In recent years, the rapid development of technologies such as computer science, artificial intelligence, and automatic control has significantly enhanced the application value of drones in modern society. The drone formation, characterized by low cost, low risk, and high efficiency, has gained widespread recognition for its potential in various fields. To ensure safe flight, drone formations often maintain electromagnetic silence during missions, minimizing the emission of electromagnetic signals to avoid external interference. Maintaining a precise formation is crucial for drone clusters, necessitating accurate positioning of each drone. The drone formation involves arranging multiple drones in a specific pattern to keep the formation unchanged throughout the task, enabling expanded视野,全方位摄像 of ground or control targets, and improving overall efficiency in reconnaissance and strike operations. Through协同完成 of drone formations, tasks such as cooperative detection, positioning, and jamming in battlefield感知 can be achieved. Research on drone formation positioning has become a hotspot in recent years, with pure bearing passive localization emerging as a key method for adjusting drone positions without emitting signals.
Pure bearing passive localization involves having some drones in the formation emit signals while others passively receive them. Based on the received signals, the receiving drones determine and adjust their positions. In a drone formation, all drones have fixed identifiers, and their relative positions remain constant. The directional information received by a drone includes the angles between the lines connecting it to any two emitting drones. This paper focuses on a circular drone formation, addressing the problem of how to locate passively receiving drones when the identifiers of the emitting drones are known and their positions are without偏差. The drone formation considered consists of 10 drones, with one drone, labeled as the reference, positioned at the center of a circle, and the other nine drones evenly distributed on a circumference at the same altitude.
To model the positioning of passively receiving drones in a drone formation, we assume three drones emit signals: one is always the central drone, and the other two are unknown but selected from the peripheral drones. For simplicity in coordinate establishment, we choose one peripheral drone as the second emitter, and the third emitter’s position is categorized into four cases based on its angular separation from the second emitter: adjacent, separated by one drone, two drones, or three drones. This corresponds to different identifiers on the circumference. Since the emitting drones have no position偏差, the distance between the central drone and any peripheral emitter is constant, denoted as R. We establish a polar coordinate system with the central drone as the pole and a ray from the pole through the second emitter as the polar axis, taking counterclockwise as the positive direction for angles. The polar coordinates of a receiving drone are represented as (ρ, θ).
In this drone formation positioning scenario, the receiving drone measures two angles: α1, the angle between the lines connecting it to the central drone and the second emitter, and α2, the angle between the lines connecting it to the central drone and the third emitter. These angles are known from the received signals. Using geometric relationships and the sine theorem, we can derive expressions for ρ and θ. The sine theorem states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Applying this to triangles formed by the receiving drone and the emitting drones allows us to establish equations for localization.
We consider the four cases for the third emitter’s position. For each case, we analyze the geometry and apply the sine theorem to triangles involving the receiving drone, the central drone, and the emitters. Through systematic derivation, we find that the positioning equations can be unified into a general form, regardless of the third emitter’s identifier. This demonstrates the robustness of the pure bearing passive localization method for drone formations.
| Symbol | Description | Unit |
|---|---|---|
| k | Identifier suffix of the third emitting drone (k=1,2,3,4,5) | N/A |
| R | Distance between the central drone and any peripheral emitting drone | m |
| (ρ, θ) | Polar coordinates of the receiving drone | m, rad |
| β | Angle between the line from the central drone to the third emitter and the polar axis | rad |
| α1 | Angle at the receiving drone between lines to the central drone and the second emitter | rad |
| α2 | Angle at the receiving drone between lines to the central drone and the third emitter | rad |
The positioning model for the drone formation is developed step by step. First, we consider the case where the third emitter is adjacent to the second emitter. In this scenario, the receiving drone’s position can be determined using triangles formed with the central drone and the emitters. For drones on the upper half of the circumference, the relationship derived from the sine theorem is:
$$ \frac{R}{\sin \alpha_1} = \frac{\rho}{\sin(\alpha_1 + \theta)} $$
This equation arises from applying the sine theorem to the triangle involving the central drone, the second emitter, and the receiving drone. For drones on the lower half of the circumference, the equation modifies to:
$$ \frac{R}{\sin \alpha_1} = \frac{\rho}{\sin(\alpha_1 – \theta)} $$
These equations account for the geometric configuration where the receiving drone is positioned relative to the polar axis. Similarly, using the angle α2 and the third emitter, we derive additional equations. For receiving drones where the angle between the line to the central drone and the line to the third emitter is less than 180 degrees, the relationship is:
$$ \frac{R}{\sin \alpha_2} = \frac{\rho}{\sin(\alpha_2 + \theta – \beta)} $$
And for angles greater than 180 degrees, it becomes:
$$ \frac{R}{\sin \alpha_2} = \frac{\rho}{\sin(\alpha_2 – \theta + \beta)} $$
Here, β is the fixed angle between the polar axis and the line from the central drone to the third emitter, which depends on the third emitter’s identifier. For example, if the third emitter is adjacent to the second emitter, β = 40° (or 2π/9 radians) in a nine-drone perimeter; if separated by one drone, β = 80° (4π/9 radians); and so on. This parameterization allows the drone formation to adapt to different emitter configurations.
