Journal of System Simulation ›› 2025, Vol. 37 ›› Issue (3): 803-821.doi: 10.16182/j.issn1004731x.joss.23-1392
• Papers • Previous Articles
Jin Xu1, Mo Yuanbin1,2
Received:2023-11-16
Revised:2024-01-26
Online:2025-03-17
Published:2025-03-21
Contact:
Mo Yuanbin
CLC Number:
Jin Xu, Mo Yuanbin. Multi-strategy Hybrid Mountain Gazelle Optimizer for Robot Path Planning[J]. Journal of System Simulation, 2025, 37(3): 803-821.
Fig. 1
Mountain gazelle optimizer flowchart
Fig. 2
Convergence factor curve diagram
Fig. 3
The comparative plot of the new population
Fig. 4
Hyperbolic tangent function and its opposite function
Fig. 5
Scaling factor function and its transition function
Fig. 6
Flowchart of HMGO
Table 1
Description comparison algorithm
| 算法 | 功能描述 |
|---|---|
| MGO | 标准山地瞪羚优化器 |
| MGO1 | HMGO去掉2.1节初始化种群策略 |
| MGO2 | HMGO去掉2.2节动态自适应密度因子 |
| MGO3 | HMGO去掉2.3节缩放因子策略改善相对位移 |
| MGO4 | HMGO去掉2.4节混合改进的算术优化算法 |
| HMGO | 混合多种策略改进的山地瞪羚优化器 |
Table 2
Ablation experiment results
| 函数 | 指标 | MGO | MGO1 | MGO2 | MGO3 | MGO4 | HMGO |
|---|---|---|---|---|---|---|---|
| F1 | 均值 | 3.177 | 5.910 | 0 | 0 | 1.196 | 0 |
| 标准差 | 1.251 | 0 | 0 | 0 | 4.292 | 0 | |
| F2 | 均值 | 1.316 | 0 | 0 | 0 | 1.869 | 0 |
| 标准差 | 3.531 | 0 | 0 | 0 | 7.750 | 0 | |
| F3 | 均值 | 3.934 | 0 | 0 | 0 | 5.941 | 0 |
| 标准差 | 4.421 | 0 | 0 | 0 | 6.552 | 0 | |
| F4 | 均值 | 3.332×10-21 | 0 | 0 | 0 | 1.116×10-36 | 0 |
| 标准差 | 1.111×10-20 | 0 | 0 | 0 | 1.420×10-71 | 0 | |
| F5 | 均值 | 5.575×10-18 | 6.867×10-4 | 5.437×10-4 | 2.193×10-22 | 1.768×10-4 | 3.491×10-4 |
| 标准差 | 9.325×10-34 | 2.278E×10-6 | 7.587E×10-7 | 1.396×10-42 | 6.319×10-8 | 2.827×10-7 | |
| F6 | 均值 | 1.811×10-8 | 3.457×10-9 | 3.874×10-9 | 4.327×10-9 | 8.421×10-9 | 1.665×10-9 |
| 标准差 | 3.690×10-15 | 3.698×10-17 | 2.604×10-16 | 1.667×10-18 | 1.263×10-16 | 9.701×10-18 | |
| F7 | 均值 | 4.539×10-4 | 4.685×10-5 | 4.872×10-5 | 7.807×10-5 | 2.100×10-4 | 4.195×10-5 |
| 标准差 | 1.309×10-7 | 1.139×10-9 | 1.305×10-9 | 3.875×10-9 | 7.905×10-8 | 1.242×10-9 | |
| F8 | 均值 | -1.257×104 | -1.259×104 | -1.259×104 | -1.259×104 | -1.259×104 | -1.260×104 |
