Li et al. proposed an R5DOS model based on the region connection calculus (RCC) theory in 2020, demonstrating the model's completeness and mutual exclusivity, the model can represent 11,038 possible topological directional relationships among three simple regions in three-dimensional space. Subsequently, they improved and applied it in UAV swarm algorithms and UAV path planning algorithms, providing detailed insights into the improvements made to the R5DOS model (Li et al., 2020, 2022, 2023). This study adopts the improved R5DOS model to define UAVs and obstacles, dividing UAVs into the body region and detection region. The UAV's body region represents the area that ensures absolute safety during flight, while the detection region is responsible for detecting obstacles and target points. As indicated by reference (Li et al., 2022), there are five types of topological relations: Discrete (DR), Partial Overlap (PO), Proper Part (PP), Equal (EQ), and Proper Part Inverses (PPI). The corresponding topological situations are illustrated in Figure 1 . Among them, RCC-8 is a boundary-sensitive topological relation model (Jonsson et al., 2021).

According to the R5DOS model, this study provides the following definitions. For the detection region B, it must contain the UAV body A, satisfying the PP topological relation, denoted as PP(B, A). As for the relationship between obstacles, the UAV, and the detection region, three possible scenarios are most likely to occur, as illustrated in Figure 2 .

In the figure, A represents the UAV body region, B₁ represents the detection region, and C₁ represents the obstacle. Figure 2A illustrates the scenario where the obstacle does not intersect with the detection region or the UAV, indicating that the UAV has not detected any obstacles and is in a relatively safe state. The corresponding topological relation matrix is denoted as R 5 a = ( 0 1 0 0 0 1 1 1 ) . Figure 2B depicts the scenario where the obstacle is within the detection region but has not yet intersected with the UAV. In this case, the obstacle is inside the detection region but has not collided with the UAV, placing the UAV in a relatively dangerous situation. The corresponding topological relation matrix is denoted as R 5 b = ( 0 1 0 0 1 1 1 1 ) . Figure 2C illustrates the scenario where the obstacle is within the detection region and intersects with the UAV. In this case, the UAV collides with the obstacle, posing a dangerous situation. The corresponding topological relation matrix is denoted as R 5 c = ( 1 1 0 0 1 1 1 1 ) .

To better represent the spatial relationships between UAVs and obstacles, an improvement to the R5DOS model is essential. The induction matrix of the R5DOS model is divided into the R5 layer and the DOS layer, which are used to express topological and spatial relationships, respectively. For detected obstacles, the UAV needs to store their information to determine the location of the obstacles and their corresponding topological spatial relationships. However, the R5DOS model can only express the topological relationships of three simple objects, which is evidently impractical for high-density obstacle maps. To better express and store the topological spatial relationships between multiple obstacles and UAVs, we have modified the DOS layer of the R5DOS model to be a 4×4 matrix, defined as follows.

The spatial relationships corresponding to each matrix element are as follows.

Wherein, 1-8 represent the eight hexagram limits in three-dimensional space, and “c” represents an obstacle. θ a c represents the dihedral angle between the UAV (a) and the obstacle (c). Therefore, regardless of the number of obstacles, as long as an obstacle appears in any octant, it can be recorded in the matrix. Thus, irrespective of how many obstacles are present in a particular octant, we can provide the following definitions.

Through the improvements made to the R5DOS model, we gain the capability to articulate the spatial relationships between UAV and any number of obstacles. Consequently, the refined R5DOS model allows for a comprehensive representation of the spatial topology including UAVs, obstacles, and target points. Building upon the improvements introduced in the D5DOS model, we can further refine the node selection process of the A-Star algorithm.

