In recent years, the application of unmanned aerial vehicles (UAVs) has expanded significantly across various fields, including photography, military operations, environmental monitoring, agriculture, and transportation. However, the capabilities of a single drone are often limited, necessitating coordinated drone formation for complex tasks. For instance, in military missions, a single drone may struggle to achieve precise strikes, requiring a drone formation to provide support. Drawing inspiration from biological systems, such as the V-shaped flight formations of geese, we explore efficient drone formation control strategies that mimic natural collective behaviors. Geese rely on visual information to maintain formation, which offers insights for developing lightweight drone systems with minimal sensor payloads. Specifically, we focus on small drones equipped with monocular fisheye lenses, which provide wide field-of-view (FOV) but introduce visual blind spots. This paper addresses the challenge of drone formation control under limited visual information, ensuring that drones avoid each other’s blind zones while achieving and maintaining desired formations.
The concept of drone formation control has been extensively studied using methods like virtual structures, behavior-based approaches, and leader-follower strategies. However, these methods often rely on external sensors or GPS, increasing payload and reducing flight endurance. Our approach leverages visual data from fisheye lenses, mimicking the visual constraints observed in flying geese. The fisheye lens offers a FOV exceeding 200 degrees, yet it still has blind areas where other drones become invisible. This limitation necessitates innovative control algorithms to prevent drones from entering these blind zones, ensuring continuous communication and collision avoidance. In this work, we propose a cost-function-based control law that integrates visual constraints and formation expectations. Using gradient descent, we optimize drone trajectories to achieve stable drone formation while avoiding visual blind spots. This method reduces dependency on heavy sensors, enhancing the practicality of small drone swarms in real-world applications.
To model the visual blind zones, we consider a scenario where each drone has a limited FOV similar to geese. The blind zone is defined relative to the drone’s heading direction. Let drone j have a heading angle $\theta_j$, and define the negative heading vector as $[-\cos \theta_j, -\sin \theta_j]^T$. The relative position vector between drone i and drone j is $[x_i – x_j, y_i – y_j]^T$. The angle $\gamma_{ij}$ between these vectors is given by:
$$\cos \gamma_{ij} = f(x_i, y_i, x_j, y_j, \theta_j) = \frac{[-\cos \theta_j, -\sin \theta_j] \cdot [x_i – x_j, y_i – y_j]^T}{\|[-\cos \theta_j, -\sin \theta_j]\| \cdot \|[x_i – x_j, y_i – y_j]^T\|}$$
Simplifying, since $\|[-\cos \theta_j, -\sin \theta_j]\| = 1$, we have:
$$\cos \gamma_{ij} = \frac{-(x_i – x_j)\cos \theta_j – (y_i – y_j)\sin \theta_j}{\sqrt{(x_i – x_j)^2 + (y_i – y_j)^2}}$$
The distance between drones is $d_{ij} = \sqrt{(x_i – x_j)^2 + (y_i – y_j)^2}$. The visual blind zone is characterized by a blind angle $\alpha$ and a maximum visual range $R_s$. For drone i to be outside drone j‘s blind zone, it must satisfy:
$$\cos \gamma_{ij} \leq \cos \alpha, \quad d_{ij} > 0, \quad d_{ij} \leq R_s$$
This defines a non-blind region $D^-$ where drones can maintain visual contact. Ensuring all drones operate within these regions is crucial for effective drone formation control.
We adopt a second-order kinematic model for the drones, which is simple yet effective for planar motion. For drone i, the state variables are position $(x_i, y_i)$, heading angle $\theta_i$, speed $v_i$, and acceleration $a_i$. The dynamics are:
$$\dot{x}_i = v_i \cos \theta_i$$
$$\dot{y}_i = v_i \sin \theta_i$$
$$\dot{\theta}_i = \omega_i$$
$$\dot{v}_i = a_i$$
where $\omega_i$ is the angular velocity. This model ignores altitude and attitude variations, focusing on 2D drone formation control. To handle visual information, we need to transform coordinates between local drone frames and a global world frame. Each drone has a local frame $F_i$ with its position as origin and heading as the y-axis. The transformation from local coordinates $(s’, t’)$ in $F_i$ to global coordinates $(s, t)$ in frame $F$ involves rotation by $\theta_i$ and translation by $(x_i, y_i)$:
$$\begin{bmatrix} s \\ t \end{bmatrix} = \begin{bmatrix} \cos \theta_i & -\sin \theta_i \\ \sin \theta_i & \cos \theta_i \end{bmatrix} \begin{bmatrix} s’ \\ t’ \end{bmatrix} – \begin{bmatrix} x_i \\ y_i \end{bmatrix}$$
This transformation allows each drone to interpret visual data relative to its own perspective, essential for decentralized drone formation control.
