A Skeleton-Line-Based Spiral Coverage Path Planning Method for UAV Inspection of Three-Dimensional Structures

Abstract

UAV-based visual inspection has become an effective approach for acquiring surface information from three-dimensional building structures. However, existing coverage path planning methods usually treat viewpoint planning and path sequencing as two separate stages, which may introduce redundant viewpoints, long connection paths, and high computational cost. To address this problem, this paper proposes a skeleton-guided spiral coverage path planning method for UAV inspection of 3D structures. The target building model is first converted into a watertight triangular mesh, from which a one-dimensional skeleton line is extracted to guide both viewpoint generation and path construction. Surface sampling points are generated using rotating radial rays along the skeleton line, and UAV viewpoints are obtained by offsetting these points according to a predefined viewing distance. The ordered viewpoints are then connected to construct spiral coverage paths, while visibility checking, safety-distance constraints, and collision detection are incorporated to ensure path feasibility. Parameter sensitivity analysis shows that the sampling interval has a dominant influence on coverage performance and path cost, while the angular increment mainly affects path compactness and construction efficiency. Comparative experiments on the Christ, Wind Turbine, and Big Ben models demonstrate that the proposed method achieves high coverage rates of 96.62%, 97.72%, and 99.67%, respectively, while generating shorter paths and requiring substantially less computation time than ACO−OPD and Zhao’s method. These simulation results indicate that the proposed method can generate compact coverage paths with substantially lower computation time for UAV coverage inspection of 3D structures.

1. Introduction

As urbanization continues to accelerate, the density of urban buildings, particularly high-rise structures, has increased substantially. Over time, building facades are prone to aging, cracking, tile spalling, and structural corrosion, all of which may pose potential safety risks [1]. Therefore, regular inspection and maintenance have become essential, and building safety has emerged as a key component of urban infrastructure management [2]. Compared with conventional manual inspection methods, unmanned aerial vehicles (UAVs) possess the advantages of high flexibility, high operational efficiency, and low cost, making them increasingly promising tools for inspection tasks in various scenarios, such as buildings, wind turbines, and power transmission lines [3]. UAVs can also be equipped with different sensors according to specific inspection requirements, including visible-light cameras [4], LiDAR [5], infrared thermal imagers, and hyperspectral sensors [6]. These sensing capabilities enable rapid data acquisition and high-resolution visualization, thereby improving the coverage and accuracy of defect detection. In addition, UAV-assisted inspection reduces safety risks for workers, helps eliminate blind spots in manual inspection, and overcomes many limitations of conventional inspection techniques.

When UAVs equipped with sensors such as visible-light cameras are used for building inspection, a critical challenge is to plan autonomous inspection routes in a rational, efficient, and safe manner. This task is commonly referred to as the UAV coverage path planning (CPP) problem [7]. Early UAV CPP studies were mainly formulated within two-dimensional spaces or simplified environments [8]. However, with the increasing demand for coverage tasks in various industrial scenarios, CPP has gradually extended from 2D planar environments to more complex three-dimensional (3D) scenarios [9]. Existing 3D CPP tasks can generally be divided into two categories: offline and online methods [10]. Offline methods rely on prior knowledge of the 3D structure to generate high-coverage paths, whereas online methods are commonly used in unknown environments, such as 3D reconstruction tasks. Since offline methods generally provide better path performance when a reliable structural model is available, this study adopts an offline framework for 3D coverage path planning.

Within the offline framework, a typical 3D coverage path planning scenario involves using a UAV to achieve near-exhaustive coverage of a three-dimensional structure represented by a mesh model. The primary objective is to generate a path that satisfies the coverage requirements of the 3D structure while minimizing the total path length, thereby reducing UAV energy consumption. In addition, the planned path must ensure the safety of both the UAV and the target structure by avoiding potential collisions.

Current research on UAV coverage path planning can generally be divided into two aspects [11,12]: viewpoint planning (VPP) and path planning, which is often formulated as a traveling salesman problem (TSP) [13,14,15]. The primary objective of VPP is to determine suitable observation positions and orientations for the UAV, so as to achieve near-exhaustive coverage of the target area while reducing the number of viewpoints and improving viewpoint quality. In contrast, path planning focuses on generating an efficient flight path based on a given set of viewpoints by determining the visiting sequence that minimizes the overall inspection cost and improves task efficiency.

Bircher et al. [16] employed a triangular mesh representation and a two-stage optimization framework for structural inspection path planning. Their method can select viewpoints that ensure high-coverage paths while reducing the connection cost between viewpoints. However, because the number of viewpoints is not explicitly optimized, redundant viewpoints may be introduced, thereby increasing the overall inspection path length. Choi et al. [17] used point clouds to represent terrain models and generated viewpoints by offsetting the surface normal at the center of each grid cell based on a planar uniform sampling strategy. The final coverage path was then derived from these candidate viewpoints using a shortest-distance objective function. However, this method is mainly applicable to CPP problems involving terrain models with approximately 2.5D planar characteristics. When the number of candidate viewpoints increases, the computational efficiency of the greedy shortest-path calculation inevitably decreases. Jing et al. [18] proposed a model-based view planning method that generates candidate viewpoints through iterative random sampling and a probabilistic potential-field strategy. Combinatorial optimization was then used to select an optimal subset of viewpoints and generate the final inspection path. Nevertheless, as the number of candidate viewpoints increases, the greedy search procedure used in this method may become the computational bottleneck of the overall framework.

