In recent years, quadrotor unmanned aerial vehicles (UAVs) have garnered significant attention due to their compact size, high maneuverability, and versatility in applications such as surveillance, inspection, and disaster response. However, single quadrotor systems face limitations in efficiency and mission coverage, prompting the exploration of multi-agent systems inspired by collective behaviors in nature. Quadrotor formation flight, as a practical embodiment of multi-agent systems, offers enhanced operational capabilities, including improved task efficiency and robust performance in dynamic environments. This paper addresses the distributed attitude cooperative control problem for a class of quadrotor UAV formation systems (QUAVFS) subject to dynamic disturbances and input saturation constraints under directed communication topologies. The primary objective is to design a control protocol that ensures all follower quadrotors accurately track the attitude of a virtual leader quadrotor, with bounded stability guarantees for the closed-loop system.
The dynamics of a quadrotor are inherently nonlinear and coupled, making attitude control a fundamental challenge, especially in formation settings. For a formation of N quadrotors, the attitude dynamics of the i-th quadrotor can be described using the Newton-Euler formulation. Let $\Theta_i = [\phi_i, \theta_i, \psi_i]^T$ represent the roll, pitch, and yaw angles, and $\Omega_i = [\dot{\phi}_i, \dot{\theta}_i, \dot{\psi}_i]^T$ denote the angular velocities. The dynamics are given by:
$$\ddot{\phi}_i = \frac{(J_{yi} – J_{zi}) \dot{\theta}_i \dot{\psi}_i}{J_{xi}} + \frac{\tau_{\phi i}}{J_{xi}}$$
$$\ddot{\theta}_i = \frac{(J_{zi} – J_{xi}) \dot{\phi}_i \dot{\psi}_i}{J_{yi}} + \frac{\tau_{\theta i}}{J_{yi}}$$
$$\ddot{\psi}_i = \frac{(J_{xi} – J_{yi}) \dot{\theta}_i \dot{\phi}_i}{J_{zi}} + \frac{\tau_{\psi i}}{J_{zi}}$$
where $J_{xi}$, $J_{yi}$, and $J_{zi}$ are the moments of inertia about the body axes, and $\tau_{\phi i}$, $\tau_{\theta i}$, $\tau_{\psi i}$ are the control torques. In a more compact form, considering external disturbances and input saturation, the dynamics can be expressed as:
$$\dot{\Theta}_i = \Omega_i$$
$$\dot{\Omega}_i = F_i(\Theta_i, \Omega_i) + G_i(\Theta_i, \Omega_i) \text{sat}(U_i) + D_i(\Theta_i, \Omega_i, t)$$
Here, $F_i$ and $G_i$ are known nonlinear functions derived from the quadrotor dynamics, $D_i$ represents the lumped dynamic disturbances, and $\text{sat}(U_i)$ denotes the saturated control input due to actuator limitations. The saturation function is 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 limits, respectively. The communication topology among the quadrotors is represented by a directed graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, A)$, where $\mathcal{V} = \{1, 2, \dots, N\}$ is the set of nodes (quadrotors), $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of edges, and $A = [a_{ij}]$ is the adjacency matrix. The Laplacian matrix $L$ is defined as $L = C – A$, where $C$ is the degree matrix. To incorporate a virtual leader quadrotor (indexed as 0) that generates the desired attitude trajectory $\Theta_0(t)$, an augmented graph $\overline{\mathcal{G}} = (\overline{\mathcal{V}}, \overline{\mathcal{E}}, A)$ is used, with a diagonal matrix $B = \text{diag}(b_{10}, \dots, b_{N0})$ indicating the communication links from the leader to followers.
The control design must address several challenges: (1) compensation of dynamic disturbances $D_i$, which are time-varying and unknown; (2) mitigation of input saturation effects to prevent performance degradation; and (3) achieving distributed coordination under directed communication topologies. To this end, a finite-time disturbance observer based on a high-order sliding-mode differentiator is developed to estimate $D_i$ accurately and ensure convergence of estimation errors to zero in finite time. Additionally, an auxiliary anti-saturation system is constructed to handle input constraints. The control protocol is designed using backstepping techniques and multi-agent consensus theory, ensuring that the attitude tracking errors converge to a small neighborhood of zero.
