基于深度学习的弹道优化方法

doi:10.12305/j.issn.1001-506X.2025.12.29

摘要/Abstract

摘要：

针对飞行器弹道优化问题，提出一种基于深度学习的最优控制求解方法。有别于现有直接法和间接法，该方法使用神经网络对控制变量进行参数化构造，使得采用往返积分求解非线性哈密顿系统的多点边值问题成为可能，避免了协态变量初值猜测问题。基于Pontryagin极大值原理构造损失函数，并通过随机梯度下降算法对控制变量进行寻优。相较于现有深度学习算法，该方法无需预先生成数据集，也不需要对飞行力学方程进行化简，适用于航天飞行器弹道设计。以美国民兵-3导弹为例，该方法通过协同优化第一子级秒耗量剖面和主动段俯仰角程序，可在满足飞行过程中各项约束条件下实现增程，证明了所提方法的有效性。

关键词: 导弹, 弹道优化, 最优控制, 深度学习

Abstract:

A deep learning based optimal control solution method is proposed for aircraft ballistic optimization. Unlike existing direct and indirect methods, this method uses neural networks to parameterize the control variables, making it possible to solve the multi-point boundary value problem of nonlinear Hamiltonian systems using back and forth integration, and avoiding the problem of guessing the initial values of covariates. Construct a loss function based on the Pontryagin maximum principle and optimize the control variables using stochastic gradient descent algorithm. Compared to the existing deep learning algorithms, this method does not require pre-generation of datasets or simplification of flight mechanics equations, making it suitable for spacecraft ballistic design. Taking Minuteman 3 missile as an example, this method achieves range extension while satisfying various constraints during flight by synergistically optimizing the first-level second consumption profile and active stage pitch angle program, demonstrating the effectiveness of the proposed method.

Key words: missile, ballistic optimization, optimal control, deep learning

中图分类号:

V 448

王哲, 李志文, 孙均政, 张增辉. 基于深度学习的弹道优化方法[J]. 系统工程与电子技术, 2025, 47(12): 4174-4185.

Zhe WANG, Zhiwen LI, Junzheng SUN, Zenghui ZHANG. Ballistic optimization method based on deep learning[J]. Systems Engineering and Electronics, 2025, 47(12): 4174-4185.

