In recent years, the rapid advancement in unmanned aerial vehicle (UAV) technology has propelled quadrotor drones into the forefront of various applications, from environmental monitoring to disaster response. As a researcher deeply immersed in the field of multi-agent systems, I have observed that single quadrotor drones often face limitations in task efficiency and robustness, especially in complex scenarios. Inspired by collective behaviors in nature, formation flight of multiple quadrotor drones has emerged as a promising solution to enhance operational capabilities. However, achieving precise attitude coordination in such formations under real-world constraints—such as dynamic disturbances and input saturation—remains a significant challenge. This article delves into the design of a distributed attitude cooperative control scheme for quadrotor drone formation flight, leveraging finite-time disturbance observers and anti-saturation techniques to ensure robust performance.
The core problem I address here involves a fleet of N quadrotor drones operating under directed communication topologies, where each drone is subjected to time-varying external disturbances and actuator saturation limits. The attitude dynamics of each quadrotor drone can be modeled using Newton-Euler principles, leading to a nonlinear system that requires careful control design. In my work, I consider the following representation for the i-th quadrotor drone (where i = 1, …, N):
$$ \dot{\Theta}_i = \Omega_i $$
$$ \dot{\Omega}_i = F_i(\Theta_i, \Omega_i) + G_i(\Theta_i, \Omega_i) \text{sat}(U_i) + D_i(\Theta_i, \Omega_i, t) $$
Here, $\Theta_i = [\phi_i, \theta_i, \psi_i]^T$ denotes the attitude angles (roll, pitch, yaw), $\Omega_i = [\dot{\phi}_i, \dot{\theta}_i, \dot{\psi}_i]^T$ represents the angular velocities, and $U_i = [\tau_{\phi i}, \tau_{\theta i}, \tau_{\psi i}]^T$ is the control input vector. The functions $F_i$ and $G_i$ capture the known nonlinear dynamics, while $D_i$ accounts for unknown lumped dynamic disturbances. The saturation function $\text{sat}(U_i)$ models actuator limits, defined as:
$$ \text{sat}(U_i) = \begin{cases}
\overline{U}_i & \text{if } U_i > \overline{U}_i \\
U_i & \text{if } \underline{U}_i \leq U_i \leq \overline{U}_i \\
\underline{U}_i & \text{if } U_i < \underline{U}_i
\end{cases} $$
where $\overline{U}_i$ and $\underline{U}_i$ are the upper and lower saturation bounds, respectively. To tackle this, I propose a comprehensive control framework that integrates finite-time disturbance observation and anti-saturation compensation, ensuring that the quadrotor drone formation can accurately track a virtual leader’s attitude trajectory.
The communication topology among quadrotor drones is represented using directed graph theory. I define an augmented graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{A})$ that includes a virtual leader (indexed as 0) and N follower quadrotor drones. The adjacency matrix $\mathcal{A}$ and Laplacian matrix $\mathcal{L}$ encode the information flow, with a diagonal matrix $\mathcal{B}$ indicating connections to the leader. Under the assumption that the graph contains a directed spanning tree rooted at the leader, the matrix $\mathcal{L} + \mathcal{B}$ is nonsingular—a critical property for achieving consensus in the quadrotor drone formation. To formalize the control objective, I aim to design a distributed controller such that the attitude tracking errors for all follower quadrotor drones converge to a small neighborhood of zero, despite disturbances and input saturation.
In my approach, I first construct an auxiliary anti-saturation system to mitigate the effects of input constraints on the quadrotor drone actuators. This system is defined as:
$$ \dot{\xi}_{\Theta i} = -P_{\Theta i} \xi_{\Theta i} + \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) \xi_{\Omega i} $$
$$ \dot{\xi}_{\Omega i} = -P_{\Omega i} \xi_{\Omega i} – \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) \xi_{\Theta i} + G_i \Delta U_i $$
where $\xi_{\Theta i}$ and $\xi_{\Omega i}$ are auxiliary states, $P_{\Theta i}$ and $P_{\Omega i}$ are positive definite design matrices, and $\Delta U_i = \text{sat}(U_i) – U_i$ represents the saturation error. This mechanism helps maintain system stability when control inputs hit their limits, which is common in practical quadrotor drone operations.
