The coordinated flight of multiple Unmanned Aerial Vehicles (UAVs), or drone formation, presents significant advantages over single-agent operations, including expanded mission coverage, enhanced system reliability, and greater operational efficiency. Among various UAV types, quadrotors are particularly suited for drone formation tasks due to their capabilities for vertical take-off and landing, hovering, and agile maneuvering. A critical challenge in achieving autonomous drone formation flight is real-time, collision-free path planning in complex, obstacle-filled environments. The Artificial Potential Field (APF) method, renowned for its conceptual simplicity and computational efficiency, has been widely adopted for this purpose. However, its classical formulation suffers from well-known pitfalls that hinder practical application in sophisticated 3D drone formation scenarios.
This article addresses the fundamental shortcomings of the traditional APF method—specifically, the Goal Non-Reachable with Obstacles Nearby (GNRON) problem, the existence of local minima, and oscillatory movements near obstacles. We propose a comprehensive framework for 3D multi-drone path planning that integrates modified potential field functions, a dynamic local minima escape mechanism, and a global path optimization technique. Our approach is designed for the dynamic interactions inherent in a moving drone formation, where each UAV generates a repulsive field perceived by others.
1. Limitations of the Classical APF Model
In the classical APF model, each agent, treated as a point mass, navigates under the influence of attractive forces from its target and repulsive forces from obstacles and other agents. For a system with n drones and m static obstacles, the traditional potential functions for drone i are defined as follows.
The attractive potential from its goal at position $\mathbf{P}_{gi}$ is:
$$ U_{att}^i = \frac{1}{2} \eta \rho_g^2 $$
where $\eta > 0$ is an attractive gain, and $\rho_g = ||\mathbf{P}_i – \mathbf{P}_{gi}||$ is the distance to the goal.
The repulsive potential from obstacle j and from another drone k are typically defined as:
$$
U_{rep, obs}^{ij} = \begin{cases}
\frac{1}{2} k_1 \left( \frac{1}{\rho_{o}^{ij}} – \frac{1}{\rho_0} \right)^2, & \text{if } \rho_{o}^{ij} \le \rho_0 \\
0, & \text{if } \rho_{o}^{ij} > \rho_0
\end{cases}
$$
$$
U_{rep, uav}^{ik} = \begin{cases}
\frac{1}{2} k_2 \left( \frac{1}{\rho_{u}^{ik}} – \frac{1}{\rho_0} \right)^2, & \text{if } \rho_{u}^{ik} \le \rho_0 \\
0, & \text{if } \rho_{u}^{ik} > \rho_0
\end{cases}
$$
Here, $k_1, k_2 > 0$ are repulsive gains, $\rho_0$ is the maximum influence distance, $\rho_{o}^{ij}=||\mathbf{P}_i – \mathbf{P}_{oj}||$, and $\rho_{u}^{ik}=||\mathbf{P}_i – \mathbf{P}_{k}||$.
The total force on drone i is the negative gradient of the combined potential:
$$ \mathbf{F}_{total}^i = -\nabla U_{att}^i – \sum_{j=1}^{m} \nabla U_{rep, obs}^{ij} – \sum_{\substack{k=1 \\ k \neq i}}^{n} \nabla U_{rep, uav}^{ik} = \mathbf{F}_{att}^i + \mathbf{F}_{rep, obs}^i + \mathbf{F}_{rep, uav}^i $$
The inherent flaws of this model are summarized in the table below:
| Problem | Description | Consequence for Drone Formation |
|---|---|---|
| GNRON | When the goal is very close to an obstacle, the repulsive force grows while the attractive force diminishes to zero, resulting in a net force of zero before the goal is reached. | The drone halts before completing the formation, leaving gaps or misalignments in the intended drone formation pattern. |
| Local Minima | Points where the vector sum of all attractive and repulsive forces is zero, but which are not the goal (e.g., symmetric obstacle configurations, narrow passages). | One or more drones in the drone formation become stuck, failing to converge to their target positions and potentially disrupting the entire fleet’s coordination. |
| Oscillations | In narrow corridors or near obstacle boundaries, force vectors can cause back-and-forth chattering motion. | Inefficient, unstable flight paths that waste energy and can lead to instability in tight drone formation maneuvers. |
2. Proposed Enhanced 3D APF Framework for Drone Formation
Our framework modifies the core APF formulation and introduces higher-level strategies to overcome its limitations for 3D drone formation path planning.
