Objective Programming Partitions and Rule-Based Spanning Tree for UAV Swarm Regional Coverage Path Planning

Highlights

What are the main findings?

• A two-stage approach combining sub-region partitionings and spanning tree generation is adopted to realize UAVs’ regional coverage path planning. A fast calculation method for the number of turns under the Spanning Tree Coverage (STC) algorithm is proposed, and a minimum turn rule is developed.
• The proposed OPP-RSTC algorithm outperforms traditional algorithms in UAV coverage path planning. Simulations show all UAVs have identical path length (150 dm) and task time, with fewer turns (26–52) and higher coverage rate (94.04% effective coverage in real flight) than traditional algorithms.

What are the implications of the main findings?

• Objective programming-based partitionings resolve unbalanced workload (Voronoibased algorithm 3 has path length gaps like 206 dm vs. 78 dm), and the rule-based spanning tree is theoretically proven to minimize UAV turns in specific conditions.
• This study provides a practical solution for multi-UAV coverage path planning, addressing traditional issues such as unbalanced allocation and excessive turns, suitable for scenarios like environmental monitoring and disaster rescue.

Abstract

To address the problem of regional coverage path planning for unmanned aerial vehicle swarms (UAVs), this study proposes an algorithm based on objective programming partitions (OPP) and rule-based spanning tree coverage (RSTC). Aiming at the shortcomings of the traditional Divide Areas based on Robots Initial Positions combined with Spanning Tree Coverage (DARP-STC) algorithm in two core stages, that is, region partitions and spanning tree generation, the proposed algorithm conducts a targeted design and optimization, respectively. In the region partition stage, an objective programming and 0–1 integer programming model are adopted to realize the balanced allocation of UAVs’ task regions. In the spanning tree generation stage, a rule is designed to construct a spanning tree of coverage paths and is proven to achieve the minimum number of turns for the UAV under certain conditions. Both simulations and physical experiments demonstrate that the proposed algorithm can not only significantly reduce the number of turns of UAVs but also enhance the efficiency and coverage degree of tasks for UAV swarms.

1. Introduction

In recent years, with the rapid iteration [1] of the unmanned aerial vehicle swarm (UAVs) and the continuous reduction in cost [2], they have been widely used in many fields, such as environmental monitoring [3], agricultural production [4], disaster management [5], power inspection, urban security [6] and so on, relying on their advantages of high flexibility [7], convenient deployment, and wide operation range [8]. It has become a key tool for improving industry efficiency and expanding operational boundaries. For example, in the field of environmental monitoring, UAVs need to collect data without omission in specific geographical areas, and the rationality of their coverage path directly affects the integrity and accuracy of monitoring data [9]. Unscientific path planning may lead to missing or repeated collection in key areas, reducing the monitoring efficiency; In agricultural production, UAV-based agricultural plant protection operations depend on the path design of fully covering farmland [10]. The inefficient path will cause the wastage of pesticides or fertilizers, and may also affect the growth of crops due to uneven coverage; In the disaster rescue scenario, UAVs need to quickly traverse the disaster area to search for trapped people and assess the disaster situation. The quality of path planning is directly related to the grasp of rescue opportunities and rescue effectiveness [11]. Currently, the unmanned aerial vehicle (UAV) coverage path planning (CPP) problem can be classified based on the characteristics from different dimensions, as follows:

• According to core requirements and coverage modes, UAV coverage path planning can be divided into partial coverage, full coverage, single coverage, multiple coverage, and periodic coverage with time stamp.
• According to the spatial dimension of the coverage environment, it can be divided into two-dimensional coverage and three-dimensional coverage.
• According to the modeling methods for reconnaissance environments, the mainstream methods include topological mapping, regional decomposition, and grid mapping [12].

Scholars have developed various approaches to UAV coverage path planning. Savvas D. Apostolidis et al. introduced a grid model with three coverage modes (GCM, BCM, CCM), improving cell labeling and path generation to prevent boundary crossing and incomplete coverage [13]. Gang Tang et al. proposed a region optimal decomposition (ROD) method, using an enhanced DFS to decompose and merge concave polygons, and a genetic algorithm to optimize sub-region sequence, effectively reducing turns and non-working distance in irregular areas [14]. Mengyang Wang et al. presented a collaborative 3D path planning framework for fixed-wing UAVs, integrating vehicle capabilities and terrain with dynamic programming and Dubins curves to achieve full coverage with minimized path length and computation time [15]. In addition, there are many related methods or frameworks, such as rotational tiling (RT) [16], an adaptive SAR framework [17], a real-time and adaptive UAV system [18], and a framework for the collaboration between UAV and USVs [19].

Although existing UAV coverage path planning approaches offer valuable technical insights, they fail to effectively address uneven task distribution and excessive flight turns in multi-UAV collaboration scenarios. This paper proposes a novel algorithm integrating OPP and RSTC to resolve these issues, with OPP ensuring balanced task allocation via 0–1 integer programming and customized rules optimizing spanning tree structure to reduce turns, and dual simulation and real-world validation confirming its superiority in task balance and coverage efficiency, thus providing a reliable solution for multi-UAV collaborative applications.

The remainder of this paper is organized as follows: In Section 2, related works are presented and discussed. Section 3 provides a detailed description of the problem, including grid map modeling and key definitions. Section 4 introduces the sub-region partition method based on objective programming, and Section 5 elaborates on the rule-based spanning tree generation algorithm along with its optimality proof. Simulation results and comparative analyses are presented in Section 6, and real-flight experiments and performance evaluation are discussed in Section 7. Finally, Section 8 concludes the study and suggests directions for future research.

2. Related Work

This section systematically reviews the research progress of UAV regional coverage path planning (CPP) from three dimensions: single UAV path planning (divided by 2D/3D modeling), multi-UAV cooperative planning (divided by 2D/3D modeling), and environmental adaptation.

2.1. Single UAV Coverage Path Planning

Regarding the impact of wind on drones, Coombes et al. [20] applied the classic boustrophedon algorithm to unmanned aerial vehicle aerial measurements, considering the influence of wind on the path.They proposed using trochoidal turning (cycloid turning) to adjust the trajectory and optimized the straight section path by calculating the wind correction angle and ground speed. Under a wind speed of 10 m/s, the proposed method yields a noticeable reduction in mission duration. For the two-dimensional visual coverage task, Mansouri et al. [21] established a planning framework that couples camera footprints, considering the camera’s 3D degrees of freedom (x, y, and psi) motion. Metaheuristic algorithms such as pattern search (PS), genetic algorithm (GA), and particle swarm optimization (PSO) were used to approximate the optimal solution, achieving coverage rates of over 96% in both convex and non-convex regions, and generating the shortest path from takeoff to landing [21].

