有限推力作用下航天器轨道优化设计方法研究
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摘要
随着空间技术的发展和成熟,外层空间的军事价值越来越明显,为保持空间优势,必然产生空间对抗。在此背景下,除了具备“进入空间”与“利用空间”的能力之外,还必须具备强有力的“控制空间”能力,才能保证空间资源不受侵犯。而轨道机动技术与先进的轨道优化设计方法是完成各种空间对抗任务的基本保障之一。因此,有必要对先进的轨道优化设计方法进行研究。
本文以近些年来美国空间对抗系统试验为背景,对有限推力作用下的航天器的轨道优化设计方法与应用进行了研究,主要的研究内容包括以下三个方面:
为了避免求解过程中的奇异,建立了改进春分点坐标形式的航天器轨道动力学方程;对地球引力线性化,建立了编队飞行相对动力学方程;利用庞特里亚金极值原理推导了基于上述两种动力学方程的最优控制模型。本文给出三种求解轨道优化设计问题的方法(间接法、混合法和高斯伪谱方法)及求解思路。
针对有限推力作用下的轨道优化设计问题,分别用间接法、混合法和高斯伪谱方法求解几类典型的轨迹优化问题。考虑到间接法收敛的困难性,先采用高斯伪谱方法求解,之后再将高斯伪谱方法得到协状态值代入间接法,以此加快间接法计算速度并改善其收敛性。利用混合法和高斯伪谱方法成功求解了轨道提升、大倾角改变、轨道根数大范围改变以及低轨道向GPS卫星轨道转移的轨道优化问题,仿真结果表明高斯伪谱方法对求解复杂轨道优化问题具有较强的适应能力。
根据相对动力学原理,针对圆轨道下编队飞行,分析和设计几种特殊的编队队形,获得了形成圆轨道下编队飞行所必须具备的初始条件。然后,利用高斯方法给出了一个空间圆编队飞行的轨道设计方法。
With the development and mature of space technology, the martial value of outer space is becoming more and more evident. In order to keep space superiority, space competition is inevitable. In the space competition background, only the ability of entering space and the ability of utilizing space are not enough. What is more important is the power of control space to protect space resources from invasion and to maintain space superiority. Advanced orbit maneuver technique is one of the most important guarantees to complete space competition missions and keep space superiority. It is necessary to study the advanced trajectory optimization techniques and methods.
Under USA’s space competition system demonstration test background, the methods and application of orbit optimization design were studied aiming at spacecraft with finite thrust. The work mainly contained the following three parts:
In order to avoid the singular phenomenon on the solution process, spacecraft orbit kinetics equations of Modified Equinoctial Elements was derived. The relative kinetics equations for formation flying was establishhed with the linear earth gravitation model. According to pontryagin minimum principle, the optimum control model was founded based on above two kinetics equations. Three kinds of trajectory optimization methods including Indirect method, Hybrid method and Gauss pseudospectral method, as well as solving thoughts were given in this paper.
Some typical trajectory optimization problems aiming at finite thrust effect by using Indirect method, Hybrid method and Gauss pseudospectral method respectively. Because it was difficult to enable Indirect Method to converge, Gauss pseudospectral method was used to get costates and then those costates were substituted in the Indirect method to speed up calculation and improve convergence. Hybrid method and Gauss pseudospectral method were used to solve following problems: orbit raising from LEO to GEO, large inclination change from LEO to SSO, orbital elements large-scale change from MEO to Molniya orbit, orbit maneuver from LEO to GPS orbit. It was indicated that Gauss Pseudospetral method had more adaptive to deal with complicated trajectory optimization problem. According to relative kinetics theory, some special formations were analyzed and designed for the formation flying in the case of circular orbit and then obtain initial conditions of circle formation flying. Finally, applying Gauss pseudospectral method give an example of orbit design for circular formation flying.
