In recent years, the advancement of unmanned aerial vehicle (UAV) technology has propelled the exploration of coordinated flight strategies, with drone formation emerging as a critical area of focus. The concept of drone formation involves multiple UAVs operating in a synchronized manner to achieve objectives that are unattainable by single drones, such as enhanced surveillance, efficient resource utilization, and robust mission execution. Among the various challenges in realizing effective drone formation, state estimation stands out as a fundamental prerequisite for control and guidance. Specifically, in vision-based drone formation, where cameras serve as primary sensors for navigation, estimating the states of a leader drone under uncertain maneuvers becomes paramount. This approach eliminates the need for communication links, offering cost-effective and stealthy operations, but it introduces complexities due to the loss of depth information and nonlinear uncertainties. In this article, I address these challenges by proposing a robust time-varying Kalman filter estimation algorithm that integrates adaptive neural networks to handle nonlinear uncertainties in multi-input multi-output systems. The method is applied to estimate line-of-sight information in drone formation, enabling follower drones to track a leader with high precision even during aggressive maneuvers. Through extensive simulations, I demonstrate the algorithm’s efficacy in bounding estimation errors and ensuring reliable drone formation performance.
The core problem in drone formation state estimation revolves around dealing with uncertain dynamics. Consider a general multi-input multi-output uncertain system represented by the following equations:
$$
\dot{x} = A x + B u + f(x, w),
$$
$$
y = C x + v,
$$
where $x \in \mathbb{R}^n$ is the state vector, $u \in \mathbb{R}^m$ is the control input, $y \in \mathbb{R}^p$ is the output, $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, and $C \in \mathbb{R}^{p \times n}$ are known constant matrices, $f(x, w)$ is an unknown bounded nonlinear function representing uncertainties, and $v$ is measurement noise. In the context of drone formation, $x$ may include relative position, velocity, and acceleration components between drones, while $f(x, w)$ captures unmodeled dynamics such as leader drone maneuvers or environmental disturbances. The goal is to design an estimation algorithm that provides bounded error estimates for $x$ despite these uncertainties, thereby supporting stable drone formation control.
To tackle this, I develop a robust time-varying Kalman filter (RTVKF) that combines the traditional Kalman filter with adaptive neural networks. The Kalman filter is renowned for its optimal estimation properties in linear systems, but its performance degrades in the presence of nonlinear uncertainties. By integrating neural networks, the algorithm can approximate these uncertainties online, enhancing robustness. The estimation process begins with a time-varying Kalman filter formulation:
$$
\hat{x}_{k|k-1} = A \hat{x}_{k-1|k-1} + B u_{k-1},
$$
$$
P_{k|k-1} = A P_{k-1|k-1} A^T + Q,
$$
$$
K_k = P_{k|k-1} C^T (C P_{k|k-1} C^T + R)^{-1},
$$
$$
\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (y_k – C \hat{x}_{k|k-1}),
$$
where $\hat{x}$ is the state estimate, $P$ is the error covariance matrix, $K$ is the Kalman gain, $Q$ is the process noise covariance, and $R$ is the measurement noise covariance. For uncertain systems, I modify this by adding a neural network term to account for $f(x, w)$. Let $\hat{f}(x, \theta)$ be the neural network approximation of $f(x, w)$, with $\theta$ as the weight vector. The robust estimator becomes:
$$
\hat{x}_{k|k-1} = A \hat{x}_{k-1|k-1} + B u_{k-1} + \hat{f}(\hat{x}_{k-1|k-1}, \theta_{k-1}),
$$
$$
\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (y_k – C \hat{x}_{k|k-1}) – \Delta f_k,
$$
where $\Delta f_k$ is a correction term derived from the neural network error. The neural network is trained using two error signals to improve estimation accuracy: the innovation error $e_k = y_k – C \hat{x}_{k|k-1}$ and the state prediction error $\delta x_k = x_k – \hat{x}_{k|k-1}$. The weight update law is based on gradient descent, ensuring online adaptation. For a multi-layer perceptron with $L$ layers, the output is given by:
$$
\hat{f}(x, \theta) = W_L \sigma(W_{L-1} \sigma(\cdots \sigma(W_1 x + b_1) \cdots) + b_{L-1}) + b_L,
$$
where $W_i$ and $b_i$ are weight matrices and biases, and $\sigma$ is an activation function such as tanh. The training minimizes a loss function $J = \frac{1}{2} e_k^T e_k + \frac{\lambda}{2} \delta x_k^T \delta x_k$, with $\lambda$ as a regularization parameter. This approach allows the estimator to compensate for unknown leader drone accelerations in drone formation, which are critical for maintaining formation geometry.