| 标准差 | 3.999×10-8 | 2.324×10-7 | 1.709×10-8 | 3.377×10-10 | 1.026×10-9 | 3.382×10-8 | |
| F9 | 均值 | 0 | 0 | 0 | 0 | 0 | 0 |
| 标准差 | 0 | 0 | 0 | 0 | 0 | 0 | |
| F10 | 均值 | 1.717×10-15 | 8.882×10-16 | 8.882×10-16 | 8.883×10-16 | 1.480×10-15 | 8.882×10-16 |
| 标准差 | 2.336×10-30 | 0 | 0 | 0 | 1.814×10-30 | 0 | |
| F11 | 均值 | 0 | 0 | 0 | 0 | 0 | 0 |
| 标准差 | 0 | 0 | 0 | 0 | 0 | 0 | |
| F12 | 均值 | 3.124×10-23 | 2.185×10-8 | 6.220×10-8 | 1.845×10-23 | 7.306×10-9 | 3.060×10-11 |
| 标准差 | 4.954×10-50 | 1.314×10-15 | 8.081×10-15 | 6.545×10-44 | 1.404×10-16 | 2.221×10-17 | |
| F13 | 均值 | 1.958×10-32 | 1.318×10-6 | 7.664×10-7 | 2.435×10-32 | 7.134×10-8 | 3.164×10-7 |
| 标准差 | 3.127×10-64 | 2.828×10-11 | 1.653×10-12 | 1.359×10-65 | 1.761×10-14 | 1.834×10-13 |
Table 3
13 benchmark Functions
| 序号 | 标准测试函数 | 搜索空间 | 最优值 |
|---|---|---|---|
| F1 | Sphere Function | [-100,100] | 0 |
| F2 | Schwefel's Problem 1.2 | [-10,10] | 0 |
| F3 | Quadnc | [-100,100] | 0 |
| F4 | Schwefel's Problem 1.22 | [-100,100] | 0 |
| F5 | Generalized Rosenbrock's Function | [-30,30] | 0 |
| F6 | Step Function | [-100,100] | 0 |
| F7 | Quartic Function i.e. Noise | [-1.28,1.28] | 0 |
| F8 | Generalized Schwefel's Problem 2.26 | [-500,500] | 0 |
| F9 | Generalized Rastrigin's Function | [-5.12,5.12] | 0 |
| F10 | Ackley's Function | [-32,32] | 0 |
| F11 | Generalized Griewank's Function | [-600,600] | 0 |
| F12 | Generalized Penalized Function1 | [-50，50] | 0 |
| F13 | Generalized Penalized Function2 | [-50，50] | 0 |
Table 4
Parameters setting of the algorithm
| 算法 | 参数设置 |
|---|---|
| PSO | F=0.3；vmax=6； |
| GSA | α=20；G0=100；R=1 |
| AOA | Min =0.2；Max =1；μ=0.499；α=5 |
| AMGO | 混沌漂移-Alpa=0.7 |
| NGO | I = [ |
| CSA | Pp=0.1； |
| GWO | a =[2,0] |
Table 5
Comparison with other swarm intelligence algorithms
| 函数 | 评价指标 | 最优值 | 最差值 | 平均值 | 标准差 | 时间/s |
|---|---|---|---|---|---|---|
| F1 | PSO | 4.996×10-2 | 6.326×10-1 | 2.551×10-1 | 1.430×10-1 | 1.615×10-1 |
| GSA | 5.033×10-17 | 4.441×10-16 | 1.248×10-16 | 8.188×10-17 | 1.877 | |
| AOA | 3.977×10-290 | 3.578×10-62 | 1.431×10-45 | 6.533×10-63 | 2.141×10-1 | |
| NGO | 5.369×10-183 | 9.526×10-179 | 2.657×10-179 | 0 | 3.149×10-1 | |
