An adaptive multi-strategy cooperative elk herd optimization algorithm for three-dimensional unmanned aerial vehicle path planning

Abstract

In the development of the low-altitude economic field, unmanned aerial vehicle (UAV) three-dimensional (3D) path planning often suffers from slow convergence and local optima problems. To address these problems, this study established a comprehensive technical framework for path planning that spanned application scenario design to algorithm performance enhancement. A parametric stochastic 3D environment model was first established to simulate real-world uncertainties. Then, a multi-dimensional performance evaluation model integrating weather conditions and terrain types was developed. This model incorporated multiple core flight indicators, including the path length, safety, flight altitude, no-fly zone constraints, and path smoothness. Moreover, the weights of these indicators were dynamically adjusted to make the path planning results more aligned with actual operational needs. Based on this, an adaptive multi-strategy cooperative elk herd optimization (AMCEHO) algorithm with better chaotic ergodicity and optimization balance was proposed. Through the synergy of hybrid chaotic initialization, a dynamic elite pool, and multi-strategy collaborative updating, it not only overcame the population diversity limitation of a single chaotic map and calibrated search directions, but also balanced exploration and exploitation. The hierarchical correlation among the three components significantly enhanced the algorithm’s rapid convergence and global optimization capabilities, solving the key difficulties of traditional algorithms. Experiments were conducted for six benchmark functions and two scenarios (medium and complex obstacles scenarios), comparing the AMCEHO algorithm with several state-of-the-art algorithms. The results demonstrated that among the six benchmark functions, the AMCEHO algorithm achieved optimal values in the 10⁻² to 10⁻⁵ range with minimal standard deviation, significantly enhancing the global optimization capabilities. Compared with other algorithms, the AMCEHO algorithm achieved up to 71.9% fewer iterations and 56.4% lower total path cost, while achieving a maximum 36.7% reduction in overall fitness. This study provides a promising solution for high-quality UAV 3D path planning in low-altitude economic field.

Graphical abstract

Keywords

elk herd optimization algorithm / chaotic mechanism / three-dimensional path planning / unmanned aerial vehicle

1 Introduction

With the rapid development of the global low-altitude economic field, unmanned aerial vehicles (UAVs), as the core method of transport for low-altitude activities, have been widely applied in diverse fields such as logistics distribution (Wen et al., 2025), emergency rescues (Mu et al., 2025), urban inspection (Gan et al., 2024), and agricultural plant protection (Bouguettaya et al., 2022). As a core technical support component for the autonomous flight of UAVs, three-dimensional (3D) path planning technology directly determines the efficiency, safety, and energy consumption performances of task execution. However, the complexity of real-world 3D scenarios has produced multiple challenges for path planning (Bo et al., 2024). First, 3D constraints from terrain and obstacles, such as urban building clusters, canyons, and high-voltage cables, require algorithms to have accurate spatial obstacle avoidance capabilities. Second, dynamic environmental factors (Wu et al., 2024) require planning results to have real-time adaptability. Third, the conflicting requirements of multi-objective optimization (Peng et al., 2025) require algorithms to achieve collaborative balance. This places higher demands on the robustness and adaptability of planning algorithms. Traditional path planning methods such as the A* algorithm (Zhang et al., 2025) and Dijkstra’s algorithm (Ahmad and Wahab, 2025) are prone to computational bottlenecks in high-dimensional complex scenarios. In contrast, intelligent optimization algorithms have gradually become the mainstream technical direction because of their twin advantages of not relying on accurate mathematical models and exhibiting strong robustness. However, traditional intelligent optimization algorithms still have shortcomings, such as insufficient population diversity, unbalanced exploration-exploitation capabilities, and poor adaptability to complex 3D scenarios. Thus, targeted improvements such as the optimization of population initialization and the design of dynamic elite mechanisms are needed to enhance algorithm performance in urban low-altitude UAV 3D path planning.

The development of intelligent optimization algorithms has progressed through distinct methodological stages, each characterized by a different source of inspiration and computational model.

The initial stage of development featured evolutionary algorithms. These methods were inspired by biological evolution. The genetic algorithm proposed by J. H. Holland (Katoch et al., 2021) was a classic example. It used selection, crossover, and mutation to evolve solutions. This landmark work not only laid the mathematical foundation for intelligent optimization but also provided new ideas for early intelligent path planning (Guo et al., 2025). However, these algorithms often converged slowly and required careful parameter tuning, especially for high-dimensional spaces.

Later, the focus shifted to swarm intelligence algorithms. These methods modeled the collective behavior of groups. Kennedy and Eberhart (1995) proposed the particle swarm optimization (PSO) algorithm, which mimicked bird flocking. Concurrently, the ant colony optimization (ACO) algorithm was introduced by Dorigo et al. (1996). This algorithm simulated ant foraging. The strengths of these algorithms were their simple rules and efficient information sharing. However, in 3D path planning, the algorithms often converged prematurely and struggled to balance global and local search.

Recently, researchers have proposed many nature-inspired algorithms. These algorithms mimic specific biological or physical processes. Mirjalili et al. (2014) proposed the gray wolf optimization (GWO) algorithm based on wolf pack hierarchy. Arora and Singh (2019) first proposed the butterfly optimization algorithm, which was inspired by butterfly foraging. These algorithms demonstrated good global search ability. Yet they often required complex parameter tuning and sometimes performed inconsistently in dynamic environments, revealing a gap between theory and practice.

Against this backdrop, Al-Betar et al. (2024) proposed the elk herd optimization (EHO) algorithm. This new swarm intelligence algorithm was inspired by elk herd (EH) behavior. Its advantages included simple parameters and strong local search capability, making it suitable for 3D path planning. In low-dimensional cases, the EHO algorithm has demonstrated higher convergence accuracy than traditional PSO and ACO algorithms. However, when applied to complex 3D urban scenarios, the EHO algorithm exhibited several limitations: low population diversity, slow convergence speed, and the tendency to fall into local optima. These shortcomings highlight the need for targeted improvements to achieve EHO’s potential in practical UAV applications for the low-altitude economy.

To address the limitations of the EHO algorithm in complex urban low-altitude scenarios, in this study, a comprehensive path planning framework was established that spanned from application scenario design to algorithm performance optimization. Based on the authenticity of scenarios and the scientific nature of evaluation, the adaptive multi-strategy cooperative elk herd optimization (AMCEHO) algorithm was proposed. This framework combined theoretical breakthroughs with practical applications. The specific research content is described below.

First, to avoid overfitting from fixed scenarios and ensure test authenticity, a 3D environment model with random obstacles was established. Within a defined spatial range, obstacles and no-fly zones were randomly generated, providing realistic, variable-complexity test benchmarks for subsequent algorithm verification.

Next, to align path optimization with actual operational needs, a multi-dimensional performance evaluation model was constructed. The weighted sum of the total path costs was calculated, integrating key indicators such as the path length, safety, flight altitude, no-fly zone constraints, and path smoothness. The weight coefficients were dynamically adjusted by weather conditions and terrain types, achieving scenario adaptability unavailable in traditional fixed-weight models.

Finally, to solve the slow convergence and local optima problems of traditional algorithms, the AMCEHO algorithm was developed, integrating hybrid chaotic initialization, dynamic elite pool, and multi-strategy collaborative updating. Validated against the above dynamic scenarios and evaluation model, the algorithm demonstrated superior convergence speed and global optimization capabilities, addressing key difficulties with conventional algorithms.

