In recent years, the quadrotor drone has emerged as a highly versatile unmanned aerial vehicle (UAV) due to its agility, vertical take-off and landing capability, and relatively low cost. These attributes make it invaluable for applications ranging from surveillance and inspection to delivery and agricultural monitoring. However, the practical deployment of quadrotor drones is often hindered by several critical challenges, including severe parameter uncertainties, completely unknown external environmental disturbances, and various actuator faults. These issues can significantly degrade control performance, leading to instability or even catastrophic failures. Therefore, developing robust and adaptive control strategies that can ensure reliable operation under such adverse conditions is of paramount importance.
Traditional control methods, such as PID, backstepping, or even standard sliding mode control, often fall short when dealing with the combined effects of model inaccuracies, disturbances, and faults. Specifically, sliding mode control (SMC) is known for its robustness against uncertainties, but it suffers from the inherent chattering phenomenon, which can cause excessive wear on actuators, reduce control precision, and increase energy consumption. Moreover, conventional sliding mode reaching laws struggle to balance fast convergence with chattering suppression. To address these limitations, we propose a comprehensive control framework that integrates a novel sliding mode reaching law (NSMRL) with an adaptive fault-tolerant mechanism. This approach not only enhances the convergence speed and reduces chattering but also effectively compensates for unknown parameters, disturbances, and actuator faults in real-time.
The core contribution of this work lies in three aspects. First, we design a novel sliding mode reaching law that dynamically adjusts its coefficients and power terms based on the sliding surface value, ensuring rapid convergence while minimizing chattering. Second, we develop a nonlinear robust adaptive sliding mode controller (NRASMC) that incorporates adaptive laws to estimate all uncertain system parameters, unknown disturbance bounds, and actuator fault information (including both bias and partial loss-of-effectiveness faults). Third, we provide rigorous stability proofs using Lyapunov theory, demonstrating that the closed-loop system achieves asymptotic tracking despite the presence of severe uncertainties and faults. Extensive simulation studies validate the superiority of our proposed method compared to existing approaches, showcasing its enhanced fault tolerance and disturbance rejection capabilities.
This paper is structured as follows. We begin by presenting the dynamic model of a quadrotor drone under parameter uncertainties, external disturbances, and actuator faults. Then, we introduce the novel sliding mode reaching law and detail the design of the adaptive fault-tolerant controller for both attitude and position subsystems. Stability analysis is conducted, followed by simulation results and discussions. Finally, conclusions are drawn.
Dynamic Model of Quadrotor Drone with Uncertainties and Faults
The quadrotor drone is modeled as a rigid body with four rotors arranged in a symmetric configuration. Let us define an inertial frame \(O_e(X_e, Y_e, Z_e)\) and a body-fixed frame \(O_B(X_B, Y_B, Z_B)\). The dynamics are derived under the assumptions of symmetrical structure, negligible aerodynamic effects other than drag, and coincident center of mass and geometry. The equations of motion are subject to parameter uncertainties, unknown external disturbances, and actuator faults.
The actuator fault model is described as:
$$U_r = \beta_\tau U_{ra} + f_r, \quad r=1,2,3,4, \quad \tau = z, \phi, \theta, \psi$$
where \(U_{ra}\) is the commanded control input under normal conditions, \(U_r\) is the actual control input under fault conditions, \(\beta_\tau\) (\(0 < \beta_\tau \leq 1\)) is the effectiveness factor representing partial loss of effectiveness, and \(f_r\) is a bias fault. The fault cases are summarized as:
| Condition | Fault Type |
|---|---|
| \(\beta_\tau = 1, f_r = 0\) | No fault |
| \(\beta_\tau = 1, f_r \neq 0\) | Bias fault |
| \(\beta_\tau \in (0,1), f_r = 0\) | Partial loss-of-effectiveness |
| \(\beta_\tau \in (0,1), f_r \neq 0\) | Combined partial loss and bias fault |
Note that complete failure (i.e., \(\beta_\tau = 0\)) is not considered in this study to ensure controllability.
