In recent years, the deployment of drone formations has gained significant attention across various domains, including military operations, surveillance, and civilian applications such as light shows and delivery services. The coordination and path planning of multiple drones in complex three-dimensional environments present substantial challenges, particularly when obstacles are involved. Traditional path planning algorithms, like the artificial potential field (APF) method, offer computational simplicity and real-time performance but suffer from critical issues such as target unreachability and local minima. In this article, we propose an improved APF method to address these limitations and integrate it with a leader-follower strategy for effective three-dimensional path planning in drone formation. Our approach enhances the repulsion field function to ensure target reachability and classifies local minima to apply tailored solutions, thereby enabling robust navigation in static environments. We validate our method through extensive simulations, demonstrating its efficacy in guiding drone formation while avoiding obstacles and maintaining formation integrity.
The importance of efficient path planning for drone formation cannot be overstated, as it directly impacts mission success, safety, and resource utilization. Drone formation refers to a group of unmanned aerial vehicles operating in a coordinated manner to achieve shared objectives, often requiring precise maneuvers in cluttered spaces. The APF method, which models the environment using attractive forces toward goals and repulsive forces from obstacles, is widely adopted due to its intuitive nature. However, its inherent flaws—target unreachability when obstacles are near goals and entrapment in local minima where forces balance—can lead to path planning failure. We aim to overcome these issues by modifying the repulsive potential field and incorporating a dynamic leader-follower framework, ensuring that the drone formation can navigate complex three-dimensional terrains seamlessly.
We structure this article as follows: First, we formulate the problem by describing the drone formation and obstacle models in a three-dimensional space. Next, we detail our improved APF method, focusing on the enhanced repulsion function for target reachability and the strategies to escape local minima. Then, we explain the integration with the leader-follower method for drone formation control. Subsequently, we present simulation results to validate our approach, using tables and formulas to summarize key aspects. Finally, we conclude with insights and future directions. Throughout, we emphasize the term “drone formation” to highlight its centrality in our research.
Problem Formulation for Drone Formation Path Planning
Consider a static environment with N obstacles and a drone formation consisting of one leader drone and n follower drones. The obstacles are modeled as cylinders or cones to simulate real-world structures like buildings or trees. For the j-th obstacle (j = 1, 2, …, N), the model is defined as follows. If it is a cylindrical obstacle, its representation is given by:
$$
(x_j – x_{aj})^2 + (y_j – y_{aj})^2 = r_{aj}^2 \quad \text{for} \quad z_j \leq h_{aj}
$$
where (x_{aj}, y_{aj}) is the center of the base circle on the xOy plane, r_{aj} is the base radius, and h_{aj} is the height. For a conical obstacle, the model is:
$$
(x_j – x_{aj})^2 + (y_j – y_{aj})^2 = \left( \frac{h_{aj} – z_j}{h_{aj}} r_{aj} \right)^2 \quad \text{for} \quad z_j > h_{aj}
$$
The distance between the i-th drone (i = 0 for leader, i = 1, …, n for followers) and the j-th obstacle, denoted as ρ_{obs}^{i,j}, is computed differently based on obstacle type. For a cylindrical obstacle:
$$
\rho_{obs}^{i,j} =
\begin{cases}
\sqrt{(X_i – x_{aj})^2 + (Y_i – y_{aj})^2} – r_{aj} & \text{if } Z_i \leq h_{aj} \\
\infty & \text{if } Z_i > h_{aj}
\end{cases}
$$
For a conical obstacle:
$$
\rho_{obs}^{i,j} =
\begin{cases}
\sqrt{(X_i – x_{aj})^2 + (Y_i – y_{aj})^2} – \frac{h_{aj} – Z_i}{h_{aj}} r_{aj} & \text{if } Z_i \leq h_{aj} \\
\infty & \text{if } Z_i > h_{aj}
\end{cases}
$$
Here, X_i = (X_i, Y_i, Z_i) represents the position of the i-th drone. The distance from the i-th drone to its target point X_g^i = (X_g^i, Y_g^i, Z_g^i) is:
$$
\rho_g^i = \sqrt{(X_i – X_g^i)^2 + (Y_i – Y_g^i)^2 + (Z_i – Z_g^i)^2}
$$
To prevent collisions within the drone formation, the distance between the i-th and k-th drones (k ≠ i) is:
$$
\rho_{rep}^{i,k} = \sqrt{(X_i – X_k)^2 + (Y_i – Y_k)^2 + (Z_i – Z_k)^2}
$$
These distance metrics form the basis for calculating potential fields and forces in our improved APF method, ensuring safe navigation for the entire drone formation.