Next, I design a finite-time disturbance observer (FTDO) to estimate the lumped disturbances $D_i$ affecting each quadrotor drone. Traditional observers often only achieve asymptotic convergence, but in dynamic environments, finite-time convergence is desirable for quicker response. My FTDO leverages a high-order sliding-mode differentiator to accurately estimate disturbances within a finite time. The observer dynamics are given by:
$$ \dot{\hat{D}}_i = -h_{1i} |\mu_{1i}|^{1/2} \text{sgn}(\mu_{1i}) – h_{2i} \int_0^t \text{sgn}(\mu_{1i}) d\tau – \lambda_{0i} $$
where $\hat{D}_i$ is the disturbance estimate, $h_{1i}$ and $h_{2i}$ are positive design parameters, $\mu_{1i}$ is an auxiliary variable derived from the tracking error, and $\lambda_{0i}$ is obtained from the differentiator. Under the assumption that the disturbance derivative is bounded, this observer ensures that the estimation error $\tilde{D}_i = \hat{D}_i – D_i$ converges to zero in finite time, significantly enhancing the robustness of the quadrotor drone formation control.
Using backstepping and multi-agent consensus theory, I then develop a distributed attitude cooperative controller for the quadrotor drone formation. The control law for each quadrotor drone is derived as:
$$ U_i = -G_i^{-1} \left[ K_{\Omega i} \tilde{E}_{\Omega i} + F_i + \hat{D}_i – \dot{\Omega}_{id} + P_{\Omega i} \xi_{\Omega i} + \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) (\tilde{E}_{\Theta i} – \xi_{\Theta i}) \right] $$
where $\tilde{E}_{\Theta i}$ and $\tilde{E}_{\Omega i}$ are transformed tracking errors incorporating the anti-saturation states, $K_{\Omega i}$ is a control gain matrix, and $\Omega_{id}$ is a virtual control law designed in the backstepping procedure. This controller ensures that all quadrotor drones in the formation synchronize their attitudes with the leader while compensating for disturbances and saturation effects.
To validate the proposed scheme, I conducted extensive simulation experiments involving a formation of four follower quadrotor drones and one virtual leader. The quadrotor drone parameters were set based on typical values, and external disturbances were modeled as sinusoidal signals. I compared three control protocols: the proposed FTDO-based anti-saturation protocol (SIDO), an FTDO-based protocol without anti-saturation (IDO), and a conventional backstepping protocol (COM). The results, summarized in the table below, demonstrate the superiority of the proposed method in terms of tracking accuracy and input constraint adherence for quadrotor drone formations.
| Control Protocol | Attitude Tracking Error (Max) | Consistency Error (Max) | Input Saturation Violation |
|---|---|---|---|
| SIDO (Proposed) | ~10^{-5} rad | ~10^{-6} rad | None |
| IDO | ~10^{-5} rad | ~10^{-6} rad | Present |
| COM | ~10^{-2} rad | ~10^{-2} rad | Present |
The simulation outcomes clearly indicate that the proposed controller achieves high-precision attitude tracking for quadrotor drone formations, with errors reduced by orders of magnitude compared to conventional methods. Moreover, the anti-saturation mechanism effectively prevents control inputs from exceeding actuator limits, which is crucial for real-world quadrotor drone applications. To further illustrate the performance, I derived key metrics from the simulations, as shown in the formulas below, where the global tracking error bound is given by:
$$ \lim_{t \to \infty} \| \Theta – \Theta_0 \| \leq \frac{\sqrt{2 \gamma / \alpha}}{\sigma_{\min}((\mathcal{L} + \mathcal{B}) \otimes I_3)} $$
Here, $\gamma$ and $\alpha$ are positive constants related to the control parameters and saturation bounds, and $\sigma_{\min}$ denotes the minimum singular value. This bound confirms that the quadrotor drone formation’s tracking errors remain uniformly ultimately bounded.
