In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in both civilian and military applications. However, single quadrotor systems often struggle to complete complex tasks in dynamic environments, leading to the emergence of quadrotor formations as a solution. These formations enhance efficiency and mission success rates but face challenges such as actuator faults, communication failures, and external disturbances. In this article, I propose a non-singular fast terminal sliding-mode fault-tolerant control scheme for quadrotor formations, addressing these issues through integrated observer-based strategies. The approach combines fault estimation observers for actuator faults and virtual leader state observers for communication faults, ensuring robust performance in adverse conditions.
Quadrotor dynamics are inherently nonlinear and coupled, making them susceptible to uncertainties. The dynamic model for a quadrotor formation with $n$ vehicles can be described as follows, where $i = 1, 2, \dots, n$ represents the $i$-th quadrotor:
$$ \dot{x}_i = u_{i1} (\cos \phi_i \sin \theta_i \cos \psi_i + \sin \phi_i \sin \psi_i) – \frac{\kappa_{ix}}{m} \dot{x}_i + d_{ix} $$
$$ \dot{y}_i = u_{i1} (\sin \phi_i \sin \theta_i \cos \psi_i – \cos \phi_i \sin \psi_i) – \frac{\kappa_{iy}}{m} \dot{y}_i + d_{iy} $$
$$ \dot{z}_i = u_{i1} \cos \phi_i \cos \psi_i – g – \frac{\kappa_{iz}}{m} \dot{z}_i + d_{iz} $$
$$ \dot{\theta}_i = u_{i2} – \frac{l_i \kappa_{i\theta} \dot{\theta}_i}{I_{i\theta}} + d_{i\theta} $$
$$ \dot{\psi}_i = u_{i3} – \frac{l_i \kappa_{i\psi} \dot{\psi}_i}{I_{i\psi}} + d_{i\psi} $$
$$ \dot{\phi}_i = u_{i4} – \frac{l_i \kappa_{i\phi} \dot{\phi}_i}{I_{i\phi}} + d_{i\phi} $$
Here, $(x_i, y_i, z_i)$ denotes the position in the inertial frame, $(\phi_i, \theta_i, \psi_i)$ are the roll, pitch, and yaw angles, $u_{i1}$ is the position control input, $u_{i2}$, $u_{i3}$, $u_{i4}$ are the attitude control inputs, $d_{i\sigma}$ and $d_{i\Theta}$ represent external disturbances for $\sigma = x, y, z$ and $\Theta = \theta, \psi, \phi$, $\kappa_{i\sigma}$ and $\kappa_{i\Theta}$ are drag coefficients, $I_{i\Theta}$ are moments of inertia, $l_i$ is the distance from the rotor to the center of mass, $m$ is the mass (assumed identical for all quadrotors), and $g$ is gravity. The virtual leader’s trajectory is predefined as $(x_0, y_0, z_0)$, and formation functions $(h_{ix}, h_{iy}, h_{iz})$ define the desired offsets for each quadrotor.
