In recent years, drone formation technology has gained significant attention due to its wide-ranging applications in areas such as environmental monitoring, logistics, and disaster response. However, the practical implementation of drone formation systems faces challenges, including external disturbances and actuator faults, which can compromise formation stability and safety. This paper addresses the trajectory tracking problem for quadrotor drone formations under actuator faults by proposing a hierarchical fault-tolerant control strategy. The approach integrates prescribed performance control, backstepping sliding mode techniques, and adaptive neural network observers to ensure robust and precise formation control.
The core innovation lies in a dual-layer architecture that separates virtual cooperative planning from physical execution. This design effectively mitigates error coupling and fault propagation issues common in traditional distributed drone formation control. The virtual formation coordination layer generates globally consistent trajectories using distributed strategies, while the physical execution layer employs localized controllers for trajectory tracking and fault compensation. For known initial states, a prescribed-performance backstepping sliding mode fault-tolerant controller (PPBSMFC) is developed, whereas an initial condition-independent prescribed-performance backstepping fault-tolerant controller (ICPPBFC) is designed for unknown initial states. Adaptive radial basis function (RBF) neural network observers are incorporated to estimate and compensate for actuator faults and external disturbances in real-time.
Extensive simulations demonstrate that the proposed strategy achieves superior dynamic response, tracking accuracy, and fault tolerance compared to existing methods. The hierarchical structure ensures that faults in individual drones do not propagate through the formation, maintaining overall system stability. This work contributes to advancing reliable drone formation control in complex environments.
Introduction
Drone formation control has emerged as a critical research area due to its potential for collaborative tasks in diverse fields. However, real-world operations often involve uncertainties such as actuator failures and environmental disturbances, which can destabilize the formation. Traditional distributed control methods suffer from error coupling and fault propagation, where issues in one drone affect others through communication links. To address these limitations, this paper introduces a hierarchical control framework that decouples coordination from execution, enhancing robustness and reliability.
The proposed approach leverages prescribed performance control to enforce error constraints, ensuring that tracking errors remain within predefined bounds. Backstepping sliding mode techniques provide robustness against uncertainties, while adaptive observers handle unknown faults and disturbances. By combining these elements, the controller achieves finite-time convergence and high precision in trajectory tracking. The following sections detail the problem formulation, control design, and validation through numerical simulations.
Preliminaries and Problem Formulation
Consider a drone formation system consisting of N follower drones and a virtual leader. The communication topology is represented by a directed graph $G = (V, E, A)$, where $V = \{1, 2, \dots, N\}$ is the node set, $E \subseteq V \times V$ is the edge set, and $A = [a_{ij}] \in \mathbb{R}^{N \times N}$ is the weighted adjacency matrix. The Laplacian matrix $L = [l_{ij}]$ is defined as $l_{ij} = -a_{ij}$ for $i \neq j$ and $l_{ii} = \sum_{k=1}^{N} a_{ik}$. The system includes a virtual leader with dynamics accessible to a subset of followers.
The dynamics of the i-th quadrotor drone with actuator faults are described by:
$$U_{ji} = (1 – \varrho_{ji}) u_{ji} + a_{ji} f_{ji}, \quad j = \phi, \theta, \psi, \text{ and } i = 1, \dots, N,$$
where $\varrho_{ji} \in [0,1]$ represents the failure factor, $a_{ji} \in \{0,1\}$ indicates bias fault status, $f_{ji}$ is the bias fault, $U_{ji}$ is the actual control input, and $u_{ji}$ is the desired input. The overall dynamics accounting for faults and disturbances are expressed as:
$$\begin{aligned}
\dot{p}_i &= v_i, \\
\dot{v}_i &= G_i u_{Ai} + F_i + D_i,
\end{aligned}$$
where $p_i = [x_i, y_i, z_i, \psi_i]^\top$ denotes the position and yaw angle, $v_i$ is the velocity vector, $G_i$ is the input matrix, $F_i$ encompasses nonlinear terms, and $D_i$ aggregates external disturbances and fault effects. The control objective is to ensure that the drone formation tracks a desired trajectory while maintaining a predefined shape, even under actuator faults. Specifically, the global consensus errors should satisfy:
$$\lim_{t \to \infty} \| p_j – p_i – (\Lambda_j – \Lambda_i) \| = 0, \quad \lim_{t \to \infty} \| p_i – p_d – \Lambda_i \| = 0,$$
for $i, j = 1, \dots, N$, where $p_d$ is the leader’s trajectory and $\Lambda_i$ is the formation offset.
