In recent years, the rapid advancement of technology has led to widespread applications of unmanned aerial vehicles (UAVs), or drones, in both military and civilian domains. The concept of drone formation, where multiple drones operate in a coordinated manner, leverages scale effects to enhance efficiency and performance in tasks such as surveillance, logistics, and environmental monitoring. However, maintaining precise formation during flight is critical for system accuracy and safety. Passive positioning, which relies on方位角 measurements without active signal transmission, offers a promising approach for adjusting drone positions in a formation. This paper addresses the problem of方位偏差 in drone formation飞行, focusing on circular formations where drones are evenly distributed on a circle with one drone at the center. The primary challenge lies in correcting deviations from ideal positions using only方位角 information from信标 drones.
The importance of drone formation maintenance cannot be overstated, as even slight deviations can disrupt协同作业 and lead to failures. Traditional methods often involve active sensing or GPS, but these may be susceptible to interference or unavailable in certain environments. Hence, passive positioning based on纯方位 measurements has gained attention. In this context, we propose a novel approach combining a drift circle model with a simulated annealing algorithm to optimize方位调整 strategies. The drift circle model quantifies deviations in terms of radius and angle, while simulated annealing provides a robust optimization framework to iteratively adjust drone positions towards their ideal locations. This study aims to contribute to the field by offering a scalable and efficient solution for drone formation control.
To set the stage, consider a circular drone formation consisting of Z drones, with one drone, denoted as FY00, positioned at the center, and the remaining Z-1 drones uniformly distributed on a circle of radius R. Each drone has a fixed identifier, and the relative方位 relationships among them remain constant during flight. We assume ideal conditions: no external干扰, real-time signal reception, and small deviations between actual and ideal positions. The coordinate system is defined with FY00 at the origin (0,0), and the line connecting FY00 and an arbitrary drone X1 as the positive x-axis. For any receiving drone Y, its position can be定位 using方位角 measurements from two transmitting drones X1 and X2. Let α1 and α2 be the方位角 from Y to X1 and X2, respectively. These angles are related to the central angles β1 and β2, as shown in the geometry:
$$ \alpha_i = \frac{180^\circ – \beta_i}{2}, \quad i=1,2 $$
where βi depends on the angular separation between drones on the circle:
$$ \beta_i = \frac{360^\circ}{Z-1} \times |X_i – Y| $$
Here, |Xi – Y| represents the absolute difference in indices, assuming a circular arrangement. The pair (α1, α2) uniquely determines Y’s position in an ideal formation. However, in practice, deviations occur due to factors like wind or sensor errors. We define the radius deviation σ and angle deviation θ as follows:
$$ \sigma = |r – r’| $$
$$ \theta = |\beta_1 – \beta_1’| $$
where r and r’ are the ideal and actual radial distances from the center, and β1 and β1′ are the ideal and actual central angles. The goal is to minimize σ and θ through controlled adjustments.
The core of our approach is the drift circle model, which visualizes deviations in a geometric manner. For a given set of three points—two transmitting drones X1, X2 and the ideal receiving drone Y—we can construct a circle passing through them. Similarly, using the actual position Y’ instead of Y, we obtain another circle. The centers of these circles, O and O’, represent the “ideal” and “actual” reference points, respectively. The distance between O and O’ quantifies the overall deviation in the drone formation. Mathematically, given coordinates (x1, y1), (x2, y2), (x3, y3) for X1, X2, Y, the center O = (x, y) can be derived using circle equations:
$$ x = \frac{(x_1^2 + y_1^2)(y_2 – y_3) + (x_2^2 + y_2^2)(y_3 – y_1) + (x_3^2 + y_3^2)(y_1 – y_2)}{2(x_1(y_2 – y_3) – y_1(x_2 – x_3) + x_2 y_3 – x_3 y_2)} $$
$$ y = \frac{(x_1^2 + y_1^2)(x_3 – x_2) + (x_2^2 + y_2^2)(x_1 – x_3) + (x_3^2 + y_3^2)(x_2 – x_1)}{2(x_1(y_2 – y_3) – y_1(x_2 – x_3) + x_2 y_3 – x_3 y_2)} $$
Similarly, replacing Y with Y’ yields O’ = (x’, y’). The Euclidean distance OO’ serves as a metric for调整 effectiveness. By designing control strategies to minimize OO’, we can ensure that the drone formation remains close to its ideal configuration.
To optimize the adjustment process, we employ the simulated annealing algorithm, a probabilistic technique inspired by thermodynamic cooling. This algorithm is well-suited for nonlinear problems like drone formation control due to its ability to escape local minima and converge to全局最优解. The energy function E, which we aim to minimize, is defined based on the deviations in drone positions:
$$ E = \sqrt{ (E_{n_X})^2 + (E_{n_Y})^2 } $$
where n denotes the iteration number, and En_X and En_Y are the Euclidean distances between actual and ideal positions for drones X and Y, respectively:
$$ E_{n_X} = \| X_{\text{real}} – X_{\text{ideal}} \| $$
$$ E_{n_Y} = \| Y_{\text{real}} – Y_{\text{ideal}} \| $$
In each iteration, drones are moved along specific trajectories towards their ideal positions with a step size. The simulated annealing process starts with an initial temperature T0 and iteratively updates the state. Let f(X) represent the current state (drone positions), and f(X+1) the state after a potential move. The Metropolis criterion determines whether to accept a new state:
$$ P(f(X+1) | f(X)) = \begin{cases} 1 & \text{if } E(f(X+1)) \leq E(f(X)) \\ \exp\left( -\frac{E(f(X+1)) – E(f(X))}{K T_n} \right) & \text{otherwise} \end{cases} $$
where K is the Boltzmann constant (often set to 1 for simplicity), and Tn is the current temperature. The temperature is gradually reduced according to a cooling schedule:
$$ T_{n+1} = \lambda \times T_n $$
with λ being the退火系数 between 0 and 1. This process continues until a termination condition is met, such as reaching a minimum temperature or sufficient convergence. The algorithm ensures that the drone formation gradually adjusts to minimize deviations, maintaining a stable formation throughout the flight.
