In recent years, quadrotor drones have gained significant attention due to their wide-ranging applications in areas such as traffic surveillance, power line inspection, and disaster relief. However, the quadrotor drone is a nonlinear, underactuated system with strong coupling, making it highly susceptible to external disturbances during flight, which complicates control design. Traditional control methods, including PID, backstepping control, active disturbance rejection control, sliding mode control, and LQR control, have been employed, but challenges remain in achieving robust performance under uncertainties and disturbances. To address these issues, we propose a robust adaptive backstepping sliding mode control scheme based on a Radial Basis Function (RBF) network for quadrotor drone attitude control. This approach combines the strengths of backstepping sliding mode control with neural network approximation and adaptive compensation, enhancing tracking accuracy, disturbance rejection, and robustness. In this article, we present the design, stability analysis, and simulation results of our method, demonstrating its superiority over conventional backstepping sliding mode control.
The dynamics of a quadrotor drone are derived from the Newton-Euler formulation, considering the vehicle as a rigid body. We define two coordinate systems: an Earth-fixed frame \(E = (O_e, x_e, y_e, z_e)\) and a body-fixed frame \(B = (O_b, x_b, y_b, z_b)\), where \(z_e\) and \(z_b\) are perpendicular to the ground. The quadrotor drone’s position is given by \([x, y, z]^T\), and its Euler angles are represented as \([\phi, \theta, \psi]^T\), where \(\phi\) is the roll angle, \(\theta\) is the pitch angle, and \(\psi\) is the yaw angle. The control inputs are derived from the thrusts \(F_i\) generated by four rotors with speeds \(\omega_i\) for \(i = 1, 2, 3, 4\). Under standard assumptions—such as the quadrotor drone being a rigid body with its center of mass coinciding with the origin of the body frame, and neglecting energy losses and friction—the six-degree-of-freedom dynamics can be expressed. For attitude control, we focus on the rotational dynamics, which, in the presence of external disturbances, are given by:
$$\dot{\phi} = \frac{l U_2}{I_x} + \dot{\theta} \dot{\psi} \frac{I_y – I_z}{I_x} – \frac{J_r \dot{\theta} \omega_r}{I_x} + d_1,$$
$$\dot{\theta} = \frac{l U_3}{I_y} + \dot{\phi} \dot{\psi} \frac{I_z – I_x}{I_y} – \frac{J_r \dot{\phi} \omega_r}{I_y} + d_2,$$
$$\dot{\psi} = \frac{U_4}{I_z} + \dot{\theta} \dot{\phi} \frac{I_x – I_y}{I_z} + d_3,$$
where \(m\) is the mass, \(g\) is gravitational acceleration, \(I_x, I_y, I_z\) are moments of inertia, \(l\) is the distance from rotor center to drone center of mass, \(J_r\) is rotor inertia, \(\omega_r = \omega_1 – \omega_2 + \omega_3 – \omega_4\) is the gyroscopic effect component, and \(U_1, U_2, U_3, U_4\) are control inputs related to rotor speeds via:
$$\begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix} = \begin{bmatrix} b & b & b & b \\ 0 & -l b & 0 & l b \\ -l b & 0 & l b & 0 \\ -k_l & k_l & -k_l & k_l \end{bmatrix} \begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix},$$
with \(b\) as the lift factor and \(k_l\) as the drag coefficient. The disturbances \(d_1, d_2, d_3\) are bounded such that \(|d_i| \leq d_0\) for \(i=1,2,3\). For controller design, we define the state vector for attitude as \([x_1, x_2, x_3, x_4, x_5, x_6]^T = [\phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}]^T\), leading to the state-space representation:
$$\dot{x}_1 = x_2,$$
$$\dot{x}_2 = \frac{l U_2}{I_x} + x_4 x_6 \frac{I_y – I_z}{I_x} – \frac{J_r x_4 \omega_r}{I_x} + d_1,$$
$$\dot{x}_3 = x_4,$$
$$\dot{x}_4 = \frac{l U_3}{I_y} + x_2 x_6 \frac{I_z – I_x}{I_y} – \frac{J_r x_2 \omega_r}{I_y} + d_2,$$
$$\dot{x}_5 = x_6,$$
$$\dot{x}_6 = \frac{U_4}{I_z} + x_2 x_4 \frac{I_x – I_y}{I_z} + d_3.$$
Our control objective is to ensure that the quadrotor drone’s attitude angles \(\phi, \theta, \psi\) track desired trajectories \(\phi_d, \theta_d, \psi_d\) despite disturbances. To achieve this, we design a robust adaptive backstepping sliding mode controller using an RBF network. The control structure integrates backstepping sliding mode control for systematic design, an RBF network to approximate and compensate for ideal control laws, and adaptive laws to estimate uncertain parameters, thereby minimizing the need for exact disturbance knowledge and enhancing robustness for the quadrotor drone.
