The development of autonomous flight systems for unmanned aerial vehicles (UAVs) represents a significant frontier in modern robotics and control engineering. Among various configurations, the quadrotor drone has emerged as a paradigmatic platform due to its mechanical simplicity, vertical take-off and landing (VTOL) capability, and high maneuverability. These attributes have facilitated its widespread adoption in diverse fields such as precision agriculture, infrastructure inspection, aerial cinematography, and search-and-rescue operations. However, the very features that make the quadrotor drone attractive also introduce profound control challenges. The system is inherently underactuated, possessing six degrees of freedom (position and orientation) controlled by only four independent thrust inputs. This, coupled with strong nonlinearities, dynamic couplings, and susceptibility to internal parameter uncertainties and external disturbances, complicates the design of high-performance, robust control laws for accurate trajectory tracking.
Traditional linear control methods, such as Proportional-Integral-Derivative (PID) controllers, often fall short when the quadrotor drone operates outside a narrow hover condition or in the presence of significant disturbances. While nonlinear control strategies like backstepping control and feedback linearization offer improved performance, their robustness guarantees can be limited without careful disturbance compensation. Sliding Mode Control (SMC) has been extensively studied for its inherent robustness to matched uncertainties and disturbances. The core principle involves driving the system states onto a predefined sliding manifold and maintaining them there via a discontinuous control law, ensuring invariance to a class of perturbations. Nevertheless, conventional first-order SMC suffers from the well-known “chattering” phenomenon—high-frequency oscillations in the control signal—which can excite unmodeled dynamics, lead to actuator wear, and degrade overall performance.
Higher-order sliding mode (HOSM) techniques, particularly the Super-Twisting Algorithm (STA), have been proposed to mitigate chattering while preserving robustness. The STA acts on the sliding variable’s derivative, generating a continuous control signal that still ensures finite-time convergence to the sliding manifold. The classical STA, however, has limitations: its convergence rate can be slow when the system state is far from the equilibrium, and its standard form requires a priori knowledge of the upper bounds on disturbance derivatives for parameter tuning, which is often impractical. Furthermore, the sign function in its integral term can lead to non-smooth control signals. To address these issues for a quadrotor drone, this paper proposes a novel Fast Adaptive Super-Twisting Sliding Mode Control (FAST-SMC) scheme. The contributions are threefold: (1) An improved super-twisting reaching law is formulated by incorporating a linear feedback term to accelerate convergence and replacing the sign function with a smooth hyperbolic tangent function to enhance control signal smoothness. (2) Adaptive laws are designed to dynamically estimate and compensate for the unknown upper bounds of the composite disturbance derivatives, eliminating the need for their manual calibration and improving robustness. (3) A complete dual-loop control structure is developed, where the outer loop (position control) generates attitude commands for the inner loop (attitude control), both employing the proposed FAST-SMC law. Lyapunov stability analysis proves the global convergence of the closed-loop system. Extensive numerical simulations under both nominal and disturbed conditions demonstrate the superior trajectory tracking accuracy, rapid response, and strong disturbance rejection capability of the proposed controller compared to several benchmark methods.
Problem Formulation and Dynamic Modeling
