidance and Control. 1979, 2(5): 442 - 443
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31白涛,袁国雄,任章.一种天基反卫星武器远程变轨方法研究.航天控制. 2007,25(2): 13-16
32王会利,和兴锁,张亚锋.空间作战拦截最优轨道设计.长春理工大学学报. 2007, 30(2): 51-53
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36 Jones. Optimal Rendezvous in the Neighborhood of a Circular Orbit. Journal of the Astronautical Science. 1976, 24(1): 55-90
37 Jezewski and Donaldson. An Analytic Approach to Optimal Rendezvous Using Clohessy-Wiltshire Equation. Journal of the Astronautical Science. 1979, 27(3):293-310
38 Carter. Fuel-Optimal Maneuvers of a Spacecraft Relative to a Point in Circular Orbit. Journal of Guidance, Control, and Dynamics. 1984, 7(6):710-716
39 John E. Prussing. Optimal Multiple-Impulse Orbital Rendezvous. PhD Thesis, MIT. 1967
40 Prussing. Optimal Four-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit. AIAA Journal. 1969, 7(5): 928-935
41 John E. Prussing. Optimal Two- and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit. AIAA Journal. 1970, 8(7): 1221-1228
42 De Vries. Elliptic Elements in Terms of Small Increments of Position and Velocity Components. AIAA Journal. 1963, 1(11): 2626-2629
43 Tschauner. Eliptic Orbit Rendezvous. AIAA Journal. 1967,5(6):1110-1113
44 Euler and Shulman. Second Order Solution to the Elliptical Rendezvous Problem. AIAA Journal. 1967, 5(5): 1033-1035
45 Euler. Optimal Low-Thrust Rendezvous Control. AIAA Journal. 1969, 7(6) : 1140-1144
46 Wolfsberger, Weiss and Rangnitt. Strategies and Schemes for Rendezvous on Geostationary Transfer Orbit. Astronautica Acta. 1983, 10: 527-538
47 Marec. Optimal Space Trajectories. Elsevier, New Yark, 1979.
48 Catrer and Humi. Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit. Journal of Guidance, Control, and Dynamics. 1987, 10(6): 567-573
49 Cater. New Form for the Optimal Rendezvous Equations Near a Keperian Orbit. Journal of Guidance, Control, and Dynamics. 1990, 13(1): 183-186
50 T. E. Carter. Optimal Impulsive Space Trajectories Based on Linear Equations. Journal of Optimization Theory and Applications. 1991, 70(2): 277-297
51 Cater and Brient. Fuel-Optimal Rendezvous for Linearized Equations of Motion. Journal of Guidance, Control, and Dynamics. 1992, 15(6): 1411-1416
52 Cater T. E., Brient J.. Linearized Impulsive Rendezvous Rendezvous Problem. Journal of Optimization Theory and Application. 1995, 86(3): 553-584
53 Carter T., Humi M. Clohessy-Wiltshire Equations modified to Include Quadratic Drag . Journal of Guidance Control and Dynamics, 2002, 25(6): 1058-1063
54 Baranov. An Algorithm for Calculating Parameters of Multi-Orbit Maneuversin Maneuvers in Remote Guidance. Cosmic Research. 1990, 28(1):61-67
55 Christopher D. Karlgaard and Frederik H. Lutze. Second-Order Equations for Rendezvous in a Circular Orbit . J. Guidance. 2004, 27(3): 499-501
56荆武兴,吴瑶华,王学孝.关于Hill方程的一点思考.哈尔滨工业大学学报. 1994, 26(2): 120-123
57陈建祥. Hill方程在远距离拦截中的应用.国防科技大学学报. 1994, 16(2): 61-66
58谌颖,黄文虎.多冲量最优交会的动态规划方法.宇航学报. 1993, 14(2):1-7
59谌颖,王旭东.水平冲量作用下共面椭圆轨道上航天器的交会.控制工程. 1993, 1(2): 9-18
60梁新刚,杨涤.应用非线性规划求解异面最优轨道转移问题.宇航学报. 2006, 27(3): 363-368
61 S. P. Hughes, L. M. Mailhe. A Comparision of Trajectory Optimization Methods for the Impulsive Minimum Fuel Rendezvous Problem. Advances in the Astronautical Sciences. 2003, 113:85-104
62赵宇,张洪华.一种多脉冲轨道规划的智能方法.控制工程. 2005, 4:1-9
63 Young Ha Kim and David B. Spencer. Optimal Spacecraft Rendezvous Using Genetic Algorithms. Jounal of Spacecraft and Rockets. 2002, 39(6): 859-865
64 Ossama Abdelkhalik, Daniele Mortari. N-Impulse Orbit Transfer Using Genetic Algorithms. J of Guidance Control and Dynamics. 2007, 44(2): 456-459
65王石等.用进化算法求解轨道转移的时间-能量优化问题.宇航学报. 2002, 23(1): 22-24
66王石,文援兰,戴金海.用进化-模拟退火算法求解Lambert方程.系统仿真学报. 2007, 19(2): 450-452
67 Dawei Zhang, Shenmin Song, Guangren Duan. Fuel and Time Optimal Transfer of Spacecrafers Rendezvous Using Lambert’s Theorem and Improved Genetic Algorithm. IEEE Xplore. 2008:1-5
68 Luo Yazhong, Lei Yongjun, Tang guojin. Remarks on a Benchmark Nonlinear Constrained Optimization Problem. Journal of System Science and Eletronics. 2006, 17(3):551-553
