基于多任务多目标的空天飞行器轨迹设计及优化研究
|
摘要
空天飞行器结合了航空航天技术,具有飞行速度快、重复使用率高和功能齐全的优点,使其不仅具有多种民用用途而且还具有较高的军事价值,它的推出势必会引起新一轮的军事革命。基于此,本文重点研究了空天飞行器的轨迹设计与优化问题。
空天飞行器轨迹优化问题本质上是一类最优控制问题,针对该问题,本文提出模型变换和最优化计算的分步求解思路,分别讨论每一步中不同方法的优缺点,并给出求解步骤。建立了空天飞行器的三自由度运动模型,并精确分析影响最优轨迹精度的环境模型、大气密度模型、声速模型和地球引力模型等。
研究了空基投放型空天飞行器的规避飞行轨迹优化问题,根据任务把飞行过程分为上升段和巡航段,提出了分段建模,分段优化的思想。针对传统的间接法最优性条件推导繁琐,最优轨迹对初值比较敏感的缺陷,首先利用直接配置法将无限维的动态优化问题转换成有限维的静态优化问题,即非线性规划问题,然后考虑到非线性规划问题约束条件复杂,可行域较窄,提出了逐步细分的优化策略,并利用序列二次规划算法对其求解,最后进行了仿真验证和误差分析。
研究了空天飞行器的滑翔轨迹优化问题,提出了基于Legendre伪谱法的模型变换方法,设计了遗传算法和序列二次规划相结合的串行组合优化策略,通过计算轨迹安全走廊得到了初始参考轨迹,加快了收敛速度,最后基于最远航程性能指标的仿真实例说明了Legendre伪谱法同样可以很好地处理轨迹优化问题,设计的串行组合优化策略计算效率高,切实有效。
As the combination of aviation technology and space technology, aerospace vehicle has the advantages of fast flight speed, high reuse rate, strong mobility and complete functions, such that it not only has many civilian uses, but also has high military, and may cause a new round of military revolution. Therefore, this paper focuses on the trajectory design and optimization of aerospace vehicle.
In essence, trajectory optimization is regarded as an optimal control problem. For this problem, the solving ideas of model transformation and optimization calculation are proposed in this paper. And the advantages and disadvantages of different methods in each step are discussed, then further the solution procedure is given.
The 3DOF motion model of aerospace vehicle is established, and they are analyzed accurately to environment model, atmospheric model, sound velocity model and gravity model affecting the the optimal trajectory precision.
In this paper, the dodge-flying trajectory optimization problem for an air-based aerospace vehicle is studied. According to mission requirements, the whole flight is divided into two segments: ascent phase and cruise phase, and the ideas of modeling and optimization by step are proposed. For the traditional indirect methods, whose optimal conditions derive cumbersome and whose optimal trajectory are sensitive to the initial values, firstly the direct collocation methods are used totransfer the infinite dimensional dynamic optimal problem into finite dimensional static optimal problem, namely nonlinear programming problem (NLP). Secendly considering that the constraints of the NLP problem are complex and the feasible region is narrow, a sequential-division optimization strategy is proposed to solve the optimal problem using sequential quadratic programming (SQP), finally then the simulation and error analysis are done.
The glide trajectory optimization problems for aerospace vehicle are studied, and the new model transforming method based on the Legendre pseudospectral method is developed, then further a serial hybrid optimization strategy combined genetic algorithm with SQP is designed. By calculating the safe corridors, the initial reference trajectory is obtained, which can speed up the convergence. Finally, the simulation results based on the performance of maximum range demonstrate that the Legendre pseudo-spectral method can also deal with trajectory optimization problems well, and the serial hybrid optimization strategy is efficient and feasible.