| CSA | 2.404×10-3 | 3.823×10-1 | 8.330×10-2 | 9.017×10-2 | 1.497×10-1 | |
| GWO | 3.443×10-62 | 5.754×10-58 | 5.327×10-59 | 1.042×10-58 | 2.761×10-1 | |
| MGO | 2.378×10-146 | 5.754×10-58 | 5.189×10-59 | 1.041×10-58 | 2.709 | |
| AMGO | 1.626×10-159 | 4.130×10-140 | 1.377×10-141 | 7.540×10-141 | 2.194 | |
| HMGO | 0 | 0 | 0 | 0 | 4.601 | |
| F2 | PSO | 5.238×10-1 | 1.825 | 9.845×10-1 | 3.407×10-1 | 1.851×10-1 |
| GSA | 3.465×10-8 | 9.244×10-8 | 5.365×10-8 | 1.388×10-8 | 1.851×10-1 | |
| AOA | 0 | 0 | 0 | 0 | 2.333×10-1 | |
| NGO | 4.560×10-94 | 5.004×10-92 | 1.223×10-92 | 1.278×10-92 | 3.729×10-1 | |
| CSA | 1.680 | 1.393×10 | 5.495 | 2.860 | 1.763×10-1 | |
| GWO | 1.003×10-35 | 3.888×10-34 | 7.661×10-35 | 8.168×10-35 | 3.041×10-1 | |
| MGO | 5.325×10-90 | 3.888×10-34 | 7.589×10-35 | 8.227×10-35 | 3.179 | |
| AMGO | 7.171×10-90 | 8.478×10-82 | 3.110×10-83 | 1.545×10-82 | 2.485 | |
| HMGO | 0 | 0 | 0 | 0 | 4.522 | |
| F3 | PSO | 3.220×10 | 1.343×102 | 7.765×10 | 2.646×10 | 1.077 |
| GSA | 1.882×102 | 7.716×102 | 4.417×102 | 1.566×102 | 1.077 | |
| AOA | 6.219×10-221 | 1.731×10-2 | 1.749×10-3 | 4.709×10-3 | 9.931×10-1 | |
| NGO | 1.400×10-58 | 7.704×10-47 | 3.104×10-48 | 1.424×10-47 | 1.954 | |
| CSA | 5.990×102 | 3.284×103 | 1.495×103 | 6.646×102 | 9.916×10-1 | |
| GWO | 1.400×10-20 | 2.875×10-14 | 3.692×10-15 | 7.895×10-15 | 1.196 | |
| MGO | 1.024×10-21 | 2.875×10-14 | 3.100×10-15 | 7.456×10-15 | 7.154 | |
| AMGO | 9.948×10-28 | 5.139×10-17 | 3.118×10-18 | 1.035×10-17 | 3.508 | |
| HMGO | 0 | 0 | 0 | 0 | 5.780 | |
| F4 | PSO | 1.021 | 1.845 | 1.535 | 2.076×10-1 | 1.606×10-1 |
| GSA | 9.517×10-9 | 3.121 | 1.096 | 1.052 | 1.606×10-1 | |
| AOA | 1.361×10-106 | 6.836×10-2 | 2.011×10-2 | 2.078×10-2 | 2.095×10-1 | |
| NGO | 5.175×10-78 | 7.663×10-76 | 1.557×10-76 | 1.859×10-76 | 3.265×10-1 | |
| CSA | 7.844 | 1.853×10 | 1.315×10 | 2.887 | 1.569×10-1 | |
| GWO | 5.176×10-16 | 2.136×10-13 | 2.503×10-14 | 4.091×10-14 | 2.899×10-1 | |
| MGO | 5.711×10-55 | 2.136×10-13 | 2.448×10-14 | 4.114×10-14 | 2.798 | |
