In recent years, unmanned aerial vehicles (UAVs), commonly known as drones, have gained significant attention due to their versatility in both civilian and military applications. Their advantages, such as adaptability, safety, cost-effectiveness, and stealth, make them ideal for tasks ranging from surveillance and delivery to disaster response. However, the real power of drones is realized when multiple units operate together in a coordinated manner, forming what is known as a drone formation. A drone formation enhances efficiency, coverage, and redundancy, but it also introduces complex challenges, particularly in avoiding collisions between drones during flight. Ensuring safety in a drone formation is paramount, as any collision can lead to catastrophic failures and mission compromise. This paper addresses the critical issue of anti-collision control within drone formations, leveraging a velocity vector field approach to enable autonomous collision avoidance while maintaining formation integrity.
The concept of drone formation involves multiple drones flying in a predefined pattern, such as a leader-follower structure, where one drone acts as the leader and others as followers. This structure simplifies control but requires robust mechanisms to prevent inter-drone collisions, especially during dynamic maneuvers or in congested environments. Traditional methods for drone formation control include leader-follower approaches, behavior-based methods, virtual structure techniques, and graph theory-based strategies. Among these, the leader-follower method is widely adopted due to its simplicity, controllability, and flexibility. In this method, the leader drone follows a pre-planned trajectory, while follower drones adjust their states—such as velocity, heading, and altitude—based on the leader’s movements. However, this approach often lacks inherent collision avoidance capabilities, necessitating supplementary strategies to ensure safe operations.
To tackle collision avoidance in drone formation, various techniques have been proposed, including artificial potential fields, velocity obstacles, and model predictive control. Artificial potential field methods, introduced by Khatib, model the environment as a potential field where goals attract and obstacles repel drones. While effective, these methods can suffer from local minima and oscillatory behaviors. Velocity vector field methods build upon this idea by using velocity fields instead of force fields, offering smoother and faster responses. In this paper, we propose an anti-collision control method for drone formation based on a velocity vector field, integrated with a bidirectional communication topology for information exchange. Our approach designs traction and threat-avoidance velocity components, dynamically adjusted via priority-based parameters, to ensure that each drone in the formation avoids others while adhering to the desired trajectory. We validate the method through simulations in a C#-based platform, demonstrating its feasibility and effectiveness.
The core of our method lies in modeling the drone formation as a dynamic system influenced by velocity fields. Each drone generates a velocity field that affects others within a defined threat region, enabling proactive collision avoidance. The traction velocity guides drones toward their targets (e.g., the leader toward waypoints, followers toward ideal formation positions), while the threat-avoidance velocity pushes drones away from potential collisions. By combining these velocities with adjustable parameters based on drone priorities, we achieve a balanced control that maintains formation cohesion without sacrificing safety. This paper is structured as follows: First, we explain the fundamental concepts of velocity vector fields and information interaction topologies. Then, we detail the mathematical modeling of the velocity fields. Next, we present simulation results and analysis. Finally, we conclude with insights and future directions.
Fundamental Concepts for Drone Formation Control
In drone formation control, understanding the underlying principles of velocity vector fields and communication topologies is essential. These concepts form the basis for our anti-collision strategy, enabling drones to navigate collaboratively while avoiding conflicts.
Velocity Vector Field Principle
The velocity vector field method is an enhancement of the artificial potential field approach. Instead of using forces to drive drones, it employs velocity vectors derived from field sources. In a drone formation, field sources include target waypoints (for attraction) and other drones (for repulsion). Each drone experiences a velocity vector at its location, which is the vector sum of contributions from all relevant sources. This resultant velocity guides the drone’s motion, allowing it to steer clear of obstacles while progressing toward its goal.
Mathematically, a velocity vector field can be represented as a function that assigns a velocity vector to every point in space. For a drone at position $\mathbf{p} = (x, y)$ in a 2D plane (we extend to 3D later), the velocity $\mathbf{V}(\mathbf{p})$ is given by:
$$
\mathbf{V}(\mathbf{p}) = \mathbf{V}_t(\mathbf{p}) + \mathbf{V}_a(\mathbf{p})
$$
where $\mathbf{V}_t(\mathbf{p})$ is the traction velocity from target sources, and $\mathbf{V}_a(\mathbf{p})$ is the avoidance velocity from threat sources. In our drone formation context, the leader drone has a traction velocity toward its next waypoint, while follower drones have traction velocities toward their ideal positions relative to the leader. Threat sources are other drones within a certain radius, each generating a repulsive velocity field to prevent collisions.
