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混沌多精英鲸鱼优化算法

汤安迪 韩统 徐登武 谢磊

汤安迪, 韩统, 徐登武, 等 . 混沌多精英鲸鱼优化算法[J]. 北京航空航天大学学报, 2021, 47(7): 1481-1494. doi: 10.13700/j.bh.1001-5965.2020.0585
引用本文: 汤安迪, 韩统, 徐登武, 等 . 混沌多精英鲸鱼优化算法[J]. 北京航空航天大学学报, 2021, 47(7): 1481-1494. doi: 10.13700/j.bh.1001-5965.2020.0585
TANG Andi, HAN Tong, XU Dengwu, et al. Chaotic multi-leader whale optimization algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics, 2021, 47(7): 1481-1494. doi: 10.13700/j.bh.1001-5965.2020.0585(in Chinese)
Citation: TANG Andi, HAN Tong, XU Dengwu, et al. Chaotic multi-leader whale optimization algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics, 2021, 47(7): 1481-1494. doi: 10.13700/j.bh.1001-5965.2020.0585(in Chinese)

混沌多精英鲸鱼优化算法

doi: 10.13700/j.bh.1001-5965.2020.0585
基金项目: 

陕西省自然科学基金 2020JQ-481

陕西省自然科学基金 2021JM224

航空科学基金 201951096002

详细信息
    通讯作者:

    韩统, E-mail: 418932433@qq.com

  • 中图分类号: TP301.6

Chaotic multi-leader whale optimization algorithm

Funds: 

Natural Science Foundation of Shaanxi Province 2020JQ-481

Natural Science Foundation of Shaanxi Province 2021JM224

Aeronautical Science Foundation of China 201951096002

More Information
  • 摘要:

    针对无人机(UAV)的航迹规划问题,提出了一种基于混沌多精英鲸鱼优化算法(CML-WOA)的航迹规划方法。首先,在已知飞行环境下,建立3D飞行空间模型和航迹代价模型。通过引入罚函数,将有约束3D航迹规划问题转化为无约束多维函数优化问题,利用CML-WOA求解模型来获得最优航迹。其次,为克服WOA易陷入局部最优的缺陷,引入立方映射混沌算子改善初始种群,增强种群多样性,并通过自适应框架融入正余弦算法(SCA),利用多精英搜索策略有效地提高了算法开发能力和探索能力。最后,使用贪婪策略保证了收敛效率。通过20个基准函数测试和航迹规划仿真实验对提出的改进WOA进行验证。结果表明:所提算法相对其他算法,寻优性能明显提升,具有较强局部最优规避能力和更高的收敛精度与收敛速度;能够稳定快速地规划出代价最少、满足约束的安全可行的飞行航迹。

     

  • 图 1  映射对比图

    Figure 1.  Mapping contrast diagram

    图 2  CML-WOA流程图

    Figure 2.  Flow chart of CML-WOA

    图 3  航迹规划原理

    Figure 3.  Principles of path planning

    图 4  坐标系转换

    Figure 4.  Coordinate system transformation

    图 5  B样条曲线航迹平滑效果

    Figure 5.  Effect of B-spline trajectory smoothing

    图 6  威胁区穿越判断

    Figure 6.  Judgment of crossing the threat zone

    图 7  算法排序雷达图

    Figure 7.  Radar chart of sort of algorithms

    图 8  8种算法在9个基准测试函数上的收敛箱式图

    Figure 8.  Convergence box diagram of eight algorithms on nine benchmark test functions