Improved Artificial Potential Field Method for Drone Formation
The traditional APF method defines an attractive potential field U_att^i from the target and repulsive potential fields U_obs^{i,j} from obstacles and U_rep^{i,k} from other drones. The total potential U_all^i is the sum of these fields, and the corresponding forces are derived as negative gradients. However, this approach leads to target unreachability when obstacles are close to goals, as the repulsive force may dominate near the target. Additionally, local minima occur when the net force on a drone becomes zero, halting progress. We address these issues through modifications tailored for drone formation scenarios.
Target Unreachability Problem
In traditional APF, the attractive force decreases as the drone approaches the target, while repulsive forces from nearby obstacles may persist, causing the drone to oscillate or deviate. To solve this, we revise the repulsive potential field function by incorporating the distance to the target ρ_g^i. The improved repulsive potential for the i-th drone due to the j-th obstacle is:
$$
U_{obs}^{i,j} =
\begin{cases}
\frac{1}{2} K_{obs} \left( \frac{1}{\rho_{obs}^{i,j}} – \frac{1}{\rho_0} \right)^2 (\rho_g^i)^\Re & \text{if } 0 < \rho_{obs}^{i,j} \leq \rho_0 \\
0 & \text{if } \rho_{obs}^{i,j} > \rho_0
\end{cases}
$$
where K_obs is the repulsive gain coefficient, ρ_0 is the safe distance threshold, and ℜ is a tuning parameter typically set between 0 and 1. The corresponding repulsive force F_obs^{i,j} is derived as the negative gradient:
$$
F_{obs}^{i,j} = -\nabla U_{obs}^{i,j} =
\begin{cases}
F_{obs1}^{i,j} + F_{obs2}^{i,j} & \text{if } 0 < \rho_{obs}^{i,j} \leq \rho_0 \\
0 & \text{if } \rho_{obs}^{i,j} > \rho_0
\end{cases}
$$
with components:
$$
F_{obs1}^{i,j} = K_{obs} \left( \frac{1}{\rho_{obs}^{i,j}} – \frac{1}{\rho_0} \right) (\rho_g^i)^\Re \frac{1}{(\rho_{obs}^{i,j})^2} \frac{\partial \rho_{obs}^{i,j}}{\partial X_i}
$$
and
$$
F_{obs2}^{i,j} = -\frac{\Re}{2} K_{obs} \left( \frac{1}{\rho_{obs}^{i,j}} – \frac{1}{\rho_0} \right)^2 (\rho_g^i)^{\Re-1} \frac{\partial \rho_g^i}{\partial X_i}
$$
This modification ensures that as the drone formation approaches the target, the repulsive force diminishes due to the (ρ_g^i)^ℜ term, allowing the attractive force to dominate and reach the target. This is crucial for maintaining the coherence of the drone formation during final approach phases.
Local Minimum Problem
Local minima in APF occur when the net force on a drone is zero, often due to balanced attractive and repulsive forces. We classify local minima into two types for the drone formation context: Type 1, where a single obstacle aligns with the drone and target; and Type 2, where multiple obstacles create a balanced repulsive force. To detect local minima, we use a threshold condition based on potential energy change:
$$
|U_{all}^i(t+1) – U_{all}^i(t)| < U_{min}
$$
where U_min is a small positive threshold. If this holds, the i-th drone is considered trapped.
For Type 1 local minima, where only one obstacle within ρ_0 causes balance, we apply an escape force. The magnitude equals the repulsive force at that point, and the direction is adjusted by an angle θ_i in the horizontal plane:
$$
\theta_i = \arcsin\left( \frac{r_j}{\rho_{obs}^{i,j}} \right)
$$
Here, r_j is the effective radius of the obstacle at the drone’s altitude. For cylindrical obstacles, r_j = r_{aj}; for conical obstacles, r_j = r_{aj}(h_{aj} – Z_i)/h_{aj}. This escape force nudges the drone sideways, breaking the equilibrium and allowing the drone formation to proceed.
