In the realm of unmanned aerial vehicles (UAVs), the concept of drone formation has emerged as a pivotal strategy for enhancing operational efficiency, mission success rates, and cost-effectiveness. As a researcher focused on navigation systems, I have observed that while individual drones rely heavily on inertial navigation systems (INS), these systems suffer from error accumulation over time, leading to divergent navigation inaccuracies. This poses a significant challenge for large-scale drone formation operations, where maintaining precise relative positioning is crucial. Traditional master-slave cooperative navigation methods, where a leader with high-precision equipment guides followers, fall short in massive formations due to limitations in data link bandwidth and the performance of relative navigation sensors. Specifically, not all followers can establish direct communication or measurement links with the leader, rendering single-layer structures ineffective for improving the overall navigation accuracy of the entire drone formation. To address these issues, I propose a hierarchical cooperative navigation system tailored for drone formation, which leverages multi-layered information sharing and relative measurements to bound navigation errors across all layers.
The core idea behind this approach is to organize the drone formation into a tree-like hierarchical structure, as illustrated in the following figure. This structure optimizes resource allocation and ensures that each drone can benefit from corrected navigation information through inter-layer links. The top layer consists of a single drone designated as the leader, equipped with a high-accuracy INS/GPS integrated navigation system. Subsequent layers comprise follower drones, each outfitted with a lower-precision INS, along with relative navigation sensors such as laser rangefinders, angle sensors, and Doppler velocimeters. These sensors enable measurements of relative distance, elevation, azimuth, and velocity relative to drones in the upper layer. Communication within the drone formation is facilitated by onboard data links, primarily transmitting navigation information from upper to lower layers. This hierarchical design mitigates the bottlenecks associated with data link constraints and sensor range limitations, allowing for scalable and efficient cooperative navigation across the entire drone formation.
To formalize this system, I developed a mathematical model based on the geographic coordinate system as the navigation frame. The state equation for a follower drone captures the error dynamics of its INS. Let the state vector X represent various error components, including platform misalignment angles, velocity errors, position errors, and sensor biases. The system is described by the continuous-time linear model:
$$ \dot{\mathbf{X}}(t) = \mathbf{F}(t) \mathbf{X}(t) + \mathbf{G}(t) \mathbf{W}(t) $$
Here, X is an 18-dimensional vector encompassing errors such as $\phi_E$, $\phi_N$, $\phi_U$ (platform misalignment angles in east, north, and up directions), $\delta v_E$, $\delta v_N$, $\delta v_U$ (velocity errors), $\delta \lambda$, $\delta \phi$, $\delta h$ (position errors in longitude, latitude, and height), $\epsilon_{bx}$, $\epsilon_{by}$, $\epsilon_{bz}$ (gyroscope constant drifts), $\epsilon_{rx}$, $\epsilon_{ry}$, $\epsilon_{rz}$ (gyroscope first-order Markov drifts), and $\nabla_x$, $\nabla_y$, $\nabla_z$ (accelerometer first-order Markov drifts). The noise vector W includes Gaussian white noise processes for gyroscopes and accelerometers, with zero mean and known covariance. This state equation forms the foundation for predicting error propagation in the INS of each follower within the drone formation.