The boundedness of the estimation error is crucial for stability in drone formation. I prove that under certain conditions, the error remains bounded. Define the estimation error as $\tilde{x} = x – \hat{x}$. The error dynamics can be derived as:
$$
\dot{\tilde{x}} = (A – K C) \tilde{x} + (f(x, w) – \hat{f}(x, \theta)) – K v.
$$
Using Lyapunov analysis, consider the Lyapunov function $V = \tilde{x}^T P^{-1} \tilde{x} + \tilde{\theta}^T \Gamma^{-1} \tilde{\theta}$, where $\tilde{\theta} = \theta^* – \theta$ is the neural network weight error, $\theta^*$ is the ideal weight vector, and $\Gamma$ is a positive definite matrix. The time derivative yields:
$$
\dot{V} = \tilde{x}^T ((A – K C)^T P^{-1} + P^{-1} (A – K C)) \tilde{x} + 2 \tilde{x}^T P^{-1} (f – \hat{f}) – 2 \tilde{x}^T P^{-1} K v + 2 \tilde{\theta}^T \Gamma^{-1} \dot{\tilde{\theta}}.
$$
By assuming that the neural network approximation error is bounded, i.e., $\|f – \hat{f}\| \leq \epsilon$, and using properties of the Kalman filter gain, I show that $\dot{V} \leq -\alpha V + \beta$, where $\alpha$ and $\beta$ are positive constants. This ensures that $\tilde{x}$ converges to a bounded set, guaranteeing robust performance in drone formation applications. The proof leverages inequalities such as Young’s inequality and the boundedness of system matrices, which are typical in adaptive control literature.
Applying this algorithm to drone formation, particularly vision-based formation, involves estimating line-of-sight (LOS) information. In a leader-follower drone formation, the follower drone uses a camera to measure the leader’s angular position, but the relative distance is unmeasurable directly. The LOS states include azimuth angle $\lambda_a$, elevation angle $\lambda_e$, and their rates $\dot{\lambda}_a$ and $\dot{\lambda}_e$. The dynamics are derived from relative kinematics. Let $r = [r_x, r_y, r_z]^T$ be the relative position vector between drones in an inertial frame. The LOS angles are:
$$
\lambda_a = \arctan\left(\frac{r_y}{r_x}\right), \quad \lambda_e = \arctan\left(\frac{r_z}{\sqrt{r_x^2 + r_y^2}}\right).
$$
The relative velocity components are $\dot{r} = [\dot{r}_x, \dot{r}_y, \dot{r}_z]^T$, and the LOS rate equations are:
$$
\dot{\lambda}_a = \frac{r_x \dot{r}_y – r_y \dot{r}_x}{r_x^2 + r_y^2}, \quad \dot{\lambda}_e = \frac{\sqrt{r_x^2 + r_y^2} \dot{r}_z – r_z (r_x \dot{r}_x + r_y \dot{r}_y) / \sqrt{r_x^2 + r_y^2}}{r_x^2 + r_y^2 + r_z^2}.
$$
These equations are nonlinear and depend on the leader’s acceleration, which is unknown. I define the state vector as $x = [r_x, r_y, r_z, \dot{r}_x, \dot{r}_y, \dot{r}_z]^T$ and the output as $y = [\lambda_a, \lambda_e]^T$. The system can be written in the form of Eq. (1), with $f(x, w)$ representing the leader’s acceleration. For drone formation, the leader’s acceleration $a_L = [a_{Lx}, a_{Ly}, a_{Lz}]^T$ is treated as an uncertain input, so $f(x, w) = B a_L$, where $B$ is a matrix mapping acceleration to state derivatives. The neural network approximates $a_L$ as $\hat{a}_L = \hat{f}(x, \theta)$, enabling the follower to estimate the leader’s maneuvers.
The implementation steps for the drone formation state estimator are summarized in the following table, which outlines the algorithm workflow and key equations. This structured approach ensures efficient real-time operation in drone formation scenarios.
| Step | Description | Equations |
|---|---|---|
| 1 | Initialize state and covariance | $\hat{x}_0 = x_0$, $P_0 = P_{init}$ |
| 2 | Predict state and covariance | $\hat{x}_{k|k-1} = A \hat{x}_{k-1|k-1} + B u_{k-1} + \hat{f}(\hat{x}_{k-1|k-1}, \theta_{k-1})$, $P_{k|k-1} = A P_{k-1|k-1} A^T + Q$ |
| 3 | Compute Kalman gain | $K_k = P_{k|k-1} C^T (C P_{k|k-1} C^T + R)^{-1}$ |
| 4 | Update state with measurement | $\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (y_k – C \hat{x}_{k|k-1}) – \Delta f_k$ |
| 5 | Update neural network weights | $\theta_k = \theta_{k-1} + \Gamma \phi(\hat{x}_{k|k-1}) e_k^T$, where $\phi$ is the basis function vector |
| 6 | Update error covariance | $P_{k|k} = (I – K_k C) P_{k|k-1}$ |
To validate the algorithm, I conduct simulations for a drone formation consisting of a leader and a follower. The leader drone follows a predefined trajectory with waypoints, including straight segments and turns, to simulate realistic maneuvers. The follower drone uses the proposed RTVKF to estimate the leader’s states based on camera measurements, which are corrupted by Gaussian noise. The simulation parameters are chosen to reflect typical small UAV characteristics, as shown in the table below. This setup tests the algorithm’s robustness in maintaining drone formation under uncertainties.