In recent years, many studies have attempted to solve the TSP-like path planning problem in UAV CPP using various AI-based algorithms, including swarm intelligence optimization, genetic algorithms (GAs), and ant colony optimization (ACO). Bui et al. [13] introduced a swarm intelligence-based cooperative path planning approach using 3D models and ACO. Zhao et al. [19] proposed a two-stage optimization framework for computationally efficient structural inspection path planning with satisfactory coverage performance. Their method first generates uniformly sampled viewpoints from a voxel model and constructs an initial path based on adjacency edges between neighboring viewpoints. The path is then iteratively optimized using a GA combined with a shortest-path objective function. Li et al. [14] proposed an improved ACO algorithm with variable pheromones (ACO−VP), together with a greedy assignment strategy, to achieve efficient UAV path planning in dynamic environments. Du et al. [10] presented an efficient UAV path planning method for three-dimensional structural coverage based on an improved ACO algorithm, where a cost function derived from Minimum Snap trajectory planning was used instead of simple waypoint-connection distance to more accurately characterize the actual flight effort. Although these AI-based path planning methods offer considerable flexibility and adaptability, they generally rely on iterative search and multi-objective optimization. As the number of candidate viewpoints increases, their computational efficiency tends to decrease significantly.

The aforementioned methods for UAV-based 3D structural coverage generally treat viewpoint planning and path planning as two separate stages. Due to the lack of an effective coupling mechanism between these two stages, the overall efficiency of the planning pipeline is often limited. Moreover, the compactness of the coverage path, the achieved coverage performance, the total path length, and the computational efficiency still require further improvement. Therefore, new research strategies and technical routes are needed to address these issues. This paper proposes a novel skeleton-guided spiral coverage path planning method that integrates viewpoint generation and path construction within a unified framework. Specifically, spiral-guided viewpoints are first generated based on the extracted skeleton lines, and the final spiral coverage path is then constructed by connecting these viewpoints in their natural sampling order. By coupling viewpoint planning with path generation, the proposed method reduces the complexity of both viewpoint selection and path sequencing. In addition, the direct mesh model-based spiral path generation strategy avoids time-consuming iterative optimization and enables efficient generation of short coverage paths. The main contributions of this paper are summarized as follows:

• A skeleton-guided coverage path planning framework is proposed for UAV inspection of three-dimensional structures. Unlike conventional methods that first generate an unordered set of candidate viewpoints and then solve a separate path sequencing problem, the proposed method uses the one-dimensional skeleton line as a geometric and topological guide to generate viewpoints and coverage paths, which to some extent reduces the dependence on global combinatorial optimization.
• A spiral-guided viewpoint generation strategy based on rotating radial sampling is developed. Sampling points are first distributed along the extracted skeleton curve. At each sampling point, rotating radial rays are constructed on the plane perpendicular to the local skeleton direction and intersected with the mesh surface to obtain surface sampling points. UAV viewpoints are then generated by offsetting these surface points according to the predefined viewing distance. Since the generated viewpoints naturally follow a spiral order, the coverage path can be constructed directly according to the sampling sequence.

The rest of this paper is organized as follows: Section 2 presents the problem statement, including the UAV inspection model, coverage requirements, safety constraints, and path length objective. Section 3 details the proposed skeleton line-based spiral coverage path planning method in detail, including model preprocessing, skeleton line extraction, spiral-guided viewpoint generation, and coverage path construction. Section 4 presents the simulation experiments and discusses the parameter sensitivity analysis, feasibility verification, and comparative results. Finally, Section 5 concludes this paper and outlines future work.

2. Problem Statement

In this study, a UAV equipped with a gimbal-mounted camera is assumed to perform coverage inspection of the surface of a 3D structure. To enable quantitative analysis of surface coverage, the 3D surface is represented as a high-resolution triangular mesh model. The mesh model consists of $N$ triangular facets, where each facet represents a potential coverage target in the set $S=\{s_1,s_2,\dots {,s}_N\}$. The CPP task is to select a sufficient set of viewpoints $V\subseteq \left\{p_1,p_2,\dots {,p}_k\right\}$ to generate paths achieving a high level of coverage over the triangles in S. Each viewpoint is characterized by $\{vp,\overrightarrow{vn}\}$, where $vp=(p_x,p_y,p_z)$ denotes the viewpoint position, and $\overrightarrow{vn}={(vn}_x,{vn}_y,{vn}_z)$ represents the viewing direction. The viewing direction is aligned with the line of sight of the gimbal-mounted camera, and the distance between the viewpoint position and the observed target point $o$ is defined as the predefined view distance $d_{view}$. The mounted camera has horizontal and vertical field-of-view (FOV) angles, denoted as ${\theta }_h$ and ${\theta }_v$, respectively. Given the view distance, the actual width and height of the camera inspection region can be acquired. The schematic diagram of the visible space covered by the viewpoint in the UAV inspection system is shown in Figure 1, and the relevant notations and parameters are listed in

Table 1. Definitions of notations and parameters used throughout this work.

Symbol	Description
S	the set of all triangular facets in the mesh model
$V$	the set of viewpoints
$p$	a 3D point in space with three coordinate components $p=(p_x,p_y,p_z)$
$vp$	the position of a viewpoint
$\overrightarrow{vn}$	the viewing direction of a viewpoint
$d_{view}$	the view distance between a camera position and a target point
${\theta }_h$ and ${\theta }_v$	the horizontal and vertical field-of-view (FOV) angles of a camera
$c$	the vertex set of a skeleton curve
$d_{curve}$	the sampling interval along the skeleton curve
$d_{angle}$	the angular sampling interval
${\theta }_{start}$	the initial sampled angular position
$N$	the number of sampling points along the skeleton curve
${sq}_k$	the sampling positions along the skeleton curve
$\overrightarrow{{sn}_k}$	the unit direction vector at sampling point with index k
$\overrightarrow{{rn}_k}$	the radial vector at the sampling point with index k
${\theta }_k$	the orientation angle at the sampling point with index k
$(\overrightarrow{u},\overrightarrow{v})$	a pair of orthogonal basis vectors at the sampling point with index k
${vp}_k$	the viewpoint position corresponding to the sampling point with index k
$\overrightarrow{{vn}_k}$	the viewpoint direction vector corresponding to the sampling point with index k

.