The controller design begins with the backstepping sliding mode approach for the roll angle subsystem, as similar steps apply to pitch and yaw. Define the tracking error \(z_1 = x_1 – x_{1d}\), where \(x_{1d} = \phi_d\). Introduce a virtual control \(x_{2v} = \dot{x}_{1d} – c_1 z_1\) with \(c_1 > 0\), and define the velocity tracking error \(z_2 = x_2 – x_{2v} = \dot{z}_1 + c_1 z_1\). Construct a sliding surface \(\sigma = k_1 z_1 + z_2\), where \(k_1 > 0\) and \(k_1 + c_1 > 0\). Taking the derivative yields:
$$\dot{\sigma} = \alpha + \beta U_2 + v + d_1,$$
where \(\alpha = \dot{\theta} \dot{\psi} (I_y – I_z)/I_x – J_r \dot{\theta} \omega_r / I_x\), \(\beta = l/I_x\), and \(v = (k_1 + c_1) \dot{z}_1 – \ddot{x}_{1d}\). Assuming no disturbance initially, the nominal control law is designed as:
$$U_2^* = -\frac{1}{\beta} \left[ \alpha + v + \left( \frac{1}{\Delta} + \frac{1}{\Delta \beta} – \frac{\dot{\beta}}{2\beta} \right) \sigma \right],$$
with \(\Delta > 0\). This nominal control ensures asymptotic stability of the closed-loop system when \(d_1 = 0\), as proven via Lyapunov theory. However, to handle disturbances and uncertainties, we employ an RBF network to approximate \(U_2^*\). According to the universal approximation property, for any continuous function \(U_2^*(k)\) on a compact set \(\Omega_k\), there exists an RBF network such that:
$$U_2^*(k) = W^{*T} h(k) + \mu_l,$$
where \(W^* \in \mathbb{R}^q\) is the ideal weight vector, \(h(k) = [h_1(k), h_2(k), \dots, h_q(k)]^T\) is the Gaussian basis function output with \(q\) neurons, and \(\mu_l\) is the approximation error bounded by \(|\mu_l| \leq \mu_0\). The Gaussian functions are defined as \(h_j = \exp\left( -\frac{\|a – c_j\|^2}{2b_j^2} \right)\), with input vector \(a\), center \(c_j\), and width \(b_j\). For the quadrotor drone attitude control, we use the neural network to directly approximate the nominal control law, compensating for disturbances.