For simulation, we implemented the model in MATLAB to validate its effectiveness. We considered a circular drone formation with 10 drones: one at the center and 9 evenly spaced on a circle of radius 100 units. Initial deviations were randomly generated with radius deviation σ ≤ 13.5% and angle deviation θ ≤ 0.65%. The simulation parameters were set as T0 = 100, λ = 0.943, and the algorithm was run for multiple iterations. The results demonstrate the dynamic adjustment of drone positions over time.
The following table summarizes the radius deviation σ for each drone during key iterations, highlighting the improvement in the drone formation:
| Drone ID | Initial State (%) | After 1st Iteration (%) | After 5th Iteration (%) | After 9th Iteration (%) |
|---|---|---|---|---|
| 2 | 10.94 | 5.63 | 0.93 | 0.42 |
| 3 | 2.69 | 2.09 | 1.28 | 0.93 |
| 4 | 1.26 | 1.08 | 0.79 | 0.48 |
| 5 | 0.79 | 0.18 | 0.00 | 0.00 |
| 6 | 8.99 | 5.13 | 1.66 | 0.62 |
| 7 | 5.23 | 2.63 | 0.66 | 0.33 |
| 8 | 4.67 | 3.28 | 1.53 | 0.78 |
| 9 | 2.63 | 2.12 | 1.38 | 0.83 |
Similarly, the angle deviation θ for key angles in the drone formation is shown below:
| Angle | Initial State (%) | After 1st Iteration (%) | After 5th Iteration (%) | After 9th Iteration (%) |
|---|---|---|---|---|
| ∠1 | 0.312 | 0.086 | 0.011 | 0.000 |
| ∠2 | 0.344 | 0.258 | 0.161 | 0.065 |
| ∠3 | 0.118 | 0.097 | 0.075 | 0.032 |
| ∠4 | 0.419 | 0.097 | 0.000 | 0.000 |
| ∠5 | 0.183 | 0.108 | 0.032 | 0.000 |
| ∠6 | 0.086 | 0.043 | 0.011 | 0.000 |
| ∠7 | 0.634 | 0.667 | 0.323 | 0.075 |
| ∠8 | 0.194 | 0.161 | 0.097 | 0.054 |
From the tables, it is evident that both radius and angle deviations decrease significantly over iterations. After 9 iterations, the maximum radius deviation is reduced to 0.93%, with a mean of 0.445%, indicating that all drones are within approximately one unit distance from their ideal positions. Similarly, angle deviations drop to a maximum of 0.075% and a mean of 0.0282%. These results confirm the efficacy of our adjustment strategy in maintaining a precise drone formation.
The simulated annealing algorithm’s performance can be further analyzed through convergence plots. The energy E decreases monotonically with iterations, demonstrating the optimization process. The cooling schedule ensures that the system avoids getting stuck in local optima, which is crucial for complex drone formation scenarios. Additionally, the drift circle model provides a geometric intuition for deviations, making it easier to visualize and control adjustments. This combination of models and algorithms offers a robust framework for real-time drone formation control.
In practice, the adjustment strategy can be extended to various formation shapes beyond circular layouts. For instance,锥形 or linear formations could be handled by adapting the drift circle model to different geometries. Moreover, the simulated annealing algorithm’s parameters, such as initial temperature and cooling rate, can be tuned based on specific requirements like convergence speed or computational resources. This flexibility makes our approach applicable to a wide range of drone formation tasks.
However, there are limitations to consider. The current study assumes a two-dimensional平面 and ignores three-dimensional effects that may arise in real-world飞行. Future work could incorporate altitude adjustments and dynamic obstacles to enhance realism. Additionally, the communication between drones in the formation must be reliable for方位角 measurements; delays or packet loss could affect performance. Integrating error-correction mechanisms or redundant信标 drones might mitigate these issues.
In conclusion, this paper presents a comprehensive method for passive positioning and adjustment in drone formation using a drift circle model and simulated annealing algorithm. The drift circle model effectively quantifies deviations in terms of radius and angle, while simulated annealing optimizes the adjustment process to minimize these deviations. Simulation results validate the model’s accuracy, showing significant reductions in both radius and angle deviations after a few iterations. This work contributes to the field of drone formation control by offering a scalable, efficient, and geometrically intuitive solution. As drone technology continues to evolve, such strategies will be essential for enabling reliable and autonomous协同作业 in diverse applications.
The importance of maintaining a stable drone formation cannot be overstated, especially in mission-critical operations. By leveraging passive positioning techniques, we reduce reliance on external systems and enhance robustness. The proposed approach lays a foundation for further research into adaptive formation control, multi-agent coordination, and real-time optimization. Ultimately, advancing drone formation capabilities will unlock new possibilities in areas like disaster response, agriculture, and defense, driving innovation in the UAV industry.