    图 9  算法收敛曲线

    Figure 9.  Algorithm convergence curves

    图 10  三维航迹图

    Figure 10.  Three-dimensional path map

    图 11  二维等高线航迹图

    Figure 11.  Two-dimensional contour path map

    图 12  转弯角的变化率

    Figure 12.  Change rate of turning angle

    图 13  爬升角的变化率

    Figure 13.  Change rate of climbing angle

    图 14  航迹代价收敛曲线

    Figure 14.  Convergence curves of path cost

    表  1  测试函数

    Table  1.   Test functions

    函数分类 测试函数 维度 范围 最优值
    单峰
    测试
    函数
    30 [-100, 100] 0
    30 [-10, 10] 0
    30 [-100, 100] 0
    30 [-100, 100] 0
    30 [-30, 30] 0
    30 [-100, 100] 0
    30 [-1.28, 1.28] 0
    多峰
    测试
    函数
    30 [-5.12, 5.12] 0
    30 [-32, 32] 0
    30 [-600, 600] 0
    30 [-50, 50] 0
    30 [-50, 50] 0
    固定
    维度
    测试
    函数
    2 [-65, 65] 0.998
    4 [-5, 5] 0.000 3
    2 [-5, 5] -1.031 6
    2 [-2, 2] 3
    3 [1, 3] -3.86
    6 [0, 1] -3.32
    4 [0, 10] -10.153 2
    4 [0, 10] -10.536 3
    下载: 导出CSV

    表  2  F1~F7算法测试结果比较

    Table  2.   Results comparison among test functions F1-F7 of algorithms

    测试函数 统计值 WOA IWOA VCS BSO PSO SSA GSA CML-WOA
    F1 最优值 4.01×10-116 1.30×10-118 7.21×10-121 1.65×10-53 2.73×10-26 0 2.57×10-19 0
    平均值 2.56×10-99 2.19×10-106 2.93×10-106 1.11×10-25 1.60×10-39 9.94×10-100 1.31×10-18 0
    标准差 1.37×10-98 8.11×10-106 1.37×10-105 5.19×10-25 3.17×10-39 5.32×10-99 6.82×10-19 0
    耗时/s 0.788 1 0.779 3 3.336 5 3.837 1 2.662 3 4.377 1 8.332 5 3.013 5
    F2 最优值 1.38×10-72 1.25×10-76 3.22×10-59 8.71×10-11 2.07×10-22 2.50×10-76 1.16×10-9 5.90×10-294
    平均值 5.92×10-65 1.50×10-69 1.02×10-52 7.58×10-3 2.86×10-21 6.25×10-53 2.36×10-9 2.60×10-160
    标准差 1.84×10-64 8.38×10-69 5.12×10-52 1.17×10-2 3.43×10-21 3.40×10-52 7.60×10-10 0
    耗时/s 1.063 2 1.061 5 3.645 4 4.700 3 2.868 0 4.773 7 8.296 9 3.305 5
    F3 最优值 1.04×10-44 4.01×10-28 4.42×10-111 7.70×10-54 3.98×10-33 0 3.02×10-19 0
    平均值 2.22×10-4 1.33×10-4 8.68×10-97 2.98×10-8 2.47×10-30 2.99×10-58 1.76×10-18 3.55×10-297
    标准差 8.10×10-4 5.71×10-4 3.67×10-96 1.63×10-7 3.92×10-30 9.64×10-58 1.31×10-18 0
    耗时/s 1.898 1 1.980 2 4.532 4 7.302 0 3.707 5 6.210 0 9.224 7 4.354 4
    F4 最优值 6.60×10-2 5.77×10-8 1.09×10-57 8.95×10-16 3.07×10-20 3.24×10-242 3.81×10-10 0
    平均值 2.13×101 8.78 1.10×10-51 3.54×10-2 8.17×10-19 1.75×10-43 8.45×10-10 0
    标准差 1.31×101 9.07 5.31×10-51 1.44×10-1 6.12×10-19 6.69×10-43 2.59×10-10 0
    耗时/s 0.784 8 0.772 6 3.434 0 3.774 7 2.624 0 4.318 3 8.101 3 3.204 7
    F5 最优值 9.96×10-1 9.76×10-1 1.94×10-13 0 4.58×10-5 1.70×10-11 1.01 9.53×10-4
    平均值 2.04×101 5.49×101 8.86×10-9 1.91 2.22 1.51×10-5 7.43 2.93×10-1
    标准差 9.19×101 2.78×102 2.74×10-8 2.02 4.91 3.90×10-5 3.21×101 1.02
    耗时/s 1.176 1 1.170 9 3.809 7 5.174 9 2.982 4 4.910 2 8.335 2 5.534 0
    F6 最优值 2.26×10-3 4.5×10-3 3.89×10-25 0 0 0 4.50×1019 1.84×10-32
    平均值 3.38×10-3 1.16×10-1 4.18×10-16 8.32×10-21 0 0 1.15×10-18 8.60×10-3
    标准差 4.00×10-2 1.14×10-1 2.13×10-15 4.42×10-20 0 0 6.32×10-19 1.89×10-2
    耗时/s 0.794 4 0.776 5 3.312 0 3.932 3 2.619 2 4.354 3 8.082 8 3.476 1
    F7 最优值 5.12×10-5 1.09×10-5 1.19×10-5 1.98×10-4 9.60×10-5 3.50×10-5 3.62×10-4 4.63×10-6
    平均值 2.42×10-3 2.04×10-3 3.42×10-4 1.27×10-3 7.21×10-4 6.46×10-4 5.38×10-3 2.47×10-4
    标准差 2.82×10-3 3.27×10-3 2.69×10-4 1.11×10-3 5.28×10-4 3.57×10-4 3.44×10-3 1.93×10-4
    耗时/s 1.130 0 1.180 0 3.691 2 4.844 1 2.927 3 4.906 7 8.551 8 3.478 1
    下载: 导出CSV