2 Literature review

As the core computational engine for UAV 3D path planning, the performance of intelligent optimization algorithms directly determines the optimality and computational efficiency of planning outcomes. With the increase in the complexity of low-altitude scenarios and the growing sophistication of multi-objective requirements, the limitations of traditional intelligent algorithms have become increasingly prominent for critical aspects such as high-dimensional low-altitude space exploration, adaptation to complex constraints, and real-time response to dynamic scenarios. To address these challenges, the academic community has conducted systematic research, focusing on enhancing the abilities of algorithms to ensure stable convergence, and achieve dynamic responses, and escape local optima. These three directions, while distinct in their focus, are mutually reinforcing. Together, they form a complete logical chain for enhancing algorithm performance. This structure will provide a comprehensive foundation for subsequent research on the AMCEHO algorithm, guiding its development from theoretical conception to practical implementation.

2.1 Population initialization

The spatial distribution and quality of the initial population fundamentally determine the convergence performance and search efficiency of an algorithm. Traditional intelligent optimization algorithms, such as the PSO and ACO algorithms, predominantly adopt uniform random initialization, which often fails to ensure population diversity and increases the risk of oscillations in the convergence process or even premature convergence. To resolve this current limitation, the academic community has introduced chaos theory with the specific goal of improving the coverage uniformity and quality superiority, ensuring a smooth and reliable convergence process. Among the chaotic maps that are extensively used in this context, the tent map (Xue, 2025) has been distinguished by its piecewise linear properties, which enable the rapid traversal of the solution space. The logistic map (Huang et al.,2025) is widely favored because of its strong chaotic behavior, an attribute that effectively enhances the randomness of initial solutions. The piecewise linear chaotic map (PWLCM) (Guo and Gu, 2024) has also earned recognition for its ability to achieve flexible distribution adjustment through a single controllable parameter. These three maps, each with its own distinct advantages, have become mainstream options for the optimization of initial population generation in intelligent optimization algorithms. Their specific working mechanisms and the ways they were applied in this study are elaborated on in subsequent sections.

The core idea behind the integration of chaos theory lies in leveraging the ergodicity, randomness, and regularity of chaotic sequences to generate initial populations with more comprehensive coverage and more uniform distribution. Alikhani Koupaei and Ebadi (2025) proposed a method based on chaotic decomposition, utilizing the ergodicity of chaotic maps to enhance optimization performance. This method consisted of two key steps: generating diverse populations through chaotic sequence initialization to facilitate global searching, and balancing the exploration-exploitation trade-off by integrating three-point operators and local improvement operators based on chaotic correction. These steps collectively helped avoid premature convergence or excessive computational costs. He and Wang (2024) proposed an improved chaotic sparrow search algorithm to address the slow convergence speed and strong tendency to fall into local optima observed in UAV path planning for complex 3D scenarios. At the population initialization stage, piecewise chaotic mapping was employed to improve the quality of the initial solutions. To balance global and local exploration capabilities, a nonlinear dynamic weighting factor was introduced into the producer update equation. For the forager update process, an enhanced sine-cosine algorithm was implemented to effectively broaden the search space. Additionally, a dynamic boundary lens imaging opposition-based learning strategy was adopted to mitigate the risk of the algorithm converging to local optima. The experimental results demonstrated that this algorithm outperformed the chaotic sparrow search algorithm, sparrow search algorithm, and PSO algorithms in terms of both convergence speed and accuracy in complex scenarios. Petavratzis et al. (2021) enhanced the performance of chaotic path planning using improved memory technology. They employed chaotic maps to generate exploratory bit sequences. Furthermore, a memory technology guided the robot toward the least-visited adjacent cells, which effectively reduced repeated visits and improved the overall coverage performance. However, existing research has exhibited a limitation in the adaptability of single chaotic maps. The reliance on only one chaotic sequence has failed to adequately cover the local solution space in complex UAV 3D path planning scenarios with highly variable obstacles.

To overcome these limitations, this study proposed a hybrid chaotic initialization strategy that integrated the advantages of multiple chaotic maps. Furthermore, through adaptive weight adjustment, the generated initial population achieved broader distribution and more uniform coverage in the solution space, significantly enhancing its diversity and randomness, thereby laying a high-quality search foundation for subsequent optimization.

2.2 Elitism selection strategy

After obtaining a high-quality initial population, how to effectively identify and utilize elite individuals to achieve dynamic response of the algorithm has become a key challenge. Specifically, “dynamic response” denotes the capability of the algorithm to adaptively adjust key parameters or strategies based on the real-time status of the search process, including the iteration stage, population diversity, and convergence trend. Within the framework of the elitism selection strategy, the dynamic response is specifically manifested as the algorithm dynamically adjusting the size and composition of the elite pool based on the current convergence state and diversity level of the population, thereby achieving an adaptive balance between exploration and exploitation in different search stages. This method is particularly important in dynamic and uncertain path planning environments, as it enables the algorithm to adjust the search strategy in real time according to environmental changes, avoiding premature convergence or excessive exploration. Traditional algorithms typically employ a fixed update strategy, which commonly results in imbalanced exploration-exploitation behavior in UAV 3D path planning scenarios. To effectively overcome these problems, the academic community has actively explored innovations and proposed the concept of a dynamic elite pool.

Numerous studies have applied dynamic elite pool strategies to regulate the number and influence of elite individuals, thereby enhancing an algorithm’s global guidance. Deng et al. (2022) proposed a “dual mutation collaboration with elite guidance and inferior elimination techniques” for the differential evolution (DE) algorithm. The innovation of this mechanism lay in the simultaneous adoption of two complementary mutation strategies. The first strategy was an explorative strategy (DE/current-to-embest) that utilized elite individuals as part of the differential vector. This strategy was responsible for expanding the solution space coverage to strengthen global exploration. The second strategy was an exploitative mutation strategy (DE/ebest-to-rand) that used elite individuals as the base vector. This strategy was focused on the area around high-quality solutions to improve local exploitation capabilities. Together, these strategies overcame the limitations of the single-strategy approach in traditional DE algorithms. Similarly, the self-adaptive cuckoo search algorithm proposed by Salgotra et al. (2021) replaced the fixed discovery probability of the traditional cuckoo search algorithm with an adaptive population reduction strategy. This strategy flexibly adjusted the population size based on the fitness difference between the current best solution and the historical optimum. When the fitness improvement was marginal, the population size was reduced proportionally to eliminate individuals with poor fitness. When a significant improvement occurred, the population size was maintained or slightly adjusted to preserve an adequate exploration capacity. The goal of this strategy was to reduce the computational complexity in later iterations while avoiding premature convergence due to over-aggressive population reduction. Essentially, in this algorithm, the traditional “elite retention” mechanism was substituted with “dynamic population sizing,” indirectly emphasizing high-quality solutions. However, most of the existing dynamic elite pool strategies have excessive reliance on elites, leading to a decline in population diversity and a tendency to fall into local optima.

To overcome these limitations, this study proposed a framework that was closely coupled with multi-strategy cooperative updating. Elite individuals were not only utilized to guide the global search but were also integrated with probabilistic strategy selection and local chaotic perturbation to prevent premature convergence caused by over-reliance on a single elite guidance. This structure preserved the guiding role of elites while maintaining population exploration capability through strategic diversity, thereby achieving more robust optimization performance in complex 3D path planning tasks.

2.3 Population update strategy

As the core component of algorithm optimization, population update strategies need to effectively address the problem of getting trapped in local optima. Traditional single update strategies exhibit obvious limitations. A single global search can maintain diversity but has a slow convergence rate, while a single local search converges quickly but tends to get stuck in local optima. To overcome these challenges, the academic community has proposed the concept of multi-strategy collaborative updating, aiming to effectively escape local optima while maintaining the convergence rate, thereby significantly enhancing the global optimization capability of the algorithm.

