Abstract: Formation cooperative control is a critical technology for the execution of missions by multiple fixed-wing drones. This technology faces significant challenges in complex, adversarial environments, particularly due to the presence of actuator failures and the limitations of asymptotic convergence, where the convergence time tends towards infinity. This paper presents a novel nonsingular finite-time fault-tolerant formation cooperative control strategy tailored for a class of six-degree-of-freedom (6-DOF) multiple fixed-wing drone systems operating under actuator failures. The proposed approach decouples the control design into velocity and attitude layers. For each layer, a finite-time formation tracking control algorithm is developed. A key innovation is the introduction of a novel switching function that effectively circumvents the singularity problem commonly encountered in finite-time controllers. Furthermore, an adaptive update law based on fault parameters is designed to estimate and compensate for the loss of actuator effectiveness and bias faults. The finite-time stability of the overall closed-loop system is rigorously established using Lyapunov stability theory. Finally, the effectiveness and feasibility of the proposed control method are validated through a combination of hardware-in-the-loop (HIL) simulations and numerical simulations, demonstrating robust performance in various mission scenarios, including turning climbs and low-altitude penetrations.
1. Introduction
In recent years, the utilization of multiple fixed-wing drones in formation has garnered widespread interest due to their advantages in improving mission execution efficiency and reducing operational costs. Distributed control strategies are particularly appealing for such systems as they do not rely on global information, enhancing scalability and robustness. Existing research has addressed various aspects of formation control, including maintaining and adjusting formation geometry, velocity control, and attitude synchronization. For instance, some studies have tackled the challenges of fault-tolerant control and obstacle avoidance under communication link failures, while others have proposed dynamic event-triggered communication mechanisms for distributed coordination. However, a common limitation of many of these methods is their reliance on simplified three-degree-of-freedom (3-DOF) models, which often neglect the inherent coupling between the velocity and attitude dynamics. To address this, a full 6-DOF model for fixed-wing drones has been established, which accounts for these couplings and provides a more accurate representation of the actual system.
Despite these advances, most existing research on formation control of fixed-wing drones focuses on asymptotic convergence, where the tracking errors approach zero only as time approaches infinity. In time-critical missions, such as target interception or coordinated attacks, a faster, finite-time convergence rate is highly desirable. Moreover, the presence of actuator failures, such as loss of effectiveness and bias faults in the thrust and rudder mechanisms, can severely degrade system performance and even lead to instability. While fault-tolerant control methods, including adaptive sliding mode control and distributed adaptive schemes, have been proposed for quadrotors and general multi-agent systems, their application to the specific challenges of 6-DOF fixed-wing drone formations remains insufficiently explored.
To simultaneously address the demand for rapid transient response, the risk of controller singularity, and the threat of actuator failures, this paper proposes a novel control framework. The primary contributions of this work are threefold:
- A Nonsingular Finite-Time Control Law: A new switching function is integrated into the finite-time formation control law. This function effectively prevents the singular value problem, which can arise when differentiating the virtual control law due to fractional power terms, a common issue in standard finite-time controllers.
- An Integrated Fault-Tolerant Scheme: The control framework incorporates an adaptive update law based on fault parameters. This allows the system to estimate the effects of actuator failures (both loss-of-effectiveness and bias) and compensate for them in real-time, ensuring robust formation tracking even under fault conditions.
- Rigorous Stability Analysis and Validation: The finite-time stability and boundedness of all signals within the closed-loop system are proven using Lyapunov stability theory. The theoretical findings are substantiated through realistic HIL simulations and comprehensive numerical simulations of a 6-DOF fixed-wing drone formation executing complex maneuvers.
2. Problem Formulation and Preliminaries
2.1 Graph Theory
The communication topology among the $N$ fixed-wing drones in the formation is represented by an undirected graph $G = (V, E, A)$. The node set $V = {v_1, v_2, \ldots, v_n}$ contains the $N$ drones. The edge set $E \subseteq V \times V$ describes the information exchange links, where $(v_i, v_j) \in E$ indicates that drone $i$ and drone $j$ can communicate bidirectionally. The weighted adjacency matrix $A = [a_{ij}] \in \mathbb{R}^{n \times n}$ is defined by $a_{ij} = 1$ if $(v_i, v_j) \in E$, and $a_{ij} = 0$ otherwise. The Laplacian matrix $L = D – A$, where $D = \text{diag}{d_{11}, d_{22}, \ldots, d_{nn}}$ is the degree matrix with $d_{ii} = \sum_{j=1}^{n} a_{ij}$.