2.1 Modified Repulsive Potential Function
To solve the GNRON problem and ensure the goal is the global minimum of the total potential field, we modify the repulsive potential functions by incorporating the distance to the goal $\rho_g$. The new repulsive potentials are:
For obstacles:
$$
U_{rep, obs}^{ij} = \begin{cases}
\frac{1}{2} k_1 \left( \frac{1}{\rho_{o}^{ij}} – \frac{1}{\rho_0} \right)^2 (\rho_g)^p, & \text{if } \rho_{o}^{ij} \le \rho_0 \\
0, & \text{if } \rho_{o}^{ij} > \rho_0
\end{cases}
$$
For other drones in the formation:
$$
U_{rep, uav}^{ik} = \begin{cases}
\frac{1}{2} k_2 \left( \frac{1}{\rho_{u}^{ik}} – \frac{1}{\rho_0} \right)^2 (\rho_g)^p, & \text{if } \rho_{u}^{ik} \le \rho_0 \\
0, & \text{if } \rho_{u}^{ik} > \rho_0
\end{cases}
$$
where $p$ is a positive constant (e.g., $p=3$). The corresponding repulsive force $\mathbf{F}_{rep}^i$ is derived from the negative gradient, $-\nabla U_{rep}^i$, and now consists of two components due to the product rule:
$$ \mathbf{F}_{rep}^i = \mathbf{F}_{rep1}^i + \mathbf{F}_{rep2}^i $$
$$ \mathbf{F}_{rep1}^i = \sum_{j} k_1 \left( \frac{1}{\rho_{o}^{ij}} – \frac{1}{\rho_0} \right) \frac{(\rho_g)^p}{(\rho_{o}^{ij})^3} (\mathbf{P}_i – \mathbf{P}_{oj}) + \sum_{k} k_2 \left( \frac{1}{\rho_{u}^{ik}} – \frac{1}{\rho_0} \right) \frac{(\rho_g)^p}{(\rho_{u}^{ik})^3} (\mathbf{P}_i – \mathbf{P}_{k}) $$
$$ \mathbf{F}_{rep2}^i = – \sum_{j} \frac{p}{2} k_1 \left( \frac{1}{\rho_{o}^{ij}} – \frac{1}{\rho_0} \right)^2 (\rho_g)^{p-2} (\mathbf{P}_i – \mathbf{P}_{gi}) – \sum_{k} \frac{p}{2} k_2 \left( \frac{1}{\rho_{u}^{ik}} – \frac{1}{\rho_0} \right)^2 (\rho_g)^{p-2} (\mathbf{P}_i – \mathbf{P}_{gi}) $$
Here, $\mathbf{F}_{rep1}^i$ points away from the obstacle/other drone, and crucially, $\mathbf{F}_{rep2}^i$ points toward the goal. This latter component ensures that as the drone approaches its goal ($\rho_g \to 0$), the total repulsive force also tends to zero, guaranteeing that the goal point is reachable.
2.2 Detection and Escape from Local Minima
Even with the modified potential, local minima can still arise from complex geometric configurations in a 3D drone formation environment. We implement a real-time detection and escape routine.
Detection: Let $\mathbf{P}_i(t)$ and $\mathbf{P}_i(t-1)$ be the current and previous positions of drone i. Let $\epsilon$ and $\xi$ be small positive thresholds. A local minimum is declared if:
$$ ||\mathbf{P}_i(t) – \mathbf{P}_i(t-1)|| < \epsilon \quad \text{AND} \quad ||\mathbf{P}_i(t) – \mathbf{P}_{gi}|| > \xi $$
This indicates the drone has stalled significantly far from its goal.
Escape Strategy – “90° to Goal Direction Move”: Upon detection, we apply an additional virtual force $\mathbf{F}_{add}^i$ to perturb the drone out of the equilibrium. The key insight is to apply this force in a direction orthogonal to the current goal direction vector $\mathbf{d}_g = \mathbf{P}_{gi} – \mathbf{P}_i(t)$. This prevents simply reinforcing the existing force balance. We generate a random unit vector $\hat{\mathbf{n}}$ lying on the plane perpendicular to $\mathbf{d}_g$ (i.e., $\hat{\mathbf{n}} \cdot \mathbf{d}_g = 0$). The additional potential and force are defined as:
$$
U_{add} = \begin{cases}
\frac{1}{2} s \left( ||\mathbf{P}_i – \mathbf{P}_{local}||^2 – \rho_a^2 \right), & \text{if } ||\mathbf{P}_i – \mathbf{P}_{local}|| \le \rho_a \\
0, & \text{if } ||\mathbf{P}_i – \mathbf{P}_{local}|| > \rho_a
\end{cases}
$$
$$ \mathbf{F}_{add}^i = -\nabla U_{add} = \begin{cases}
-s (\mathbf{P}_i – \mathbf{P}_{local}), & \text{if } ||\mathbf{P}_i – \mathbf{P}_{local}|| \le \rho_a \\
0, & \text{if } ||\mathbf{P}_i – \mathbf{P}_{local}|| > \rho_a
\end{cases}
$$
where $\mathbf{P}_{local}$ is the detected local minimum position, $s>0$ is a gain, and $\rho_a$ is the influence radius. The force $\mathbf{F}_{add}^i$ is then projected onto the random orthogonal direction $\hat{\mathbf{n}}$ to create the final escape force. This method effectively “pushes” the drone sideways, allowing it to break symmetry and find a new path.