Considering severe parameter uncertainties and unknown disturbances, the dynamic model of the quadrotor drone is expressed as:
$$
\begin{aligned}
\ddot{x} &= \frac{\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi}{m} U_1 – \frac{k_x}{m} \dot{x} + d_x, \\
\ddot{y} &= \frac{\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi}{m} U_1 – \frac{k_y}{m} \dot{y} + d_y, \\
\ddot{z} &= \frac{\cos\phi \cos\theta}{m} U_1 – g – \frac{k_z}{m} \dot{z} + d_z, \\
\ddot{\phi} &= \frac{I_{yy} – I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} + \frac{U_2}{I_{xx}} – \frac{k_2}{I_{xx}} \dot{\phi} + d_\phi, \\
\ddot{\theta} &= \frac{I_{zz} – I_{xx}}{I_{yy}} \dot{\phi} \dot{\psi} + \frac{U_3}{I_{yy}} – \frac{k_3}{I_{yy}} \dot{\theta} + d_\theta, \\
\ddot{\psi} &= \frac{I_{xx} – I_{yy}}{I_{zz}} \dot{\phi} \dot{\theta} + \frac{U_4}{I_{zz}} – \frac{k_4}{I_{zz}} \dot{\psi} + d_\psi,
\end{aligned}
$$
where \(x, y, z\) are position coordinates; \(\phi, \theta, \psi\) are roll, pitch, and yaw angles; \(m\) is the mass; \(I_{xx}, I_{yy}, I_{zz}\) are moments of inertia; \(k_x, k_y, k_z\) are drag coefficients; \(k_2, k_3, k_4\) are gyroscopic coefficients; \(g\) is gravitational acceleration; and \(d_x, d_y, d_z, d_\phi, d_\theta, d_\psi\) represent unknown external disturbances (e.g., wind gusts).
To facilitate controller design, we redefine parameters and lumped disturbances. Let:
$$
\begin{aligned}
\alpha_{1i} &= m \ (i=x,y,z), \quad \alpha_{2i} = k_i, \quad \alpha_3 = I_{xx}, \quad \alpha_4 = I_{yy} – I_{zz}, \\
\alpha_5 &= I_{yy}, \quad \alpha_6 = I_{zz} – I_{xx}, \quad \alpha_7 = I_{zz}, \quad \alpha_8 = I_{xx} – I_{yy}.
\end{aligned}
$$
These parameters are assumed unknown but constant or slowly varying. The lumped disturbances including bias faults and external disturbances are denoted as \(\bar{d}_\mu\) for \(\mu = x,y,z,\phi,\theta,\psi\), and are assumed bounded by unknown constants \(\sigma_\mu\), i.e., \(\|\bar{d}_\mu\| \leq \sigma_\mu\).
Design of Novel Sliding Mode Reaching Law (NSMRL)
Conventional sliding mode reaching laws, such as the exponential reaching law or power rate reaching law, often face a trade-off between convergence speed and chattering reduction. To overcome this, we propose a novel sliding mode reaching law (NSMRL) that adapts its structure based on the sliding surface value.
Define the sliding surface for each control channel as \(s_j\), where \(j = x,y,z,\phi,\theta,\psi\). The NSMRL is given by:
$$
\dot{s}_j = -p_j \tanh\left(\frac{s_j}{\Delta_j}\right) – \sigma_j |s_j|^{q_j} \text{sgn}(s_j),
$$
with adaptive coefficients:
$$
p_j = \frac{2\chi_j}{\varepsilon_j + (1-\varepsilon_j) e^{(-n_j |s_j|^{-1})}}, \quad q_j = \begin{cases} r_j, & |s_j| \geq 1, \\ 1, & |s_j| < 1, \end{cases}
$$
where \(\chi_j, \varepsilon_j, n_j, \Delta_j, \sigma_j, r_j\) are positive constants, with \(0 < \varepsilon_j < 1\) and \(r_j > 1\). The function \(\tanh(\cdot)\) is used to replace the sign function, softening the control signal near the sliding surface and reducing chattering. The term \(|s_j|^{q_j}\) ensures fast convergence when far from the surface, while \(p_j\) adapts to provide smooth reaching.