2.2 6-DOF Fixed-Wing Drone Model
Consider a group of $N$ identical fixed-wing drones. The dynamics of the $i$-th drone are described by a 6-DOF model, partitioned into velocity and attitude subsystems.
Velocity Dynamics: The velocity dynamics of the $i$-th fixed-wing drone in the body frame is given by $\mathbf{v}_i = [u_i, v_i, w_i]^T$. The position $\mathbf{p}_i = [x_i, y_i, z_i]^T$ in the inertial frame is governed by:
$$ \dot{\mathbf{p}}_i = \mathbf{R}_1(\phi_i) \mathbf{v}_i $$
$$ \dot{\mathbf{v}}_i = -\boldsymbol{\omega}_i \times \mathbf{v}_i + \frac{\mathbf{T}_i}{m_i} + \mathbf{R}_1^T(\phi_i) \mathbf{g} + \frac{\mathbf{F}_i}{m_i} + \mathbf{d}_{vi} $$
where $\phi_i = [\phi_i, \theta_i, \psi_i]^T$ are the Euler angles, $\boldsymbol{\omega}_i = [p_i, q_i, r_i]^T$ is the angular velocity vector, $\mathbf{T}_i = [T_{xi}, 0, 0]^T$ is the thrust vector, $\mathbf{g} = [0, 0, g]^T$ is the gravity vector, $\mathbf{d}_{vi}$ represents external disturbances, and $\mathbf{F}_i$ is the aerodynamic force vector. $\mathbf{R}_1$ is the rotation matrix from the body frame to the inertial frame.
Attitude Dynamics: The attitude dynamics are described by:
$$ \dot{\boldsymbol{\phi}}_i = \mathbf{R}_2^{-1}(\phi_i) \boldsymbol{\omega}_i $$
$$ \mathbf{J}_i \dot{\boldsymbol{\omega}}_i = -\boldsymbol{\omega}_i \times \mathbf{J}_i \boldsymbol{\omega}_i + \mathbf{N}_i + \mathbf{C}(\delta_i) \boldsymbol{\delta}_i + \mathbf{J}_i \mathbf{d}_{\omega i} $$
where $\mathbf{J}_i$ is the inertia tensor, $\mathbf{N}_i$ is the gyroscopic and aerodynamic moment vector, $\mathbf{C}(\delta_i)$ is the control effectiveness matrix, $\boldsymbol{\delta}_i = [\delta_{ai}, \delta_{ei}, \delta_{ri}]^T$ is the control surface deflection vector (aileron, elevator, rudder), and $\mathbf{d}_{\omega i}$ is the external disturbance torque. $\mathbf{R}_2$ is the transformation matrix from angular velocity to Euler angle rates.
2.3 Actuator Fault Model
Two primary types of actuator failures are considered: thrust and control surface faults. The thrust fault for the $i$-th fixed-wing drone is modeled as:
$$ T_{xi} = \rho_{Ti} T_{xi0} + T_{xif} $$
where $0 \le \rho_{Ti} \le 1$ is the unknown efficiency factor of the thrust actuator, $T_{xi0}$ is the commanded thrust, and $T_{xif}$ is the unknown additive bias fault.
Similarly, the control surface fault model is:
$$ \boldsymbol{\delta}_i = \boldsymbol{\rho}_{\delta i} \boldsymbol{\delta}_{i0} + \boldsymbol{\delta}_{if} $$
where $\boldsymbol{\rho}_{\delta i} = \text{diag}{\rho_{\delta ai}, \rho_{\delta ei}, \rho_{\delta ri}}$ is the unknown efficiency matrix for the control surfaces, $\boldsymbol{\delta}_{i0}$ is the commanded deflection, and $\boldsymbol{\delta}_{if}$ is the unknown bias fault vector.
2.4 Assumptions and Lemmas
Assumptions:
- The model uncertainties and fault parameters ensure the control direction is known and non-singular.
- The minimum singular value of the control effectiveness matrix is bounded from below by a positive constant.
- The desired velocity $V_r$, attitude $\boldsymbol{\phi}_r$, and their derivatives are smooth and bounded.
Useful Lemmas: The following lemmas are integral to the proof of finite-time stability.
