In recent years, China UAV technology has been widely adopted in civilian fields, demonstrating significant potential and application prospects. Among various configurations, the tri-rotor China UAV exhibits unique actuator coupling characteristics, which enhance maneuverability but introduce severe control challenges — especially when the tail servo suffers from stuck faults and the system is subject to unknown external disturbances. To address these issues, we propose an adaptive gain-based super-twisting sliding mode (STSM) disturbance rejection fault-tolerant control strategy. By integrating adaptive estimation with super-twisting sliding mode, our approach achieves online estimation of the uncertain control allocation matrix, effectively compensates for disturbances, modeling uncertainties, and sudden servo faults, and ensures finite-time convergence of the closed-loop system. Hardware-in-the-loop experiments validate the superiority of the proposed method over conventional super-twisting algorithm (STA) in terms of fault tolerance and robustness.
1. Problem Formulation and Modeling
1.1 Attitude Dynamics of Tri-Rotor China UAV
Consider a tri-rotor China UAV operating in three-dimensional space. The rotational dynamics are described by Euler’s equation in the body-fixed frame. Let $\omega = [p, q, r]^T$ be the angular velocity, $\eta = [\phi, \theta, \psi]^T$ be the Euler angles (roll, pitch, yaw), $J = \text{diag}\{J_x, J_y, J_z\}$ the inertia matrix, $\tau = [\tau_\phi, \tau_\theta, \tau_\psi]^T$ the control torque, $d(t)$ the external disturbance, and $D(\eta,\omega)$ the modeling uncertainty. The attitude dynamics are given by:
$$
\dot{\omega} = -J^{-1}S(\omega)J\omega + J^{-1}\tau + d(t) + D(\eta,\omega),
$$
$$
\dot{\eta} = \Phi(\eta) \omega,
$$
where $S(\omega)$ is the skew-symmetric matrix and $\Phi(\eta)$ the angular velocity transformation matrix. Define the lumped disturbance $\rho(t) = d(t) + D(\eta,\omega)$, which is assumed to satisfy $\|\rho(t)\| \le a_0 + a_1\|\eta\| + a_2\|\omega\|^2 = \iota$, with $a_0,a_1,a_2 > 0$.
1.2 Servo Fault Model
The tri-rotor China UAV employs three rotors and one tail servo. The control torque is generated by the rotor thrusts $f_i$ ($i=1,2,3$) and servo deflection $\delta$. Define $\lambda_1 = l_3\cos\delta – \mu\sin\delta$ and $\lambda_2 = l_3\sin\delta + \mu\cos\delta$, where $l_i$ are moment arms and $\mu$ the torque coefficient. The control allocation matrix $A(\lambda_1,\lambda_2)$ becomes:
$$
A(\lambda_1,\lambda_2) = \begin{bmatrix}
-l_1 & l_1 & 0 \\
-l_2 & -l_2 & \lambda_1 \\
\mu & -\mu & \lambda_2
\end{bmatrix}.
$$
When a servo stuck fault occurs, $\delta$ becomes frozen at an unknown constant value, rendering $\lambda_1,\lambda_2$ unknown. The attitude system under fault becomes:
$$
\dot{\omega} = -J^{-1}S(\omega)J\omega + J^{-1}A(\lambda_1,\lambda_2)f + \rho(t).
$$
2. Controller Design with Adaptive Gain Super-Twisting
2.1 Sliding Surface and Control Law
Define the attitude error $e = \eta – \eta_d$ and the sliding surface $s = ce + \dot{e}$, with $c = \text{diag}\{c_1,c_2,c_3\} > 0$. Taking the derivative and substituting the dynamics yields:
$$
\dot{s} = c\dot{e} – \ddot{\eta}_d + \dot{\Phi}\omega – \Phi J^{-1}S(\omega)J\omega + \Phi J^{-1}A(\lambda_1,\lambda_2)f + \Phi\rho(t).