The measurement equation for cooperative navigation is derived from the relationship between the follower’s INS outputs, the corrected navigation information from an upper-layer drone, and the relative navigation measurements. Specifically, the measurement vector Z is constructed as the difference between computed values (based on received and INS data) and measured relative values. In the navigation frame (n-frame), this can be expressed as:
$$ \mathbf{Z} = \begin{bmatrix} \mathbf{p}_A – \mathbf{p}_{IB} – \mathbf{p}_m \\ \mathbf{v}_A – \mathbf{v}_{IB} – \mathbf{v}_m \end{bmatrix} = \begin{bmatrix} \mathbf{p}_c – \mathbf{p}_m \\ \mathbf{v}_c – \mathbf{v}_m \end{bmatrix} $$
where $\mathbf{p}_A$ and $\mathbf{v}_A$ are the position and velocity of the upper-layer drone (assumed accurate after correction), $\mathbf{p}_{IB}$ and $\mathbf{v}_{IB}$ are the follower’s INS outputs (containing errors $\delta \mathbf{p}_B$ and $\delta \mathbf{v}_B$), and $\mathbf{p}_m$ and $\mathbf{v}_m$ are the relative measurements from onboard sensors. By decomposing these into true values and errors, we relate Z directly to the state vector:
$$ \mathbf{Z} = -\begin{bmatrix} \delta \mathbf{p}_B \\ \delta \mathbf{v}_B \end{bmatrix} + \begin{bmatrix} \delta \mathbf{p}_m \\ \delta \mathbf{v}_m \end{bmatrix} $$
However, a critical aspect involves coordinate transformations, as relative measurements are taken in the body frame (b-frame) of the follower drone. For instance, the relative velocity measured by sensors includes components in the b-frame: $v_{mbx}$, $v_{mby}$, $v_{mbz}$, derived from measured magnitude $v_m$, elevation $\alpha_m$, and azimuth $\beta_m$. Assuming small measurement errors, these can be linearized to express error terms $\delta v_{bx}$, $\delta v_{by}$, $\delta v_{bz}$ as functions of sensor noise parameters. Transformation to the n-frame is achieved using the direction cosine matrix $\mathbf{C}_b^n$, which is approximated using the INS-derived attitude angles. Thus, the measurement noise component becomes:
$$ \begin{bmatrix} \delta \mathbf{p}_m \\ \delta \mathbf{v}_m \end{bmatrix} = \begin{bmatrix} \hat{\mathbf{C}}_b^n & \mathbf{0} \\ \mathbf{0} & \hat{\mathbf{C}}_b^n \end{bmatrix} \begin{bmatrix} \delta \mathbf{p}_b \\ \delta \mathbf{v}_b \end{bmatrix} $$
where $\hat{\mathbf{C}}_b^n$ is the estimated transformation matrix. The covariance matrix R of the measurement noise is computed based on the statistical properties of sensor errors, such as variances for distance ($\sigma_p^2$), angle ($\sigma_\alpha^2$, $\sigma_\beta^2$), and velocity ($\sigma_v^2$). For a three-dimensional measurement, R is a 6×6 matrix capturing correlations between east, north, and up components.
Additionally, the follower’s INS position output, given in geodetic coordinates (longitude $\lambda$, latitude $L$, height $h$), must be converted to relative distance in the n-frame. For small inter-drone distances, the Earth can be approximated as flat, leading to:
$$ \begin{bmatrix} p_{ce} \\ p_{cn} \\ p_{cu} \end{bmatrix} = \begin{bmatrix} (R + h_A) \cos L_A (\lambda_A – \lambda_{IB}) \\ (R + h_A) (L_A – L_{IB}) \\ h_A – h_{IB} \end{bmatrix} $$
where $R$ is the Earth’s radius, and subscript $A$ denotes the upper-layer drone. Expressing the follower’s INS outputs as true values plus errors ($\lambda_{IB} = \lambda_B^t + \delta \lambda_B$, etc.), we can separate the true relative position from error terms. Combining this with the coordinate-transformed relative measurements, the final measurement equation in discrete form is:
$$ \mathbf{Z}_k = \mathbf{H}_k \mathbf{X}_k + \mathbf{V}_k $$
Here, H is the measurement matrix that maps state errors to the measurement vector, and V is the noise vector with covariance R. For a follower drone, H includes elements like $-(R + h_A) \cos L_A$ for longitude error coupling, ensuring that all relevant error sources are accounted for in the drone formation context.
To validate this hierarchical cooperative navigation system, I conducted extensive simulations using MATLAB, focusing on a three-layer drone formation with sequential links. The simulation parameters were carefully chosen to reflect realistic scenarios, as summarized in the table below. Each follower drone was equipped with an INS having moderate precision, and relative navigation sensors with specified noise characteristics. The flight trajectory involved variable-speed curvilinear motion over 1000 seconds, with data updates at 1-second intervals. This setup allowed me to compare the performance of the proposed method against a baseline where drones rely solely on INS without cooperative navigation.