Since the imaging sensor of a camera is typically rectangular, the geometric model of the camera’s visible space at a viewpoint is actually a quadrilateral pyramid, as shown in Figure 1a. In the algorithmic implementation, to facilitate the visibility calculation of triangular facets in the coverage path planning procedure, the geometric structure of the camera’s visible space is further simplified as a spherical cone, as shown in Figure 1b. The horizontal and vertical FOV angles are simplified into an equivalent field-of-view angle, which is used as the aperture angle of the spherical cone. The viewing direction is defined as the direction of its central axis, and the viewing distance is defined as its radius. The region covered by this spherical cone represents the visible space corresponding to the viewpoint. Triangular facets located within this viewing spherical cone and not occluded by other parts of the surface can be considered visible from the current viewpoint. In this study, the simplified spherical-cone structure is used as the geometric representation of the visible space.

For mature commercial UAV systems, inspection missions are commonly specified by a sequence of waypoints with prescribed positions and viewing orientations. The underlying flight-control system is usually responsible for stabilization, motion execution, and low-level trajectory tracking, while limited low-level control interfaces are exposed to the mission-planning layer. Therefore, in this study, the UAV is assumed to hover or stabilize at each selected viewpoint and capture images with the prescribed camera orientation. Based on this assumption, the main scope of the paper is geometry-oriented viewpoint generation and waypoint connection for coverage inspection. Although stable hovering of a UAV at spatial positions relies on low-level control strategies [20], trajectory-tracking techniques, and UAV speed, practical UAV route-planning missions generally place greater emphasis on the geometric waypoint planning. Based on this assumption, low-level control strategies and trajectory-tracking techniques are not explicitly considered in this study.

In addition, to simplify the UAV energy consumption model, the total coverage path length is adopted as the primary indicator of energy expenditure [9]. Accordingly, in this study, the overall path length is quantified as the cumulative Euclidean distance between sequentially ordered viewpoints, excluding the takeoff, landing, and return-to-home segments. Thus, both the viewpoint generation strategy and the connectivity among viewpoints play critical roles in reducing the total coverage path length. In this study, the UAV coverage path is defined as the waypoint generation problem, which focuses on the geometry-oriented viewpoint generation and connections among waypoints. The safety of the planned UAV route is ensured by two main requirements. First, the line segments connecting consecutive viewpoints must not intersect with the mesh model, thereby avoiding collisions or interference between the UAV path and the structure. Second, the UAV route must maintain a prescribed safe Cartesian distance from the model throughout the flight mission.

3. Skeleton-Guided Spiral Coverage Path Planning Method

The overall framework of the proposed method is shown in Figure 2. Given a 3D structural model, the proposed method first performs model preprocessing to obtain a watertight triangular mesh with a uniform facet distribution, which provides a reliable geometric basis for subsequent visibility analysis and collision checking. Then, a skeleton line is extracted from the volumetric representation of the mesh model. Since the skeleton line compactly describes the main geometric shape and topological structure of the target object, it is used as the central guide for both viewpoint generation and coverage path construction.

Based on the extracted skeleton line, the proposed method generates spiral-guided sampling rays around the model. Sampling points are first distributed along the skeleton curve, and rotating radial directions are constructed on the plane perpendicular to the local skeleton direction. The intersections between these radial rays and the mesh surface are regarded as surface sampling points. UAV viewpoints are then obtained by offsetting the surface sampling points outward according to the predefined viewing distance, and the camera viewing directions are set toward the corresponding surface points. In this way, the generated viewpoints are arranged in a natural spiral order around the structure.

After viewpoint generation, the candidate viewpoints are refined through visibility evaluation, safe-distance checking, and collision detection. Invalid viewpoints that cannot observe any triangular facets or violate safety constraints are removed. For structures with branch skeletons, the primary skeleton curve and branch skeleton curves are separately converted into spiral path segments, and these segments are connected through their endpoints to form a continuous coverage path. Therefore, the proposed method transforms the conventional separate viewpoint planning and path sequencing problem into a unified skeleton-guided spiral path generation process, which reduces computational complexity while maintaining a high level of coverage.

3.1. Model Preprocessing

To facilitate visibility calculation for triangular facets on the surface of a 3D structural model, the target digital model is first converted into a watertight triangular mesh [21]. For mesh models containing excessively slender facets or highly non-uniform facet sizes, remeshing is required to obtain a more regular triangular mesh with improved facet uniformity. The classical Bubble Mesh method [22] can be adopted for this purpose, as it generates well-shaped triangular facets while preserving the original geometry of the model. This preprocessing step provides the geometric basis for the proposed algorithm and improves the accuracy and stability of subsequent visibility calculations. In practice, such preprocessing can also be performed using advanced mesh-processing software, such as Magics (21.0), Meshmixer (2.9.4), and MeshLab (2025.07). In this work, the target 3D structural model is processed using Magics, and the remeshed result is shown in Figure 3.