In the multi-strategy coordination method, the multiple strategies complement each other, avoiding the limitation of a single strategy, and skillfully balancing the exploration and development stages. The elitist fruit fly optimization algorithm proposed by He et al. (2023) employed multi-strategy collaboration to address the limitations of the traditional fruit fly optimization algorithm. Within a single iteration, this algorithm alternately executed the three stages of “random guidance exploration, elite guidance exploitation, and boundary guidance diversity restart.” These stages formed a closed loop integrating exploration, exploitation, and diversity maintenance. The exploration stage expanded the search range through random individuals and a sigmoid dynamic step size. The exploitation stage utilized dual-mode elite guidance to screen high-quality regions. The diversity stage leveraged boundary information and low-probability single-boundary updates to prevent premature convergence. At the same time, an elite retention mechanism reused high-quality information, and dynamic parameters were used to control the search intensity across the stages. The complementarity of these multi-strategies was adapted to different problems, balancing solution accuracy and efficiency. Similarly, the hybrid multi-strategy improvement dung beetle optimizer proposed by Wang et al. (2025) used multi-strategy collaboration to improve the traditional dung beetle optimizer, which enhanced the balance between global exploration and local exploitation. Compared with other algorithms, this algorithm demonstrated superior iteration speed and enhanced global exploration capabilities. However, the existing multi-strategy cooperation methods still had room for improvement regarding the depth of strategy integration. The probability distribution among strategies often relied on preset experience and lacked an adaptive connection with the state of the search process. Specifically, there were three fundamental challenges:

(1) Static strategy weights: many methods used fixed probabilities when choosing exploration and exploitation strategies, which prevented them from adjusting to dynamically evolving search conditions.

(2) Absence of feedback mechanisms: the strategy selection process often ignored real-time feedback from population convergence, diversity, and fitness trends.

(3) Inadequate balancing in complex environments: in highly constrained 3D path planning problems, such static selection mechanisms often failed to sustain an effective trade-off between global exploration and local refinement, resulting in premature convergence or unnecessarily high computational expense.

To address these challenges, the AMCEHO algorithm employed an adaptive strategy coordination approach that utilized a dynamic elite pool and real-time learning rate adaptation. The dynamic elite pool adjusted the number of guiding elite individuals based on convergence progress, while the adaptive learning rate fine-tuned the interaction between elite guidance and chaotic perturbation. These methods enabled the algorithm to dynamically recalibrate strategy weights during the search process, leading to a more responsive and well-balanced optimization procedure.

The respective advantages of these strategies provide core ideas for algorithm design, while their existing flaws point out key directions for improvement. The AMCEHO algorithm first built a high-quality and high-coverage initial population through mixed chaotic initialization, laying the foundation for global optimization. Subsequently, with the aid of the dynamic elite pool strategy, the evolution direction of the population was adaptively screened and guided to form the core guiding information in the iterative process. Ultimately, the elite guidance information was transformed into specific search behaviors through a multi-strategy collaboration method. Global exploration and local development were carried out in a probabilistic manner in a coordinated way, and a new generation of population was output to enter the next round of optimization cycle. The three in sequence formed a closed-loop feedback from high-quality initialization to adaptive guidance and then to intelligent search, systematically enhancing the convergence efficiency and solution robustness of the algorithm in complex three-dimensional path planning problems.

3 Methodology

In this study, a comprehensive technical framework for UAV path planning was established, with the highlights not only reflected in fundamental optimization of the core algorithms but also in the thorough consideration of engineering applicability. First, by establishing a 3D environment model containing random obstacles, it provided realistic and complex testing benchmarks for algorithm validation, effectively bridging simulation and practical requirements. Furthermore, through the development of a multi-dimensional performance evaluation model that integrated weather conditions and terrain type weights, it offered a precise quantitative basis for intelligent path balancing and optimization, ensuring the practical feasibility of planning results. Building upon these solid application foundations, in this study, the inherent limitations of the EHO algorithm were addressed by deeply integrating hybrid chaotic initialization, a dynamic elite pool, and multi-strategy collaborative updating, and the AMCEHO algorithm was proposed for path optimization. These three interconnected components collectively demonstrated that this study represented a systematic solution rooted in practical application needs, with algorithm improvement at its core, ultimately significantly enhancing comprehensive path planning performance. The study framework is illustrated in Fig. 1.

3.1 Establishment of 3D environment model

This section introduced the dynamic generation of test scenarios containing obstacles and no-fly zones of different complexities through parametric random strategies. This method broke through the limitation of algorithm overfitting caused by traditional fixed scenarios, and enhanced the generalization ability and robustness of the algorithm in real low-altitude environments. By adopting a dynamic random generation strategy, this environment model could simulate the random distribution characteristics of various types of obstacles such as buildings and temporary no-fly zones in urban environments, thereby providing a more realistic high-fidelity test benchmark for path planning algorithms. In addition, this method supported flexible configuration of the number, size and distribution density of obstacles, facilitating the construction of multi-level complexity scenarios and systematically evaluating the performance of the algorithm under different environmental conditions. Therefore, the dynamic random generation strategy not only enhanced the authenticity and configurability of the simulation experiment, but also provided a reliable and scalable environmental foundation for the subsequent algorithm verification.

3.1.1 Space definition

The spatial definition was used to construct a standardized 3D testing environment for UAV path planning. First, a 3D airspace with dimensions of 1000 × 1000 × 500 m was established to simulate the airspace range of a typical urban area. Based on this, strict altitude constraints were imposed, confining the UAV operations to a vertical corridor in the range of 50–450 m. This design not only complied with the safety management requirements for low-altitude flight but also provided clear vertical boundaries for path planning. The mission path started at (50, 50, 100) and ended at (950, 950, 250) in diagonal coordinates, ensuring full coverage of the test area to evaluate the algorithm’s global navigation capabilities. Based on the above spatial parameters, all of the generated path points were kept within feasible domains by defining explicit solution space boundaries for optimization algorithms, establishing a stable testing framework for subsequent path planning algorithms.

3.1.2 Scenarios configuration

In this study, a multi-level and configurable challenge environment was subsequently constructed within the established space using a parameterized random strategy. Adopting the concept of scenario-based design, two scenario configurations were presented to meet different testing requirements, including medium obstacles and complex obstacles. The urban environment was simulated by precisely controlling the obstacle quantity, size range, and distribution density to replicate diverse complexity levels. In terms of obstacle modeling, the cylindrical geometry represented urban buildings, creating diverse building groups by randomly generating parameters such as position and height. Additionally, spherical models were employed to represent no-fly zones, simulating sensitive airspace restrictions in reality. The overall visualization effect of the 3D urban scenarios is shown in Fig. 2.

3.2 Construction of a multi-dimensional performance evaluation model

To enhance the practicality and adaptability of UAV 3D path planning in complex low-altitude environments, a multi-dimensional performance evaluation model was constructed that comprehensively considered the path length, safety, flight altitude, no-fly zone constraints, and path smoothness. In contrast to traditional static evaluation models, our model introduced a dynamic weight adjustment strategy that could respond to terrain types and weather conditions. This ensured that the planned path was not only theoretically optimal, but also feasible in ever-changing actual scenarios.

3.2.1 Path length costs

In UAV path planning, shorter trajectory lengths contributed to reduced flight times and improved path quality. The cost function for the path length is formulated as shown in Eq. (1).

(1)F_1(X_i)=\sum _{i=1}^{n-1}\sqrt{{(x_{i+1}-x_i)}^2+{(y_{i+1}-y_i)}^2+{({\mathit{\text{z}}}_{i+1}-{\mathit{\text{z}}}_i)}^2},

where n is the number of track points, and (x_i,y_i,{\mathit{\text{z}}}_i) is the coordinate of the i-th track point.