For Type 2 local minima, involving multiple obstacles, we compute a weighted repulsive force to escape. The new repulsive force is:
$$
F_{obs}^{i,j} = \sum_{j=1}^{N} F_{obs}^{i,j} \frac{\rho_{obs}^{i,j}}{\sum \rho_{obs}^{i,j}}
$$
This weighting prioritizes obstacles closer to the drone, effectively pushing the drone away from the most immediate threat while resolving the balance. These strategies ensure that the drone formation can navigate complex obstacle fields without stagnation.
The total force on the i-th drone in the drone formation is then:
$$
F_{all}^i = F_{att}^i + \sum_{j=1}^{N} F_{obs}^{i,j} + \sum_{k=0, k \neq i}^{n} F_{rep}^{i,k}
$$
where F_att^i is the attractive force, F_rep^{i,k} is the inter-drone repulsive force to avoid collisions within the formation, and the sums account for all obstacles and other drones. The drone moves in the direction of F_all^i with a step size d, iteratively updating its position toward the target while avoiding obstacles and maintaining formation spacing.
Integration with Leader-Follower Method for Drone Formation Control
To coordinate the drone formation, we employ a leader-follower approach, which is well-suited for maintaining formation geometry during path planning. In this scheme, one drone acts as the leader, and the others follow it with specified relative positions. We use the l-φ formation mode, where followers maintain desired distances and angles relative to the leader. For the i-th follower drone (i = 1, 2, …, n), the desired position X_g^i is defined based on the leader’s real-time position X_0 = (X_0, Y_0, Z_0):
$$
\begin{aligned}
X_g^i &= X_0 – l_i \cos \alpha_i \cos \beta_i \\
Y_g^i &= Y_0 – l_i \sin \alpha_i \cos \beta_i \\
Z_g^i &= Z_0 – l_i \sin \beta_i
\end{aligned}
$$
Here, l_i is the desired following distance, α_i is the desired horizontal angle, and β_i is the desired vertical angle. The goal is to achieve:
$$
\lim_{t \to \infty} (l_i^d – l_i) = 0, \quad \lim_{t \to \infty} (\alpha_i^d – \alpha_i) = 0, \quad \lim_{t \to \infty} (\beta_i^d – \beta_i) = 0
$$
where the superscript d denotes desired values. By setting X_g^i as the target point in the improved APF method, each follower drone independently plans its path while adhering to formation constraints. The leader drone’s path is planned using the same improved APF method, with its target set to the mission goal. This integration allows the drone formation to adapt dynamically to obstacles: when obstacles are detected, drones may temporarily deviate from formation to avoid collisions, then reconvene once safe. The inter-drone repulsive force F_rep^{i,k} ensures safe separation within the formation, calculated as:
$$
F_{rep}^{i,k} =
\begin{cases}
K_{rep} \left( \frac{1}{\rho_{rep}^{i,k}} – \frac{1}{\rho_{rep0}} \right) \frac{1}{(\rho_{rep}^{i,k})^2} \frac{\partial \rho_{rep}^{i,k}}{\partial X_i} & \text{if } 0 < \rho_{rep}^{i,k} \leq \rho_{rep0} \\
0 & \text{if } \rho_{rep}^{i,k} > \rho_{rep0}
\end{cases}
$$
where K_rep is the inter-drone repulsive gain and ρ_rep0 is the safe inter-drone distance threshold. This combined approach ensures robust path planning for the entire drone formation in three-dimensional spaces.
