The proliferation of unmanned aerial vehicles (UAVs) has revolutionized numerous fields, from aerial photography and precision agriculture to search and rescue and infrastructure inspection. Among the various configurations, the quadrotor drone stands out due to its mechanical simplicity, ability to hover, and high maneuverability. However, these advantages come with significant control challenges. The quadrotor drone is an underactuated, highly coupled, and nonlinear system. Its flight performance, especially its ability to accurately track desired attitudes and positions, is critically dependent on the robustness and responsiveness of its control system. External disturbances like wind gusts and internal model uncertainties are ever-present threats to stability and tracking precision. Therefore, the design of high-performance flight controllers that guarantee fast, accurate, and robust tracking remains a central research focus in aerial robotics.
Traditional linear control methods, such as the ubiquitous Proportional-Integral-Derivative (PID) controller, are widely used due to their simplicity and ease of implementation. For a quadrotor drone operating near a hover state (small angles), PID controllers can provide adequate performance. However, their efficacy diminishes rapidly when the system deviates from its linearized operating point or when subjected to significant disturbances. Their fixed-gain structure lacks the inherent adaptability to counteract unmodeled dynamics or persistent external forces effectively.
To address these limitations, advanced nonlinear control strategies have been extensively investigated. Sliding Mode Control (SMC) is renowned for its robustness against bounded disturbances and parameter variations. Once the system state is driven to a predefined sliding surface, it becomes invariant to matched disturbances. However, conventional SMC suffers from the well-known “chattering” phenomenon—high-frequency oscillations in the control signal—caused by the discontinuous sign function. This chattering can excite unmodeled high-frequency dynamics and lead to increased actuator wear. Backstepping control offers a systematic, recursive design procedure for nonlinear systems in strict-feedback form, making it structurally ideal for the cascaded dynamics of a quadrotor drone. It provides a framework for stabilizing each subsystem step-by-step. Nevertheless, its performance is highly sensitive to model accuracy and lacks inherent robustness against disturbances unless combined with other techniques like disturbance observers.
This work proposes a novel synthesis that leverages the strengths of both methodologies while mitigating their individual weaknesses. We design a Backstepping Super-Twisting Sliding Mode Controller (BSTC) for the attitude and altitude control of a quadrotor drone. The backstepping technique is employed to recursively design virtual control laws and stabilize the lower-order subsystems. A Super-Twisting Algorithm (STA)-based sliding mode controller is then integrated at the final step. The Super-Twisting algorithm, a second-order sliding mode technique, generates a continuous control signal by integrating a discontinuous term. This fundamental characteristic allows it to achieve the robustness of standard SMC while drastically reducing or even eliminating chattering. The combined approach ensures finite-time convergence, strong disturbance rejection, and smooth control action. We begin by deriving a nonlinear dynamic model, proceed to the detailed controller design and stability proof via Lyapunov theory, and finally present comprehensive simulation studies comparing the proposed BSTC against a classic PID and a standard Backstepping Sliding Mode Controller (BSMC) under various operating conditions.
Nonlinear Dynamic Modeling of the Quadrotor Drone
Accurate modeling is the cornerstone of model-based control design. We consider a standard “plus+” configuration quadrotor drone. Two coordinate frames are defined: the Earth-fixed inertial frame \( E(O_E, x_E, y_E, z_E) \) and the body-fixed frame \( B(O_B, x_B, y_B, z_B) \) attached to the drone’s center of mass. The position of the drone in the inertial frame is given by the vector \( \boldsymbol{p} = [x, y, z]^T \), and its orientation is described by the Euler angles \( \boldsymbol{\Theta} = [\phi, \theta, \psi]^T \), representing roll, pitch, and yaw, respectively. The following assumptions are commonly made to simplify the model: 1) The drone structure is rigid and symmetrical. 2) The center of mass coincides with the geometric center of the body frame. 3) Propeller thrust and drag are proportional to the square of the rotor’s angular speed.
