1. Introduction
An unmanned aerial vehicle (UAV) is an unmanned aerial vehicle that is operated by a remote control or automatic control system. In recent years, UAVs have been increasingly employed in agriculture due to their great flexibility, mobility, low safety risk factor, and low cost, and agricultural UAVs are unmanned aerial vehicles specifically designed for use in agriculture, which can significantly boost agricultural output, decrease expenses, and provide environmental benefit. For instance, UAVs may precisely spray crops using path planning, ensuring uniform coverage [
1]. UAVs can fly along prearranged routes and gather data on crop growth to assist farmers in making more informed management choices [
2]. UAVs can use path planning to conduct regular inspections of fields, monitor soil moisture and crop water storage with on-board sensors, and direct the irrigation system for precision irrigation [
3]. UAVs can make 3D maps of farms to help farmers understand the topography of their farmland and plan irrigation systems and agricultural planting [
4]. UAVs can be used to assess crop damage and offer data support to agricultural insurers, helping farmers obtain compensation more quickly [
5].
Path planning is a key function for UAV mission execution and is a major challenge for autonomous UAVs in engineering applications. The current UAV path planning algorithms fall into two broad categories: classical and meta-heuristic algorithms. Classical algorithms include the Rapid Random Number Method based on random sampling (RRT) [
6], the A* algorithm [
7], and so on; these algorithms are less efficient in searching and converge slowly in complex situations. Path planning problems are classic NP hard problems with intricate constraints that make it difficult for general solvers to discover accurate solutions. Meta-heuristic algorithms can effectively solve complex combinatorial optimization problems. Common meta-heuristic algorithms include genetic algorithms (GAs) [
8], ant colony optimization algorithms (ACOs) [
9], gray wolf optimization algorithms (GWOs) [
10], Harris Hawk optimization algorithms (HHOs) [
11], whale optimization algorithms (WOAs) [
12], sine cosine algorithms (SCAs) [
13], and slime mold optimization algorithms (SMAs) [
14].
Meta-heuristic algorithms are widely used in the study of UAV path planning. For example, Chen et al. [
15] proposed an enhanced version of the chimp optimization algorithm to address the UAV path planning problem in a 3D environment. Zhang et al. [
16] proposed an improved adaptive gray wolf optimization algorithm to address the three-dimensional path planning problem of UAVs in the complicated environment of material conveying in earthquake disaster zones. Wang et al. [
17] proposed an updated bionic tuna swarm optimization algorithm for planning UAV flight paths safely and efficiently in complicated obstacle situations. Li et al. [
18] proposed an improved particle swarm optimization algorithm for balancing the development and exploration capabilities of UAVs. He and Wang [
19] proposed an improved chaos sparrow search algorithm to address the issue of delayed convergence and falling into local optimums in path planning in three-dimensional complicated environments for UAVs. Yu et al. [
20] used a hybrid gray wolf optimization and differential evolutionary algorithm to tackle the UAV path planning problem. These algorithms can swiftly identify effective pathways by employing various search techniques across several iterations.
However, one of the most important difficulties remains: how to keep the algorithm from falling into a local optimum [
21]. To enhance the path smoothness of agricultural UAVs in complex environments and achieve a reasonable and safe flight path, this study recognizes the agricultural UAV path planning problem as a complex optimization problem that necessitates an effective solution. It then suggests a multi-strategy improvement-based differentiated creative search algorithm (MSDCS) for agricultural UAV path planning. The differentiated creative search algorithm (DCS) is a novel meta-heuristic optimization algorithm proposed by Duankhan et al. [
22] in 2024. The DCS is a model for iterative optimization of team performance that uses differential knowledge acquisition and creative realism strategies to solve complex optimization problems. While it is effective at solving complex optimization problems, it suffers from a lack of search accuracy and does not adequately explore unknown regions in the search space. This may result in agricultural UAVs not being reasonably fast or smooth for path planning in complex environments. Therefore, in this paper, a set of good points is used to resolve the uneven distributional problem in the initial population of the original DCS, and the oppositional learning strategy is used to improve the quality of the population and strengthen the algorithm’s search ability. To bring low performers closer to the region near the optimal solution, an adaptive fitness–distance balance reset strategy is proposed, which helps the algorithm search meticulously near the optimal solution to find the potential optimal solution. To address the issue that the DCS algorithm aggregates population individuals to the optimal individuals in later iterations, which can easily lead to search stagnation and the population falling into the local optimum, a vertical and horizontal crossover strategy with random dimensions based on the stochastic dimension is proposed to improve the algorithm’s computational accuracy, as well as its ability to jump out of the local optimum. A comparison with different algorithms through 29 test functions from the CEC2017 test set shows that the MSDCS possesses better convergence, and the stronger comprehensive performance of the MSDCS is further verified using the Wilcoxon rank-sum test and Friedman test. Finally, different agricultural UAV environment models were established to carry out simulation experiments on the trajectories of agricultural UAVs, and the applicability of the MSDCS in the agricultural UAV path planning problem was verified [
23]. The main contributions of the novel algorithm proposed in this study are as follows:
- (1)
The performance of the algorithm is enhanced, and a new optimization method for path planning of agricultural UAVs is developed by utilizing a good point set and oppositional learning strategy, an adaptive fitness–distance balance reset strategy, and a vertical and horizontal crossover strategy with random dimensions based on the original DCS algorithm.