Lemma 1 (Practical Finite-Time Stability): For a continuous function $V(\mathbf{x})$, if $\dot{V}(\mathbf{x}) \le -\lambda_1 V(\mathbf{x}) – \lambda_2 V^{\gamma}(\mathbf{x}) + \eta$, where $\lambda_1,\lambda_2 > 0$, $0<\gamma<1$, and $0<\eta<\infty$, then the origin is practically finite-time stable.
Lemma 2 (Bound on tanh): For any $\varepsilon>0$ and $\vartheta \in \mathbb{R}$, the inequality $0 \le |\vartheta| – \vartheta\tanh(\vartheta/\varepsilon) \le 0.2785\varepsilon$ holds.
Lemma 3 (Young’s Inequality): For vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, if $c > 0$ and $p,q \ge 1$ satisfying $1/p + 1/q = 1$, then $\mathbf{x}^T\mathbf{y} \le \frac{c^p}{p} \|\mathbf{x}\|^p + \frac{1}{q c^q} \|\mathbf{y}\|^q$.
Lemma 4 (Bound on sign function): For any $\varepsilon > 0$ and $z \in \mathbb{R}$, the inequality $0 \le |z| – \frac{z^2}{\sqrt{z^2 + \varepsilon}} \le \sqrt{\varepsilon}$ holds.
3. Controller Design and Stability Proof
The control architecture is hierarchical, with separate designs for the velocity and attitude loops. The goal is to force the velocity $V_i$ and attitude $\boldsymbol{\phi}_i$ of each fixed-wing drone to track a desired trajectory, while the formation is maintained through a distributed communication topology.
3.1 Velocity Loop Controller Design
Controller Design: The velocity tracking error is defined as $\tilde{V}_i = V_i – V_r$. The distributed velocity error for the $i$-th fixed-wing drone is:
$$ e_{vi} = \lambda_{i1} \tilde{V}_i + \lambda_{i2} \sum_{j \in N_i} a_{ij} (\tilde{V}_i – \tilde{V}_j) $$
The dynamics of $e_{vi}$ are derived. To handle the model uncertainty $\Delta_{vi}$, an RBF neural network is used to approximate the term $\mathbf{v}_i^T (\Delta_{vi} + \mathbf{d}_{vi})$. The adaptive thrust control law for the $i$-th fixed-wing drone is designed as:
$$ T_{xi0} = -\frac{\alpha_{vi} \hat{\beta}_{vi}}{\chi_{vi}} \tanh\left(\frac{e_{vi}\alpha_{vi} \hat{\beta}_{vi}}{c_{vi}}\right) – \hat{\theta}_{vi} \tanh\left(\frac{e_{vi}\chi_{vi} \hat{\theta}_{vi}}{c_{vi}}\right) $$
where $\hat{\beta}_{vi}$, $\hat{\Phi}_{vi}$, and $\hat{\theta}_{vi}$ are estimates of the fault-related parameters.
Adaptive Laws: The parameter adaptive laws are:
$$ \dot{\hat{\beta}}_{vi} = \gamma_{vi1} e_{vi} \alpha_{vi} – 2\gamma_{vi1} \sigma_{vi1} \hat{\beta}_{vi} $$
$$ \dot{\hat{\Phi}}_{vi} = \frac{\Lambda_1 \gamma_{vi2} e_{vi} \mathbf{v}_i^T \boldsymbol{\Psi}_{vi}}{2h_{v1}^2 V_i^2} – 2\gamma_{vi2} \sigma_{vi2} \hat{\Phi}_{vi} $$
$$ \dot{\hat{\theta}}_{vi} = \gamma_{vi3} e_{vi} \chi_{vi} – 2\gamma_{vi3} \sigma_{vi3} \hat{\theta}_{vi} $$
Theorem 1 (Velocity Loop Stability): For the velocity dynamics of the fixed-wing drone system, under the control law and adaptive update laws, the velocity tracking error $e_{vi}$ converges to a small neighborhood of the origin in finite time. All signals in the closed-loop system remain bounded.
Proof (Sketch): Consider the Lyapunov function candidate $L_2 = \frac{1}{2} e_{vi}^2 + \frac{\rho_{Ti}}{2\gamma_{vi1}} \tilde{\beta}_{vi}^2 + \frac{1}{2\gamma_{vi2}} \tilde{\Phi}_{vi}^2 + \frac{\rho_{Ti}}{2\gamma_{vi3}} \tilde{\theta}_{vi}^2$. By applying the virtual control law and adaptive laws, and using Lemmas 1-4, it can be shown that:
$$ \dot{L}_2 \le -\lambda_1 L_2 – \lambda_2 L_2^{m_{vi}} + \zeta_2 $$
where $\lambda_1, \lambda_2 > 0$ and $\zeta_2$ is a bounded constant. According to Lemma 1, this proves the finite-time boundedness and convergence of the velocity tracking error.