| AMGO | 1.937×10-57 | 1.896×10-46 | 1.335×10-47 | 4.097×10-47 | 2.114 | |
| HMGO | 0 | 0 | 0 | 0 | 4.667 | |
| F5 | PSO | 9.487×10 | 5.793×102 | 2.456×102 | 1.196×102 | 2.000×10-1 |
| GSA | 2.576×10 | 8.237×10 | 2.821×10 | 1.023×10 | 2.000×10-1 | |
| AOA | 2.692×10 | 2.878×10 | 2.810×10 | 4.325×10-1 | 2.445×10-1 | |
| NGO | 2.194×10 | 2.452×10 | 2.359×10 | 5.229×10-1 | 4.403×10-1 | |
| CSA | 2.940×10 | 1.388×103 | 2.483×102 | 3.014×102 | 1.918×10-1 | |
| GWO | 2.522×10 | 2.875×10 | 2.685×10 | 8.174×10-1 | 3.181×10-1 | |
| MGO | 1.294×10-30 | 2.875×10 | 2.592×10 | 4.959 | 3.116 | |
| AMGO | 0 | 2.360×10-29 | 1.800×10-30 | 4.887×10-30 | 2.169 | |
| HMGO | 6.727×10-10 | 8.679×10-4 | 1.602×10-4 | 1.988×10-4 | 4.871 | |
| F6 | PSO | 7.924×10-2 | 7.766×10-1 | 3.559×10-1 | 1.985×10-1 | 1.585×10-1 |
| GSA | 5.274×10-10 | 3.484×10-5 | 3.227×10-6 | 6.860×10-7 | 1.585×10-1 | |
| AOA | 2.238 | 3.326 | 2.785 | 2.591×10-1 | 2.112×10-1 | |
| NGO | 9.245×10-9 | 2.078×10-5 | 7.110×10-5 | 5.558×10-6 | 3.317×10-1 | |
| CSA | 4.252×10-3 | 2.291×10-1 | 6.376×10-2 | 6.886×10-2 | 1.662×10-1 | |
| GWO | 3.361×10-5 | 1.246 | 5.895×10-1 | 3.096×10-1 | 3.219×10-1 | |
| MGO | 5.571×10-17 | 1.246 | 5.561×10-1 | 3.175×10-1 | 2.890 | |
| AMGO | 5.542×10-8 | 2.282×10-5 | 2.199×10-6 | 4.143×10-6 | 2.008 | |
| HMGO | 8.000×10-22 | 2.728×10-12 | 1.023×10-13 | 4.982×10-13 | 5.281 | |
| F7 | PSO | 4.985×10-1 | 8.132 | 2.711 | 1.937 | 3.109×10-1 |
| GSA | 2.709×10-2 | 1.268×10-1 | 5.499×10-2 | 2.283×10-2 | 3.109×10-1 | |
| AOA | 8.382×10-7 | 1.127×10-4 | 2.896×10-5 | 3.040×10-5 | 3.735×10-1 | |
| NGO | 3.233×10-5 | 5.632×10-4 | 2.741×10-4 | 1.323×10-4 | 6.027×10-1 | |
| CSA | 1.202×10-1 | 9.775×10-1 | 5.537×10-1 | 2.573×10-1 | 2.899×10-1 | |
| GWO | 2.849×10-4 | 3.500×10-3 | 1.028×10-3 | 7.190×10-4 | 4.190×10-1 | |
| MGO | 3.164×10-5 | 3.500×10-3 | 1.019×10-3 | 7.294×10-4 | 3.307 | |
| AMGO | 4.101×10-5 | 6.953×10-4 | 2.226×10-4 | 1.670×10-4 | 2.856 | |
| HMGO | 1.826×10-7 | 7.703×10-5 | 1.921×10-5 | 1.766×10-5 | 6.559 | |
| F8 | PSO | -8.503×103 | -3.902×103 | -6.788×103 | 1.232×103 | 2.234×10-1 |
| GSA | -3.702×103 | -2.020×103 | -2.569×103 | 4.028×102 | 2.234×10-1 | |