The convergence properties of NSMRL are analyzed via Lyapunov theory. Consider the Lyapunov candidate \(V_N = \frac{1}{2} s_j^2\). Its derivative is:
$$
\dot{V}_N = s_j \dot{s}_j = -p_j s_j \tanh\left(\frac{s_j}{\Delta_j}\right) – \sigma_j |s_j|^{q_j+1} \leq 0.
$$
Thus, the sliding surface converges to zero asymptotically. Moreover, the finite-time convergence can be estimated. For instance, if \(|s_j(0)| > 1\), the convergence time is bounded by:
$$
t_{s_j} < \frac{1 – |s_j(0)|^{1-r_j}}{(r_j-1)\sigma_j} – \frac{\ln(v)}{\sigma_j},
$$
where \(v\) is a small positive value representing a neighborhood of zero. This demonstrates that NSMRL achieves faster convergence than traditional laws while minimizing chattering.
Adaptive Fault-Tolerant Controller Design
We now design the adaptive fault-tolerant controller for the quadrotor drone using the NSMRL. The control system is divided into attitude (roll, pitch, yaw) and position (x, y, z) subsystems. For each, we define tracking errors, sliding surfaces, and derive control laws with adaptive parameters.
Attitude Controller
Define attitude tracking errors as \(\phi_e = \phi_d – \phi\), \(\theta_e = \theta_d – \theta\), \(\psi_e = \psi_d – \psi\), where subscript \(d\) denotes desired trajectories. The sliding surfaces are chosen as:
$$
s_\phi = \dot{\phi}_e + c_\phi \phi_e, \quad s_\theta = \dot{\theta}_e + c_\theta \theta_e, \quad s_\psi = \dot{\psi}_e + c_\psi \psi_e,
$$
with \(c_\phi, c_\theta, c_\psi > 0\). Taking derivatives and substituting the dynamics, we obtain expressions for \(\dot{s}_j\). For roll angle \(\phi\), the control law \(U_{2a}\) (commanded input for actuator 2) is designed as:
$$
U_{2a} = \frac{1}{\hat{\beta}_\phi} \left( \hat{\alpha}_3 (c_\phi \dot{\phi}_e + \ddot{\phi}_d) + \hat{\sigma}_\phi |s_\phi|^{q_\phi} \text{sgn}(s_\phi) + p_\phi \tanh\left(\frac{s_\phi}{\Delta_\phi}\right) – \hat{\alpha}_4 \dot{\theta} \dot{\psi} \right),
$$
where \(\hat{\alpha}_3, \hat{\alpha}_4, \hat{\sigma}_\phi, \hat{\beta}_\phi\) are estimates of \(\alpha_3, \alpha_4, \sigma_\phi, \beta_\phi\), respectively. The adaptive laws for these estimates are:
$$
\begin{aligned}
\dot{\hat{\alpha}}_3 &= \gamma_{\phi1} s_\phi (c_\phi \dot{\phi}_e + \ddot{\phi}_d), \\
\dot{\hat{\alpha}}_4 &= -\gamma_{\phi2} s_\phi \dot{\theta} \dot{\psi}, \\
\dot{\hat{\sigma}}_\phi &= \lambda_{\phi1} |s_\phi|, \\
\dot{\hat{\beta}}_\phi &= -\lambda_{\phi2} \frac{1}{\hat{\beta}_\phi} \left( \hat{\alpha}_3 (c_\phi \dot{\phi}_e + \ddot{\phi}_d) + \hat{\sigma}_\phi |s_\phi|^{q_\phi} \text{sgn}(s_\phi) + p_\phi \tanh\left(\frac{s_\phi}{\Delta_\phi}\right) – \hat{\alpha}_4 \dot{\theta} \dot{\psi} \right) s_\phi,
\end{aligned}
$$
with \(\gamma_{\phi1}, \gamma_{\phi2}, \lambda_{\phi1}, \lambda_{\phi2} > 0\). Similar controllers and adaptive laws are derived for pitch and yaw:
$$
\begin{aligned}
U_{3a} &= \frac{1}{\hat{\beta}_\theta} \left( \hat{\alpha}_5 (c_\theta \dot{\theta}_e + \ddot{\theta}_d) + \hat{\sigma}_\theta |s_\theta|^{q_\theta} \text{sgn}(s_\theta) + p_\theta \tanh\left(\frac{s_\theta}{\Delta_\theta}\right) – \hat{\alpha}_6 \dot{\phi} \dot{\psi} \right), \\
U_{4a} &= \frac{1}{\hat{\beta}_\psi} \left( \hat{\alpha}_7 (c_\psi \dot{\psi}_e + \ddot{\psi}_d) + \hat{\sigma}_\psi |s_\psi|^{q_\psi} \text{sgn}(s_\psi) + p_\psi \tanh\left(\frac{s_\psi}{\Delta_\psi}\right) – \hat{\alpha}_8 \dot{\phi} \dot{\theta} \right).