- (2)
Comparing the proposed algorithm with nine other algorithms, simulation experiments based on the CEC2017 test set validated the effectiveness of the MSDCS and further validated its comprehensive performance using the Wilcoxon rank-sum test and Friedman test.
- (3)
The MSDCS is applied to the agricultural UAV path planning problem to minimize the objective function of the problem, which has complicated constraints. The results show that the MSDCS can generate high-quality solutions for agricultural UAV path planning, even in complex environmental conditions. Thus, a reasonable and safe flight path was obtained.
The remainder of this paper is organized as follows.
Section 2 introduces the basic differentiated creative search algorithm;
Section 3 describes, in detail, the three improvement strategies of the MSDCS;
Section 4 calculates the time complexity of the MSDCS;
Section 5 describes the environment, parameters, and result analysis of the simulation experiments;
Section 6 models and experiments the agricultural UAV path planning problem;
Section 7 discusses the results of the MSDCS in agricultural UAV path planning; and
Section 8 summarizes the research of this paper.
2. Differentiated Creative Search Algorithm
2.1. Population Initialization
The process of the DCS algorithm is as follows. A set of candidate solutions
(team members) is randomly generated with upper and lower bounds. Equation (1) is shown as follows:
where
denotes the
th candidate solution,
denotes the
th location of the candidate solution,
represents the number of candidate solutions, and
represents the dimension of the optimization problem. Each element of
is randomly generated by Equation (2) as follows:
and are the lower and upper bounds of the optimization problem, respectively, and represents a uniform distribution in the interval . After initialization, each individual is evaluated to determine their fitness value. Then, the individuals in are ordered based on their fitness value, with a smaller fitness value indicating higher performance.
2.2. Differentiated Knowledge Acquisition
The stages of differentiated knowledge acquisition are rooted in the different potentials for knowledge acquisition and can be categorized into three types, depending on the individual’s ability and level of performance. The first type of individual is the high performer, and these members serve as team idea generators and have a mindset that generates a diverse range of ideas; the second type of individual is the average performer, who serve as concept refiners, examining and comparing several ideas to determine the optimal answer; and the third type of individual is the low performer, and these members contribute to the team’s diversity and exploration by offering new ideas and approaches to problem solving. The process of updating the position of each individual can be executed using the following Equations (3)–(6), in which the poorer performers need to assimilate new knowledge or experience more than the better performers:
As stated in Equation (3), denotes a random number that ranges between 0 and 1 and is generated independently for each dimension of each particle. If the condition is met, the position is updated by Equation (7), otherwise, the position remains unchanged. Where parameter is the quantitative knowledge acquisition rate of the th individual at the th iteration, and the uniformly distributed random numbers generated by the two ’s in Equation (4) are different, with the first half indicating rounding to the nearest whole number, while the second half compares the random number with , and, if true, the value is 1; otherwise, the value is 0. represents the variable ’s value at the th iteration of an individual, while represents the th individual’s ranking at the start of the th iteration. The coefficient measures the degree of imperfection in an individual’s knowledge. Higher values indicate larger gaps in knowledge and the need for additional learning and experience to compensate; smaller values indicate a more comprehensive and solid foundation in the relevant knowledge area. is an integer chosen randomly from 1 to and is generated once per individual.
