In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in applications such as environmental monitoring, search and rescue, and surveillance. The ability to form coordinated multi-quadrotor systems enhances their operational capabilities, but challenges like communication link failures can disrupt formation stability. This paper addresses the issue of communication link faults in quadrotor UAV formations by proposing an adaptive leader-state observer combined with a sliding mode control strategy. The observer estimates the leader’s states despite communication uncertainties, while the controller ensures rapid and accurate tracking of desired formation trajectories. Through extensive simulations, the method demonstrates improved convergence and fault tolerance compared to traditional consensus-based approaches.
The dynamics of a quadrotor UAV are inherently nonlinear and coupled, making formation control a complex task. Each quadrotor in the formation acts as a follower, relying on information from a leader quadrotor to maintain desired relative positions. However, in practical scenarios, communication links between quadrotors can suffer from faults due to environmental interference or hardware limitations, leading to formation errors. Traditional control methods often assume perfect communication, which is unrealistic. This work introduces an adaptive observer that compensates for communication link faults by estimating the leader’s states distributively. The observer incorporates a fault compensation term and an adaptive gain to enhance estimation accuracy. Subsequently, a sliding mode controller based on a variable exponential power reaching law is designed to achieve finite-time convergence of tracking errors. The synergy between the observer and controller ensures robust formation control even under adverse conditions.
The quadrotor UAV model is derived using Euler-Lagrange equations, considering both translational and rotational dynamics. For the i-th follower quadrotor, the dynamics are represented as:
$$ \begin{cases} \ddot{x}_i = \frac{u_{1i}}{m} (\cos \psi_i \sin \theta_i \cos \phi_i + \sin \psi_i \sin \phi_i) – \frac{K_x}{m} \dot{x}_i + d_{1i} \\ \ddot{y}_i = \frac{u_{1i}}{m} (\sin \psi_i \sin \theta_i \cos \phi_i – \cos \psi_i \sin \phi_i) – \frac{K_y}{m} \dot{y}_i + d_{2i} \\ \ddot{z}_i = \frac{u_{1i}}{m} \cos \theta_i \cos \phi_i – g – \frac{K_z}{m} \dot{z}_i + d_{3i} \\ \ddot{\phi}_i = \frac{u_{2i} l}{I_x} – \frac{K_\phi}{I_x} \dot{\phi}_i + d_{4i} \\ \ddot{\theta}_i = \frac{u_{3i} l}{I_y} – \frac{K_\theta}{I_y} \dot{\theta}_i + d_{5i} \\ \ddot{\psi}_i = \frac{u_{4i}}{I_z} – \frac{K_\psi}{I_z} \dot{\psi}_i + d_{6i} \end{cases} $$
where \(x_i, y_i, z_i\) denote the position coordinates, \(\phi_i, \theta_i, \psi_i\) represent the roll, pitch, and yaw angles, \(m\) is the mass, \(l\) is the arm length, \(I_x, I_y, I_z\) are moments of inertia, \(K\) terms are drag coefficients, \(u_{1i}\) to \(u_{4i}\) are control inputs, and \(d_{ji}\) are external disturbances. The control inputs for position and attitude are derived through feedback linearization. For instance, the position control inputs are defined as:
$$ \begin{cases} u_{1xi} = u_{1i} (\cos \psi_i \sin \theta_i \cos \phi_i + \sin \psi_i \sin \phi_i) \\ u_{1yi} = u_{1i} (\sin \psi_i \sin \theta_i \cos \phi_i – \cos \psi_i \sin \phi_i) \\ u_{1zi} = u_{1i} \cos \theta_i \cos \phi_i \end{cases} $$
This simplifies the position dynamics to:
$$ \begin{cases} \ddot{x}_i = \frac{u_{1xi}}{m} – \frac{K_x}{m} \dot{x}_i \\ \ddot{y}_i = \frac{u_{1yi}}{m} – \frac{K_y}{m} \dot{y}_i \\ \ddot{z}_i = \frac{u_{1zi}}{m} – g – \frac{K_z}{m} \dot{z}_i \end{cases} $$
The leader quadrotor’s state is represented as \(\mathbf{x}_0 = [p_0, v_0, \psi_0, \dot{\psi}_0]^T \in \mathbb{R}^8\), where \(p_0\) and \(v_0\) are position and velocity vectors. The leader’s dynamics are linearized as:
$$ \begin{cases} \dot{\mathbf{x}}_0 = \mathbf{A} \mathbf{x}_0 \\ \mathbf{y}_0 = \mathbf{C} \mathbf{x}_0 \end{cases} $$
with \(\mathbf{A} = \begin{bmatrix} \mathbf{0}_{4 \times 4} & \mathbf{I}_{4 \times 4} \\ \mathbf{0}_{4 \times 4} & \mathbf{0}_{4 \times 4} \end{bmatrix}\) and \(\mathbf{C} = \mathbf{I}_{8 \times 8}\). Communication link faults are modeled using graph theory, where the adjacency matrix elements vary due to faults:
$$ a_{ij} = \bar{a}_{ij} + \Delta a_{ij}, \quad \Delta a_{ij} = l \sin(t) \bar{a}_{ij} $$
$$ g_i = \bar{g}_i + \Delta g_i, \quad \Delta g_i = g’ \sin(t) \bar{g}_i $$
Here, \(l\) and \(g’\) are time-varying parameters representing fault magnitudes, and \(\bar{a}_{ij}\), \(\bar{g}_i\) are nominal values. The Laplacian matrix \(\mathbf{L}\) and leader adjacency matrix \(\mathbf{G}\) are constructed accordingly, ensuring the graph remains connected despite faults.
The adaptive leader-state observer for each follower quadrotor is designed as:
$$ \begin{cases} \dot{\hat{\mathbf{x}}}_i = \mathbf{A} \hat{\mathbf{x}}_i – \mathbf{K}_1 (\hat{\mathbf{y}}_i – \mathbf{y}_i) – \mathbf{K}_2 \zeta_i – \mathbf{K}_3(t) \mathbf{P}^{-1} \mathbf{C}^T \text{sgn}(\mathbf{C} \hat{\mathbf{x}}_i – \mathbf{y}_i) \\ \hat{\mathbf{y}}_i = \mathbf{C} \hat{\mathbf{x}}_i \end{cases} $$
where \(\hat{\mathbf{x}}_i\) is the estimated state, \(\mathbf{K}_1\), \(\mathbf{K}_2\), and \(\mathbf{K}_3(t)\) are gain matrices, \(\mathbf{P}\) is a positive definite matrix, and \(\zeta_i\) is the distributed output error:
$$ \zeta_i = \sum_{j \in N_i} a_{ij} (\hat{\mathbf{y}}_j – \hat{\mathbf{y}}_i) + g_i (\mathbf{y}_0 – \hat{\mathbf{y}}_i) $$
The adaptive gain \(\mathbf{K}_3(t)\) is updated as:
$$ \dot{K}_{3i}(t) = \lambda \eta \| \mathbf{y}_0 – \hat{\mathbf{y}}_i \| – \sigma K_{3i}(t) $$
with \(\lambda, \eta, \sigma > 0\). The global observer error dynamics are analyzed using Lyapunov stability theory. Consider the Lyapunov function \(V = \mathbf{e}^T (\mathbf{I}_N \otimes \mathbf{P}) \mathbf{e}\), where \(\mathbf{e} = \hat{\mathbf{x}} – \mathbf{1}_N \otimes \mathbf{x}_0\). The derivative \(\dot{V}\) is shown to be negative definite under certain conditions, ensuring asymptotic convergence of the estimation error.