本文以近些年来美国空间对抗系统试验为背景,对有限推力作用下的航天器的轨道优化设计方法与应用进行了研究,主要的研究内容包括以下三个方面:
为了避免求解过程中的奇异,建立了改进春分点坐标形式的航天器轨道动力学方程;对地球引力线性化,建立了编队飞行相对动力学方程;利用庞特里亚金极值原理推导了基于上述两种动力学方程的最优控制模型。本文给出三种求解轨道优化设计问题的方法(间接法、混合法和高斯伪谱方法)及求解思路。
针对有限推力作用下的轨道优化设计问题,分别用间接法、混合法和高斯伪谱方法求解几类典型的轨迹优化问题。考虑到间接法收敛的困难性,先采用高斯伪谱方法求解,之后再将高斯伪谱方法得到协状态值代入间接法,以此加快间接法计算速度并改善其收敛性。利用混合法和高斯伪谱方法成功求解了轨道提升、大倾角改变、轨道根数大范围改变以及低轨道向GPS卫星轨道转移的轨道优化问题,仿真结果表明高斯伪谱方法对求解复杂轨道优化问题具有较强的适应能力。
根据相对动力学原理,针对圆轨道下编队飞行,分析和设计几种特殊的编队队形,获得了形成圆轨道下编队飞行所必须具备的初始条件。然后,利用高斯方法给出了一个空间圆编队飞行的轨道设计方法。
With the development and mature of space technology, the martial value of outer space is becoming more and more evident. In order to keep space superiority, space competition is inevitable. In the space competition background, only the ability of entering space and the ability of utilizing space are not enough. What is more important is the power of control space to protect space resources from invasion and to maintain space superiority. Advanced orbit maneuver technique is one of the most important guarantees to complete space competition missions and keep space superiority. It is necessary to study the advanced trajectory optimization techniques and methods.
Under USA’s space competition system demonstration test background, the methods and application of orbit optimization design were studied aiming at spacecraft with finite thrust. The work mainly contained the following three parts:
In order to avoid the singular phenomenon on the solution process, spacecraft orbit kinetics equations of Modified Equinoctial Elements was derived. The relative kinetics equations for formation flying was establishhed with the linear earth gravitation model. According to pontryagin minimum principle, the optimum control model was founded based on above two kinetics equations. Three kinds of trajectory optimization methods including Indirect method, Hybrid method and Gauss pseudospectral method, as well as solving thoughts were given in this paper.
Some typical trajectory optimization problems aiming at finite thrust effect by using Indirect method, Hybrid method and Gauss pseudospectral method respectively. Because it was difficult to enable Indirect Method to converge, Gauss pseudospectral method was used to get costates and then those costates were substituted in the Indirect method to speed up calculation and improve convergence. Hybrid method and Gauss pseudospectral method were used to solve following problems: orbit raising from LEO to GEO, large inclination change from LEO to SSO, orbital elements large-scale change from MEO to Molniya orbit, orbit maneuver from LEO to GPS orbit. It was indicated that Gauss Pseudospetral method had more adaptive to deal with complicated trajectory optimization problem. According to relative kinetics theory, some special formations were analyzed and designed for the formation flying in the case of circular orbit and then obtain initial conditions of circle formation flying. Finally, applying Gauss pseudospectral method give an example of orbit design for circular formation flying.
引文
1杨乐平.空间攻防对抗系统体系与概念研究. 863航天航空技术. 2002, 5: 20~27
2于小红.空间平台在空间对抗中的应用研究.装备指挥技术学院学报. 2003, 14(4):35~38
3程芳,管清波,杨勇.空间攻防作战武器分配优化模型研究.装备指挥技术学院学报. 2005, 16(2): 48~50
4闻新,李东江.美国自主交会技术试验卫星.中国航天. 2006, 29(12):31~34.