2.3. Creative Realism
The differential knowledge acquisition stated above divides teams into low, high, and average performers, and creative realism is based on the divergent thinking of high performers and the convergent thinking of average performers. Creative ideas and problem solving are both key characteristics of high performers. Not only do high performers serve as a knowledge basis, but each member of the team is required to constantly draw on fresh knowledge and broaden their divergent thinking. For the average performer, convergent thinking relies on the knowledge of the best individual, combined with random contributions from two different team members. The position update of an individual can be performed using the following Equations (7)–(9):
represents the
th location of a randomly picked individual in
, and
is the Lévy flying random number generator with control parameters
and
.
stands for the number of high performers according to the population’s golden ratio,
, which will have a value of an integer larger than or equal to six.
is the
th position of the best individual and
is the dth position of the individual randomly selected from
.
denotes the learning intensity state of an individual, with a default value of 1.
is the coefficient of the individual at the
th iteration,
is the current number of iterations, and
is the maximum number of iterations. The
coefficient measures how much a team’s social dynamics influence an individual’s views. Higher values indicate a greater reliance on team members, while lower values indicate a greater degree of independence. The DCS algorithm replaces underperforming members with new ones to increase the diversity of ideas generated by the team. The formula for producing new members is given by Equation (10) as follows:
When is 0.5, the method defaults to the last team member, and is an inefficiency and uses random initialization to reset the individual.
3. Improvement Strategies
3.1. Population Initialization Based on Good Point Set and Oppositional Learning Strategies
The distribution location of the initial population and the quality of the solution have a significant impact on the effectiveness of the metaheuristic algorithm [
24]; if the initial population is too centralized or too dispersed, the algorithm may underutilize the search space, and if the quality of the solution in the population is low, the algorithm’s convergence accuracy will suffer. The original DCS algorithm’s random initialization is too unpredictable to ensure the algorithm’s population diversity and convergence accuracy. This study provides enhancements to its initialization procedure.
3.1.1. Good Point Set Strategy
In the optimization process of meta-heuristic algorithms, the diversity of the algorithm can be enhanced if the initialized population is made to be uniformly distributed over the spatial range of the objective function. The good point set [
25] is an effective uniform point selection method with good distribution, so we use it to initialize the population in the MSDCS. Let
be a
-dimensional Euclidean space unit cube, with
. Then, the set of good points is formed in Equations (11) and (12) as follows:
where
is the number of populations,
is the set of good points,
is the value of good points, and
is the smallest prime number that satisfies
. Therefore, the initialization based on the set of good points is given by Equation (13) as follows:
Theoretically, weighted sums formed with
good points have smaller errors than those obtained with any other
points, making them appropriate for approximate computations in higher dimensional spaces. Generate 500 points in a two-dimensional unit search space and compare the randomized points. The points from the good point set approach are shown in
Figure 1.
Figure 1 shows that random points are not uniformly distributed, resulting in lower utilization of the algorithm’s search space, whereas points generated using the good point set method are more uniformly distributed in the search space, resulting in broader coverage of the population and improving the efficiency of the global search.
3.1.2. Oppositional Learning Strategy
Meta-heuristic algorithms typically save high-quality individuals from the previous generation population to the next generation population. The number of high-quality individuals in the initial population has a direct impact on the algorithm’s convergence rate; if there are fewer high-quality individuals in the initial population, the algorithm’s convergence speed and accuracy will suffer. Therefore, in this paper, we use an oppositional learning strategy to boost the number of quality individuals in the generated initial population. To be more precise, the algorithm simultaneously generates an individual who opposes the individual in the good point set. Finally, the individuals with the best fitness values are chosen as the beginning population, increasing the algorithm’s search efficiency. Their opposing positions (
) can be expressed by Equation (14) as follows:
The lower and upper bounds of the individual positions are
and
, respectively. The implementation process of population initialization based on the good point set and oppositional learning strategy is as follows: generate an initial population based on the good point set, compute the oppositional population using Equation (14), rank the fitness values, and select the top
individuals with better fitness values as the new initial population. As shown in
Table 1, the best fitness values of four representative test functions were selected from the CEC2017 test set with 500 iterations of the algorithm and 30 independent runs. DCS1 uses the good point set strategy, and DCS2 uses the good point set and oppositional learning strategy. The results show that population initialization based on a good point set and an oppositional learning strategy can effectively improve the algorithm’s ability to find the optimal value.