3.2. Skeleton Line Extraction

Skeleton lines provide a compact representation of 3D structures and effectively characterize both the geometric shape and topological connectivity of models. Therefore, they are used in this study to guide viewpoint generation and coverage path construction, serving as the geometric basis of the proposed algorithm. Skeleton line extraction is a well-established problem in computational geometry, and various algorithms have been developed for this purpose [23,24]. In this work, a volumetric model-based skeleton line extraction method [24] is adopted due to its simplicity and robustness. The proposed framework assumes that the input model is a singly connected structure without internal cavities, with a relatively continuous surface and no overly complex local concave regions. These assumptions help reduce the sensitivity of the skeleton extraction algorithm to surface noise and local geometric irregularities, while enabling the generation of a relatively smooth one-dimensional (1D) skeleton line that remains reasonably close to the medial axis of the model.

The mesh model used in this study is first voxelized into a volumetric representation. According to the 6-neighborhood definition of voxels, the generated voxels are classified into two categories: boundary voxels and interior voxels. Interior voxels are located inside the model and have six neighboring voxels, whereas boundary voxels are distributed on the model surface. Initially, the edge lengths between neighboring voxels are identical. To contract the original volumetric model, the edges connecting neighboring voxels are iteratively shortened, while the boundary voxels are constrained to remain close to their original positions. This contraction process is formulated as an optimization problem, in which the voxel positions are continuously updated during iteration. Subsequently, a thinning operation is performed to remove boundary voxels and obtain the final 1D skeleton. The skeleton lines generated by this extraction procedure are shown in Figure 2, where the mesh models are rendered semi-transparently for visualization, and the brown curves represent the extracted 1D skeleton lines.

3.3. Spiral Coverage Path Planning Based on Skeleton Line

The spiral path is generated based on the extracted skeleton lines. In this process, viewpoint generation and coverage path construction are tightly coupled, allowing the final spiral coverage path to be directly constructed from the mesh model. A skeleton line generally consists of one or more curves, and the convergence point of multiple curves is defined as a branching node. Such branching nodes divide the skeleton line into several simple, loop-free skeleton curves, thereby simplifying the coverage path planning process. Among these skeleton curves, the curve originating from the bottom of the structure is defined as the primary curve, while the remaining curves are classified as branch curves.

3.3.1. Spiral-Guided Viewpoint Generation

Given a skeleton curve $c$, represented as an ordered sequence of vertices $c=\{p_1,p_2,\dots ,p_n\}$, a sampling interval along the skeleton curve $d_{curve}$, an angular sampling interval $d_{angle}$, and an initial sampled angular position ${\theta }_{start}$ (defaulting to 0), the sampling-based viewpoint generation method is described as follows:

(1)

Sampling point generation along a skeleton curve.

The curve is first treated as a polyline; the total length is computed using Equations (1) and (2), where $n$ is the number of curve points. Specifically, the Euclidean distances between consecutive vertices are calculated and accumulated sequentially. Once the total length is obtained, the number $m$ of sampling points is determined according to the sampling interval, resulting in an approximately uniform distribution of sampling points along the curve, as indicated by Equation (3). The sampling positions ${sq}_k$ are then sequentially determined along the curve with a fixed step length $d_{curve}$. When a sampling point falls within the $i$-th line segment, linear interpolation between the two endpoints of this segment is used to compute its spatial coordinates, as indicated by Equation (4). Instead of directly using the local direction vectors of the original skeleton curve, the unit direction vectors $\overrightarrow{{sn}_k}$ at sampling point k is computed using the central difference method. For boundary points, a forward or backward difference is adopted, as indicated by Equation (5). To avoid abrupt directional variations caused by local excessive curvature or irregularities in the original skeleton curve, a sliding-average filter is then applied within a window of radius $w=1$, the smoothed direction vector $\overrightarrow{{sn}_k}$ at the sampling point k can be calculated using Equation (6), and normalization is performed to obtain the unit direction vector.

$$l_i=\left|p_{i+1}-p_i\right|,i=1,2,\dots ,n-1$$

(1)

$$L={\sum }_1^{n-1}{l}_i$$

(2)

$$m=⌊{\displaystyle \frac{L}{d_{curve}}}⌋+1$$

(3)

$${sq}_k=p_k+\left(p_{k+1}-p_k\right)\times {\displaystyle \frac{k\times d_{curve}-{\sum }_1^{k-1}{l}_j}{l_k}},k=1,2,\dots ,m$$

(4)

$${∆D}_k={sq}_{k+1}-{sq}_{k-1}$$

(5)

$$\overrightarrow{{sn}_k}={\displaystyle \frac{1}{2w+1}}{\sum }_{i=k-w}^{k+w}{∆D}_i$$

(6)

(2)

Sampling point generation on the mesh surface.

Based on the ordered sampling points generated along the skeleton curve in the previous step, each sampling point is processed according to the predefined initial rotation angle ${\theta }_{start}$ and increment angle $d_{angle}$. For the current sampling point with index $k$, the corresponding radial vector $\overrightarrow{{rn}_k}$ lies in the plane perpendicular to the unit direction vector $\overrightarrow{{sn}_k}$. Its orientation angle ${\theta }_k$ is determined according to Equation (7). A pair of orthogonal basis vectors $(\overrightarrow{u},\overrightarrow{v})$ is then constructed. The normalized vector of $\overrightarrow{{sn}_k}$ is defined as $\overrightarrow{u}$, and a direction vector perpendicular to $\overrightarrow{{sn}_{\mathit{k}}}$ is chosen for the initial rotation angle ${\theta }_{start}=0$ along the rotation direction, and its normalized form serves as the base vector $\overrightarrow{\mathit{v}}$. Subsequently, the radial vector $\overrightarrow{{rn}_k}$ at the sampling point can be determined from the specified angle ${\theta }_k$, as defined in Equation (8).