Hierarchical Fault-Tolerant Control System
The control system is divided into two layers: the virtual formation coordination layer and the physical execution layer. This separation prevents error coupling and fault propagation, as each layer operates independently with limited interaction.
Virtual Formation Coordination Layer
This layer generates reference trajectories for the physical drones using a distributed approach. The virtual leader and followers communicate to compute consistent paths based on the desired formation pattern. The dynamics of the virtual layer are given by:
$$\begin{aligned}
\dot{p}_{oi} &= v_{oi}, \\
\dot{v}_{oi} &= u_{oi} + \delta_{oi} p_{oi},
\end{aligned}$$
where $p_{oi}$ and $v_{oi}$ are the position and velocity of the i-th virtual drone, $u_{oi}$ is the control input, and $\delta_{oi}$ is a constant. The formation errors are defined as:
$$\begin{aligned}
e_{poi} &= \sum_{j=1}^{N} a_{ij} [(p_{oj} – p_{oi}) – (\Lambda_j – \Lambda_i)] + b_i (p_d – p_{oi} – \Lambda_0), \\
e_{voi} &= \sum_{j=1}^{N} a_{ij} (\dot{p}_{oj} – \dot{p}_{oi}) + b_i (\dot{p}_d – \dot{p}_{oi}),
\end{aligned}$$
where $a_{ij}$ and $b_i$ are weighting coefficients. A sliding surface is designed as $s_{oi} = \zeta_{oi} e_{poi} + \dot{e}_{poi}$, and the control law is derived using a reaching law approach:
$$u_{oi} = (L \otimes I_3)^{-1} \left( \dot{p}_d – \delta_{oi} p_{oi} – \zeta_{oi} e_{poi} – k_{oi} \text{sgn}(s_{oi}) \right),$$
where $\zeta_{oi}$ and $k_{oi}$ are positive constants. This ensures finite-time convergence of the virtual formation errors.
Physical Execution Layer
This layer comprises the actual drones that track the trajectories provided by the virtual layer. Controllers are designed for different subsystems based on initial state knowledge.
PPBSMFC Controller for Known Initial States
For position and yaw control (x, y, z, ψ axes), where initial states are known, a prescribed-performance backstepping sliding mode controller is developed. The tracking error $e_{pi} = p_i – p_{di}$ is transformed using a performance function $\rho(t) = (\rho_0 – \rho_\infty) e^{-\kappa t} + \rho_\infty$, which constrains the error as $-\delta^- \rho(t) < e < \delta^+ \rho(t)$. The transformed error is defined as:
$$\epsilon = \frac{e}{\rho(t)}, \quad \lambda = \frac{1}{2} \ln\left( \frac{\epsilon + \delta^-}{\delta^+ – \epsilon} \right).$$
A sliding surface is constructed as:
$$s = c_0 \ln(\cosh(c_1 \lambda)) + c_2 \tanh(\lambda) + \tau,$$
where $\tau$ is an intermediate error, and $c_0, c_1, c_2$ are positive constants. The control law incorporates an adaptive RBF neural network observer to estimate disturbances and faults:
$$\hat{D} = \hat{w}^\top h + u_d, \quad u_d = \hat{v} \tanh(s / b),$$
where $\hat{w}$ and $\hat{v}$ are adaptive weights updated by:
$$\dot{\hat{w}} = \eta_1 s h, \quad \dot{\hat{v}} = \eta_2 s \tanh(s / b).$$
The final control input ensures that the system trajectories converge to the sliding surface with prescribed performance.
ICPPBFC Controller for Unknown Initial States
For roll and pitch angles (φ, θ axes), where initial states are unknown, a modified prescribed-performance controller is designed. The performance function is redefined as:
$$\rho(t) = (\rho_0 – \rho_\infty) e^{-\xi (t_f – t)} + \rho_\infty,$$
and the error transformation uses a hyperbolic tangent function to handle unknown initial conditions. The control law is derived using backstepping with a tracking differentiator to avoid derivative explosions. The stability is proven via Lyapunov analysis, ensuring that errors remain within bounds without initial state knowledge.
Simulation Validation and Analysis
Numerical simulations are conducted to evaluate the proposed control strategy. The drone formation consists of four quadrotors with communication topology as shown in the figure below.