The key advantage of using velocity fields over force fields is the direct control over drone speed and direction, leading to more predictable and stable trajectories. This is crucial in drone formation, where sudden accelerations or decelerations can disrupt the formation and increase collision risks. The velocity vector field method ensures smooth adjustments, as drones continuously adapt their velocities based on the evolving field.
To illustrate, consider a simple scenario with two drones: a leader and a follower. The leader’s traction velocity pulls it toward a waypoint, while the follower’s traction velocity pulls it toward a position behind the leader. If the follower gets too close to the leader, the leader’s threat field generates an avoidance velocity for the follower, pushing it away. Simultaneously, the follower may generate a threat field affecting the leader, but in our priority-based system, the leader may have higher priority, so its response is modulated. This dynamic interaction forms the essence of our anti-collision control.
Information Interaction Topology for Drone Formation
Effective communication is vital for coordinated drone formation. Drones must exchange state information—such as position, velocity, and heading—to maintain formation and avoid collisions. The information interaction topology defines the communication links between drones, determining how data flows within the formation. In our approach, we use a bidirectional communication topology between the leader and each follower, enabling two-way data exchange.
This topology is depicted as a graph where nodes represent drones and edges represent communication links. For a formation with one leader and two followers, the leader communicates directly with each follower, and followers can indirectly infer each other’s states via the leader. This structure balances simplicity and robustness, as it minimizes direct inter-follower communication while ensuring all drones have sufficient information for collision avoidance.
The communication protocol involves periodic broadcasting of state data. The leader sends its GPS coordinates, altitude, speed, and heading to followers, while followers reply with their own states. This allows the leader to monitor followers and adjust its path if necessary, and followers to update their target positions based on the leader’s movement. Additionally, to mitigate error propagation in leader-follower chains—where position errors accumulate in downstream followers—error correction data is also shared. This helps followers correct their relative positions, maintaining formation accuracy.
In practice, the topology must account for communication delays, packet loss, and bandwidth limitations. However, for our simulation, we assume ideal communication to focus on the control algorithm. The bidirectional topology enhances safety by enabling proactive threat detection: if a follower enters the leader’s threat zone, both drones can adjust simultaneously, reducing collision probability.
Table 1 summarizes the key parameters of our information interaction topology for a three-drone formation.
| Parameter | Description | Value/Example |
|---|---|---|
| Topology Type | Bidirectional leader-follower | Leader ↔ Follower1, Leader ↔ Follower2 |
| Communication Frequency | Rate of data exchange | 10 Hz (assumed) |
| Data Shared | Position, velocity, heading, error | GPS coordinates, speed, yaw |
| Indirect Communication | Followers via leader | Follower1 and Follower2 infer states through leader |
| Error Correction | Mitigates error propagation | Position offset messages |
This topology supports our velocity vector field method by providing real-time data for field computation. Each drone calculates its velocity vector based on the latest states of other drones, ensuring responsive anti-collision behavior.
Mathematical Modeling of Velocity Vector Field for Drone Formation
To implement anti-collision control, we develop detailed mathematical models for the velocity vector field components. These models define how traction, threat-avoidance, and leader-avoidance velocities are computed and combined. We consider a 2D plane for simplicity, but the models can be extended to 3D by adding altitude components.
Traction Velocity Model
The traction velocity guides a drone toward its target, which varies based on drone role. For the leader drone, the target is a waypoint; for followers, the target is an ideal formation position relative to the leader.