3.2. Adaptive Fitness–Distance Balance Reset Strategy
The DCS algorithm categorizes team members into three types: low, high, and average performers. For the high performer, combining the existing knowledge and creative elements can provide better solutions to the innovation of the problem; for the average performer, refinement of the knowledge from the best team members promotes the diversity of solutions. However, for low performers, acquiring new knowledge only affects one dimension of an individual’s knowledge structure, as in Equation (10). This local variation contributes to diversity through random initialization and cannot ensure the quality of the solution. To encourage low performers to migrate closer to the optimal solution zone, this research presents an adaptive fitness–distance balance reset strategy. By dynamically adjusting the balance between randomized and targeted exploration, the strategy efficiently identifies the candidates with the highest potential from the population while maintaining diversity, enabling the algorithm to perform a detailed search on the area of good fitness to find a potentially better solution, as described in the following steps.
The first step is to compute the distance (
) between the present solution (
) and the best solution (
) using the Euclidean distance metric, which is mathematically modeled by Equation (15) as follows:
The second step is to compute the candidate solution’s score (
), which is obtained by weighted summation of the normalized fitness value (
) and the normalized distance value (
) using Equations (16)–(18) as follows:
where
and
are the maximum and minimal fitness values, respectively, and the fitness value in the minimization problem is defined in Equation (18). The weighting coefficient
is used to modify the effect of the fitness and distance values. In this study,
is an adaptive operator with the following Equation (19):
The value of
decreases as the number of iterations increases, allowing for a fast exploration of the space based on the fitness value in the early stages, followed by a meticulous search as approaching the optimum in the later stages. Then, using Equation (20), the candidate with the highest score (
) is chosen to update the inefficient individual (
).
In the original DCS algorithm, the last individual is defined as inefficient and updated using random initialization, resulting in low-quality solutions. In this study, the algorithm designates the last ten individuals in the fitness ranking list as inefficient and then employs an adaptive fitness–distance balance reset technique to enhance their quality.
Table 2 shows the result of the best fitness value obtained by the DCS algorithm based on the adaptive fitness–distance balance reset strategy (DCS3) and the original algorithm (DCS) (the experimental setup is found in
Section 3.1.2). It can be seen that the DCS algorithm based on the adaptive fitness–distance balance reset strategy has a better value of the best fitness in these test functions.
3.3. Vertical and Horizontal Crossover Strategy with Random Dimensions
In later iterations of the DCS, the individuals gradually converge to the optimal individual, and the fitness values of the individuals tend to stabilize. As a result, new and better solutions are difficult to emerge, which can easily lead to the search becoming stagnant and the population falling into the local optimum. To improve the convergence accuracy of the DCS and enhance the algorithm’s ability to jump out of the local optimum, this paper uses the vertical and horizontal crossover strategy based on random dimensions to improve the current individuals. Specifically, the horizontal crossover is used for searching to reduce the search blind spots and improve the algorithm’s global search ability; the vertical crossover is used to enhance population variety while decreasing the risk of the algorithm falling into the local optimum. The dimensions involved are selected by
, i.e.,
determines the number of dimensions involved in the vertical and horizontal crossings. Initially, a comprehensive random perturbation is performed, which is conducive to quickly jumping out of the local optimum, and with the increase in the number of iterations, the search is gradually centralized, which is conducive to fine optimization. The calculation process is given by Equation (21) as follows:
The horizontal crossover refers to the exchange of individual information between two different members in a randomly chosen dimension so that different individuals can learn from each other. This operation can also be viewed as a mutating process of generating a new position based on two random individuals. The horizontal crossover of two unduplicated individuals of the team on
randomly chosen dimensions
generates offspring with Equations (22) and (23) as follows:
In this equation, and are the individuals generated by and after the horizontal crossover; in this way, each mutation operation affects only the values on the selected dimension, while the other dimensions remain unchanged. and are random numbers of used for balancing the individual’s current position in each mutation operation, and and are random numbers of used for regulating the distance between the individual’s current position and crossover point position in each mutation operation. The resulting children compete with the parent generation, and the best individuals are finally retained.
The vertical crossover of this strategy implies that the best individuals in the population exchange information across randomly selected dimensions, resulting in a new generation of the best individuals to compete with the parent generation, allowing different dimensions to learn from one another and avoiding premature convergence of one dimension. The individual formulae for the offspring obtained from the vertical crossover are given by Equation (24) as follows:
The offspring individual
is created via the vertical crossover of the present optimal individual
on
randomly chosen dimensions
and
(
).
is used to control the magnitude of the shift of the new position during the vertical crossover.