    表  3  F8~F12算法测试结果比较

    Table  3.   Results comparison among test functions F8-F12 of algorithms

    测试函数 统计值 WOA IWOA VCS BSO PSO SSA GSA CML-WOA
    F8 最优值 0 0 0 0 0 0 0 0
    平均值 0 2.36×10-16 0 5.67 4.64×10-1 0 2.08 0
    标准差 0 1.29×10-15 0 4.34 5.04×10-1 0 1.59 0
    耗时/s 0.824 4 0.812 5 3.379 4 4.112 4 2.686 8 4.472 2 7.798 9 3.097 1
    F9 最优值 8.88×10-16 8.88×10-16 8.88×10-16 4.44×10-15 8.88×10-16 8.88×10-16 1.09×10-9 8.88×10-16
    平均值 3.84×10-15 3.37×10-15 8.88×10-16 6.80×10-1 3.01×10-15 8.88×10-16 2.29×10-9 8.88×10-16
    标准差 2.30×10-15 2.31×10-15 0 8.56×10-1 1.77×10-15 8.88×10-16 6.82×10-10 0
    耗时/s 1.139 8 1.118 9 3.660 8 5.039 3 2.990 2 4.954 5 8.690 0 3.366 7
    F10 最优值 0 0 0 2.71×10-2 0 0 9.36×10-2 0
    平均值 4.13×10-2 4.47×10-2 0 1.75×10-1 3.88×10-2 0 9.05×10-1 0
    标准差 9.08×10-2 1.12×10-1 0 1.03×10-1 2.67×10-2 0 6.32×10-1 0
    耗时/s 1.396 5 1.374 9 4.066 0 5.773 5 3.227 2 5.491 3 8.750 0 3.699 0
    F11 最优值 2.41×10-3 5.57×10-3 1.61×10-24 9.42×10-32 9.42×10-32 9.42×10-32 1.07×10-20 9.42×10-32
    平均值 1.61×10-1 2.27×10-1 4.91×10-18 2.44×10-1 9.42×10-32 9.42×10-32 5.29×10-20 4.25×10-5
    标准差 3.86×10-1 4.19×10-1 2.14×10-17 1.02 3.34×10-47 3.53×10-34 3.56×10-20 2.97×10-4
    耗时/s 3.155 6 3.140 2 5.822 1 10.999 5 4.959 2 8.483 5 10.533 0 6.326 9
    F12 最优值 3.35×10-3 2.29×10-2 3.02×10-23 1.34×10-32 3.44×10-32 1.34×10-32 2.74×10-20 1.34×10-32
    平均值 9.82×10-2 1.71×10-1 1.48×10-17 3.84×10-3 2.21×10-3 1.38×10-32 1.57×10-19 1.34×10-32
    标准差 6.99×10-2 1.11×10-1 4.88×10-17 1.04×10-2 4.20×10-3 1.25×10-33 1.05×10-19 5.56×10-48
    耗时/s 3.149 3 3.145 4 5.768 7 11.117 3 4.993 7 8.603 5 10.547 6 6.234 5
    下载: 导出CSV