2.3 Global Path Optimization via Regression Search
The APF method, even improved, often generates suboptimal, meandering paths. To enhance the overall efficiency of the drone formation flight, we apply a regression search algorithm as a post-processing step (or an intermittent optimization during planning). The process for a single drone’s path, consisting of a sequence of via-points $\{\mathbf{Q}_1, \mathbf{Q}_2, …, \mathbf{Q}_N\}$ generated by the APF, is as follows:
- Start with the first point $\mathbf{Q}_1$ (start position).
- For $j$ from 2 to $N$, check if the straight-line segment $\overline{\mathbf{Q}_1 \mathbf{Q}_j}$ intersects the “safety zone” (a buffer of radius $l_0$) around any obstacle.
- Find the largest $j$ such that $\overline{\mathbf{Q}_1 \mathbf{Q}_j}$ is collision-free, but $\overline{\mathbf{Q}_1 \mathbf{Q}_{j+1}}$ is not. Replace the original path from $\mathbf{Q}_1$ to $\mathbf{Q}_j$ with the direct segment $\overline{\mathbf{Q}_1 \mathbf{Q}_j}$.
- Set $\mathbf{Q}_j$ as the new starting point and repeat the process until the goal $\mathbf{Q}_N$ is connected.
This algorithm guarantees a shorter, globally collision-free path that respects the same safety margins. In a dynamic drone formation setting, this optimization can be run periodically on the recent path history.
3. Simulation Analysis and Results
We simulated a 5-drone formation in a MATLAB environment with multiple spherical obstacles to validate our framework. The key parameters used are listed below:
| Parameter | Symbol | Value |
|---|---|---|
| Attractive Gain | $\eta$ | 20 |
| Obstacle Repulsive Gain | $k_1$ | 9 |
| Inter-Drone Repulsive Gain | $k_2$ | 8 |
| Goal-Distance Exponent | $p$ | 3 |
| Maximum Influence Distance | $\rho_0$ | 3 units |
| Drone Safety Radius | – | 0.5 units |
Stage 1: Classical APF Failure. The simulation confirmed the GNRON problem. Drones assigned goals near obstacles became trapped in local minima, oscillating indefinitely without reaching their final positions, which would completely disrupt the intended drone formation.
Stage 2: Modified APF. Using the modified repulsive potential from Section 2.1, all drones successfully navigated to their target positions, forming the desired drone formation. This solved the GNRON issue. However, visual analysis of the trajectories revealed noticeable oscillatory segments in the paths of some drones, indicative of residual local minima effects in the dynamic multi-agent field.
Stage 3: Integrated Local Minima Escape. Implementing the detection and escape mechanism from Section 2.2 allowed drones to identify and break free from these trapping regions. The characteristic maneuver showed a drone deviating laterally (the “90° move”) from its stalled position before reorienting and proceeding smoothly to its goal. This ensured robust convergence for the entire drone formation.
Stage 4: Path Optimization. Applying the regression search algorithm to the successful paths from Stage 3 produced significantly shorter and smoother trajectories. The optimized paths, represented as straight-line segments between key via-points, reduced the total travel distance and energy expenditure for the drone formation without compromising safety.
4. Discussion and Conclusion
This work presents a holistic enhancement of the Artificial Potential Field method tailored for 3D multi-drone path planning and formation control. The core improvements address the classical APF’s most critical flaws, making it viable for practical drone formation applications.
- The modified repulsive potential mathematically ensures the target is the global minimum of the field, solving the goal unreachability problem.
- The dynamic local minima escape strategy provides a simple yet effective heuristic for drones to exit stagnant equilibrium points, crucial for navigating complex 3D spaces where symmetric force configurations are common.
- The regression-based path optimizer adds a layer of global efficiency, straightening the often circuitous paths generated by potential field methods.
The framework operates effectively in the dynamic potential field generated by a moving drone formation, managing both static obstacles and inter-agent collision avoidance. The “90° escape” maneuver is particularly suitable for agile platforms like quadrotors. Future work will focus on integrating explicit communication or sensing topologies within the drone formation to improve coordination robustness, extending the method to handle non-spherical and moving obstacles, and implementing the algorithm on physical hardware for real-world validation. This refined APF approach offers a robust and computationally efficient solution for autonomous 3D drone formation navigation.