The quadrotor drone is a versatile aerial vehicle prized for its Vertical Take-Off and Landing (VTOL) and hovering capabilities. Its agility makes it suitable for a wide array of applications. However, controlling a quadrotor drone presents significant challenges. The system is underactuated, meaning it has six degrees of freedom (position and orientation) but only four independent control inputs (rotor thrusts). Its dynamics are inherently nonlinear and coupled. Furthermore, real-world flight is complicated by parametric uncertainties, external disturbances, and, critically, actuator faults which can lead to catastrophic failure.
My research focuses on addressing these challenges by designing a robust and adaptive control scheme. Specifically, I propose an Adaptive Non-singular Fast Terminal Sliding Mode Controller (ANFTSMC). This method is designed to handle uncertainties in system parameters, reject unknown external disturbances, and compensate for actuator faults in real-time, ensuring precise trajectory tracking and stable flight.
System Modeling and Problem Formulation
To design an effective controller, an accurate dynamic model of the quadrotor drone is essential. We define two coordinate frames: an Earth-fixed inertial frame \( O_e (x_e, y_e, z_e) \) and a body-fixed frame \( O_B (x_B, y_B, z_B) \) attached to the drone’s center of mass. The four rotors generate lift forces \( F_1, F_2, F_3, F_4 \). The dynamic model, incorporating external disturbances, actuator faults, and key parametric uncertainties, is given by:
$$ \begin{align}
\ddot{x} &= \frac{U_1}{m} (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) – \frac{k_x}{m} \dot{x} + d_x \\
\ddot{y} &= \frac{U_1}{m} (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi) – \frac{k_y}{m} \dot{y} + d_y \\
\ddot{z} &= \frac{U_1}{m} \cos\phi \cos\theta – g – \frac{k_z}{m} \dot{z} + d_z \\
\ddot{\phi} &= \dot{\theta} \dot{\psi} \left( \frac{I_{yy} – I_{zz}}{I_{xx}} \right) + \frac{U_2}{I_{xx}} – \frac{k_1}{I_{xx}} \dot{\phi} + d_\phi \\
\ddot{\theta} &= \dot{\phi} \dot{\psi} \left( \frac{I_{zz} – I_{xx}}{I_{yy}} \right) + \frac{U_3}{I_{yy}} – \frac{k_2}{I_{yy}} \dot{\theta} + d_\theta \\
\ddot{\psi} &= \dot{\phi} \dot{\theta} \left( \frac{I_{xx} – I_{yy}}{I_{zz}} \right) + \frac{U_4}{I_{zz}} – \frac{k_3}{I_{zz}} \dot{\psi} + d_\psi
\end{align} $$
Where \( (x, y, z) \) and \( (\phi, \theta, \psi) \) denote position and attitude (roll, pitch, yaw) in the inertial frame. \( m \) is the mass, \( g \) is gravity, and \( I_{xx}, I_{yy}, I_{zz} \) are moments of inertia. The terms \( k_x, k_y, k_z \) represent aerodynamic drag coefficients, and \( k_1, k_2, k_3 \) model gyroscopic effects; these parameters are often uncertain. The terms \( d_i \) represent bounded external disturbances. The control inputs \( U_r \) (\( r = 1,2,3,4 \)) are the total thrust, roll moment, pitch moment, and yaw moment, respectively.
A critical aspect of this work is modeling actuator faults. We consider a bias fault model where the actual control input deviates from the commanded input:
$$ U_r = U_{r_a} + f_r $$
Here, \( U_{r_a} \) is the healthy (commanded) control input, and \( f_r \) is an unknown, bounded fault signal representing a constant or time-varying bias. When \( f_r \neq 0 \), the corresponding actuator (motor/ESC/propeller) is faulty.
To streamline the controller design, we define lumped disturbances that encompass external disturbances, fault effects, and the influence of parametric uncertainties:
$$ \begin{align}
F_x &= \frac{(\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)}{m} f_1 + d_x \\
F_y &= \frac{(\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi)}{m} f_1 + d_y \\
F_z &= \frac{\cos\phi \cos\theta}{m} f_1 + d_z \\
F_\phi &= \frac{f_2}{I_{xx}} + d_\phi, \quad F_\theta = \frac{f_3}{I_{yy}} + d_\theta, \quad F_\psi = \frac{f_4}{I_{zz}} + d_\psi
\end{align} $$
We make the following standard assumptions:
Assumption 1: The lumped disturbances \( F_i \) and their first derivatives are bounded, i.e., \( |F_i| \leq \sigma_i \) and \( |\dot{F}_i| \leq \varrho_i \), where \( \sigma_i \) and \( \varrho_i \) are unknown positive constants.
