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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。
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关键词:
- 航迹规划 /
- 鲸鱼优化算法(WOA) /
- 混沌算子 /
- 多精英搜索 /
- 正余弦算法(SCA) /
- 全局优化
Abstract:Aimed at the path planning problem of Unmanned Aerial Vehicle (UAV), a path planning method based on a Chaotic Multi-Leader Whale Optimization Algorithm (CML-WOA) is proposed. In the known flight environment, 3D model of flight area and a flight path cost model are established. By introducing penalty functions, the constrained 3D path planning problem is transformed into an unconstrained multi-dimensional function optimization problem, which is solved using CML-WOA to obtain the optimal flight path. To overcome the defect that the WOA is easy to fall into local optimum, this paper introduces the cubic mapping chaos operator to improve the initial population and enhance the population diversity. And the Sine Cosine Algorithm (SCA) is integrated through an adaptive framework. A multi-leader search strategy is used to effectively improve the algorithm exploitation and exploration capability. Finally, a greedy strategy is used to ensure the convergence efficiency. The proposed improved CML-WOA is tested and validated by 20 benchmark functions test and path planning simulation experiments. The results show that the algorithm has significantly improved performance compared to other algorithms, with strong local optimal avoidance capability, higher convergence accuracy and convergence speed. Also, it is able to provide stable and fast planning of safe and feasible flight path with minimum cost and satisfying constraints.
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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
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 
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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
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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
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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
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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
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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。
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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
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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
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