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参考文献 31

1	赵汉元. 航天飞行器再入动力学和制导[M]. 北京: 科学出版社, 2024: 143−201.
	ZHAO H Y. Re-entry dynamics and guidance of spacecraft[M]. Beijing: Science Press, 2024: 143−201.
2	程国采. 航天飞行器最优控制理论与方法[M]. 北京: 国防工业出版社, 1991: 305−364.
	CHENG G C. Optimal control theory and method for aerospace vehicles[M]. Beijing: National Defense Industry Press, 1991: 305−364.
3	包为民, 朱建文, 张洪波, 等. 高超声速飞行器全程制导方法[M]. 北京: 科学出版社, 2021: 12−52.
	BAO W M, ZHU J W, ZHANG H B, et al. Full range guidance method for hypersonic aircraft[M]. Beijing: Science Press, 2021: 12−52.
4	BETTS J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998, 21 (2): 193- 207.
5	GILL P E, MURAY W, SAUNDERS M A. SNOPT: an SQP algorithm for large-scale constrained optimization[J]. SIAM Review, 2005, 47 (1): 99- 131. doi: 10.1137/S0036144504446096
6	HARGRAVES C R, PARIS S W. Direct trajectory optimization using nonlinear programming and collocation[J]. Journal of Guidance, Control, and Dynamics, 1987, 10 (4): 338- 342.
7	ELNAGAR G, KAZEMI M A, RAZZAGHI M. The pseudospectral method for discretizing optimal control problems[J]. IEEE Trans. on Automatic Control, 1995, 40 (10): 1793- 1796. doi: 10.1109/9.467672
8	冯浩阳, 汪雪川, 岳晓奎, 等. 航天器轨道递推及Lambert问题计算方法综述[J]. 航空学报, 2023, 44 (13): 028027.
	FENG H Y, WANG X C, YUE X K, et al. A survey of computational methods for spacecraft orbit ropagation and Lambert problems[J]. Acta Aeronautica et Astronautica Sinica, 2023, 44 (13): 028027.
9	雍恩米, 唐国金, 陈磊. 基于Gauss伪谱法的高超声速飞行器再入轨迹快速优化[J]. 宇航学报, 2008, 29 (6): 1766- 1772.
	YONG E M, TANG G J, CHEN L. Rapid trajectory optimization for hypersonic reentry vehicle via Gauss pseudospectral method[J]. Journal of Astronautics, 2008, 29 (6): 1766- 1772.
10	马宗占, 许志, 唐硕, 等. 一种改进的运载火箭迭代制导方法[J]. 航空学报, 2021, 42 (2): 324218.
	MA Z Z, XU Z, TANG S, et al. Improved iterative guidance method for launch vehicles[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42 (2): 324218.
11	BRAUER G L, CORNICK D E, STEVENSON R. Capabilities and applications of the program to optimize simulated trajectories (POST). Program summary document[R]. Washington, D. C. : NASA, 1977.
12	KRAFT D. On the choice of minimization algorithms in parametric optimal control problems[M]. Berlin: Springer, 1981: 219−231.
13	BRYSON A E, HO Y C. Applied optimal control[M]. New York: Taylor & Francis Goup, 1975: 42−127.
14	SPURLOCK O F, WILLIAMS C H. DUKSUP: a computer program for high thrust launch vehicle trajectory design and optimization[C]//Proc. of the 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, 2014.
15	BERTRAND R, EPENOY R. New smoothing techniques for solving bang-bang optimal control problems numerical results and statistical interpretation[J]. Optimal Control Applications & Methods, 2002, 23 (4): 171- 197.
16	彭坤, 彭睿, 黄震, 等. 求解最优月球软着陆轨道的隐式打靶法[J]. 航空学报, 2019, 40 (7): 322641.
	PENG K, PENG R, HUANG Z, et al. Implicit shooting method to solve optimal lunar soft landing trajectory[J]. Acta Aeronautica et Astronautica Sinica, 2019, 40 (7): 322641.
17	廖宇新, 李惠峰, 包为民. 基于间接Radau伪谱法的滑翔段轨迹跟踪制导律[J]. 宇航学报, 2015, 36 (12): 1398- 1405.
	LIAO Y X, LI H F, BAO W M. Gliding trajectory tracking guidance law based on indirect Radau psedo-spectral method[J]. Journal of Astronautics, 2015, 36 (12): 1398- 1405.
18	IZZO D, MARTENS M, PAN B. A survey on artificial intelligence trends in spacecraft guidance dynamics and control[J]. Astrodynamics, 2019, 3, 287- 299. doi: 10.1007/s42064-018-0053-6
19	GORBAN A N, WUNSCH D C. The general approximation theory[C]//Proc. of the International Joint Conference on Neural Networks, Anchorage, 1998: 1271−1274.
20	刘宇航, 杨洪伟, 李爽. 小推力最优轨迹协态估计的高效机器学习方法[J]. 宇航学报, 2022, 43 (5): 594- 602.
	LIU Y H, YANG H W, LI S. Efficient machine learning method for costate estimation of low thrust optimal trajectories[J]. Journal of Astronautics, 2022, 43 (5): 594- 602.
21	GEIGER B R, SCHMIDT E M, HORN J F. Use of neural networks approximation in multiple-unmanned aerial vehicle trajectory optimization[C]//Proc. of the AIAA Guidance, Navigation, and Control Conference, 2009.
22	TAILOR D, IZZO D. Learning the optimal stage-feedback via supervised imitation learning[J]. Astrodynamics, 2019, 3, 361- 374. doi: 10.1007/s42064-019-0054-0
23	IZZO D, OZTURK E. Real-time guidance for low-thrust transfers using neural networks[J]. Journal of Guidance, Control, and Dynamics, 2021, 44 (2): 315- 327.
24	陈克俊, 刘鲁华, 孟云鹤. 远程火箭飞行动力学与制导[M]. 北京: 国防工业出版社, 2014: 201−254.
	CHEN K J, LIU L H, MENG Y H. Launch vehicle flight dynamics guidance[M]. Beijing: National Defense Industry Press, 2014: 201−254.
25	MCFALL K S, MAHAN J R. Artificial neural network method for of boundary value problems with exact satisfaction of arbitrary boundary conditions[J]. IEEE Trans. on Neural Networks, 2009, 20(8): 1221−1233.
26	WANG Z, GUET C. Deep learning in physics: a study of dielectric quasi-cubic particles in a uniform electric field[J]. IEEE Trans. on Emerging Topics in Computational Intelligence, 2022, 6 (3): 429- 438.
27	BAYDIN A G, PEARLMUTTER B A, RADUL A A, et al. Automatic differentiation in machine learning: a survey[J]. Journal of Machine Learning Research, 2017, 18(1): 5595−5637.
28	RAISSI M, PERDIKARIS P, KARNIADAKIS G E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378, 686- 707. doi: 10.1016/j.jcp.2018.10.045
29	KARNIADAKIS G E, KEVREKIDIS I G, LU L, et al. Physics-informed machine learning[J]. Nature Review Physics, 2021, 3, 422- 440. doi: 10.1038/s42254-021-00314-5
30	WANG Z, GUET C. Self-Consistent learning of neural dynamical systems from noisy time series[J]. IEEE Trans. on Emerging Topics in Computational Intelligence, 2022, 6 (5): 1103- 1112.
31	ANAGNOSTOPOULOS S J, TOSCANO J D, STERGIOPULOS N. Residual-based attention in physics-informed neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 421: 116805.

参数	第1级	第2级	第3级
起飞质量/kg	34730	11930	4780
装药量/kg	20752	6246	3321
工作时间/s	61.6	65.2	59.6
真空比冲/（Ns/kg）	3030	2823	2793
横截面直径/m	1.67	1.32	1.32

参数	优化前	优化后
射程/km	9450.96	9715.88
关机点速度/（m/s）	6812.63	6868.95
关机点当地倾角/（$^\circ $）	21.03	21.05
关机点航程/km	448.14	461.51
关机点高度/km	225.29	212.88

参数	优化前	优化后
第一子级推力贡献	2515.44	2540.01
第一子级气动损失	−140.49	−127.07
第一子级重力损失	−461.00	−456.92
第二子级推力贡献	2089.90	2091.12
第二子级气动损失	−7.25	−6.68
第二子级重力损失	−307.59	−276.86
第三子级推力贡献	3309.31	3306.35
第三子级气动损失	0.00	0.00
第三子级重力损失	−212.47	−199.57

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