Assumption 2: The attitude angles remain within \( (-\pi/2, \pi/2) \) to avoid singularities in the control transformation.
Controller Design: Adaptive Non-singular Fast Terminal Sliding Mode
The core objective is to design control laws \( U_{1_a}, U_{2_a}, U_{3_a}, U_{4_a} \) such that the quadrotor drone tracks a desired trajectory \( (x_d, y_d, z_d, \psi_d) \) despite the lumped disturbances \( F_i \). The proposed solution has two layers: an outer-loop position controller and an inner-loop attitude controller.
1. Position Controller Design
Define the position tracking error as \( e_p = p_d – p \), where \( p = [x, y, z]^T \). The virtual control inputs are extracted from the translational dynamics:
$$ \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix} = \frac{U_1}{m} \begin{bmatrix}
\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi \\
\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi \\
\cos\phi \cos\theta
\end{bmatrix} $$
We design the controllers for \( u_x, u_y, u_z \). For the x-axis, we define a Non-singular Fast Integral Terminal Sliding Surface (NFITSS):
$$ s_x = \int_0^t e_x(\tau) d\tau + e_x + \eta_x e_x^{p_x} + \mu_x \dot{e}_x^{n_x} $$
where \( \eta_x, \mu_x > 0 \), \( 1 < p_x < 2 \), and \( n_x > 1 \). This surface combines the benefits of integral action (steady-state error elimination), fast terminal convergence, and non-singularity (as the exponent on \( \dot{e}_x \) is \( n_x > 1 \)).
The proposed adaptive NFTSM controller for the x-direction is:
$$ u_x = \ddot{x}_d + \frac{\hat{k}_x}{m} \dot{x} + M_x \frac{(e_x + \dot{e}_x)}{\dot{e}_x^{n_x-1}} + N_x e_x^{p_x-1} \dot{e}_x^{2-n_x} + \hat{\sigma}_x \tanh\left(\frac{s_x}{\epsilon_x}\right) + \alpha_x s_x $$
where \( M_x = 1/(\mu_x n_x) \), \( N_x = \eta_x p_x / (\mu_x n_x) \), and \( \epsilon_x, \alpha_x > 0 \). The function \( \tanh(\cdot) \) is used to approximate the sign function, reducing chattering. The adaptive laws for estimating the unknown bound \( \sigma_x \) and parameter \( k_x \) are:
$$ \dot{\hat{\sigma}}_x = \lambda_{x1} |s_x|, \quad \dot{\hat{k}}_x = -\frac{\gamma_{x1}}{m} \dot{x} s_x $$
where \( \lambda_{x1}, \gamma_{x1} > 0 \) are adaptation gains. The stability is proven using the Lyapunov function \( V_x = \frac{1}{2}s_x^2 + \frac{1}{2\lambda_{x1}} \tilde{\sigma}_x^2 + \frac{1}{2\gamma_{x1}} \tilde{k}_x^2 \), where \( \tilde{\sigma}_x = \sigma_x – \hat{\sigma}_x \) and \( \tilde{k}_x = k_x – \hat{k}_x \). Its derivative can be shown to be negative semi-definite, guaranteeing convergence. The controllers for \( u_y \) and \( u_z \) are designed analogously.
From the computed virtual controls \( (u_x, u_y, u_z) \) and the desired yaw \( \psi_d \), the total thrust \( U_{1_a} \) and desired roll/pitch angles \( (\phi_d, \theta_d) \) for the inner loop are derived:
$$ \begin{align}
U_{1_a} &= m \sqrt{u_x^2 + u_y^2 + (u_z + g)^2} \\
\phi_d &= \arctan\left( \frac{\sin\psi_d \cdot u_x – \cos\psi_d \cdot u_y}{u_z + g} \right) \\
\theta_d &= \arctan\left( \frac{\cos\psi_d \cdot u_x + \sin\psi_d \cdot u_y}{u_z + g} \right)
\end{align} $$
2. Attitude Controller Design
The inner loop must track the desired attitudes \( (\phi_d, \theta_d, \psi_d) \). Defining the attitude error as \( e_\phi = \phi_d – \phi \), we use a similar NFITSS:
$$ s_\phi = \int_0^t e_\phi(\tau) d\tau + e_\phi + \eta_\phi e_\phi^{p_\phi} + \mu_\phi \dot{e}_\phi^{n_\phi} $$
The proposed adaptive NFTSM controller for the roll channel is:
$$ U_{2_a} = I_{xx} \left[ \ddot{\phi}_d – \dot{\theta}\dot{\psi}\left(\frac{I_{yy}-I_{zz}}{I_{xx}}\right) + \frac{\hat{k}_1}{I_{xx}} \dot{\phi} + M_\phi \frac{(e_\phi + \dot{e}_\phi)}{\dot{e}_\phi^{n_\phi-1}} + N_\phi e_\phi^{p_\phi-1} \dot{e}_\phi^{2-n_\phi} + \hat{\sigma}_\phi \tanh\left(\frac{s_\phi}{\epsilon_\phi}\right) + \alpha_\phi s_\phi \right] $$
With adaptive laws:
$$ \dot{\hat{\sigma}}_\phi = \lambda_{\phi1} |s_\phi|, \quad \dot{\hat{k}}_1 = -\frac{\gamma_{\phi1}}{I_{xx}} \dot{\phi} s_\phi $$
The controllers for pitch \( U_{3_a} \) and yaw \( U_{4_a} \) are derived following the same principle. The stability of each attitude loop is proven independently using Lyapunov analysis similar to the position loop.
