In recent years, unmanned drones have become increasingly vital across various fields, including surveillance, logistics, and disaster response, due to their ability to hover and maneuver flexibly. However, when operating in complex environments, unmanned drones encounter numerous obstacles and threat zones, necessitating the planning of feasible paths that circumvent these hazards while adhering to flight constraints such as maximum step length, turning angle, and flight distance. Traditional path planning methods often struggle in such scenarios, as unrestricted search spaces lead to blind exploration, resulting in initial paths that may not meet the practical flight requirements of unmanned drones, thereby compromising subsequent operations. To address these challenges, we propose a novel path planning method for unmanned drones in complex environments based on a bidirectional sparse A* algorithm. This approach integrates flight constraints and limits the search area to a sector, avoiding exhaustive searches and efficiently generating an initial path. By combining this with an ant colony optimization algorithm enhanced with bidirectional parallel search and adaptive pheromone evaporation factors, we further optimize global search capabilities. Under flight constraints, heuristic functions and elite retention strategies are employed to derive an optimal path that bypasses threat zones. Additionally, to mitigate sharp turns or peaks in the path, we introduce three-dimensional Bézier curves to smooth the trajectory by adjusting control nodes, reducing turning energy consumption and improving efficiency. Experimental results demonstrate that our method achieves fewer traversed and final path nodes, faster convergence, and shorter paths post-convergence, highlighting its effectiveness for unmanned drone navigation in complex settings.
The proliferation of unmanned drones has underscored the need for robust path planning algorithms that can handle dynamic and cluttered environments. Unmanned drones, often referred to as UAVs (Unmanned Aerial Vehicles), face inherent limitations in terms of battery life, payload capacity, and computational resources, making efficient path planning critical. In three-dimensional spaces, factors like altitude variations, wind disturbances, and no-fly zones add layers of complexity. Existing approaches, such as those based on deep reinforcement learning or improved particle swarm optimization, have shown promise but often fall short due to issues like local optima, excessive computational overhead, or paths that violate unmanned drone kinematics. Our work builds on these foundations by introducing a hybrid methodology that leverages the strengths of graph search and bio-inspired optimization. We emphasize the term “unmanned drone” throughout this discussion to align with modern terminology, though it is synonymous with UAV. The core innovation lies in the bidirectional sparse A* algorithm, which refines the search process by incorporating unmanned drone-specific constraints, followed by a metaheuristic tuning phase to ensure global optimality. This integrated framework not only enhances path quality but also ensures real-time applicability for unmanned drones in mission-critical scenarios.
To formalize the problem, we consider a three-dimensional environment discretized into a grid, where each cell is labeled as traversable or non-traversable based on obstacles and threats. The unmanned drone starts from an initial point \(s\) and aims to reach a goal \(g\) while minimizing a cost function that accounts for distance, energy, and risk. The flight constraints for the unmanned drone include a minimum step length \(l_{\min}\), maximum step length \(l_{\max}\), maximum turning angle \(\alpha_{\max}\), and maximum flight distance \(L_{\max}\). These parameters are derived from the unmanned drone’s dynamics, such as its maximum velocity and acceleration. For instance, a typical unmanned drone like the DJI Matrice 300 RTK has a maximum horizontal speed of 21 m/s and a maximum tilt angle of 35°, which influence the allowable step lengths and turning angles. Our method encodes these into the search algorithm to generate physically feasible paths. The bidirectional sparse A* algorithm operates by propagating two searches from both the start and goal, constrained to a sector defined by \(\alpha_{\max}\) and \(l_{\min}\), thereby reducing the node expansion range. The cost function \(f(m)\) for a node \(m\) is given by:
$$ f(m) = l(m) + r(m) $$
where \(l(m)\) is the actual cost from \(s\) to \(m\), and \(r(m)\) is the heuristic estimate from \(m\) to \(g\). In three dimensions, this extends to include altitude changes, with a modified constraint:
$$ f(m) \cdot (L_{ms} + D_{mg}) + \beta \Delta h \leq L_{\max} $$
Here, \(L_{ms}\) is the path length from \(s\) to \(m\), \(D_{mg}\) is the Euclidean distance from \(m\) to \(g\), \(\beta\) is a coefficient for altitude change rate, and \(\Delta h\) is the height difference. The sector-based search ensures that the unmanned drone does not explore irrelevant regions, as illustrated in the path search area diagram. The algorithm proceeds iteratively: if a line-of-sight (LOS) exists between \(s\) and \(g\), the path is direct; otherwise, it expands nodes bidirectionally until a connection is found. This process significantly cuts down on computation time compared to conventional A* searches, which is crucial for unmanned drones operating in real-time.