Using the Newton-Euler formalism, the translational and rotational dynamics of the quadrotor drone can be derived. The equations of motion are governed by the following set of nonlinear differential equations:
Rotational Dynamics (Body Frame):
$$
\begin{align}
\dot{p} &= \frac{I_{yy} – I_{zz}}{I_{xx}} q r + \frac{J_r}{I_{xx}} q \Omega + \frac{l}{I_{xx}} U_2 \\
\dot{q} &= \frac{I_{zz} – I_{xx}}{I_{yy}} p r – \frac{J_r}{I_{yy}} p \Omega + \frac{l}{I_{yy}} U_3 \\
\dot{r} &= \frac{I_{xx} – I_{yy}}{I_{zz}} p q + \frac{1}{I_{zz}} U_4
\end{align}
$$
Translational Dynamics (Inertial Frame, z-axis):
$$
\begin{align}
\ddot{z} &= -g + \frac{\cos\phi \cos\theta}{m} U_1
\end{align}
$$
Kinematic Relationship (Euler Angle Rates):
$$
\begin{align}
\begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} =
\begin{bmatrix}
1 & \sin\phi \tan\theta & \cos\phi \tan\theta \\
0 & \cos\phi & -\sin\phi \\
0 & \sin\phi / \cos\theta & \cos\phi / \cos\theta
\end{bmatrix}
\begin{bmatrix} p \\ q \\ r \end{bmatrix}
\end{align}
$$
Where \( p, q, r \) are the angular velocities in the body frame. The control inputs \( U_1, U_2, U_3, U_4 \) are generated by the four rotors and are related to the individual rotor speeds \( \omega_i \) by:
$$
\begin{align}
U_1 &= C_T (\omega_1^2 + \omega_2^2 + \omega_3^2 + \omega_4^2) \\
U_2 &= l C_T (\omega_4^2 – \omega_2^2) \\
U_3 &= l C_T (\omega_3^2 – \omega_1^2) \\
U_4 &= C_D (\omega_2^2 + \omega_4^2 – \omega_1^2 – \omega_3^2)
\end{align}
$$
Here, \( \Omega = \omega_2 + \omega_4 – \omega_1 – \omega_3 \) is the sum of rotor speed imbalances, and \( J_r \) is the total rotational inertia of the rotors. For controller design, we assume near-hover conditions (\( \phi, \theta \) are small), which allows the common simplification \( [\dot{\phi}, \dot{\theta}, \dot{\psi}]^T \approx [p, q, r]^T \). The system can then be expressed in a state-space form suitable for the backstepping procedure. Let us define the state vector as \( \boldsymbol{x} = [x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8]^T = [\phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}, z, \dot{z}]^T \). The simplified model becomes:
$$
\begin{align}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= a_1 x_4 x_6 + a_2 x_4 \Omega + b_1 U_2 + \Delta_\phi \\
\dot{x}_3 &= x_4 \\
\dot{x}_4 &= a_3 x_2 x_6 + a_4 x_2 \Omega + b_2 U_3 + \Delta_\theta \\
\dot{x}_5 &= x_6 \\
\dot{x}_6 &= a_5 x_2 x_4 + b_3 U_4 + \Delta_\psi \\
\dot{x}_7 &= x_8 \\
\dot{x}_8 &= -g + \frac{\cos x_1 \cos x_3}{m} U_1 + \Delta_z
\end{align}
$$
where \( a_i \) and \( b_i \) are constants composed of the inertial parameters \( I_{xx}, I_{yy}, I_{zz}, J_r, l, m \), and \( \Delta_{(\cdot)} \) represent lumped bounded disturbances (e.g., wind, unmodeled dynamics). The parameters for a typical mid-size quadrotor drone used in our study are summarized in Table 1.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | \( m \) | 0.65 | kg |
| Gravity | \( g \) | 9.81 | m/s² |
| Arm Length | \( l \) | 0.23 | m |
| Thrust Coefficient | \( C_T \) | \( 3.13 \times 10^{-5} \) | N·s²/rad² |
| Drag Coefficient | \( C_D \) | \( 7.50 \times 10^{-7} \) | N·m·s²/rad² |
| Inertia (x-axis) | \( I_{xx} \) | \( 7.50 \times 10^{-3} \) | kg·m² |
| Inertia (y-axis) | \( I_{yy} \) | \( 7.50 \times 10^{-3} \) | kg·m² |
| Inertia (z-axis) | \( I_{zz} \) | \( 1.30 \times 10^{-2} \) | kg·m² |
| Rotor Inertia | \( J_r \) | \( 6.50 \times 10^{-5} \) | kg·m² |
Controller Design: Integrating Backstepping and Super-Twisting Sliding Mode
The control objective is to force the quadrotor drone states \( (z, \phi, \theta, \psi) \) to track desired reference trajectories \( (z_d, \phi_d, \theta_d, \psi_d) \) with high accuracy and robustness. We design four separate controllers: one for altitude \( z \) and three for the attitude angles \( \phi, \theta, \psi \). The overall control architecture is depicted in Figure 1. The design philosophy follows a two-stage process for each channel: first, a backstepping procedure to define stable error dynamics and virtual controls; second, the use of the Super-Twisting Algorithm to synthesize the final robust control law.
