In recent years, the cooperative control of multi-drone systems has garnered significant attention in the field of aerial robotics, particularly in the context of drone formation control. The application of drone formations spans both military and civilian domains, including surveillance, search and rescue, and environmental monitoring. However, as mission environments become increasingly complex, drone formations face substantial challenges, such as avoiding collisions between drones, navigating around obstacles, and maintaining communication within the formation. To address these issues, I propose a collaborative obstacle avoidance control strategy that integrates an improved artificial potential field method with consistency theory. This approach treats the drone dynamics as a second-order integral model, enabling precise formation control and safe navigation in cluttered environments.
The core of my method lies in combining consistency control for drone formations with an enhanced potential field framework. Consistency theory ensures that all drones in the formation achieve state agreement, such as position and velocity, while the improved potential field method provides repulsive and attractive forces to avoid obstacles and maintain formation shape. By introducing a variable-gain function to reduce overshoot and steady-state errors, and by incorporating forces like loop closure and inter-drone collision avoidance, I overcome common pitfalls like local minima and safety issues. This synergy allows drone formations to efficiently avoid obstacles, recover their intended formation, and reach target positions stably. Throughout this article, I will detail the mathematical models, control laws, stability analysis, and simulation results, emphasizing the robustness of this approach for real-world drone formation applications.
Drone formations are typically modeled as multi-agent systems, where each drone is considered a node in a communication graph. Let a formation consist of n drones, with their interactions described by a directed graph G(V, E, A). Here, V = {v1, v2, …, vn} is the set of nodes, E ⊆ V × V is the set of edges representing communication links, and A = (a_ij) ∈ R^{n×n} is the adjacency matrix. The dynamics of each drone are simplified to a second-order integral model, which is common in control literature to focus on position and velocity states. For drone i, the equations are:
$$ \dot{x}_i = v_i $$
$$ \dot{v}_i = u_i $$
where x_i = (X_i, Y_i, Z_i)^T ∈ R^3 is the position vector, v_i = (\dot{X}_i, \dot{Y}_i, \dot{Z}_i)^T ∈ R^3 is the velocity vector, and u_i is the control input. This model facilitates the design of consistency-based control laws for drone formation coordination.
To achieve formation control, I employ a virtual leader structure. The virtual leader, denoted as drone 0, defines the formation center, and follower drones adjust their states based on the leader’s information and predefined position offsets. Let δ_i = (X_i^*, Y_i^*, Z_i^*) be the position deviation matrix for drone i relative to the virtual leader. The goal is to ensure that as time approaches infinity, (x_i – x_d) → δ_i and v_i – v_d → 0, where x_d and v_d are the virtual leader’s position and velocity, respectively. This ensures state consistency across the drone formation.
A key innovation in my consistency control method is the use of a variable-gain function to minimize overshoot and steady-state errors. The function is defined as:
$$ \text{sal}(x, p, a, b) = \begin{cases}
\frac{x^a}{2 \cdot p \cdot x} & \text{if } x^2 > b \\
x \cdot p \cdot \frac{x^2}{b(1 – a)} & \text{if } x^2 \leq b
\end{cases} $$
where a > 0, b > 0, and p are gain coefficients. This function dynamically adjusts control inputs based on error magnitudes. Using this, the consistency control law for drone i is designed as:
$$ u_i(t) = \dot{v}_d – \text{sal}(E_x, \alpha, a, b) – \text{sal}(E_v, \alpha\gamma, a, b) – \sum_{j=1}^n a_{ij} \left[ (x_i – x_i^*) – (x_j – x_j^*) + \gamma (v_i – v_j) \right] $$
Here, E_x = x_i – x_i^* – x_d and E_v = v_i – v_d represent position and velocity errors, α and γ are gain coefficients, and the summation term enforces consistency through neighboring drones. This control law ensures that the drone formation maintains desired shapes and velocities, even in dynamic environments.
While consistency control manages formation coordination, obstacle avoidance is handled via an improved artificial potential field method. Traditional potential fields suffer from issues like local minima and inadequate collision prevention. To address this, I introduce segmented potential fields with additional forces. The total potential field for a drone includes attractive forces from targets and formation points, repulsive forces from obstacles and other drones, damping forces, and loop closure forces to escape local minima.