For 2D scenarios with obstacles, Zhang et al. [22] combined the sub-region coverage sequence with the Laguerre diagram and reduced the number of search nodes through local grid division. In the 600 × 600 grid scenario, the search efficiency was improved compared to the traditional A* algorithm, which significantly enhanced the efficiency of obstacle avoidance path planning between regions [22].

For 3D region modeling and path optimization, Cankun Xie et al. [8] proposed an elite reverse learning population selection strategy based on piecewise mapping, a stochastic factor-controlled elite pool exploration strategy, and a hard frost puncture exploitation mechanism based on the sine-cosine function, constructing the ELRIME algorithm, and verifying its effectiveness in 3D UAV path planning.

For dynamic adjustment of 3D complex environments, Melo et al. proposed the 3DD-CPP method [23], which realizes online coverage path planning for UAVs in 3D environments by combining linear optimization and heuristics, enabling dynamic adjustment in unknown environments. They also constructed an energy cost estimation model [23] considering UAV power usage and implemented distributed execution of the algorithm via fog-edge computing to support path re-optimization in dynamic environments.

2.2. Multi UAV Cooperative Coverage Path Planning

For 2D task allocation strategies, Zhao et al. proposed a clustering-based hyper-heuristic algorithm (CBHHA) [24]. They first cut the fully connected graph formed by target regions [24] into multiple subgraphs using spectral clustering, then allocated tasks based on the UAV capability assessment results to achieve load balancing among heterogeneous UAVs. Meanwhile, they utilized graph neural networks [24] to parameterize the heuristic space and automatically design and optimize heuristic metrics through reinforcement learning, improving task allocation efficiency and path planning performance. Xiao et al. proposed a coverage-based scanning clustering algorithm (CSCA) [25]. By taking UAV bases as initial cluster centers, they performed regional clustering based on spatiotemporal similarity and designed a region transfer strategy to balance the task completion time of each UAV [25]. They also adopted the nearest-to-end policy to optimize the regional visiting order, improving the allocation efficiency and balance of multi-region coverage tasks for heterogeneous UAVs [25].

For 2D collaborative coverage algorithms, Li et al. [26] proposed a weighted target sweep coverage (WTSC) algorithm for path planning of multiple UAVs to solve the Min-Time Max-Coverage (MTMC) problem. The algorithm [26] considers target weights and UAV performance constraints, constructs an objective function including task time and coverage weight, and uses a DFS strategy to assign targets to UAVs, maximizing coverage while shortening task time, with performance superior to that of the existing CycleSplit and G-MSCR algorithms [26].

For multi-UAV minimum time coverage of ground areas and remote sensing, Avellar et al. [27] proposed a solution that splits the task into two parts: modeling the coverage area as a graph (with vertices as geographic coordinates for single-UAV minimum-time coverage) and solving a mixed-integer linear programming problem based on this graph to route UAV teams. This method [27] explicitly considers practical constraints like UAV maximum flight time, setup time (for UAV preparation and launch), and the number of operators, enabling automatic selection of the optimal number of UAVs while minimizing total mission time, and its effectiveness has been verified through real-world experiments with fixed-wing UAVs.

For optimizing the collaborative efficiency of multi-UAV coverage in urban environments, Muñoz et al. [28] proposed two coverage path planning algorithms: one adopts a back-and-forth flight pattern to cover target areas delimited by perimeters, and the other calculates smooth paths to pass through predefined waypoints. The algorithms [28] employ a deformable triangular leader-follower formation (with a virtual leader for even-numbered UAVs) and integrate the Fast Marching Square (FM²) algorithm for collision avoidance. A hybrid approach [28] is introduced to reduce computation time by activating FM² only when approaching obstacles. Furthermore, 2D paths are combined with terrain floor level data to generate smooth 3D paths, significantly enhancing the collaborative efficiency and operational safety of multi-UAVs in cluttered urban scenarios [28].

2.3. Environmental Adaptation and Energy Efficiency Optimization

For the problem of environmental adaptation during UAV landing in maritime environments, Wang et al. [29] proposed a solution combining LSTM network and adaptive fuzzy control. They used an LSTM network [29] to predict the deck motion (roll, pitch, heave) of a USV affected by wind and waves, determining a safe landing time window, and designed an adaptive fuzzy fixed-time tracking control strategy, utilizing fuzzy logic systems to approximate unknown dynamics, and handling external disturbances and unmodeled dynamics, ensuring stable landing of UAVs in complex maritime environments.

For energy efficiency optimization in multi-UAV multi-region coverage scenarios, Ahmed et al. [30] proposed an energy-efficient multi-UAV multi-region coverage path planning approach (E²M²CPPA). They generated intra-region paths [30] using a back-and-forth strategy to reduce the number of turns, and combined a smoothing turns approach (STA) based on Bezier curves to reduce energy consumption from turns. They modeled inter-region path planning as a mixed integer linear programming problem, using the CPLEX solver (for small-scale) and heuristic methods (for large-scale) to optimize the region visiting order, and proposed a region allocation optimization strategy to balance energy consumption among UAVs, significantly reducing overall energy consumption [30].

3. Problem Description

3.1. Grid Map Modeling

The grid (equivalent to the cell in this study) mapping method divides the environmental area into several grid units of equal size, with a single grid serving as the basic unit for modeling. It features a simple and efficient modeling process, thus becoming one of the most widely used environmental modeling methods. This method is also frequently employed in research and practical application scenarios of UAV aerial photography technology [31], and its specific illustration is shown in Figure 1.

