In recent years, quadcopters have gained significant attention due to their versatility in applications such as aerial photography, search and rescue, and geographic mapping. However, their flight performance is often compromised by parameter uncertainties, external disturbances, and actuator faults. These challenges arise from factors like gyroscopic effects, aerodynamic drag, and motor failures, which can lead to instability or even catastrophic failures. To address these issues, we propose an adaptive non-singular fast integral terminal sliding mode control (ANFITSMC) strategy. This approach ensures robust trajectory tracking and stability under adverse conditions by integrating adaptive laws to estimate unknown parameters and disturbances. In this paper, we present the dynamic modeling of the quadcopter, controller design, stability analysis, and validation through simulations and experiments. The results demonstrate the superiority of our method over traditional terminal sliding mode control (TSMC) and fast terminal sliding mode control (FTSMC) in terms of accuracy and resilience.
The dynamic model of a quadcopter is derived under the assumption of a rigid body with coincident geometric and mass centers, neglecting air resistance. The system is subject to external disturbances and actuator faults, which are incorporated into the model. Let the inertial frame be denoted as \( O_e(x_e, y_e, z_e) \) and the body frame as \( O_B(x_B, y_B, z_B) \). The forces generated by the four rotors are \( F_1, F_2, F_3, F_4 \). The equations of motion are given by:
$$ \begin{aligned}
\dot{x} &= \frac{U_1 (c_\phi s_\theta c_\psi + s_\phi s_\psi)}{m} – \frac{k_x}{m} \dot{x} + d_x \\
\dot{y} &= \frac{U_1 (c_\phi s_\theta s_\psi – s_\phi c_\psi)}{m} – \frac{k_y}{m} \dot{y} + d_y \\
\dot{z} &= \frac{U_1 c_\phi c_\theta}{m} – g – \frac{k_z}{m} \dot{z} + d_z \\
\dot{\phi} &= H_\phi \dot{\theta} \dot{\psi} + \frac{U_2}{I_{xx}} – \frac{k_1}{I_{xx}} \dot{\phi} + d_\phi \\
\dot{\theta} &= H_\theta \dot{\phi} \dot{\psi} + \frac{U_3}{I_{yy}} – \frac{k_2}{I_{yy}} \dot{\theta} + d_\theta \\
\dot{\psi} &= H_\psi \dot{\phi} \dot{\theta} + \frac{U_4}{I_{zz}} – \frac{k_3}{I_{zz}} \dot{\psi} + d_\psi
\end{aligned} $$
where \( x, y, z \) represent the position in the inertial frame; \( \phi, \theta, \psi \) are the roll, pitch, and yaw angles; \( m \) is the mass; \( I_{xx}, I_{yy}, I_{zz} \) are the moments of inertia; \( g \) is the gravitational acceleration; \( k_x, k_y, k_z \) are air drag coefficients; \( k_1, k_2, k_3 \) represent gyroscopic effects; and \( d_i \) (for \( i = x, y, z, \phi, \theta, \psi \)) denote external disturbances. The control inputs \( U_r \) (for \( r = 1, 2, 3, 4 \)) account for actuator faults as follows:
$$ U_r = U_{ra} + f_r $$
where \( U_{ra} \) is the fault-free control input and \( f_r \) is the fault deviation. To simplify the analysis, we define lumped disturbances that combine external disturbances and fault effects, such as \( f_x = \frac{c_\phi s_\theta c_\psi + s_\phi s_\psi}{m} f_1 + d_x \). We assume these disturbances are bounded, i.e., \( |f_x| \leq \sigma_x \), where \( \sigma_x \) is an unknown positive constant. Similar bounds apply to other states.