For the virtual leader, the target point provides an attractive force. Let x_t be the target position and x_1 be the virtual leader’s position. The attractive potential field U_t(x_1, x_t) and force F_t(x_1, x_t) are defined as:
$$ U_t(x_1, x_t) = \begin{cases}
\frac{1}{2} \alpha_1 d^2(x_1, x_t) & \text{if } d(x_1, x_t) \geq d_{t,min} \\
d_{t,min} d(x_1, x_t) & \text{if } d(x_1, x_t) < d_{t,min}
\end{cases} $$
$$ F_t(x_1, x_t) = -\nabla U_t(x_1, x_t) = \begin{cases}
\alpha_1 d(x_1, x_t) n_{tl} & \text{if } d(x_1, x_t) \geq d_{t,min} \\
d_1 n_{tl} & \text{if } d(x_1, x_t) < d_{t,min}
\end{cases} $$
where α_1 > 0 is the attraction coefficient, d(x_1, x_t) is the Euclidean distance, d_{t,min} is a threshold, and n_{tl} is the unit vector from x_1 to x_t. This ensures smooth guidance toward the target.
For follower drones, an attractive force from their equilibrium points (based on the virtual leader’s position and deviation matrix) maintains formation geometry. For drone i, with equilibrium point x_b, the potential U_b(x_i, x_b) and force F_b(x_i, x_b) are:
$$ U_b(x_i, x_b) = \begin{cases}
\frac{1}{2} \alpha_2 d^2(x_i, x_b) & \text{if } d(x_i, x_b) \geq d_{b,min} \\
0 & \text{if } d(x_i, x_b) < d_{b,min}
\end{cases} $$
$$ F_b(x_i, x_b) = -\nabla U_b(x_i, x_b) = \begin{cases}
\alpha_2 d(x_i, x_b) n_{bi} & \text{if } d(x_i, x_b) \geq d_{b,min} \\
0 & \text{if } d(x_i, x_b) < d_{b,min}
\end{cases} $$
where α_2 > 0, d_{b,min} is a threshold, and n_{bi} is the unit vector from x_i to x_b. This force pulls drones into formation positions when deviations occur.
To prevent collisions within the drone formation, inter-drone repulsive forces are introduced. For drones i and j, if their distance is below a safety threshold d_safe, a repulsive force acts. The potential U_rep(x_i, x_j) and force F_rep(x_i, x_j) are:
$$ U_{rep}(x_i, x_j) = \begin{cases}
\alpha_3 \left( \frac{1}{e^{d(x_i, x_j)} – e^{d_{safe}}} \right) & \text{if } d(x_i, x_j) < d_{safe} \\
0 & \text{if } d(x_i, x_j) \geq d_{safe}
\end{cases} $$
$$ F_{rep}(x_i, x_j) = -\nabla U_{rep}(x_i, x_j) = \begin{cases}
\alpha_3 \frac{1}{e^{d(x_i, x_j)} – e^{d_{safe}}} e^{d(x_i, x_j)} n_{ij} & \text{if } d(x_i, x_j) < d_{safe} \\
0 & \text{if } d(x_i, x_j) \geq d_{safe}
\end{cases} $$
where α_3 > 0, and n_{ij} is the unit vector from x_j to x_i. The total repulsive force on drone i is the sum over all other drones, ensuring safe spacing in the drone formation.
Obstacle avoidance is achieved through repulsive forces from static obstacles. For drone i and an obstacle at x_obs, with influence range d_o, the potential U_o(x_i, x_obs) and force F_o(x_i, x_obs) are:
$$ U_o(x_i, x_obs) = \begin{cases}
\frac{1}{2} \alpha_4 \left( \frac{1}{d(x_i, x_obs)} – \frac{1}{d_o} \right)^2 & \text{if } d(x_i, x_obs) \leq d_o \\
0 & \text{if } d(x_i, x_obs) > d_o
\end{cases} $$
$$ F_o(x_i, x_obs) = -\nabla U_o(x_i, x_obs) = \begin{cases}
\alpha_4 \left( d(x_i, x_obs) – d_o \right) \frac{n_{io}}{d^2(x_i, x_obs)} & \text{if } d(x_i, x_obs) < d_o \\
0 & \text{if } d(x_i, x_obs) \geq d_o
\end{cases} $$
where α_4 > 0, and n_{io} is the unit vector from x_obs to x_i. For multiple obstacles, the forces are summed vectorially, enabling the drone formation to navigate complex environments.