$${\theta }_k={\theta }_{start}+k\times d_{angle},k=1,2,\dots ,m$$

(7)

$$\overrightarrow{{rn}_k}=cos{\theta }_k\times \overrightarrow{u}+\sin {\theta }_k\times \overrightarrow{v},k=1,2,\dots ,m$$

(8)

For the primary skeleton curve, a spatial ray is constructed from the current sampling point along the corresponding radial vector. The intersection between this ray and the mesh model is efficiently calculated using the double-voxel-based interference search tool (DVIST) [25], and the resulting intersection point is taken as the corresponding sampling point on the mesh surface. For branch skeleton curves, a spatial line is defined using the position of each sampling point and its associated radial direction vector. The intersections between the constructed line and the mesh surface are then calculated and regarded as the corresponding surface sampling points. The generated mesh surface sampling points are shown in Figure 4, where the red spheres represent the sampling points distributed on the mesh surface.

(3)

Viewpoint generation based on mesh surface sampling points.

Based on the mesh surface sampling points generated in the previous step, the viewpoint generation procedure is described as follows. First, each sampling point ${sq}_k$ on the mesh surface is taken as the starting position. Then, a displacement is applied along the corresponding unit radial direction according to the predefined viewing distance $d_{view}$ of the gimbal-mounted camera. The resulting position is regarded as the spatial coordinate of the corresponding UAV viewpoint ${vp}_k$, as formulated in Equation (9). The generated viewpoints are illustrated in Figure 5, where the green spheres denote the viewpoints and the light green line segments indicate the offset distance along the radial directions, corresponding to the viewing distance of the gimbal-mounted camera. Accordingly, the viewing direction $\overrightarrow{{vn}_k}$ of the gimbal-mounted camera is opposite to the corresponding radial vector $\overrightarrow{{rn}_k}$, as defined in Equation (10). This viewing direction configuration keeps the camera nearly perpendicular to the local surface, thereby reducing image distortion.

$${vp}_k={sq}_k+\overrightarrow{{rn}_k}\times d_{view},k=1,2,\dots ,m$$

(9)

$$\overrightarrow{{vn}_k}=-\overrightarrow{{rn}_k},k=1,2,\dots ,m$$

(10)

3.3.2. Spiral Coverage Path Construction

The preceding step generates not only UAV viewpoints but also their corresponding spiral-guided order, which facilitates efficient construction of the spiral coverage path. However, these generated viewpoints are still regarded as candidate viewpoints. To determine the final viewpoint set and the corresponding spiral coverage path, these candidates must be further refined through UAV path safety verification, triangular facet coverage assessment, and effective connection of multiple spiral curves.

For the candidate viewpoints, the path safety check first removes viewpoints that lie inside the model or violate the constraint of prescribed safe distance from the model surface. It also ensures that the connection segments between adjacent spiral curves do not intersect the interior of the model. For triangular facet coverage assessment, a visibility matrix is constructed, where each element represents the visibility relationship between a viewpoint and a triangular facet [26]. The rows and columns of this matrix correspond to the candidate viewpoints and the triangular facets of the mesh model, respectively, and all elements are initialized as invisible. By traversing the triangular facets visible from each viewpoint, the corresponding matrix entries are updated as visible. The visible region of each viewpoint is computed based on the simplified spherical cone space introduced in Section 2. If a triangular facet lies entirely within the viewing cone space of a viewpoint, and the rays from the viewpoint to the three vertices of the triangular facet are not occluded by other mesh elements, the triangular facet is considered visible from the current viewpoint. Otherwise, it is considered invisible. If no triangular facet is visible from a given viewpoint, this viewpoint is removed from the candidate viewpoint set. By adjusting the sampling interval $d_{curve}$ along the skeleton curves and the angular increment $d_{angle}$ of the rotating sampling rays, the visible triangular facet sets associated with the sampling points can be dynamically regulated. By selecting appropriate sampling interval and angular increment, this process enables the coverage ratio of triangular facets represented in the visibility matrix to reach a prescribed threshold.

Since the sampling points generated along the skeleton curves naturally preserve their inherent order, the viewpoints corresponding to each skeleton curve can be directly connected to construct individual spiral coverage path segments. Therefore, the remaining UAV coverage path planning problem is transformed into an endpoint connection problem, in which multiple spiral path segments are linked into a continuous collision-free path while satisfying UAV safety constraints.

The primary skeleton curve generates a single spiral path segment, whereas each branch skeleton curve generates two double-spiral paths. The two endpoints of the double-spiral path near the terminal point of each branch skeleton curve are disconnected. Therefore, a connection point is introduced by extending a distance $d_{view}$ along the direction vector at the terminal point of the skeleton curve. This point connects the two endpoints, thereby constructing the two double-spiral paths into a bidirectional spiral path with forward and return motions.

Subsequently, the endpoints of the constructed bidirectional spiral curves near the skeleton branching node are further connected to form a continuous path. Since each branch skeleton curve produces two open endpoints, the spiral endpoint of the primary skeleton curve is taken as the starting point. The branch skeleton curves around the branching node are then searched in counterclockwise order, and their corresponding spiral curves are connected sequentially. During this process, the endpoint of the current spiral curve is connected to the closer of the two endpoints of the next bidirectional spiral curve, measured by Euclidean distance. A path safety constraint check is then performed. If the connection satisfies the safety constraints, the terminal point of the current spiral curve is connected to the matched endpoint, and the other endpoint of the bidirectional spiral curve is assigned as the new terminal point of the current path. Otherwise, if the connection violates the safety constraints, the initial rotation angle used to generate the spiral path of the current branch is incrementally adjusted with a fixed step size, which is set to 10° default. The spiral path is then regenerated, and the terminal connection is checked again. This process is repeated until the spiral path corresponding to the current branch skeleton curve is successfully connected.