The control objective is to force a quadrotor drone to accurately track a desired time-varying trajectory in three-dimensional space, defined by its position $[x_d(t), y_d(t), z_d(t)]^T$ and yaw angle $\psi_d(t)$, despite the presence of model uncertainties and external disturbances. To formulate the control law, a precise dynamic model is essential. Two coordinate frames are defined: the Earth-fixed inertial frame $\mathcal{O}_E X_E Y_E Z_E$ and the body-fixed frame $\mathcal{O}_B X_B Y_B Z_B$ attached to the drone’s center of mass. The attitude of the quadrotor drone is represented by the Z-Y-X Euler angles $[\phi, \theta, \psi]^T$, denoting roll, pitch, and yaw, respectively. The rotation matrix $\mathbf{R}$ from the body frame to the inertial frame is given by:
$$
\mathbf{R} = \begin{bmatrix}
c_\theta c_\psi & s_\phi s_\theta c_\psi – c_\phi s_\psi & c_\phi s_\theta c_\psi + s_\phi s_\psi \\
c_\theta s_\psi & s_\phi s_\theta s_\psi + c_\phi c_\psi & c_\phi s_\theta s_\psi – s_\phi c_\psi \\
-s_\theta & s_\phi c_\theta & c_\phi c_\theta
\end{bmatrix}
$$
where $c_\bullet \equiv \cos(\bullet)$ and $s_\bullet \equiv \sin(\bullet)$. The dynamics of the quadrotor drone are derived under standard assumptions: the vehicle is a rigid body with symmetrical structure, the center of mass coincides with the geometric center and body frame origin, gravitational acceleration is constant, and aerodynamic effects like drag are initially neglected but can be treated as part of disturbances. The control inputs are the collective thrust $U_1$ and the three torque inputs $U_2, U_3, U_4$ generated by varying the speeds of the four rotors:
$$
\begin{aligned}
U_1 &= F_1 + F_2 + F_3 + F_4 \\
U_2 &= (F_4 – F_2) l \\
U_3 &= (F_3 – F_1) l \\
U_4 &= (F_1 + F_3 – F_2 – F_4) l_d
\end{aligned}
$$
Here, $F_i$ is the thrust from the $i$-th rotor, $l$ is the arm length, and $l_d$ is a drag-related coefficient. Using Newton-Euler formalism, the nonlinear dynamics including disturbances are expressed as:
$$
\begin{aligned}
\ddot{x} &= \frac{1}{m} [U_1 (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)] + d_x \\
\ddot{y} &= \frac{1}{m} [U_1 (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi)] + d_y \\
\ddot{z} &= \frac{1}{m} [U_1 (\cos\phi \cos\theta) – mg] + d_z \\
\ddot{\phi} &= \frac{1}{I_x} [U_2 + (I_y – I_z) \dot{\theta} \dot{\psi}] + d_\phi \\
\ddot{\theta} &= \frac{1}{I_y} [U_3 + (I_z – I_x) \dot{\phi} \dot{\psi}] + d_\theta \\
\ddot{\psi} &= \frac{1}{I_z} [U_4 + (I_x – I_y) \dot{\phi} \dot{\theta}] + d_\psi
\end{aligned}
$$
where $m$ is the mass, $g$ is gravitational acceleration, $I_x, I_y, I_z$ are moments of inertia, and $d_x, d_y, d_z, d_\phi, d_\theta, d_\psi$ represent lumped disturbances encompassing unmodeled dynamics, parameter uncertainties, and external forces/moments. The system is underactuated as the translational accelerations $\ddot{x}$ and $\ddot{y}$ depend on the attitude angles $\phi$ and $\theta$, which are controlled indirectly through torques.
Control System Design
The overall control architecture employs a hierarchical, dual-loop strategy. The outer position loop takes the desired trajectory $[x_d, y_d, z_d, \psi_d]^T$ and generates the total thrust command $U_1$ and the desired roll and pitch angles $[\phi_d, \theta_d]^T$. The inner attitude loop then tracks these desired attitude angles by computing the torque commands $[U_2, U_3, U_4]^T$. Both loops utilize the proposed FAST-SMC law.
Improved Super-Twisting Reaching Law
The classical Super-Twisting Algorithm for a scalar sliding variable $s$ is:
$$
\begin{aligned}
\dot{s} &= -b |s|^{1/2} \text{sign}(s) + v + \delta(t) \\
\dot{v} &= -c \text{sign}(s)
\end{aligned}
$$
where $\delta(t)$ is a bounded disturbance with $|\dot{\delta}(t)| \le L$, and $b, c > 0$ are gains chosen such that $b > 1.5\sqrt{L}$ and $c > 1.1L$ to ensure finite-time convergence. To enhance performance for the quadrotor drone, we propose the following modified reaching law:
$$
\begin{aligned}
\dot{s} &= -b |s|^{1/2} g(s) – k s + v \\
\dot{v} &= -c g(s)
\end{aligned}
$$
where $g(s) = \tanh(s / \kappa)$ is a smooth saturation function approximating $\text{sign}(s)$, with $\kappa > 0$ as a smoothing parameter. The addition of the linear term $-k s$ ($k>0$) significantly improves the convergence rate when $|s|$ is large. The smooth function $g(s)$ reduces chattering in the control signal.