69 Luo Yazhong, Li Haiyang, Tang Guojin. Hybrid Approach to Optimize a Rendezvous Phasing Strategy. Journal of Guidance, Control and Dynamics. 2007, 30(1):185-191
70 Luo Yazhong, Tang Guojin, Li Haiyang. Optimization of Multi-Impulse Minimum-Time Rendezvous Using a Hybrid Genetic Algorithms. Aerospace Science and Technoloty. 2006, 10(6): 534-540
71 Luo Yazhong, Tang Guojin, Wang Zhonggui, etc. Optimization of Perturbedand Constrained Fuel-Optimal Impulse Rendezvous Using a Hybrid Approach. Engineering Optimization. 2006, 38(8): 959-973
72 Luo Yazhong, Tang Guojin, Zhou Lini. Simulated Annealing for Solving Near-Optimal Low-Thrust Orbit Transfer. Engineering Optimization. 2005, 37(2): 201-216
73 Luo Yazhong, Tang Aerospace Science and Technoloty. Spacecraft Optimal Rendezvous Controller Design Using Simulated Annealing. Aerospace Scinence and Technology. 2005, 9(8): 632-637
74 Luo Yazhong, Tang Guojin. Guojin. Parallel Simulated Annealing Using Simplex Mehtod. AIAA Journal. 2006, 44(12): 3143-3146
75 Luo Yazhong, Tang Guojin. Rendezvous Phasing Special-Point Maneuver Mixed Constinuous-Discrete Optimization Using Simulated Annealing. Aerospace Science and Technology. 2006, 10(7): 652-658
76 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimal Multi-Objective Linearzed Impulsive Rendezvsou. J. of Guidance Control and Dynamics. 2007, 30(2): 383-389
77 Ya-Zhong Luo, Guo-Jin Tang, Geoff Parks. Multi-Objective Optimization of Perturbed Impulsive Rendezvous Trajectories Using Physical Programming. J. of Guidance Control and Dynamics. 2008, 31(6): 1829-1832
78 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimal Multi-Objective Nonlinear Impulsive Rendezvous. J of Guidance Control and Dynamics. 2007, 30(4): 994-1002
79 Ya-Zhong Luo, Guo-Jin Tang, and Yong-Jun Lei. Optimization of Multiple-Impulse Multiple-Renvolution Rendezvsou-Phasing Maneuvers. J. of Guidance Control and Dynamics. 2007, 30(4): 946-952
80李晨光,肖业伦.多脉冲C-W交会的优化方法.宇航学报. 2006, 27(2): 172-177
81卢山,陈统,徐世杰.基于自适应模拟退火遗传算法的最优Lambert转移.北京航空航天大学学报. 2007, 33(10): 1191-1195
82陈统,徐世杰.基于遗传算法的最优Lambert双脉冲转移.北京航空航天大学学报. 2007, 33(3): 273-277
83范淑梅.基于自适应遗传算法的航天器快速轨道机动研究.硕士学位论文.山东大学. 2008
84秦帅.遗传算法在航天器轨道机动中的应用研究.硕士学位论文.哈尔滨工业大学. 2007
85张洪波,郑伟,汤国建.混合遗传算法在远程交会轨道设计中的应用.航天控制. 2006, 24(2): 34-37
86齐映红,曹喜滨.三脉冲最优交会问题的解法.吉林大学学报. 2006, 36(4): 608-612
87张秋华,韩琦,孙毅.基于模糊推力的基向量曲线求解方法.宇航学报. 2007, 28(4): 1002-1006
88 D. F. Lawden. Optimal Trajectories for Space Navigation . London Butter Worths. 1963
89 P. M. Lion, M. Handelsman. Primer Vector on Fixed-Time Impulsive Trajectories . AIAA Journal. 1968, 6(1): 127-132
90 David R. Glandorf. Lagrange Multipliers and the State Transition Matrix for Coasting Arcs . AIAA Journal. 1969, 7(2): 363-365
91 David R. Glandorf. Primer Vector Theory for Matched-Conic Trajectories. AIAA Journal. 1970, 8(1):155-156
92 J. E. Prussing. Optimal Impulsive Linear Systems: Sufficient Conditions and Maximum Number of Impulse . Journal of the Astronautical Science. 1995, 43(2): 195-206
93 Thomas E. Carter. Necessary and Sufficient Conditions for Optimal Impulsive Rendezvous with Linear Equations of Motion. Dynamics and Control. 2000, 10: 219-227
94 D.J. Jezewski and H. L. Rozenkdal. An Efficient Method for Calculating Optimal Free-Space N-Impulse Trajecteries. AIAA Journal. 1968, 6(11): 2160-2165
95 Larry R. Gross, John E. Prussing. Optimal Multiple-Impulse Direct Ascent Fixed-Time Rendezvous. AIAA Journal. 1974,12(7): 885 - 890
96 John E. Prussing, Jeng-Hua Chiu. Optimal Multiple-Impulse Time-Fixed Rendezvous Between Circular Orbits . J.Guidance. 1986, 9(1): 17-25
97 Der-Ren Taur, Victoria coverstone Carroll, John E. Prussing. Optimal Impulsive Time-Fixed Orbital Rendezvous and Interception with Path Constraints. Journal of Guidance, Control, and Dynamics. 1995, 18(1): 54–60
98 Der-Ren Taur. Optimal Impulsive Time-Fixed Orbital Rendezvous and Interception with Path Constraints. Ph. D. Thesis, The University of Illinois at Urbana-Champaign, 1989
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100 John E. Prussing, Linda J. Wellnitz, Willian G. Hechathorn. Optimal impulsive time-fixed direct-ascent interception . J. Guidance. 1989, 12(4): 487–494
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