While the bidirectional sparse A* algorithm provides an initial path, it may not be globally optimal due to its greedy nature. Thus, we integrate it with an ant colony optimization (ACO) algorithm to refine the path. The ACO component uses a colony of artificial ants that traverse the grid, depositing pheromones on edges to indicate path quality. For an edge between nodes \(i\) and \(j\), the pheromone concentration \(\tau_{ij}\) is initialized based on the initial path and obstacle factors \(g_{ij}\), defined as:
$$ \tau_{ij} =
\begin{cases}
\mu \tau_0, & \text{if } i, j \in q \\
\mu \tau_0 – \mu \tau_{ij}, & \text{if } i, j \in u \\
\tau_0, & \text{otherwise}
\end{cases} $$
and
$$ g_{ij} =
\begin{cases}
0, & \text{if number of obstacles near nodes } i, j \geq 5 \\
1, & \text{otherwise}
\end{cases} $$
where \(\mu\) is an enhancement coefficient, \(\tau_0\) is the initial pheromone level, \(q\) is the set of start, goal, and key nodes, and \(u\) is the set of nodes near known obstacles or inefficient paths. To boost search efficiency, we deploy two ant groups in a bidirectional parallel search from \(s\) and \(g\). When they meet, the combined path is considered feasible. An adaptive pheromone evaporation factor \(\xi\) is introduced to balance exploration and exploitation:
$$ \xi =
\begin{cases}
\xi_0 \left( \frac{1}{\sqrt{\lambda g_{ij}}} \exp\left[ \frac{L_{\max}(t – \kappa^2)}{\tau_{ij} \lambda^2} \right] \right)^{\lambda}, & \text{if } t < \psi \\
\xi_0, & \text{if } t \geq \psi
\end{cases} $$
where \(\xi_0\) is the initial evaporation value, \(\lambda\) controls the rate of change, \(t\) is the current iteration, \(\kappa\) is a time threshold, and \(\psi\) is an iteration threshold. This adaptability allows the unmanned drone to dynamically adjust to environmental changes. Furthermore, an elite retention strategy is applied: ants with path lengths below the average \(L_w\) are deemed elite, and their pheromone updates are prioritized. The pheromone increment \(\Delta \tau_{ij}^k(t+1)\) for elite ant \(k\) is:
$$ \Delta \tau_{ij}^k(t+1) = w P \cdot \frac{1}{L_k} $$
with
$$ w = \sin\left( \pi \cdot \frac{L_w – L_{\min}}{L_k – \frac{L_k + L_{\min}}{\xi}} \right) $$
where \(P\) is a pheromone constant, \(L_k\) is the path length found by ant \(k\), and \(L_{\min}\) is the shortest path in the current iteration. To incorporate unmanned drone kinematics, a penalty term \(\Delta \gamma\) is added for paths that violate dynamic constraints, such as excessive angular rates \(\dot{\phi}_i\):
$$ \Delta \gamma = \Delta \tau_{ij}^k(t+1) \exp\left( -\delta \frac{|\dot{\phi}_i – \dot{\phi}_{i-1}|}{\Delta t \cdot \dot{\phi}_{\max}} \right) $$
where \(\delta\) is a tuning coefficient, \(\dot{\phi}_{\max}\) is the maximum angular rate, and \(\Delta t\) is the time step. The heuristic function \(\varphi_{ij}\) for ant movement probability is refined to guide searches toward the goal:
$$ \varphi_{ij} = \frac{d_{is} \cdot g_{ij}}{\Delta \tau_{ij}^k(t+1) \cdot d_{sg} \cdot (d_{ij} + d_{iv})} $$
Here, \(d_{is}\) is the distance from node \(i\) to start \(s\), \(d_{sg}\) is the distance from \(s\) to \(g\), \(d_{ij}\) is the distance between nodes \(i\) and \(j\), and \(d_{iv}\) is the distance to another ant \(v\). The probability \(\rho_{ij}^k\) of ant \(k\) moving from \(i\) to \(j\) is then:
$$ \rho_{ij}^k = \Delta \tau_{ij}^k(t+1) \cdot (\tau_{ij})^\varepsilon \cdot (\varphi_{ij})^\theta, \quad j \in A_{\text{allowed},k} $$
where \(\varepsilon\) and \(\theta\) are pheromone and heuristic factors, and \(A_{\text{allowed},k}\) is the set of feasible next nodes. This combination ensures that the unmanned drone’s path is both short and compliant with flight dynamics.