By applying the same procedure to the spiral curves of all remaining branch paths, a near-exhaustive spiral coverage path is finally obtained, as shown in Figure 6.

A concise computational complexity assessment of the main algorithmic stages is provided as follows. The proposed method is mainly governed by skeleton line extraction, spiral-guided viewpoint generation, visibility evaluation, safety checking, and path construction. Skeleton line extraction primarily depends on voxel resolution and model complexity. Viewpoint generation scales with the number of skeleton sampling points, whereas visibility evaluation and safety checking are mainly affected by the number of generated viewpoints and triangular facets. Since the viewpoints are generated in an ordered spiral sequence, the final path can be constructed directly without an additional global TSP-like sequencing step, thereby reducing the overall computational burden.

4. Experiment and Discussion

4.1. Simulation Setup, Parameter Sensitivity Analysis, and Feasibility Verification

The proposed skeleton line-based spiral coverage path planning method is implemented in Python (3.13.0). The simulation experiments are conducted on a Windows 11 workstation with 16 GB RAM, and the current implementation does not employ multi-core acceleration. Three representative three-dimensional structural models, namely Wind Turbine, Christ, and Big Ben, were selected to evaluate the feasibility and generality of the proposed method. These benchmark models were also used for simulation validation in the ACO−OPD method [10], and the model information is listed in

Table 2. The model information statistics in simulation experiments.

Models	$\mathbf{Model}\mathbf{Size}(\mathbf{x}\times$$\mathbf{y}\times$ z mm3)	Triangle Number
Christ	$10\times$$36\times$ 38	19,383
Wind Turbine	$75\times$$10\times$ 120	8564
Big Ben	$20\times$$20\times$ 110	8668

, and the rendered models are shown in the first column of Figure 8.

These models exhibit different geometric characteristics, including slender tower-like structures, compact free-form surfaces, and structures with multiple local details, thereby effectively testing whether the proposed spiral path generation strategy is applicable to various 3D inspection scenarios. To reduce computational cost, the geometric dimensions of the 3D structural models were scaled down in the simulation experiments. The scale ratio was set such that 1 mm in the simulation model corresponds to 1 m in the actual structure, that is, a scale factor of 1:1000. Therefore, the model sizes and path lengths reported in millimeters can be interpreted as meter-level values in real-world inspection scenarios. The key parameters used in the simulation were set as follows: The basic voxel size and safe distance were set to 1 mm and 2 mm, respectively. The sampling interval along the skeleton curve was set to $d_{curve}$= 2.0 mm, the angular increment of the rotating sampling rays was set to $d_{angle}=40$°. In this study, the camera FOV angles were set to ${\theta }_h={\theta }_v=120$° and the viewing distance was set to 20.0 mm, consistent with the parameter settings reported for the ACO−OPD method in the literature [10]. Since the coverage threshold for mesh triangular facets was not specified in Ref. [10], it was set to 95%, following the coverage criterion used in the comparative study [19].

First of all, a parameter sensitivity analysis was conducted on the Christ model to investigate the effects of the sampling interval and angular increment on the simulation results. The analysis focused on the mesh triangular-facet coverage ratio, as well as the number of viewpoints, path length, and construction time. The Christ model was used as the test case, and orthogonal experiments were conducted by varying the sampling interval and angular increment. The statistical results are summarized in

Table 3. Statistical results of the sensitivity analysis for orthogonal experiments on the sampling interval and angular increment.

Index Label	${\mathit{d}}_{\mathit{c}\mathit{u}\mathit{r}\mathit{v}\mathit{e}}$ (mm)	${\mathit{d}}_{\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}}$ (°)	Covered Facets	Uncovered Facets	Coverage Rate	Viewpoints	Path Length (mm)	Time (s)
a	1	30	18,865	518	97.33%	130	535.53	13.52
b	1	40	18,890	493	97.46%	123	629.45	12.43
c	1	50	18,883	500	97.42%	125	764.47	13.66
d	2	30	18,796	587	96.97%	83	346.42	10.19
e	2	40	18,728	655	96.62%	78	334.32	8.73
f	2	50	18,700	683	96.48%	75	432.39	11.37
g	3	30	17,406	1977	89.80%	48	282.01	8.45
h	3	40	17,193	2190	88.70%	45	294.48	7.23
i	3	50	17,552	1831	90.55%	48	354.96	7.91

, with the labels in Figure 7 corresponding one-to-one to the test groups listed in the table. In the visualized coverage results, mesh triangular facets identified as visible are rendered in green, whereas uncovered facets are rendered in red.

The statistical results of the parameter sensitivity analysis show that the sampling interval $d_{curve}$ has a dominant influence on the experimental results. As $d_{curve}$ increases from 1 mm to 3 mm, the coverage rate decreases markedly, from approximately 97.33%~97.46% to 88.70%~90.55%. This trend is also reflected in Figure 7, where the upper-row results exhibit better coverage than the lower-row results. Since $d_{curve}$ increases from top to bottom, a larger sampling interval reduces the sampling density along the skeleton curve, resulting in fewer generated viewpoints and more uncovered triangular facets. In folded surface regions of the model, a reduced distribution of viewpoints often causes the triangular facets in these areas to become invisible, as shown in the last row of Figure 7. Specifically, when $d_{curve}$ = 1 mm, the number of viewpoints ranges from 123 to 130, whereas it decreases to 45~48 when $d_{curve}$ = 3 mm. Accordingly, smaller sampling intervals improve mesh triangular-facet coverage, but at the cost of more viewpoints, longer path lengths, and higher construction time.