\end{aligned}
$$
The corresponding adaptive laws follow the same structure as for roll.
Position Controller
Define position tracking errors: \(e_x = x_d – x\), \(e_y = y_d – y\), \(e_z = z_d – z\). The sliding surfaces are:
$$
s_x = \dot{e}_x + c_x e_x, \quad s_y = \dot{e}_y + c_y e_y, \quad s_z = \dot{e}_z + c_z e_z,
$$
with \(c_x, c_y, c_z > 0\). The virtual control inputs are defined from the position dynamics:
$$
u_x = \frac{\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi}{m} U_1, \quad u_y = \frac{\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi}{m} U_1, \quad u_z = \frac{\cos\phi \cos\theta}{m} U_1.
$$
Then, the actual control laws for \(u_x, u_y, u_z\) are designed using NSMRL. For the \(z\)-subsystem (altitude), which involves actuator fault \(\beta_z\), the control law is:
$$
u_z = \frac{1}{\hat{\beta}_z} \left( \hat{\alpha}_{1z} (c_z \dot{e}_z + \ddot{z}_d + g) + \hat{\alpha}_{2z} \dot{z} + \hat{\sigma}_z |s_z|^{q_z} \text{sgn}(s_z) + p_z \tanh\left(\frac{s_z}{\Delta_z}\right) \right),
$$
with adaptive laws:
$$
\begin{aligned}
\dot{\hat{\alpha}}_{1z} &= \gamma_{z1} s_z (c_z \dot{e}_z + \ddot{z}_d + g), \\
\dot{\hat{\alpha}}_{2z} &= \gamma_{z2} s_z \dot{z}, \\
\dot{\hat{\sigma}}_z &= \lambda_{z1} |s_z|, \\
\dot{\hat{\beta}}_z &= -\lambda_{z2} \frac{1}{\hat{\beta}_z} \left( \hat{\alpha}_{1z} (c_z \dot{e}_z + \ddot{z}_d + g) + \hat{\alpha}_{2z} \dot{z} + \hat{\sigma}_z |s_z|^{q_z} \text{sgn}(s_z) + p_z \tanh\left(\frac{s_z}{\Delta_z}\right) \right) s_z.
\end{aligned}
$$
For \(x\) and \(y\) subsystems, since they do not directly involve effectiveness faults (faults are in the actuators that primarily affect thrust and torques), the control laws are:
$$
\begin{aligned}
u_x &= \hat{\alpha}_{1x} (c_x \dot{e}_x + \ddot{x}_d) + \hat{\alpha}_{2x} \dot{x} + \hat{\sigma}_x |s_x|^{q_x} \text{sgn}(s_x) + p_x \tanh\left(\frac{s_x}{\Delta_x}\right), \\
u_y &= \hat{\alpha}_{1y} (c_y \dot{e}_y + \ddot{y}_d) + \hat{\alpha}_{2y} \dot{y} + \hat{\sigma}_y |s_y|^{q_y} \text{sgn}(s_y) + p_y \tanh\left(\frac{s_y}{\Delta_y}\right),
\end{aligned}
$$
with similar adaptive laws for parameter estimates.