空天飞行器轨迹优化问题本质上是一类最优控制问题,针对该问题,本文提出模型变换和最优化计算的分步求解思路,分别讨论每一步中不同方法的优缺点,并给出求解步骤。建立了空天飞行器的三自由度运动模型,并精确分析影响最优轨迹精度的环境模型、大气密度模型、声速模型和地球引力模型等。
研究了空基投放型空天飞行器的规避飞行轨迹优化问题,根据任务把飞行过程分为上升段和巡航段,提出了分段建模,分段优化的思想。针对传统的间接法最优性条件推导繁琐,最优轨迹对初值比较敏感的缺陷,首先利用直接配置法将无限维的动态优化问题转换成有限维的静态优化问题,即非线性规划问题,然后考虑到非线性规划问题约束条件复杂,可行域较窄,提出了逐步细分的优化策略,并利用序列二次规划算法对其求解,最后进行了仿真验证和误差分析。
研究了空天飞行器的滑翔轨迹优化问题,提出了基于Legendre伪谱法的模型变换方法,设计了遗传算法和序列二次规划相结合的串行组合优化策略,通过计算轨迹安全走廊得到了初始参考轨迹,加快了收敛速度,最后基于最远航程性能指标的仿真实例说明了Legendre伪谱法同样可以很好地处理轨迹优化问题,设计的串行组合优化策略计算效率高,切实有效。
As the combination of aviation technology and space technology, aerospace vehicle has the advantages of fast flight speed, high reuse rate, strong mobility and complete functions, such that it not only has many civilian uses, but also has high military, and may cause a new round of military revolution. Therefore, this paper focuses on the trajectory design and optimization of aerospace vehicle.
In essence, trajectory optimization is regarded as an optimal control problem. For this problem, the solving ideas of model transformation and optimization calculation are proposed in this paper. And the advantages and disadvantages of different methods in each step are discussed, then further the solution procedure is given.
The 3DOF motion model of aerospace vehicle is established, and they are analyzed accurately to environment model, atmospheric model, sound velocity model and gravity model affecting the the optimal trajectory precision.
In this paper, the dodge-flying trajectory optimization problem for an air-based aerospace vehicle is studied. According to mission requirements, the whole flight is divided into two segments: ascent phase and cruise phase, and the ideas of modeling and optimization by step are proposed. For the traditional indirect methods, whose optimal conditions derive cumbersome and whose optimal trajectory are sensitive to the initial values, firstly the direct collocation methods are used totransfer the infinite dimensional dynamic optimal problem into finite dimensional static optimal problem, namely nonlinear programming problem (NLP). Secendly considering that the constraints of the NLP problem are complex and the feasible region is narrow, a sequential-division optimization strategy is proposed to solve the optimal problem using sequential quadratic programming (SQP), finally then the simulation and error analysis are done.
The glide trajectory optimization problems for aerospace vehicle are studied, and the new model transforming method based on the Legendre pseudospectral method is developed, then further a serial hybrid optimization strategy combined genetic algorithm with SQP is designed. By calculating the safe corridors, the initial reference trajectory is obtained, which can speed up the convergence. Finally, the simulation results based on the performance of maximum range demonstrate that the Legendre pseudo-spectral method can also deal with trajectory optimization problems well, and the serial hybrid optimization strategy is efficient and feasible.
引文
[1]顾诵芬,史超礼.世界航天发展史[M].郑州:河南科学技术出版社, 2000
[2] Kahler D N. A ground environment to support aerospace vehicle testing in the mid-1970's[C]. AIAA, Aircraft Design for 1980 Operations Meeting, Washington, D.C. 1968.
[3] Wilhite A.W. The aerospace vehicle interactive design system [C]. American Institute of Aeronautics and Astronautics, Aerospace Sciences Meeting, 19th, St. Louis.1981.
[4]陈劲松.超音速和高超音速翼型非定常气动力的一种近似计算方法.空气动力学学报, 1990, 8(8): 339-343.
[5]杨炳渊,宋伟力.前缘激波脱体的大迎角翼面颤振工程计算方法.振动与冲击, 2002, 21(4): 100-103.