Foundations: The Super-Twisting Algorithm (STA)
The STA is a powerful second-order sliding mode algorithm. Consider a scalar system with a bounded disturbance:
$$
\begin{align}
\dot{s} = u + \delta(t), \quad |\delta(t)| \leq L
\end{align}
$$
The standard STA control law is:
$$
\begin{align}
u &= -k_1 |s|^{1/2} \text{sign}(s) + v \\
\dot{v} &= -k_2 \text{sign}(s)
\end{align}
$$
where \( k_1 \) and \( k_2 \) are positive gains. The remarkable property of STA is that it drives both \( s \) and \( \dot{s} \) to zero in finite time without requiring the derivative of \( s \), and it produces a continuous control signal \( u \), thereby eliminating chattering. Finite-time convergence is guaranteed if the gains satisfy \( k_2 > L \) and \( k_1^2 \geq 4L \frac{k_2+L}{k_2-L} \).
Altitude Controller Design
Let us define the altitude tracking error: \( e_z = z_d – z \). The first step in backstepping is to treat \( \dot{z} = x_8 \) as a virtual control for stabilizing \( e_z \). Define a Lyapunov function candidate \( V_1 = \frac{1}{2} e_z^2 \). Its derivative is \( \dot{V}_1 = e_z (\dot{z}_d – x_8) \). We choose the desired virtual control \( \alpha_z \) to make \( \dot{V}_1 \) negative definite:
$$
\begin{align}
\alpha_z &= \dot{z}_d + c_z e_z, \quad c_z > 0
\end{align}
$$
This yields \( \dot{V}_1 = -c_z e_z^2 + e_z (\alpha_z – x_8) \). Define the velocity error \( z_8 = x_8 – \alpha_z \). The new error dynamics are \( \dot{e}_z = -c_z e_z + z_8 \). Now, we aim to drive \( z_8 \) to zero using STA. Define a sliding surface \( s_z = z_8 + \lambda_z \int_0^t z_8(\tau)d\tau \), where \( \lambda_z > 0 \). Taking its derivative and substituting the \( \ddot{z} \) dynamics from the model:
$$
\begin{align}
\dot{s}_z &= \ddot{z} – \ddot{z}_d + c_z \dot{e}_z + \lambda_z z_8 + \Delta_z \\
&= \left(-g + \frac{\cos\phi \cos\theta}{m} U_1 \right) – \ddot{z}_d + c_z(-c_z e_z + z_8) + \lambda_z z_8 + \Delta_z
\end{align}
$$
We design \( U_1 \) to enforce \( \dot{s}_z = -k_{z1}|s_z|^{1/2}\text{sign}(s_z) – k_{z2} \int \text{sign}(s_z) dt \). Solving for \( U_1 \):
$$
\begin{align}
U_1 &= \frac{m}{\cos\phi \cos\theta} \Bigg[ g + \ddot{z}_d – c_z(-c_z e_z + z_8) – \lambda_z z_8 \notag \\
&\quad – k_{z1}|s_z|^{1/2}\text{sign}(s_z) – k_{z2} \int_0^t \text{sign}(s_z(\tau))d\tau \Bigg]
\end{align}
$$
To further smooth the control signal, the discontinuous \( \text{sign}(\cdot) \) function is replaced by a saturation function \( \text{sat}(s/\Phi) \) with boundary layer \( \Phi \). With proper gain selection (\( k_{z2} > |\Delta_z|, k_{z1} \) sufficiently large), the altitude tracking errors \( e_z \) and \( z_8 \) converge to a small neighborhood of zero in finite time.
