﻿ UKF参数估计在三体Lambert问题中的应用
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UKF参数估计在三体Lambert问题中的应用

1. 北京航空航天大学 宇航学院, 北京 100191;
2. 北京空天技术研究所, 北京 100074

Application of UKF parameter estimation in the three-body Lambert problem
ZHANG Hongli1, LUO Qinqin2, HAN Chao1
1. School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China;
2. Beijing Aerospace Technology Institute, Beijing 100074, China
Abstract:A new algorithm based on unscented Kalman filter (UKF) parameter estimation was proposed for the fast and efficient solution of the three-body Lambert problem. The algorithm was divided into two steps, guessing the initial solution and searching the exact solution. The initial solution of the three-body Lambert problem was generated using the two-body model of the Earth-Moon system. Then the two-point boundary value problem corresponding to the original three-body Lambert problem was converted to a parameter estimation problem. Through solving the converted problem using UKF, the converged exact solution was found. The algorithm was based on the theory of probability, so the derivation of the gradient matrixes required by traditional numerical methods was omitted. Moreover, the demand for the accuracy of the initial solutions for the three-body Lambert problem was modified. Therefore, the difficulty of solving the three-body Lambert problem was greatly reduced. Numerical examples indicate that the algorithm is of high efficiency and robustness and obtains a larger convergence domain compared with the differential-correction method and the second order differential-correction method.
Key words: three-body systems     Lambert problem     two-point boundary value problem     unscented Kalman filter     parameter estimation

1 问题描述

 图 1 三体Lambert问题示意图Fig. 1 The three-body Lambert problem
2 基于二体模型的初值猜测

 图 2 初值猜测流程图Fig. 2 Initial guess flowchart

1) 将近月点时刻记为tp,令tp=(t1+t2)/2,计算tp时刻月球的位置和速度向量 R m,V m.

2) 求解从 R 1R m、转移时间为tpt1和从 R mR 2、转移时间为t2tp的两段地心圆锥曲线轨道,由此可以得到 R m处的地心速度矢量 V 1,V2,相应的月心速度矢量为 v 1= V 1V m,v 2= V2Vm.

3) 若 ||v 1v 2||大于或等于给定阈值,则采用牛顿-拉弗逊方法调整tp,返回步骤2).若 ||v 1v2||小于给定阈值,则迭代终止.

4) 计算 R 1处速度矢量V1,即求得猜测初值,作为下一步精确解求解的基础.

3 UKF参数估计及其求解方法

 图 3 UKF参数估计流程图Fig. 3 UKF parameter estimation flow chart
Nw的维数；η是尺度参数；常量ε决定了无损变换(UT)的σ点相对于w 当前均值的分布范围,一般设为小量,取值范围为[10－4,1]；常量κ一般取为0或者3－Nβ是与 w 的先验分布相关的常量,对于高斯分布,β=2是最优的.ρRLS是遗忘因子,用于防止因模型误差较大造成的滤波发散,其取值范围为(0,1].α是权重因子,取值范围为[0,1].

4 UKF参数估计算法求解精确解

 参数 取值 系统噪声协方差阵Rr 对角线元素为10－4 尺度参数常量ε 5×10－4或8×10－4 遗忘因子λRLS 0.1 权重因子α 0.5
5 算例与分析

 迭代次数 Δx Δy Δz 1 -4.82×106 -3.06×108 3.11×106 2 -1.17×109 -7.17×107 3.87×108 3 -5.36×107 -1.01×107 3.69×107 4 -1.53×107 -5.33×106 1.85×107 5 -1.03×108 1.56×107 8.10×107 6 -1.25×109 3.71×108 -7.42×108 7 -1.28×108 3.35×107 -4.29×107 8 -1.72×108 3.64×107 1.10×108

 迭代次数 Δx Δy Δz 1 -4.82×106 -3.06×108 3.11×106 2 -1.10×108 9.77×107 -5.58×106 3 -4.13×106 3.51×107 6.43×107 4 -5.69×106 1.27×106 3.98×106 . . . . . . . . . . . . 9 1.62×103 -7.83×102 3.39×103 10 1.06×102 -5.03×101 2.22×102 11 4.22×100 -1.90×100 8.67×100 12 1.10×10-1 -3.95×10-2 2.09×10-1

 图 4 三体Lambert问题的飞行轨迹Fig. 4 Trajectory of the three-body Lambert problem

 扰动分量 收敛域 微分修正 二阶微分修正 UKF参数估计 V1,x 6.3 6.9 29.2 V1,y 1.0 10.2 22.9 V1,z 0.3 0.4 7.6

6 结 论

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#### 文章信息

ZHANG Hongli, LUO Qinqin, HAN Chao
UKF参数估计在三体Lambert问题中的应用
Application of UKF parameter estimation in the three-body Lambert problem

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(2): 228-233.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0120