In recent years, the concept of drone formation flight has garnered significant attention globally. As a researcher in autonomous systems, I have been deeply involved in exploring high-precision navigation solutions for cooperative drone formations. The ability of multiple drones to fly in coordinated patterns offers immense advantages, such as expanded surveillance coverage, enhanced reliability through redundancy, and improved aerodynamic efficiency. However, achieving and maintaining precise geodetic positioning for all members in a drone formation remains a critical challenge. In this paper, I propose an innovative navigation algorithm that leverages a combination of inertial navigation systems (INS), global positioning systems (GPS), and relative measurements to address this issue. My focus is on ensuring that even if only one drone in the formation has reliable GPS access, the entire drone formation can achieve accurate localization, thereby enhancing system robustness. Throughout this discussion, I will emphasize the importance of drone formation coordination and detail the mathematical models, simulations, and results that validate my approach.
The core problem in drone formation flight lies in the high-precision positioning of each member. Traditional methods often rely on individual GPS units for each drone, but GPS signals can be unstable or lost in certain environments, leading to degraded navigation accuracy. To overcome this, my method designates one drone as the leader, equipped with an INS/GPS integrated navigation system, while the follower drones utilize INS, data links like JTIDS for communication, and vision-based sensors for relative measurements. This setup allows the drone formation to maintain accuracy even under GPS-denied conditions for some members. The key innovation is the use of Kalman filtering to estimate INS errors based on relative distance and angle measurements between drones, correcting inertial outputs to obtain precise geodetic navigation information. This approach not only improves reliability but also ensures that all drones in the formation benefit from the leader’s GPS when available.
To lay the groundwork, let me define the state model for the inertial navigation system. For both the leader and follower drones in the drone formation, I consider an 18-dimensional state vector that encompasses various error sources. This vector is represented as:
$$ X = [\phi_E, \phi_N, \phi_U, \delta v_E, \delta v_N, \delta v_U, \delta L, \delta \lambda, \delta h, \varepsilon_{bx}, \varepsilon_{by}, \varepsilon_{bz}, \varepsilon_{rx}, \varepsilon_{ry}, \varepsilon_{rz}, \nabla_x, \nabla_y, \nabla_z]^T $$
Here, the components include platform error angles (e.g., $\phi_E$, $\phi_N$, $\phi_U$), velocity errors (e.g., $\delta v_E$, $\delta v_N$, $\delta v_U$), position errors (e.g., $\delta L$, $\delta \lambda$, $\delta h$ for latitude, longitude, and height), gyroscope drift errors (e.g., $\varepsilon_{bx}$, $\varepsilon_{by}$, $\varepsilon_{bz}$ for constant biases and $\varepsilon_{rx}$, $\varepsilon_{ry}, \varepsilon_{rz}$ for random components), and accelerometer errors (e.g., $\nabla_x$, $\nabla_y$, $\nabla_z$). This comprehensive state vector allows me to model the INS imperfections that accumulate over time during drone formation flight. The state equation governing these errors is a continuous-time linear model:
$$ \dot{X}(t) = F(t) X(t) + G(t) W(t) $$
where $F(t)$ is the system matrix that relates the state variables based on INS dynamics, $G(t)$ is the noise coupling matrix, and $W(t)$ represents process noise assumed to be zero-mean Gaussian white noise. The matrices $F(t)$ and $G(t)$ are derived from standard INS error propagation equations, accounting for Earth’s rotation and gravity effects. For instance, $F(t)$ includes terms like:
$$ F(t) = \begin{bmatrix}
F_{11} & F_{12} & F_{13} \\
F_{21} & F_{22} & F_{23} \\
F_{31} & F_{32} & F_{33}
\end{bmatrix} $$
with submatrices corresponding to error couplings between attitude, velocity, and position. To summarize the state variables and their meanings, I present the following table:
| State Variable | Description | Unit |
|---|---|---|
| $\phi_E$, $\phi_N$, $\phi_U$ | Platform error angles in east, north, up directions | rad |
| $\delta v_E$, $\delta v_N$, $\delta v_U$ | Velocity errors in east, north, up directions | m/s |
| $\delta L$, $\delta \lambda$, $\delta h$ | Position errors in latitude, longitude, height | rad, rad, m |
| $\varepsilon_{bx}$, $\varepsilon_{by}$, $\varepsilon_{bz}$ | Constant gyro drift errors | rad/s |
| $\varepsilon_{rx}$, $\varepsilon_{ry}$, $\varepsilon_{rz}$ | Random gyro drift errors | rad/s |
| $\nabla_x$, $\nabla_y$, $\nabla_z$ | Accelerometer errors | m/s² |
For the leader drone in the drone formation, I employ an INS/GPS integrated navigation approach. The Kalman filter uses the difference between INS outputs and GPS measurements as the observation to estimate the INS errors. Once estimated, these errors are corrected to obtain the leader’s precise navigation information, which is then treated as a reference for the followers. This process is well-established in literature, but its integration into a drone formation context enhances overall coordination. The observation model for the leader involves position and velocity differences, such as:
$$ Z_{\text{leader}} = \begin{bmatrix} L_I – L_{\text{GPS}} \\ \lambda_I – \lambda_{\text{GPS}} \\ h_I – h_{\text{GPS}} \\ v_{E,I} – v_{E,\text{GPS}} \\ v_{N,I} – v_{N,\text{GPS}} \\ v_{U,I} – v_{U,\text{GPS}} \end{bmatrix} $$
where subscript $I$ denotes INS outputs and $\text{GPS}$ denotes GPS measurements. The Kalman filter then updates the state estimate to minimize these differences, ensuring the leader’s accuracy.
