﻿ 基于分段常值推力的水滴悬停构型控制策略<sup>*</sup>
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Teardrop hovering configuration control strategy based on piecewise constant thrust
BAI Shengzhou, WANG Huijiang, HAN Chao, ZHANG Sihang
School of Astronautics, Beihang University, Beijing 100083, China
Received: 2018-07-09; Accepted: 2018-10-17; Published online: 2018-11-22 09:08
Corresponding author. HAN Chao, E-mail:hanchao@buaa.edu.cn
Abstract: In order to meet the requirements of relative motion control for spacecraft formation flying missions, design and control of spacecraft's forced fly-around formation under the control of piecewise constant thrust are investigated. For teardrop hovering configuration under pulse control, a multi-level constant thrust control strategy is proposed. The shooting equation is transformed into the solution of the extreme value problem, and the least square method is used to solve it. In this paper, constant thrust feasibility is also analyzed. Based on continuous constant small-thrust control equations, the small neighborhood theorem is deduced, and the feasibility of two-segment constant thrust control is analyzed in near-distance relative motion. In addition, a small-thrust increment equation is proposed to improve the solution accuracy, and it is proved that multiple iterations can precisely approximate the ideal solution. Finally, numerical simulations show that a constant small-thrust control strategy is feasible for the teardrop hovering relative motion. The theory of spacecraft forced fly-around design and control is enriched, and the results provide a reference for engineering applications.
Keywords: formation flying     teardrop hovering     least squares optimization     shooting equation     configuration design

1 水滴悬停构型

 图 1 水滴悬停构型三维示意图 Fig. 1 Schematic diagram of 3D teardrop hovering configuration

1.1 水滴悬停打靶方程

 (1)

 (2)

 (3)
1.2 最小二乘法求解常值推力

 (4)

2 单段常值推力与多段常值推力

 (5)

 (6)

 图 2 小邻域定理示意图 Fig. 2 Schematic diagram of small neighborhood theorem

 (7)

 (8)

 (9)
3 修正公式

L为有界常数，仅与领域B(x0, δ)有关, 当xy0, L→0时，必，若x, yδ1, 有，其中ρ < 1。

 (10)

 (11)

4 数值仿真

 图 3 五段常值推力下仿真结果 Fig. 3 Simulation results of five-segment constant thrust

 推力/(m·s-2) μ1 μ2 ft 0.006 796 -0.006 789 fn -0.018 311 -0.018 306 8 fh 0 0

 推力/(m·s-2) μ1 μ2 ft 0.005 074 -0.005 080 fn -0.021 028 -0.021 026 fh 0 0

 图 4 未修正的两段常值推力解 Fig. 4 Solution of unmodified two-segment constant thrust

 图 5 修正后的两段常值推力解 Fig. 5 Solution of modified two-segment constant thrust

5 结论

1) 多段常值推力控制问题可以转化为求解水滴悬停构型的打靶方程，最小二乘法是一种求解此类问题的实用方法。

2) 小邻域定理为近距离相对运动条件下两段常值推力控制提供了可行性，悬停构型采用两段常值推力，一定程度上保证了解的存在性，也缩小了优化变量的数量，提高优化精度并减少优化时间。

3) 若最小二乘解的精度不够，但距离理想解不太远，那么采用迭代方法对最小二乘解进行修正，可以提高精度，甚至收敛到理想解。

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#### 文章信息

BAI Shengzhou, WANG Huijiang, HAN Chao, ZHANG Sihang

Teardrop hovering configuration control strategy based on piecewise constant thrust

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(3): 560-566
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0408