Leader Traction Velocity: Let $\mathbf{p}_l = (x_l, y_l)$ be the leader’s current position, and $\mathbf{p}_t = (x_t, y_t)$ be the target waypoint. The distance between them is $d_1 = \|\mathbf{p}_t – \mathbf{p}_l\| = \sqrt{(x_t – x_l)^2 + (y_t – y_l)^2}$. The traction velocity $\mathbf{V}_{TL}(t)$ is directed from $\mathbf{p}_l$ to $\mathbf{p}_t$, with magnitude adjusted based on $d_1$ to ensure smooth approach:
$$
\mathbf{V}_{TL}(t) =
\begin{cases}
k_1 \frac{S_m^2}{S_1 d_1} (\mathbf{p}_t – \mathbf{p}_l), & \text{if } d_1 \in (0, S_1] \\
k_1 \frac{S_m^2}{d_1^3} (\mathbf{p}_t – \mathbf{p}_l), & \text{if } d_1 \in (S_1, S_m] \\
k_1 \frac{1}{d_1} (\mathbf{p}_t – \mathbf{p}_l), & \text{if } d_1 \in (S_m, +\infty)
\end{cases}
$$
where $k_1$ is a tunable parameter dependent on mission and environment, $S_m$ is the maximum detection distance for the waypoint, and $S_1$ is a threshold less than $S_m$. This piecewise function ensures that the velocity is high when far away, reduces as the drone approaches, and becomes gentle near the target to avoid overshoot. In our simulation, we set $k_1 = 4$, $S_1 = 100$ m, and $S_m = 300$ m based on typical drone dynamics.
Follower Traction Velocity: For a follower drone at position $\mathbf{p}_f = (x_f, y_f)$ with ideal formation position $\mathbf{p}_{id} = (x_{id}, y_{id})$ (computed from leader’s position and formation geometry), the distance is $d_2 = \|\mathbf{p}_{id} – \mathbf{p}_f\|$. The traction velocity $\mathbf{V}_{TF}(t)$ is:
$$
\mathbf{V}_{TF}(t) =
\begin{cases}
k_2 \frac{1}{d_2} (\mathbf{p}_{id} – \mathbf{p}_f), & \text{if } d_2 \in (0, d_{Tmax}] \\
\mathbf{V}_{TFmax}, & \text{if } d_2 \in (d_{Tmax}, +\infty)
\end{cases}
$$
where $k_2$ is a tunable parameter, $d_{Tmax} = 50$ m is a limit to prevent excessive speed, and $\mathbf{V}_{TFmax}$ is the maximum traction velocity when $d_2 = d_{Tmax}$. We set $k_2 = 4$ for consistency. This model ensures followers converge to their ideal positions without oscillating.
Table 2 summarizes the parameters for traction velocity models in drone formation.
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Leader traction gain | $k_1$ | 4 | Adjusts leader’s attraction to waypoint |
| Follower traction gain | $k_2$ | 4 | Adjusts follower’s attraction to ideal position |
| Detection threshold | $S_1$ | 100 m | Distance for velocity transition in leader |
| Max detection distance | $S_m$ | 300 m | Leader’s maximum waypoint sensing range |
| Max traction distance | $d_{Tmax}$ | 50 m | Distance for follower’s maximum traction speed |
Threat-Avoidance Velocity Model
Threat-avoidance velocity prevents collisions by repelling drones from each other. Each drone is modeled as a threat source with a circular region of influence. The region has two parts: a警戒区域 (warning zone) of radius $b$ (e.g., twice the wingspan) where collision is imminent, and a threat region of radius $R_{max}$ where avoidance is active. If another drone enters this threat region, it experiences a repulsive velocity.
For drone $i$ at position $\mathbf{p}_i = (x_i, y_i)$ and a nearby drone $c$ at position $\mathbf{p}_c = (x_c, y_c)$, the distance is $d_3 = \|\mathbf{p}_c – \mathbf{p}_i\|$. The avoidance velocity $\mathbf{V}^i_p(t)$ exerted by drone $i$ on drone $c$ is:
$$
\mathbf{V}^i_p(t) =
\begin{cases}
\frac{\omega_{pi}}{d_3} (\mathbf{p}_c – \mathbf{p}_i), & \text{if } d_3 \in (0, R_{max}] \\
\mathbf{0}, & \text{if } d_3 \in (R_{max}, +\infty)
\end{cases}
$$
where $\omega_{pi}$ is a control parameter that varies with $d_3$ and drone priority:
$$
\omega_{pi} =
\begin{cases}
k_3 \left( \frac{R_{max}}{b_1} \right)^2, & \text{if } d_3 \in (0, b_1] \\
k_3 \left( \frac{R_{max}}{d_3} \right)^2, & \text{if } d_3 \in (b_1, R_{max}] \\
0, & \text{if } d_3 \in (R_{max}, +\infty)
\end{cases}
$$
Here, $k_3$ is a tunable parameter related to the priority of the affected drone (higher priority means larger $k_3$), $R_{max}$ is the threat radius (set to 80 m in simulation), and $b_1$ is the warning radius (e.g., 20 m based on wingspan). The velocity direction is from drone $i$ to drone $c$, pushing them apart.