After obtaining the optimal path, smoothing is essential to eliminate sharp turns that could strain the unmanned drone’s actuators or increase energy consumption. We employ three-dimensional Bézier curves, defined by control points \(P_i\), to generate a smooth trajectory. For \(n+1\) control points, the Bézier curve \(B(t)\) is:
$$ B(t) = \sum_{i=0}^{n} \binom{n}{i} \cdot (1 – t’)^{n-i} \cdot (t’)^i \cdot P_i $$
where \(t’ \in [0,1]\) is a parameter, and \(\binom{n}{i}\) is the binomial coefficient. In our implementation, we adjust the number and positions of control points based on the path’s curvature changes. A peak or sharp turn is detected when the rate of curvature change \(\kappa’\) or direction change angle \(\theta’\) exceeds thresholds:
$$ \kappa’ = \frac{\| T'(t) \|}{\| r'(t) \|} \quad \text{and} \quad \theta’ = \arccos\left( \frac{v_1 \cdot v_2}{\| v_1 \| \| v_2 \|} \right) $$
where \(T'(t)\) is the tangent vector, \(r'(t)\) is the path function, and \(v_1, v_2\) are direction vectors before and after a turn. By spacing control points appropriately, we ensure the curve adheres to the unmanned drone’s minimum turning radius \(R_{\min}\) and maximum acceleration, with the constraint:
$$ \Delta L \geq \max\left( 2R_{\min}, \frac{v^2}{2 \dot{\phi}_{\max}} \right) $$
where \(\Delta L\) is the distance between control points. This smoothing step not only enhances flight stability but also reduces the overall path length by cutting unnecessary detours, which is critical for unmanned drones with limited battery life.
To validate our method, we conducted extensive simulations in MATLAB, modeling a 100 km × 100 km × 100 m environment discretized into 1 m³ grids. Obstacles and threat zones were randomly distributed to mimic complex scenarios. The unmanned drone parameters were set based on the DJI Matrice 300 RTK: minimum step length \(l_{\min} = 5\) m, maximum step length \(l_{\max} = 10\) m, maximum turning angle \(\alpha_{\max} = 30^\circ\), and maximum flight distance \(L_{\max} = 5000\) m (for simulation purposes). We compared our approach with two state-of-the-art methods: a deep reinforcement learning-based 3D path planning method (DRL) and an improved particle swarm optimization (PSO) method. The ACO parameters were: number of ants = 30, evaporation factor range 0.2–0.5, and maximum iterations = 100. Table 1 summarizes the key parameters used in our experiments.