The space complexity of the traditional A-star exhibits exponential growth, showing a noticeable increase in computational requirements as the map size expands (Li et al., 2022). This issue primarily manifests during the execution of the A-star algorithm, where a substantial number of nodes are visited without considering whether these nodes are relevant to the final path. Consequently, this further increases the algorithm’s runtime. To address this challenge, this study introduces a custom feature attention mechanism to enhance the A-star algorithm. In the A-star algorithm, its cost function, typically denoted as ‘ F c o t ( n ) ’, is commonly expressed through the expression ‘ F c o t ( n ) = H ( n ) + G ( n ) ’. In this study, we employed Euclidean distance to calculate the movement cost, which is particularly suitable for the representation of three-dimensional space using the grid-based approach. Here, ‘ H ( n ) ’ represents the cost function for estimating the path from the nth feature point to the target, while ‘ G ( n ) ’ denotes the movement cost function for the shortest path from the starting point to the nth feature point. For the grid-based representation of three-dimensional space, the formula for calculating the movement cost ‘ f ( n ) ’ is given by Equation 4.

Here, ‘ ( x n , y n , z n ) ’ and ‘ ( x n − 1 , y n − 1 , z n − 1 ) ’ represent the coordinates of nodes ‘n’ and ‘n-1’, respectively. This study redefines the nodes for UAV path search. In three-dimensional space, the number of nodes that a UAV can choose for its next move is significantly greater than the cardinal directions available in two-dimensional space. To better apply the A-star algorithm, this study redivides the space into 16 regions based on the R5DOS model. However, this partitioning method is not very friendly to the node evaluation of the A-star algorithm. To improve this, this study is based on eight octants, assuming the orange point represents the UAV’s position. In this case, all vertices of the neighboring octants can be considered as neighboring nodes for the current node, resulting in a total of 26 potential search nodes. Having too many nodes can impact the efficiency of the algorithm. Therefore, based on the UAV’s next move direction, the study filters out the closest four nodes (yellow points). After the first round of searching does not yield the optimal node, it then searches for the neighbors of the yellow nodes (blue points), as illustrated in Figure 3 .

Simultaneously, based on the definition of the improved DOS layer, we can further enhance the search mechanism. By retrieving the spatial topology matrix of the current UAV, the R5 layer is divided into two scenarios.

In case R 5 = ( 0 1 0 0 0 1 1 1 ) is satisfied, employ the search method illustrated in Figure 3 to filter nodes.

In case R 5 = ( 0 1 0 0 1 1 1 1 ) is satisfied, indicating the detection of an obstacle, under the search mechanism depicted in Figure 3 , proceed to eliminate nodes with non-zero DOS. By redefining the method for searching nodes, it is possible to significantly reduce unnecessary node searches, thereby enhancing the search efficiency of the improved algorithm in three-dimensional environments. This approach facilitates the more effective identification of the optimal path.

Although the R5DOS model-based search mechanism defined in Section 2.2 can eliminate most unnecessary nodes, there are still a significant number of invalid nodes in three-dimensional space that cannot be filtered out using this method. Therefore, this study incorporates a feature attention mechanism module into the A-Star algorithm, which is activated when the UAV detects obstacles. After integrating this module, the UAV focuses more on the local features of obstacles during path planning. The dual-node filtering algorithm, combining the R5DOS model with the feature attention mechanism, effectively removes a large number of unnecessary nodes, as shown in Figure 4 .

In the illustration, green nodes represent the UAV ( x U A V , y U A V , z U A V ) , orange nodes represent the target point ( x T , y T , z T ) and black blocks represent obstacles. When a UAV needs to avoid obstacles in a three-dimensional space, it must focus on the primary characteristics of the obstacles, particularly the vertices and special points along the edges. By concentrating on these features and ignoring other insignificant nodes, the UAV can significantly reduce its computational load, thus alleviating much of the processing burden. For obstacles in three-dimensional space, their features include vertices and edges. Therefore, based on these features, the following definitions are made. The obstacle grid representation is denoted as Gobs(n) ( x G , y G , z G ) , with a value of 1 for obstacle grids, and the non-obstacle grid representation is denoted as GN-obs(n), with a value of 0. This study defines a feature attention estimation function, A(n), which represents the number of adjacent obstacles to the nth node. Typically, a node has 8 adjacent grids, and A(n) is expressed as formula 5.

