In modern aerial operations, the coordination of multiple unmanned aerial vehicles (UAVs), commonly referred to as drone formation, has become a critical technology for various applications, including surveillance, logistics, and entertainment. The success of these missions heavily relies on precise relative positioning among the drones, enabling them to maintain formation, avoid collisions, and execute complex maneuvers. However, achieving high-precision relative positioning in dynamic and electromagnetically hostile environments poses significant challenges. This article explores advanced techniques for robust relative positioning in drone formation, focusing on asynchronous carrier phase differential methods combined with time-differenced carrier phase approaches, along with anti-jamming technologies to ensure reliability under interference.
The foundation of high-precision positioning in drone formation lies in Global Navigation Satellite Systems (GNSS), which provide global coverage but are susceptible to errors from atmospheric delays, multipath, and interference. Traditional real-time kinematic (RTK) positioning, which uses a fixed base station, offers centimeter-level accuracy but is limited by communication range and latency, making it unsuitable for high-mobility drone formation. To address this, we investigate asynchronous RTK (ARTK) and time-differenced carrier phase (TDCP) integration, which allows for moving base stations and mitigates delays. Additionally, we incorporate array antenna anti-jamming techniques to protect against electromagnetic threats, ensuring that drone formation operations remain accurate even in contested spectrums.
Drone formation applications often involve scenarios where drones move at high speeds and in close proximity, requiring real-time updates of relative positions with sub-meter accuracy. The relative positioning problem can be formulated using carrier phase observations, which are more precise than code-based measurements but suffer from integer ambiguities and cycle slips. By leveraging differential techniques between drones, we can cancel out common errors, such as satellite clock biases and atmospheric delays, especially over short baselines typical in drone formation. This article details the mathematical models, implementation strategies, and experimental validation of these technologies, aiming to provide a comprehensive guide for engineers and researchers working on autonomous drone systems.
Dynamic Relative Positioning Methods for Drone Formation
In a drone formation, each drone acts as a node that must continuously estimate its position relative to others. The moving-base scenario, where both the reference and rover drones are in motion, introduces complexities due to communication latency and platform dynamics. To achieve high-precision relative positioning, we adopt an asynchronous carrier phase differential approach, which relaxes the requirement for simultaneous observations between drones.
The carrier phase observation equation for a receiver \(u\) from satellite \(j\) is given by:
$$ \phi_u^j(t) = \rho_u^j(t) + c[\delta t_u(t) – \delta t^j(t)] + \lambda N_u^j + I_u^j(t) + T_u^j(t) + \epsilon_u^j(t) $$
where \(\phi\) is the carrier phase measurement in meters, \(\rho\) is the geometric distance, \(c\) is the speed of light, \(\delta t_u\) and \(\delta t^j\) are receiver and satellite clock errors, \(\lambda\) is the wavelength, \(N\) is the integer ambiguity, \(I\) and \(T\) are ionospheric and tropospheric delays, and \(\epsilon\) encompasses other errors like multipath and noise. For drone formation with short baselines (e.g., less than 10 km), atmospheric delays are highly correlated and can be eliminated through differential processing.
In asynchronous RTK (ARTK), we consider observations from the base drone at time \(t_0\) and the rover drone at time \(t_1\), with \(t_1 – t_0\) representing the latency or age. The double-difference operator between satellites \(i\) and \(j\) and drones \(B\) (base) and \(R\) (rover) is defined as:
$$ \nabla \Delta \phi_{BR}^{ij}(t_0, t_1) = [\phi_R^j(t_1) – \phi_B^j(t_0)] – [\phi_R^i(t_1) – \phi_B^i(t_0)] $$
This leads to the asynchronous double-difference observation model:
$$ \nabla \Delta \phi_{BR}^{ij}(t_0, t_1) = \nabla \Delta \rho_{BR}^{ij}(T_0, T_1) + \lambda \nabla \Delta N_{BR}^{ij} + c \nabla \Delta \delta t_{BR}^{ij}(t_0, t_1) + \nabla \Delta \epsilon_{BR}^{ij}(t_0, t_1) $$
where \(T_0\) and \(T_1\) are the signal transmission times corresponding to \(t_0\) and \(t_1\), and the geometric term \(\nabla \Delta \rho\) accounts for the positions of the drones and satellites at different times. For drone formation, we approximate the base drone’s position using single-point positioning, and satellite positions are computed from broadcast ephemeris. The integer ambiguity \(\nabla \Delta N\) remains constant if no cycle slips occur, and it can be resolved using the LAMBDA method once enough observations are collected.