[6]刘桐林.国外高超声速技术发展探析.飞航导弹, 2002(6):30～40.
[7]陈予恕,郭虎伦,钟顺.高超声速飞行器若干问题研究进展.飞航导弹, 2009, 8: 25-33.
[8] Moses L P, Rausch L V , Luat T N, et al. NASA hypersonic flight demonstrators-overview: status, and future plans. Astronautic, 2004, 55: 619-631.
[9] Talay A T. The HL-20 personnel launch system [C]. AIAA, Space Programs and Technology Conference, AIAA-92-1416, 1992.
[10] Shaughnessy J D, Pinckney S Z, McMinn J D, et al. Hypersonic vehicle simulation model: winged-cone configuration[R]. NASA TM-102610, 1990.
[11] Keshmiri S, Colgren R, Mirmirani M. Development of an aerodynamic database for a generic hypersonic air vehicle [R]. AIAA-2005-35352, 2005
[12] Hank M J, Murphy S J. The X-51A scream jet engine flight demonstration program. AIAA 2008-2540.
[13] Shachtman N. Hypersonic cruise missile: America's new global strike weapon Popluar Mecheanics, 1, 2007(1)
[14]刘桐林.俄罗斯高超声速技术飞行试验计划(一至四).飞航导弹, 2000, 4-7.
[15]陈延辉.日本的高超声速吸气式发动机试验设备及试验技术.飞航导弹, 2006, 2: 41-48.
[16] Gilmore J F. Autonomous vehicle planning analysis methodology [C]. The Proceedings of Association for Unmanned Vehicles Systems Conference, Washington, D.C.,1991.
[17]吴强.任务规划系统关键技术的研究[D].北京:北京航空航天大学, 2002.
[18]闵昌万,袁建平.军用飞行器航迹规划综述.飞行力学, 1998,16(4): 14-19.
[19] Mcdonough W G.. Tactical flight management-total mission capability [C]. The proceeding of IEEE NAECON,1984.
[20] Hura M, Mchleod G. Route planning issues for low observable aircraft and cruise missiles: Implications for the Intelligence Community[R]. RAND, Santa Monica, CA, 1993.
[21] James L. Pittman F. McNeil C. Hypersonic project overview[C]. Fundamental Aeronautics Program 2008 Annual Meeting. 2008.
[22] Chuang H C, Morimoto H. Periodic optimal cruise for a hypersonic vehicle with constraints, Journal of Spacecraft and Rockets. 1997, Vol. 34, No. 2.
[23] Speyer J L. Periodic optimal flight [J]. Journal of Guidance, 1996, 19(4): 745-75.
[24] Carter P H, Pines D J, Rudd L V. Approximate performance of periodic hypersonic cruise trajectories for global Reach. AIAA 98-1644.
[25] Youssef H. Hypersonic Skipping Trajectory, AIAA 2003--5498.
[26] Hui T L, Ping Y J, Qun F, et al. Recentry skipping trajectory optimization using direct parameter optimization method[C], Space Plans and Hypersonic Systems and Technologies Conference, 2006.
[27] Jorris R T, Cobb G R. Three-dimensional trajectory optimization satisfying waypoint and no-fly zone constraints. Journal of Guidance, Control and Dynamics. 2009, 2(32): 551-572.
[28]黄国强,南英,陆宇平.二级入轨空天飞机上升轨迹优化,宇航学报, 2010,3(31): 641-647.
[29] George R. The common aero vehicle-space delivery system of the future the common aero vehicle-space delivery system of the future[C] AIAA, Space Technology Conference Exposition , Albuquerque ,NM, 1999:1-11.
[30] Richie G. The Common Aero Vehicle: Space Delivery System of the Future [C]. AIAA, Space Technology Conference and Exposition, 1999, (9): 35-44.