Additionally, a guidance velocity $\mathbf{V}^i_g(t)$ is introduced to steer the affected drone around the threat. It is perpendicular to $\mathbf{V}^i_p(t)$ and oriented toward the traction velocity direction. If rotating $\mathbf{V}^i_p(t)$ by $-\pi/2$ aligns with the traction velocity, then:
$$
\mathbf{V}^i_g(t) = \mathbf{R}(-\pi/2) \cdot \mathbf{V}^i_p(t)
$$
otherwise:
$$
\mathbf{V}^i_g(t) = \mathbf{R}(\pi/2) \cdot \mathbf{V}^i_p(t)
$$
where $\mathbf{R}(\theta)$ is the rotation matrix:
$$
\mathbf{R}(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}
$$
The total threat-avoidance velocity for a drone due to multiple threats is the sum of avoidance and guidance components from all threatening drones:
$$
\mathbf{V}_a(t) = \sum_{i \in \text{threats}} \left( \mathbf{V}^i_p(t) + \mathbf{V}^i_g(t) \right)
$$
This combined velocity helps drones navigate away from threats while still progressing toward their targets.
Leader-Avoidance Velocity Model
In a leader-follower drone formation, followers must give way to the leader due to its higher priority. When a follower enters the leader’s threat region, it experiences an additional leader-avoidance velocity. Let the leader be at $\mathbf{p}_L = (x_L, y_L)$ and a follower at $\mathbf{p}_F = (x_F, y_F)$, with distance $d_4 = \|\mathbf{p}_F – \mathbf{p}_L\|$. The leader-avoidance velocity $\mathbf{V}_{LF}(t)$ is:
$$
\mathbf{V}_{LF}(t) = \frac{\omega_n}{d_4} (\mathbf{p}_F – \mathbf{p}_L)
$$
where $\omega_n$ is a control parameter:
$$
\omega_n =
\begin{cases}
k_4 \left( \frac{R_{max}}{b_2} \right)^2, & \text{if } d_4 \in (0, b_2] \\
k_4 \left( \frac{R_{max}}{d_4} \right)^2, & \text{if } d_4 \in (b_2, R_{max}] \\
0, & \text{if } d_4 \in (R_{max}, +\infty)
\end{cases}
$$
Here, $k_4$ is a tunable parameter for follower avoidance, $R_{max}$ is the leader’s threat radius (same as before), and $b_2$ is the leader’s warning radius. This velocity is added to the follower’s threat-avoidance velocity, ensuring it actively steers clear of the leader.
Total Velocity Model
The overall velocity for each drone in the formation is the vector sum of traction, threat-avoidance, and leader-avoidance components. For the leader drone, which only avoids followers when necessary, the total velocity $\mathbf{V}_L(t)$ is:
$$
\mathbf{V}_L(t) = \mathbf{V}_{TL}(t) + \sum_{i \in \text{followers}} \left( \mathbf{V}^i_p(t) + \mathbf{V}^i_g(t) \right)
$$
For a follower drone, the total velocity $\mathbf{V}_F(t)$ is:
$$
\mathbf{V}_F(t) = \mathbf{V}_{TF}(t) + \sum_{i \in \text{threats}} \left( \mathbf{V}^i_p(t) + \mathbf{V}^i_g(t) \right) + \mathbf{V}_{LF}(t)
$$
The drones’ trajectories are then computed by integrating the velocity over time:
$$
\mathbf{S}(t) = \int_{t_0}^{t} \mathbf{V}(t) \, dt
$$
where $\mathbf{S}(t)$ is the path. In discrete time for simulation, we use:
$$
\mathbf{p}(t+\Delta t) = \mathbf{p}(t) + \mathbf{V}(t) \Delta t
$$
with a small time step $\Delta t$. This model ensures that each drone in the drone formation dynamically adjusts its flight path to avoid collisions while maintaining formation structure.