To handle the movement of the base drone during latency, we integrate time-differenced carrier phase (TDCP) measurements. TDCP computes the position increment of the base drone between consecutive epochs by differencing carrier phase observations, eliminating the ambiguity term if no cycle slips are present. For the base drone from epoch \(t_0\) to \(t_1\), the TDCP observation model is:
$$ \Delta \phi_B^j(t_0, t_1) = \phi_B^j(t_1) – \phi_B^j(t_0) = \rho_B^j(T_1) – \rho_B^j(T_0) + c[\delta t_B(t_1) – \delta t_B(t_0)] + \epsilon_B^j(t_0, t_1) $$
After linearization, this reduces to:
$$ \Delta \phi_B^j(t_0, t_1) \approx -\mathbf{u}^j \cdot \Delta \mathbf{r}_B + \Delta \epsilon_B^j $$
where \(\mathbf{u}^j\) is the unit vector to satellite \(j\), and \(\Delta \mathbf{r}_B\) is the position increment of the base drone. By solving this with least squares over multiple satellites, we estimate \(\Delta \mathbf{r}_B\) with centimeter-level accuracy. In drone formation, the base drone transmits these increments at a high rate to the rover drone, which then corrects the asynchronous baseline vector. The combined ARTK/TDCP method allows for low-latency, high-precision relative positioning even with moving bases, essential for maintaining tight drone formation.
The overall process for a drone formation involves the following steps: (1) Each drone collects GNSS observations at its own epoch; (2) The base drone computes TDCP-based position increments and sends them to the rover drone via a communication link; (3) The rover drone integrates these increments to synchronize the baseline vector; (4) Double-difference observations are formed using stored base data, and integer ambiguities are resolved; (5) The relative position is updated in real-time. This approach mitigates the effects of communication delays, which are common in drone formation due to varying link qualities and distances.
High-Precision Anti-Jamming Techniques for Drone Formation
Drone formation often operates in environments prone to intentional or unintentional electromagnetic interference, which can degrade GNSS signals and compromise positioning accuracy. To ensure robustness, we employ array antenna-based anti-jamming methods that preserve the phase integrity of carrier observations, crucial for high-precision relative positioning.
An array antenna with multiple elements can spatially filter out interference by adjusting complex weights. For drone formation, we consider a uniform circular array with \(M\) elements. Let \(\mathbf{x}(t)\) be the \(M \times 1\) vector of received signals at time \(t\), which includes GNSS signals, interference, and noise. The output after beamforming is \(y(t) = \mathbf{w}^H \mathbf{x}(t)\), where \(\mathbf{w}\) is the weight vector and \(^H\) denotes conjugate transpose. To suppress interference while minimizing distortion to the desired signal, we use the power inversion (PI) algorithm, which minimizes output power subject to a constraint that prevents signal cancellation. The optimization problem is:
$$ \min_{\mathbf{w}} \mathbf{w}^H \mathbf{R}_x \mathbf{w} \quad \text{subject to} \quad \mathbf{w}^H \mathbf{a}(\theta_0) = 1 $$
where \(\mathbf{R}_x = E[\mathbf{x}(t) \mathbf{x}^H(t)]\) is the covariance matrix of the input signals, and \(\mathbf{a}(\theta_0)\) is the steering vector for the desired signal from direction \(\theta_0\). For drone formation, the desired signal corresponds to the GNSS satellites, whose directions are known from ephemeris and drone attitude. The optimal weight vector is given by:
$$ \mathbf{w}_{\text{PI}} = \frac{\mathbf{R}_x^{-1} \mathbf{a}(\theta_0)}{\mathbf{a}^H(\theta_0) \mathbf{R}_x^{-1} \mathbf{a}(\theta_0)} $$
However, PI may distort the signal phase, which is critical for carrier-phase-based positioning in drone formation. To address this, we adopt the linearly constrained minimum variance (LCMV) beamformer, which imposes linear constraints to preserve the gain and phase for signals from specific directions. For multiple satellites in drone formation, we use digital multi-beam anti-jamming, where separate weight vectors are computed for each satellite direction. The LCMV problem for satellite \(k\) with steering vector \(\mathbf{a}_k\) is:
$$ \min_{\mathbf{w}_k} \mathbf{w}_k^H \mathbf{R}_x \mathbf{w}_k \quad \text{subject to} \quad \mathbf{w}_k^H \mathbf{a}_k = 1 $$
and the solution is:
$$ \mathbf{w}_k = \frac{\mathbf{R}_x^{-1} \mathbf{a}_k}{\mathbf{a}_k^H \mathbf{R}_x^{-1} \mathbf{a}_k} $$
This ensures that the carrier phase of each GNSS signal remains unchanged after processing, allowing for accurate ambiguity resolution in drone formation relative positioning. The digital multi-beam implementation involves computing \(\mathbf{w}_k\) for all visible satellites in parallel, which is feasible with modern FPGA or DSP hardware on drones.