Meanwhile, to ensure information accuracy in large areas, the aerial images captured by the UAV-mounted camera need to have a certain amount of overlap with adjacent images. In regional coverage tasks, to ensure that the images captured by the UAV in each grid area completely include all objects in the target area, the actual geographical area corresponding to each grid should be slightly smaller than the field of view of the UAV. Therefore, the scanning width of the UAV is used to represent the side length of the rectangular area corresponding to the grid in the actual geographical environment [32]. This study assumes that the aspect ratio of the UAV’s operation range is 1:1, that is, $a=b$. UAV reconnaissance range equations are shown in Figure 2 and Equations (1) and (2).

$$d=htan\left({\displaystyle \frac{\theta }{2}}\right)$$

(1)

$$b=2d\times {\displaystyle \frac{1}{\sqrt{2}}}=2htan\left({\displaystyle \frac{\theta }{2}}\right){\displaystyle \frac{1}{\sqrt{2}}}$$

(2)

The length and width of the actual task region to be covered are denoted by L and W, respectively, and the scanning width of the UAV at the current altitude is denoted by b. Then, the number of rows and columns after rasterizing the task region is given by

$$rows=⌈{\displaystyle \frac{L}{b}}⌉$$

(3)

$$cols=⌈{\displaystyle \frac{W}{b}}⌉$$

(4)

Among them, L and W represent the length and width of the region, respectively, and $\lceil \rceil$ represents rounding upwards. Therefore, the set of grids can be represented as a set [32] represented by grid (in this study, it refers to subcell) $(x,y)$, as shown in Equation (5).

$$U=\left\{\right(x,y)\mid x\in [0,\mathrm{rows}-1],y\in [0,\mathrm{cols}-1\left]\right\}$$

(5)

3.2. Explanation of Key Definition

The explanation of some terms is shown in Figure 3, and the others are shown in definitions.

Definition 1.

A region is an area that UAVs need to survey and cover.

Definition 2.

Sub-region refers to the area that a single drone needs to achieve full coverage.

Definition 3.

When modelling the environment, the size of the subcell was set based on the reconnaissance range of the drone (as shown in Equation (2)), that is, the reconnaissance range of the drone is the size of the subcell.

Definition 4.

Drones must pass through the subcell center (in simulations) to achieve full coverage within their region (detecting all subcells).

Definition 5.

When dividing sub-regions, the cell is the smallest unit of division, meaning that each sub-region detected by the drone was composed of an integer number of cells.

Definition 6.

Continuous points refer to the adjacent points of cell centers within the same sub-region (if continuous points are not connected by adjacent line segments, these points become discrete points).

Definition 7.

Discrete points refer to non-adjacent cell centers within the same sub-region.

Definition 8.

The sub-regions that each drone needs to detect are connected, which means that there is always a cell adjacent to any cell in the region.

4. Sub-Region Partitions Based on Objective Programming

After completing the grid modeling, it is necessary to use an appropriate algorithm to subdivide the region to meet the needs of UAV cluster cooperative operation. Reasonable sub-region division is an indispensable core component of an efficient CPP algorithm. In traditional DARP (Divide Areas based on Robots Initial Positions) algorithms [34], the Voronoi diagram method is commonly used for sub-region division. This method only divides the area based on the distance between the to-be-reconnoitered region and the initial position of the drone, making it difficult to achieve a balanced workload allocation among multiple drones. To address this limitation, this study proposes a sub-region division method based on objective programming: while retaining the consideration of distance factors, it further incorporates the goal of balanced workload allocation among drones, thereby achieving a more optimal regional division effect.

In practical deployment, the Ground Control Station (GCS) can act as the core decision-making entity to conduct centralized global optimization via a 0–1 integer programming model and the spanning tree-based UAV flight path strategy, thus formulating static pre-mission planning schemes for partitioned UAV coverage tasks.

4.1. Principles of Sub-Regional Partitions

A scientific CPP algorithm should meet the following three requirements when conducting sub-region partitions:

• Minimize the space distance between the reconnaissance region and the initial position of each UAV.
• Ensure that each subarea is a connected graph.
• The reconnaissance load of each UAV is evenly distributed so that the number of grids borne by each UAV is as close as possible.

By meeting the above conditions, the path efficiency, regional coverage connectivity and cluster operation fairness of UAVs can be optimized simultaneously.

4.2. Establishment of Sub-Regional Partitions Model

To solve this problem, this study proposes a sub-region partition model that integrates objective programming and 0–1 integer programming theory. By introducing 0–1 decision variables to build a mathematical model, the sub-region partition problem is transformed into a standard 0–1 integer programming model. The model has formed a mature theoretical system and solution algorithm in the academic community, which can effectively address combinatorial optimization problems under multi-objective constraints. The specific objective function is as follows:

$$\begin{array}{ccc}\hfill & minz=w_1\times d_1^{+}+w_2\times |d_2^{+}-d_2^{-}|\hfill & \end{array}$$

(6)

$$\begin{array}{ccc}\hfill & minz=P_1\times d_1^{+}+P_2\times |d_2^{+}-d_2^{-}|\hfill & \hfill P_1\gg P_2\end{array}$$

(7)

$$\begin{array}{ccc}\hfill & minz=P_1\times d_1^{+}+P_2\times |d_2^{+}-d_2^{-}|\hfill & \hfill P_2\gg P_1\end{array}$$

(8)

z is the objective function, which needs to be minimized, and $P_1$ is used as the penalty coefficient to penalize the mismatch between performance and reconnaissance range. $P2$ is another penalty coefficient used to penalize the distance that exceeds the target constraint. $w1$ and $w2$ are weights, and different weights represent different focuses of consideration.

$$minz\mathrm{s}.\mathrm{t}.x_{ij}\in D_{t}1\le t\le k$$

(9)

Among them, $D_t$ is the t-th sub-region, and the objective function z is one of the three objective functions mentioned above. Its purpose is to adopt different objective functions for different requirements. Hard constraints (conditional constraints) are shown in Equations (10)–(12).

$$\sum _{k=1}^5{x}_{ijk}=1\forall i,j$$

(10)

$$f\left(i\right)=1i=1,2,\dots ,5$$

(11)

$$x_{ijk}=0,1$$

(12)

The decision variable $x_{ijk}$: $x_{ijk}=1$ indicates that the subgrid at row i and column j is reconnoitered by sub-UAV k, $x_{ijk}=0$ indicates that the subgrid is not reconnoitered.

The function $f\left(i\right)$ represents the connected component of the i-th region (in a connected graph, its connected component is counted as 1).

To ensure the connectivity of sub-regions assigned to each unmanned aerial vehicle (UAV) (avoiding isolated cells) while balancing computational feasibility, since global connectivity is an NP-hard problem, this study adopts a local heuristic neighborhood constraint scheme. It is important to note that this local constraint cannot guarantee that each sub-region has exactly one connected component.

$$2x_{i,j,k}\le \sum _{(i^{\prime },j^{\prime })\in N(i,j)}{x}_{i^{\prime },j^{\prime },k}\forall (i,j)\in V, k\in K$$

(13)

where $N(i,j)$ denotes the set of valid neighboring cells of cell $(i,j)$ (i.e., the four orthogonal adjacent cells (up, down, left, right) that are within the study area and not in the no-fly zone); K is the set of drone indices; V is the set of points.