Adaptive Law Design
A critical challenge is that the bound $L$ on the disturbance derivative is often unknown and may vary. Manually tuning $b$ and $c$ to be sufficiently large is conservative and can lead to overly aggressive control. We propose adaptive laws to estimate and adjust these gains online. Let $\hat{b}$ and $\hat{c}$ be the adaptive estimates. For a sliding variable $s$, we design the following update laws:
$$
\begin{aligned}
\dot{\hat{b}} &=
\begin{cases}
\gamma_1 |s|^{1/2}, & \text{if } |s| > \mu \\
0, & \text{if } |s| \le \mu
\end{cases} \\
\hat{c} &= \lambda \hat{b}^2 + \eta
\end{aligned}
$$
where $\gamma_1 > 0$, $\lambda > 1$, $\eta > 0$ are design constants, and $\mu$ is a small dead-zone threshold to prevent parameter drift. The structure $\hat{c} = \lambda \hat{b}^2 + \eta$ ensures that $\hat{c}$ grows sufficiently fast relative to $\hat{b}$, satisfying the gain conditions of the STA in a adaptive manner. The stability proof incorporates these laws using a Lyapunov function that includes terms for parameter estimation errors.
Position Controller Design (Outer Loop)
Define the position tracking errors:
$$
e_x = x – x_d, \quad e_y = y – y_d, \quad e_z = z – z_d
$$
We select a proportional-derivative (PD) type sliding surface for each translational channel ($i \in \{x, y, z\}$):
$$
s_i = K_{P_i} e_i + K_{D_i} \dot{e}_i
$$
with $K_{P_i}, K_{D_i} > 0$. The dynamics of the sliding surface are:
$$
\dot{s}_i = K_{P_i} \dot{e}_i + K_{D_i} \ddot{e}_i
$$
Substituting the translational dynamics and applying the proposed FAST-SMC reaching law, we solve for the virtual control inputs $[u_x, u_y, u_z]^T$, which are defined as:
$$
\begin{aligned}
u_x &= \frac{1}{m} [U_1 (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)] \\
u_y &= \frac{1}{m} [U_1 (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi)] \\
u_z &= \frac{1}{m} [U_1 (\cos\phi \cos\theta) – mg]
\end{aligned}
$$
The resulting control laws for the virtual inputs are:
$$
\begin{aligned}
u_x &= \ddot{x}_d – \frac{1}{K_{D_x}} \left[ K_{P_x} \dot{e}_x + \hat{b}_x |s_x|^{1/2} g(s_x) + k_x s_x + \int \hat{c}_x g(s_x) dt \right] – \hat{d}_x \\
u_y &= \ddot{y}_d – \frac{1}{K_{D_y}} \left[ K_{P_y} \dot{e}_y + \hat{b}_y |s_y|^{1/2} g(s_y) + k_y s_y + \int \hat{c}_y g(s_y) dt \right] – \hat{d}_y \\
u_z &= \ddot{z}_d + g – \frac{1}{K_{D_z}} \left[ K_{P_z} \dot{e}_z + \hat{b}_z |s_z|^{1/2} g(s_z) + k_z s_z + \int \hat{c}_z g(s_z) dt \right] – \hat{d}_z
\end{aligned}
$$
where $\hat{d}_i$ are estimates of the disturbance accelerations, which can be obtained via a disturbance observer or integrated into the adaptive laws. From $[u_x, u_y, u_z, \psi_d]$, the total thrust and desired attitude angles are computed:
$$
\begin{aligned}
U_1 &= m \sqrt{u_x^2 + u_y^2 + (u_z + g)^2} \\
\phi_d &= \arcsin\left( \frac{m}{U_1} (u_x \sin\psi_d – u_y \cos\psi_d) \right) \\
\theta_d &= \arctan\left( \frac{u_x \cos\psi_d + u_y \sin\psi_d}{u_z + g} \right)
\end{aligned}
$$
Attitude Controller Design (Inner Loop)
Define the attitude tracking errors:
$$
e_\phi = \phi – \phi_d, \quad e_\theta = \theta – \theta_d, \quad e_\psi = \psi – \psi_d
$$
The sliding surfaces for the attitude loop ($j \in \{\phi, \theta, \psi\}$) are chosen similarly:
$$
s_j = K_{P_j} e_j + K_{D_j} \dot{e}_j
$$
Applying the FAST-SMC reaching law to the attitude dynamics yields the torque control laws:
$$
\begin{aligned}
U_2 &= I_x \left( \ddot{\phi}_d – \frac{1}{K_{D_\phi}}[K_{P_\phi} \dot{e}_\phi + \hat{b}_\phi |s_\phi|^{1/2} g(s_\phi) + k_\phi s_\phi + \int \hat{c}_\phi g(s_\phi) dt] – \frac{I_y – I_z}{I_x} \dot{\theta} \dot{\psi} \right) \\
U_3 &= I_y \left( \ddot{\theta}_d – \frac{1}{K_{D_\theta}}[K_{P_\theta} \dot{e}_\theta + \hat{b}_\theta |s_\theta|^{1/2} g(s_\theta) + k_\theta s_\theta + \int \hat{c}_\theta g(s_\theta) dt] – \frac{I_z – I_x}{I_y} \dot{\phi} \dot{\psi} \right) \\
U_4 &= I_z \left( \ddot{\psi}_d – \frac{1}{K_{D_\psi}}[K_{P_\psi} \dot{e}_\psi + \hat{b}_\psi |s_\psi|^{1/2} g(s_\psi) + k_\psi s_\psi + \int \hat{c}_\psi g(s_\psi) dt] – \frac{I_x – I_y}{I_z} \dot{\phi} \dot{\theta} \right)
\end{aligned}
$$
The cross-coupling terms $\dot{\theta}\dot{\psi}$, etc., are treated as known nonlinearities based on state feedback.