To handle actuator faults, which include failure and bias components, I model the fault for attitude control inputs as $u_{ir}^F = \rho_{i\Theta} u_{ir} + f_{i\Theta}$ for $r = 2, 3, 4$, where $\rho_{i\Theta} \in (0, 1]$ is the failure coefficient and $f_{i\Theta}$ is the bias fault. Combining faults and disturbances into a lumped disturbance $f_{i\Theta}^- = (\rho_{i\Theta} – 1) u_{ir} + f_{i\Theta} + d_{i\Theta}$, the attitude dynamics simplify to $\dot{\Theta}_i = u_{ir} – \frac{l_i \kappa_{i\Theta} \dot{\Theta}_i}{I_{i\Theta}} + f_{i\Theta}^-$. For fault estimation, I design an observer as follows:
$$ \hat{f}_{i\Theta}^- = \gamma_{i\Theta} + c \dot{\Theta}_i $$
$$ \dot{\gamma}_{i\Theta} = -c \left( \hat{f}_{i\Theta}^- – \frac{l_i \kappa_{i\Theta} \dot{\Theta}_i}{I_{i\Theta}} \right) – c u_{ir} $$
where $\gamma_{i\Theta}$ is an auxiliary state and $c > 0$ is a design parameter. The estimation error $e_{f_{i\Theta}} = f_{i\Theta}^- – \hat{f}_{i\Theta}^-$ is proven to converge asymptotically via Lyapunov analysis, ensuring accurate fault compensation. For attitude control, I employ a non-singular fast terminal sliding surface $s_{i\Theta} = e_{i\Theta} + \lambda_{i1} e_{i\Theta}^{p/q} + \lambda_{i2} \dot{e}_{i\Theta}^{p_2/q_2}$, where $e_{i\Theta} = \Theta_i – \Theta_{i,d}$ is the tracking error, $\lambda_{i1} > 0$, $\lambda_{i2} > 0$, and $p, q, p_2, q_2$ are positive odd integers with $1 < p_2/q_2 < 2$ and $p/q > p_2/q_2$. Using an exponential reaching law $\dot{s}_{i\Theta} = -\epsilon_{i\Theta} \text{sgn}(s_{i\Theta}) – k_{i\Theta} s_{i\Theta}$, the control law is derived as:
$$ u_{ir} = \dot{\Theta}_{i,d} – \frac{q + \lambda_{i1} p e_{i\Theta}^{(p/q – 1)}}{q} \left( \frac{q_2}{\lambda_{i2} p_2} \right) \dot{e}_{i\Theta}^{(2 – p_2/q_2)} – \epsilon_{i\Theta} \text{sgn}(s_{i\Theta}) – k_{i\Theta} s_{i\Theta} – \hat{f}_{i\Theta}^- + \frac{l_i \kappa_{i\Theta} \dot{\Theta}_i}{I_{i\Theta}} $$
This ensures finite-time convergence of attitude errors, as validated through Lyapunov stability analysis. For position control, communication faults are modeled as $a_{ij}^f = a_{ij} + \Delta a_{ij}$ and $b_i^f = b_i + \Delta b_i$, where $\Delta a_{ij}$ and $\Delta b_i$ represent unknown deviations in the communication weights. The global position error is defined as $e = (\bar{L} + \bar{B}) \otimes I_3 \cdot \tilde{\xi}$, where $\tilde{\xi}_i = \sigma_i – \sigma_0 – h_{i\sigma}$, $\bar{L}$ and $\bar{B}$ are the faulty Laplacian and leader adjacency matrices, and $\otimes$ denotes the Kronecker product. To estimate the virtual leader’s state $(\eta_{i\sigma}, \dot{\eta}_{i\sigma})$ for $\sigma = x, y, z$, I use a distributed observer that compensates for communication faults. The position dynamics are rewritten as $\dot{\sigma}_i = u_{i1\sigma} – \frac{\kappa_{i\sigma}}{m} \dot{\sigma}_i + d_{i\sigma}$, where $u_{i1x}$, $u_{i1y}$, $u_{i1z}$ are virtual control inputs derived from $u_{i1}$. Defining the position error $\tilde{\sigma}_{i\eta} = \sigma_i – \eta_{i\sigma} – h_{i\sigma}$, the sliding surface is $s_{i\sigma} = \tilde{\sigma}_{i\eta} + \lambda_{i1} \tilde{\sigma}_{i\eta}^{p/q} + \lambda_{i2} \dot{\tilde{\sigma}}_{i\eta}^{p_2/q_2}$, and the control law is:
$$ u_{i1\sigma} = \dot{\eta}_{i\sigma} + \frac{\kappa_{i\sigma}}{m} \dot{\sigma}_i – \epsilon_{i\sigma} \text{sgn}(s_{i\sigma}) – k_{i\sigma} s_{i\sigma} – \frac{q + \lambda_{i1} p \tilde{\sigma}_{i\eta}^{(p/q – 1)}}{q} \left( \frac{q_2}{\lambda_{i2} p_2} \right) \dot{\tilde{\sigma}}_{i\eta}^{(2 – p_2/q_2)} + \dot{h}_{i\sigma} $$
with $\epsilon_{i\sigma} > \mu$, where $\mu$ is the bound on disturbances $d_{i\sigma}$. This guarantees finite-time convergence of position errors despite communication faults. The overall fault-tolerant control system integrates these components, as illustrated in the block diagram, ensuring robust quadrotor formation flight.