The soft constraints (objective constraints) are shown in Equations (14)–(16).

$$\overline{P}={\displaystyle \frac{{\sum }_{i=1}^k{\sum }_j{x}_{ij}}{{\sum }_{i=1}^k{P}_i}}$$

(14)

$$\sum _i\sum _j\left|\sum _{k=1}^5{x}_{ijk}-P_i\times \overline{P}\right|\le 0$$

(15)

$$\sum _i\sum _j\sum _{k}D\left(G_k,x_{ijk}\right)\le D$$

(16)

where D represents the Euclidean distance between two points, $G_i$ denotes the initial point of the UAV in the i-th sub-region, $\overline{P}$ indicates the average performance indicator of the UAV, and $P_i$ represents the performance indicator of the UAV. To formulate this as a goal programming problem that reflects different objectives under positive and negative deviation variables, Equations (15) and (16) are modified into Equations (17) and (18).

$$\sum _i\sum _j\left|\sum _{k=1}^5{x}_{ijk}-P_i\times \overline{P}\right|+d_{2-k}^{-}-d_{2-k}^{+}=0$$

(17)

$$\sum _i\sum _{j}D\left(G_i,x_{ij}\right)+d_1^{-}-d_1^{+}=D$$

(18)

4.3. Connectivity Verification Process

Since heuristic local constraints (Equation (13)) cannot guarantee global connectivity, in the actual solution process, if any sub-region has more than one connected component (i.e., it is not a connected graph), the solution is defined as infeasible. Subsequently, we fine-tune parameters, including the input data order and optimal gap, which prompts the solver to adopt different cutting plane strategies. This enables the model to be resolved and ultimately yields a feasible solution. The verification process is illustrated in Figure 4.

5. The Optimal Generation Rule for Spanning Trees and Its Proof

5.1. Recursive Formula for the Number of Spanning-Tree Turns

It can be seen from Figure 5 that in the area coverage path planning, the path formed by the flying spanning tree can be split line by line (main direction) by adopting the above rules. Each arrow represents a number of turns. It can be found that the total number of turns of the path before splitting is equal to the sum of the turns of the path after splitting.

For a spanning tree that is a tree when adding layers line by line, we have the following Equation (19):

$$f^n(L_{\mathrm{total}},R_{\mathrm{total}})=f(L_1,L_2,R_{1-2})+f^{n-1}(L_{\mathrm{rest}},R_{\mathrm{rest}})$$

(19)

where $f^n(L_{\mathrm{total}},R_{\mathrm{total}})$ denotes the function for calculating the number of turns in n rows (columns) for the full point set $L_{\mathrm{total}}$ and the inter-point relationship $R_{\mathrm{total}}$; $L_{total}$ represents the set of all points; $R_{total}$ denotes the relationship among all point sets (i.e., the relationship of connected edges); $L_i$ indicates the set of points in the i-th main direction layer; $R_{i-i+1}$ refers to the relationship between the i-th main direction layer and the $(i+1)$-th main direction layer (i.e., the relationship of connected edges); $L_{\mathrm{rest}}$ denotes the set of remaining points; and $R_{\mathrm{rest}}$ represents the relationship among the remaining point sets.

5.2. Method for Quickly Calculating the Number of Turns

Firstly, define three concepts—aligning at both ends, not aligning at both ends, and aligning at one end—as follows:

• Aligning at both ends: The point set is aligned at both ends between the top and bottom rows.
• Not aligning at both ends: The set of points between the top and bottom lines is not aligned at both ends.
• Aligning at one end: Between the top and bottom rows, the point set has two ends, and only one end is aligned.

For aligning at both ends, the relationships between layers and the increase in the number of turns $f(l_i,l_{i+1},r_{i-i+1})$ for this layer are illustrated in Figure 6.

For not aligning at both ends, the relationships between layers and the increase in the number of turns $f(l_i,l_{i+1},r_{i-i+1})$ for this layer are illustrated in Figure 7.

For aligning at one end, the relationships between layers and the increase in the number of turns $f(l_i,l_{i+1},r_{i-i+1})$ for this layer are illustrated in Figure 8.

In fact, the above three cases, their corresponding strategies, and the number of turns can be simplified to a core principle, namely, the number of corners principle.

$$f\left(l_i,l_{i+1},R_{i-i+1}\right)=2\times T$$

(20)

$$f^n(L,R)=\sum _{i=1}^{n-1}f\left(l_i,l_{i+1},R_{i-i+1}\right)+4$$

(21)

Among them, T denotes the number of corners of the relationship $R_{i-i+1}$ connecting $l_i$ and $l_{i+1}$. From this, the number of turns in a path can be quickly calculated. Here, 4 is the correction amount for supplementing the first layer, and $f(l_i,l_{i+1},r_{i-i+1})$ is the additional number of turns between each layer in the figure, which can be selected from 4, 6, or 8 according to Figure 6, Figure 7 and Figure 8 and can be calculated by Equation (20).

From this, it can be concluded that when there are no discrete points between layers and only continuous points, there exists a selectable optimal relationship $r_{i-i+1}$ for each layer that minimizes the number of turns $f(l_i,l_{i+1},r_{i-i+1})$. The range of $f_{optimal}(l_i,l_{i+1},r_{i-i+1})$ and $f_{else}(l_i,l_{i+1},r_{i-i+1})$ is shown in the Equations (22) and (23).

$$f_{\mathrm{optimal}}(l_i,l_{i+1},r_{i-i+1})\in \{4,6\}$$

(22)

$$f_{\mathrm{else}}(l_i,l_{i+1},r_{i-i+1})\in \{4,6,8\}.$$

(23)

Based on the above analysis, we obtained the optimal strategy for spanning trees and the calculation method for the number of turns under the optimal strategy, as shown in Equation (22). The total number of turns $f^n(L,R)$ of the spanning tree can be computed via the recursive Formula (19) and the interlayer strategy selection Equations (22) and (23) quickly. This optimal strategy can be extended to broader scenarios, such as the inclusion of discrete points and discretization of continuous points. A detailed proof is provided in the next subsection.

5.3. Proof of Optimal Generation Rule for Spanning Trees

Conditions: Let $T^n$ be a spanning tree with n rows. For any given regional partitions, if it satisfies the following conditions:

• For a given deterministic sub-region.
• Edge relationships are only within one’s own region.
• In the main direction, for all k ($1\le k\le n$), the substructure formed by taking the first k layers of $T^n$ remains a tree (denoted as $T^k$).