$$
Since $A(\lambda_1,\lambda_2)$ is unknown under servo fault, we use an estimate $\hat{A}(\hat{\lambda}_1,\hat{\lambda}_2)$ with entries $\hat{\lambda}_1,\hat{\lambda}_2$ adapted online. The control input $f$ is designed as:
$$
f = (\Phi J^{-1}\hat{A})^{-1}\Bigl(\Phi J^{-1}S(\omega)J\omega – c\dot{e} + \ddot{\eta}_d – \dot{\Phi}\omega – \hat{L} – k_1\|s\|^{\frac12}\text{sign}(s) – k_2\int\text{sign}(s)\,dt\Bigr),
$$
where $k_1,k_2$ are adaptive gains, and $\hat{L}$ is the adaptive estimate of the lumped disturbance effect. Substituting into $\dot{s}$ gives:
$$
\dot{s} = \Phi J^{-1}\tilde{A} f + \Phi\rho(t) – \hat{L} – k_1\|s\|^{\frac12}\text{sign}(s) – k_2\int\text{sign}(s)\,dt,
$$
with $\tilde{A} = A – \hat{A}$.
2.2 Adaptive Laws for Fault and Disturbance
To estimate $\lambda_1,\lambda_2$, we define auxiliary variables $Z_1 = \|s\|^{1/2}\text{sign}(s)$ and $Z_2 = -k_2\int\text{sign}(s)\,dt$. The adaptive laws are:
$$
\dot{\hat{\lambda}}_1 = \text{Proj}\left( \frac{\sigma_1}{\|Z_1\|} B C \Phi J^{-1} [0, f_3, 0]^T \right),
$$
$$
\dot{\hat{\lambda}}_2 = \text{Proj}\left( \frac{\sigma_2}{\|Z_1\|} B C \Phi J^{-1} [0, 0, f_3]^T \right),
$$
where $B = [4\xi^2+1,\,-1]$, $C = [Z_1^T, Z_2^T]^T$, $\sigma_1,\sigma_2,\xi > 0$, and Proj is the projection operator ensuring boundedness. The disturbance estimate $\hat{L}$ is updated by:
$$
\dot{\hat{L}} = \Lambda\bigl((4\xi^2+1)Z_1 – Z_2\bigr),\quad \Lambda > 0.
$$
The controller gains $k_1,k_2$ are adapted based on the tracking error magnitude $\|e\|$:
$$
\dot{k}_1 = \begin{cases}
\beta_1\sqrt{\frac{\gamma_1}{2}}\,\text{sign}(\|e\| – b), & k_1 \ge k_{1m},\\
a, & k_1 < k_{1m},
\end{cases}
$$
$$
\dot{k}_2 = \begin{cases}
\beta_2\sqrt{\frac{\gamma_2}{2}}\,\text{sign}(\|e\| – b), & k_2 \ge k_{2m},\\
a, & k_2 < k_{2m},
\end{cases}
$$
with $\beta_i,\gamma_i,a,b,k_{im} > 0$ and the condition $k_1(4\xi^2+1) – 2k_2 – \bigl(k_2 – \frac{k_1}{2} – \frac{1}{2}(4\xi^2+1)\bigr)^2 > 0$.
3. Stability Analysis
Consider the Lyapunov candidate:
$$
V = Z^T P Z + \frac12 \tilde{L}^T \Lambda^{-1} \tilde{L} + \frac12 \sigma_1^{-1} \tilde{\lambda}_1^2 + \frac12 \sigma_2^{-1} \tilde{\lambda}_2^2 + \frac{1}{2\gamma_1}(k_1 – k_1^*)^2 + \frac{1}{2\gamma_2}(k_2 – k_2^*)^2,
$$
where $P = \begin{bmatrix} 4\xi^2+1 & -1 \\ -1 & 1 \end{bmatrix}$ and $\tilde{L} = L – \hat{L}$. Taking the time derivative and using the adaptive laws, we obtain:
$$
\dot{V} \le -\frac{1}{\|Z_1\|} Z^T Q Z + \frac{1}{\gamma_1}(k_1 – k_1^*)\dot{k}_1 + \frac{1}{\gamma_2}(k_2 – k_2^*)\dot{k}_2,
$$
with $Q = \begin{bmatrix} k_1(4\xi^2+1)-2k_2 & k_2 – \frac{k_1}{2} – \frac12(4\xi^2+1) \\ k_2 – \frac{k_1}{2} – \frac12(4\xi^2+1) & 1 \end{bmatrix}$. Under condition (31), $Q>0$, leading to:
$$
\dot{V} \le -\kappa V^{1/2} + \nu^*(t),
$$
where $\nu^*(t)$ vanishes when $\|e\| > b$. Consequently, finite-time convergence is proven by Lyapunov theory. The detailed analysis shows that the closed-loop system converges in finite time to a residual set around zero.