For feature points, the following definitions can be obtained:

‘ f G − U ( n ) ’ represents the path cost function between obstacles and the UAV, while ‘ f T − G ( n ) ’ denotes the path cost function between the target point and obstacles. According to the above definitions, the feature points that can be filtered by this attention mechanism include the vertices of obstacles and the points on the edges of obstacles that are closest to the UAV or the target point. The characteristic points that meet the conditions are stored in List A l ( n ) . By employing this filtering method, the UAV can focus on the critical features of the obstacles that need to be avoided, thereby reducing the interference of irrelevant nodes in the algorithm. During path planning, the characteristic points in List A l ( n ) are prioritized for visitation. First, the R5DOS model filters out most of the nodes, followed by the feature attention mechanism, which further filters the nodes around the obstacles. This approach allows the UAV to focus on the key features of obstacles that need to be avoided, thereby reducing the interference of irrelevant nodes in the algorithm.

Finally, we conduct a two-sample t-test on the algorithm results to analyze the simulated experimental outcomes, considering whether the differences between the proposed algorithm and other algorithms are significant. The formula for the t-test statistic for independent samples is shown below:

In which, S 1 2 and S 2 2 represent the sample variances of two sets of data, while n 1 and n 2 denote the sample sizes.

Building on the preceding definitions, this paper provides the definition of the RFA-Star algorithm. Detection is performed within the exploration area, and the environmental and obstacle detection results are stored in the feature matrix of R5DOS. If the UAV detects an obstacle, a custom feature attention mechanism is introduced to search for the feature points of the obstacle. This scenario is defined as condition A.

‘ ϵ ( D O S ) ’ represents the DOS layer of the R5DOS model (Li et al., 2023). When the UAV encounters conditions corresponding to Condition A, the local map of the detection area is gridded. An improved A-star algorithm and a feature attention module are then utilized to filter feature points. Subsequently, guiding the UAV through obstacle avoidance involves a process outlined by the RFA-Star algorithm, as illustrated in the following steps.

Based on the above process, the corresponding pseudocode of the algorithm is provided in Table 1 .

Figure 5 illustrates the flowchart of the algorithm.

To assess the efficacy of the RFA-Star algorithm in path planning within high-density obstacle environments and to compare its performance with other algorithms, this study designed two experiments:

This study undertook a comparative analysis of the RFA-Star algorithm against other state-of-the-art spatial reasoning-based path planning methods, namely RJA-Star (Li et al., 2023) Improved A-Star (Li et al., 2022)., and the A-Star. The RJA-Star integrates an enhanced jump point search algorithm into the A-Star, effectively diminishing the number of nodes, computation time, and computational complexity. In contrast, the Improved A-Star algorithm enhances the A-Star by incorporating the R5DOS model to reduce search nodes. Furthermore, given the similar path-searching approaches employed by the RFA-Star and RJA-Star algorithms, this study extends the comparison to include A-Star, RJA-Star, and RFA-Star across five distinct map sizes. Detailed map information is available in Table 2 .

For better comparison, this study set the obstacle density of the map to 0.8 obs./m², with the starting point designated as (0, 0, 0). MATLAB was employed in this research to randomly generate obstacle maps, and all simulations were conducted on a 13th Gen Intel(R) Core(TM) i5-13600KF 3.50 GHz CPU and NVIDIA GeForce RTX 4080 GPU.

For each obstacle density level, this simulation experiment generated 10 distinct maps randomly, with the recording of three experimental outcomes: average flight distance, computation time, and the number of search nodes. To maintain variable consistency, this study set the starting coordinates at (0,0,0) and the target coordinates at (90,90,15).