5吴勤,高雁翎.独霸太空—美国空间对抗装备发展迅速.太空探索. 2007, (28)6:16~19
6美空军XSS-11微卫星通过首次关键性系列试验. http://news.163.com/05/091 2/13/1TF1BEU400011235.html
7 M. Martin and M. J. Stallard. Distributed Satellite Missions and Technologies– The TechSat21 Program. Space Technology Conference and Exposition, Albuquerque, New Mexico, Sep. 1999: AIAA 99-4479
8 F. W Gobetz, J. R. Doll. A Survey of Impulsive Trajectories. AIAA Journal, 1969,7(5):801~834
9 D. F. Lawden. Optimal Trajectories for Space Navigation. Butterworths, 1963
10 J. T. Betts. Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control, and Dynamics. 1998, 21(2):193~207
11 J. T. Betts. Optimal Interplanetary Orbit Transfers by Direct Transcription. The Journal of the Astronautical Sciences. 1994:42(3):247~268
12雍恩米,陈磊,唐国金.飞行器轨迹优化数值方法综述.宇航学报. 2008, 29(2): 2~5
13 T. Goodson. Fuel-Optimal Control and Guidance for Low-and Medium-Thrust Orbit Transfer. Ph.D Dissertation of Georgia Institute of Techonlogy. 1995:2-16
14 C. H Chuang and J. L. Speyer. Periodic Optimal Hypersonic SCRAMjet Cruise. Optimal Control Applications and Models. 1987, 8(3):231~242
15 C. H. Chuang, T. Goodson and L. A. Ledsinger. The Second Variation and Neighboring Optimal Feedback Guidance for Multiple Burn Orbit Transfers. Proceedings of the 1995 AIAA Conference on Guidance, Navigation, andControl. Baltimore, Maryland, USA
16 R. Roberto and F. Topputo. A sitth-order accurate scheme for solving two-point boundary value problems in astrodynamics. Celestial Mechanics and Dynamical Astronomy. 2006, 96:289~309
17 P. N. Desai and B.A. Conway. A Two-Timescale Discretization Scheme for Collocation. Advances in the Astronautical Sciences. 2005, 119(2):2053~2063
18 P. J. Enright and B. A. Conway. Optimal Finite-Thrust Spacecraft Trajectores Using Collocation and Nonlinear Programming. Journal of Guidance, Control and Dyanmics. 1991, 14(5):981~985
19 K. P. Zondervan, L. J. Wood and T. K. Caughey. Optimal Low-Thrust Three-Burn Orbit Transfer with Large Plane Changes. Journal of Astronautical Science. 1984, 32(3):407~427
20 M. R. Ilgen. Hybrid Method for Computing Optimal Low Thrust OTV Trajectories. Advances in the Astronautical Sciences. 1999, 87(2):941~958
21 B. Wall, B. A. Conway. Near-Optimal Low-Thrust Earth-Mars Trajectories via a Genetic Algorithm. Journal of Guidance, Control and Dynamics. 2005, 28(5):1027~1031
22王明春,荆武兴等.能量最省有限推力同平面轨道转移.宇航学报,1992,13(3):24~31
23王小军等.最省燃料消耗的固定推力共面轨道变轨研究.宇航学报,1995,16(4):9~15
24王小军.地球同步卫星远地点最省燃料小推力多次变轨研究.中国空间科学技术,1995,15(4):1~9
25严辉,吴宏鑫等.小推力轨道优化研究.中国空间科学技术,1998,18(2):8~12
26荆武兴,吴瑶华.卫星轨道圆化的有限推力点火控制策略.宇航学报,1997 18 (4):2~7
27荆武兴,吴瑶华.基于交会概念的最省燃料异面有限推力转移轨道研究.哈尔滨工业大学学报,1998,44(2):124~128
28王劼等.小推力登月飞行器轨道初步研究.飞行力学,2002,20(3):46~49
29王志刚等.形成三星星座的小推力变轨的时间最短控制.宇航学报,2001, 22(6):35~39
30王志刚等.伴飞卫星最偶小推力释放与回收控制.宇航学报,2003,24(3): 294~297
31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
32任远,崔平远等.基于退火遗传算法的小推力轨道优化问题研究.宇航学报,2007,28(1):162~166
33 R. A. Broucke, P. J. Cefola. On the Equinoctial Orbit Elements. Celestial Mechanics. 1972, 5: 303~310
34 R. H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics. Revised edition. AIAA, 1999: 490~492
35 M. J. H. Walker, B. Ireland, J. Owens. A Set of Modified Equinoctial Orbit Elements. Celestial Mechanics. 1985, 36: 409~419
36 J. A. Kechichian. Minimum-Time Low-Thrust Rendezvous and Transfer Using Epoch Mean Longitude Formulation. Journal of Guidance, Control and Dynamcis. 1999, 22(3):421~432
