## 3 Materials and Methods
### Working Mechanism of the Method
### Analysis and Evaluation (Convergence Curve)

Figure 9 presents the convergence curve of each proposed path planning algorithm. As mentioned before, the number of obstacles and the boundary sizes of the map are listed in Figure 3. The three metaheuristic algorithms used have different structures in the exploration and exploitation phases. Figure 9 illustrates the convergence curve of each method with various iterations and population sizes. In the I-GWO algorithm, the transition from exploration to the exploitation phase is faster than the other two metaheuristic algorithms (GWO and Ex-GWO). As a result of the observations, it was concluded that 50 iterations were enough to analyze of convergence rate because the results achieved did not display remarkable differences [20]. Figure 8: Taken time for each method on the relevant map. In Figure 10, the statistical results of the path planning methods with the boxplot graph are presented. Boxplots are a standard method for displaying data distribution based on statistical indicators such as minimum, first quartile (Q1), median, third quartile (Q3), and maximum. This diagram also provides information regarding the existence of outlier data. In addition, the symmetry ratio in the data can be analyzed from this graph. The values were obtained from three metaheuristic algorithms with a population size of 30 and 50 iterations after 15 runs. The box plot graph analysis describes the maximum and minimum values of the obtained best cost and the frequency of the values. The \(x\)-axis of each figure indicates the name of the respective algorithm, whereas the \(y\)-axis indicates the average of best cost obtained. From Figure 10, it can be observed that the results obtained using the Ex-GWO algorithm are near to the best solution, whilst the algorithm tries to find the best solution. As well as this, after initial iterations in the exploitation phase the Ex-GWO obtained results near to the best cost. Figure 11 illustrates the distributions of costs in 15 runs. While the UAV\({}_{2}\) has an almost uniform distribution, both UAV\({}_{1}\) and UAV\({}_{3}\) have lower cost densities. Note that

Figure 10: Boxplot graph analysis for each UAV in population of 30 and 50 iterations. Figure 9: Convergence analysis for each UAV on the relevant map in population of 30 and 50 iterations. obstacles are employed as a marker for these metaheuristic algorithms, and paths with an appropriate number of obstacles help to improve the performances. For this purpose, a Student's t-test was applied for all combinations of UAVs. The \(p\)-value of UAV\({}_{1}\) and UAV\({}_{2}\) for all combinations of algorithms, population, and iterations count in 15 runs is 9.3 \(\times\) 10\({}^{-12}\), the \(p\)-value of UAV\({}_{1}\), and UAV\({}_{3}\) is 0.201, and the \(p\)-value of UAV\({}_{2}\) and UAV\({}_{3}\) is 9.8 \(\times\) 10\({}^{-6}\). Generally, \(p\)-values less than 0.05 are accepted for hypothesis rejection. The null hypothesis is that all UAV s do not have a meaningful relationship. The significant difference between UAV\({}_{2}\) with both UAV\({}_{1}\) and UAV\({}_{3}\) is proved. Therefore, the null hypothesis is rejected. ## 5 Conclusions

The focus of the paper was to solve the NP-hard problem of efficient crop harvesting by finding the most suitable and optimal paths for UAVs. This study presented adaptive 3D path planning methods using metaheuristic algorithms (I-GWO and Ex-GWO) for autonomous agricultural UAVs. Therefore, in this study, maximum profit was achieved by consuming the least energy by harvesting the most crops in the shortest possible time. In addition, the use of resources such as human and natural resources was carried out efficiently by creating sustainable and smart agriculture. In other words, the method allows farmers to monitor crop variability and stress conditions continuously and harvest the best crops, resulting in efficient resource consumption and an increase in profits.