Compared with $d_{curve}$, the angular increment $d_{angle}$ has a relatively limited effect on the coverage rate. For a fixed sampling interval, changing $d_{angle}$ only causes small fluctuations in coverage, suggesting that the coverage performance is mainly determined by the density of sampling points along the skeleton curve. For each row in Figure 7, the left-column results generally show slightly better coverage than the right-column results. This is because a larger angular increment produces a more dispersed angular distribution of viewpoints around the skeleton curve, which may weaken local coverage continuity. For example, when $d_{curve}$ = 2 mm, increasing $d_{angle}$ from 30° to 50° changes the coverage rate only slightly, from 96.97% to 96.48%, whereas changing $d_{curve}$ from 1 mm to 3 mm leads to a much larger decrease in coverage. The number of viewpoints is also only slightly affected by $d_{angle}$.

The path distribution further confirms these trends. The paths in the upper row of Figure 7 are denser than those in the lower row, leading to longer path lengths and higher construction costs. When $d_{curve}$ = 1 mm, the path length ranges from 535.53 mm to 764.47 mm, whereas it decreases to 282.01~354.96 mm when $d_{curve}$ = 3 mm. In contrast, the path length generally increases as $d_{angle}$ increases from 30° to 50°, indicating that a larger angular increment may lead to less compact spiral connections. For construction time, $d_{angle}$ = 40° yields the shortest time in each $d_{curve}$ group, suggesting that this setting provides a relatively efficient balance between angular sampling and path construction complexity. Overall, $d_{curve}$ mainly controls the trade-off between coverage quality and computational/path cost, while $d_{angle}$ mainly affects the compactness and construction efficiency of the generated spiral path. The middle-row results provide a more balanced performance: the path distribution is relatively compact, the coverage rate remains higher than the prescribed threshold of 95%, and the number of viewpoints, path length, and construction time are all reduced compared with the upper-row results. Therefore, the parameter setting corresponding to label (e), namely $d_{curve}$ = 2 mm and $d_{angle}$ = 40°, provides a reasonable trade-off between coverage performance and path construction efficiency, achieving a coverage rate of 96.62% with 69 viewpoints, a path length of 397.65 mm, and a construction time of 9.84 s

In addition, the proposed method was also applied to the Wind Turbine and Big Ben models, and the generated coverage paths are shown in the second column of Figure 8. The generated spiral paths are distributed around the surfaces of the target structures and follow the overall geometry of the extracted skeleton lines. For the Wind Turbine and Christ models, the extracted skeletons consist of a primary skeleton curve and several branch skeleton curves. Accordingly, the generated spiral coverage paths effectively wrap around the model surfaces. In the regions corresponding to the branch skeleton curves, connection points are inserted to link the double-spiral curves, thereby forming bidirectional spiral trajectories with forward and return motions. For the Big Ben model, only one primary skeleton curve is extracted; therefore, the final coverage path consists of a single spiral path.

The feasibility simulation results indicate that the proposed algorithm can effectively generate compact coverage paths for 3D building structure models with primary or primary-branch skeleton topologies, relatively continuous surfaces, and moderately complex local concave regions. Moreover, the algorithm can generate different spiral path patterns according to the skeleton-line morphology of the target structure. Instead of first generating a large unordered set of candidate viewpoints and then solving a complex TSP-like sequencing problem, the proposed method directly obtains an ordered coverage path through the skeleton-guided spiral sampling process. Therefore, the simulation results verify the feasibility of the proposed method for generating continuous and collision-free coverage paths for 3D structures.

4.2. Comparison with Existing Methods

To further evaluate the performance of the proposed method, two representative baseline methods, the ACO−OPD method [10] and Zhao’s method [19], were selected for comparison. To ensure a fair comparison, Zhao’s method and the proposed method were tested on the same hardware platform using the same model data, viewing distance configurations. The remaining method-specific parameters of Zhao’s method were also set according to the optimized values reported in the corresponding paper (Grid interval = 1 m, IER = 0.75, GER = 0.10, Tournament size = 25, Rule-based initialization proportion = 0.5, Population size = 250, Generations = 500, Coverage requirement = 95%, Visible inclination angle = 45°).

The results of the ACO−OPD method were taken directly from its original publication, in which the specific hardware configuration used for the simulation experiments was not provided. In addition, the exact coverage rate of ACO−OPD was not explicitly reported. Based on the statement in the original publication, its coverage rate was listed as 100% in this comparison. It should also be noted that the test models used in the ACO−OPD study were processed by mesh smoothing, which significantly reduces model details and may facilitate higher triangular-facet coverage. The experimental results of the three methods are summarized in Figure 8. The quantitative comparative experiments mainly focus on three indicators: the coverage rate, the number of viewpoints, path length, and computation time. The comparison results are summarized in

Table 4. Model information and evaluation statistics of different coverage planning methods for 3D structures.

