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Space interception orbit optimization design based on hybrid optimal algorithm
GAO Xiaoguang , TANG Hong, DUAN Junhong
College of Electronic and Information, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: Based on a hybrid algorithm combining genetic algorithm (GA) with improved Gauss method (IGM), a design method of space interception orbit was proposed for solving time-fuel-optimal trajectory planning problem of interceptor. First, classical Gauss method was improved by applying Newton-Raphson iteration, solving the problem of the classical Gauss method of slow convergence speed and small transfer angle. Then, a theorem on the necessary and sufficient condition for the existence of unique solution was proved. When the initial orbital parameters were given, this condition could be used to judge whether elliptical orbit could be introduced as the interception orbit. After that, constraints of transfer time and maximum pulse rate were given, as well as the calculation steps of hybrid optimal algorithm, and way of coding was improved. Finally taking optimization problem of space interception orbit as an example, simulation was carried out. Simulation result shows that the hybrid algorithm has fewer generations and shorter consuming time compared with conventional optimal algorithm, indicating the algorithm is applicable in determining interception orbit in space.
Key words: optimization     space interception     Gauss method     genetic algorithm (GA)     hybrid algorithm

1 空间拦截问题

 图 1 空间拦截问题示意图 Fig. 1 Schematic diagram of space interception
2 改进高斯法

1) 当转移角大于π/2时,方法失效.

2) 原始高斯法的迭代初值可能不在函数迭代收敛域内.

3) 函数迭代收敛时,线性收敛速度慢.

 图 2 改进高斯法流程图 Fig. 2 low chart of IGM

 比较参数 原始高斯法 改进高斯法 求解时间/ms 54.4 44.8

3 迭代方程有唯一解的充要条件

x∈[1/2,1)时,由式(13)可知dXdx>0.

x∈［0,1/2],ΔE∈[0,π]时,令

X(0)=4/3和dXdx≥0,可知X(x)≥4/3,x∈[0,1).经过前面分析可知,F(x)G(Y)分别在它们的定义域上单调增.

G(Y)的单调性可知

4.1 约束条件

1) 开始转移时刻应小于目标轨道周期的1/2,即tk<1/2tp,拦截时刻应小于目标轨道周期,即te<tp,从而使目标航天器预警机动的时间减少,提高命中概率.

2) 拦截器变轨是在一次速度脉冲的作用下瞬间完成的.轨道转移能量消耗应有上限约束,即Δv≤vmax,本文将拦截器最大变轨脉冲限制为vmax=8.5km/s.

4.2 变量的编码

4.3 适应度函数

4.4 混合优化算法设计步骤

 图 3 混合优化算法流程图 Fig. 3 Flow chart of hybrid optimal algorithm

5 仿真算例

 图 4 霍曼转移示意图 Fig. 4 Schematic diagram of Homan transfer

 算法 Δt/s Δf/(°) Δv/(km·s-1) 霍曼转移 6136.6 180.0 1.0029 文献[5]算法 6102.8 178.9 0.9961 本文算法 6124.8 179.8 1.00291

 图 5 适应度值随遗传代数的变化 Fig. 5 Variation of fitness value via generation
 图 6 共面圆间转移的运动轨迹 Fig. 6 Trajectory of coplanar transfer

 组别 权重因子 Δt/s |Δf|/(°) Δv/(km·s－1) e 1 CT=0,Cv=1 13697.1 241.3 1.4853 0.465 2 CT=0.3,Cv=0.7 4272.9 112.4 2.5007 0.806 3 CT=1,Cv=0 2685.4 24.3 8.1571 0.958

 图 7 CT=0,Cv=1时目标器与拦截器的运动轨迹 Fig. 7 Trajectory of target and interceptor when
 图 8 CT=0.3,Cv=0.7时目标器与拦截器的运动轨迹 Fig. 8 Trajectory of target and interceptor when CT=0.3,Cv=0.7
 图 9 CT=1,Cv=0时目标器与拦截器的运动轨迹 Fig. 9 Trajectory of target and interceptor when CT=1,Cv=0

6 结 论

1) 此方法不需要初始猜测,避免了传统轨道优化方法中初始猜测的困难.

2) 可以求得全局最优解,克服了某些传统优化方法容易收敛于局部最优解的困难.

3) 与传统优化方法相比,收敛精度和收敛速度有所提高.

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#### 文章信息

GAO Xiaoguang, TANG Hong, DUAN Junhong

Space interception orbit optimization design based on hybrid optimal algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(9): 1574-1581.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0673