[31]雍恩米,陈磊,唐国金.基于物理规划的高超声速飞行器滑翔式再入轨迹优化.航空学报, 2008, 29(5): 1092-1097.
[32] Istratie V. Optimal profound entry into atmosphere with minimum heat and constraints [C]. AIAA, Atmospheric Flight Mechanics Conference and Exhibit, Montreal, Canada, 2001.
[33] Betts J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(2): 193 - 206.
[34] Bellman R E. Dynamic programing [M].Princeton, USA: Princeton University Press, 1957.
[35]张之胚,李建德.动态规划及其应用[M].北京:国防工业出版社, 1994.
[36] Elnagar J, Kazemi M A.. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems [J]. Computational Optimization and Applications, 1998 (11): 195-217.
[37] Hitoshi Morimoto, Chuang Jason C H, Minimum-fuel trajectory along entire profile for a hypersonic vehicle with constraint[R]. AIAA, 3998-4122, 1998.
[38]周浩,陈万春.高超声速飞行器滑行轨迹优化[J].北京航空航天大学学报, 2009, 32(5): 513-517.
[39]刘同仁.用参数最优化方法计算最优飞行轨迹[J].航空学报, 1994, 15(11): 1298-1305.
[40] Hull G D, Speyer L J. Hypersonic vehicle trajectory optimization [R], AFNAL-TR-81-3034 (ADA101538), 1981, 5.
[41] Robert T. Trajectory optimization for a fixed-trim reentry vehicle using direct collocation and nonlinear programming [C]. AIAA, Guidance, Navigation, and Control Conference and Exhibit, Denver, USA, 2000
[42] David G. Conversion of optimal control problems into parameter optimization problems [J]. Journal of Guidance, Control, and Dynamics, 1997, 20(1): 57-60.
[43] Hargraves C R. Direct trajectory optimization using nonlinear programming and collocation [J]. Journal of Guidance, Control, and Dynamics, 1987, 10(4): 338-342.
[44] Albert L, Bruce A. Direct optimization using collocation based on high-order gauss-lobatto Quadrature rules[J]. Journal of Guidance, Control, and Dynamics, 1996, 19(3): 592-599.
[45] Paul J, Bruce A. Optimal spacecraft trajectories using collocation and nonlinear programming [J]. Journal of Guidance, Control, and Dynamics, 1991, 14(5): 981-985.
[46]涂良辉,袁建平,罗建军等.基于直接配点法的月球软着陆轨道快速优化[J].中国空间科学技术, 2008,4: 19-24.
[47] Fahroo F, Ross M I. Costate estimation by a legendre pesudospectral method [J]. Journal of Guidance, Control and Dynamics, 2001, 24(2): 270-277.
[48] Yan H, Ross M I, Alfriend T F. Pseudospectral feedback Control for Three-Axis Magnetic Attitude Stabilization in Elliptic Orbits [J]. Journal of Guidance, Control and Dynamics, 2007, 30(4): 1107-1115.
[49]赵汉元.飞行器再入动力学和制导[M].长沙:国防科技大学出版社, 1997.
[50] Stoer J, Bulisch R. Introduction to numerical analysis[M]. Springer-Verlag, New Work, Heidelberg 1980:382-496.
[51] Hopfield J J. Neural networks and physical systems with emergent collective computational abilities [J]. Proc. of the national academy of science, 1982, 179(8): 2554-2558.
[52]高隽.人工神经网络原理与仿真实例[M].北京:机械工业出版社, 2003.
[53] Kirkpatrick S, Gelatt J C, Vecchi P. Optimization by simulated annealing [J]. Science, 1983: 670-671
[54] Ansari N, Hou E. Computational intelligence for optimization [M]. Kluwer Academic Publishers, 1997.
[55] Hullavid G, Speyerson L. Optimal reentry and plane-change trajectory [J]. The journal of the astronautical sciences, 1982, (2): 117-130.
[56]孙文瑜.最优化方法[M].北京,高等教育出版社, 2004.