37 R. H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics. Revised edition. AIAA, 1999: 494
38郗晓宁,任萱等著.近地航天器轨道基础.长沙:国防科技大学出版社,2003年
39 T. Goodson. Fuel-Optimal Control and Guidance for Low-and Medium-Thrust Orbit Transfer. Ph.D. Dissertation, Georgia Institute of Technology. 1995:2
40 R. R. Bate, D. D. Mueller, J. E. White. Fundamentals of astrodynamics. Dover publications Inc., 1971:40～43
41程国采.航天飞行器最优控制理论与方法.北京:国防工业出版社,1999年
42 M. R. Ilgen. Hybrid Method for Computing Optimal Low Thrust OTV Trajectories. Advances in the Astronautical Sciences. 1994, 87(2): 941~958
43 D. A. Benson. A Gauss Pseudospectral Transcription for Optimal Control. Ph.D. Dissertation, Massachusetts Institute of Technology, 2005
44 D. A. Benson, G. T. Huntington, T. P. Thorvaldsen and A. V. Rao. Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method. Journal of Guidance, Control and Dynamics, 2006, 29(6):1435~1440
45 F. Fahroo, M. Ross. Costate Estimation by a Legendre Pseudospectral Method. Journal of Guidance, Control and Dynamics, 2001, 24(2):270~277
46 I. M. Ross, F. Fahroo. A Pseudospectral Transformation of the Covectors of Optimal Control Systems. Proceedings of the 1st IFAC/IEEE Symposium on Structure and Control, Prague, Czech Republic, August 2001
47 I. M. Ross, F. Fahroo. A Perspective on Methods for Trajectory Optimization. Proceedings of the AIAA/AAS Astrodynamics Conference, Monterey, CA., August 2002
48梁新刚.面向两类空间对抗任务的轨道优化设计方法研究.哈尔滨工业大学博士论文. 2008: 33~48
49王鹏基.空间飞行器编队飞行相对动力学与队形控制方法及应用研究.哈尔滨工业大学博士论文. 2004: 34~38
2于小红.空间平台在空间对抗中的应用研究.装备指挥技术学院学报. 2003, 14(4):35~38
3程芳,管清波,杨勇.空间攻防作战武器分配优化模型研究.装备指挥技术学院学报. 2005, 16(2): 48~50
4闻新,李东江.美国自主交会技术试验卫星.中国航天. 2006, 29(12):31~34.
5吴勤,高雁翎.独霸太空—美国空间对抗装备发展迅速.太空探索. 2007, (28)6:16~19
6美空军XSS-11微卫星通过首次关键性系列试验. http://news.163.com/05/091 2/13/1TF1BEU400011235.html
7 M. Martin and M. J. Stallard. Distributed Satellite Missions and Technologies– The TechSat21 Program. Space Technology Conference and Exposition, Albuquerque, New Mexico, Sep. 1999: AIAA 99-4479
8 F. W Gobetz, J. R. Doll. A Survey of Impulsive Trajectories. AIAA Journal, 1969,7(5):801~834
9 D. F. Lawden. Optimal Trajectories for Space Navigation. Butterworths, 1963
10 J. T. Betts. Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control, and Dynamics. 1998, 21(2):193~207
11 J. T. Betts. Optimal Interplanetary Orbit Transfers by Direct Transcription. The Journal of the Astronautical Sciences. 1994:42(3):247~268
12雍恩米,陈磊,唐国金.飞行器轨迹优化数值方法综述.宇航学报. 2008, 29(2): 2~5
13 T. Goodson. Fuel-Optimal Control and Guidance for Low-and Medium-Thrust Orbit Transfer. Ph.D Dissertation of Georgia Institute of Techonlogy. 1995:2-16
14 C. H Chuang and J. L. Speyer. Periodic Optimal Hypersonic SCRAMjet Cruise. Optimal Control Applications and Models. 1987, 8(3):231~242
15 C. H. Chuang, T. Goodson and L. A. Ledsinger. The Second Variation and Neighboring Optimal Feedback Guidance for Multiple Burn Orbit Transfers. Proceedings of the 1995 AIAA Conference on Guidance, Navigation, andControl. Baltimore, Maryland, USA
16 R. Roberto and F. Topputo. A sitth-order accurate scheme for solving two-point boundary value problems in astrodynamics. Celestial Mechanics and Dynamical Astronomy. 2006, 96:289~309