| Parameter | Value | Description |
|---|---|---|
| Gyro Constant Drift | 0.15 °/h | Constant bias in gyroscope measurements |
| Gyro Markov Drift | 0.15 °/h | First-order Markov process drift for gyroscopes |
| Accelerometer Markov Drift | 150 μg | First-order Markov process drift for accelerometers |
| Distance Measurement Variance ($\sigma_p^2$) | (1 m)$^2$ | Variance in laser rangefinder distance error |
| Elevation Angle Variance ($\sigma_\alpha^2$) | (0.005°)$^2$ | Variance in elevation angle measurement error |
| Azimuth Angle Variance ($\sigma_\beta^2$) | (0.005°)$^2$ | Variance in azimuth angle measurement error |
| Velocity Measurement Variance ($\sigma_v^2$) | (0.1 m/s)$^2$ | Variance in Doppler velocimeter error |
| Simulation Time | 1000 s | Total duration of the flight simulation |
| Update Interval | 1 s | Time step for data fusion and Kalman filtering |
The Kalman filter was employed to estimate the INS errors of follower drones in real-time. By fusing the relative measurements with corrected information from upper layers, the filter continuously updated the state vector, thereby refining the navigation solution. The results demonstrated a significant improvement over the INS-only scenario. In the baseline case, where drones operated independently, position and velocity errors diverged over time due to the inherent drift of INS. For example, position errors grew unbounded, exceeding tens of meters after 1000 seconds, which is unacceptable for precise drone formation maneuvers. In contrast, under the hierarchical cooperative navigation system, errors remained bounded across all layers, as shown in the error analysis table below.
| Layer | Latitude Error (m) | Longitude Error (m) | Height Error (m) | East Velocity Error (m/s) | North Velocity Error (m/s) | Up Velocity Error (m/s) |
|---|---|---|---|---|---|---|
| Layer 2 (Followers) | 1.8145 | 1.3157 | 2.2774 | -0.0309 | -0.0216 | -0.0176 |
| Layer 3 (Followers) | 3.4025 | -2.3660 | 4.8798 | -0.0767 | -0.0603 | -0.0377 |
From these results, it is evident that the hierarchical approach effectively contains navigation errors, ensuring that the entire drone formation maintains acceptable accuracy for prolonged missions. However, I observed a slight degradation in performance from Layer 2 to Layer 3; for instance, height errors approximately doubled. This can be attributed to the assumptions in the model, such as treating upper-layer navigation information as perfectly accurate and the linear approximations in coordinate transformations. These factors introduce cumulative uncertainties as information propagates down the hierarchy. Nonetheless, the overall system performance is robust, with errors staying within a few meters for position and centimeters per second for velocity, which is sufficient for most drone formation applications.
To delve deeper into the system dynamics, let’s consider the mathematical formulation of the error propagation. The state transition matrix F incorporates the effects of Earth’s rotation, gravity, and sensor biases. For a drone moving in a formation, the misalignment angles evolve according to:
$$ \dot{\boldsymbol{\phi}} = -\boldsymbol{\omega}_{in}^n \times \boldsymbol{\phi} + \delta \boldsymbol{\omega}_{ib}^n – \boldsymbol{\epsilon}^n $$
where $\boldsymbol{\omega}_{in}^n$ is the angular rate of the n-frame relative to the inertial frame, $\delta \boldsymbol{\omega}_{ib}^n$ is the error in measured angular rate, and $\boldsymbol{\epsilon}^n$ represents gyro drifts transformed to the n-frame. Similarly, velocity error dynamics are influenced by accelerometer errors and gravity anomalies:
$$ \delta \dot{\mathbf{v}}^n = \mathbf{f}^n \times \boldsymbol{\phi} + \mathbf{C}_b^n \nabla^b – (2\boldsymbol{\omega}_{ie}^n + \boldsymbol{\omega}_{en}^n) \times \delta \mathbf{v}^n + \delta \mathbf{g}^n $$
Here, $\mathbf{f}^n$ is the specific force, $\nabla^b$ are accelerometer errors in the b-frame, and $\delta \mathbf{g}^n$ is gravity error. These equations highlight the complex interactions within an INS, necessitating external corrections via cooperative navigation in a drone formation.
The measurement matrix H plays a crucial role in linking these state errors to observable quantities. For a follower drone measuring relative position and velocity to an upper-layer drone, H can be partitioned as:
$$ \mathbf{H} = \begin{bmatrix} \mathbf{0}_{3\times3} & -\mathbf{I}_{3\times3} & -\text{diag}\left( (R+h_A)\cos L_A, R+h_A, 1 \right) & \mathbf{0}_{3\times9} \\ \mathbf{0}_{3\times3} & -\mathbf{I}_{3\times3} & \mathbf{0}_{3\times3} & \mathbf{0}_{3\times9} \end{bmatrix} $$
where the top row corresponds to position measurements and the bottom to velocity measurements. This structure ensures that errors in latitude, longitude, and height are properly scaled according to geodetic transformations, a key aspect for accurate drone formation navigation over large areas.