The drone formation control strategy must satisfy two main constraints: avoiding visual blind zones and achieving a desired formation shape. For avoidance, each drone must remain in the non-blind region of all other drones. For formation, we define desired relative positions $h_i – h_j$ for drones i and j, where $h_i = [x_i^d, y_i^d]^T$ is the desired position. The actual relative position is $P_i – P_j = [x_i – x_j, y_i – y_j]^T$. The formation error is minimized when:
$$\sum_{i=1}^N \sum_{j=1}^N (h_i – h_j)^2 = \sum_{i=1}^N \sum_{j=1}^N (P_i – P_j)^2$$
To integrate these constraints, we design a cost function $J$ that penalizes deviations from desired states and entries into blind zones. For drone i, the cost function is:
$$J_i = (\theta_i – \theta_0)^2 + E$$
where $\theta_0$ is a reference heading, and $E$ is a position error term. The overall cost for the drone formation is:
$$J_{\sum} = \sum_{i=1}^N J_i$$
The term $E$ is defined using a ratio function $V_{ij}$ that measures the discrepancy between actual and desired relative positions:
$$V_{ij} = \begin{cases}
\frac{\| (P_i – P_j) – (h_i – h_j) \|^2}{\| P_i – P_j \|^2}, & \text{if } \| P_i – P_j \|^2 \in (0, \| h_i – h_j \|) \\
\frac{\| (P_i – P_j) – (h_i – h_j) \|^2}{(R_s – \| P_i – P_j \|)^2}, & \text{if } \| P_i – P_j \|^2 \in (\| h_i – h_j \|, R_s)
\end{cases}$$
To smooth the cost function near the maximum range $R_s$, we introduce a smoothing function $\rho(d_{ij})$:
$$\rho(d_{ij}) = \begin{cases}
1, & d_{ij} \in (0, r) \\
\frac{1}{2} \left(1 + \cos \left( \pi \frac{d_{ij} – r}{R_s – r} \right) \right), & d_{ij} \in [r, R_s] \\
0, & d_{ij} > R_s
\end{cases}$$
where $r$ is a threshold distance. This ensures continuous gradients for optimization.
We propose a blind-avoiding formation controller that minimizes the cost function using gradient descent. The control inputs for drone i are acceleration $u_i$ and angular velocity $\omega_i$:
$$u_i = -\left( \frac{\partial J}{\partial x_i} \cos \theta_i + \frac{\partial J}{\partial y_i} \sin \theta_i \right) – \sum_{j \in \mathcal{N}} (v_i – v_j) – (v_i – v_0)$$
$$\omega_i = -\frac{\partial J}{\partial \theta_i}$$
Here, $J = \sum_{i=1}^N \sum_{j=1}^N J_{ij}$, with $J_{ij}$ incorporating blind zone and formation costs. The term $\sum_{j \in \mathcal{N}} (v_i – v_j)$ enforces speed consensus among neighbors, and $v_0$ is a desired steady-state speed. The partial derivatives are computed from the cost function, driving the drone formation toward optimal states. For example, $J_{ij}$ can be expressed as:
$$J_{ij} = \frac{\| P_i – P_j \|^2 (\theta_i – \theta_0)^2 + \| (P_i – P_j) – (h_i – h_j) \|^2}{\| P_i – P_j \|^2 \cos \alpha + \| P_i – P_j \| \left[ (x_i – x_j) \cos \theta_j + (y_i – y_j) \sin \theta_j \right]}$$
This formulation balances avoidance and formation goals, enabling decentralized control for drone formation.
To validate our approach, we conducted numerical simulations with a drone formation of three drones. Parameters were set as blind angle $\alpha = 30^\circ$, maximum visual range $R_s = 200$ m, and sampling time $0.01$ s. Initial states are shown in Table 1.
| Drone ID | $x$ (m) | $y$ (m) | $v$ (m/s) | $\theta$ (rad) |
|---|---|---|---|---|
| 1 | -100 | -100 | 10 | $\pi/2$ |
| 2 | -60 | -60 | 20 | $\pi/2$ |
| 3 | 0 | 0 | 15 | $\pi/2$ |
The desired drone formation states are specified in Table 2.
| Drone ID | $x$ (m) | $y$ (m) | $v$ (m/s) | $\theta$ (rad) |
|---|---|---|---|---|
| 1 | -120 | -100 | 15 | $\pi/2$ |
| 2 | -60 | -50 | 15 | $\pi/2$ |
| 3 | 0 | 0 | 15 | $\pi/2$ |
Simulation results demonstrate that the drone formation successfully avoids blind zones and converges to the desired formation. The trajectories show smooth paths without collisions, and speeds align to $v_0 = 15$ m/s. Final states after convergence are listed in Table 3.
| Drone ID | $x$ (m) | $y$ (m) | $v$ (m/s) | $\theta$ (rad) |
|---|---|---|---|---|
| 1 | -100.26 | -439.55 | 15.01 | 1.571 |
| 2 | -40.68 | -389.27 | 14.97 | 1.572 |
| 3 | 20.59 | -339.73 | 14.98 | 1.574 |
The cost function decreases over time, indicating optimization progress. We also tested various formation shapes, such as V-shape and line formations, all while maintaining blind zone avoidance. This underscores the versatility of our method for different drone formation configurations.
Our drone formation control strategy offers several advantages. By relying on visual information from fisheye lenses, it reduces sensor payload and extends flight endurance. The cost-function-based approach seamlessly integrates avoidance and formation constraints, ensuring robust performance. The use of gradient descent provides a computationally efficient way to achieve optimal drone formation. However, limitations exist, such as the assumption of perfect visual data and 2D motion. Future work could extend this to 3D environments, incorporate more realistic sensor models, and explore deep learning techniques for enhanced prediction. Additionally, larger drone formations with dynamic obstacles present interesting challenges for scalability.
In conclusion, we have presented a novel method for drone formation control based on limited visual information, inspired by geese flight patterns. The mathematical modeling of visual blind zones, combined with a carefully designed cost function and gradient-based controller, enables drones to maintain formations while avoiding blind spots. Simulations confirm the feasibility and effectiveness of this approach, paving the way for practical applications in autonomous drone swarms. As drone technology evolves, such bio-inspired strategies will play a crucial role in advancing coordinated drone formation capabilities.