| AOA | -6.378×103 | -4.978×103 | -5.709×103 | 4.098×102 | 2.517×10-1 | |
| NGO | -9.385×103 | -6.968×103 | -8.145×103 | 6.393×102 | 4.287×10-1 | |
| CSA | -1.235×104 | -5.776×103 | -8.043×103 | 2.835×103 | 1.983×10-1 | |
| GWO | -7.840×103 | -3.610×103 | -6.090×103 | 8.241×102 | 3.277×10-1 | |
| MGO | -1.257×104 | -3.610×103 | -6.296×103 | 1.442×103 | 2.920 | |
| AMGO | -3.764×106 | -1.933×104 | -9.315×105 | 1.101×106 | 2.239 | |
| HMGO | -1.257×104 | -1.257×104 | -1.257×104 | 7.027×10-4 | 4.947 | |
| F9 | PSO | 5.444×10 | 2.122×102 | 1.070×102 | 3.777×10 | 1.850×10-1 |
| GSA | 1.194×10 | 4.378×10 | 2.623×10 | 6.550 | 1.850×10-1 | |
| AOA | 0 | 0 | 0 | 0 | 2.009×10-1 | |
| NGO | 0 | 0 | 0 | 0 | 3.246×10-1 | |
| CSA | 1.592×10 | 5.572×10 | 3.303×10 | 1.057×10 | 1.647×10-1 | |
| GWO | 0 | 5.836 | 4.140×10-1 | 1.359 | 2.798×10-1 | |
| MGO | 0 | 5.836 | 4.140×10-1 | 1.359 | 2.724 | |
| AMGO | 0 | 0 | 0 | 0 | 2.292 | |
| HMGO | 0 | 0 | 0 | 0 | 4.791 | |
| F10 | PSO | 2.775×10-1 | 2.950 | 1.033 | 5.554×10-1 | 2.104×10-1 |
| GSA | 4.747×10-9 | 1.202×10-8 | 8.146×10-9 | 1.972×10-9 | 2.104×10-1 | |
| AOA | 8.882×10-16 | 8.882×10-16 | 8.882×10-16 | 0 | 2.377×10-1 | |
| NGO | 4.441×10-15 | 7.994×10-15 | 5.744×10-15 | 1.741×10-15 | 3.938×10-1 | |
| CSA | 1.997×10 | 1.997×10 | 1.997×10 | 2.308×10-13 | 2.069×10-1 | |
| GWO | 1.155×10-14 | 2.931×10-14 | 1.605×10-14 | 3.605×10-15 | 3.090×10-1 | |
| MGO | 8.882×10-16 | 2.931×10-14 | 1.557×10-14 | 4.545×10-15 | 2.912 | |
| AMGO | 8.882×10-16 | 4.441×10-15 | 1.007×10-15 | 6.486×10-16 | 2.415 | |
| HMGO | 8.882×10-16 | 8.882×10-16 | 8.882×10-16 | 0 | 4.822 | |
| F11 | PSO | 2.213×10-3 | 6.051×10-2 | 2.433×10-2 | 1.742×10-2 | 2.331×10-1 |
| GSA | 2.616 | 2.065×10 | 9.090 | 4.308 | 2.331×10-1 | |
| AOA | 1.315×10-4 | 3.848×10-1 | 9.922×10-2 | 8.343×10-2 | 2.581×10-1 | |
| NGO | 0 | 0 | 0 | 0 | 4.307×10-1 | |
| CSA | 1.302×10-1 | 1.100 | 6.446×10-1 | 2.690×10-1 | 2.150×10-1 | |
| GWO | 0 | 1.896×10-2 | 2.185×10-3 | 5.774×10-3 | 3.287×10-1 | |
| MGO | 0 | 1.896×10-2 | 2.185×10-3 | 5.774×10-3 | 2.985 | |