17 P. N. Desai and B.A. Conway. A Two-Timescale Discretization Scheme for Collocation. Advances in the Astronautical Sciences. 2005, 119(2):2053~2063
18 P. J. Enright and B. A. Conway. Optimal Finite-Thrust Spacecraft Trajectores Using Collocation and Nonlinear Programming. Journal of Guidance, Control and Dyanmics. 1991, 14(5):981~985
19 K. P. Zondervan, L. J. Wood and T. K. Caughey. Optimal Low-Thrust Three-Burn Orbit Transfer with Large Plane Changes. Journal of Astronautical Science. 1984, 32(3):407~427
20 M. R. Ilgen. Hybrid Method for Computing Optimal Low Thrust OTV Trajectories. Advances in the Astronautical Sciences. 1999, 87(2):941~958
21 B. Wall, B. A. Conway. Near-Optimal Low-Thrust Earth-Mars Trajectories via a Genetic Algorithm. Journal of Guidance, Control and Dynamics. 2005, 28(5):1027~1031
22王明春,荆武兴等.能量最省有限推力同平面轨道转移.宇航学报,1992,13(3):24~31
23王小军等.最省燃料消耗的固定推力共面轨道变轨研究.宇航学报,1995,16(4):9~15
24王小军.地球同步卫星远地点最省燃料小推力多次变轨研究.中国空间科学技术,1995,15(4):1~9
25严辉,吴宏鑫等.小推力轨道优化研究.中国空间科学技术,1998,18(2):8~12
26荆武兴,吴瑶华.卫星轨道圆化的有限推力点火控制策略.宇航学报,1997 18 (4):2~7
27荆武兴,吴瑶华.基于交会概念的最省燃料异面有限推力转移轨道研究.哈尔滨工业大学学报,1998,44(2):124~128
28王劼等.小推力登月飞行器轨道初步研究.飞行力学,2002,20(3):46~49
29王志刚等.形成三星星座的小推力变轨的时间最短控制.宇航学报,2001, 22(6):35~39
30王志刚等.伴飞卫星最偶小推力释放与回收控制.宇航学报,2003,24(3): 294~297
31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
32任远,崔平远等.基于退火遗传算法的小推力轨道优化问题研究.宇航学报,2007,28(1):162~166
33 R. A. Broucke, P. J. Cefola. On the Equinoctial Orbit Elements. Celestial Mechanics. 1972, 5: 303~310
34 R. H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics. Revised edition. AIAA, 1999: 490~492
35 M. J. H. Walker, B. Ireland, J. Owens. A Set of Modified Equinoctial Orbit Elements. Celestial Mechanics. 1985, 36: 409~419
36 J. A. Kechichian. Minimum-Time Low-Thrust Rendezvous and Transfer Using Epoch Mean Longitude Formulation. Journal of Guidance, Control and Dynamcis. 1999, 22(3):421~432
37 R. H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics. Revised edition. AIAA, 1999: 494
38郗晓宁,任萱等著.近地航天器轨道基础.长沙:国防科技大学出版社,2003年
39 T. Goodson. Fuel-Optimal Control and Guidance for Low-and Medium-Thrust Orbit Transfer. Ph.D. Dissertation, Georgia Institute of Technology. 1995:2
40 R. R. Bate, D. D. Mueller, J. E. White. Fundamentals of astrodynamics. Dover publications Inc., 1971:40～43
41程国采.航天飞行器最优控制理论与方法.北京:国防工业出版社,1999年
42 M. R. Ilgen. Hybrid Method for Computing Optimal Low Thrust OTV Trajectories. Advances in the Astronautical Sciences. 1994, 87(2): 941~958
43 D. A. Benson. A Gauss Pseudospectral Transcription for Optimal Control. Ph.D. Dissertation, Massachusetts Institute of Technology, 2005
44 D. A. Benson, G. T. Huntington, T. P. Thorvaldsen and A. V. Rao. Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method. Journal of Guidance, Control and Dynamics, 2006, 29(6):1435~1440
45 F. Fahroo, M. Ross. Costate Estimation by a Legendre Pseudospectral Method. Journal of Guidance, Control and Dynamics, 2001, 24(2):270~277
46 I. M. Ross, F. Fahroo. A Pseudospectral Transformation of the Covectors of Optimal Control Systems. Proceedings of the 1st IFAC/IEEE Symposium on Structure and Control, Prague, Czech Republic, August 2001
47 I. M. Ross, F. Fahroo. A Perspective on Methods for Trajectory Optimization. Proceedings of the AIAA/AAS Astrodynamics Conference, Monterey, CA., August 2002
48梁新刚.面向两类空间对抗任务的轨道优化设计方法研究.哈尔滨工业大学博士论文. 2008: 33~48
49王鹏基.空间飞行器编队飞行相对动力学与队形控制方法及应用研究.哈尔滨工业大学博士论文. 2004: 34~38