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Optimization of double-impulse rendezvous using gradient-splitting interval optimization algorithm
LIU Qi , ZHU Hongyu
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-05-20; Accepted: 2015-07-31; Published online: 2016-05-25 12: 00
Foundation item: National Natural Science Foundation of China (11272028)
Corresponding author. Tel.:010-82339753 E-mail:09190@buaa.edu.cn
Abstract: The optimal problem of time-open double-impulse rendezvous was studied and the gradient-splitting interval optimization algorithm (GIOA) was introduced. Considering the characteristics of the problem, GIOA utilized the interval selection strategy which selected a finite number of subintervals to compute, the interval splitting strategy based on the result of the gradient optimization algorithm, the interval contraction strategy based on monotonicity, the test of constraints and the updating strategy of target estimated value based on gradient, etc. As the gradient-algorithm was only used for the interval splitting strategy and the updating strategy of target estimated value, it had no negative effect on GIOA's inheriting of the global characteristic and convergence of the interval optimization algorithm. Simultaneously it accelerated the appearance of an interval containing the optimal value with small width and the updating rate of target estimated value. Thereby the operation efficiency was improved. By the interval selection strategy, the increase of subinterval numbers has been controlled, and the storage costs have been reduced. In the simulation, GIOA solves the optimal problem of time-open double-impulse rendezvous successfully, and shows the advantages of the algorithm.
Key words: impulse rendezvous     interval optimization     gradient optimization     interval splitting     global optimization

1 优化模型

2个航天器之间的相对运动可采用Clohessy-Wiltshire(CW)方程近似描述:

 (1)

 (2)

 (3)

 (4)
 (5)

 (6)

J取最小值,则双脉冲交会燃料消耗最低。

2 基于梯度分割区间优化算法

2.1 算法预处理

2.1.1 决策变量区间的选取

2.1.2 约束条件的转化

2.2 区间选择

2.3 区间分割

1) 结合模型特性的符号分割法

2) 结合梯度优化算法的梯度分割法

2.4 区间紧缩

 (7)

 (8)

 (9)

2.5 区间取舍和目标优化估计值更新

1) 约束条件测试

2) 基于梯度的目标优化估计值更新

2.6 算法终止条件

2.6.1 决策变量区间运算终止条件

2.6.2 运算终止条件

 图 1 基于梯度分割区间优化算法流程 Fig. 1 Flowchart of GIOA
3 算法的适用性、全局性和收敛性 3.1 适用性

GIOA是针对非固定时间的双脉冲交会优化问题而设计的,但是依然具有一定的普适性。在算法设计过程中由于采用了梯度优化算法、区间紧缩策略使用了单调性等,因此该算法针对在定义域某一局部范围内线性或非线性的目标函数和约束条件对决策变量连续可微的优化问题理论上均具有较好的应用效果。显然,本文要解决的非固定时间的双脉冲交会优化问题是符合的。

3.2 全局性和收敛性

GIOA是在区间优化算法的基础上,引入梯度优化算法,但不直接使用梯度优化算法结果,只是将其作为区间分割和目标优化估计值更新的一种手段,与其他模块设计一样,并没有破坏区间优化算法整体框架,因此整个算法实质上仍属于区间优化算法,其全局性和收敛性继承了区间优化算法的全局性和收敛性[19]

4 仿真算例

4.1 仿真条件

 组次 初始位置/km 初始速度/(m·s－1) [［Δv1x];[Δv1y];[Δv1z]］/(m·s－1) [［t1→f];[tf]］/s [J]/(m·s－1) 设计结果区间队列中区间个数 运算时间/min 1 (-7,-1,5) (2.820 3,7.438 5,-7.93) [7.416 511 185,7.416 511 187][-6.420 169 446,-6.420 169 444][4.491 909 435,4.491 909 437] [2 371.772 220 606,2 371.772 220 608][2 568.085 214 200,2 568.085 214 202] [12.424 959 620 4,12.424 959 620 7] 3 056 64 2 (2,-1,-5) (-2.831 1,7.433 0,2.260 3) [-7.197 484 639,-7.197 484 637][-5.160 281 956,-5.160 281 954][1.498 190 084,1.498 190 086] [2 887.638 905 907,2 887.638 905 909][2 986.254 943 073,2 986.254 943 075] [12.890 903 408 4,12.890 903 408 8] 4 363 63 3 (-8,5,10) (5.650 6,-7.441 2,-9.071 3) [13.439 968 090,13.439 968 092][9.178 826 090,9.178 826 092][-3.568 758 729,-3.568 758 727] [1 525.128 810 862,1 525.128 810 864][1 746.702 977 286,1 746.702 977 288] [22.017 747 598 8,22.017 747 599 3] 4 164 85