4. Experimental Validation
4.1 Hardware-in-the-Loop Platform
We conducted experiments on a UAV flight control hardware-in-the-loop (HIL) platform, as shown in the figure. The platform consists of a real-time simulator, ground station, three-axis turntable, remote controller, and flight controller board. The tri-rotor China UAV parameters are listed below:
| Parameter | Value |
|---|---|
| $l_1 = l_2$ | 0.15 m |
| $l_3$ | 0.20 m |
| $\mu$ | 0.05 |
| $J$ | diag{$2.0\times10^{-3}$, $8.3\times10^{-3}$, $8.2\times10^{-3}$} kg·m² |
| $\xi$ | 1 |
| $\Lambda$ | diag{0.3, 0.3, 0.3} |
| $c$ | diag{6, 5, 6} |
| $\sigma_1,\sigma_2$ | 0.005 |
| $k_{1m},k_{2m}$ | 12, 14 |
| $\beta_1,\beta_2$ | 2, 1 |
| $\gamma_1,\gamma_2$ | 3, 1 |
| $a,b$ | 0.7, 0.04 |
4.2 Results and Comparison with STA
The initial attitude is $\eta_0 = [5.1^\circ, -14.5^\circ, 28.8^\circ]^T$ and the desired trajectory is $\eta_d = [0.20\sin(0.5t), -0.25\cos(0.5t), 0.45\sin(0.4t)]^T$. An external disturbance $\rho(t) = 3\sin(0.5t)I + 0.4\eta + 0.3\omega^2$ is injected at $t=14$ s, and a servo stuck fault occurs at $t=19$ s. We compared our proposed adaptive STSM (ASTSM) with the conventional super-twisting algorithm (STA) from reference [6].
The attitude tracking errors are plotted in the experimental videos. Quantitative results for the post-fault 31 s window are summarized below:
| Channel | Controller | Mean | Mean Absolute Error | RMSE |
|---|---|---|---|---|
| Roll | ASTSM | 0.0416 | 0.7416 | 0.7104 |
| STA | 0.0221 | 0.8698 | 0.9629 | |
| Pitch | ASTSM | -0.0272 | 0.4043 | 0.2703 |
| STA | 0.2152 | 1.0173 | 1.3606 | |
| Yaw | ASTSM | -0.0057 | 0.4092 | 0.2294 |
| STA | 0.0262 | 0.3592 | 0.2000 |
The results clearly demonstrate that the proposed ASTSM achieves smaller mean absolute errors and lower RMSE values in roll and pitch channels compared to STA, indicating stronger robustness and disturbance rejection. The adaptive gains and estimated parameters converge to bounded values, further confirming the effectiveness of the adaptive fault-tolerant mechanism.
5. Conclusion
This paper presents an adaptive gain-based super-twisting sliding mode fault-tolerant control strategy for a tri-rotor China UAV subject to servo stuck faults and unknown external disturbances. By integrating adaptive estimation of the uncertain control allocation matrix and disturbance effects, the proposed method effectively attenuates the impact of faults and disturbances while ensuring finite-time convergence of the attitude tracking errors. Hardware-in-the-loop experiments validate the superior performance of the approach over conventional super-twisting algorithm. Future work will extend the framework to address position control and multi-actuator failures in China UAV systems.