In addition to the controller design, I explored the theoretical foundations of the approach. Using Lyapunov stability theory, I proved that the closed-loop system for the quadrotor drone formation is globally uniformly ultimately bounded. The Lyapunov function candidate was chosen as:
$$ V = \sum_{i=1}^N \left( \frac{1}{2} \tilde{E}_{\Theta i}^T \tilde{E}_{\Theta i} + \frac{1}{2} \xi_{\Theta i}^T \xi_{\Theta i} + \frac{1}{2} \tilde{E}_{\Omega i}^T \tilde{E}_{\Omega i} + \frac{1}{2} \xi_{\Omega i}^T \xi_{\Omega i} \right) $$
Through rigorous analysis, I showed that its derivative satisfies $\dot{V} \leq -\alpha V + \gamma$, ensuring stability. This analysis underscores the robustness of the proposed method for quadrotor drone formations operating under adverse conditions.
Another critical aspect is the finite-time convergence of the disturbance observer. By employing the high-order sliding-mode differentiator, the observer error dynamics can be expressed as:
$$ \dot{\mu}_{1i} + h_{1i} |\mu_{1i}|^{1/2} \text{sgn}(\mu_{1i}) + h_{2i} \int_0^t \text{sgn}(\mu_{1i}) d\tau = \Xi_i $$
where $\Xi_i$ is a bounded perturbation. According to the principles of finite-time stability, if the gains $h_{1i}$ and $h_{2i}$ are chosen appropriately, the error converges to zero within a finite time $T$, enhancing the responsiveness of the quadrotor drone control system.
To provide a comprehensive overview, I have summarized the key components of the control design in the following table, which highlights the roles of each element in the quadrotor drone formation control scheme.
| Component | Function | Mathematical Form | Impact on Quadrotor Drone Performance |
|---|---|---|---|
| Auxiliary Anti-Saturation System | Mitigates input saturation effects | $\dot{\xi}_{\Theta i} = -P_{\Theta i} \xi_{\Theta i} + (\sum a_{ij} + b_{i0}) \xi_{\Omega i}$ | Prevents actuator damage and maintains stability |
| Finite-Time Disturbance Observer | Estimates unknown disturbances | $\dot{\hat{D}}_i = -h_{1i} |\mu_{1i}|^{1/2} \text{sgn}(\mu_{1i}) – h_{2i} \int_0^t \text{sgn}(\mu_{1i}) d\tau – \lambda_{0i}$ | Enhances robustness and accuracy in dynamic environments |
| Distributed Cooperative Controller | Ensures attitude synchronization | $U_i = -G_i^{-1}[K_{\Omega i} \tilde{E}_{\Omega i} + F_i + \hat{D}_i – \dot{\Omega}_{id} + P_{\Omega i} \xi_{\Omega i} + (\sum a_{ij} + b_{i0}) (\tilde{E}_{\Theta i} – \xi_{\Theta i})]$ | Achieves precise formation tracking under directed topologies |
| Virtual Control Law | Facilitates backstepping design | $\Omega_{id} = \frac{1}{\sum a_{ij} + b_{i0}} (-K_{\Theta i} \tilde{E}_{\Theta i} + \sum a_{ij} \Omega_j + b_{i0} \dot{\Theta}_0 – P_{\Theta i} \xi_{\Theta i})$ | Simplifies control synthesis and improves tracking performance |
Throughout this work, I have emphasized the importance of addressing real-world challenges in quadrotor drone formation flight. The integration of finite-time disturbance observation and anti-saturation control represents a significant step forward in achieving reliable and efficient multi-quadrotor drone operations. Future research could extend this framework to include trajectory tracking, obstacle avoidance, and adaptive learning mechanisms for further enhancing the autonomy of quadrotor drone formations. In conclusion, the proposed distributed attitude cooperative control scheme offers a viable solution for robust formation flight of quadrotor drones, paving the way for more advanced applications in surveillance, logistics, and collaborative missions.