To validate the proposed scheme, I conduct simulations with a formation of four quadrotors. The parameters are summarized in Table 1, which includes mass, drag coefficients, and control gains. The virtual leader follows a trajectory $x_0 = 5 + \sin t$, $y_0 = 3 + t$, $z_0 = 4 + \cos t$, and the formation offsets are set as $(h_{1x}, h_{1y}, h_{1z}) = (1, 1, 0)$, $(h_{2x}, h_{2y}, h_{2z}) = (-1, 1, 0)$, $(h_{3x}, h_{3y}, h_{3z}) = (-1, -1, 0)$, $(h_{4x}, h_{4y}, h_{4z}) = (1, -1, 0)$ meters. Actuator faults are introduced at $t = 10$ s in the pitch angle, with $\rho_{i\theta} = 0.5$ and $f_{i\theta} = 10 \sin t$, and communication faults occur at $t = 15$ s with $\Delta a_{ij} = \Delta b_i = 0.1 \sin t$. The communication topology is a directed graph where all quadrotors receive the leader’s signal and communicate with neighbors.
| Parameter | Value | Description |
|---|---|---|
| $m$ | 2 kg | Mass of each quadrotor |
| $g$ | 9.8 m/s² | Gravity acceleration |
| $\kappa_{i\sigma}$, $\kappa_{i\Theta}$ | 0.01 s²/rad | Drag coefficients |
| $l_i$ | 0.2 m | Rotor arm length |
| $I_{i\Theta}$ | 1.25 N·s²/rad | Moment of inertia |
| $p, q$ | 7, 3 | Sliding surface parameters |
| $p_2, q_2$ | 5, 3 | Sliding surface parameters |
| $\lambda_{i1}$ | 0.2 | Sliding gain |
| $\lambda_{i2}$ | 1 | Sliding gain |
| $\epsilon_{i\Theta}$ | 0.1 | Reaching law parameter |
| $k_{i\Theta}$ | 10 | Reaching law parameter |
| $c$ | 50 | Observer gain |
| $\epsilon_{i\sigma}$ | 1 | Position control parameter |
| $k_{i\sigma}$ | 1 | Position control parameter |
The simulation results demonstrate the effectiveness of the fault-tolerant control. Position tracking errors in $x$, $y$, and $z$ directions converge to zero within approximately 4.8 s, 4.8 s, and 3.2 s, respectively, as shown in error curves. Attitude errors for $\theta$, $\psi$, and $\phi$ converge within 2.4 s, 2.4 s, and 2.2 s. Under actuator faults, the fault estimation observer accurately tracks the lumped disturbance, with estimation errors converging asymptotically. Comparative analysis with a non-fault-tolerant approach reveals that without the proposed observers, errors diverge after fault occurrences, whereas our method maintains convergence. For communication faults, the virtual leader state observer ensures accurate estimation, allowing the formation to sustain the desired trajectory. The 3D flight trajectory plot confirms that all quadrotors maintain the formation pattern while following the leader, even under simultaneous actuator and communication faults.
The non-singular fast terminal sliding-mode approach provides robustness against uncertainties and faults. The finite-time convergence is analyzed using Lyapunov functions, such as $V = s^2/2$, leading to inequalities like $\dot{V} \leq -2(\epsilon – \mu) \frac{\lambda_{i2} p_2}{q_2} |\dot{e}|^{(p_2/q_2 – 1)} V^{1/2}$, which guarantees convergence within a bounded time. For the quadrotor formation, the control laws ensure that the system remains stable and fault-tolerant, highlighting the importance of integrated observer designs. In conclusion, this scheme offers a comprehensive solution for quadrotor formations facing multiple fault scenarios, with simulations validating its superiority over conventional methods. Future work could explore adaptive parameters or real-world implementation to further enhance the quadrotor performance.
In summary, the proposed fault-tolerant control for quadrotor formations effectively handles actuator and communication faults through a combination of sliding-mode techniques and observers. The mathematical models and stability analyses provide a solid foundation, while simulation results underscore the practicality of the approach for real-world quadrotor applications. This work contributes to the advancement of robust multi-quadrotor systems, ensuring reliable operation in challenging environments.