Conclusion: The each UAV flight paths composed of $f^n(L,R)(i=1,2,\dots ,n)$ for such spanning trees satisfy

$$f_{\mathrm{optimal}}^n(L,R)\le f_{\mathrm{other}}^n(L^{\prime },R^{\prime })$$

(24)

where n is the number of layers of the spanning tree; $(L,R)$ corresponds to the structure of the optimal spanning tree $T_{\mathrm{optimal}}^n$; $(L^{\prime },R^{\prime })$ corresponds to the structure of other spanning trees $T_{\mathrm{other}}^n$; and $f^n(·)$ denotes the number of turns of a single spanning tree with n rows.

The nested property of the above tree structures can be expressed as: for any $2\le n$ and $1\le k\le n$,

$$T^k\subseteq T^n$$

(25)

and $T^k$ is a tree (maintaining connectivity and acyclicity).

Base case ($n=2$): When the number of rows is 2, the first layer $T_{\mathrm{optimal}}^1$ of $T_{\mathrm{optimal}}^2$ is a tree (guaranteed by connectivity), and the first layer $T_{\mathrm{other}}^1$ of $T_{\mathrm{other}}^2$ is a tree (guaranteed by connectivity). The number of turns of the tree generated according to the optimal rule is less than or equal to that of the tree generated by any other method. Therefore, we have

$$f_{\mathrm{optimal}}^2(L,R)\le f_{\mathrm{other}}^2(L^{\prime },R^{\prime })$$

(26)

when $n=2$ The proposition holds.

Inductive hypothesis: Assume the proposition holds when $n=k$, i.e., for the tree structure with k rows,

$$f_{\mathrm{optimal}}^k(L,R)\le f_{\mathrm{other}}^k(L^{\prime },R^{\prime })$$

(27)

Inductive step ($n=k+1$):

According to the conclusion, the first k layers of $T_{\mathrm{optimal}}^{k+1}$ are $T_{\mathrm{optimal}}^k$ (which is a subtree of $T_{\mathrm{optimal}}^{k+1}$), and the first k layers of $T_{\mathrm{other}}^{k+1}$ are $T_{\mathrm{other}}^k$ (which is a subtree of $T_{\mathrm{other}}^{k+1}$ ). After expansion, the following holds true:

$$T^{k+1}=T^k\cup S^{k+1}$$

(28)

where $S^{k+1}$ is the structure of the $(k+1)$-th layer, and $T^{k+1}$ is a tree (maintaining connectivity and acyclicity).

Decomposition of the $(k+1)$-th layer structure: The structure $S^{k+1}$ of the $(k+1)$-th layer consists of two parts: n discrete points (non-adjacent isolated points or adjacent points without a line segment to connect) and m line segments (composed of adjacent points, where the i-th segment contains $N_{m_i}$ points, denoted as $C_{m_i}$).

Number of turns for discrete points: Discrete points need to be connected to the underlying original tree structure to maintain the connectivity of the tree (horizontal connections can ensure connectivity but cross other UAVs’ regions, which is inconsistent with the premise of the conclusion). Thus, the additional number of turns generated is fixed as N, i.e.,

$$N_{\mathrm{optimal}}=N_{\mathrm{other}}=N$$

(29)

Number of turns for the line segments: For m line segments, let the total additional number of turns under the optimal connection method be M, and that under other methods be $M_{\mathrm{other}}$. Then,

$$M=\sum _{i=1}^m{f}_{\mathrm{optimal}}\left(C_{m_i}\right)$$

(30)

$$M_{\mathrm{other}}=\sum _{i=1}^m{f}_{\mathrm{other}}\left(C_{m_i}\right)$$

(31)

From Equations (22) and (23), we have

$$f_{\mathrm{optimal}}\left(C_{m_i}\right)\in \{4,6\}$$

(32)

$$f_{\mathrm{other}}\left(C_{m_i}\right)\in \{4,6,8\}$$

(33)

Verification of inequality: It is necessary to prove $M\le M_{\mathrm{other}}$, which is discussed for the two cases below.

No additional segment division: For a single segment $C_{m_i}$, we directly have

$$f_{\mathrm{optimal}}\left(C_{m_i}\right)\le f_{\mathrm{other}}\left(C_{m_i}\right)$$

(34)

Summing these yields $M\le M_{\mathrm{other}}$.

Division into x additional segments: If $C_{m_i}$ is divided into $x\ge 2$ segments $C_{m_i-1},\dots ,C_{m_i-x}$, then

$$f_{\mathrm{optimal}}\left(C_{m_i}\right)\le 6$$

(35)

$$\sum _{t=1}^x{f}_{\mathrm{other}}\left(C_{m_i-t}\right)\ge 4x$$

(36)

When $x\ge 2$, $4x\ge 8>6$, so

$$f_{\mathrm{optimal}}\left(C_{m_i}\right)\le \sum _{t=1}^x{f}_{\mathrm{other}}\left(C_{m_i-t}\right)$$

(37)

Summing these still gives $M\le M_{\mathrm{other}}$.

Combining the contributions of discrete points and line segments, we have

$$f_{\mathrm{optimal}}^{k+1}(L,R)=f_{\mathrm{optimal}}^k(L,R)+N+M$$

(38)

$$f_{\mathrm{other}}^{k+1}(L^{\prime },R^{\prime })=f_{\mathrm{other}}^k(L^{\prime },R^{\prime })+N+M_{\mathrm{other}}$$

(39)

From the inductive hypothesis $f_{\mathrm{optimal}}^k\le f_{\mathrm{other}}^k$ and the above-proven $M\le M_{\mathrm{other}}$, we obtain

$$f_{\mathrm{optimal}}^{k+1}(L,R)\le f_{\mathrm{other}}^{k+1}(L^{\prime },R^{\prime })$$

(40)

Conclusion: By mathematical induction, the proposition holds for all $n\ge 2$, i.e.,

$$f_{\mathrm{optimal}}^n(L,R)\le f_{\mathrm{other}}^n(L^{\prime },R^{\prime })$$

(41)

5.4. Construction Method of Spanning Tree Based on Rule

In order to reduce the number of turns and the length of the actual flight path, this paper designs a spanning tree construction strategy, which is implemented as follows: After determining the geometric boundary of the target detection point set, calculating its horizontal and vertical spans (defined as width and length, respectively), and selecting the direction corresponding to the larger span as the main connection direction along which parallel lines are drawn to connect adjacent points, the parallel lines along the main connection direction are subsequently traversed in a sequential row-wise manner. Next, the alignment status of two adjacent rows is evaluated: if both ends of the two rows are aligned, either the leftmost or rightmost edge is randomly selected for inter-row connection; if only one end of the two rows is aligned, vertical connection between the two rows is performed along the aligned end; if neither end is aligned, the inter-row connection is still implemented at the leftmost or rightmost position. Following the completion of the above inter-row connection operations, connectivity verification is conducted on the constructed graph: if the graph is disconnected, the process returns to the row-wise traversal step of the main direction and repeats the aforementioned connection operations until the graph meets the connectivity requirement; once a connected graph is formed, the spanning tree construction is completed, and the final spanning tree structure is outputted; the flowchart is shown in Figure 9.