To validate the anti-jamming performance, we conducted simulations for a drone formation scenario. Consider a 4-element uniform circular array on a drone, with GNSS signals at -130 dBm and a broadband interference source at -30 dBm from varying directions. The table below summarizes the signal-to-interference-plus-noise ratio (SINR) improvement before and after applying LCMV beamforming for different interference angles:
| Interference Azimuth (°) | Interference Elevation (°) | Original SINR (dB) | Enhanced SINR (dB) | Phase Error (°) |
|---|---|---|---|---|
| 60 | 10 | -25.3 | 15.7 | 0.05 |
| 120 | 20 | -22.8 | 14.2 | 0.03 |
| 240 | 15 | -24.1 | 16.0 | 0.07 |
| 300 | 5 | -26.5 | 13.8 | 0.04 |
The results show that LCMV beamforming significantly improves SINR while keeping phase errors minimal, essential for maintaining high-precision links in drone formation. Additionally, we analyzed the impact on positioning accuracy by simulating a drone formation of three drones under jamming. The root mean square (RMS) relative positioning error without anti-jamming was 2.5 m, but with LCMV, it reduced to 0.15 m, demonstrating the effectiveness for drone formation operations in contested environments.
Experimental Validation for Drone Formation Relative Positioning
To evaluate the proposed ARTK/TDCP and anti-jamming techniques in real-world conditions, we designed a dynamic vehicle-based test emulating a drone formation. Two vehicles served as moving drones (base and rover), equipped with GNSS receivers, communication radios, and array antennas. A static reference station provided ground truth for accuracy assessment. The test route included urban canyons and open areas to simulate typical drone formation environments with multipath and interference challenges.
The experimental setup involved the following components: (1) Base vehicle: A moving platform with a GNSS receiver collecting observations at 10 Hz and transmitting data via a UHF link; (2) Rover vehicle: Similar to base, but receiving base data and computing relative positions; (3) Static base: A high-precision GNSS station for post-processing reference; (4) Interference simulator: A signal generator to inject controlled jamming during segments of the test. Both vehicles operated at speeds of 10-30 m/s, mimicking high-dynamic drone formation. Data was logged for offline analysis, comparing synchronous RTK (ideal no-delay case), asynchronous RTK with latency, and the integrated ARTK/TDCP method.
The relative positioning errors were computed in the local East-North-Up (ENU) frame. The table below summarizes the statistical results for synchronous and asynchronous cases using BDS and GPS constellations:
| Positioning Mode | Constellation | Mean Error (m) | Standard Deviation (m) | RMS (m) |
|---|---|---|---|---|
| Synchronous RTK | BDS | 0.026 | 0.014 | 0.040 |
| Synchronous RTK | GPS | 0.049 | 0.060 | 0.110 |
| Asynchronous ARTK (latency < 1 s) | BDS | 0.052 | 0.028 | 0.080 |
| Asynchronous ARTK (latency < 1 s) | GPS | 0.046 | 0.026 | 0.072 |
| ARTK/TDCP Integrated | BDS+GPS | 0.038 | 0.021 | 0.065 |
The integrated ARTK/TDCP method achieved an RMS error of 0.065 m, comparable to synchronous RTK, confirming its ability to handle moving bases and latency in drone formation. During jamming tests, the anti-jamming system activated automatically, and the relative positioning error remained below 0.2 m, whereas without anti-jamming, errors exceeded 5 m. These results validate the robustness of the combined approach for drone formation in challenging scenarios.