Table 3 depicts the results of the random dimension-based vertical and horizontal crossover strategy (DCS4) and the original algorithm (DCS) in the test function (the experimental setup is the same as above). The test results of DCS4 are significantly better than the DCS.
4. Time Complexity
The basic DCS algorithm has a total time complexity of , where is the population size, is the dimension, and is the maximum number of iterations. The time complexity of the MSDCS algorithm is evaluated as follows.
The process consists mostly of population initialization, fitness evaluation, and updating candidate solutions. The temporal complexity of population initialization is , and fitness evaluation is . The candidate solution is updated in four steps. First, when the team is inefficient, the time complexity is , where is a probability constant. Second, the time complexity of updating the candidate solution when the team is efficient is , where denotes the number of high-performance candidates and denotes the knowledge acquisition rate. Then, the time complexity of updating the candidate solution when the team is average is . Finally, the time complexity of updating the candidate solutions using the vertical and horizontal crossover strategy with random dimensions is . In summary, excluding the low-order terms, the time complexity of the MSDCS is , which is the same as the original algorithm.
The pseudo-code for the improved Algorithm 1 is as follows.
Algorithm 1: MSDCS |
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7. Conclusions
The simulation experiments based on the CEC2017 test set in
Section 5 lead to the conclusion that the MSDCS has a fast convergence speed and high accuracy compared to the other nine comparison algorithms, and the comprehensive performance of the MSDCS ranks first in Friedman’s test, which proves the effectiveness of the improved algorithm.
Section 6 applies the improved algorithms to the agricultural UAV path planning problem. It can be seen in
Figure 5,
Figure 6,
Figure 7 and
Figure 8 that the PSO, SAO, BKA, IMODE, and LSHADE algorithms have poorer performance in solving the agricultural UAV path planning, and the flight paths obtained are prone to incur high flight costs and even higher risks during the flight process, whereas the MSDCS has a faster convergence speed and higher convergence accuracy. Compared to other algorithms, the MSDCS can obtain better flight paths for agricultural UAVs with a safer flight process, thus reducing the flight cost.
Table 8 and
Table 9 show that the MSDCS produces high-quality solutions in agricultural UAV path planning in both simple and complex scenarios with good stability.
This study not only confirms the efficacy of the MSDCS but also its practical worth in agricultural UAV path planning. The comparative study demonstrates that the MSDCS exhibits notable superiority in effectively navigating various intricate situations. When compared to the classical PSO method, the PSO exhibits a higher rate of convergence. However, it demonstrates limited stability and adaptability in the presence of impediments. The MSDCS, in this study, demonstrates superior capability in guaranteeing that agricultural UAVs locate the most advantageous route while evading obstacles, hence enhancing the quality and efficiency of UAV flight pathways for path planning.
The cost function of agricultural UAV path planning is flexible, encompassing factors like fuel consumption and collision risk. In the future, the algorithm can be modified to achieve more comprehensive optimization of agricultural UAV path planning. This will cater to a variety of agricultural application scenarios, ensuring the efficient and safe operation of agricultural UAVs. Furthermore, it will further advance the development of agricultural UAVs in the field of agriculture.
8. Summary
For the agricultural UAV path planning problem, a multi-strategy improved differentiated creative search algorithm is proposed. The MSDCS employs the good point set and the oppositional learning strategy to increase the diversity of the population and improve the quality of the solution; the adaptive fitness–distance balanced reset strategy prompts the low performers to converge to the optimal solution region. Additionally, the vertical and horizontal crossover strategy with random dimensions is proposed to help the algorithm jump out of the local optimum and improve the convergence accuracy.
To demonstrate the MSDCS’s effectiveness, 29 test functions from the CEC2017 test set are used for comparison experiments with the original algorithms, the classical algorithms, the most recent modified algorithms, and the competition algorithms. The experimental results reveal that the MSDCS outperforms the other comparative algorithms in terms of convergence speed, accuracy, and stability. Further validation is performed using the Wilcoxon rank-sum test and the Friedman test, and the findings confirmed the MSDCS’s excellent performance.
Two sets of experiments are conducted using relatively simple terrain maps and relatively complex terrain maps to test the performance of the MSDCS in agricultural UAV path planning problems. The simulation experiments show that the MSDCS algorithm can converge quickly and generate efficient and collision-free agricultural UAV flight paths in both simple and complex scenarios, which verifies the applicability and superiority of the MSDCS.