For the follower drones in the drone formation, the observation model is more complex and constitutes the core of my contribution. Each follower drone is equipped with vision sensors that measure the relative elevation angle $\alpha_i$ and azimuth angle $\beta_i$ to the leader, along with a time-of-arrival (TOA) system that estimates the distance $r_i$. These measurements are combined with the follower’s own INS outputs and the leader’s corrected navigation data to form an observation vector. Specifically, the measured distance $r_i$ between the leader and follower $i$ is given by:
$$ r_i = r_i^t + v_i $$
where $r_i^t$ is the true distance and $v_i$ is measurement noise with variance $\sigma_v^2$. The angles are measured as:
$$ \alpha_i = \alpha_i^t + w_{i1}, \quad \beta_i = \beta_i^t + w_{i2} $$
with $w_{i1}$ and $w_{i2}$ being angle noise terms with variance $\sigma_w^2$. Using these, the measured relative distances in the Earth-fixed coordinate system are:
$$ d_{ix}^m = r_i \cos \alpha_i \cos \beta_i, \quad d_{iy}^m = r_i \cos \alpha_i \sin \beta_i, \quad d_{iz}^m = r_i \sin \alpha_i $$
Meanwhile, the computed relative distances based on INS outputs are:
$$ d_{ix}^c = x_I^i – x, \quad d_{iy}^c = y_I^i – y, \quad d_{iz}^c = z_I^i – z $$
where $(x_I^i, y_I^i, z_I^i)$ is the follower’s INS position in Earth-fixed coordinates, and $(x, y, z)$ is the leader’s corrected position. The observation for the Kalman filter is then the difference between computed and measured distances:
$$ Z_i = \begin{bmatrix} d_{ix}^c – d_{ix}^m \\ d_{iy}^c – d_{iy}^m \\ d_{iz}^c – d_{iz}^m \end{bmatrix} $$
Through linearization and error propagation, this can be expressed as:
$$ Z_i = H_i X_i + V_i $$
where $H_i$ is the observation matrix that relates the state vector to the distance differences, and $V_i$ is the measurement noise with covariance matrix $R_k$. The elements of $R_k$ are derived based on the noise statistics of $v_i$, $w_{i1}$, and $w_{i2}$. For example, the variance for the first component is:
$$ R_k(1,1) = \sigma_v^2 (\cos \alpha_i \cos \beta_i)^2 + \sigma_w^2 \left[ (r_i \sin \alpha_i \cos \beta_i)^2 + (r_i \cos \alpha_i \sin \beta_i)^2 \right] $$
This observation model enables the follower drones to estimate their own INS errors by comparing their relative position estimates with sensor-based measurements, all within the framework of the drone formation. To clarify the coordinate transformations involved, note that the Earth-fixed coordinates $(x, y, z)$ are related to geodetic coordinates $(L, \lambda, h)$ by:
$$ x = (R + h) \cos L \cos \lambda, \quad y = (R + h) \cos L \sin \lambda, \quad z = (R + h) \sin L $$
where $R$ is Earth’s radius. This relationship is crucial for converting between coordinate systems in the drone formation navigation algorithm.