The finite-time disturbance observer for the quadrotor formation is designed as follows. Define an auxiliary variable $\mu_{0i} = Z_{\Omega i} – \tilde{E}_{\Omega i}$, where $\tilde{E}_{\Omega i}$ is a tracking error term. Then, the observer dynamics are:
$$\dot{\hat{D}}_i = -h_{1i} |\mu_{1i}|^{1/2} \text{sgn}(\mu_{1i}) – h_{2i} \int_0^t \text{sgn}(\mu_{1i}) d\tau – \lambda_{0i}$$
where $\hat{D}_i$ is the estimate of $D_i$, $h_{1i}$ and $h_{2i}$ are positive design parameters, and $\lambda_{0i}$ is obtained from a sliding-mode differentiator. This observer guarantees that the estimation error $\tilde{D}_i = \hat{D}_i – D_i$ converges to zero in finite time under suitable conditions.
The auxiliary anti-saturation system for each quadrotor is given by:
$$\dot{\xi}_{\Theta i} = -P_{\Theta i} \xi_{\Theta i} + \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) \xi_{\Omega i}$$
$$\dot{\xi}_{\Omega i} = -P_{\Omega i} \xi_{\Omega i} – \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) \xi_{\Theta i} + G_i \Delta U_i$$
where $\xi_{\Theta i}$ and $\xi_{\Omega i}$ are internal states, $P_{\Theta i}$ and $P_{\Omega i}$ are positive definite matrices, and $\Delta U_i = \text{sat}(U_i) – U_i$ represents the saturation error. This system helps to compensate for the effects of input saturation.
The distributed attitude cooperative control protocol is derived using backstepping. First, define the coordinated tracking errors:
$$\tilde{E}_{\Theta i} = \sum_{j \in \mathcal{N}_i} a_{ij} (\Theta_i – \Theta_j) + b_{i0} (\Theta_i – \Theta_0) – \xi_{\Theta i}$$
$$\tilde{E}_{\Omega i} = \Omega_i – \Omega_{id} – \xi_{\Omega i}$$
where $\Omega_{id}$ is a virtual control law designed as:
$$\Omega_{id} = \frac{1}{\sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0}} \left( -K_{\Theta i} \tilde{E}_{\Theta i} + \sum_{j \in \mathcal{N}_i} a_{ij} \Omega_j + b_{i0} \dot{\Theta}_0 – P_{\Theta i} \xi_{\Theta i} \right)$$
with $K_{\Theta i} > 0$ being a gain matrix. The actual control law for the quadrotor is then:
$$U_i = -G_i^{-1} \left[ K_{\Omega i} \tilde{E}_{\Omega i} + F_i + \hat{D}_i – \dot{\Omega}_{id} + P_{\Omega i} \xi_{\Omega i} + \left( \sum_{j \in \mathcal{N}_i} a_{ij} + b_{i0} \right) (\tilde{E}_{\Theta i} – \xi_{\Theta i}) \right]$$
where $K_{\Omega i} > 0$ is another gain matrix. This control protocol ensures that the closed-loop system is globally uniformly ultimately bounded, and the attitude tracking errors converge to a small region around zero.
To validate the proposed approach, numerical simulations are conducted for a formation of four follower quadrotors and one virtual leader under a directed communication topology. The quadrotor parameters are standardized across all agents, with moments of inertia $J_{xi} = J_{yi} = 6.23 \times 10^{-3} \, \text{N·m·s}^2/\text{rad}$ and $J_{zi} = 1.12 \times 10^{-3} \, \text{N·m·s}^2/\text{rad}$. The external disturbances are set as $D_i = [0.3 \sin(\phi_i t) + 1.5 \sin(0.5t), 0.3 \sin(\theta_i t) + 1.5 \sin(0.5t), 0.3 \sin(\psi_i t) + 1.5 \sin(0.5t)]^T$. The desired attitude trajectory from the leader is $\Theta_0 = [0.2 \sin(0.5t), 0.2 \sin(0.5t), 0.2 \sin(0.5t)]^T$. The control parameters are chosen as $K_{\Theta i} = \text{diag}(500, 500, 500)$, $K_{\Omega i} = \text{diag}(500, 500, 500)$, $h_{1i} = 0.01$, $h_{2i} = 0.001$, $P_{\Theta i} = \text{diag}(3, 3, 3)$, and $P_{\Omega i} = \text{diag}(1, 1, 1)$. The saturation limits are $\overline{\tau}_{\phi i} = \overline{\tau}_{\theta i} = 0.04 \, \text{N·m}$, $\underline{\tau}_{\phi i} = \underline{\tau}_{\theta i} = -0.04 \, \text{N·m}$, $\overline{\tau}_{\psi i} = 0.008 \, \text{N·m}$, and $\underline{\tau}_{\psi i} = -0.008 \, \text{N·m}$.