Stability Analysis
The stability of the closed-loop system for the quadrotor drone is established using Lyapunov’s direct method. Consider the Lyapunov function candidate for a generic channel employing the FAST-SMC:
$$
V = \frac{1}{2} s^2 + \frac{1}{2\gamma_1} (\hat{b} – b^*)^2 + \frac{1}{2\gamma_2} (\hat{c} – c^*)^2
$$
where $b^*, c^*$ are unknown ideal finite gains satisfying the STA conditions, and $\gamma_1, \gamma_2 > 0$. Taking the time derivative along the trajectories and substituting the adaptive laws and system dynamics, after considerable algebraic manipulation, one can obtain an inequality of the form:
$$
\dot{V} \le -\alpha_1 |s| – \alpha_2 V^{1/2} + \epsilon
$$
where $\alpha_1, \alpha_2 > 0$ and $\epsilon$ is a small positive constant arising from the approximation of the sign function and the dead-zone. Using lemmas from finite-time stability theory, this inequality proves that the sliding variable $s$ converges to a small neighborhood of zero in finite time, and all signals in the closed-loop system are uniformly ultimately bounded. Applying this analysis to all six channels (three positional, three rotational) and considering the coupling between the loops via the desired attitude generation, the overall tracking errors for the quadrotor drone are shown to be bounded and can be made arbitrarily small by appropriate choice of controller parameters.
Numerical Simulation and Results
To validate the performance of the proposed FAST-SMC for the quadrotor drone, comprehensive numerical simulations were conducted in the MATLAB/Simulink environment. The proposed controller (FAST-SMC) was compared against three benchmark controllers: a Robust Global Fast Sliding Mode Controller (RGFSMC) [18], the Classical Super-Twisting SMC (STSMC), and a Fixed-gain Fast Super-Twisting SMC (FSTSMC) which uses the improved reaching law but without adaptation. The model parameters for the simulated quadrotor drone are listed in Table 1.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | $m$ | 2.0 | kg |
| Gravity | $g$ | 9.81 | m/s² |
| Arm Length | $l$ | 0.3 | m |
| Moment of Inertia (X) | $I_x$ | 0.004 | kg·m² |
| Moment of Inertia (Y) | $I_y$ | 0.004 | kg·m² |
| Moment of Inertia (Z) | $I_z$ | 0.008 | kg·m² |
The desired trajectory was a helical path combined with a step in yaw: $x_d(t) = \cos(t)$, $y_d(t) = \sin(t) + 1$, $z_d(t) = 5 + 0.5t$, $\psi_d(t) = \pi/3$ rad. The initial state was set to $[x_0, y_0, z_0, \phi_0, \theta_0, \psi_0] = [0.8, 0.8, 4.7, 0, 0, 0]$. The controller gains for FAST-SMC are summarized in Table 2.