Attitude Controller Design (Roll Channel Example)
The design for the roll angle \( \phi \) follows an identical procedure. Define \( e_\phi = \phi_d – \phi \). Step 1: Virtual control \( \alpha_\phi = \dot{\phi}_d + c_\phi e_\phi \). Define \( z_2 = \dot{\phi} – \alpha_\phi \). The sliding surface is \( s_\phi = z_2 + \lambda_\phi \int_0^t z_2(\tau)d\tau \). Its derivative involves the \( \ddot{\phi} \) dynamics:
$$
\begin{align}
\dot{s}_\phi &= \ddot{\phi} – \ddot{\phi}_d + c_\phi \dot{e}_\phi + \lambda_\phi z_2 + \Delta_\phi \\
&= \left( \frac{I_{yy}-I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} + \frac{J_r}{I_{xx}} \dot{\theta} \Omega + \frac{l}{I_{xx}} U_2 \right) – \ddot{\phi}_d + c_\phi(-c_\phi e_\phi + z_2) + \lambda_\phi z_2 + \Delta_\phi
\end{align}
$$
Applying the STA principle, the roll control input \( U_2 \) is derived as:
$$
\begin{align}
U_2 &= \frac{I_{xx}}{l} \Bigg[ \ddot{\phi}_d – \frac{I_{yy}-I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} – \frac{J_r}{I_{xx}} \dot{\theta} \Omega – c_\phi(-c_\phi e_\phi + z_2) – \lambda_\phi z_2 \notag \\
&\quad – k_{\phi 1}|s_\phi|^{1/2}\text{sat}(s_\phi/\Phi) – k_{\phi 2} \int_0^t \text{sat}(s_\phi(\tau)/\Phi)d\tau \Bigg]
\end{align}
$$
The controllers for pitch \( \theta \) (\( U_3 \)) and yaw \( \psi \) (\( U_4 \)) are designed analogously. The final control structure for the quadrotor drone consists of these four independent but coupled control laws.
Stability Analysis
The closed-loop stability of the entire quadrotor drone system under the proposed BSTC can be proven using Lyapunov’s direct method. Consider a composite Lyapunov function candidate for the four subsystems:
$$
\begin{align}
V = \frac{1}{2} e_z^2 + \frac{1}{2} e_\phi^2 + \frac{1}{2} e_\theta^2 + \frac{1}{2} e_\psi^2
\end{align}
$$
Taking its time derivative and substituting the first-step virtual control laws (\( \alpha_z, \alpha_\phi, \alpha_\theta, \alpha_\psi \)):
$$
\begin{align}
\dot{V} &= e_z \dot{e}_z + e_\phi \dot{e}_\phi + e_\theta \dot{e}_\theta + e_\psi \dot{e}_\psi \\
&= -c_z e_z^2 + e_z z_8 – c_\phi e_\phi^2 + e_\phi z_2 – c_\theta e_\theta^2 + e_\theta z_4 – c_\psi e_\psi^2 + e_\psi z_6
\end{align}
$$
The terms \( e_i z_j \) are cross terms arising from the backstepping procedure. The subsequent design of the STA-based control laws (\( U_1, U_2, U_3, U_4 \)) guarantees that the sliding surfaces \( s_z, s_\phi, s_\theta, s_\psi \) are reached in finite time. Upon reaching the sliding surface \( s=0 \), it follows that \( z_j = -\lambda_j \int z_j dt \), which implies \( z_j \to 0 \) exponentially. Therefore, the cross terms vanish. Consequently, after the finite-time transient of the STA, the derivative simplifies to:
$$
\begin{align}
\dot{V} \approx -c_z e_z^2 – c_\phi e_\phi^2 – c_\theta e_\theta^2 – c_\psi e_\psi^2 \leq 0
\end{align}
$$
Since \( V > 0 \) and \( \dot{V} \leq 0 \), by Lyapunov’s stability theorem, the tracking errors \( e_z, e_\phi, e_\theta, e_\psi \) are globally uniformly bounded and converge asymptotically to zero. The finite-time convergence of the STA ensures that the system states reach and remain on the sliding manifolds, providing robust and precise tracking for the quadrotor drone.
Simulation Results and Performance Evaluation
To validate the effectiveness of the proposed Backstepping Super-Twisting Controller (BSTC), extensive simulations were conducted in the MATLAB/Simulink environment. The performance was compared against two benchmark controllers: a well-tuned PID controller and a standard Backstepping Sliding Mode Controller (BSMC). The BSMC uses a linear sliding surface and a sign function with a saturation layer for chattering reduction. The controller gains for the BSTC were tuned based on the stability conditions and are listed in Table 2.
| Control Channel | Gains |
|---|---|
| Altitude (z) | \( c_z=4, \lambda_z=10, k_{z1}=33, k_{z2}=5, \Phi=0.01 \) |
| Roll (\(\phi\)) | \( c_\phi=5, \lambda_\phi=12, k_{\phi 1}=8, k_{\phi 2}=5, \Phi=0.01 \) |
| Pitch (\(\theta\)) | \( c_\theta=8, \lambda_\theta=18, k_{\theta 1}=12, k_{\theta 2}=18, \Phi=0.01 \) |
| Yaw (\(\psi\)) | \( c_\psi=8, \lambda_\psi=10, k_{\psi 1}=15, k_{\psi 2}=18, \Phi=0.01 \) |
Case 1: Nominal Performance (Disturbance-Free)
The quadrotor drone was commanded to climb to 20 meters and track varying attitude references. The initial conditions were zero. The tracking performance is summarized in Figure 2 and Table 3. The BSTC demonstrated superior transient performance. For altitude, it reached within a 5 cm error band in only 0.97 seconds, which is 32.6% faster than the BSMC (1.44 s) and 65.8% faster than the PID controller (2.84 s). All three attitude channels under BSTC showed fast, accurate, and smooth tracking with virtually no overshoot, outperforming the other controllers.