3.2 Attitude Loop Controller Design
Controller Design: The attitude tracking error is $\tilde{\boldsymbol{\phi}}_i = \boldsymbol{\phi}_i – \boldsymbol{\phi}_r$, and the distributed attitude error is:
$$ \mathbf{e}_{\phi i} = \lambda_{i3} \tilde{\boldsymbol{\phi}}_i + \lambda_{i4} \sum_{j \in N_i} a_{ij} (\tilde{\boldsymbol{\phi}}_i – \tilde{\boldsymbol{\phi}}_j) $$
To address the singularity issue, a novel switching function $\boldsymbol{\Omega}_i = [\Omega_{\phi i}, \Omega_{\theta i}, \Omega_{\psi i}]^T$ is introduced. For each component $m \in {\phi, \theta, \psi}$, it is defined as:
$$ \Omega_{mi} = \begin{cases} e_{mi}^{2m_{\phi i} – 1}, & |e_{mi}| \ge \tau_1 \\ \iota_1 e_{mi} + o_1 e_{mi}^3, & |e_{mi}| < \tau_1 \end{cases} $$
This ensures that the derivative $\dot{\boldsymbol{\Omega}}_i$ is bounded and avoids the singularity problem. A virtual control law $\boldsymbol{\omega}_{di}$ is designed to stabilize the kinematics:
$$ \boldsymbol{\omega}_{di} = \frac{\mathbf{R}_2}{\Lambda_2} (-\boldsymbol{\mu}_{\phi 1} \mathbf{e}_{\phi i} – \boldsymbol{\mu}_{\phi 2} \mathbf{E}_{\phi i} + \Lambda_2 \dot{\boldsymbol{\phi}}_r + \lambda_{i4} \sum_{j \in N_i} a_{ij} \dot{\tilde{\boldsymbol{\phi}}}_j – \boldsymbol{\kappa}_{\phi i} \boldsymbol{\Omega}_{\phi i}) $$
The actual control surface deflection command is designed to be a robust adaptive law:
$$ \boldsymbol{\delta}_{i0} = -(\mathbf{J}_{i0}^{-1} \mathbf{C}_0)^T \mathbf{e}_{\omega i} \frac{\hat{\beta}_{\omega i}^2 \boldsymbol{\eta}_{\omega i}^T \boldsymbol{\eta}_{\omega i}}{\hat{\beta}_{\omega i}^2 \mathbf{e}_{\omega i}^T \mathbf{e}_{\omega i} \boldsymbol{\eta}_{\omega i}^T \boldsymbol{\eta}_{\omega i} + c_{\omega i}^2} $$
Adaptive Laws for Attitude Loop:
$$ \dot{\hat{\beta}}_{\omega i} = \gamma_{\omega i1} \mathbf{e}_{\omega i}^T \boldsymbol{\eta}_{\omega i} – 2\gamma_{\omega i1} \sigma_{\omega i1} \hat{\beta}_{\omega i} $$
$$ \dot{\hat{\Phi}}_{\omega i} = \frac{\gamma_{\omega i2} \mathbf{e}_{\omega i}^T \boldsymbol{\Psi}_{\omega i}}{2h_{\omega 1}^2} – 2\gamma_{\omega i2} \sigma_{\omega i2} \hat{\Phi}_{\omega i} $$
Theorem 2 (Attitude Loop Stability): For the attitude dynamics of the fixed-wing drone system, under the control law and adaptive update laws, the attitude tracking error $\mathbf{e}_{\phi i}$ converges to a small neighborhood of the origin in finite time. All signals in the closed-loop system remain bounded.
Proof (Sketch): Consider the Lyapunov function $L_4 = \frac{1}{2} \mathbf{e}_{\phi i}^T \mathbf{e}_{\phi i} + \frac{1}{2} \mathbf{e}_{\omega i}^T \mathbf{e}_{\omega i} + \frac{1}{2\gamma_{\omega i1}} \tilde{\beta}_{\omega i}^2 + \frac{1}{2\gamma_{\omega i2}} \tilde{\Phi}_{\omega i}^2$. The proof involves showing that the derivative $\dot{L}_4$ satisfies a practical finite-time stability condition similar to that in Theorem 1, leveraging the nonsingular switching function and the fault compensation provided by the adaptive laws.