By combining these equations, we can solve for ρ and θ. For instance, from the equations involving α1 and α2, we can eliminate ρ or θ to find explicit expressions. The general solution for the receiving drone’s position in the drone formation is given by:
$$ \rho = R \cdot \frac{\sin(\alpha_1 \pm \theta)}{\sin \alpha_1} \quad \text{and} \quad \rho = R \cdot \frac{\sin(\alpha_2 \pm |\theta – \beta|)}{\sin \alpha_2} $$
Where the signs depend on the quadrant of the receiving drone. These equations can be solved simultaneously to determine ρ and θ uniquely, provided the angles are measured accurately. This method ensures precise localization within the drone formation without requiring active emission from all drones, aligning with the goal of electromagnetic silence.
To illustrate the applicability across different third emitter positions, we summarize the equations for each case in a table. This highlights the consistency of the positioning model for the drone formation.
| Third Emitter Identifier | β Value (radians) | Equation from α1 | Equation from α2 for θ relative to β |
|---|---|---|---|
| Adjacent (e.g., FY02) | 2π/9 | $$ \frac{R}{\sin \alpha_1} = \frac{\rho}{\sin(\alpha_1 + \theta)} $$ for upper half, $$ \frac{R}{\sin \alpha_1} = \frac{\rho}{\sin(\alpha_1 – \theta)} $$ for lower half | For |θ – β| < π: $$ \frac{R}{\sin \alpha_2} = \frac{\rho}{\sin(\alpha_2 + \theta – \beta)} $$, else: $$ \frac{R}{\sin \alpha_2} = \frac{\rho}{\sin(\alpha_2 – \theta + \beta)} $$ |
| Separated by one drone (e.g., FY03) | 4π/9 | Same as above | Same form, with β adjusted |
| Separated by two drones (e.g., FY04) | 6π/9 | Same as above | Same form, with β adjusted |
| Separated by three drones (e.g., FY05) | 8π/9 | Same as above | Same form, with β adjusted |
The drone formation positioning model relies on accurate angle measurements. In practice, drones are equipped with sensors to measure the angles α1 and α2. Using the known distance R and the derived equations, the polar coordinates (ρ, θ) can be computed. This enables each receiving drone to determine its position relative to the central drone and adjust accordingly to maintain the formation. The method is scalable for larger drone formations, as the same principles apply regardless of the number of drones, as long as the geometry is defined.
To further elaborate, let’s derive the general solution mathematically. From the equation involving α1, we have:
$$ \rho = R \cdot \frac{\sin(\alpha_1 \pm \theta)}{\sin \alpha_1} $$
And from the equation involving α2:
$$ \rho = R \cdot \frac{\sin(\alpha_2 \pm |\theta – \beta|)}{\sin \alpha_2} $$
Setting these equal, we get:
$$ \frac{\sin(\alpha_1 \pm \theta)}{\sin \alpha_1} = \frac{\sin(\alpha_2 \pm |\theta – \beta|)}{\sin \alpha_2} $$
This equation can be solved for θ numerically or analytically depending on the signs. For instance, if we assume the receiving drone is in the upper half (using + for α1), and |θ – β| < π (using + for α2), then:
$$ \frac{\sin(\alpha_1 + \theta)}{\sin \alpha_1} = \frac{\sin(\alpha_2 + \theta – \beta)}{\sin \alpha_2} $$
Rearranging:
$$ \sin(\alpha_1 + \theta) \sin \alpha_2 = \sin(\alpha_2 + \theta – \beta) \sin \alpha_1 $$
Using trigonometric identities, such as the product-to-sum formulas, we can simplify to find θ. For example:
$$ \sin(\alpha_1 + \theta) \sin \alpha_2 = \frac{1}{2}[\cos(\alpha_1 + \theta – \alpha_2) – \cos(\alpha_1 + \theta + \alpha_2)] $$
And similarly for the right side. This leads to a transcendental equation that can be solved iteratively in real-time by the drone’s onboard processor. Once θ is found, ρ can be computed directly from one of the equations.