4.2 算法正确性验证

 图 2 表 1解区间任意取值的相对轨迹 Fig. 2 Relative path of value in interval solution of Table 1

 图 3 不同区间宽度精度下的可行解[t1→f]-[tf] Fig. 3 Feasible solution [t1→f]-[tf] of various interval width precision

 次数 Jmin/(m·s－1) 次数 Jmin/(m·s－1) 1 12.909 036 593 4 6 12.909 036 594 8 2 12.890 905 501 5 7 12.909 042 299 3 3 12.909 036 606 6 8 12.909 050 776 2 4 12.909 036 593 1 9 12.909 036 860 0 5 12.890 903 762 3 10 12.909 036 655 4

4.3 算法有效性验证

 算法模块 算法A 算法B 算法C 算法D “分而治之”区间选择 × √ √ √ 符号分割法 √ √ √ √ 梯度分割法 × × × √ 中点分割法 √ √ √ √ 基于单调性的区间紧缩 √ √ √ √ 约束条件测试 √ √ √ √ 基于梯度目标优化估计值更新 × × √ √ 注:“√”表示有;“×”表示没有。

 迭代次数 区间群队列L中决策变量区间数目 算法A 算法B 算法C 算法D 1 138 138 88 88 2 188 188 84 126 3 444 444 128 166 4 929 732 147 318 ┇ ┇ ┇ ┇ ┇ 14 14 261 1 223 389 1 838 15 21 437 1 485 501 1 668 16 15 723 1 262 685 1 509 17 3 435 1 001 885 1 646 18 0 932 940 1 583 ┇ ┇ ┇ ┇ 71 561 585 244 72 494 421 133 73 606 366 6 74 661 317 0 ┇ ┇ ┇ 102 980 211 103 1 028 63 104 728 0 ┇ ┇ 142 122 143 83 144 0 区间数目最大值 21 437 1 782 1 662 2 208 运算时间/min 316 83 72 64

 迭代次数 Jmin/(m·s－1) 算法B 算法C 算法D 1 100 100 100 2 100 12.424 959 7 12.424 959 7 ┇ ┇ ┇ ┇ 16 66.036 284 9 12.424 959 7 12.424 959 7 ┇ ┇ ┇ ┇

5 结论

1) GIOA将梯度优化算法作为区间分割方式和目标优化估计值更新的重要手段,并对区间优化算法其他模块予以了改进,依然作为一种区间优化算法,继承了区间优化算法的全局性和收敛性。成功求解了非固定时间双脉冲交会问题不同精度下的全局优化解区间,并通过对比,全局性上明显优于遗传算法。

2) 梯度分割的区间优化算法中“分而治之”区间选择策略的使用,有效控制了决策变量区间数量的增长,降低算法运行的存储需求,由内存的降低而导致运算效率较算法A增长近3倍。

3) 由优化模型的约束条件和目标函数在局部连续可微而引入梯度优化算法,衍生的结合梯度优化算法区间分割方式和基于梯度的目标优化估计值更新,大幅提高了目标优化估计值的更新速率,在实例中分别提高了12%和13%的运算效率,整体提高了25%的运算效率。