3.2.2 Safety costs

In path planning, ensuring the safe operation of a UAV is critical. This entails avoiding obstacles in the environment to prevent collisions. As illustrated in Fig. 3, C_k represents the central projection coordinates, and R_k denotes the obstacle radius. For a given path segment T_k(\overrightarrow{P_{ij}^{\prime }{P}_{i,j+1}^{\prime }}), its associated cost F₂ with the threat depends on the distance d_k between the path segment and the threat center C_k. D represents the diameter of the UAV, while S represents the safe distance to the collision zone. Threat cost F₂ along the path point P_ij can be calculated in Eqs. (2) and (3).

(2)F_2(X_i)={\sum }_{j=1}^{n-1}{\sum }_{k=1}^K{T}_k(\overrightarrow{P_{ij}^{\prime }{P}_{i,j+1}^{\prime }}),

(3)

Among them, the symbol M represents a sufficiently large normal number (M = 10¹⁰), which is used as an infeasibility penalty term. Any path segment that violates the constraints or ranges will be subject to this penalty, which effectively prevents it from being selected as the optimal solution during the optimization process. The use of a finite constant ensures the computability of the calculation while retaining the expected prohibitive effect of the original infinity.

3.2.3 Flight altitude costs

The flight altitude of a UAV must be confined within a specific range. As illustrated in Fig. 4, excessively low altitudes increase vulnerability to ground obstacles such as buildings, while excessively high altitudes lead to greater UAV energy consumption. Letting the flight altitude be h_{ij}, when the flight altitude is between the minimum altitude h_{min} and the maximum altitude h_{max}, the cost is calculated based on the deviation from the average altitude (h_{min}+h_{max})/2. The cost becomes infinite if h_{ij} exceeds this range.

The formula for calculating the flight altitude is presented in Eq. (4):

(4)H_{ij}=\{\begin{array}{l}\left|h_{ij}-{\displaystyle \frac{\left(h_{\text{max}}+{\mathrm{h}}_{\text{min}}\right)}{2}}\right|,h_{min}\le h_{ij}\le h_{\text{max}}\\ M,\text{otherwise}\end{array}.

The total altitude cost is shown in Eq. (5):

(5)F_3\left(X_i\right)=\sum _{j=1}^n{H}_{ij}.

3.2.4 No-fly zone constraints costs

Considering the no-fly zone constraints in actual UAV flight situations, in this study, it was assumed that these types of zones could be represented by specific 3D geometric shapes. Taking a 3D sphere as a typical example, letting the center coordinates of the spherical no-fly zone be (x₀, y₀, z₀), the radius be d, and the position coordinate of the UAV individual be (x, y, z), the calculation of the no-fly zone constraint cost is provided in Eq. (6):

(6)F_4(X_i)=\{\begin{array}{l}k\times \sqrt{{(x-x_0)}^2+{(y-y_0)}^2+{(\mathit{\text{z}}-{\mathit{\text{z}}}_0)}^2},\\ \sqrt{{(x-x_0)}^2+{(y-y_0)}^2+{(\mathit{\text{z}}-{\mathit{\text{z}}}_0)}^2}\le d\\ 0,\sqrt{{(x-x_0)}^2+{(y-y_0)}^2+{(\mathit{\text{z}}-{\mathit{\text{z}}}_0)}^2}>d\end{array},

where k denotes the penalty coefficient. This coefficient is used to adjust the penalty intensity for violations of no-fly zone constraints. In cases involving multiple no-fly zones, the penalty terms corresponding to each individual no-fly zone are summed to obtain the total no-fly zone penalty.

3.2.5 Path smoothness costs

The UAV flight track should minimize large-scale altitude changes and large-angle directional adjustments to form a smooth path. Path smoothness is evaluated using the turning angle {\omega }_{ij} and the climbing angle {\phi }_{ij}. Equations (7) and (8) represent the calculation formulas for the turning angle {\omega }_{ij} and climbing angle {\phi }_{ij}. The terms {\beta }_1 and {\beta }_2 serve as the penalty coefficients corresponding to the turning angle and the climbing angle. The cost function for path smoothness is presented in Eq. (9):

(7){\omega }_{ij}=\arctan \left(\frac{\left|\left|\overrightarrow{P_{ij}^{\prime }{P}_{i,j+1}^{\prime }}\times \overrightarrow{P_{i,j+1}^{\prime }{P}_{i,j+2}^{\prime }}\right|\right|}{\overrightarrow{P_{ij}^{\prime }{P}_{i,j+1}^{\prime }}\cdot \overrightarrow{P_{i,j+1}^{\prime }{P}_{i,j+2}^{\prime }}}\right),

(8){\phi }_{ij}=\arctan \left(\frac{Z_{i,j+1}-Z_{ij}}{\left|\left|\overrightarrow{P_{ij}^{\prime }{P}_{i,j+1}^{\prime }}\right|\right|}\right),

(9)F_5\left(X_i\right)={\beta }_{\text{1}}\sum _{j=1}^{n-2}{\omega }_{ij}+{\beta }_{\text{2}}\sum _{j=1}^{n-1}\left|{\phi }_{ij}-{\phi }_{i,j-1}\right|.

3.2.6 Total costs of the path

Considering the multiple factors of the UAV flight path, including the path length, safety, flight altitude, no-fly zone constraints, and path smoothness, in this study, the costs of each indicator were quantified and normalized to eliminate dimensional differences. Then, weights were dynamically assigned to each indicator based on scene characteristics, and finally, a total flight cost function was constructed, as shown in Eq. (10). The function value ranges from 0 to 1, which is negatively correlated with the path quality, meaning that the smaller the function value is, the higher the quality of the path generated by the algorithm is.

(10)F\left(X_i\right)=b_1{F}_1+b_2{F}_2+b_3{F}_3+b_4{F}_4+b_5{F}_5,

where b_1,b_2,b_3,b_4 and b_5 are the weight coefficients of these indicators. Letting the basic weights be b_{\text{base,}1},b_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e},2},b_{\text{base,3}},b_{\text{base,4}} and b_{\text{base,5}}, ensuring \sum _{i=1}^5{b}_{\text{base},i}=1. In this study, considering the factors such as weather conditions and terrain types, the dynamic adjustment amount was introduced to the basic weight to achieve adaptive optimization of weight coefficients. By intelligently optimizing the priority of various performance indicators in the objective function, the path planning of the UAV achieves the optimal effect across diverse scenarios. This paper set two terrain types and three weather conditions for dynamic weight adjustment.

In terms of terrain types, both moderately obstructed terrain and complex obstructed terrain were considered. In the moderately obstructed terrain, the obstacles were sparsely distributed but still posed threats. This was necessary to balance safety and efficiency. Therefore, the safety weight needed to be moderately increased, allowing the path to extend appropriately while reducing the smoothness requirement. In the complex obstructed terrain, obstacles were densely distributed, leading to a significantly increased risk of collision. Safety became the primary concern. Accordingly, the safety weight needed to be substantially raised, making full use of vertical space for obstacle avoidance, while significantly relaxing the requirements for both the path length and smoothness.

In terms of weather conditions, sunny, rainy, and foggy conditions were considered. In sunny conditions, the basic weight coefficients were maintained. During rainy weather, the visibility decreased, and the sensor performance was degraded. Therefore, the safety weight was increased to adopt a more conservative flight strategy and enhance the avoidance of no-fly zones. In foggy conditions, visibility was severely restricted, leading to increased navigation uncertainty. Consequently, the safety weight was significantly raised, allowing for frequent heading corrections to ensure reliable obstacle avoidance with visual limitations.