Table 3 provides a comprehensive list of all parameters used in the velocity vector field models for drone formation.
| Component | Parameter | Symbol | Typical Value | Role in Drone Formation |
|---|---|---|---|---|
| Traction | Leader gain | $k_1$ | 4 | Controls leader’s attraction to waypoints |
| Follower gain | $k_2$ | 4 | Controls follower’s attraction to ideal positions | |
| Max detection | $S_m$ | 300 m | Limits leader’s waypoint sensing range | |
| Threat-Avoidance | Avoidance gain | $k_3$ | Varies by priority | Adjusts repulsion strength based on drone priority |
| Threat radius | $R_{max}$ | 80 m | Defines zone where avoidance is active | |
| Warning radius | $b_1, b_2$ | 20 m | Inner zone for imminent collision warning | |
| Rotation angle | $\theta$ | $\pm \pi/2$ | Sets guidance velocity direction | |
| Leader-Avoidance | Avoidance gain | $k_4$ | Tunable | Sets follower’s avoidance of leader |
| Priority factor | Implicit in $k_3, k_4$ | Higher for leader | Ensures leader has right of way in drone formation | |
| General | Time step | $\Delta t$ | 0.1 s | Discrete integration interval for simulation |
Simulation Validation and Analysis for Drone Formation
To validate our anti-collision control method, we developed a simulation platform in C# that models a drone formation with one leader and two followers. The platform implements the velocity vector field models and bidirectional communication topology, allowing us to test various scenarios and measure performance metrics such as inter-drone distances and collision occurrences.
Simulation Setup
The simulation environment is a 2D Cartesian coordinate system, with drones represented as points moving according to the velocity equations. We extend to 3D conceptually by adding altitude, but for simplicity, we focus on planar motion. The initial formation is an isosceles triangle: the leader at position (0, 0), follower 1 at (-100, 0), and follower 2 at (100, 0), so initial distances are 100 m from leader to each follower, and 140 m between followers. All drones have a threat radius $R_{max} = 80$ m and warning radius $b = 20$ m. The leader’s target waypoint is set along a straight-line path, and followers’ ideal positions are computed based on the leader’s current position and desired formation geometry (e.g., triangle, line).
The simulation runs at a time step of $\Delta t = 0.1$ s, with communication updates every step. We consider two key test cases: (1) formation transformation where followers are directed to the same target point, and (2) line formation where followers align in front and behind the leader. Each case is tested with and without the anti-collision control method to highlight its impact.
Results and Discussion
Case 1: Formation Transformation to Overlapping Targets. In this scenario, the drone formation initially flies in a triangle. At time $t = 113$ s, we send a command to change the formation by setting both followers’ ideal positions to the same point (e.g., directly behind the leader). Without anti-collision control, the followers immediately move toward that point, causing their distance to decrease rapidly. As shown in Figure 5 (simulated plot), the inter-follower distance drops to zero by around $t = 130$ s, indicating a collision. This demonstrates the inherent risk in drone formation when collision avoidance is not integrated.
With our anti-collision control enabled, the same command triggers a different response. The followers still move toward the target point, but as they approach each other, their threat-avoidance velocities activate. The distance between followers decreases but stabilizes at approximately 66 m, well above the warning radius. The minimum distance observed is 61 m at $t = 138$ s, after which it oscillates slightly due to velocity adjustments. No collision occurs, and the followers eventually settle near the target point while maintaining a safe separation. This confirms that our method effectively prevents collisions even when targets overlap, a common issue in dynamic drone formation reconfiguration.
Case 2: Line Formation Maneuver. Here, the drone formation transitions from a triangle to a vertical line, with follower 1 moving to a position behind the leader and follower 2 moving to a position in front. Without anti-collision control, follower 2 accelerates toward the leader’s front, and the distance between them shrinks. At $t = 210$ s, it reaches zero, resulting in a collision. This highlights how aggressive maneuvers in drone formation can lead to accidents if not properly managed.
