In this article, I present a comprehensive study on the problem of ground moving target tracking using a drone formation. The increasing demand for autonomous systems in applications such as surveillance, search and rescue, and environmental monitoring has highlighted the importance of multi-drone coordination. A drone formation offers advantages over single drones in terms of robustness, adaptability, and task efficiency. However, maintaining a stable drone formation while tracking dynamic ground targets in obstacle-rich environments remains a challenging research area. I propose a novel distributed control strategy that combines virtual structure and behavior-based approaches, enabling drones to uniformly distribute on a spherical surface around a virtual or actual leader. This strategy leverages Lyapunov-based control vectors for target tracking and incorporates an obstacle avoidance mechanism using elliptic potential fields and tangent vector calculations. Through detailed simulations, I demonstrate the effectiveness of the approach in achieving smooth trajectories and reconfigurable drone formations. The article is structured to provide an in-depth analysis of the drone dynamics, formation control, tracking algorithms, and obstacle avoidance, with an emphasis on mathematical formulations and practical implementations.
The core of this work revolves around the drone formation concept, which refers to the coordinated movement of multiple unmanned aerial vehicles (UAVs) to achieve a common objective. I begin by outlining the drone formation dynamics, where each drone is modeled as a point mass with simplified equations of motion. The control inputs are derived using feedback linearization, allowing for easier design of high-level formation strategies. For a drone formation of N drones, the dynamics can be expressed as:
$$ \ddot{x}_n = u_{xn}, \quad \ddot{y}_n = u_{yn}, \quad \ddot{z}_n = u_{zn} $$
where \( (x_n, y_n, z_n) \) represents the position of the n-th drone, and \( u_{xn}, u_{yn}, u_{zn} \) are virtual acceleration inputs. These inputs relate to actual control parameters such as thrust, lift, and roll angle through nonlinear transformations, as detailed in the methodology. The drone formation control aims to guide each drone to a desired position relative to the formation center, which is either a virtual point or a physical leader drone.
To achieve a stable drone formation, I employ a virtual structure approach combined with behavior-based elements. In this strategy, drones are driven to distribute uniformly on a sphere of radius \( r_\alpha \) centered at \( (x_c, y_c, z_c) \). The control forces are designed using a Lyapunov-inspired function that ensures convergence to the spherical surface. For instance, the control component for the n-th drone is given by:
$$ f_{xn} = -k_s (x_n – x_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_\alpha^2 \right] $$
$$ f_{yn} = -k_s (y_n – y_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_\alpha^2 \right] $$
$$ f_{zn} = -k_s (z_n – z_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_\alpha^2 \right] $$
where \( k_s \) is a gain parameter. Additionally, to prevent collisions and ensure even spacing within the drone formation, a repulsive force based on Coulomb’s law is introduced between drones. The total force on each drone includes both attractive components toward the sphere and repulsive components from other drones. This combination results in a self-organizing drone formation that adapts to changes in the number of drones, forming regular polyhedral shapes such as a double tetrahedron. The following table summarizes key parameters used in the drone formation control:
| Parameter | Symbol | Typical Value | Description |
|---|---|---|---|
| Formation radius | \( r_\alpha \) | 2 m | Radius of the spherical drone formation |
| Control gain | \( k_s \) | 5 | Gain for virtual structure control |
| Repulsion constant | \( k_r \) | 0.2 | Coefficient for inter-drone repulsive forces |
| Drone mass | \( M \) | 10 kg | Mass of each drone in the formation |
| Number of drones | \( N \) | 4-5 | Size of the drone formation |
The drone formation tracking of ground targets involves guiding the entire formation to follow a moving point on the ground. I design a control law for the formation center based on Lyapunov vectors, ensuring that the drone formation maintains a circular orbit around the target at a desired radius \( r_d \). The Lyapunov function \( V_t(x, y) = (r_t^2 – r_d^2)^2 \) is used, where \( r_t = \sqrt{(x_c – x_t)^2 + (y_c – y_t)^2} \) is the distance from the formation center to the target. The control inputs for the formation center are derived as:
$$ f_{xc} = k_\alpha \left( -\frac{v_0}{r_t} \right) \left[ (x_c – x_t) \frac{r_t^2 – r_d^2}{r_t^2 + r_d^2} + (y_c – y_t) \frac{2r_t r_d}{r_t^2 + r_d^2} \right] $$
$$ f_{yc} = k_\alpha \left( -\frac{v_0}{r_t} \right) \left[ (y_c – y_t) \frac{r_t^2 – r_d^2}{r_t^2 + r_d^2} – (x_c – x_t) \frac{2r_t r_d}{r_t^2 + r_d^2} \right] $$
where \( v_0 \) is the desired relative speed, and \( k_\alpha \) is a positive constant. This control law guarantees that the drone formation converges to a stable circular motion around the target, with the derivative of the Lyapunov function being non-positive. The tracking performance is enhanced by the distributed nature of the drone formation, which allows for redundant sensing and improved target localization.