The sliding mode controller for each quadrotor employs a variable exponential power reaching law. For the x-position subsystem, the tracking error is \(e_i = x_i – x_{id}\), and the sliding surface is \(s_i = \dot{e}_i + c_1 e_i\). The reaching law is:
$$ \dot{s}_i = -k_1 |s_i|^\gamma \text{sgn}(s_i) – k_2 \text{fal}(s_i, a, \delta) – k_3 s_i $$
where \(\gamma = \frac{p}{q}\), \(p\) and \(q\) are positive integers, and \(\text{fal}(s_i, a, \delta)\) is a nonlinear function defined as:
$$ \text{fal}(s_i, a, \delta) = \begin{cases} |s_i|^a \text{sgn}(s_i), & |s_i| > \delta \\ \frac{s_i}{\delta^{1-a}}, & |s_i| \leq \delta \end{cases} $$
The control law for the x-position is derived as:
$$ u_{1xi} = m \left( \ddot{x}_{id} – c_1 \dot{e}_i + \frac{K_x}{m} \dot{x}_i + k_1 |s_i|^\gamma \text{tanh}(s_i) + k_2 \text{fal}(s_i, a, \delta) + k_3 s_i \right) $$
Similar control laws are derived for y and z positions, and for attitude angles. The desired roll and pitch angles are computed from the position control inputs:
$$ \theta_{id} = \arctan \left( \frac{u_{1xi} \cos \psi_i + u_{1yi} \sin \psi_i}{u_{1zi}} \right) $$
$$ \phi_{id} = \arcsin \left( \frac{u_{1xi} \sin \psi_i – u_{1yi} \cos \psi_i}{u_{1zi}} \right) $$
The overall control structure ensures that each quadrotor tracks its desired trajectory while compensating for communication faults.
Simulation results validate the proposed method. The parameters for the quadrotor model and controller are listed in the following tables:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Mass (\(m\)) | 2 kg | Arm length (\(l\)) | 0.2 m |
| Gravity (\(g\)) | 9.8 m/s² | Moment of inertia (\(I_x, I_y\)) | 1.25 kg·m² |
| Moment of inertia (\(I_z\)) | 2.5 kg·m² | Drag coefficients (\(K_x, K_y, K_z\)) | 0.01 |
| Drag coefficients (\(K_\phi, K_\theta, K_\psi\)) | 0.012 | Initial positions | Varied |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(c_1, c_2, c_3\) | 5 | \(c_4, c_5, c_6\) | 30 |
| \(k_1\) | 3 | \(k_2\) | 2 |
| \(k_3\) | 7 | \(k_4\) | 15 |
| \(k_5\) | 15 | \(k_6\) | 70 |
| \(p\) | 2 | \(q\) | 1 |
| Parameter | Value |
|---|---|
| \(\delta\) | 1 |
| \(\lambda\) | 10 |
| \(\eta\) | 2 |
| \(\sigma\) | 1 |
| \(\mathbf{K}_1\) | \(\begin{bmatrix} 3.5\mathbf{I}_{4 \times 4} & 1.7962\mathbf{I}_{4 \times 4} \\ -0.7962\mathbf{I}_{4 \times 4} & 3.5\mathbf{I}_{4 \times 4} \end{bmatrix}\) |
| \(\mathbf{K}_2\) | \(0.8862 \mathbf{I}_{8 \times 8}\) |
The formation consists of one leader and four follower quadrotors. The communication topology is directed, with the leader transmitting to follower 1, and followers connected in a cycle. The desired relative positions are:
| Follower | X (m) | Y (m) | Z (m) |
|---|---|---|---|
| 1 | -0.3 | -0.4 | -0.3 |
| 2 | -0.8 | -0.3 | -0.6 |
| 3 | -0.8 | +0.3 | -0.6 |
| 4 | -0.3 | +0.4 | -0.3 |
The leader’s trajectory is defined as \(x_0 = 0.5t + 0.1\), \(y_0 = t + 1.5\), \(z_0 = 1.5\), and \(\psi_0 = \pi/3\). Simulation results show that the adaptive observer accurately estimates the leader’s states despite communication faults, with estimation errors converging to zero within seconds. The sliding mode controller achieves fast tracking, with position errors reducing to near zero in approximately 1 second. A performance index \(J = \frac{1}{N} \sum_{i=1}^N \| \mathbf{p}_i – \mathbf{p}_0 \|^2\) is used to compare with consensus-based methods. The proposed method reduces \(J\) to zero in about 2 seconds, outperforming traditional approaches by 40% in convergence speed. The three-dimensional formation trajectory demonstrates stable and precise tracking.
In conclusion, this paper presents a robust formation control strategy for quadrotor UAVs that addresses communication link faults through an adaptive leader-state observer and a variable exponential power reaching law-based sliding mode controller. The observer ensures accurate state estimation under faulty conditions, while the controller guarantees rapid error convergence. Future work will explore more complex communication topologies and real-time implementation on quadrotor platforms.