The simulation results demonstrate the effectiveness of the proposed control protocol. The following table summarizes the key performance metrics for the quadrotor formation, including the maximum attitude tracking errors and control input magnitudes under different scenarios:
| Quadrotor Index | Max Roll Error (rad) | Max Pitch Error (rad) | Max Yaw Error (rad) | Control Effort (N·m) |
|---|---|---|---|---|
| 1 | 2.1e-5 | 1.8e-5 | 3.2e-5 | 0.038 |
| 2 | 1.9e-5 | 2.3e-5 | 2.9e-5 | 0.039 |
| 3 | 2.4e-5 | 2.0e-5 | 3.1e-5 | 0.037 |
| 4 | 2.2e-5 | 2.1e-5 | 3.0e-5 | 0.040 |
The attitude tracking errors for all quadrotors converge to values on the order of $10^{-5}$ rad, indicating high precision. The control inputs remain within the saturation limits, validating the anti-saturation design. The consistency errors, defined as the disagreements among quadrotors, are also minimized, as shown in the table below:
| Error Type | Max Value | Convergence Time (s) |
|---|---|---|
| Attitude Tracking Error | 3.2e-5 rad | 2.5 |
| Angular Velocity Error | 4.1e-4 rad/s | 3.0 |
| Consistency Error | 5.6e-6 | 2.8 |
The finite-time disturbance observer performance is analyzed through the estimation error dynamics. The observer ensures that $\tilde{D}_i$ converges to zero within approximately 1.5 seconds, as per the following equation derived from the Lyapunov analysis:
$$\dot{V}_D \leq -\alpha_D V_D + \gamma_D$$
where $V_D$ is a Lyapunov function for the disturbance estimation error, $\alpha_D$ and $\gamma_D$ are positive constants. The convergence time $T_c$ satisfies $T_c \leq \frac{7.6 V_D(0)}{\alpha_D – \gamma_D}$.
The stability of the closed-loop system is proven using Lyapunov theory. Consider the Lyapunov function candidate:
$$V = \sum_{i=1}^N \left( \frac{1}{2} \tilde{E}_{\Theta i}^T \tilde{E}_{\Theta i} + \frac{1}{2} \xi_{\Theta i}^T \xi_{\Theta i} + \frac{1}{2} \tilde{E}_{\Omega i}^T \tilde{E}_{\Omega i} + \frac{1}{2} \xi_{\Omega i}^T \xi_{\Omega i} \right)$$
Its derivative along the trajectories of the system is bounded by:
$$\dot{V} \leq -\alpha V + \gamma$$
where $\alpha = \min\{ \lambda_{\min}(K_{\Theta i}), \lambda_{\min}(K_{\Omega i} – 0.5 I_3), \lambda_{\min}(P_{\Theta i}), \lambda_{\min}(P_{\Omega i} – 0.5 I_3) \}$ and $\gamma = 0.5 \sum_{i=1}^N \| G_i \|^2 \Delta \overline{U}_i^2$. This ensures that all signals are globally uniformly ultimately bounded, and the tracking errors satisfy:
$$\lim_{t \to \infty} \| \Theta – \Theta_0 \| \leq \frac{\sqrt{2 \gamma / \alpha}}{\sigma_{\min}((L + B) \otimes I_3)}$$
where $\sigma_{\min}$ denotes the minimum singular value.
In comparison with traditional backstepping control (COM) and a method using an infinite-time disturbance observer (IDO), the proposed approach (SIDO) shows superior performance in handling disturbances and input saturation. The following table highlights the key differences:
| Control Method | Disturbance Rejection | Input Saturation Handling | Convergence Time (s) |
|---|---|---|---|
| COM | Limited | Poor | >5.0 |
| IDO | Moderate | Moderate | 3.5 |
| SIDO | Excellent | Excellent | 2.5 |
The proposed control strategy for quadrotor formation flight effectively addresses the challenges of dynamic disturbances and input saturation under directed communication topologies. The finite-time disturbance observer ensures rapid and accurate estimation of uncertainties, while the auxiliary anti-saturation system mitigates the effects of actuator limitations. The distributed control protocol, based on backstepping and consensus theory, guarantees bounded stability and precise attitude tracking. Future work will explore trajectory tracking cooperative control and multi-constraint coordination for quadrotor formations, extending the approach to more complex scenarios such as obstacle avoidance and fault tolerance.
In summary, this paper contributes to the field of quadrotor UAV formation control by providing a comprehensive solution that integrates finite-time disturbance observation, anti-saturation mechanisms, and distributed coordination. The theoretical analysis and simulation results confirm the feasibility and effectiveness of the proposed method, paving the way for advanced applications in autonomous systems. The repeated emphasis on quadrotor technologies underscores their importance in modern aerospace engineering, and the developed framework can be adapted to various multi-agent systems beyond UAVs.