    表  4  F13~F20算法测试结果比较

    Table  4.   Results comparison among test functions F13-F20 of algorithms

    测试函数 统计值 WOA IWOA VCS BSO PSO SSA GSA CML-WOA
    F13 最优值 9.98×10-1 9.98×10-1 9.98×10-1 9.98×10-1 9.98×10-1 9.98×10-1 9.98×10-1 9.98×10-1
    平均值 5.89 5.09 1.42 3.35 9.98×10-1 5.22 5.2 6.01
    标准差 4.93 3.95 2.13 3.45 5.82×10-17 5.49 3.79 5.12
    耗时/s 15.793 1 15.603 5 19.221 3 48.949 5 17.548 9 30.023 6 22.608 2 16.261 7
    F14 最优值 3.49×10-4 4.58×10-4 3.07×10-4 3.07×10-4 3.07×10-4 3.07×10-4 1.88×10-3 3.11×10-4
    平均值 3.59×10-3 5.08×10-3 5.65×10-4 7.71×10-3 5.95×10-4 3.37×10-4 5.51×10-3 4.08×10-4
    标准差 4.70×10-3 3.07×10-4 3.45×10-4 1.92×10-2 3.88×10-4 8.26×10-5 3.97×10-3 3.31×10-4
    耗时/s 0.974 8 0.953 1 3.491 5 4.506 8 2.796 9 4.593 2 8.054 0 3.124 3
    F15 最优值 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03
    平均值 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03 -1.03
    标准差 3.70×10-9 7.35×10-10 6.04×10-16 5.92×10-16 6.45×10-16 5.04×10-16 4.72×10-16 3.06×10-16
    耗时/s 0.651 0 0.573 5 3.118 7 3.270 6 2.408 0 4.057 0 6.912 4 3.552 3
    F16 最优值 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
    平均值 1.21×101 1.28×101 3.00 3.00 3.00 3.00 3.00 3.00
    标准差 1.93×101 1.77×101 1.716×10-15 2.28×10-15 1.56×10-15 1.37×10-15 3.94×10-15 1.06×10-15
    耗时/s 0.443 3 0.428 9 2.929 4 2.821 0 2.297 8 3.679 1 6.846 8 2.344 8
    F17 最优值 -3.86 -3.86 -3.86 -3.71 -3.86 -3.86 -3.86 -3.86
    平均值 -3.75 -3.76 -3.86 -3.18 -3.86 -3.86 -3.86 -3.86
    标准差 1.02×10-1 1.54×10-1 2.69×10-15 4.20×10-1 2.68×10-15 2.32×10-15 2.31×10-15 2.17×10-15
    耗时/s 1.716 9 1.714 9 4.247 6 6.941 1 3.571 1 6.145 1 8.411 6 3.473 2
    F18 最优值 -3.27 -3.14 -3.32 -2.73 -3.32 -3.32 -3.32 -3.32
    平均值 -3.01 -2.70 -3.29 -1.35 -3.27 -3.27 -3.31 -3.30
    标准差 1.77×10-1 4.83×10-1 5.34×10-2 7.33×10-1 5.83×10-2 5.92×10-2 2.22×10-2 5.79×10-2
    耗时/s 1.773 1 1.170 8 4.322 7 6.934 4 3.630 3 6.176 7 9.296 6 4.463 4
    F19 最优值 -9.81 -10.12 -10.15 -1.72 -10.15 -10.15 -10.15 -10.15
    平均值 -5.86 -4.71 -10.15 -6.8×10-1 -5.97 -8.18 -6.94 -10.15
    标准差 2.30 2.42 6.56×10-15 3.38×10-1 3.55 2.49 3.72 6.35×10-15
    耗时/s 3.210 0 3.190 9 5.737 1 11.526 2 5.126 5 8.617 4 10.264 8 5.079 3
    F20 最优值 -10.11 -10.42 -10.53 -4.57 -10.53 -10.53 -10.53 -10.53
    平均值 -4.17 -3.68 -10.53 -1.00 -7.79 -8.88 -9.68 -10.53
    标准差 2.37 2.19 2.91×10-15 8.32×10-1 3.71 2.65 2.34 2.51×10-15
    耗时/s 6.000 4 6.001 4 8.826 4 19.921 0 7.758 9 13.714 2 12.980 1 12.753 2
    下载: 导出CSV