5.5. Generation of Planning Path

After using the rule-based Minimum Spanning Tree (STC) algorithm to generate the topological structure of the target reconnaissance area, each grid point can be divided into four equally large sub-regions, and flight coverage trajectory planning can be performed based on the topological structure of the spanning tree [33]. This algorithm can generate reconnaissance trajectories without path redundancy and full coverage, ensuring that all reconnaissance points within each sub-region are fully covered while avoiding path intersections and duplicate detections between different sub-regions.

6. Simulation Result and Discussion

A simulation experiment was designed and conducted to verify the effectiveness of the improved algorithm. In this simulation experimental scenario, the spatial scale of the region to be reconnoitered was set to 20 dm × 20 dm. A grid-based modeling method was adopted: the region was divided into several 1 dm × 1 dm grids (cells), and each grid unit (cell) was further subdivided into four subgrid units (subcells) of 0.25 dm × 0.25 dm to meet the requirements of the reconnaissance range of the UAV. Meanwhile, to simulate the practical application scenario of multi-UAV collaborative reconnaissance, five UAV sorties were initialized, and the flight speed of each UAV sortie was set to 1 dm/s. The algorithm was validated using the aforementioned simulation scheme.

To validate the effectiveness of the proposed algorithm, three different algorithms were employed for comparison, as detailed in the

Table 1. Algorithm combination table.

Algorithm Serial Number	Partition Algorithm	Spanning Tree Generation Algorithm
Algorithm 1	Objective Planning	Based on Rule
Algorithm 2	Objective Planning	Based on DFS
Algorithm 3	DARP (Voronoi Diagram)	Based on DFS

.

As shown in

Table 2. Objective planning parameters table.

Parameters	Values
$P_1$	10
$P_2$	100
Solver	CBC
Presolve	True
Cuts	True
$P_i(i=1,2,\dots ,5)$	1
D	1805
${\mathrm{Initial-Position}}_{drone1}$	(0.25, 0.25)
${\mathrm{Initial-Position}}_{drone2}$	(8.25, 13.25)
${\mathrm{Initial-Position}}_{drone3}$	(10.25, 18.25)
${\mathrm{Initial-Position}}_{drone4}$	(17.25, 5.25)
${\mathrm{Initial-Position}}_{drone5}$	(13.25, 13.25)

, Solver uses the CBC solver; Cuts represents the generation of cutting planes; Presolve represents whether to use pre-solving; D represents the total distance; $P_i$ is 1 for all drones to represent consistent performance indicators; and P1 and P2 represent the weights of the objective function. ${\mathrm{Initial-Position}}_{\mathrm{drone}i}$ represents the initial position of drone i, which corresponds to $G_i$ in Equation (18).

6.1. Simulation Result for Algorithm 1

Algorithm 1 employs an objective-based partitioning approach and a rule-based spanning tree construction method. Based on the above algorithm, the trajectory route of the UAVs and results in the simulation environment can be obtained, as shown in

Table 3. Regional partitions results under objective planning.

Indicator Results	Values
objective function	119.99
$d_1^{+}$/$d_1^{-}$	12/0
$d_{2-1}^{+}$/$d_{2-1}^{-}$	0/0
$d_{2-2}^{+}$/$d_{2-2}^{-}$	0/0
$d_{2-3}^{+}$/$d_{2-3}^{-}$	0/0
$d_{2-4}^{+}$/$d_{2-4}^{-}$	0/0
$d_{2-5}^{+}$/$d_{2-5}^{-}$	0/0

and

Table 4. Region coverage indicators of UAVs based on algorithm 1—spanning tree 1.

Drone Sorties	Path Length (dm)	Number of Turns	Time (s)
drone 1	150	40	150
drone 2	150	48	150
drone 3	150	26	150
drone 4	150	40	150
drone 5	150	52	150

and Figure 10.

Since there are $2^x$ possible spanning trees after the completion of region partitions in algorithm 1, spanning tree 1 is a manually fixed one that complies with the rules of algorithm 1. The flight path of the UAV depends on this spanning tree, while the number of turns and path length do not rely on it. The remaining spanning trees that comply with the rules will also be provided in the Appendix A.

The different colored blocks in Figure 10 represent the drone reconnaissance zone partitions, and the central gray area denotes the no-fly zone. The red lines represent the spanning tree generated by the algorithm. The green route shows the flight path of the drone circling the spanning tree.

To verify the optimality of the minimum spanning tree and the presence of random contradictions in the generation rules, different spanning tree paths are obtained based on the same region partitioning, as shown in Figure 11.

As demonstrated earlier, the path coverage metrics obtained from this spanning tree is consistent with Table 4, as shown in

Table 5. Region coverage indicators of UAVs based on algorithm 1—spanning tree 2.

Drone Sorties	Path Length (dm)	Number of Turns	Time (s)
drone 1	150	40	150
drone 2	150	48	150
drone 3	150	26	150
drone 4	150	40	150
drone 5	150	52	150

:

Figure 10 and Figure 11 illustrate the continuous, non-overlapping UAV sub-regions and regular paths that avoid no-fly zones resulting from the combination of objective planning and rule-based spanning trees in algorithm 1. As demonstrated in Table 4, all the UAVs exhibited equivalent path lengths (150 dm) and task times (150 s), thereby substantiating the load balance. The range of turns varied from 26 to 52. Table 3 presents an objective function value of 119.99, a small total distance deviation ($d_1^{+}$ = 12), and zero load-related deviations ($d_{2-k}^{+/-}$). These data prove that the model effectively balances the distance minimization and load equilibrium.

6.2. Simulation Result for Algorithm 2

Algorithm 2 uses the objective planning algorithm for region partitions and the spanning tree generation algorithm in the DFS mode. Since the region partitions results are the same as Table 3, they will not be repeated here. The indicators and trajectory of the algorithm are shown in

Table 6. Region coverage indicators of UAVs based on algorithm 2.

Drone Sorties	Path Length (dm)	Number of Turns	Time (s)
drone 1	150	50	150
drone 2	150	54	150
drone 3	150	42	150
drone 4	150	46	150
drone 5	150	72	150

and Figure 12.