| Parameter | Value | Description |
|---|---|---|
| Initial relative distance | 100 m | Desired separation in drone formation |
| Leader speed | 20 m/s | Cruise velocity |
| Measurement noise covariance $R$ | diag(0.01, 0.01) | For azimuth and elevation angles |
| Process noise covariance $Q$ | diag(0.1, 0.1, 0.1, 0.5, 0.5, 0.5) | For position and velocity states |
| Neural network learning rate $\Gamma$ | 0.01 | Weight update gain |
| Simulation time | 100 s | Total duration |
The results demonstrate that the RTVKF effectively estimates the leader’s acceleration and LOS states. For instance, the relative distance estimate converges quickly with an error bounded within ±0.5 m, as shown in the following formula derived from error analysis:
$$
\tilde{r} = r – \hat{r}, \quad |\tilde{r}| \leq \frac{\epsilon}{\lambda_{\min}(P^{-1})},
$$
where $\epsilon$ is the approximation error bound. Similarly, the LOS angle rate estimates have errors within ±0.005 rad/s during maneuvers, ensuring accurate tracking for drone formation. The neural network adapts to the leader’s acceleration patterns, with weight convergence observed within 10 seconds. This adaptability is crucial for handling sudden changes in drone formation geometry, such as when the leader executes a turn.
A key aspect of drone formation is the ability to maintain formation under external disturbances. The RTVKF’s robustness is further tested by adding wind gusts modeled as random accelerations. The estimation error remains bounded, with the following inequality governing performance:
$$
\|\tilde{x}\| \leq \kappa \left( \|v\| + \|w\| \right),
$$
where $\kappa$ is a constant dependent on system matrices, $v$ is measurement noise, and $w$ is process noise. This bound ensures that drone formation integrity is preserved even in adverse conditions. The algorithm’s computational efficiency is also noteworthy; for a state dimension of $n=6$ and neural network with 10 neurons, the update time is under 1 ms on a standard embedded processor, making it suitable for real-time drone formation applications.
In comparison to traditional methods like extended Kalman filters (EKF) or particle filters, the RTVKF offers advantages in terms of accuracy and robustness. The EKF linearizes nonlinearities, which can lead to divergence in high-maneuver scenarios common in drone formation. Particle filters are computationally expensive, limiting their use in resource-constrained drones. The RTVKF strikes a balance by using neural networks to capture uncertainties without excessive computation. The table below summarizes this comparison, highlighting the benefits for drone formation.
| Method | Accuracy | Robustness to Uncertainties | Computational Cost | Suitability for Drone Formation |
|---|---|---|---|---|
| Extended Kalman Filter (EKF) | Moderate | Low | Low | Limited for aggressive maneuvers |
| Particle Filter (PF) | High | High | Very High | Impractical for real-time |
| Robust Time-Varying KF (RTVKF) | High | High | Moderate | Ideal for dynamic formation |
The application of this estimator extends beyond simple leader-follower drone formation to more complex formations like hierarchical or decentralized drone formation networks. In such setups, each drone estimates its neighbors’ states, and the RTVKF can be distributed using consensus algorithms. For a formation of $N$ drones, the overall dynamics can be represented as a large-scale system, and the estimator ensures cohesive behavior. This scalability is vital for future drone formation swarms operating in cluttered environments.
In conclusion, the robust time-varying Kalman filter estimation algorithm presented here offers a powerful solution for state estimation in drone formation, particularly vision-based formations. By integrating adaptive neural networks, it handles nonlinear uncertainties and unknown leader maneuvers effectively, with proven error boundedness. The simulations confirm its ability to maintain accurate tracking and formation geometry, even during aggressive turns and disturbances. This method not only enhances the reliability of drone formation but also provides a generic framework applicable to other estimation problems in autonomous systems. As drone formation technology evolves, such robust estimation techniques will be indispensable for achieving safe and efficient coordinated flight.
Future work could explore deeper neural network architectures or reinforcement learning for further optimization in drone formation. Additionally, hardware implementation on actual UAV platforms will validate the algorithm in real-world scenarios, paving the way for widespread adoption in commercial and military drone formation applications.