As illustrated in Figure 7 , the RFA-Star algorithm’s flight distance is only 0.06% to 0.49% longer than that of the RJA-Star algorithm, but 3.88% to 6.83% shorter than that of the Improved A-Star algorithm. The number of explored nodes is 19.42% to 36.51% higher than that of RJA-Star, while it accounts for only 0.59% to 1.17% of the nodes explored by Improved A-Star. The computation time for RFA-Star ranges from 84% to 94% of that required by RJA-Star, and from 51% to 96% of that required by Improved A-Star. While RFA-Star shows minimal differences from RJA-Star in terms of flight distance and explored nodes, it significantly outperforms RJA-Star in computation time. For UAVs, the ability to quickly and efficiently generate safe paths is crucial. Compared to the other three path planning algorithms, RFA-Star demonstrates the capability to swiftly generate high-quality paths while maintaining relatively shorter routes.

This study presents distribution plots of the experimental results for flight distance, computation time, and search nodes. It is important to note that, due to the significantly higher computation time and search nodes associated with the A-Star and Improved A-Star algorithms compared to the RJA-Star and RFA-Star algorithms, corresponding distribution plots for A-Star and Improved A-Star were not generated for clarity, as shown in Figure 8 . The box plots of the experimental results reveal that the RFA-Star algorithm demonstrates more concentrated outcomes across various obstacle densities compared to the other three path planning algorithms, with less fluctuation in results due to changes in the environment. However, as observed in Figure 8A , the A-Star algorithm maintains a nearly consistent flight distance across maps of the same size. Although it does not achieve the shortest distances in this study, its stability is one of the reasons why the A-Star algorithm has become a classic.

We conducted a t-test to analyze the search node results of the RJA-Star algorithm and the RFA-Star algorithm, with the findings presented in Table 3 . The t-test was performed with a significance level of 0.05 to determine if there were statistically significant differences in the number of search nodes between the two algorithms across various obstacle densities.

As shown in Table 3 , at obstacle densities of 0.4, 0.5, 0.6, and 0.8, no significant difference was observed (H = 0), indicating that the two algorithms perform similarly in these conditions. However, at an obstacle density of 0.7, a significant difference was detected (H = 1, p = 0.04), suggesting that the algorithms differ in their efficiency or behavior under this specific condition. The t-test thus highlights where the algorithms diverge in performance, particularly in their handling of search nodes.

Since the RFA-Star, RJA-Star, and A-Star can all utilize grid maps for search, this study evaluated the effectiveness of these three algorithms under different map sizes. Under the precondition that the obstacle density across all maps is set to 0.8 obs./m², the paper compares the flight distance, computation time, and search nodes of these three algorithms across five different map sizes. Ten maps were randomly generated for each size to conduct experimental comparisons. As depicted in Figure 9 , the path planning scenarios for Map 1 and Map 5 are illustrated.

In Figure 9 , it is shown that among the results of Map1 and Map5, the A-star algorithm produces the longest path compared to the other three algorithms. Additionally, as depicted in Figure 9B , the A-star algorithm generates a wavy path for obstacle avoidance, which is not in line with typical UAV motion patterns. Although the RJA-star algorithm results in the shortest path, it does so by closely hugging obstacles during avoidance. This approach compromises the UAV’s safety in favor of a shorter route. However, in practical UAV operations, this close proximity to obstacles is highly dangerous, increasing the risk of collision. Therefore, considering both safety and flight distance, the RFA-star algorithm offers the most balanced and optimal solution by planning a relatively short path while ensuring safety.

From the t-test results and the findings presented in Table 4 , it is evident that influenced by the map size, the values of all three experimental metrics exhibit an increasing trend. Particularly noteworthy is the exponential increase observed in the average computation time and average search nodes for the A-Star algorithm. From the results, it can be observed that the average flight distance of the RFA-Star algorithm proposed in this study is comparable to that of the RJA-Star algorithm. Although the average search nodes for RFA-Star are higher than those for RJA-Star, it outperforms the other two algorithms in the crucial aspect of computation time. Moreover, in different map sizes, the average computation time of the RFA-Star algorithm is 85% to 92% of that of the RJA-Star algorithm, indicating a significant improvement. This suggests that the RFA-Star algorithm, even with an increase in map size, can maintain stable computation performance, providing rapid and stable path planning.