To validate my proposed method for drone formation navigation, I conducted extensive simulations in MATLAB. The simulation scenario involved a leader drone and multiple follower drones flying in a formation pattern over 300 seconds. The leader used a low-grade INS with GPS integration, while the followers used INS with relative measurements. The process noise covariance matrix $Q_k$ was set based on typical INS error characteristics, and the measurement noise parameters were chosen to reflect realistic sensor accuracies. For the leader, the INS/GPS filter estimated errors with high precision, as shown in the results. For the followers, the filter leveraged the relative observations to suppress INS drift. The simulation parameters are summarized in the table below:
| Parameter | Value for Leader | Value for Follower |
|---|---|---|
| INS gyro bias stability | $0.1^\circ/\text{h}$ | $0.1^\circ/\text{h}$ |
| INS accelerometer bias | $0.0001g$ | $0.0001g$ |
| GPS position error (1σ) | 10 m | N/A |
| GPS velocity error (1σ) | 0.2 m/s | N/A |
| Distance measurement error $\sigma_v$ | N/A | 1 m |
| Angle measurement error $\sigma_w$ | N/A | 0.0005 rad |
| Filter update rate | 1 Hz | 1 Hz |
The simulation results demonstrated the effectiveness of my navigation approach for drone formation flight. For the leader drone, the Kalman filter converged within approximately 50 seconds, yielding average position errors of around 2 meters and velocity errors of about 0.03 m/s. Specifically, the mean errors were as follows:
| Error Type | East Direction | North Direction | Up Direction |
|---|---|---|---|
| Position Error (m) | 1.38 | 1.95 | 1.01 |
| Velocity Error (m/s) | 0.024 | 0.017 | 0.020 |
For a representative follower drone in the drone formation, the filter also converged within 50 seconds, with slightly larger errors due to the reliance on the leader’s position as a reference. The average errors were:
| Error Type | East Direction | North Direction | Up Direction |
|---|---|---|---|
| Position Error (m) | 2.34 | 3.84 | 5.54 |
| Velocity Error (m/s) | 0.080 | 0.090 | 0.083 |
These results indicate that both the leader and followers achieve high geodetic navigation accuracy, with errors remaining within acceptable limits for typical drone formation operations. The follower’s errors are marginally higher, as expected, because the leader’s own errors propagate into the relative measurement process. However, the use of precise relative sensors (TOA and vision) ensures that the degradation is minimal, keeping all drones in the formation well-localized. This robustness is a key advantage in scenarios where GPS availability is limited, making the drone formation more resilient to individual sensor failures.
To further analyze the performance, I examined the error dynamics over time. The position error curves for both leader and follower showed exponential decay after the initial convergence period, stabilizing around the mean values mentioned above. The velocity errors exhibited similar behavior, with rapid reduction post-convergence. This demonstrates the Kalman filter’s ability to continuously estimate and correct INS biases, leveraging the relative measurements in the drone formation. The mathematical foundation for this can be expressed through the filter update equations. For discrete-time implementation, the state estimate is propagated as:
$$ \hat{X}_{k|k-1} = \Phi_{k-1} \hat{X}_{k-1|k-1} $$
where $\Phi_{k-1}$ is the state transition matrix derived from $F(t)$. The covariance propagation is:
$$ P_{k|k-1} = \Phi_{k-1} P_{k-1|k-1} \Phi_{k-1}^T + Q_{k-1} $$
Then, the Kalman gain is computed:
$$ K_k = P_{k|k-1} H_k^T (H_k P_{k|k-1} H_k^T + R_k)^{-1} $$
followed by the state update:
$$ \hat{X}_{k|k} = \hat{X}_{k|k-1} + K_k (Z_k – H_k \hat{X}_{k|k-1}) $$
and covariance update:
$$ P_{k|k} = (I – K_k H_k) P_{k|k-1} $$
These equations were implemented in the simulation, ensuring optimal estimation of INS errors for each drone in the formation. The effectiveness of this process is evident in the low steady-state errors, which are crucial for maintaining tight formation geometry during drone formation flight.
In discussing the implications, my method enhances the scalability of drone formations. By relying on a single GPS-equipped leader, the system reduces cost and complexity while maintaining accuracy. The relative measurement approach allows followers to calibrate their INS without direct GPS access, which is beneficial in environments with signal obstructions. Moreover, the use of vision sensors adds redundancy, as they can operate independently of radio-frequency systems. This multi-sensor fusion is key to robust drone formation navigation. I also considered the impact of formation size on performance. As the number of followers increases, the communication load and data processing requirements grow, but the decentralized nature of the Kalman filters (each follower runs its own filter) ensures scalability. However, the leader’s accuracy sets a bound on overall performance, emphasizing the need for reliable GPS on at least one drone.
From a practical standpoint, the integration of this navigation method into real-world drone formations requires attention to sensor calibration and synchronization. The vision sensors must be accurately aligned to measure angles relative to the leader, and the TOA system must account for signal propagation delays. In my simulation, I assumed ideal synchronization, but in practice, time-stamping and clock synchronization protocols would be necessary. Additionally, the Earth-fixed coordinate system used in the observation model assumes a spherical Earth, which is sufficient for short-range formations but may require ellipsoidal corrections for large-scale drone formations. These considerations are part of ongoing work to refine the algorithm.
In conclusion, I have presented a comprehensive navigation method for drone formation flight that combines INS, GPS, and relative measurements to achieve high-precision geodetic positioning. The leader drone uses INS/GPS integration to obtain accurate reference data, while follower drones use INS corrected by relative distance and angle measurements via Kalman filtering. Simulation results confirm that this approach yields position errors within a few meters and velocity errors below 0.1 m/s for all members of the drone formation, even when only the leader has GPS access. This enhances the reliability and robustness of drone formations in diverse operational environments. Future work will focus on experimental validation with physical drones and extension to dynamic formation reconfiguration. The continuous emphasis on drone formation coordination throughout this study underscores its importance in advancing autonomous aerial systems.