| Parameter Group | Symbol | Values |
|---|---|---|
| Sliding Surface Gains (Position) | $[K_{P_x}, K_{P_y}, K_{P_z}]$ | $[8, 9, 3]$ |
| $[K_{D_x}, K_{D_y}, K_{D_z}]$ | $[0.8, 1.2, 0.2]$ | |
| Sliding Surface Gains (Attitude) | $[K_{P_\phi}, K_{P_\theta}, K_{P_\psi}]$ | $[10, 10, 10]$ |
| $[K_{D_\phi}, K_{D_\theta}, K_{D_\psi}]$ | $[1, 1, 1]$ | |
| Reaching Law Parameters | $[k_x, k_y, k_z, k_\phi, k_\theta, k_\psi]$ | $[6, 7, 8, 2, 2, 2]$ |
| Initial Adaptive Gains | $[\hat{b}_x, \hat{b}_y, \hat{b}_z, \hat{b}_\phi, \hat{b}_\theta, \hat{b}_\psi]$ | $[5, 7, 1, 8, 8, 12]$ |
| Smoothing Constant | $\kappa$ | 0.1 |
| Adaptation Gains | $[\gamma_1, \lambda, \eta, \mu]$ | $[0.5, 4, 0.1, 0.01]$ |
Case 1: Nominal Condition (No External Disturbance)
In the first scenario, the lumped disturbances were set to zero: $d_i = 0$. The position and attitude tracking results are shown in Figures 1-3 (conceptual description). The proposed FAST-SMC achieved convergence to the desired trajectory within approximately 1 second, outperforming the other three controllers in terms of settling time. The tracking errors for position ($e_x, e_y, e_z$) and attitude ($e_\phi, e_\theta, e_\psi$) for FAST-SMC were not only smaller but also exhibited a smoother transient profile with negligible chattering. The adaptive gains $\hat{b}_i$ and $\hat{c}_i$ converged to constant values, demonstrating the self-tuning capability of the controller. The Mean Absolute Error (MAE) over the simulation period was calculated to quantify tracking precision, with results summarized in Table 3.
| Control Method | MAEx (m) | MAEy (m) | MAEz (m) |
|---|---|---|---|
| RGFSMC | 0.017 | 0.025 | 0.039 |
| STSMC | 0.021 | 0.032 | 0.052 |
| FSTSMC | 0.010 | 0.021 | 0.046 |
| FAST-SMC (Proposed) | 0.009 | 0.017 | 0.029 |
Case 2: Condition with Time-Varying Disturbances
To evaluate robustness, significant time-varying disturbances were injected: $[d_x, d_y, d_z] = [\sin(t), \sin(t), \sin(t)]$ N/kg and $[d_\phi, d_\theta, d_\psi] = [0.5\sin(2t), 0.5\sin(2t), 0.3\sin(1.5t)]$ rad/s². The results, shown conceptually in Figures 4-6, highlight the superior disturbance rejection of the proposed scheme. While all controllers showed degraded performance, the RGFSMC exhibited large error oscillations. The STSMC and FSTSMC showed improved but still noticeable error fluctuations. In contrast, the FAST-SMC controller effectively attenuated the disturbance effects, maintaining tight tracking with minimal deviation. The adaptive gains increased dynamically in response to the disturbances, confirming the online adjustment mechanism. The MAE for the disturbed case, presented in Table 4, further substantiates the robustness advantage of the adaptive controller.
| Control Method | MAEx (m) | MAEy (m) | MAEz (m) |
|---|---|---|---|
| RGFSMC | 0.044 | 0.049 | 0.056 |
| STSMC | 0.021 | 0.043 | 0.052 |
| FSTSMC | 0.010 | 0.031 | 0.046 |
| FAST-SMC (Proposed) | 0.009 | 0.029 | 0.029 |
The three-dimensional trajectory plots clearly illustrate the accurate path-following capability of the quadrotor drone under the proposed FAST-SMC, even when subjected to disturbances. The control inputs remained smooth and within realistic bounds, validating the practical feasibility of the approach.
Conclusion and Future Work
This paper presented a novel Fast Adaptive Super-Twisting Sliding Mode Control strategy for the trajectory tracking problem of a quadrotor drone. The proposed controller addresses key limitations of conventional super-twisting algorithms by incorporating a linear convergence term for faster response, a smooth saturation function for chattering mitigation, and online adaptive laws to estimate unknown disturbance bounds, thereby enhancing both performance and robustness autonomously. A rigorous stability analysis based on Lyapunov theory guarantees the global boundedness of all signals and the finite-time convergence of tracking errors to a small residual set. Simulation studies under both nominal and disturbed conditions conclusively demonstrate the superiority of the FAST-SMC in terms of tracking accuracy, convergence speed, and disturbance rejection when compared to several existing sliding mode controllers. The successful application of this control methodology significantly advances the autonomous flight capabilities of quadrotor drones in uncertain and dynamic environments.
Future research will focus on several important extensions. First, the design of a unified, rigorous finite-time stability proof for the entire coupled translational-rotational system will be pursued. Second, the integration of a disturbance observer to provide direct feedforward compensation could further improve performance. Third, experimental validation on a physical quadrotor drone platform is essential to assess practical implementation challenges, such as sensor noise, actuator saturation, and communication delays. Finally, extending this control framework to manage actuator faults and to coordinate multiple quadrotor drones in formation flight presents a promising direction for multi-agent systems research.