The robustness of this drone formation positioning method is enhanced by considering multiple receiving drones. In a typical drone formation, all non-emitting drones can use the same signals to compute their positions independently. This decentralized approach reduces the risk of single points of failure and improves the resilience of the formation. Moreover, the pure bearing passive localization minimizes electromagnetic interference, making the drone formation less detectable and more secure in sensitive applications.
In terms of implementation, the drone formation requires precise synchronization and calibration. The emitting drones must transmit signals at known times, and the receiving drones must have accurate clocks to measure angles based on signal arrival times or phase differences. Additionally, the distance R must be known or estimated from prior calibration. In a dynamic drone formation, R might change due to formation scaling, but it can be communicated or derived from additional measurements.
To validate the model, simulation studies can be conducted. For a circular drone formation with 10 drones, we can simulate the angles α1 and α2 for each receiving drone given their true positions. Then, using the positioning equations, we compute the estimated positions and compare them to the true ones. The error can be analyzed in terms of ρ and θ, and sensitivity to measurement noise can be assessed. This helps in optimizing the drone formation for real-world conditions.
Furthermore, the drone formation positioning method can be extended to other formation shapes, such as linear, triangular, or锥形 formations. The key is to adapt the coordinate system and geometric relationships. For example, in a锥形 formation, the central drone might be at the apex, and others along lines radiating outward. Similar pure bearing passive localization can be applied by defining appropriate reference frames and angles. This versatility makes the method valuable for various drone formation applications, from aerial shows to military surveillance.
Another aspect to consider is the impact of environmental factors on the drone formation. Weather conditions like wind, rain, or electromagnetic interference can affect signal propagation and angle measurements. To mitigate this, the drone formation can incorporate filtering algorithms, such as Kalman filters, to fuse bearing measurements with other sensors like IMUs or GPS when available. This hybrid approach enhances the reliability of the drone formation in challenging environments.
The computational complexity of the positioning algorithm is also important for real-time operation in a drone formation. The equations involve trigonometric functions and potentially iterative solutions. However, modern drone processors are capable of handling such computations efficiently. For large drone formations with hundreds of drones, distributed computing can be employed, where each drone calculates its own position based on local measurements, reducing the central processing burden.
In summary, the pure bearing passive localization method provides an effective way to position drones in a formation without relying on active emission from all units. By leveraging geometric relationships and the sine theorem, we derive concise equations that allow receiving drones to determine their coordinates. This approach supports the maintenance of precise drone formations, which is essential for协同 tasks. The method is mathematically sound and can be implemented with standard sensors and processors.
For future work, the drone formation positioning can be enhanced in several ways. First, integrating machine learning algorithms to predict and compensate for measurement errors could improve accuracy. Second, exploring three-dimensional formations where drones operate at different altitudes would require extending the model to spherical coordinates. Third, investigating the use of multiple frequency signals or编码 signals could enhance the robustness against interference. Fourth, developing adaptive formation control algorithms that use the positioning data to dynamically adjust drone positions in response to obstacles or mission changes. These advancements will further solidify the role of drone formations in modern technology.
In conclusion, this paper presents a comprehensive model for drone formation positioning based on pure bearing passive localization. The use of polar coordinates and the sine theorem allows for elegant derivations of positioning equations. The method is scalable, secure, and suitable for various formation shapes. As drone technology continues to evolve, such positioning techniques will be crucial for enabling complex and reliable drone formations in diverse applications, from entertainment to defense. The emphasis on electromagnetic silence aligns with the need for stealth and efficiency, making this approach a valuable contribution to the field of drone formation research.