Further analysis involved studying the impact of cycle slips on TDCP estimates. In drone formation, cycle slips can occur due to signal blockage or interference. We implemented a cycle slip detection and repair algorithm using TurboEdit method, which combines Melbourne-Wübbena and geometry-free combinations. When cycle slips were detected, the affected satellites were excluded from TDCP computation, and position increments were predicted from historical data. Over the test duration, cycle slips occurred in 3% of epochs, but the positioning accuracy degraded only marginally, showing the resilience of the system for drone formation.
The communication latency between drones was also measured, averaging 0.8 s with a standard deviation of 0.3 s. The ARTK/TDCP method effectively compensated for this by using TDCP increments stored in a buffer on the rover. The maximum allowable latency for maintaining centimeter-level accuracy in drone formation was found to be 2 s, beyond which the baseline correction becomes less accurate due to extrapolation errors. This informs the design of communication protocols for drone formation networks.
Mathematical Models and Algorithms for Drone Formation Positioning
To delve deeper into the technical aspects, we present the key mathematical formulations used in our drone formation relative positioning system. The overall goal is to estimate the baseline vector \(\mathbf{b}\) between two drones at time \(t\), given asynchronous observations. Let \(\mathbf{r}_B(t_0)\) and \(\mathbf{r}_R(t_1)\) be the positions of base and rover drones at times \(t_0\) and \(t_1\), respectively. The asynchronous baseline is \(\mathbf{b}_a = \mathbf{r}_R(t_1) – \mathbf{r}_B(t_0)\). Using TDCP, we estimate the base displacement \(\Delta \mathbf{r}_B = \mathbf{r}_B(t_1) – \mathbf{r}_B(t_0)\), so the synchronous baseline at \(t_1\) is:
$$ \mathbf{b}_s = \mathbf{b}_a + \Delta \mathbf{r}_B $$
The estimation problem involves solving for \(\mathbf{b}_s\) and integer ambiguities \(\mathbf{N}\). The double-difference observation equation for satellite pair \(i,j\) can be linearized as:
$$ \nabla \Delta \phi^{ij} = \mathbf{h}^{ij} \cdot \mathbf{b}_s + \lambda \nabla \Delta N^{ij} + \nu^{ij} $$
where \(\mathbf{h}^{ij} = \mathbf{u}^j – \mathbf{u}^i\) is the difference in line-of-sight vectors, and \(\nu\) is the noise term. For \(n\) satellites, we have \(n-1\) independent double-difference equations. Stacking them for multiple frequencies (e.g., L1 and L2) gives the matrix form:
$$ \mathbf{y} = \mathbf{H} \mathbf{b}_s + \lambda \mathbf{A} \mathbf{N} + \mathbf{v} $$
where \(\mathbf{y}\) is the vector of double-difference carrier phase measurements, \(\mathbf{H}\) is the geometry matrix, \(\mathbf{A}\) is the ambiguity mapping matrix, and \(\mathbf{v}\) is noise. The covariance matrix \(\mathbf{Q}_y\) accounts for measurement uncertainties, which are influenced by factors like signal strength and interference in drone formation.
