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Analysis of trajectory deviation for spacecraft relative motion in close-range
SHI Hao, ZHAO Yushan, SHI Peng
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-08-08; Accepted: 2016-11-04; Published online: 2016-12-08
Foundation item: National Natural Science Foundation of China (11102007); the Fundamental Research Funds for the Central Universities (YWF-14-YHXY-012)
Corresponding author. ZHAO Yushan, E-mail:yszhao@buaa.edu.cn
Abstract: The trajectory deviation analysis of relative motion in closed-loop control system was conducted with the concept of relative reachable domain (RRD), which is a geometric description for the collection of all the possible relative positions. Given that the state error of spacecraft is subjected to arbitrary Gauss distribution, the RRD could be obtained by assembling the time variable error ellipsoids of position. A closed-loop control system for close-range relative motion based on the linearized dynamical model with measurement and control errors was considered in the problem. The covariance matrix of spacecraft state, which defined the error ellipsoid, was analyzed by covariance analysis describing function technique (CADET). An algorithm for solving the envelope of RRD with the state covariance matrix was proposed subsequently. Comparison between the RRDs, solved in both open-loop and closed-loop systems, and the simulation result of 1 000 Monte Carlo runs demonstrates the feasibility and validity of the proposed method.
Key words: trajectory deviation     closed-loop control     relative reachable domain (RRD)     error ellipsoid     covariance analysis describing function technique (CADET)

1 描述轨迹偏差的相对可达区

1.1 航天器状态误差椭球

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6维状态误差分布的等概率密度面可表示为

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 图 1 误差椭球 Fig. 1 Error ellipsoid

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1.2 相对可达区的包络

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 图 2 相对可达区包络 Fig. 2 Envelope of RRD

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2 闭环控制系统协方差分析

2.1 偏差预测

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tk时刻，航天器实际状态的预测方程为

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2.2 测量更新

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tk时刻的测量过程并不会改变航天器的实际状态，如用上标-m和+m分别代表航天器在测量更新前后的状态量，则有

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tk时刻的测量过程会更新导航系统的估计值，航天器估计状态的更新关系式可写为[12]

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2.3 控制修正

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2.4 协方差矩阵的计算

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3 相对可达区包络计算

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 图 3 坐标系 Fig. 3 Coordinate systems

S0系到S1系的坐标转换矩阵可写为

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y2=R10y1表示y1S1系中的投影，则

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R10n=[001]T=ez，将式 (56) 代入式 (54) 得到

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4 仿真算例

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 图 4 开环系统相对可达区包络 Fig. 4 Envelope of RRD for open-loop system

 图 5 开环系统相对可达区包络与Monte Carlo轨迹的x-y平面比较 Fig. 5 Comparison of RRD envelope and Monte Carlo trajectories in x-y plane for open-loop system

t=0.8T时小卫星的标称位置处取垂直于标称轨迹的横截面，将相对可达区包络和Monte Carlo轨迹的比较结果在此平面上投影，如图 6所示。此横截面在标称轨迹上的位置在图 5中以点粗点画线标出。在上述截面建立截面坐标系XYZ，原点位于标称位置处，X向沿着此刻的标称速度方向，Z向垂直于标称轨迹所在的平面，Y轴构成右手坐标系。如图 6所示，绝大部分的小方块均位于可达包络之内，只有11个位于包络之外，概率为1.1%，小于给定概率2.93%。

 图 6 开环系统相对可达区包络与Monte Carlo轨迹的横截面比较 Fig. 6 Comparison of RRD envelope and Monte Carlo trajectories in cross-section plane for open-loop system

 图 7 闭环系统相对可达区包络 Fig. 7 Envelope of RRD for closed-loop system

 图 8 闭环系统相对可达区包络与Monte Carlo轨迹的x-y平面比较 Fig. 8 Comparison of RRD envelope and Monte Carlo trajectories in x-y plane for closed-loop system

 图 9 闭环系统相对可达区包络与Monte Carlo轨迹的横截面比较 Fig. 9 Comparison of RRD envelope and Monte Carlo trajectories in cross-section plane for closed-loop system

5 结论

1) 本文所述方法能切实地反映闭环控制系统对轨迹的修正效果，可以有效地对航天器近距离相对运动中轨迹偏差散布的区域在几何上进行量化地描述。

2) 通过结合相对可达区理论与协方差分析描述函数法，航天器轨迹偏差散布区域的计算十分迅速，本文中相对可达区包络的计算时间仅需要1 s左右。

3) 通过仿真发现，航天器很小的速度误差都会导致很大的轨迹偏差，因此使用本文所述方法研究近距离相对运动的安全问题时不能忽略相对速度误差。

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#### 文章信息

SHI Hao, ZHAO Yushan, SHI Peng

Analysis of trajectory deviation for spacecraft relative motion in close-range

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(3): 636-644
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0641