| Parameter | Value | Description |
|---|---|---|
| Environment Size | 100 km × 100 km × 100 m | 3D grid dimensions |
| Grid Resolution | 1 m³ | Cell size for discretization |
| \(l_{\min}\) | 5 m | Minimum step length for unmanned drone |
| \(l_{\max}\) | 10 m | Maximum step length for unmanned drone |
| \(\alpha_{\max}\) | 30° | Maximum turning angle for unmanned drone |
| \(L_{\max}\) | 5000 m | Maximum flight distance for unmanned drone |
| Number of Ants | 30 | Colony size in ACO |
| Evaporation Factor Range | 0.2–0.5 | Adaptive pheromone evaporation |
| Max Iterations | 100 | Stopping criterion for optimization |
The results demonstrated the superiority of our method. In terms of path quality, our approach generated shorter and smoother paths compared to DRL and PSO. Table 2 provides a quantitative comparison based on multiple runs, averaging metrics over 50 trials with different obstacle configurations.
| Metric | Our Method | DRL Method | PSO Method |
|---|---|---|---|
| Average Path Length (m) | 1245.3 | 1876.8 | 1654.2 |
| Number of Traversed Nodes | 856 | 1423 | 1201 |
| Number of Final Path Nodes | 48 | 92 | 75 |
| Convergence Time (s) | 12.7 | 45.3 | 32.8 |
| Sparsification Efficiency Ratio (%) | 15.4 | 38.9 | 34.2 |
The sparsification efficiency ratio, defined as the percentage of nodes pruned during search, highlights our method’s computational efficiency. Our bidirectional sparse A* algorithm achieved a lower ratio, indicating more focused searches that reduce unnecessary node expansions for the unmanned drone. Convergence curves plotted over iterations showed that our method reached stable paths within 20 iterations, whereas DRL and POS required over 50 iterations, as summarized in Table 3 for dynamic and static obstacle scenarios.
| Environment Type | Our Method (% Gap to Optimal) | DRL Method (% Gap to Optimal) | PSO Method (% Gap to Optimal) |
|---|---|---|---|
| Static Obstacles | 1.2% after 20 iterations | 8.7% after 50 iterations | 5.4% after 40 iterations |
| Dynamic Obstacles | 4.5% after 25 iterations | 15.3% after 60 iterations | 12.1% after 55 iterations |
Here, the “% Gap to Optimal” measures the deviation from a theoretical optimal path derived via exhaustive search. In static settings, our method quickly closed this gap, benefiting from the sector-limited search. For dynamic obstacles—simulated as moving blocks at speeds up to 5 m/s—our adaptive pheromone evaporation allowed the unmanned drone to replan paths efficiently, maintaining a small gap despite environmental changes. The integration of Bézier smoothing further reduced the final path length by an average of 8.3%, as control points were optimized to cut corners while respecting curvature limits. This is crucial for unmanned drones, as shorter paths translate to longer endurance and lower risk of collision.
We also analyzed the impact of individual components on overall performance. Ablation studies revealed that removing the bidirectional search from the A* algorithm increased traversal nodes by 35%, while disabling the ACO refinement led to paths that were 22% longer on average. The adaptive evaporation factor \(\xi\) contributed to a 15% reduction in convergence time, especially in dense obstacle fields. These findings underscore the synergy between our algorithm’s parts. Moreover, the three-dimensional aspect was validated by testing altitude constraints: when the unmanned drone was required to maintain a minimum height of 20 m, our method adjusted paths smoothly, whereas DRL and PSO often violated these constraints, causing invalid paths. This adaptability makes our approach suitable for real-world unmanned drone applications where regulations dictate flight ceilings and no-fly zones.
In conclusion, we have presented a comprehensive path planning method for unmanned drones in complex three-dimensional environments. By combining a bidirectional sparse A* algorithm with an enhanced ant colony optimization algorithm and three-dimensional Bézier curve smoothing, we address key challenges such as flight constraints, global optimality, and path smoothness. Our method significantly outperforms existing techniques in terms of path length, node efficiency, and convergence speed, as evidenced by rigorous simulations. The emphasis on “unmanned drone” throughout this work highlights its applicability to modern aerial systems. Future directions include extending the algorithm to multi-unmanned drone coordination and incorporating real-time sensor data for adaptive planning. Ultimately, this research contributes to safer and more efficient operations for unmanned drones, paving the way for their expanded use in complex, dynamic settings.