| Controller | Rise Time (seconds) | Improvement vs. PID |
|---|---|---|
| PID | 2.840 | – |
| BSMC | 1.444 | 49.2% faster |
| BSTC (Proposed) | 0.972 | 65.8% faster |
A critical advantage of the BSTC is evident in the control effort signals, shown in Figure 3. The BSMC, despite using a saturation function, exhibits noticeable high-frequency chattering in all control inputs \( U_1 \) to \( U_4 \). This chattering is energetically inefficient and detrimental to actuators. In stark contrast, the control signals generated by the proposed BSTC are smooth and continuous, thanks to the integral action of the Super-Twisting Algorithm. This demonstrates the BSTC’s superior chattering suppression capability.
Case 2: Performance Under Disturbances
The robustness of the controllers was tested by introducing significant disturbances. At t = 8 seconds, a step wind gust disturbance of 2 m/s was applied vertically. Additionally, continuous internal disturbances \( \Delta = \sin(t) + 1.5\cos(t+\pi/4) \) were injected into all four channels. The results, shown in Figure 4, highlight the robustness of the proposed method. Upon the wind gust, the PID-controlled quadrotor drone exhibited a large deviation and slow recovery. The BSMC performed better but still showed a noticeable dip and oscillation. The BSTC showed the smallest deviation and the fastest, smoothest recovery to the desired altitude. The attitude tracking under combined disturbances further confirms the superiority of the BSTC. While all controllers maintained stability, the BSTC maintained the smallest tracking error bounds, as quantified in Table 4.
| Controller | \( \max(|\phi_{err}|) \) (rad) | \( \max(|\theta_{err}|) \) (rad) | \( \max(|\psi_{err}|) \) (rad) |
|---|---|---|---|
| PID | \( 4.21 \times 10^{-2} \) | \( 3.98 \times 10^{-2} \) | \( 3.05 \times 10^{-2} \) |
| BSMC | \( 1.87 \times 10^{-2} \) | \( 1.65 \times 10^{-2} \) | \( 1.22 \times 10^{-2} \) |
| BSTC (Proposed) | \( 8.50 \times 10^{-3} \) | \( 7.90 \times 10^{-3} \) | \( 6.10 \times 10^{-3} \) |
Discussion and Conclusion
This article presented a novel robust flight control strategy for a quadrotor drone. The proposed Backstepping Super-Twisting Sliding Mode Controller (BSTC) successfully merges the systematic design procedure of backstepping with the robustness and finite-time convergence of the Super-Twisting Algorithm. The key achievement is the effective resolution of the chattering problem inherent in conventional sliding mode control, while preserving and even enhancing robust tracking performance.
The simulation studies conclusively demonstrate the advantages of the BSTC. Compared to a classical PID controller, it offers dramatically faster response and superior disturbance rejection. Compared to a standard Backstepping Sliding Mode Controller (BSMC), it provides smoother control action (no chattering) and slightly better tracking accuracy and speed, particularly in the presence of aggressive disturbances. The continuous nature of the BSTC’s control output is a significant practical benefit for real-world implementation on a physical quadrotor drone, as it prevents actuator stress and avoids exciting unmodeled structural resonances.
Future work will focus on several important extensions. First, the gain parameters \( (k_1, k_2) \) for the STA were manually tuned. Implementing adaptive or optimization-based techniques (e.g., Particle Swarm Optimization) to auto-tune these gains could further optimize performance across different flight regimes. Second, while the controller handles bounded disturbances well, the integration of a disturbance observer (DOB) could provide explicit estimation and feedforward compensation of disturbances, potentially allowing for lower controller gains. Finally, experimental validation on a real quadrotor drone platform is the essential next step to confirm the simulation findings and address practical challenges such as sensor noise, actuator dynamics, and real-time computational constraints.
In conclusion, the BSTC presents a compelling solution for high-performance, robust attitude and altitude control of quadrotor drones, effectively balancing tracking precision, response speed, and control smoothness.