Models	Methods	Coverage Rate	Number of Viewpoints	Path Length (mm)	Time (s)
Christ	Ours	96.62%	78	334.32	8.73
	ACO−OPD a	100.00%	80	337.63	322.36
	Zhao’s algorithm b	95.20%	54	536.72	250.43
Wind Turbine	Ours	97.72%	102	560.81	13.64
	ACO−OPD a	100.00%	45	735.65	2570.97
	Zhao’s algorithm b	95.18%	155	1607.84	490.47
Big Ben	Ours	99.67%	41	524.04	5.12
	ACO−OPD a	100.00%	70	630.84	311.75
	Zhao’s algorithm b	95.22%	198	2414.70	688.30

Note: ^a the coverage rate of the ACO−OPD method is not explicitly reported in the literature and is deduced based on the statement in the original publication, other values are reproduced from Ref. [10]. ^b Zhao’s results are obtained by running the corresponding implementation.

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The rendered models used for simulation experiments on the proposed method and Zhao’s method are illustrated in the first column in Figure 8. The second and third columns present the coverage paths generated by the proposed method and Zhao’s method, respectively. The last column shows the results of the ACO−OPD method, reproduced from its original publication. A comparison between the first and last columns clearly indicates that the models used in the ACO−OPD experiments were processed by mesh smoothing. As observed from Figure 8, compared with the ACO−OPD method and Zhao’s method, the spiral coverage paths generated by the proposed method conform better to the geometric shapes of the target models while maintaining good path continuity and compactness. Moreover, the proposed method avoids frequent long-distance transitions between discrete viewpoints, because the viewpoints are naturally arranged along skeleton-guided spiral trajectories. For the Big Ben model, the coverage path generated by the proposed approach consists of a continuous single-spiral path, resulting in a more compact and smoother path around the model surface.

As shown in Table 4, all three methods achieve coverage rates higher than the prescribed threshold for the tested models. The coverage rates of the proposed method are 96.62%, 97.72%, and 99.67% for the Christ, Wind Turbine, and Big Ben models, respectively. The ACO−OPD method is listed with a coverage rate of 100.00% according to the complete coverage statement in its original publication, although the exact triangular-facet coverage rate was not explicitly reported. It should also be noted that the test models used in the ACO−OPD study were processed by mesh smoothing, which reduces geometric details and may facilitate higher triangular-facet coverage. Therefore, the coverage comparison should be interpreted together with these differences in model preprocessing.

In terms of path length, the proposed method achieves the shortest paths for all three models. Specifically, the path lengths generated by the proposed method are 334.32 mm, 560.81 mm, and 524.04 mm for the Christ, Wind Turbine, and Big Ben models, respectively. In comparison, ACO−OPD produces path lengths of 337.63 mm, 735.65 mm, and 630.84 mm, while Zhao’s method produces longer paths of 536.72 mm, 1607.84 mm, and 2414.70 mm for the same three models. The proposed method also shows a clear advantage in computation time, requiring only 8.73 s, 13.64 s, and 5.12 s, respectively. These values are substantially lower than those of ACO−OPD (322.36 s, 2570.97 s, and 311.75 s) and Zhao’s method (250.43 s, 490.47 s, and 688.30 s).

In terms of the number of viewpoints, the proposed method does not always generate the fewest viewpoints. For example, Zhao’s method uses fewer viewpoints for the Christ model, and ACO−OPD uses fewer viewpoints for the Wind Turbine model. However, both methods produce longer paths and require substantially more computation time in these cases. This indicates that reducing the number of viewpoints alone does not necessarily lead to a shorter or more efficient inspection path. By arranging viewpoints according to the skeleton-guided spiral order, the proposed method can maintain compact path connectivity while avoiding expensive global route optimization.

Overall, the comparison results demonstrate that the proposed skeleton line-based spiral coverage path planning method can generate compact coverage paths with high coverage rates and much lower computational cost. This advantage mainly comes from the skeleton-guided spiral sampling strategy, which naturally arranges adjacent viewpoints along continuous spiral trajectories and avoids repeated global search or iterative route optimization. These results confirm the suitability of the proposed method for efficient UAV coverage inspection of 3D structures.

5. Conclusions

This paper proposes a skeleton-guided spiral coverage path planning method for UAV inspection of three-dimensional building structures. In the proposed method, the skeleton line of the target model is used as a geometric guide to integrate viewpoint generation and path construction, thereby avoiding the conventional process of generating unordered candidate viewpoints and solving a separate path sequencing problem. After the target building model is converted into a watertight triangular mesh, a one-dimensional skeleton line is extracted from its volumetric representation. Surface sampling points are then generated using rotating radial rays along the skeleton line, and UAV viewpoints are obtained by offsetting these points according to the predefined viewing distance. The ordered viewpoints are further connected to construct spiral coverage paths, while visibility checking, safety-distance constraints, and collision detection are incorporated to improve path feasibility. Simulation experiments on the Christ, Wind Turbine, and Big Ben models demonstrate that the proposed method can generate continuous and compact coverage paths for the tested 3D structures. Compared with ACO−OPD and Zhao’s method, the proposed method achieves high coverage rates while producing shorter paths and requiring substantially less computation time. These results indicate that the skeleton-guided spiral sampling strategy can effectively reduce the path sequencing complexity in simulation-based 3D structural inspection scenarios.

However, several limitations remain. The proposed method has not yet been validated through real UAV flight experiments, and simplified camera and energy models are adopted. In addition, UAV dynamics are not explicitly considered, including acceleration and deceleration profiles, flight-direction changes, turning angles, vertical maneuvers, flight speed, and hovering time. The method also depends on model preprocessing and skeleton quality, while the radial-sampling strategy relies on geometric assumptions that may become restrictive for complex concave structures. Future work will focus on real-world inspection validation, trajectory smoothing under UAV dynamic constraints, uncertainty-aware safety margins, and further tests on complex concave structures.