Finally, the total thrust \(U_1\) and desired roll/pitch angles are computed from the virtual inputs:
$$
U_1 = \frac{u_z}{\cos\phi_d \cos\theta_d}, \quad \phi_d = \arctan\left(\frac{\sin\psi_d u_x – \cos\psi_d u_y}{u_z}\right) \cos\theta_d, \quad \theta_d = \arctan\left(\frac{\cos\psi_d u_x + \sin\psi_d u_y}{u_z}\right),
$$
where \(\psi_d\) is the desired yaw angle. The motor speeds can be derived from \(U_1, U_{2a}, U_{3a}, U_{4a}\) using the allocation matrix.
Stability Analysis
The stability of the closed-loop system is analyzed using Lyapunov theory. For each subsystem, we construct a Lyapunov function that includes sliding surface terms and parameter estimation errors.
Consider the roll subsystem. Define the Lyapunov candidate:
$$
V_\phi = \frac{1}{2} I_{xx} s_\phi^2 + \frac{1}{2\gamma_{\phi1}} \tilde{\alpha}_3^2 + \frac{1}{2\gamma_{\phi2}} \tilde{\alpha}_4^2 + \frac{1}{2\lambda_{\phi1}} \tilde{\sigma}_\phi^2 + \frac{1}{2\lambda_{\phi2}} \tilde{\beta}_\phi^2,
$$
where \(\tilde{\alpha}_3 = \alpha_3 – \hat{\alpha}_3\), etc., are estimation errors. Taking derivative and substituting the control and adaptive laws, after simplifications, we obtain:
$$
\dot{V}_\phi \leq -p_\phi |s_\phi| \leq 0.
$$
Thus, by Lyapunov’s direct method, \(s_\phi\) converges to zero asymptotically, implying that the tracking error \(\phi_e\) converges to zero. Similar proofs apply to pitch, yaw, and position subsystems. The boundedness of all signals is ensured, and the adaptive parameters remain bounded. The use of NSMRL guarantees that chattering is suppressed while maintaining fast convergence.
Simulation Studies and Results
To validate the proposed NRASMC with NSMRL, we conduct simulation studies under various fault and disturbance scenarios. The quadrotor drone parameters are listed in the table below, which are assumed unknown to the controller and estimated online.
| Parameter | Value | Unit |
|---|---|---|
| Mass (\(m\)) | 1.5 | kg |
| Moment of inertia \(I_{xx}\) | 0.016 | kg·m² |
| Moment of inertia \(I_{yy}\) | 0.026 | kg·m² |
| Moment of inertia \(I_{zz}\) | 0.032 | kg·m² |
| Drag coefficients \(k_x, k_y, k_z\) | 0.008 | N·s/m |
| Gyroscopic coefficients \(k_2, k_3, k_4\) | 0.0022 | N·m·s/rad |
| Gravity \(g\) | 9.81 | m/s² |
The controller parameters are tuned as follows:
| Parameter | Value |
|---|---|
| \(c_j\) (for all channels) | 1.5 |
| \(\gamma_{\phi1}, \gamma_{\phi2}\) | 0.05, 0.2 |
| \(\lambda_{\phi1}, \lambda_{\phi2}\) | 0.01, 0.5 |
| \(p_\phi, p_\theta, p_\psi\) | 5 |
| \(p_x, p_y, p_z\) | 1.5 |
| \(\Delta_j\) | 0.5 |
| \(r_j\) | 1.5 |
We simulate the quadrotor drone performing a trajectory tracking task. The desired trajectory is set as:
$$
x_d(t) = 2\sin(0.5t), \quad y_d(t) = 2\cos(0.5t), \quad z_d(t) = 5 + 0.5t, \quad \psi_d(t) = 0.
$$
The following fault and disturbance scenario is imposed:
- At \(t = 5\) s, external disturbances are applied: \(d_\mu = 0.5\sin(3t)\) for \(\mu = x,y,z\) and \(d_\xi = 0.5\cos(3t)\) for \(\xi = \phi,\theta,\psi\).