To reduce computational complexity, we adopt the minimal parameter learning method, estimating the upper bound of the neural network weights rather than the weights themselves. Define \(O = \|W^*\|_F^2\), where \(\|\cdot\|_F\) is the Frobenius norm, and let \(\hat{O}\) be the estimate of \(O\) with estimation error \(\tilde{O} = \hat{O} – O\). The control law is designed as:
$$U_2 = -\frac{1}{2\sigma} \hat{O} h^T h,$$
for the roll channel, with similar expressions for pitch and yaw. Substituting into the \(\dot{\sigma}\) equation and manipulating yields:
$$\dot{\sigma} = \beta \left( -\frac{1}{2\sigma} \hat{O} h^T h – W^{*T} h – \mu_l \right) – \left( \frac{1}{\Delta} + \frac{1}{\Delta \beta} – \frac{\dot{\beta}}{2\beta} \right) \sigma + d_1.$$
The adaptive law for updating \(\hat{O}\) is designed as:
$$\dot{\hat{O}} = \frac{\gamma}{2} \sigma^2 h^T h – k \gamma \hat{O},$$
where \(\gamma > 0\) and \(k > 0\) are tuning parameters. This adaptive mechanism adjusts the estimate based on sliding surface activity, ensuring robustness without requiring precise disturbance knowledge. The stability of the closed-loop system for the quadrotor drone is proven using Lyapunov theory. Consider the Lyapunov function candidate for the roll subsystem:
$$V_2(t) = \frac{1}{2} z_1^2 + \frac{1}{2\beta} \sigma^2 + \frac{1}{2\gamma} \tilde{O}^2.$$
Taking the derivative and substituting the control and adaptive laws, we obtain after manipulations:
$$\dot{V}_2(t) \leq z_1 \dot{z}_1 – \frac{k}{2} \tilde{O}^2 – \frac{\sigma^2}{2\Delta \beta^2} + \frac{\Delta \mu_0^2}{2\beta} + \frac{\Delta d_0^2}{4} + \frac{1}{2} + \frac{k}{2} O^2.$$
Using the fact that \(z_1 \dot{z}_1 – \sigma^2/(2\Delta \beta^2) \leq 0\) from the nominal stability analysis, and setting \(k = \eta / \gamma\) with \(\eta > 0\), we derive:
$$\dot{V}_2(t) \leq -\frac{\eta}{2\gamma} \tilde{O}^2 + c_2 \leq -V_2(t) + c_2,$$
where \(c_2 = \Delta \mu_0^2/(2\beta) + \Delta d_0^2/4 + 1/2 + (k/2) O^2\). Solving this inequality shows that \(V_2(t)\) is uniformly ultimately bounded, implying that the tracking error \(z_1\) converges to a small region around zero. Specifically, we have:
$$\lim_{t \to \infty} |z_1| \leq \frac{\sqrt{2\beta c_2}}{k_1 + c_1}.$$
By appropriately selecting parameters, the tracking error can be made arbitrarily small, ensuring that the quadrotor drone’s roll angle \(\phi\) converges to the desired trajectory \(\phi_d\). Similar stability proofs apply to the pitch and yaw channels, guaranteeing overall attitude stability for the quadrotor drone.
To validate the proposed RBF network robust adaptive backstepping sliding mode control, we conduct simulations in MATLAB/Simulink and compare it with conventional backstepping sliding mode control. The physical parameters for the quadrotor drone are set as: mass \(m = 0.8 \, \text{kg}\), gravity \(g = 9.8 \, \text{m/s}^2\), moments of inertia \(I_x = I_y = 5.5 \times 10^{-3} \, \text{kg} \cdot \text{m}^2\), \(I_z = 1.1 \times 10^{-2} \, \text{kg} \cdot \text{m}^2\), distance \(l = 0.165 \, \text{m}\), lift factor \(b = 2.98 \times 10^{-6} \, \text{N} \cdot \text{s}^2\), drag coefficient \(k_l = 2 \times 10^{-7} \, \text{N} \cdot \text{m} \cdot \text{s}^2\). The controller parameters are tuned as follows for the proposed method: \(c_{1\phi} = 5\), \(k_{1\phi} = 10\), \(\Delta_\phi = 0.25\), \(\gamma_\phi = 0.05\), \(k_\phi = 0.1\); \(c_{1\theta} = 7.5\), \(k_{1\theta} = 12.5\), \(\Delta_\theta = 0.25\), \(\gamma_\theta = 0.1\), \(k_\theta = 0.05\); \(c_{1\psi} = 3\), \(k_{1\psi} = 9\), \(\Delta_\psi = 0.3\), \(\gamma_\psi = 0.01\), \(k_\psi = 0.2\). For the conventional backstepping sliding mode control, the parameters are: \(k_{es1\phi} = 1\), \(c_{1\phi} = 1\), \(k_{1\phi} = 3\); \(k_{es1\theta} = 0.5\), \(c_{1\theta} = 1.5\), \(k_{1\theta} = 5\); \(k_{es1\psi} = 1.5\), \(c_{1\psi} = 1\), \(k_{1\psi} = 10\).