6.3. Simulation Result for Algorithm 3

Algorithm 3 uses the DARP (Voronoi diagram) algorithm for region partitions and the spanning tree generation algorithm in DFS mode. The indicators and trajectory diagram of the algorithm are presented in

Table 7. Region coverage indicators of algorithm 3.

Drone Sorties	Path Length (dm)	Number of Turns	Time (s)
drone 1	138	42	138
drone 2	206	78	206
drone 3	78	22	78
drone 4	198	52	198
drone 5	130	50	130

and Figure 13.

6.4. Discussion

The analysis of the above indicators Table 4, Table 6 and Table 7 show that in the simulation environment, because the drone flies at a fixed constant speed of 1 dm/s, the path length is independent of the generation method of the spanning tree, provided that the region partitioning method is consistent. The number of turns was significantly correlated with the method of generating the spanning tree. The number of turns in the generated tree under the designed optimal rules is generally less than that of the tree generated by the dfs algorithm.

The above indicators in Table 7 indicate that there are differences between the regions divided by objective planning and those divided by the Voronoi diagram. The traditional Voronoi diagram only divides regions based on distance, resulting in uneven task allocation among UAVs. The proposed target planning considers balance, which can balance the task burden of each drone flight, thereby enabling the efficient completion of the regional coverage tasks of the drone cluster.

Algorithm Complexity Analysis

The ILP-based partitioning model proposed in this study exhibits an approximate quadratic growth ($\mathcal{O}\left(N^2\right)$) in the number of variables and constraints with the number of effective grids (cells). As demonstrated in the 20 × 20 grid experiment, the model can be solved to a 4% optimality gap within 0.45 wallclock seconds with 4 threads (Gomory and Knapsack cuts enabled), achieving efficient performance. Nevertheless, it must be emphasized that integer programming is an NP-hard problem. When grid resolution significantly increases (i.e., the number of cells surges), the solution time will grow nonlinearly, potentially failing to meet real-time requirements. Thus, the proposed method is currently more suitable for scenarios with controllable cell scales and strict requirements on partitioning quality (e.g., refined regional inspection). Future work will explore more efficient approximation algorithms or distributed solution strategies to expand its application scale.

7. Real Flight Experiment

Experimental  Preparation

To further validate the effectiveness of the proposed algorithm, real flight experiments were conducted on the basis of simulation experiments. The actual flight experiment used a small unmanned aerial vehicle, Crazyfile2.1, as shown in Figure 14.

Owing to the small detection range of the motion capture equipment used, the detection accuracy of the motion capture equipment decreases when it exceeds the detection range. Therefore, the experiment was limited to a range of 2 m × 2 m to compare various indicators under three different algorithms used during the actual flight.

Through the implementation of actual flight experiments, performance metrics were obtained for the three algorithms, as demonstrated in

Table 8. Evaluation indicators of regional coverage based on the algorithm of objective planning and rule-based spanning tree generation (algorithm 1).

Drone Sorties	Length (m)	$\cos \mathit{\theta }$ < 0	$\mathbf{cos}\mathit{\theta }\mathsf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\sqrt{\mathbf{3}}\mathbf{/}\mathbf{2}\mathbf{)}$	Coverage Area (${\mathbf{m}}^2$)	Coverage Rate	Time (s)
Drone 1	15.45	61	121	0.6944	18.52%	151.00
Drone 2	15.60	44	87	0.7278	19.41%	155.30
Drone 3	15.38	44	113	0.7129	19.01%	157.40
Drone 4	15.38	50	131	0.7125	19.00%	155.50
Drone 5	15.56	64	137	0.7388	19.70%	157.80
All drones	77.38	263	589	3.5266 ($S_{\mathrm{effective}}$) 3.5864 ($S_{\mathrm{total}}$)	94.04% ($R_{\mathrm{effective}}$) 95.64% ($R_{\mathrm{total}}$)	157.80

,

Table 9. Evaluation indicators of regional coverage based on the algorithm of objective planning and dfs-based spanning tree generation (algorithm 2).

Drone Sorties	Length (m)	cos$\mathit{\theta }$ < 0	$\mathbf{cos}\mathit{\theta }\mathsf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\sqrt{\mathbf{3}}\mathbf{/}\mathbf{2}\mathbf{)}$	Coverage Area (${\mathbf{m}}^2$)	Coverage Rate	Time (s)
Drone 1	16.07	43	167	0.6954	18.54%	164.50
Drone 2	16.09	52	138	0.6868	18.32%	161.90
Drone 3	15.62	76	214	0.7219	19.25%	177.70
Drone 4	15.52	49	135	0.7113	18.97%	159.80
Drone 5	15.67	28	150	0.7085	18.89%	156.60
All drones	78.97	248	804	3.4708 ($S_{\mathrm{effective}}$) 3.5239 ($S_{\mathrm{total}}$)	92.55% ($R_{\mathrm{effective}}$) 93.97% ($R_{\mathrm{total}}$)	177.70

and

Table 10. Evaluation indicators of regional coverage based on the algorithm of the Voronoi diagram and dfs-based spanning tree generation (algorithm 3).

Drone Sorties	Length (m)	cos$\mathit{\theta }$ < 0	$\mathbf{cos}\mathit{\theta }\mathsf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\sqrt{\mathbf{3}}\mathbf{/}\mathbf{2}\mathbf{)}$	Coverage Area (${\mathbf{m}}^2$)	Coverage Rate	Time (s)
Drone 1	14.75	29	137	0.6305	16.81%	151.30
Drone 2	21.81	57	206	0.9625	25.67%	219.80
Drone 3	8.55	77	157	0.3669	9.78%	99.30
Drone 4	21.19	49	186	0.9181	24.48%	209.00
Drone 5	14.25	46	164	0.6090	16.24%	147.20
All drones	80.56	258	850	3.4334 ($S_{\mathrm{effective}}$) 3.4870 ($S_{\mathrm{total}}$)	91.56% ($R_{\mathrm{effective}}$) 92.99% ($R_{\mathrm{total}}$)	219.80

.