To further validate the effectiveness and advancement of the RFA-Star algorithm, we conducted a series of comparative experiments with other state-of-the-art path planning algorithms. Although the unique nature of RFA-Star made it challenging to find directly comparable algorithms, this approach allowed us to place RFA-Star within the broader context of modern path planning techniques. By comparing RFA-Star with diverse algorithms designed for different environments and operational requirements, we could better understand its strengths and limitations. This comparison not only highlights the robustness and efficiency of RFA-Star in various scenarios but also provides a comprehensive perspective on its relative performance against other leading methods.

Castro et al. (Castro et al., 2023) combined Rapidly-exploring Random Trees (RRT) with Deep Reinforcement Learning (DRL) to generate and control UAV trajectories during the inspection of olive fly traps. Their proposed solution was tested in a simulated environment with 10 dynamic obstacles within a 300 cubic meter area. The RRT+DQN algorithm achieved an average runtime of 8.2 milliseconds, outperforming traditional algorithms like Genetic Algorithm (GA) and Dijkstra, which had runtimes of 8.7 milliseconds and 2.4 milliseconds, respectively. The pure RRT algorithm had a runtime of 6.5 milliseconds. Souto et al. (Souto et al., 2023) developed a novel reinforcement learning-based method aimed at reducing power consumption during UAV missions in disaster scenarios to mitigate the negative effects of changing wind directions. Compared to simpler heuristic methods, the power-saving effect was reduced by 15.93%. The study showed that Q-learning using an ϵ-greedy decay method was the most efficient, resulting in shorter mission durations compared to SARSA and basic Q-learning. While the main focus of this study was on energy efficiency rather than path planning speed, it highlighted the importance of algorithm efficiency in extending UAV mission life. Xu et al. (Xu et al., 2024) proposed a bionic 3D path planning algorithm for agricultural UAVs, designed to optimize safe flight paths between work plots obstructed by multiple obstacle zones. The algorithm was tested in a 100 cubic meter irregular hilly space with several randomly placed obstacles. The experimental results showed that the bionic 3D path planning reduced path length by 75.15%, and energy consumption decreased by 13.91% to 27.35% compared to other algorithms, including Ant Colony Optimization and Artificial Bee Colony algorithms. The specific results are presented in Table 5 .

The RFA-Star algorithm integrates the enhanced A-Star algorithm with the R5DOS model and incorporates a feature attention mechanism. Despite the addition of extra computational steps, its time complexity remains at O ( n l o g ( n ) ) , demonstrating high computational efficiency and stability in high-density obstacle environments. In contrast, the RRT+DQN algorithm combines Rapidly-exploring Random Trees (RRT) with Deep Q-Network (DQN), making it suitable for path planning in dynamic environments. Its time complexity is O ( n l o g ( n ) ) , and it shows better flexibility when handling dynamic obstacles.

The Q-learning-based energy-efficient path planning algorithm primarily focuses on reducing computational complexity to extend UAV mission life, with a time complexity of O ( m · n ) , where mmm is the number of states and nnn is the number of actions. This method is appropriate for scenarios requiring high energy efficiency. The bionic 3D path planning algorithm, which simulates krill swarm behavior, achieves dual optimization of path length and energy consumption. Although its time complexity is relatively high (approximately O ( n 2 ) , it provides superior path planning and energy management in complex terrains.

In contrast, the RFA-Star algorithm consistently demonstrates shorter computation times across various obstacle densities, underscoring its efficiency in handling complex environments. Especially in high-density obstacle scenarios, the RFA-Star algorithm not only maintains a rapid computation speed but also successfully generates stable, safe, and shorter paths, further confirming its exceptional performance in complex settings. Although RFA-Star excels in computation speed, the potential advantages of the RRT+DQN method in dynamic obstacle scenarios should not be overlooked, providing a direction for future improvements in the adaptability of RFA-Star. Additionally, while RFA-Star may not directly compete with the bionic 3D algorithm in terms of energy and path length optimization, its rapid computation and pathfinding capabilities in highly complex environments showcase its robust and stable solution.