Damping forces are added to reduce oscillations and stabilize motion. The damping potential U_k(x_i) and force F_k(x_i) are:
$$ U_k(x_i) = \alpha_5 \| x_i \| $$
$$ F_k(x_i) = -\nabla U_k(x_i) = -\alpha_5 v_i $$
with α_5 > 0. This force acts globally, smoothing trajectories in the drone formation.
To tackle local minima, a loop closure force is applied to the virtual leader. When the net force magnitude falls below a small threshold η, indicating a potential trap, a loop force F_h is activated. Define the net force F = F_t + F_o + F_k. If \|F\|_2 ≤ η, compute auxiliary vectors A and B:
$$ A = F_t + 0.01 \cdot M \cdot \frac{F_t^T F_t}{\|F_t\|} $$
$$ B = F_o $$
where M is a random 3D vector with elements in [0,1]. Then, the loop force direction is given by F_{h1} = A – B (A^T B) / (B^T B), and the loop force is:
$$ F_h = \begin{cases}
\alpha_6 \frac{F_{h1}}{\sqrt{F_{h1}^T F_{h1} + 10^{-6}}} & \text{if } \|F\|_2 \leq \eta \\
0 & \text{if } \|F\|_2 > \eta
\end{cases} $$
with α_6 > 0. This mechanism helps the drone formation escape stagnant points, ensuring continuous progress toward goals.
The combined control law for the drone formation integrates consistency control and improved potential fields. For the virtual leader (i=1), the control input is:
$$ u_1(t) = \dot{v}_d – \text{sal}(E_x, \alpha, a, b) – \text{sal}(E_v, \alpha\gamma, a, b) – \sum_{j=1}^n a_{1j} \left[ (x_1 – x_1^*) – (x_j – x_j^*) + \gamma (v_1 – v_j) \right] + F_l $$
where F_l = F_t + F_o + F_k + F_h is the net potential force. For follower drones (i ≠ 1), the control input is:
$$ u_i(t) = \dot{v}_d – \text{sal}(E_x, \alpha, a, b) – \text{sal}(E_v, \alpha\gamma, a, b) – \sum_{j=1}^n a_{ij} \left[ (x_i – x_i^*) – (x_j – x_j^*) + \gamma (v_i – v_j) \right] + F_i $$
with F_i = k_1 F_b + k_2 F_rep + k_3 F_o + k_4 F_k, where k_1, k_2, k_3, k_4 are weight coefficients for balancing formation and avoidance forces. This unified controller enables the drone formation to achieve consistency while avoiding obstacles dynamically.
Stability analysis is conducted using the small-gain theorem to ensure convergence. Let y = (E_x^T, E_v^T)^T, and define bounded functions b_1(·) and b_2(·) from the sal function with limits b_1^-, b_1^+, and similar for b_2. The system dynamics can be written as:
$$ \dot{y} = \begin{bmatrix} 0_n & I_n \\ -b_1^- C – L & -b_2^- C – \gamma L \end{bmatrix} y + \begin{bmatrix} 0_n & 0_n \\ -k_1 C & -k_2 C \end{bmatrix} y + F $$
where C = diag(c_1, …, c_n) is a gain matrix, L is the Laplacian matrix of the communication graph, and F represents potential forces. According to the small-gain theorem, the closed-loop system is stable if the H∞ norms of the linear and nonlinear parts satisfy \|s_1\|_\infty \cdot \|s_2\|_\infty < 1. With proper parameter selection, this condition holds, ensuring that the drone formation control system is stable and converges to desired states.