These tables present the regional coverage metrics of the UAVs under the three distinct algorithms, facilitating a comparative analysis of their performance in achieving effective area coverage. The “Drone Sorties” column distinguishes between individual UAVs (drone 1 to drone 5) and the system-level aggregate (“All drones”), serving as the basis for categorizing single-UAV and system-wide metrics. The “Length (m)” column reports path lengths: for individual UAVs, these correspond to $L_1$–$L_n$; for the system, the value under “All drones” represents the total path length $L_{\mathrm{total}}=\sum L_i$, quantifying the overall path control overhead. The “$\cos \theta <0$” column records the count of instances where the velocity direction cosine is negative, which indicates the severe turn of the UAV, reflecting the flight stability level of the algorithm. The “$cos\theta \in (0,\sqrt{3}/2)$” column documents the number of cases with small velocity direction cosine values, indicating a turn in the UAV, thus further evaluating the path control performance. The “Coverage Area (${\mathrm{m}}^2$)” column presents coverage areas: for individual UAVs, these are the effective coverage areas $S_1$–$S_n$; for the system, “All drones” lists both the total area with overlaps $S_{\mathrm{total}}=\sum S_i$ and the effective area excluding overlaps $R_{\mathrm{effective}}$, reflecting coverage efficiency at both levels. The “Coverage Rate” column provides coverage ratios: for individual UAVs, these are $R_1\%$–$R_n\%$; for the system, “All drones” includes the ratio corresponding to the total overlapping area $R_{\mathrm{total}}$ and the effective coverage rate $R_{\mathrm{effective}}$. The “Time (s)” column reports task durations: for individual UAVs, these are $T_1$–$T_n$; for the system, the value under “All drones” is the total task duration $T_{\mathrm{total}}=max\left(T_i\right)$, which measures overall synergy.

To demonstrate the collaborative efficiency of drone swarms in completing tasks, we utilized motion capture equipment to collect position and velocity data, and the sample time stamp was 0.1 s. This process generated a two-dimensional flight path map for the UAVs, as shown in Figure 15 and Figure 16.

These figures illustrate the swarm flight trajectories of multiple drones in cooperative regional detection. As indicated by the legend, the colored lines represent the trajectories of drone 1 to drone 5, the dashed line denotes the task boundary $[-1,1]\times [-1,1]$, the gray region is the no-fly zone area, the scattered points are the center of the cell, and the light blue region represents the effective covered area (union), collectively demonstrating the path planning and coverage performance of the drone swarm.

To analyze the turn severity of UAVs, motion capture data were employed to calculate the velocity direction cosine value between consecutive timestamps, and based on this, a time-varying plot of the UAV’s velocity direction cosine value [35] was generated.

It can be clearly observed from Table 8, Table 9 and Table 10 and Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 that the number of cosine values less than 0 under the three algorithms is roughly equivalent, while the number of cosine values between 0 and $\sqrt{3}/2$ shows an increasing trend, with algorithm 1 having the smallest number and algorithm 3 the largest. This result indicates that in practical applications, algorithm 1 exhibits superior capability in controlling unmanned aerial vehicles, which can reduce the turn severity of the UAV.

To further analyze how the task completion rate of UAVs varies over time under different algorithms, the following graph was plotted, as shown in Figure 23.

The effective coverage rate (task completion rate) $R_{\mathrm{effective}}$ defined in this study specifically refers to the ratio of the union of all drone reconnaissance coverage areas to the total area of the mission zone. As shown in Figure 23, the task completion rates of the three algorithms remained broadly comparable. This phenomenon is primarily attributed to the spanning tree flight path mechanism, which achieves 100% theoretical coverage efficiency in the simulation environment. Consequently, in practical flight scenarios, the task completion rate remained consistently above 90%.

It is crucial to emphasize that the core advantage of the proposed algorithms lies not merely in enhancing task completion rates but in significantly optimizing the balance of task distribution among UAVs while effectively reducing the number of turns executed. Typically, as the number of turns decreases and the task distribution becomes more balanced, UAVs can complete missions in a shorter time and have a higher coverage rate.

Further analysis of the quantitative results revealed that the Objective-Rule algorithm combination achieved the shortest task completion time and the highest task completion rate of 94.04%. The Objective-dfs combination yields the second shortest completion time, although its task completion rate drops to 92.55%. Conversely, the traditional Voronoi–dfs combination exhibited the longest completion time and the lowest task completion rate of only 91.56%.

8. Conclusions and Future Work

8.1. Key Conclusions and Algorithm Superiority

This study focuses on efficient multi-UAV regional coverage path planning (CPP) for small-to-medium-scale structured environments (e.g., regular grids with limited obstacles), and proposes an innovative two-stage optimization framework integrating sub-region partitioning and spanning tree path generation. Aiming to address the inefficiencies of traditional CPP methods (e.g., unbalanced task allocation, excessive flight turns, and simulation-reality performance discrepancies), the proposed OPP-RSTC algorithm optimizes the dual-stage process of sub-region partitioning and spanning tree construction, and its theoretical models are fully validated via both simulation experiments and real-flight tests, serving as a reliable and efficient coverage solution for small-to-medium-scale structured CPP scenarios. The three core conclusions and contributions of this study are summarized as follows:

First, we propose a rule-based spanning tree construction method, and theoretically prove its optimality in minimizing the number of flight turns for a single UAV, which fundamentally reduces redundant turns and improves flight stability during coverage missions.

Second, we develop a 0–1 integer programming sub-region partitioning method based on objective programming, which effectively balances the task load among multiple UAVs and the distance from each UAV to its assigned coverage area; additionally, a lightweight heuristic constraint is designed as a necessary but not sufficient condition for sub-region connectivity to ensure computational feasibility with minimal overhead.

Third, under simulation conditions with 1 × 1 grid cells, we derive a definitive conclusion that the coverage path length of a sub-region divided into x cells is invariably 2x regardless of spanning tree construction strategies, as this fixed path length is an inherent requirement for UAVs to achieve full coverage of all subcells within the assigned region.

8.2. Limitations and Future Research Directions

Despite the validated effectiveness of the improved STC algorithm (algorithm 1) in multi-UAV coverage path planning, it has limitations that guide future research, as follows:

1.

As 0–1 integer programming is an NP-hard problem, ILP-based partitioning models face significant computational challenges with the sharp increase in the number of cells, making it difficult to directly scale to ultra-large-scale scenarios. Additionally, current experiments are limited to regular grids and simple no-fly zones, failing to meet the requirements of complex practical applications; future validation should be conducted in more complex unstructured environments.

2.

This study adopts a fixed 5-UAV configuration. Subsequent research will test adaptability to varying UAV quantities and design real-time reallocation strategies for UAV failures or emergency tasks.

3.

Future work will explore turn-reduction strategies via non-adjacent inter-layer connections, particularly when the spanning tree fails to ensure tree-structured sublayers.

4.

The current heuristic constraints serve as a necessary but not sufficient condition for global connectivity, leaving a theoretical possibility of forming multiple connected components. Future work will investigate how to achieve global connectivity constraints with low computational complexity.