In terms of implementation, the hierarchical cooperative navigation system requires synchronized data exchange among drones. Each follower must receive corrected navigation data from its designated upper-layer drone, typically via wireless data links with limited bandwidth. To optimize this, I assumed a tree-structured communication protocol where data flows unidirectionally downward, minimizing overhead. The simulation accounted for perfect communication without delays, but in real-world drone formation operations, link reliability and latency could impact performance. Future work could integrate robust communication models to address these challenges.
Another important consideration is the calibration of relative navigation sensors. The noise parameters used in the simulation, such as $\sigma_p = 1$ m for distance measurements, are based on typical commercial-grade laser sensors. However, for high-precision drone formation tasks, such as tight swarm maneuvers, these values might need refinement. Sensor fusion techniques, like combining multiple measurement types, could further reduce uncertainties. For example, using vision-based relative positioning alongside laser data might enhance accuracy, especially in GPS-denied environments common to drone formation operations.
The scalability of the hierarchical approach is a significant advantage for large drone formation. As the number of drones increases, adding more layers can maintain system performance without overwhelming the data links. The tree structure allows each upper-layer drone to support a limited number of followers (e.g., up to a threshold based on link capacity), ensuring that the communication load is distributed. In my simulation with three layers, the performance degradation from Layer 2 to Layer 3 was manageable, but for deeper hierarchies, error accumulation might become more pronounced. This can be mitigated by incorporating periodic absolute updates, such as GPS fixes for selected drones, or by optimizing the topology of the drone formation to minimize path lengths.
From a control perspective, the bounded navigation errors enabled by this cooperative system facilitate stable formation flying. Controllers for drone formation often rely on accurate relative states to maintain desired geometries, such as V-shapes or grids. With the proposed method, the estimated errors from the Kalman filter can be fed back to adjust control inputs, reducing drift and enhancing cohesion. This integration of navigation and control is essential for autonomous drone formation in dynamic environments.
To further illustrate the benefits, let’s examine the error dynamics through a simplified model. Suppose a drone formation consists of $N$ drones arranged in a hierarchy. The overall navigation error for the $i$-th drone can be expressed as a function of its layer depth $d_i$, sensor noise, and update rate. Empirically, from the simulation, the position error variance $\sigma_{p,i}^2$ scales approximately with:
$$ \sigma_{p,i}^2 \approx \sigma_0^2 + k \cdot d_i $$
where $\sigma_0^2$ is the base error variance for the leader (Layer 1), and $k$ is a constant dependent on sensor quality and model approximations. For my setup, $k$ was relatively small, leading to the modest increase seen in the results. This linear relationship underscores the importance of minimizing layer depth in drone formation design.
In conclusion, the hierarchical cooperative navigation system presents a viable solution for enhancing the navigation accuracy of large-scale drone formation. By structuring the formation into layers and leveraging relative measurements, the method bounds INS errors, addressing the limitations of traditional master-slave approaches. The simulation results confirm that position and velocity errors remain within acceptable limits over extended durations, meeting the requirements for long-duration drone formation flights. However, challenges such as error propagation across layers and communication constraints warrant further investigation. Future directions include refining the mathematical models to reduce approximations, integrating advanced sensor fusion algorithms, and testing in real-world drone formation scenarios with variable environmental conditions. Ultimately, this research contributes to the advancement of autonomous systems, paving the way for more reliable and efficient drone formation operations in diverse applications, from surveillance to logistics.
Throughout this study, the term “drone formation” has been emphasized to highlight the collective nature of the problem. The hierarchical approach not only improves individual drone navigation but also fosters synergy within the formation, enabling complex missions that single drones cannot accomplish. As drone technology evolves, cooperative navigation systems like this will become increasingly critical for scaling up formations while maintaining precision and robustness. I believe that continued innovation in this area will unlock new potentials for drone formation in both civilian and military domains, making them more adaptable and resilient in the face of growing operational demands.