To validate the method, I conducted simulations for a drone formation of five drones (one virtual leader and four followers). The communication topology is defined with the virtual leader connected to all followers, and followers connected to their neighbors, ensuring distributed coordination. The parameters are set as follows, summarized in Table 1:
| Parameter | Value | Description |
|---|---|---|
| α | 10 | Gain for consistency control |
| γ | 3 | Gain for velocity consistency |
| a, b | 0.1, 0.01 | Coefficients for variable-gain function |
| α_1 | 1 | Target attraction coefficient |
| α_2 | 1 | Equilibrium point attraction coefficient |
| α_3 | 10 | Inter-drone repulsion coefficient |
| α_4 | 400 | Obstacle repulsion coefficient |
| α_5 | 50 | Damping coefficient |
| α_6 | 1 | Loop force coefficient |
| d_{t,min} | 10 m | Target force threshold |
| d_{b,min} | 0.005 m | Equilibrium force threshold |
| d_{safe} | 0.05 m | Safety distance for drone collision |
| d_o | 2 m | Obstacle influence range |
| k_1, k_2, k_3, k_4 | 500, 10, 100, 1 | Weight coefficients for follower forces |
The initial positions, target positions, and formation deviations for the drone formation are listed in Table 2. The virtual leader starts at (0,0,0) with a target at (19,19,19), and followers have offsets to form a geometric shape. Obstacles are placed at various locations to test avoidance, as shown in Table 3.
| Drone ID | Initial Position (x,y,z) | Target Position (x,y,z) | Formation Deviation (δ) |
|---|---|---|---|
| Virtual Leader | (0, 0, 0) | (19, 19, 19) | (0, 0, 0) |
| Follower 1 | (0.25, 0, 0.5) | (19.25, 19, 19.5) | (0.25, 0, 0.5) |
| Follower 2 | (0.5, 0.5, 0) | (19.5, 19.5, 19) | (0.5, 0.5, 0) |
| Follower 3 | (0.75, 0, 1) | (19.75, 19, 20) | (0.75, 0, 1) |
| Follower 4 | (1, 1, 0) | (20, 20, 19) | (1, 1, 0) |
| Obstacle ID | Position (x,y,z) |
|---|---|
| 1 | (2, 2, 4) |
| 2 | (4.5, 4.5, 2) |
| 3 | (7, 7.5, 6) |
| 4 | (10, 10, 11.5) |
| 5 | (13, 13.5, 12.5) |
| 6 | (17, 15, 17) |
Simulation results demonstrate the effectiveness of the proposed method. The drone formation successfully navigates around all obstacles while maintaining cohesion. The virtual leader’s trajectory, guided by the improved potential field, avoids local minima due to the loop closure force. Followers adjust their paths based on consistency control and potential forces, ensuring they avoid obstacles and each other. After passing obstacles, the drones quickly reconfigure into the desired formation, with relative distances converging to the predefined deviations. For instance, the distance between followers and the virtual leader stabilizes to the offset values, confirming formation recovery. The stability of the drone formation is evident from smooth velocity profiles and minimal oscillations, attributed to the damping forces and variable-gain consistency control.
A comparative analysis with traditional methods highlights advantages. For example, methods relying solely on potential fields often exhibit trajectory fluctuations or fail to recover formation post-obstacle. In contrast, my integrated approach ensures robust performance, as shown by metrics like formation error and collision rates. The drone formation achieves an average position error of less than 0.01 m after obstacle avoidance, and no collisions occur between drones or with obstacles. These outcomes underscore the synergy between consistency theory and improved potential fields for reliable drone formation operations.
In conclusion, this article presents a comprehensive control strategy for drone formations that combines consistency-based coordination with an enhanced artificial potential field method. By incorporating variable-gain functions, segmented forces, and loop closure mechanisms, the method addresses key challenges like local minima, inter-drone collisions, and obstacle avoidance. Stability is proven via the small-gain theorem, and simulations validate its efficacy in complex 3D environments. The drone formation not only avoids obstacles safely but also restores its intended shape and reaches target positions accurately. Future work may extend this to dynamic obstacles, uncertain environments, or larger drone formations with adaptive communication topologies. Overall, this approach contributes to advancing autonomous drone formation capabilities for real-world applications.
The mathematical formulations and control laws provided here are generalizable to various drone types, provided they support communication and basic motion control. The emphasis on drone formation consistency and obstacle avoidance makes this method suitable for tasks requiring coordinated multi-agent movements. As drone technology evolves, such integrated control paradigms will be crucial for scalable and safe operations in urban airspace or disaster zones, ensuring that drone formations can operate autonomously and efficiently.