[57]袁亚湘.非线性优化计算方法[M].北京:科学出版社, 2008
[58] Ross I M, Sekhavat P Fleming A, et al. Optimal feedback control: Foundations, examples and experimental results for a new approach [J]. Journal of Guidance, Control, and Dynamics, 2008, 31(2): 307-321
[59] Hawkins A M, Fill T R, Proulx R J, et al. Constrained trajectory optimization for lunar landing [C]. AAS, 06152, Jan. 2006
[60] Williams P. Application of pseudospectral methods for receding horizon control[J]. Journal of Guidance, Control, and Dynamics. 2004, 27(2): 310-314
[61] Ross I M, Fahroo F. Issues in the real-time computation of optimal control [J]. Mathematical and Computer Modeling. 2006,43(9-10):1172-1188.
[62].Richie G. The Common aero vehicle: space delivery system of the future[C]. AIAA Space Technology Conference and Exposition, 1999, (9): 352-441.
[63] Zimmerman C. , Dukeman G. , Hanson J . An automated method to compute orbital entry reentry trajectories with heating constraints [J]. AIAA, 2002-4454, 2002.
[64] Shen Z, Lu P. On-board generation of three-dimensional constrained entry trajectories [J]. Journal of Guidance, Control, and Dynamics. 2003, 26(1): 110-121.
[65] Roenneke A. Adaptive on-board guidance for entry vehicles [C]. Washington DC: American Institute of Aeronautics and Astronautics, 2001.
[66] Rajech K A. Reentry trajectory optimization: evolutionary approach[C]. AIAA, 2002-5466.
[2] Kahler D N. A ground environment to support aerospace vehicle testing in the mid-1970's[C]. AIAA, Aircraft Design for 1980 Operations Meeting, Washington, D.C. 1968.
[3] Wilhite A.W. The aerospace vehicle interactive design system [C]. American Institute of Aeronautics and Astronautics, Aerospace Sciences Meeting, 19th, St. Louis.1981.
[4]陈劲松.超音速和高超音速翼型非定常气动力的一种近似计算方法.空气动力学学报, 1990, 8(8): 339-343.
[5]杨炳渊,宋伟力.前缘激波脱体的大迎角翼面颤振工程计算方法.振动与冲击, 2002, 21(4): 100-103.
[6]刘桐林.国外高超声速技术发展探析.飞航导弹, 2002(6):30～40.
[7]陈予恕,郭虎伦,钟顺.高超声速飞行器若干问题研究进展.飞航导弹, 2009, 8: 25-33.
[8] Moses L P, Rausch L V , Luat T N, et al. NASA hypersonic flight demonstrators-overview: status, and future plans. Astronautic, 2004, 55: 619-631.
[9] Talay A T. The HL-20 personnel launch system [C]. AIAA, Space Programs and Technology Conference, AIAA-92-1416, 1992.
[10] Shaughnessy J D, Pinckney S Z, McMinn J D, et al. Hypersonic vehicle simulation model: winged-cone configuration[R]. NASA TM-102610, 1990.
[11] Keshmiri S, Colgren R, Mirmirani M. Development of an aerodynamic database for a generic hypersonic air vehicle [R]. AIAA-2005-35352, 2005
[12] Hank M J, Murphy S J. The X-51A scream jet engine flight demonstration program. AIAA 2008-2540.
[13] Shachtman N. Hypersonic cruise missile: America's new global strike weapon Popluar Mecheanics, 1, 2007(1)
[14]刘桐林.俄罗斯高超声速技术飞行试验计划(一至四).飞航导弹, 2000, 4-7.
[15]陈延辉.日本的高超声速吸气式发动机试验设备及试验技术.飞航导弹, 2006, 2: 41-48.
[16] Gilmore J F. Autonomous vehicle planning analysis methodology [C]. The Proceedings of Association for Unmanned Vehicles Systems Conference, Washington, D.C.,1991.