The specific weight adjustment range is presented in Table 1. In Table 1, b_n (where n = 1,2,3,4,5) represents the weight coefficient of the n-th performance indicator: path length (b₁), safety (b₂), flight altitude (b₃), no-fly zone constraint (b₄), and path smoothness (b₅).

According to the specific environment characteristics, the weight of each sub-item in the performance evaluation model was adjusted dynamically so that the planned path could adapt to the needs of different scenarios intelligently.

3.3 Development of the AMCEHO algorithm

Traditional EHO algorithm has a simple parameter structure and meets the need for balanced global guidance and local optimization in path planning. However, when directly applied to UAV 3D path planning, the algorithm has several limitations including insufficient population diversity, slow convergence speed, and a tendency to fall into local optima. To address these shortcomings, this section, on the basis of elucidating the core mechanisms of the EHO algorithm, introduced an enhanced form of an EHO algorithm called the AMCEHO algorithm, which incorporated three key improvements.

(1) The hybrid chaotic initialization combined three different chaotic maps with dynamic weighting. This could generate a large and diverse initial population in the solution space and laid a high-quality search foundation for the whole optimization process.

(2) A dynamic elite pool dynamically selected a set of high-quality elite individuals at the beginning of each iteration cycle with periodically fluctuating quantities from the current population. This improvement enhanced the global guidance capability while maintaining the population quality, thus effectively addressing the problem of insufficient global exploration capability.

(3) The multi-strategy collaborative updating used the elite individuals selected from the dynamic elite pool to update the population. Its core concept lied in the directional global search guided by the elites, while the remaining individuals conduct local fine-tuning, thereby achieving a dynamic balance between exploration and exploitation to ensure convergence stability.

Collectively, these improvements constituted a robust and efficient algorithm framework that was specifically designed for UAV 3D path planning. The AMCEHO algorithm effectively overcame the inherent limitations of the EHO algorithm while maintaining its fundamental strengths.

3.3.1 EHO algorithm

The EHO algorithm initializes a solution matrix consisting of two distinct gender roles, including bull elks (representing candidate solutions with enhanced exploration capabilities) and cow elks (representing solutions focused on local refinement). It simulates natural behavioral patterns to perform optimization, namely the rutting, calving, and selection seasons. The reproductive process of an EH can be regarded as an optimization process. Elk reproduce from one generation to the next to develop a more robust herd that is capable of confronting environmental challenges. Their breeding process of the population is therefore mapped to an optimization algorithm.

(1) Population initialization

Initially, the population of the EH utilized in the algorithm is divided into family groups based on the number of bulls. During the rutting season, the elk herd splits into families of various sizes. This division is based on the competition for dominance among bulls, for which the stronger bull establishes a family with a larger number of harems. To adapt the EHO algorithm to solve specific optimization problems, the key component must be defined: an objective function for evaluating solutions. The general form of the objective function can be expressed in Eq. (11):

(11){min}_{x}f(x),x\in \left[lb,ub\right],

where f(x) denotes the objective function, which is utilized to evaluate the fitness of each elk individual. The variable x_i represents the i-th individual in the elk population. In the expression, l{b}_i represents the lower bound of the attribute x_i and u{b}_i denotes the upper bound. N represents the total number of solutions in the population.

The EH is initially generated as a set of elk solutions, comprising both bulls and harems. The EH is represented as an \text{n}\times \text{EHS} matrix, and its specific form is presented in Eq. (12):

(12)EH=\left[\begin{array}{ccc}{x}_1^1& \cdots & x_n^1\\ ⋮& ⋮& ⋮\\ x_1^{EHS}& \cdots & x_n^{EHS}\end{array}\right].

(2) Rutting season

During the rutting season, the population is partitioned into families based on the male ratio (Br). The total number of families is calculated as B = |Br × EHS|. Within the EH, the elk with minimum fitness value (optimal fitness) are identified as the bulls representing the strongest individuals. These bulls are then allocated a larger number of harems. The corresponding selection criterion can be expressed in Eq. (13):

(13)\varpi =\arg {min}_{j\in (1,2,...B)}f(x^j).

Subsequently, B families are established. To assign harems to each bull in B, a roulette wheel selection method is employed, for which harems are allocated to bulls based on the ratio of their fitness value to the total fitness value, as shown in Eq. (14).

(14)P_j=\frac{f(x^j)}{{\displaystyle \sum _{k=1}^{B}f(x^k)}}.

(3) Calving season

During the calving season, the reproduction of elk calves within each family is primarily determined by the attributes of their sire and dam. If the index i of an elk calf matches that of the sire in the family, the elk calf’s reproduction follows Eq. (15):

(15)x_i^j(t+1)=x_i^j(t)+\alpha \cdot (x_i^k(t)-x_i^j(t)),

where \alpha denotes the inheritance coefficient, which regulates the intensity of parental characteristic transmission.

If the elk calf’s index coincides with that of the dam, the algorithm introduces an individual-specific perturbation term to enhance its exploration capability, as shown in Eq. (16):

(16)x_i^j(t+1)=x_i^j(t)+\beta \cdot (x_i^{h_j}(t)-x_i^j(t))+\gamma \cdot (x_i^r(t)-x_i^j(t)).

(4) Selection season

After the reproduction stage, all of the family members are merged into a temporary population E{H}_{\text{temp}}; subsequently, the elk in E{H}_{\text{temp}} are sorted in ascending order based on their fitness values, and the optimal individuals are retained to form a new generation of the population in a process that ensures the convergence of the population toward the optimal solution, as shown in Eqs. (17) and (18):

(17)E{H}_{\text{temp}}=sort(E{H}_{\text{temp}},key=f(x)),

(18)E{H}^j=E{H}_{\text{temp}},j=(1,...,EHS).

The rutting, calving, and selection seasons are repeated until the termination criterion is satisfied. Throughout the iterative process, the EHO algorithm continuously explores the optimal solution to the problem.

3.3.2 AMCEHO algorithm

(1) Hybrid chaotic initialization

Different chaotic maps exhibit distinct characteristics. This section discusses the integration of three different maps, including the logistic map, tent map, and PWLCM, to generate an initial population with a more uniform distribution and wider coverage range.

The logistic map has a simple structure but is sensitive to parameters, with a non-uniform distribution and poor chaotic performance, as shown in Eq. (19):

(19)x_{n+1}=\mu x_n(1-x_n).

The tent map has the ideal uniform distribution and strong ergodicity; however, it is prone to periodicity with finite precision because of its piecewise linear nature, as shown in Eq. (20):

(20)

The PWLCM is uniform in distribution, but its linear piecewise structure may result in its chaotic dynamic characteristics being less complex than those of nonlinear maps, as shown in Eq. (21):

(21)

Hybrid chaotic initialization fuses these three chaotic maps through the dynamic weights w_1,w_2 and w_3 (Xue et al., 2025), integrating the initial ergodicity of the tent map, the uniform distribution of PWLCM, and the continuous chaos of the logistic map. In addition, the weight coefficients satisfy the conditions of w_1+w_2+w_3=\text{1}. The formula for generating the hybrid chaotic sequence is prevented in Eq. (22).

(22)ChaoSeq=w_1\cdot Logistic+w_2\cdot Tent+w_3\cdot PWLCM.

To verify the improvement of the hybrid chaotic map in terms of the uniformity and ergodicity of the sequence distribution, in this study, the temporal distribution plots of the chaotic sequences were presented (Fig. 5). The analysis revealed that the logistic map exhibited periodic windows within specific parameter ranges, leading to compromised chaotic characteristics. However, the tent map and PWLCM might generate short-period sequences due to finite precision limitations. In contrast, the hybrid chaotic sequence proposed in this paper demonstrated superior uniformity and stochasticity across the entire parameter space, with no discernible periodicity or clustering patterns. This robust performance established a solid foundation for generating initial populations that exhibited extensive and uniform distribution throughout the solution space.