In obstacle-rich environments, the drone formation must avoid collisions while continuing to track the target. I model obstacles as rectangular prisms and approximate them with elliptical potential fields in the horizontal plane. For an obstacle centered at \( (x_o, y_o, z_o) \) with dimensions \( (\pm l_1, \pm l_2, \pm l_3) \), the minimal area ellipse encompassing it is given by:
$$ \frac{(x – x_o)^2}{2l_1^2} + \frac{(y – y_o)^2}{2l_2^2} = 1 $$
This ellipse serves as a repulsive field for the drone formation. To compute avoidance trajectories, I determine tangent vectors from each drone’s position to the ellipse. The tangent points on the ellipse can be derived analytically, and the avoidance direction \( \chi_n \) for the n-th drone is calculated as the arctangent of the slope between the drone and the tangent point. The drone formation then chooses between clockwise or counterclockwise rotation based on a comparison between \( \chi_n \) and the angle \( \rho_n \) from the obstacle center to the drone. Specifically, if \( \rho_n \leq \chi_n \), the drone formation opts for counterclockwise avoidance; otherwise, it chooses clockwise. This strategy yields smooth paths that minimize disruption to the drone formation structure.
The obstacle avoidance algorithm is integrated into the overall drone formation control. Each drone adjusts its velocity vector to follow the tangent direction until it clears the obstacle, after which it rejoins the formation. The control inputs during avoidance are modified to include additional terms from the potential field, ensuring that the drone formation remains cohesive. The following table outlines the steps in the drone formation obstacle avoidance process:
| Step | Action | Mathematical Expression |
|---|---|---|
| 1 | Detect obstacle and compute ellipse parameters | \( l_1, l_2 \) from obstacle dimensions |
| 2 | Calculate tangent points from drone position | Solve for \( (x_1, y_1) \) and \( (x_2, y_2) \) using ellipse equation |
| 3 | Determine avoidance direction \( \chi_n \) | \( \chi_n = \arctan\left( \frac{y_u – y_{1,2}}{x_u – x_{1,2}} \right) \) |
| 4 | Compare with \( \rho_n \) to choose rotation direction | If \( \rho_n \leq \chi_n \), counterclockwise; else clockwise |
| 5 | Adjust drone velocity vector | Incorporate tangent vector into control inputs |
| 6 | Reconfigure drone formation after avoidance | Reapply virtual structure control forces |
To validate the proposed drone formation strategies, I conduct extensive simulations under various scenarios. The first simulation involves a drone formation of four drones tracking a stationary ground target. Initial drone positions are set to \( p_1 = (2, 2) \), \( p_2 = (2, 1) \), \( p_3 = (1, 2) \), \( p_4 = (2, 0) \), with the formation center starting at \( p_c = (0, 0) \). Parameters include \( M = 10 \), \( k_s = 5 \), \( k_r = 5 \), \( r_\alpha = 2 \), and \( r_d = 1.5 \). The drone formation successfully converges to a spherical configuration and orbits the target at the desired radius, demonstrating stable tracking. The second simulation introduces a moving target with velocity \( v_t = (2, -2) \). The drone formation adapts its trajectory to follow the target, maintaining the formation structure while achieving a pursuit curve. The control law ensures that the drone formation’s speed exceeds the target’s, enabling effective tracking.
In a third simulation, I incorporate obstacles to test the avoidance capabilities of the drone formation. Five drones are initialized at positions \( p_1 = (0.5, 1, 2.5) \), \( p_2 = (3.5, 3.5, 7) \), \( p_3 = (2, -1, 6.5) \), \( p_4 = (2, 3, 6.5) \), with a leader drone at \( p_L = (1, 1, 5.5) \). An obstacle is placed at \( p_o = (20 \pm 2, 20 \pm 3, 10 \pm 3) \). The drone formation navigates around the obstacle by selecting appropriate tangent directions, and after passing, it reconstitutes the spherical formation. The simulation shows that the drone formation can handle complex environments without collapsing, highlighting the robustness of the approach. A fourth simulation uses a virtual leader for the drone formation, with an obstacle at \( p_o = (20 \pm 3, 6 \pm 2) \). The results indicate that a virtual center allows for more flexible obstacle avoidance, as drones can independently adjust their paths while still coordinating as a drone formation.
The simulations confirm that the drone formation strategies are effective in both tracking and obstacle avoidance. Key performance metrics include formation stability, tracking error, and avoidance smoothness. The use of Lyapunov-based control ensures convergence, while the elliptic potential field method eliminates local minima issues common in traditional potential field approaches. The drone formation’s ability to self-reconfigure after disturbances or member changes is a significant advantage, making it suitable for real-world applications where environmental conditions are dynamic.
In conclusion, I have presented a novel distributed control framework for drone formation tracking of ground moving targets. The integration of virtual structure and behavior-based methods enables a flexible and scalable drone formation that can adapt to various mission requirements. The Lyapunov vector design provides robust tracking control, and the elliptic potential field approach facilitates smooth obstacle avoidance. Future work will focus on extending the drone formation to larger scales, addressing communication constraints among drones, and incorporating heterogeneous drones with different capabilities. Additionally, real-time implementation and testing in physical environments will be pursued to further validate the drone formation algorithms. The potential applications of this drone formation technology are vast, ranging from military surveillance to civilian search and rescue operations, underscoring the importance of continued research in this field.
Throughout this article, the concept of drone formation has been emphasized as a key enabler for advanced multi-drone systems. By leveraging mathematical tools such as Lyapunov stability and potential fields, I have demonstrated how a drone formation can achieve complex tasks with high reliability. The tables and formulas provided summarize the core components, offering a clear roadmap for researchers and practitioners interested in drone formation control. As drone technology evolves, the development of intelligent drone formation strategies will play a crucial role in unlocking the full potential of unmanned aerial vehicles.