[17]吴强.任务规划系统关键技术的研究[D].北京:北京航空航天大学, 2002.
[18]闵昌万,袁建平.军用飞行器航迹规划综述.飞行力学, 1998,16(4): 14-19.
[19] Mcdonough W G.. Tactical flight management-total mission capability [C]. The proceeding of IEEE NAECON,1984.
[20] Hura M, Mchleod G. Route planning issues for low observable aircraft and cruise missiles: Implications for the Intelligence Community[R]. RAND, Santa Monica, CA, 1993.
[21] James L. Pittman F. McNeil C. Hypersonic project overview[C]. Fundamental Aeronautics Program 2008 Annual Meeting. 2008.
[22] Chuang H C, Morimoto H. Periodic optimal cruise for a hypersonic vehicle with constraints, Journal of Spacecraft and Rockets. 1997, Vol. 34, No. 2.
[23] Speyer J L. Periodic optimal flight [J]. Journal of Guidance, 1996, 19(4): 745-75.
[24] Carter P H, Pines D J, Rudd L V. Approximate performance of periodic hypersonic cruise trajectories for global Reach. AIAA 98-1644.
[25] Youssef H. Hypersonic Skipping Trajectory, AIAA 2003--5498.
[26] Hui T L, Ping Y J, Qun F, et al. Recentry skipping trajectory optimization using direct parameter optimization method[C], Space Plans and Hypersonic Systems and Technologies Conference, 2006.
[27] Jorris R T, Cobb G R. Three-dimensional trajectory optimization satisfying waypoint and no-fly zone constraints. Journal of Guidance, Control and Dynamics. 2009, 2(32): 551-572.
[28]黄国强,南英,陆宇平.二级入轨空天飞机上升轨迹优化,宇航学报, 2010,3(31): 641-647.
[29] George R. The common aero vehicle-space delivery system of the future the common aero vehicle-space delivery system of the future[C] AIAA, Space Technology Conference Exposition , Albuquerque ,NM, 1999:1-11.
[30] Richie G. The Common Aero Vehicle: Space Delivery System of the Future [C]. AIAA, Space Technology Conference and Exposition, 1999, (9): 35-44.
[31]雍恩米,陈磊,唐国金.基于物理规划的高超声速飞行器滑翔式再入轨迹优化.航空学报, 2008, 29(5): 1092-1097.
[32] Istratie V. Optimal profound entry into atmosphere with minimum heat and constraints [C]. AIAA, Atmospheric Flight Mechanics Conference and Exhibit, Montreal, Canada, 2001.
[33] Betts J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998, 21(2): 193 - 206.
[34] Bellman R E. Dynamic programing [M].Princeton, USA: Princeton University Press, 1957.
[35]张之胚,李建德.动态规划及其应用[M].北京:国防工业出版社, 1994.
[36] Elnagar J, Kazemi M A.. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems [J]. Computational Optimization and Applications, 1998 (11): 195-217.
[37] Hitoshi Morimoto, Chuang Jason C H, Minimum-fuel trajectory along entire profile for a hypersonic vehicle with constraint[R]. AIAA, 3998-4122, 1998.
[38]周浩,陈万春.高超声速飞行器滑行轨迹优化[J].北京航空航天大学学报, 2009, 32(5): 513-517.
[39]刘同仁.用参数最优化方法计算最优飞行轨迹[J].航空学报, 1994, 15(11): 1298-1305.
[40] Hull G D, Speyer L J. Hypersonic vehicle trajectory optimization [R], AFNAL-TR-81-3034 (ADA101538), 1981, 5.
[41] Robert T. Trajectory optimization for a fixed-trim reentry vehicle using direct collocation and nonlinear programming [C]. AIAA, Guidance, Navigation, and Control Conference and Exhibit, Denver, USA, 2000
[42] David G. Conversion of optimal control problems into parameter optimization problems [J]. Journal of Guidance, Control, and Dynamics, 1997, 20(1): 57-60.