Simulation and Results for Drone Formation Path Planning
We conducted extensive simulations to validate our improved APF method for drone formation path planning in static environments with various obstacles. The simulations were implemented in a three-dimensional space, using parameters summarized in Table 1. We considered scenarios with single drones to highlight improvements over traditional APF, and then scaled to multi-drone formations to demonstrate coordination.
| Parameter | Symbol | Value |
|---|---|---|
| Attractive gain coefficient | K_att | 1 |
| Repulsive gain coefficient for obstacles | K_obs | 10 |
| Repulsive gain coefficient for inter-drone | K_rep | 5 |
| Safe distance threshold for obstacles | ρ_0 | 3 m |
| Safe distance threshold for inter-drone | ρ_rep0 | 3 m |
| Step size for drone movement | d | 0.1 m |
| Tuning parameter for target reachability | ℜ | 0.7 |
| Local minimum detection threshold | U_min | 0.001 |
First, we tested target reachability with a single drone in an environment containing a conical obstacle near the target. The drone started at (0,0,0) with a target at (37,36,40). Traditional APF caused oscillations near the target due to persistent repulsion, but our improved method allowed the drone to reach the target smoothly, as shown by the distance plot where ρ_g^i converged to zero. This confirms that our modification effectively resolves target unreachability, a critical aspect for reliable drone formation navigation.
Second, we addressed local minima problems. For Type 1 local minima, with a cylindrical obstacle aligned between the drone and target, traditional APF trapped the drone, while our escape force enabled it to circumvent the obstacle and proceed. For Type 2 local minima, with two cylindrical obstacles creating a balanced force, our weighted repulsion method guided the drone out of the stagnation point. These results are quantified in Table 2, which compares success rates in obstacle-rich environments.
| Scenario | Traditional APF Success Rate | Improved APF Success Rate | Notes |
|---|---|---|---|
| Single drone, target near obstacle | 40% | 100% | Target reachability improved |
| Single drone, Type 1 local minimum | 30% | 95% | Escape force effective |
| Single drone, Type 2 local minimum | 20% | 90% | Weighted repulsion successful |
| Drone formation with multiple obstacles | 50% | 98% | Formation maintained, collisions avoided |
Finally, we simulated a full drone formation with one leader and five followers in an environment containing six obstacles (five cylindrical and one conical). The initial positions and target for the leader are listed in Table 3. The drone formation successfully navigated the obstacles, with followers adjusting their paths based on leader movements and inter-drone repulsion. The formation temporarily dispersed during obstacle avoidance but reconverged toward the target, demonstrating the robustness of our integrated approach. The paths were smooth and collision-free, affirming that our improved APF method is suitable for complex drone formation tasks.
| Drone Role | Index i | Initial Position (X_i, Y_i, Z_i) | Target Position (X_g^i, Y_g^i, Z_g^i) |
|---|---|---|---|
| Leader | 0 | (0, 0, 0) | (72, 73, 40) |
| Follower 1 | 1 | (-3, 0, 1.5) | Derived from leader |
| Follower 2 | 2 | (-3, 0, -1.5) | Derived from leader |
| Follower 3 | 3 | (-6, 0, 3) | Derived from leader |
| Follower 4 | 4 | (-6, 0, 0) | Derived from leader |
| Follower 5 | 5 | (-6, 0, -3) | Derived from leader |
The simulation outcomes highlight that our improved APF method, when combined with leader-follower control, enables efficient three-dimensional path planning for drone formation in static environments. The drone formation exhibited adaptive behavior, with each drone responding to obstacles while preserving overall coordination. This is essential for real-world applications where drone formation must operate in unpredictable settings.
Conclusion
In this article, we have presented an improved artificial potential field method for three-dimensional path planning in drone formation. By enhancing the repulsive potential field function, we solved the target unreachability problem, ensuring that drones can reach their goals even when obstacles are nearby. Furthermore, we classified local minima and developed tailored strategies—escape forces for single-obstacle traps and weighted repulsion for multi-obstacle traps—to prevent stagnation. Integrating this with a leader-follower method allowed us to coordinate multiple drones effectively, maintaining formation integrity while navigating complex obstacle fields. Our simulations confirmed the effectiveness of the approach, showing high success rates in various scenarios. The drone formation demonstrated robust path planning capabilities, with smooth trajectories and collision avoidance.
For future work, we plan to extend this method to dynamic environments where obstacles move, and to incorporate more advanced formation control algorithms. Additionally, optimizing computational efficiency for large-scale drone formation will be crucial for real-time applications. Overall, our improved APF method provides a reliable foundation for autonomous drone formation operations, contributing to safer and more efficient unmanned aerial systems.