The chaotic sequence distribution alone was not sufficient to fully characterize the dynamic properties of a chaotic system. To further evaluate the randomness, complexity, and unpredictability of chaotic maps, their phase space diagrams are presented in this paper, as shown in Fig. 6. A sequence with highly chaotic characteristics should exhibit an irregular, unstructured, and uniform point cloud in its phase space diagram. Indeed, the phase space of the hybrid chaotic mappings exhibited the most dispersed and homogeneous point cloud distribution, with no discernible trajectories or voids (Fig. 6). This indicated that the generated sequences possessed greater complexity and randomness, effectively preventing algorithms from developing biases toward regular or structured patterns during initialization. Consequently, this approach ensured deeper diversity and randomness in the initial population.

(2) Dynamic elite pool

After an initially generated population completes a performance evaluation, it is necessary to screen out high-quality solutions from the current population to guide the search direction, thereby accelerating convergence and ensuring optimization quality. The elitist strategy is therefore crucial in evolutionary algorithms. However, an elite pool with a fixed size often leads to insufficient population diversity and premature convergence. To address this problem, we chose to introduce a dynamic elite pool, for which the capacity of the elite pool adaptively oscillated during the iteration process, utilizing a sine wave model (Gecili et al., 2021). This was specifically defined by the following mathematical model, as shown in Eq. (23):

(23)EliteSi\mathit{\text{z}}e(t)=(E_{\text{base}}+A\times [1-{(t/T)}^2]\times \sin (2\text{π}t/30))\cdot N,

where t denotes the current iteration number, T represents the maximum number of iterations, and N represents the population size. This formula enabled the elite pool capacity to fluctuate periodically between 25% and 40% of the population size. In the early stage of the search, the capacity of the elite pool was expanded to enhance exploration, while in the later stage, the capacity was reduced to focus on exploitation. This avoided the simplification of the search process and prevented falling into local optima.

(3) Multi-strategy collaboration update

While the dynamic elite pool could intelligently filter high-quality solutions, relying solely on a single update mechanism might lead to insufficient global exploration or excessive local development. To avoid the inherent flaws of a single update strategy, we introduced a multi-strategy collaborative method that balanced exploration and development through global exploration and local optimization. In each iteration, each solution had a probability of K to execute the global search strategy. Otherwise, the local optimization search was performed.

The global search strategy was guided by the elite individuals in the dynamic elite pool. By integrating the guidance information from the global optimal solution, elite individuals, and historical memory solutions. Purposeful and directional exploration was conducted, thereby improving the efficiency and quality of exploration. The expression for the global search strategy is given by Eq. (24).

(24)\begin{array}{rl}\overrightarrow{x_{\text{new}}}=& \overrightarrow{x_{\text{current}}}+\alpha \cdot \xi \cdot [w_g\cdot (\overrightarrow{g_{\text{best}}}-\overrightarrow{x})+\\ & w_e\cdot (\overrightarrow{e_{\text{lite}}}-\overrightarrow{x})+w_m\cdot (\overrightarrow{m_{\text{memory}}}-\overrightarrow{x})],\end{array}

where \xi is a random number following the N (0,1) distribution, and w_g,w_e,w_m are the weight coefficients for the three types of information. These coefficients satisfy the constraint of summing to one. The α term represents the adaptive learning rate, and its adjustment rule is specified in Eq. (25):

(25)\alpha (t)={\alpha }_{\text{init}}\cdot {\left(1-\frac{t}{T}\right)}^2+{\alpha }_{min}.

The local search strategy adopted the chaotic disturbance strategy to carry out fine searching near the existing optimal solution to improve the quality of the solution. The expression for the global search strategy is given by Eqs. (26) and (27).

(26)\overrightarrow{x_{\text{new}}}=\overrightarrow{x_{\text{base}}}+\beta \cdot (UB-LB)\cdot ChaoSeq,

(27)\beta (t)=0.1\times (1-t/T),

where β denotes the chaotic intensity coefficient, and ChaoSeq represents the hybrid chaos sequence. Additionally, the chaotic intensity coefficient decreased linearly, which meant that the disturbance was smaller and the search was more refined in the later stage of the algorithm.

The proposed algorithm was based on the EHO algorithm and incorporated hybrid chaotic initialization, a dynamic elite pool, and multi-strategy coordination updating. These three components were closely connected and jointly drove the population to evolve efficiently toward the global optimal solution. The algorithm steps are presented in Table 2, and the flowchart of the algorithm is illustrated in Fig. 7.

4 Case study

To systematically evaluate the performance advantages of the AMCEHO algorithm in UAV 3D path planning, in this study, clear validation objectives were defined, and experiments were conducted based on two different perspectives. This approach provided reliable experimental support for assessing the algorithm’s effectiveness.

(1) Validation of comprehensive algorithm performance: Through comparative experiments with multiple benchmark and test functions, the AMCEHO algorithm’s advantages in solution accuracy, convergence stability, and generalization ability were verified. Five representative intelligent optimization algorithms were selected as benchmark algorithms to fully characterize the optimization performance in high-dimensional spaces, including the EHO, PSO, great wall construction algorithm (GWCA), DE, and ACO algorithms. These algorithms respectively represented the mainstream technical routes in the current field of intelligent optimization: EHO was a new swarm intelligence algorithm proposed in recent years, PSO was a classic swarm intelligence optimization method, GWCA integrated the advantages of two bionic algorithms, DE belonged to an efficient evolutionary algorithm framework, and ACO performed excellently in combinatorial optimization problems. By comparing algorithms that covered different design concepts, it was possible to more objectively and comprehensively evaluate the comprehensive performance advantages of the AMCEHO algorithm.

(2) Validation of practical path planning performance: In 3D urban scenarios with varying obstacle complexities, the AMCEHO algorithm’s practicality and superiority in actual path planning tasks were verified. The EHO algorithm and the GWO algorithm were selected for comparison.

4.1 AMCEHO algorithm performance verification

To verify the performance of the AMCEHO algorithm proposed in this paper, the AMCEHO algorithm was compared with the EHO, PSO, GWCA (Guan et al., 2023), DE (Fu et al., 2013), and ACO algorithms utilizing six test functions, as shown in Table 3. The F₁–F₃ functions were high-dimensional unimodal test functions, and F₄–F₆ were high-dimensional multimodal test functions.

All of the comparison algorithms were uniformly configured with a population size of 30 and a maximum number of iterations of 500. Additionally, each test function was independently executed 30 times, and the optimal value, average value, and standard deviation for each algorithm were computed. The optimization values and convergence curves of the six algorithms are presented in Table 4 and Fig. 8.

The AMCEHO algorithm exhibited significant advantages across the six test functions. In the unimodal functions F₁–F₃, its optimal value reached the order of 10⁻² to 10⁻⁵, which was far superior to that of the comparison algorithms and demonstrated strong local search capability. For the multimodal functions F₄–F₆, the average values of the algorithm in the F₄ function and F₅ function were only 2.01 and 0.654 with stable standard deviations, indicating excellent global convergence and robustness. Additionally, as observed from the convergence curve (Fig. 8), the AMCEHO algorithm featured a fast convergence speed and low fitness value, and its optimization performance was significantly superior to that of the comparison algorithms. Overall, the AMCEHO algorithm proposed in this paper achieved remarkable improvements in solution accuracy, stability, and generalization ability.