The overall control architecture for the quadrotor drone is summarized below:
- Outer Loop (Position): Uses ANFTSMC to compute virtual controls \( (u_x, u_y, u_z) \) and desired attitudes \( (\phi_d, \theta_d) \).
- Inner Loop (Attitude): Uses ANFTSMC to compute actual torque commands \( (U_{2_a}, U_{3_a}, U_{4_a}) \) and the thrust command \( U_{1_a} \).
- Adaptation: Online estimation of disturbance bounds \( \hat{\sigma}_i \) and uncertain parameters \( \hat{k}_i \) runs concurrently in all channels.
- Allocation: The four commands are converted to individual motor speeds \( \omega_i \) via the mixing matrix:
$$ \begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix} = \begin{bmatrix} c & c & c & c \\ 0 & -d\,c & 0 & d\,c \\ -d\,c & 0 & d\,c & 0 \\ k_p & -k_p & k_p & -k_p \end{bmatrix} \begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix} $$
where \( c \) is the thrust coefficient, \( d \) is the arm length, and \( k_p \) is the drag coefficient.
Stability Analysis
The closed-loop stability of the quadrotor drone system under the proposed ANFTSMC is rigorously established using Lyapunov’s direct method for each subsystem. The proof follows a standard procedure:
Step 1: Choose a positive definite Lyapunov candidate function for a channel (e.g., x-position):
$$ V_x = \frac{1}{2}s_x^2 + \frac{1}{2\lambda_{x1}} \tilde{\sigma}_x^2 + \frac{1}{2\gamma_{x1}} \tilde{k}_x^2 $$
Step 2: Compute its time derivative \( \dot{V}_x \) along the trajectories of the system under the designed controller and adaptive laws.
Step 3: Show that \( \dot{V}_x \leq -\alpha_x s_x^2 \leq 0 \). This inequality proves that \( s_x, \tilde{\sigma}_x, \tilde{k}_x \) are bounded. Using Barbalat’s lemma or similar invariance principles, it can be concluded that the sliding surface \( s_x \to 0 \) as \( t \to \infty \).
Step 4: Once on the sliding surface \( s_x = 0 \), the dynamics of the error system are governed by \( \int_0^t e_x d\tau + e_x + \eta_x e_x^{p_x} + \mu_x \dot{e}_x^{n_x} = 0 \), which is a stable differential equation guaranteeing that the tracking error \( e_x \to 0 \) in finite time.
This process is repeated identically for the \( y, z, \phi, \theta, \) and \( \psi \) channels. Since the subsystems are proven stable independently and the inner-outer loop structure is well-posed (with bounded desired commands \( \phi_d, \theta_d \)), the overall quadrotor drone closed-loop system is asymptotically stable.
Simulation and Experimental Validation
To validate the performance of the proposed ANFTSMC for the quadrotor drone, comprehensive simulations and real-world flight tests were conducted, comparing it against conventional Terminal SMC (TSMC) and Fast Terminal SMC (FTSMC).
Simulation Setup and Results
The simulated quadrotor drone had the following parameters: \( m = 2.0 \, \text{kg} \), \( I_{xx} = I_{yy} = 5.745 \times 10^{-3} \, \text{kg·m}^2 \), \( I_{zz} = 1.175 \times 10^{-2} \, \text{kg·m}^2 \), \( g = 9.81 \, \text{m/s}^2 \). The drag and gyroscopic coefficients were set with nominal values but treated as unknown for the adaptive controller. The desired trajectory was a helical path: \( x_d = 3\cos(t) \), \( y_d = 3\sin(t) \), \( z_d = 2 + 0.5t \), with \( \psi_d = 0.5 \, \text{rad} \).
Disturbance and Fault Scenario:
- t = 5 s: External wind disturbances \( \sin(3\pi t) \) were applied to all channels.