The model parameters are listed in Table 1.
| Parameter | Value | Description |
|---|---|---|
| m (kg) | 2.00 | Mass |
| g (m/s²) | 9.80 | Gravity |
| l (m) | 0.20 | Rotor distance |
| J (kg·m²) | 0.01 | Rotor inertia |
| K (Ns/m) | 0.01 | Drag coefficient |
| I_x, I_y (kg·m²) | 1.25 | Body inertia |
| I_z (kg·m²) | 2.50 | Body inertia |
The desired trajectory is set as $p_d = [\sin(\pi t/5), \cos(\pi t/5), t/5 + 1]^\top$, and the formation offsets are $\Lambda_1 = [0.5, 0, 0]^\top$, $\Lambda_2 = [-0.5, 0, 0]^\top$, $\Lambda_3 = [0, 0.5, 0]^\top$, $\Lambda_4 = [0, -0.5, 0]^\top$. External disturbances are modeled as $d_i = 0.6 \sin(t) e^{-0.001t} + 0.5 \cos(t)$. Actuator faults are injected at t = 8s and t = 12s, including partial failures and bias faults.
Two scenarios are tested: ideal conditions (no faults or disturbances) and faulty conditions. The proposed method is compared against robust global fast terminal sliding mode control (RGFTSMC), adaptive barrier function fast terminal sliding mode control (AFBFTSMC), and fuzzy adaptive sliding mode control (FASMC). Performance metrics include root mean square error (RMSE) for position and attitude tracking.
Under ideal conditions, the proposed controller achieves faster convergence and smaller errors than comparative methods. For example, the RMSE for x-position is 0.0294, compared to 0.1192–0.1304 for others. In faulty conditions, the hierarchical structure prevents fault propagation, maintaining stability while other methods exhibit significant performance degradation. The RBF observer accurately estimates disturbances and faults, enabling effective compensation.
| Method | RMSE_x | RMSE_y | RMSE_z | RMSE_ψ |
|---|---|---|---|---|
| RGFTSMC | 0.1258 | 0.2254 | 0.1188 | 0.0256 |
| AFBFTSMC | 0.1304 | 0.2900 | 0.1247 | 0.0369 |
| FASMC | 0.1192 | 0.2244 | 0.0849 | 0.0302 |
| Proposed | 0.0294 | 0.0454 | 0.0757 | 0.0229 |
| Method | RMSE_x | RMSE_y | RMSE_z | RMSE_ψ |
|---|---|---|---|---|
| RGFTSMC | 0.1444 | 0.2545 | 0.2253 | 0.0258 |
| AFBFTSMC | 0.1608 | 0.3211 | 0.1953 | 0.0371 |
| FASMC | 0.1368 | 0.2280 | 0.1777 | 0.0302 |
| Proposed | 0.0296 | 0.0454 | 0.0761 | 0.0229 |
The results confirm that the proposed strategy enhances tracking accuracy, fault tolerance, and robustness in drone formation control. The hierarchical architecture effectively decouples errors and isolates faults, ensuring reliable performance in challenging scenarios.
Engineering Feasibility Analysis
The proposed control strategy is designed with practical implementation in mind. The hierarchical structure reduces computational burden by localizing control tasks, making it suitable for real-time applications. The RBF neural network observer requires only five neurons, minimizing memory usage and enabling deployment on embedded platforms like ARM Cortex-M7 or NVIDIA Jetson Nano. The algorithms are compatible with common flight control software such as PX4 or ArduPilot, facilitating integration into existing systems.
Simulation results demonstrate that the controller maintains stability under actuator faults and disturbances, indicating high reliability. The prescribed performance bounds ensure safe operation by constraining errors, while the adaptive mechanisms handle uncertainties without requiring extensive parameter tuning. This approach is scalable to larger drone formations and adaptable to various mission profiles, offering a viable solution for real-world applications.
Conclusion
This paper presents a hierarchical fault-tolerant control strategy for quadrotor drone formations subject to actuator faults. The dual-layer architecture separates coordination and execution, preventing error coupling and fault propagation. Controllers with prescribed performance guarantees are developed for both known and unknown initial states, incorporating adaptive observers for disturbance compensation. Simulations validate the effectiveness of the approach, showing superior performance in tracking accuracy and fault tolerance compared to existing methods.
Future work will explore extensions to dynamic communication topologies and real-world testing under noisy environments. The integration of machine learning techniques for fault prediction and adaptive topology management could further enhance the robustness of drone formation systems in practical scenarios.