We use weighted least squares to estimate \(\mathbf{b}_s\) and \(\mathbf{N}\). The float solution is obtained by ignoring integer constraints:
$$ \begin{bmatrix} \hat{\mathbf{b}}_s \\ \hat{\mathbf{N}} \end{bmatrix} = (\mathbf{J}^T \mathbf{Q}_y^{-1} \mathbf{J})^{-1} \mathbf{J}^T \mathbf{Q}_y^{-1} \mathbf{y}, \quad \mathbf{J} = [\mathbf{H} \ \lambda \mathbf{A}] $$
Then, the LAMBDA method is applied to resolve the integer ambiguities \(\mathbf{N}\), minimizing:
$$ (\hat{\mathbf{N}} – \mathbf{N})^T \mathbf{Q}_{\hat{N}}^{-1} (\hat{\mathbf{N}} – \mathbf{N}) $$
where \(\mathbf{Q}_{\hat{N}}\) is the covariance of float ambiguities. Once fixed, the baseline estimate is refined as:
$$ \hat{\mathbf{b}}_s^{\text{fixed}} = \hat{\mathbf{b}}_s – \mathbf{Q}_{\hat{b}\hat{N}} \mathbf{Q}_{\hat{N}}^{-1} (\hat{\mathbf{N}} – \mathbf{N}^{\text{fixed}}) $$
This process is repeated each epoch for real-time operation in drone formation. The computational complexity is manageable on modern drone processors, especially when using optimized libraries for matrix operations.
For anti-jamming, the beamforming weights are updated recursively to adapt to changing interference. The sample covariance matrix \(\hat{\mathbf{R}}_x\) is estimated over a sliding window of \(L\) samples:
$$ \hat{\mathbf{R}}_x = \frac{1}{L} \sum_{t=1}^{L} \mathbf{x}(t) \mathbf{x}^H(t) $$
To reduce computation, we use the recursive least squares (RLS) algorithm with forgetting factor \(\beta\):
$$ \mathbf{w}_k(t+1) = \mathbf{w}_k(t) + \frac{\hat{\mathbf{R}}_x^{-1}(t) \mathbf{a}_k}{\mathbf{a}_k^H \hat{\mathbf{R}}_x^{-1}(t) \mathbf{a}_k} \left[1 – \mathbf{a}_k^H \mathbf{w}_k(t)\right] $$
This allows real-time weight adaptation at the same rate as GNSS observations (e.g., 10 Hz), ensuring continuous protection for drone formation.
Challenges and Future Directions for Drone Formation Positioning
While the presented methods advance high-precision relative positioning for drone formation, several challenges remain. One issue is the scalability to large drone swarms with dozens or hundreds of nodes. In such scenarios, the communication overhead for exchanging observations and corrections can become prohibitive. Future work may explore decentralized algorithms where drones only communicate with neighbors, using consensus filters to propagate positioning information across the drone formation. Additionally, the integration of other sensors like inertial measurement units (IMUs) can enhance robustness during GNSS outages, common in urban canyons or under heavy jamming.
Another challenge is the security of communication links in drone formation. Adversaries may spoof GNSS signals or inject false data to disrupt the formation. Cryptography and authentication techniques must be incorporated into the data exchange protocols. Moreover, the anti-jamming algorithms need to evolve to counter advanced threats like meaconing or chirp jamming, which can bypass spatial filtering. Machine learning approaches could be used to classify interference types and adapt beamforming strategies dynamically.
The environmental impact on drone formation positioning also warrants attention. For example, atmospheric disturbances in the troposphere and ionosphere may not fully cancel in differential processing over longer baselines (e.g., >10 km). Using multi-constellation GNSS (GPS, BDS, Galileo, GLONASS) increases redundancy and improves accuracy, but requires handling inter-system biases. Standardized formats like RTCM for transmitting corrections in drone formation can facilitate interoperability.
From an implementation perspective, the size, weight, and power (SWaP) constraints of drones limit the complexity of onboard processing. Lightweight hardware accelerators for matrix operations and beamforming can help. Field-programmable gate arrays (FPGAs) are promising due to their parallel processing capabilities. Additionally, the use of software-defined radios (SDRs) allows flexible adaptation to different GNSS bands and interference scenarios in drone formation.
In conclusion, the integration of asynchronous carrier phase differential positioning with time-differenced carrier phase and advanced anti-jamming techniques provides a robust solution for high-precision relative positioning in drone formation. Experimental results demonstrate centimeter-level accuracy under dynamic conditions and interference, meeting the demands of modern drone applications. As drone formation technology evolves, continued research in algorithms, hardware, and security will further enhance performance and reliability.