4.2 UAV 3D path planning simulation analysis

To evaluate the performance of the AMCEHO algorithm in UAV 3D path planning, comparative simulation experiments were conducted among the AMCEHO, the EHO, and the GWO algorithms. Two simulated task scenarios with varying complexities were designed, each featuring distinct environmental characteristics. Obstacles were randomly generated in both scenarios to better reflect real uncertainties. Path planning was performed by selecting 12 waypoints, including 10 intermediate navigation points as well as the starting and ending points. The mission space was defined as 1000 × 1000 × 500 m. Each algorithm was independently run five times to ensure reliability. The population size was set to 30, and the maximum number of iterations was 200. The chaos parameter p was set to 0.31. The start point was (50, 50, 100), and the end point was (950, 950, 250).

First, a medium obstacles map was established, and the weather condition was set as sunny. The obstacles and the spherical no-fly zone were generated randomly. Three-dimensional path diagrams, convergence curve comparison diagrams, and performance comparison diagrams of the three algorithms were shown in Figs. 9 and 10.

Second, a complex obstacle map was established with a significant increase in the number and complexity of obstacles. Additionally, the weather condition was set as foggy. Three-dimensional path diagrams, convergence curve comparison diagrams, and performance comparison diagrams of the three algorithms were presented in Figs. 11 and 12. Moreover, a comparison diagram of the scenario performance indicators was output. This diagram is shown in Fig. 13.

To clearly demonstrate the optimization performance of the AMCEHO algorithm across various obstacle scenarios, the comparative analysis of experimental data revealed the final fitness values and number of iterations for different algorithms, as shown in Table 5.

As shown in Figs. 9(b), 11(b), and Table 5, the AMCEHO algorithm exhibited superior performance in both global optimization capability and convergence speed. In both scenarios, the AMCEHO algorithm achieved the lowest final fitness value, indicating significantly higher precision in multi-objective optimization compared to the other two algorithms. This enabled the algorithm to approach theoretical optimal solutions more accurately. Notably, in the medium obstacle scenario, the AMCEHO algorithm required only 46 iterations to converge, representing a 71.9% reduction compared to the EHO algorithm and a 69.5% decrease from the GWO algorithm. This indicated that under relatively simple constraint conditions, the AMCEHO algorithm could rapidly approach the global optimal solution with lower iteration consumption, effectively avoiding redundant operations. In the complex obstacle scenario, despite the increased intensity of environmental constraints, the AMCEHO algorithm still maintained stable convergence performance with 115 iterations, which was 35.0% and 29.4% less than those of the EHO and GWO algorithms, respectively. These results demonstrated that the search strategy of the AMCEHO algorithm exhibited stronger adaptability across varying complexity levels, maintaining higher iteration efficiency even when facing upgraded constraint conditions.

Path quality is a core reflection of the practicality of an algorithm. As shown in Figs. 10, 12, and 13, the AMCEHO algorithm demonstrated balanced and superior performance across multiple dimensions including safety, compliance, and path smoothness. In both scenarios, the safety costs of AMCEHO algorithm were 0, while the safety costs of EHO and GWO algorithms in the complex obstacle scenario reached 1695.53 and 1271.27 respectively, indicating that the path planned by AMCEHO algorithm could effectively avoid collision risks and meet fundamental safety requirements. The no-fly zone constraints costs of AMCEHO algorithm remained 0 in both scenarios, whereas EHO and GWO algorithms exhibited prohibitive costs of 8961.67 and 6960.01 in the complex obstacle scenario, demonstrating AMCEHO algorithm’s strict adherence to no-fly zone regulations and outstanding compliance performance. Notably, the flight altitude costs of AMCEHO algorithm were lower than that of EHO and GWO algorithms in both scenarios, and the path smoothness costs remained within a reasonable range. Besides, the normalized final costs of the AMCEHO algorithm were the lowest among the three in both scenarios, achieving a multi-dimensional balance of path length, altitude control, and smoothness.

5 Conclusions

5.1 Research findings

To address the challenges of slow convergence and susceptibility to local optima in 3D path planning for UAVs in urban low-altitude environments, in this study, a comprehensive technical framework for path planning was constructed that spanned environmental modeling and performance evaluation to algorithm optimization. The framework not only provided a realistic testing benchmark for algorithm validation through high-fidelity scenario modeling and adaptive multi-dimensional evaluation, but also proposed the AMCEHO algorithm with better chaotic ergodicity and optimization balance. Ultimately, the study achieved a significant improvement in key performance indicators in quantitative experiments.

To validate the basic optimization performance of the algorithm, this study compared it with several state-of-the-art algorithms, including the EHO, PSO, GWCA, DE, and ACO algorithms on six benchmark functions. The AMCEHO algorithm achieved optimal values ranging from 10⁻² to 10⁻⁵ in the unimodal functions. Among them, it reached 2.9020 × 10⁻² in the F₁ function, which represented a 99.73% improvement over EHO algorithm’s 1.0651 × 10², a 92.99% enhancement over PSO algorithm’s 4.1461 × 10⁻¹, and notable improvements of 99.89%, 99.95%, and 99.72% over GWCA, DE, and ACO algorithms respectively, demonstrating its exceptional local search capabilities. Additionally, the standard deviation of the AMCEHO algorithm in all function tests were small, demonstrating excellent stability and robustness. In two types of 3D path planning scenarios for UAVs with medium obstacles and complex obstacles, the AMCEHO algorithm required only 46 iterations to converge in the medium obstacle scenario (71.9% fewer than the EHO algorithm and 69.5% fewer than the GWO algorithm) and only 115 iterations in the complex obstacle scenario (35.0% fewer than the EHO algorithm and 29.4% fewer than the GWO algorithm). In terms of path quality, the safety costs and zero no-fly zone constraint costs of the AMCEHO algorithm were both zero in both scenarios. In contrast, the EHO and GWO algorithms incurred substantial safety costs of 1695.53 and 1271.27 respectively, with no-fly zone constraint costs reaching 8961.67 and 6960.01 in the complex obstacle scenario, indicating significant safety and compliance risks. Additionally, the normalized total costs of the AMCEHO algorithm in the medium and complex obstacle scenarios were reduced by 15.2% and 56.4% compared to the EHO algorithm, and by 23.15% and 52.5% compared to the GWO algorithm, achieving the coordinated optimization of convergence speed and path quality.

In summary, the results of this study have not only achieved technical optimization at the algorithm level but also held significant application value in the practice of the low-altitude economy. In the scenario of logistics and distribution, the characteristics of rapid convergence could reduce the energy consumption and delivery time of UAVs, and improve the efficiency of end-to-end logistics. In the scenario of emergency rescue, the high safety compliance rate and adaptability to complex environments could ensure the reliability and timeliness of rescue paths, gaining time for life-saving efforts. In scenarios such as urban inspection and agricultural plant protection, the balanced path quality could reduce equipment wear and tear and improve operational accuracy. This study provided key technical support for the large-scale application of low-altitude UAVs and possessed certain practical significance and promotion value for promoting the large-scale and safe development of the low-altitude economy.

5.2 Limitations and future work

While current research has achieved significant results, there is still room for further improvement from the perspective of engineering application and scenario expansion. Current validation mainly relies on simulation scenarios. Although these simulated scenarios could replicate uncertainties of real urban obstacles, they differ from actual low-altitude environments with practical factors like airflow disturbances, sensor noise, and temporary airspace control. In addition, current research focuses on the optimization of path planning for a single UAV. With the increasing application demands of UAV swarms in fields such as logistics distribution and collaborative inspection, how to extend the AMCEHO algorithm to multi-UAV collaborative scenarios to achieve airspace conflict avoidance and resource collaborative allocation among multiple UAVs still needs further exploration. Moreover, the current adaptation of the algorithm to dynamic environments mainly relies on preset static obstacle scenarios, and it cannot handle sudden scenarios such as sudden movement of obstacles and the designation of temporary no-fly zones.