[43] Hargraves C R. Direct trajectory optimization using nonlinear programming and collocation [J]. Journal of Guidance, Control, and Dynamics, 1987, 10(4): 338-342.
[44] Albert L, Bruce A. Direct optimization using collocation based on high-order gauss-lobatto Quadrature rules[J]. Journal of Guidance, Control, and Dynamics, 1996, 19(3): 592-599.
[45] Paul J, Bruce A. Optimal spacecraft trajectories using collocation and nonlinear programming [J]. Journal of Guidance, Control, and Dynamics, 1991, 14(5): 981-985.
[46]涂良辉,袁建平,罗建军等.基于直接配点法的月球软着陆轨道快速优化[J].中国空间科学技术, 2008,4: 19-24.
[47] Fahroo F, Ross M I. Costate estimation by a legendre pesudospectral method [J]. Journal of Guidance, Control and Dynamics, 2001, 24(2): 270-277.
[48] Yan H, Ross M I, Alfriend T F. Pseudospectral feedback Control for Three-Axis Magnetic Attitude Stabilization in Elliptic Orbits [J]. Journal of Guidance, Control and Dynamics, 2007, 30(4): 1107-1115.
[49]赵汉元.飞行器再入动力学和制导[M].长沙:国防科技大学出版社, 1997.
[50] Stoer J, Bulisch R. Introduction to numerical analysis[M]. Springer-Verlag, New Work, Heidelberg 1980:382-496.
[51] Hopfield J J. Neural networks and physical systems with emergent collective computational abilities [J]. Proc. of the national academy of science, 1982, 179(8): 2554-2558.
[52]高隽.人工神经网络原理与仿真实例[M].北京:机械工业出版社, 2003.
[53] Kirkpatrick S, Gelatt J C, Vecchi P. Optimization by simulated annealing [J]. Science, 1983: 670-671
[54] Ansari N, Hou E. Computational intelligence for optimization [M]. Kluwer Academic Publishers, 1997.
[55] Hullavid G, Speyerson L. Optimal reentry and plane-change trajectory [J]. The journal of the astronautical sciences, 1982, (2): 117-130.
[56]孙文瑜.最优化方法[M].北京,高等教育出版社, 2004.
[57]袁亚湘.非线性优化计算方法[M].北京:科学出版社, 2008
[58] Ross I M, Sekhavat P Fleming A, et al. Optimal feedback control: Foundations, examples and experimental results for a new approach [J]. Journal of Guidance, Control, and Dynamics, 2008, 31(2): 307-321
[59] Hawkins A M, Fill T R, Proulx R J, et al. Constrained trajectory optimization for lunar landing [C]. AAS, 06152, Jan. 2006
[60] Williams P. Application of pseudospectral methods for receding horizon control[J]. Journal of Guidance, Control, and Dynamics. 2004, 27(2): 310-314
[61] Ross I M, Fahroo F. Issues in the real-time computation of optimal control [J]. Mathematical and Computer Modeling. 2006,43(9-10):1172-1188.
[62].Richie G. The Common aero vehicle: space delivery system of the future[C]. AIAA Space Technology Conference and Exposition, 1999, (9): 352-441.
[63] Zimmerman C. , Dukeman G. , Hanson J . An automated method to compute orbital entry reentry trajectories with heating constraints [J]. AIAA, 2002-4454, 2002.
[64] Shen Z, Lu P. On-board generation of three-dimensional constrained entry trajectories [J]. Journal of Guidance, Control, and Dynamics. 2003, 26(1): 110-121.
[65] Roenneke A. Adaptive on-board guidance for entry vehicles [C]. Washington DC: American Institute of Aeronautics and Astronautics, 2001.
[66] Rajech K A. Reentry trajectory optimization: evolutionary approach[C]. AIAA, 2002-5466.