    表  5  算法性能排序结果

    Table  5.   Results of sort of algorithm performance

    算法 平均排序 算法 平均排序
    WOA 5.9 PSO 4.35
    IWOA 6.4 SSA 3.1
    VCS 3.2 GSA 5
    BSO 6.2 CML-WOA 1.85
    下载: 导出CSV

    表  6  Wilcoxon统计检验结果

    Table  6.   Results of statistical tests of Wilcoxon

    算法 WOA IWOA VCS BSO PSO SSA GSA
    F1 - - - - - - -
    F2 - - - - - - -
    F3 - - - - - - -
    F4 - - - - - - -
    F5 - - + - = = -
    F6 - - = + + + =
    F7 - - - - - - -
    F8 = = = - - - -
    F9 - - = - = - -
    F10 = = - - - = -
    F11 - - - - + + -
    F12 - - - = = - =
    F13 = = = = + = =
    F14 - - = = = = -
    F15 - - - = - - -
    F16 - - - - - - -
    F17 - - = - = - -
    F18 = = + = + + +
    F19 - - = - - - -
    F20 - - = - - - -
    (-/=/+)
    16/4/0 16/4/0 10/8/2 14/5/1 11/5/4 13/4/3 16/3/1
    注:表中“+”表示该算法性能明显优于CML-WOA;“=”表明该算法性能与CML-WOA无明显差别;“-”表示该算法性能明显劣于CML-WOA。
    下载: 导出CSV

    表  7  威胁源参数

    Table  7.   Parameters of threat source

    威胁源 威胁类型 位置/km 作用半径/km 高度/km
    Threat 1 探测雷达 (25,20) 9.8 2.8
    Threat 2 地空导弹 (55,50) 11.7 2.9
    Threat 3 防空高炮 (85,45) 8.9 3.0
    Threat 4 地空导弹 (45,70) 8.0 3.0
    Threat 5 探测雷达 (24.5,50) 10.0 2.8
    Threat 6 防空高炮 (75,80) 12.0 3.1
    下载: 导出CSV

    表  8  20维下的规划代价值

    Table  8.   Planning cost in 20 dimensions

    航迹节点数 结果 WOA IWOA PSO GWO CML-WOA
    20 平均适应值 2.31×104 2.22×104 2.00×104 2.01×104 8.32×101
    标准差 8.84×103 7.35×103 6.35×103 8.21×103 2.92
    最大适应值 5.29×104 5.16×104 5.58×104 6.08×104 86.237 8
    最小适应值 80.709 8 79.061 2 76.668 6 75.486 7 73.773 7
    成功率 0.65 0.60 0.55 0.60 1.00
    耗时/s 10.477 7 11.153 1 10.606 1 11.082 9 13.256 1
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-10-15
  • 录用日期:  2021-01-22
  • 网络出版日期:  2021-07-20

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