4. Simulation Analysis
4.1 Hardware-in-the-Loop (HIL) Simulation
An HIL simulation platform was used to validate the proposed controller. The platform uses a master-slave architecture where the control algorithm runs on a flight controller, and the 6-DOF model runs on a real-time simulator. The formation consists of three fixed-wing drones with the Laplacian matrix:
$$ L = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} $$
Scenario 1: Turning Climb Mission. The drones are commanded to increase speed to 50 m/s and track a desired attitude profile involving a climbing turn. Follower 1 experiences a control surface fault at 10s, and Follower 2 at 18s.
The simulation results show that the formation successfully executes the maneuver. The plots of velocity and attitude illustrate that despite the actuator faults, the drones are able to maintain the formation and track the reference in finite time. The key controller parameters used in the simulation are summarized in Table 1.
Table 1: Key Controller Parameters for HIL Simulation
| Parameter | Leader | Follower 1 | Follower 2 |
|---|---|---|---|
| $\boldsymbol{\mu}_{\phi 1}$ | diag[10,12,15] | diag[10,12,12] | diag[8,10,11] |
| $\boldsymbol{\mu}_{\phi 2}$ | diag[11,14,14] | diag[10,11,10] | diag[10,11,10] |
| $\boldsymbol{\mu}_{\omega 1}$ | diag[5,10,10] | diag[5,10,10] | diag[5,10,10] |
| $\boldsymbol{\mu}_{\omega 2}$ | diag[5,10,9] | diag[5,10,10] | diag[5,10,10] |
| $m_{vi}, m_{\phi i}$ | 0.8, 0.9 | 0.8, 0.9 | 0.8, 0.9 |
| $c_{\omega i}, c_{vi1}$ | 1.5, 0.0001 | 1.5, 0.0001 | 1.5, 0.0001 |
| $\boldsymbol{\kappa}_{\phi i}$ | diag[-0.01] | diag[-0.01] | diag[-0.01] |
To demonstrate the effectiveness of the switching function in handling singularity, a comparison was made. Without the switching function, the control system exhibited divergent behavior when the fixed-wing drone’s velocity and attitude errors approached zero. With the switching function, the system remained stable and achieved precise tracking, confirming the superiority of the proposed nonsingular approach for fixed-wing drone formations.
4.2 Numerical Simulation
Scenario 2: Low-Altitude Penetration Mission. A numerical simulation was performed for a low-altitude penetration scenario. The three fixed-wing drones must maintain a constant speed of 60 m/s and perform a pitch-down maneuver. The faults are introduced such that Follower 2 has a fault at 10s and Follower 1 at 18s.
The results for this scenario, as shown in Figure 8 of the original text, successfully demonstrate the finite-time tracking performance. The speed and attitude tracking errors converge quickly to a small neighborhood of the origin even after the onset of actuator faults. The figure shows that the controller effectively compensates for the loss of effectiveness and bias faults in the control surfaces. The nonsingular controller handled zero-error conditions without any instability, further validating its singularity-free nature for fixed-wing drone formation control.
A comparison with a control system lacking the switching function was also conducted for this scenario. The results, shown in Figure 9 of the original text, confirm that the switching function is necessary to prevent system divergence at the moment of error convergence to zero, underscoring the practical importance of the nonsingular design for fixed-wing drone control in complex missions.
5. Conclusion
This paper has successfully developed a novel nonsingular finite-time fault-tolerant formation cooperative control strategy for a group of 6-DOF fixed-wing drones. The control system was designed hierarchically, featuring separate controllers for velocity and attitude, each enhanced with a specially designed switching function to eliminate the singularity problem inherent in conventional finite-time schemes. By integrating adaptive laws to estimate and compensate for unknown actuator faults (both thrust and control surface), the proposed method ensures robust performance and reliable formation tracking. The finite-time stability of the entire closed-loop system was rigorously proven. Extensive validation through both hardware-in-the-loop and numerical simulations confirmed the effectiveness of the proposed method in different mission profiles (turning climb and low-altitude penetration), demonstrating its ability to achieve rapid, accurate, and fault-tolerant formation control. Future work will focus on extending this approach to handle more complex scenarios, such as multiple simultaneous failures and severe environmental disturbances, as well as optimizing the computational efficiency for onboard implementation on fixed-wing drones.