| AMGO | 0 | 0 | 0 | 0 | 2.492 | |
| HMGO | 0 | 0 | 0 | 0 | 4.459 | |
| F12 | PSO | 3.879×10-4 | 1.053×10-1 | 5.434×10-3 | 1.891×10-2 | 6.203×10-1 |
| GSA | 2.993×10-19 | 2.220 | 2.581×10-1 | 4.896×10-1 | 6.203×10-1 | |
| AOA | 3.252×10-1 | 5.285×10-1 | 3.893×10-1 | 4.583×10-2 | 6.240×10-1 | |
| NGO | 7.953×10-10 | 1.863×10-8 | 4.013×10-9 | 4.298×10-9 | 1.173 | |
| CSA | 2.603 | 8.923 | 5.923 | 1.586 | 5.721×10-1 | |
| GWO | 1.316×10-2 | 1.236×10-1 | 4.258×10-2 | 2.726×10-2 | 7.194×10-1 | |
| MGO | 1.571×10-32 | 1.236×10-1 | 4.125×10-2 | 2.835×10-2 | 4.692 | |
| AMGO | 5.460×10-31 | 2.744×10-9 | 1.522×10-9 | 3.217×10-9 | 4.547 | |
| HMGO | 1.571×10-32 | 1.571×10-32 | 1.571×10-32 | 5.567×10-48 | 6.841 | |
| F13 | PSO | 1.057×10-2 | 2.911×10-1 | 8.818×10-2 | 5.618×10-2 | 5.827×10-1 |
| GSA | 5.470×10-18 | 2.592 | 1.652×10-1 | 5.180×10-1 | 5.827×10-1 | |
| AOA | 2.379 | 2.945 | 2.803 | 1.382×10-1 | 6.031×10-1 | |
| NGO | 1.269×10-7 | 4.733×10-1 | 1.174×10-1 | 1.219×10-1 | 1.108 | |
| CSA | 2.578 | 7.534×10 | 4.304×10 | 1.950×10 | 5.422×10-1 | |
| GWO | 9.656×10-2 | 1.021 | 5.431×10-1 | 2.475×10-1 | 6.588×10-1 | |
| MGO | 1.350×10-32 | 1.021 | 5.363×10-1 | 2.597×10-1 | 4.197 | |
| AMGO | 1.159×10-8 | 1.148×10-5 | 1.809×10-6 | 2.874×10-6 | 4.378 | |
| HMGO | 1.350×10-32 | 1.350×10-32 | 1.350×10-32 | 5.567×10-48 | 6.881 |
Fig. 7
Comparison of high-dimensional optimization curves of 9 algorithms
Table 6
Wilcoxon rank sum test results
| 函数 | PSO | GSA | AOA | NGO | CSA | GWO | AMGO | MGO |
|---|---|---|---|---|---|---|---|---|
| F1 | -38.7 (<) | -38.7 (<) | -37.4 (<) | -38.4 (<) | -38.8 (<) | -38.7 (<) | -34.8 (<) | -35.4 (<) |
| F2 | -13.5 (<) | -38.7 (<) | -38.7 (<) | -38.7 (<) | -38.7 (<) | -38.6 (<) | -38.7 (<) | -38.7 (<) |
| F3 | -38.7 (<) | -38.7 (<) | -30.2 (<) | -6.83 (>) | -38.8 (<) | -38.7 (<) | -37.4 (<) | -0.583 (<) |
| F4 | -38.7 (<) | -38.7 (<) | -40.4 (<) | -38.7 (<) | -38.7 (<) | -38.8 (<) | -38.7 (<) | -38.4 (<) |
| F5 | -38.7 (<) | -38.7 (<) | -38.8 (<) | -38.7 (<) | -38.8 (<) | -38.7 (<) | -14.6 (<) | -37.9 (<) |