 [1] 王华, 唐国金. 用遗传算法求解双冲量最优交会问题[J]. 中国空间科学技术,2003, 23 (1) : 26 –30. WANG H, TANG G J. Solving optimal rendezvous using two impulses based on genetic algorithms[J]. Chinese Space Science and Technology,2003, 23 (1) : 26 –30. (in Chinese). Cited By in Cnki (0) | Click to display the text [2] 戴光明, 李晖. DE算法在空间交会中的应用[J]. 上海航天,2007, 24 (3) : 46 –49. DAI G M, LI H. Study on application of differential evolution algorithm in space rendezvous[J]. Aerospace Shanghai,2007, 24 (3) : 46 –49. (in Chinese). Cited By in Cnki (0) | Click to display the text [3] 梁静静, 解永春. 基于粒子群算法优化双脉冲绕飞问题[J]. 空间控制技术与应用,2013, 39 (5) : 43 –47. LIANG J J, XIE Y C. Double-impulsive fly-around problem based on particle swarm optimization algorithm[J]. Aerospace Control and Application,2013, 39 (5) : 43 –47. (in Chinese). Cited By in Cnki (0) | Click to display the text [4] 姬晓琴, 肖利红, 陈文辉. 基于T-H方程的多脉冲最优交会方法[J]. 北京航空航天大学学报,2014, 40 (7) : 905 –909. JI X Q, XIAO L H, CHEN W H. Optimal multi-impulse rendezvous based on T-H equations[J]. Journal of Beijing University of Aeronautics and Astronautics,2014, 40 (7) : 905 –909. (in Chinese). Cited By in Cnki (0) | Click to display the text [5] 李晨光, 肖业伦. 多脉冲C-W交会的优化方法[J]. 宇航学报,2006, 27 (2) : 172 –176. LI C G, XIAO Y L. Optimization methods of multi-pulse C-W rendezvous[J]. Journal of Astronautics,2006, 27 (2) : 172 –176. (in Chinese). Cited By in Cnki (0) | Click to display the text [6] LUO Y Z, TANG G J, LEI Y J. Optimal multi-objective linearized impulsive rendezvous[J]. Journal of Guidance,Control,and Dynamics,2007, 30 (2) : 383 –389. Click to display the text [7] GAO X,LIANG B,QIU Y.A PSO algorithm of multiple impulses guidance and control for GEO space robot[C]//Proceedings of the 13th ICARCV Conference.Piscataway,NJ:IEEE Press,2014:1560-1565. Click to display the text [8] XU L M,LIU H,ZHANG T.Optimal transfer orbit design based on multi-pulse thrust[C]//Proceedings of the 32nd Chinese Control Conference.Piscataway,NJ:IEEE Press,2013:5193-5197. Click to display the text [9] 付磊, 安效民, 覃曌华, 等. 基于混合遗传算法的多冲量最优变轨[J]. 航天控制,2013, 31 (3) : 15 –19. FU L, AN X M, QIN Z H, et al. The optimal multiple-impulse orbit transfer by using hybrid genetic algorithm[J]. Aerospace Control,2013, 31 (3) : 15 –19. (in Chinese). Cited By in Cnki (0) | Click to display the text [10] MOORE R E, KEARFOTT R B, CLOUD M J. Introduction to interval analysis .2nd ed.[M]. Philadelphia : Society for Industrial & Applied Mathematics , 2009 : 7 -35. Click to display the text [11] JULIANA S, CHU Q P, MULDER J A. Reentry flight clearance using interval analysis[J]. Journal of Guidance,Control,and Dynamics,2008, 31 (5) : 1295 –1307. Click to display the text [12] DE WEERDT E,CHU Q P,MULDER J A.Global fuel optimization for constrained spacecraft formation rotations[C]//Proceedings of AIAA Guidance,Navigation,and Control Conference. Reston:AIAA,2009:1-21. Click to display the text [13] KAMPEN E V.Global optimization using interval analysis[D].Delft:Technische Universiteit Delft,2010:65-89. Click to display the text [14] 高东迎, 岳晓奎. 基于区间算法的航天器再入轨迹优化[J]. 科学技术与工程,2012, 20 (4) : 852 –856. GAO D Y, YUE X K. Trajectory optimization for reentry vehicle via interval algorithm[J]. Science Technology and Engineering,2012, 20 (4) : 852 –856. (in Chinese). Cited By in Cnki (0) | Click to display the text [15] CHEN T, KAMPEN E V, YU H, et al. Optimization of time-open constrained Lambert rendezvous using interval analysis[J]. Journal of Guidance,Control,and Dynamics,2013, 36 (1) : 175 –184. Click to display the text [16] CASADO L G, GARCIA I, CSENDES T. A new multisection technique in interval methods for global optimization[J]. Computing,2000, 65 (3) : 263 –269. Click to display the text [17] NATARAY P S V, KOTECHA K. An algorithm for global optimization using the Taylor-Bernstein form as inclusion function[J]. Journal of Global Optimization,2002, 24 (4) : 417 –436. Click to display the text [18] 陈诚.基于区间数学的并行全局寻优算法的研究与系统实现[D].上海:上海大学,2014:25-36. CHEN C.Research and system implementation of parallel global optimal algorithm based on interval mathematics[D].Shanghai:Shanghai University,2014:25-36. (in Chinese). Cited By in Cnki (0) | Click to display the text [19] RATSCHEK H, ROKNE J. New computer methods for global optimization[M]. Chichester: Ellis Horwood Ltd, 1988 : 85 -89. Click to display the text [20] KAMPEN E V,CHU Q P,MULDER J A,et al.Nonlinear aircraft trim using interval analysis[C]//Proceedings of AIAA Guidance,Navigation,and Control Conference and Exhibit.Reston:AIAA,2007,4:4073-4087. Click to display the text

#### 文章信息

LIU Qi, ZHU Hongyu

Optimization of double-impulse rendezvous using gradient-splitting interval optimization algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(5): 1071-1078
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0324