We perform two sets of simulations to evaluate tracking accuracy and disturbance rejection for the quadrotor drone. First, under external disturbances, the initial attitude is set to \([0.3, 0.3, 0.3]^T\) radians, and the desired trajectories are constant: \([\phi_d, \theta_d, \psi_d]^T = [0, 1, 0]^T\) radians over 10 seconds. Disturbances are applied as \(d_1 = 0.5 \sin(t)\), \(d_2 = 0.1 \cos(t)\), \(d_3 = 0.5 \sin(2t)\). The tracking performance is summarized in the table below, showing settling times for both methods.
| Attitude Angle | Settling Time (Proposed Method) [s] | Settling Time (Backstepping Sliding Mode) [s] |
|---|---|---|
| Roll (\(\phi\)) | 0.878 | 5.429 |
| Pitch (\(\theta\)) | 2.092 | 4.931 |
| Yaw (\(\psi\)) | 1.229 | 5.465 |
The results indicate that the proposed RBF-based controller achieves faster convergence and smaller errors, demonstrating better tracking accuracy for the quadrotor drone. The settling times are significantly reduced compared to the conventional method, highlighting the effectiveness of the adaptive neural network compensation.
Second, to test robustness against sudden disturbances, we set the desired trajectories as sinusoidal: \([\phi_d, \theta_d, \psi_d]^T = [\sin(t), \sin(t), \sin(t)]^T\) radians over 10 seconds, with white noise injected during the interval 3–4 seconds. The tracking performance is evaluated, and settling times after disturbance onset are compared in the following table.
| Attitude Angle | Settling Time (Proposed Method) [s] | Settling Time (Backstepping Sliding Mode) [s] |
|---|---|---|
| Roll (\(\phi\)) | 4.079 | 6.442 |
| Pitch (\(\theta\)) | 4.079 | 6.066 |
| Yaw (\(\psi\)) | 4.128 | 6.168 |
The proposed controller shows shorter settling times and better recovery, confirming its enhanced disturbance rejection capabilities for the quadrotor drone. The RBF network’s ability to approximate uncertainties and the adaptive laws’ quick parameter adjustment contribute to this robust performance.
In terms of control effort, the proposed method also reduces chattering compared to pure sliding mode control, as the neural network smooths the control signal. The control input for the roll channel, for instance, can be expressed as:
$$U_2 = -\frac{1}{2\sigma} \hat{O} h^T h,$$
where \(\sigma\) is the sliding surface, and \(h\) depends on the system states. This formulation avoids singularity issues that may arise in traditional backstepping sliding mode control when \(\sigma = 0\), as the adaptive mechanism ensures boundedness. The overall control scheme ensures that the quadrotor drone maintains stable attitude tracking even in challenging environments.
For further analysis, we can examine the Lyapunov stability conditions. The key inequalities used in the proof are based on standard lemmas. Lemma 1: For any \(a_1 > 0, b_1 > 0\), we have \(a_1 + b_1 \geq 2 \sqrt{a_1 b_1}\), with equality if and only if \(a_1 = b_1\). Lemma 2: For any \(a_2 > 0, b_2 > 0\), it holds that \(\sqrt{a_2 + b_2} \leq \sqrt{a_2} + \sqrt{b_2}\), with equality if \(a_2 = 0\) or \(b_2 = 0\). These lemmas assist in bounding terms in the Lyapunov derivative, ensuring negative definiteness. The stability proof culminates in the bound:
$$\lim_{t \to \infty} |z_1| \leq \frac{\sqrt{2\beta c_2}}{k_1 + c_1},$$
which can be minimized by tuning parameters like \(k_1\), \(c_1\), and \(\gamma\). This analytical guarantee underpins the reliability of the proposed controller for quadrotor drone applications.
In conclusion, we have developed a robust adaptive backstepping sliding mode control scheme based on an RBF network for quadrotor drone attitude control. The method integrates backstepping sliding mode control for systematic design, uses an RBF network to approximate and compensate for ideal control laws, and employs adaptive laws to estimate uncertain parameters via the minimal learning parameter technique. This approach addresses disturbances and uncertainties while avoiding singularity issues. Simulation results demonstrate that the proposed controller offers shorter settling times, better tracking accuracy, and improved disturbance rejection compared to conventional backstepping sliding mode control. Future work may extend this method to position control or multi-agent coordination for quadrotor drones. The robustness and adaptability of this controller make it suitable for real-world quadrotor drone operations in dynamic environments.