With anti-collision control, the transition proceeds safely. As follower 2 approaches the leader, the leader-avoidance velocity $\mathbf{V}_{LF}(t)$ and threat-avoidance velocities from both leader and follower 1 push it away. The leader-follower distance reaches a minimum of 43 m at $t = 186$ s, then increases as follower 2 adjusts its path. Eventually, all drones achieve the line formation without collisions. This shows that our priority-based system allows followers to yield to the leader, ensuring safe reorganization of the drone formation.
Table 4 summarizes the key metrics from these simulations, illustrating the effectiveness of our anti-collision control in drone formation.
| Scenario | Control Method | Min Inter-Drone Distance | Collision Occurrence | Formation Achieved |
|---|---|---|---|---|
| Overlapping Targets | Without anti-collision | 0 m (collision) | Yes | No, due to crash |
| Overlapping Targets | With anti-collision | 61 m | No | Yes, with safe spacing |
| Line Formation | Without anti-collision | 0 m (collision) | Yes | No, due to crash |
| Line Formation | With anti-collision | 43 m | No | Yes, successfully |
These results validate that our velocity vector field method enables robust anti-collision control for drone formation. The dynamic adjustment of velocity components based on real-time distances and priorities ensures that drones can navigate complex maneuvers while avoiding conflicts. The simulation also reveals that the method is computationally efficient, as the velocity calculations are straightforward and scale linearly with the number of drones. This makes it suitable for real-time applications in drone formation systems.
Further Analysis and Sensitivity
We conducted additional simulations to explore the sensitivity of our method to parameter choices. For instance, varying $k_3$ and $k_4$ affects how aggressively drones avoid each other. Higher values lead to larger avoidance velocities, which can increase safety margins but may cause excessive deviations from the desired path. We found that $k_3 = 2$ for followers and $k_4 = 3$ for leader avoidance provided a good balance for our drone formation. Similarly, changing $R_{max}$ influences the onset of avoidance behavior; a larger $R_{max}$ promotes earlier reactions but may unnecessarily restrict formation tightness. Our choice of 80 m aligns with typical drone speeds and reaction times.
Another aspect is the impact of communication delays. In real-world drone formation, data exchange may suffer from latency. We modeled delays of up to 0.5 s and observed that the system remains stable, though collisions become more likely if delays exceed 1 s. This underscores the importance of low-latency communication for safe drone formation operations.
Overall, the simulations confirm that our method is feasible and effective for anti-collision control in drone formation. It complements existing leader-follower strategies by adding a reactive layer that autonomously handles threat avoidance without requiring centralized control.
Conclusion and Future Work
In this paper, we have presented an anti-collision control method for drone formation based on a velocity vector field approach. By integrating traction and threat-avoidance velocities with a bidirectional communication topology, we enable drones to autonomously avoid collisions while maintaining formation integrity. The mathematical models provide a framework for computing velocity components dynamically, with tunable parameters to prioritize safety and performance. Simulation results demonstrate that the method effectively prevents collisions during formation transformations and maneuvers, validating its practicality for real-world drone formation applications.
The key contributions of this work include: (1) a comprehensive velocity vector field model tailored for drone formation, incorporating leader-follower dynamics; (2) a priority-based parameter adjustment mechanism that ensures higher-priority drones (e.g., the leader) are given right of way; and (3) a simulation platform that verifies the method under various scenarios. Our approach addresses a critical gap in drone formation control by adding an embedded anti-collision layer, which enhances safety without compromising formation objectives.
For future work, we plan to extend the method in several directions. First, we will incorporate 3D considerations, such as altitude control and terrain avoidance, to handle more realistic environments. Second, we will test the method with larger drone formations (e.g., 10+ drones) to assess scalability and potential swarm behaviors. Third, we will integrate machine learning techniques to optimize the tunable parameters in real-time based on environmental conditions and mission requirements. Finally, we aim to implement the algorithm on physical drone hardware and conduct field trials, addressing practical challenges like sensor noise and wind disturbances.
In conclusion, anti-collision control is essential for the safe operation of drone formation systems. Our velocity vector field method offers a promising solution that balances simplicity, efficiency, and robustness. As drone technologies advance, such methods will play a crucial role in enabling complex collaborative missions, from aerial shows to search-and-rescue operations. By continuing to refine these approaches, we can unlock the full potential of drone formation for a wide range of applications.