- t = 10 s: Simultaneous actuator bias faults (\( f_2 = 0.5, f_3 = 0.5 \)) and increased disturbances \( 2\sin(3\pi t) \) were introduced.
The tracking performance is summarized in the table below, comparing Root Mean Square Error (RMSE):
| Control Method | Position RMSE (x, y, z) [m] | Attitude RMSE (\(\phi\), \(\theta\)) [rad] | Remarks |
|---|---|---|---|
| TSMC | (0.452, 0.478, 0.321) | (0.175, 0.188) | Large overshoot, sustained oscillations after faults. |
| FTSMC | (0.198, 0.210, 0.154) | (0.082, 0.091) | Better than TSMC, but shows clear deviation post-fault. |
| Proposed ANFTSMC | (0.057, 0.062, 0.043) | (0.023, 0.026) | Minimal overshoot, fast recovery, accurate tracking throughout. |
The key observations were:
- Trajectory Tracking: The ANFTSMC-enabled quadrotor drone adhered precisely to the 3D path, while TSMC and FTSMC showed significant deviation, especially after t=10s.
- Disturbance Rejection: At t=5s, the ANFTSMC quadrotor drone exhibited only a minor, transient deviation, quickly returning to the track due to its adaptive disturbance estimation. The other controllers showed larger and more persistent errors.
- Fault Tolerance: At t=10s, the performance of TSMC and FTSMC degraded severely with large oscillations. In contrast, the ANFTSMC scheme, through its adaptive laws, successfully estimated and compensated for the combined effect of the faults and increased disturbances, maintaining stable and accurate flight.
- Parameter Estimation: The adaptive estimates for parameters like \( \hat{k}_z \) and disturbance bounds \( \hat{\sigma}_z \) converged to constant values (not necessarily the true values) that were sufficient to guarantee stability and performance, demonstrating the self-tuning capability of the controller for the quadrotor drone.
Experimental Flight Test
A real-world flight test was performed using an F450 frame quadrotor drone equipped with a Pixhawk flight controller. The ANFTSMC algorithm was implemented onboard. The test involved a 100-second hover and maneuver sequence at an altitude of 2 meters. A high-powered fan was used to generate persistent, unmodeled wind disturbances. Midway through the flight, a simulated actuator fault was injected by commanding a constant bias to one motor’s control signal.
Results: The physical quadrotor drone managed to maintain stable flight and accurate position hold despite the aggressive fan wind. Upon the injection of the simulated fault, a momentary deviation was observed, but the drone rapidly recovered and resumed stable hovering. The logged flight data confirmed the smooth adaptation of control signals, validating the practical robustness and fault-tolerant capability of the proposed ANFTSMC scheme for real quadrotor drone applications.
Conclusion and Discussion
In this work, I have presented a comprehensive adaptive robust control solution for the trajectory tracking problem of a quadrotor drone subjected to parametric uncertainties, external disturbances, and actuator faults. The proposed Adaptive Non-singular Fast Terminal Sliding Mode Controller (ANFTSMC) successfully addresses these challenges through several key features:
- Robust Sliding Surface: The NFITSS ensures fast, finite-time convergence of tracking errors without singularity issues.
- Integrated Adaptation: The online adaptive laws estimate the upper bounds of lumped disturbances and uncertain system parameters, eliminating the need for their prior knowledge.
- Fault Tolerance: The actuator fault is treated as part of the lumped disturbance, which is estimated and compensated for in real-time, providing inherent fault-tolerant capabilities.
- Chattering Reduction: The use of the \( \tanh(\cdot) \) function instead of the discontinuous \( \text{sgn}(\cdot) \) function significantly reduces control chattering.
The mathematical stability of the entire closed-loop quadrotor drone system was rigorously proven via Lyapunov analysis. Both numerical simulations and physical flight tests demonstrated the superior performance of ANFTSMC compared to traditional TSMC and FTSMC. The proposed controller enabled the quadrotor drone to maintain precise trajectory tracking with minimal overshoot and excellent recovery characteristics in the face of combined disturbances and actuator faults.
Future Work: While this study focuses on bias faults and lumped disturbance estimation, future research directions for enhancing quadrotor drone resilience include:
- Developing dedicated fault detection and isolation (FDI) observers to identify and isolate specific actuator or sensor faults before they become critical.
- Extending the control scheme to manage partial loss of effectiveness and total failure of actuators, potentially requiring control reallocation strategies.
- Implementing and testing the algorithm on more complex quadrotor drone platforms in extreme outdoor environments with GPS-denied navigation.
In summary, the ANFTSMC provides a powerful, practical, and theoretically sound framework for achieving robust and fault-tolerant autonomous flight of quadrotor drones under realistic operating conditions.