| F6 | -38.7 (<) | -38.7 (<) | -38.7 (<) | -38.7 (<) | -38.8 (<) | -38.7 (<) | -22.2 (<) | -34.5 (<) |
| F7 | -39.3 (<) | -39.2 (<) | -14.6 (<) | -39.0 (<) | -38.8 (<) | -38.8 (<) | -40.1 (<) | -39.6 (<) |
| F8 | -38.7 (<) | -39.9 (<) | -39.9 (<) | -38.7 (<) | -38.8 (<) | -38.7 (<) | -39.9 (>) | -38.7 (<) |
| F9 | -38.8 (<) | -38.7 (<) | -38.3 (<) | -8.85 (<) | -38.8 (<) | -40.0 (<) | -10.4 (<) | -11.1 (<) |
| F10 | -38.7 (<) | -38.7 (<) | -38.6 (<) | -42.9 (<) | -38.9 (<) | -39.7 (<) | -39.3 (<) | -15.8 (<) |
| F11 | -38.7 (<) | -38.7 (<) | -41.2 (<) | -9.74 (<) | -38.8 (<) | -40.7 (<) | -12.0 (<) | -9.49 (<) |
| F12 | -38.7 (<) | -38.7 (<) | -40.2 (<) | -38.7 (<) | -38.7 (<) | -38.7 (<) | -21.7 (<) | -33.0 (<) |
| F13 | -38.7 (<) | -38.7 (<) | -39.2 (<) | -38.7 (<) | -38.7 (<) | -38.8 (<) | -24.7 (=) | -38.8 (<) |
Fig. 8
20 m×20 m map
Fig. 9
40 m×40 m map
Table 7
Statistical results of path planning for a 20 m×20 m map
| 算法 | 最小值 | 最大值 | 平均值 | 标准差 |
|---|---|---|---|---|
| PSO | 28.0 | 36.3 | 29.5 | 0.528 |
| AOA | 30.4 | 32.4 | 30.8 | 0.626 |
| SSA | 28.0 | 35.2 | 28.2 | 0.902 |
| GWO | 28.6 | 35.8 | 29.9 | 2.150 |
| IAFO | 28.0 | 48.0 | 28.1 | 0.895 |
| AMGO | 26.9 | 30.3 | 28.8 | 0.800 |
| MGO | 27.5 | 35.2 | 28.0 | 0.730 |
| HMGO | 26.9 | 26.9 | 26.9 | 0 |
Fig. 10
Path planning in an environment of 20 m×20 m
Fig. 11
20 m×20 m Iterative curves
Table 8
Statistical results of path planning for a 40 m×40 m map
| 算法 | 最小值 | 最大值 | 平均值 | 标准差 |
|---|---|---|---|---|
| PSO | 82.2 | 134 | 101 | 6.47 |
| AOA | 83.3 | 112 | 96.5 | 8.04 |
| SSA | 57.5 | 57.5 | 57.5 | 1.14×10⁻13 |
| GWO | 71.2 | 145 | 83.3 | 18.7 |
| IAFO | 70 | 211 | 71.9 | 14 |
| AMGO | 68.9 | 211 | 74.4 | 13.6 |
| MGO | 69.5 | 211 | 75.3 | 13.4 |
| HMGO | 56.9 | 57.5 | 57.1 | 0.275 |
Fig. 12
Path planning in an environment of 40 m×40 m
Fig. 13
40 m×40 m Iterative curves
Fig. 14
3-dimensional space meshing
Table 9
Environmental parameters
| 编号 | 障碍物坐标 |
|---|---|
| 1 | [0, -2, 0][0, -2, 1.5][0, -3.5, 0][0, -3.5, 1.5][10, -2, 0][10, -2, 1.5][10, -3.5, 0][10, -3.5, 1.5] |
| 2 | [0, -2, 3][0, -2, 9][0, -3.5, 3][0, -3.5, 9][10, -2, 3][10, -2, 9][10, -3.5, 3][10, -3.5, 9] |