To address the limitations in current research regarding engineering applications and scenario expansion, future research will promote the AMCEHO algorithm toward practical and large-scale implementation through multi-dimensional technical upgrades. To narrow the gap between simulation and real applications, emphasis will be placed on strengthening real-scenario verification and data calibration. Combined with data collected from actual flights, such as coordinates of obstacles in real cities and UAV flight trajectory records, the reliability and robustness of the algorithm in real scenarios will be further verified. To meet the needs of large-scale applications of UAV swarms in fields such as logistics distribution and collaborative inspection, the multi-UAV collaborative planning capability will be expanded based on the core optimization of the AMCEHO algorithm, solving key problems such as airspace conflict avoidance and resource collaborative scheduling among multiple UAVs, and improving the efficiency of swarm operations. To deal with emergency scenarios, in the future, real-time perception technologies such as visual sensors and radars carried by UAVs can be integrated to build a closed-loop mechanism for dynamic obstacle perception and path re-planning, enhancing the algorithm’s response capability to dynamic factors such as temporary no-fly zones and sudden obstacles.

References

[1] Ahmad J, Nadhir Ab Wahab M, (2025). Enhancing the safety and smoothness of path planning through an integration of Dijkstra’s algorithm and piecewise cubic bezier optimization. Expert Systems with Applications, 289( 1): 128315

[2] Arora S, Singh S, (2019). Butterfly optimization algorithm: a novel approach for global optimization. Soft Computing, 23( 3): 715–734

[3] Al-Betar M A, Awadallah M A, Braik M S., (2024). Elk herd optimizer: A novel nature-inspired metaheuristic algorithm. Artificial Intelligence Review, 57( 3): 48

[4] Alikhani Koupaei J, Ebadi M J, (2025). A chaotic decomposition-based approach for enhanced multi-objective optimization. Mathematics, 13( 5): 817

[5] Bo L, Zhang T, Zhang H, Hong J, Liu M, Zhang C, Liu B., (2024). 3D UAV path planning in unknown environment: A transfer reinforcement learning method based on low-rank adaption. Advanced Engineering Informatics, 62( PD): 102920

[6] Bouguettaya A, Zarzour H, Kechida A, Taberkit A M, (2022). Deep learning techniques to classify agricultural crops through UAV imagery: A review. Neural Computing & Applications, 34( 12): 9511–9536

[7] Deng L B, Li C L, Sun H L, Qiao L Y, Miao X D, (2022). Dual mutations collaboration mechanism with elites guiding and inferiors eliminating techniques for differential evolution. Soft Computing, 26( 4): 1923–1940

[8] Dorigo M, Maniezzo V, Colorni A, (1996). Ant system: optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics, 26( 1): 29–41

[9] Fu Y G, Ding M Y, Zhou C P, Hu H P, (2013). Route planning for Unmanned Aerial Vehicle(UAV) on the sea using hybrid differential evolution and quantum-behaved particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics. Systems, 43( 6): 1451–1465

[10] Gan T, Dong S, Wang S Y, Li J X, (2024). Distributed resource allocation in dispersed computing environment based on UAV track inspection in urban rail transit. Computers, Materials & Continua, 80( 1): 643–660

[11] Gecili E, Palipana A, Brokamp C, Huang R, Andrinopoulou E R, Pestian T, Rasnick E, Keogh R H, Ni Y, Clancy J P, Ryan P, Szczesniak R D, (2021). Seasonality, medi ation and comparison (SMAC) methods to identify influences on lung function decline. MethodsX, 8: 101313

[12] Guan Z Y, Ren C J, Niu J T, Wang P X, Shang Y Z, (2023). Great wall construction algorithm: A novel meta-heuristic algorithm for engineer problems. Expert Systems with Applications, 233( 1): 120905

[13] Guo B B, Sun Y, Chen Y Y, (2025). Safe path planning of mobile robot based on improved particle swarm optimization. Transactions of the Institute of Measurement and Control, 47( 9): 1715–1724

[14] Guo L Y, Gu W, (2024). PMSOMA: optical microscope algorithm based on piecewise linear chaotic mapping and sparse adaptive exploration. Scientific Reports, 14( 1): 20849

[15] He J G, Peng Z P, Qiu J B, Cui D L, Li Q R, (2023). A novel elitist fruit fly optimization algorithm. Soft Computing, 27( 8): 4823–4851

[16] He Y, Wang M, (2024). An improved chaos sparrow search algorithm for UAV path planning. Scientific Reports, 14( 1): 366

[17] Huang Y H, Zhang Q L, Zhao Y B, (2025). A novel color image encryption algorithm based on infinite collapse map and hierarchical strategy. Digital Signal Processing, 167: 105428

[18] Katoch S, Chauhan S S, Kumar V, (2021). A review on genetic algorithm: past, present, and future. Multimedia Tools and Applications, 80( 5): 8091–8126

[19] Kennedy JEberhart R (1995). Particle swarm optimization. In: Proceedings of International Conference on Neural Networks, 4(0), 1942–1948

[20] Mirjalili S, Mirjalili S M, Lewis A, (2014). Grey wolf optimizer. Advances in Engineering Software, 69: 46–61

[21] Mu L X, Liu W H, Wang H C, Zhang Y, (2025). Research of UAV 3D path planning based on improved Dwarf mongoose algorithm with multiple strategies. Scientific Reports, 15( 1): 26979

[22] Petavratzis E, Moysis L, Volos C, Stouboulos I, Nistazakis H, Valavains K, (2021). A chaotic path planning generator enhanced by a memory technique. Robotics and Autonomous Systems, 143( 1): 103826

[23] Peng Y, Zhang C X, Tanksale A, Yu D Z, (2025). Multi-objective optimization for time-dependent vehicle routing problem with drones. Expert Systems with Applications, 291: 128503

[24] Salgotra R, Singh U, Saha S, Gandomi A H, (2021). Self adaptive cuckoo search: Analysis and experimentation. Swarm and Evolutionary Computation, 60( 1): 100751

[25] Wang J W, Huang Z H, Song Y H, (2025). Intelligent planning of safe and economical construction sites: Theory and practice of hybrid multi objective decision making. Frontiers of Engineering Management, 12( 3): 487–509

[26] Wen J F, Wang F, Su Y B, (2025). A Bi-Layer collaborative planning framework for multi-UAV delivery tasks in multi-depot urban logistics. Drones (Basel), 9( 7): 512

[27] Wu D, Shi Z Y, Zhang Y B, Huang M X, (2024). Autonomous navigation for cellular-connected UAV in highly dynamic environments: A deep reinforcement learning approach. Journal of Aerospace Engineering, 37( 5): 04024063

[28] Xue W, Wu Z, Xu H, Wang H, Liang Q, Ma C, Zhou Y, Xu S., (2025). Comprehensive risk assessment of urban flood process based on dynamic weights and lumped impact parameters. Journal of Hydrology, 662( PA): 133903

[29] Xue Y G, (2025). Multi-strategy improved snake optimizer for library robot path planning problems. Journal of Supercomputing, 81( 10): 1155

[30] Zhang B, Cai X F, Li G Q, Li X M, Peng M J, Wang H W, (2025). Path planning in radioactive environment of nuclear facilities based on modified A-star algorithm and search neighborhood optimization. Nuclear Engineering and Technology, 57( 10): 103711

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