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1. 北京航空航天大学 宇航学院, 北京 100083;
2. 上海航天控制技术研究所 研发中心, 上海 201109;
3. 上海市空间智能控制技术重点实验室, 上海 201109

An improved indirect method for in plane orbit optimization under constant thrust
ZHAO Guowei1, LI Dejin1, SONG Ting2,3, WU Hailei2,3
1. School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Research and Development Centre, Shanghai Institute of Spaceflight Control Technology, Shanghai 201109, China;
3. Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai 201109, China
Received: 2016-05-10; Accepted: 2016-09-21; Published online: 2016-10-10 17:04
Foundation item: National Natural Science Foundation of China (11572016)
Corresponding author. ZHAO Guowei, E-mail:zhaoguowei@buaa.edu.cn
Abstract: The transfer of a spacecraft between coplanar orbits under continuous constant thrust with minimum fuel consumption was investigated. A control equation of the optimal trajectory, which the steering angle must satisfy, was derived by using maximum principle. Combining the control equation with dynamic equations, we establish an improved indirect method to design optimal coplanar transfer orbit and propose an approximate application method for the condition when thrust direction adjustment ability was limited. Due to avoiding solving the Lagrange costate differential equations, the improved indirect method reduces difficulty of initial value estimation and calculation a lot than traditional indirect method. Compared with Gauss pseudospectral method, the improved indirect method can get higher precision and better numerical smoothness. Simulations show that a limitation on the magnitude of the second derivative of thrust angle can improve change law of thrust angle and reduce change range of thrust angle; as to the fuel consumption, the greater the thrust magnitude is, the more the fuel consumption is, and the optimal transfer orbit can save fuel consumption a lot under a certain large magnitude of thrust.
Key words: orbit transfer     optimal control     constant thrust     maximum principle     coplanar orbit

1 轨道优化模型

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 图 1 变量示意图 Fig. 1 Illustration of variables

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2 最优控制模型

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3 数值求解

 图 2 改进间接法求解流程 Fig. 2 Flowchart of improved indirect method
 图 3 Gauss伪谱法求解流程 Fig. 3 Flowchart of Gauss pseudospectral method
4 算例验证

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GEO废星离轨过程轨道优化问题的实际参数为：航天器质量为5t，初始轨道为GEO轨道，在连续常值推力作用下近地点升高Δh=350km。

4.1 ϕ″不受限的轨道优化

 图 4 不同推力加速度下推力方向角变化曲线 Fig. 4 Thrust angle vs time under different thrust acceleration
4.2 ϕ″受限的轨道优化

 图 5 ϕ″不受限时推力方向角变化规律 Fig. 5 Change law of thrust angle when ϕ″ is free

 图 6 |ϕ″|≤0.008 (°)/s2时推力方向角变化规律 Fig. 6 Change law of thrust angle when |ϕ″|≤0.008 (°)/s2

 推力加速度/(10-3m·s-2) 速度变化量/(m·s-1) 切向推力转移轨道 优化轨道 Gauss伪谱法求解不限ϕ″ Gauss伪谱法求解|ϕ″|≤0.008 (°)/s2 改进间接法求解不限ϕ″ 改进间接法求解|ϕ″|≤0.008 (°)/s2 0.02 13.205 12.757 12.757 12.757 12.757 0.10 14.862 14.005 14.007 14.005 14.005 0.15 12.858 12.713 12.713 12.713 12.713 0.20 14.995 14.254 14.257 14.254 14.254 0.40 22.588 20.645 20.722 20.645 20.670 1.00 40.315 34.428 34.447 34.309 34.421 4.00 100.270 72.974 73.007 72.856 72.937 10.00 185.780 117.660 117.770 117.560 117.690

4.3 燃耗分析

5 结论

1) 所提出的改进间接法由于避免了协态变量微分方程组的求解，相对于传统间接法降低了初值猜测的难度和计算量; 与Gauss伪谱法相比，改进间接法求解精度更高；在推力方向角调节能力受限时，所提出的近似优化方法求解结果与Gauss伪谱法求解结果一致并且精度更高，有工程应用价值。

2) 推力方向角的二阶导数受限能够改变推力方向角变化规律，降低推力方向角变化范围(姿态机动范围)和变化速率，增加航天器的稳定性。

3) 总体上推力加速度越大燃耗越多，当推力加速度较小时，最优转移轨道燃耗与切向推力转移轨道燃耗相差很小；当推力加速度较大时，最优转移轨道比切向推力转移轨道显著节省燃耗，所以在大推力场合，更适合选用优化方法计算转轨方案。

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#### 文章信息

ZHAO Guowei, LI Dejin, SONG Ting, WU Hailei

An improved indirect method for in plane orbit optimization under constant thrust

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(5): 894-901
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0387