- At \(t = 12\) s, actuator bias faults occur: \(f_r = 0.3\) for all actuators (\(r=1,2,3,4\)).
- At \(t = 20\) s, partial loss-of-effectiveness faults are introduced: \(\beta_z = \beta_\phi = \beta_\theta = \beta_\psi = 0.5\).
The performance of NRASMC is compared with two existing methods: a super-twisting terminal sliding mode control (STTSMC) from literature and a disturbance observer-based nonsingular fast terminal sliding mode control (DO-NFTSMC). The comparison metrics include tracking error norms, control effort, and chattering magnitude.
The simulation results demonstrate that NRASMC achieves superior tracking accuracy and robustness. The table below summarizes the root-mean-square (RMS) tracking errors over the simulation period for the position and attitude channels.
| Channel | NRASMC | STTSMC | DO-NFTSMC |
|---|---|---|---|
| \(x\) (m) | 0.032 | 0.089 | 0.071 |
| \(y\) (m) | 0.028 | 0.095 | 0.068 |
| \(z\) (m) | 0.025 | 0.102 | 0.081 |
| \(\phi\) (rad) | 0.008 | 0.025 | 0.019 |
| \(\theta\) (rad) | 0.007 | 0.022 | 0.017 |
| \(\psi\) (rad) | 0.006 | 0.018 | 0.014 |
Clearly, NRASMC yields the lowest errors across all channels. Moreover, the control inputs generated by NRASMC exhibit significantly reduced chattering compared to the other methods, as evidenced by the smoothness of the control signals. The adaptive parameters converge to steady-state values that effectively compensate for uncertainties and faults. For instance, the estimates \(\hat{\beta}_\phi\) and \(\hat{\sigma}_\phi\) adapt quickly to the fault and disturbance changes, maintaining stability.
To further illustrate, we provide key formulas from the simulation. The evolution of sliding surfaces is governed by the NSMRL equation. For example, the sliding surface for altitude is:
$$
s_z = \dot{e}_z + c_z e_z, \quad \dot{s}_z = -p_z \tanh\left(\frac{s_z}{\Delta_z}\right) – \hat{\sigma}_z |s_z|^{q_z} \text{sgn}(s_z) + \text{perturbation terms},
$$
which converges rapidly to near zero. The adaptive law for disturbance bound estimate \(\hat{\sigma}_z\) is:
$$
\dot{\hat{\sigma}}_z = \lambda_{z1} |s_z|,
$$
ensuring that \(\hat{\sigma}_z\) increases when the sliding surface magnitude is large, thus providing sufficient switching gain to reject disturbances.
Conclusion
In this paper, we have addressed the challenging problem of controlling a quadrotor drone under severe parameter uncertainties, unknown external disturbances, and actuator faults. We proposed a novel adaptive fault-tolerant control scheme based on a newly designed sliding mode reaching law. The NSMRL dynamically adjusts its parameters to achieve fast convergence and chattering reduction, overcoming limitations of traditional reaching laws. The controller incorporates adaptive laws to estimate unknown system parameters, disturbance bounds, and fault parameters online, ensuring robustness without requiring prior knowledge of the fault or disturbance characteristics. Stability of the closed-loop system is rigorously proven using Lyapunov theory. Simulation results confirm that the proposed NRASMC outperforms existing methods in terms of tracking accuracy, disturbance rejection, and fault tolerance. Future work will focus on extending the approach to handle complete actuator failures and experimental validation on a physical quadrotor drone platform.
The quadrotor drone, with its increasing applications in complex environments, demands such advanced control strategies to ensure safe and reliable operation. Our contribution provides a robust framework that can be adapted to various UAV systems facing similar challenges.