| 3 | [0, 2, 0][0, 2, 1.5][0, 4.5, 0][0, 4.5, 1.5][10, 2, 0][10, 2, 1.5][10, 4.5, 0][10, 4.5, 1.5] |
| 4 | [0, 2, 4.5][0, 2, 10.5][0, 4.5, 4.5][0, 4.5, 10.5][10, 2, 4.5][10, 2, 10.5][10, 4.5, 4.5][10, 4.5, 10.5] |
| 5 | [0, 7, 0][0, 7, 0.5][0, 14.5, 0][0, 14.5, 0.5][10, 7, 0][10, 7, 0.5][10, 14.5, 0][10, 14.5, 0.5] |
| 6 | [0, 11, 0][0, 11, 2.5][0, 22.5, 0][0, 22.5, 2.5][10, 11, 0][10, 11, 2.5][10, 22.5, 0][10, 22.5, 2.5] |
| 7 | [0, 11, 4][0, 11, 10][0, 22.5, 4][0, 22.5, 10][10, 11, 4][10, 11, 10][10, 22.5, 4][10, 22.5, 10] |
| 8 | [0, -2, 0][0, -2, 1.5][0, -3.5, 0][0, -3.5, 1.5][10, -2, 0][10, -2, 1.5][10, -3.5, 0][10, -3.5, 1.5] |
| 9 | [0, -2, 3][0, -2, 9][0, -3.5, 3][0, -3.5, 9][10, -2, 3][10, -2, 9][10, -3.5, 3][10, -3.5, 9] |
| 10 | [0, 2, 1.5][0, 2, 6][0, 4.5, 1.5][0, 4.5, 6][3, 2, 1.5][3, 2, 6][3, 4.5, 1.5][3, 4.5, 6] |
| 11 | [7, 2, 1.5][7, 2, 6][7, 4.5, 1.5][7, 4.5, 6][17, 2, 1.5][17, 2, 6][17, 4.5, 1.5][17, 4.5, 6] |
| 12 | [3, 0, 2.4][3, 0, 6.9][3, 0.5, 2.4][3, 0.5, 6.9][10, 0, 2.4][10, 0, 6.9][10, 0.5, 2.4][10, 0.5, 6.9] |
| 13 | [0, 15, 0][0, 15, 1][0, 35, 0][0, 35, 1][10, 15, 0][10, 15, 1][10, 35, 0][10, 35, 1] |
| 14 | [0, 15, 1][0, 15, 4.5][0, 31, 1][0, 31, 4.5][10, 15, 1][10, 15, 4.5][10, 31, 1.0][10, 31, 4.5] |
| 15 | [0, 18, 4.5][0, 18, 10.5][0, 37, 4.5][0, 37, 10.5][10, 18, 4.5][10, 18, 10.5][10, 37, 4.5][10, 37, 10.5] |
Fig. 15
3D space simulation results
Table 10
Statistical results of 3D space path planning
| 算法 | 最小值 | 最大值 | 平均值 | 标准差 | 平均时间 |
|---|---|---|---|---|---|
| AOA | 3.28×10² | 3.68×10² | 3.32×10² | 7.50×10⁰ | 3.28×10² |
| HMGO | 3.09×10² | 3.75×10² | 3.15×10² | 7.03×10⁰ | 6.23×102 |
| AMGO | 3.11×10² | 4.33×10² | 3.17×10² | 9.50×10⁰ | 2.25×10² |
| IAFO | 3.15×10² | 3.79×10² | 3.19×10² | 1.24×10¹ | 3.15×10² |
| GWO | 3.33×10² | 3.82×10² | 3.34×10² | 5.38×10⁰ | 2.19×10¹ |
| PSO | 3.34×10² | 1.12×10³ | 3.55×10² | 5.21×10¹ | 3.34×10² |
| MGO | 3.14×10² | 4.06×10² | 3.22×10² | 1.30×10¹ | 3.14×10² |
| SSA | 3.77×10² | 3.97×10² | 3.78×10² | 3.62×10⁰ | 3.77×10² |
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