针对空天飞行器大气层内上升段实时轨迹优化问题，提出一种基于Proximal-Newton-Kantorovich凸规划的轨迹优化方法。首先，应用Newton-Kantorovich迭代方法将轨迹优化问题转化为一系列的子问题，每个子问题都是一个线性最优控制问题；其次，针对Newton-Kantorovich迭代方法忽略运动方程中的高阶信息，导致难以收敛这一问题，提出Proximal-Newton-Kantorovich迭代方法，在子问题的性能指标中加入邻近规则化项，改善了Newton-Kantorovich迭代方法的收敛性；最后，将子问题离散为二阶锥规划问题，并应用内点法进行求解。提出的Proximal-Newton-Kantorovich凸规划方法是一种求解非线性轨迹规划问题的可行途径。理论分析表明，Proximal-Newton-Kantorovich迭代方法的收敛结果一定是轨迹优化问题的局部最优解。数值实验表明，此方法的计算时间在毫秒级。
This paper proposes a trajectory optimization approach for the real-time atmospheric ascent trajectory optimization problem based on Proximal-Newton-Kantorovich convex programming. The Newton-Kantorovich iteration approach casts the trajectory optimization problem into subproblems with each being a linear optimal control problem. However, the Newton-Kantorovich iteration approach ignores higher order terms in motion equations, making it hard to converge. This paper proposes a Proximal-Newton-Kantorovich iteration approach. A Proximal term is introduced in the performance index of the subproblems to improve the convergence. The subproblems are then casted into second-order cone programming problems and solved by the interior-point methods. The proposed Proximal-Newton-Kantorovich iteration approach is an efficient approach to solve nonlinear trajectory optimization problems. It is proved that the convergence results of the Proximal-Newton-Kantorovich iteration approach always satisfy the necessary conditions of the original trajectory optimization problem. Numerical results show that this approach can be executed in milliseconds.
[1] 张远龙, 谢愈. 滑翔飞行器弹道规划与制导方法综述[J]. 航空学报, 2020, 41(1):023377. ZHANG Y, XIE Y. Review of trajectory planning and guidance methods for gliding vehicles[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(1):023377(in Chinese).
[2] DUKEMAN G. Atmospheric ascent guidance for rocket-powered launch Vehicles[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Reston:AIAA, 2002.
[3] 宗群, 田栢苓, 窦立谦. 基于Gauss伪谱法的临近空间飞行器上升段轨迹优化[J]. 宇航学报, 2010, 31(7):1775-1781. ZONG Q, TIAN B, DOU L. Ascent phase trajectory optimization for near space vehicle based on Gauss Pseudospectral method[J]. Journal of Astronautics, 2010, 31(7):1775-1781(in Chinese).
[4] 崔乃刚, 黄盘兴, 路菲, 等. 基于混合优化的运载器大气层内上升段轨迹快速规划方法[J]. 航空学报, 2015, 36(6):1915-1923. CUI N G, HUANG P X, LU F, et al. A hybrid optimization approach for rapid endo-atmospheric ascent trajectory planning of launch vehicles[J]. Acta Aeronautica et Astronautica Sinica, 2015, 36(6):1915-1923(in Chinese).
[5] 方群, 刘怡思, 王雪峰. 空天飞行器弹道/轨道一体化设计[J]. 航空学报, 2018, 39(4):121398. FANG Q, LIU Y S, WANG X F. Trajectory-orbit unified design for aerospace vehicles[J]. Acta Aeronautica et Astronautica Sinica, 2018, 39(4):121398(in Chinese).
[6] BEN-TAL A, NEMIROVSKI A. Lectures on modern convex optimization:Analysis, algorithms, and engineering applications[M]. Philadelphia:SIAM, 2001.
[7] BOYD S, VANDENBERGHE L. Convex optimization[M]. Cambridge:Cambridge University Press, 2004.
[8] ACIKMESE B, PLOEN S. A powered descent guidance algorithm for Mars pinpoint landing[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Reston:AIAA, 2005.
[9] ACIKMESE B, PLOEN S. Convex programming approach to powered descent guidance for Mar landing[J]. Journal of Guidance, Control, and Dynamics, 2007, 30(5):1353-1366.
[10] ACIKMESE B, SCHARF D, BLACKMORE L, et al. Enhancements on the convex programming based powered descent guidance algorithm for Mars landing[C]//AIAA/AAS Astrodynamics Specialist Conference and Exhibit. Reston:AIAA, 2008.
[11] LIU X, LU P, PAN B. Survey of convex optimization for aerospace applications[J]. Astrodynamics, 2017, 1(1):23-40.
[12] CHENG X, LI H, ZHANG R. Autonomous trajectory planning for space vehicles with a Newton-Kantorovich/convex programming approach[J]. Nonlinear Dynamics, 2017, 89(4):2795-2814.
[13] 宋征宇, 王聪, 巩庆海. 运载火箭上升段推力下降故障的自主轨迹规划方法[J]. 中国科学:信息科学, 2019, 49(11):1472-1487. SONG Z Y, WANG C, GONG Q H. Autonomous trajectory planning for launch vehicle under thrust drop failure[J]. Scientia Sinica Informationis, 2019, 49(11):1472-1487(in Chinese).
[14] SZMUK M, ACIKMESE B, BERNING A W. Successive convexification for fuel-optimal powered landing with aerodynamic drag and non-convex constraints[C]//AIAA Guidance, Navigation, and Control Conference. Reston:AIAA, 2016.
[15] CHENG X, LI H, ZHANG R. Efficient ascent trajectory optimization using convex models based on the Newton-Kantorovich/Pseudospectral approach[J]. Aerospace Science and Technology, 2017, 66:140-151.
[16] 王劲博, 崔乃刚, 郭继峰, 等. 火箭返回着陆问题高精度快速轨迹优化算法[J]. 控制理论与应用, 2018, 35(3):389-398. WANG J B, CUI N G, GUO J F, et al. High precision rapid trajectory optimization algorithm for launch vehicle landing[J]. Control Theory & Applications, 2018, 35(3):389-398(in Chinese).
[17] YANG R, LIU X. Fuel-optimal powered descent guidance with free final-time and path constraints[J]. Acta Astronautica, 2020,172:70-81.
[18] LIU X. Fuel-optimal rocket landing with aerodynamic controls[C]//AIAA Guidance, Navigation, and Control Conference. Reston:AIAA, 2017.
[19] LIU X. Fuel-optimal rocket landing with aerodynamic controls[J]. Journal of Guidance, Control, and Dynamics, 2019, 42(1):65-77.
[20] KANTOROVICH L V. On Newton's method for functional equations[J]. Proceedings of the USSR Academy of Sciences, 1948, 59(7):1237-1240.
[21] BOYD J P. Chebyshev and Fourier spectral methods[M]. New York:Courier Corporation, 2001.
[22] BETTS J T. Practical methods for optimal control and estimation using nonlinear programming[M]. Philadelphia:SIAM, 2010.
[23] PARIKH N. Proximal algorithms[J]. Foundations and Trends in Optimization, 2014, 1(3):127-239.
[24] DOMAHIDI A, CHU E, BOYD S. ECOS:An SOCP solver for embedded systems[C]//2013 European Control Conference (ECC). Piscataway:IEEE Press, 2013.
[25] NOAA, NASA. USAF U.S. Standard atmosphere, 1976[M]:Washington, D.C.:NASA. 1976.
[26] RAO A V, BENSON D A, DARBY C, et al. Algorithm 902:GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the Gauss Pseudospectral method[J]. ACM Transactions on Mathematical Software, 2010, 37(2):1-39.
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