The control objective is to achieve accurate trajectory tracking for the quadcopter despite uncertainties and faults. We define the tracking errors for position and attitude as \( E_p = P_d – P \) and \( \Theta_e = \Theta_d – \Theta \), where \( P = [x, y, z]^T \) and \( \Theta = [\phi, \theta, \psi]^T \) are the actual states, and \( P_d \) and \( \Theta_d \) are the desired trajectories. The non-singular fast integral terminal sliding mode surfaces are designed as:
$$ \begin{aligned}
s_i &= \int_0^t E_{p,i} \, dt + E_{p,i} + T_i E_{p,i}^{p_i} + P_i \dot{E}_{p,i}^{n_i} \quad \text{for } i = x, y, z \\
s_j &= \int_0^t \Theta_{e,j} \, dt + \Theta_{e,j} + T_j \Theta_{e,j}^{p_j} + P_j \dot{\Theta}_{e,j}^{n_j} \quad \text{for } j = \phi, \theta, \psi
\end{aligned} $$
where \( T_i, P_i, T_j, P_j \) are positive constants, and \( p_i, p_j < 2 \), \( n_i, n_j > 1 \) to avoid singularities. For example, the sliding surface for the x-position is:
$$ s_x = \int_0^t e_x \, dt + e_x + \eta_x e_x^{p_x} + \mu_x \dot{e}_x^{n_x} $$
where \( e_x = x_d – x \). The adaptive control laws are derived to estimate unknown parameters and disturbance bounds. For the roll angle \( \phi \), the controller is designed as:
$$ U_{2a} = \frac{M_\phi (\phi_e + \dot{\phi}_e)}{\dot{\phi}_e^{n_\phi – 1}} + N_\phi \phi_e^{p_\phi – 1} \dot{\phi}_e^{2 – n_\phi} + \dot{\phi}_d – \left[ H_\phi \dot{\theta} \dot{\psi} + \frac{\hat{k}_1 \dot{\phi}}{I_{xx}} + a_\phi s_\phi + \hat{\sigma}_\phi \tanh\left(\frac{s_\phi}{b_\phi}\right) \right] I_{xx} $$
where \( \hat{k}_1 \) and \( \hat{\sigma}_\phi \) are estimates of \( k_1 \) and \( \sigma_\phi \), updated by the adaptive laws:
$$ \begin{aligned}
\dot{\hat{k}}_1 &= -\frac{\gamma_{\phi 1}}{I_{xx}} \dot{\phi} s_\phi \\
\dot{\hat{\sigma}}_\phi &= \lambda_{\phi 1} s_\phi
\end{aligned} $$
Here, \( a_\phi, b_\phi, \gamma_{\phi 1}, \lambda_{\phi 1} \) are positive constants. The tanh function is used to reduce chattering. Similar controllers and adaptive laws are derived for other states. The stability of the closed-loop system is proven using Lyapunov theory. For instance, for the roll subsystem, consider the Lyapunov function:
$$ V_\phi = \frac{1}{2} s_\phi^2 + \frac{1}{2\gamma_{\phi 1}} \tilde{k}_1^2 + \frac{1}{2\lambda_{\phi 1}} \tilde{\sigma}_\phi^2 $$
where \( \tilde{k}_1 = k_1 – \hat{k}_1 \) and \( \tilde{\sigma}_\phi = \sigma_\phi – \hat{\sigma}_\phi \). The time derivative yields:
$$ \dot{V}_\phi \leq -a_\phi s_\phi^2 \leq 0 $$
ensuring asymptotic convergence of the tracking error. The same approach applies to position control. For the z-position, the controller is:
$$ u_z = \frac{M_z (e_z + \dot{e}_z)}{\dot{e}_z^{n_z – 1}} + N_z e_z^{p_z – 1} \dot{e}_z^{2 – n_z} + \dot{z}_d + \frac{\hat{k}_z \dot{z}}{m} + \hat{\sigma}_z \tanh\left(\frac{s_z}{b_z}\right) + h_z s_z $$
with adaptive laws:
$$ \begin{aligned}
\dot{\hat{k}}_z &= -\frac{\gamma_{z1}}{m} \dot{z} s_z \\
\dot{\hat{\sigma}}_z &= \lambda_z s_z
\end{aligned} $$
The virtual control inputs for position are derived from the model and used to compute the actual control inputs \( U_1, U_2, U_3, U_4 \). The relationship between control inputs and motor speeds is given by:
$$ \begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix} = \begin{bmatrix} c & c & c & c \\ 0 & -dc & 0 & dc \\ -dc & 0 & dc & 0 \\ k_p & -k_p & k_p & -k_p \end{bmatrix} \begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix} $$
where \( c \) is the thrust coefficient, \( d \) is the distance from the center, and \( k_p \) is the torque coefficient. This ensures that the quadcopter can achieve the desired motions even under faults.
To validate the proposed ANFITSMC, we conducted simulations and experiments. The quadcopter parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Mass \( m \) | 2 kg |
| \( I_{xx} \) | 5.745e-3 kg·m² |
| \( I_{yy} \) | 5.745e-3 kg·m² |
| \( I_{zz} \) | 1.175e-2 kg·m² |
| \( k_x, k_y, k_z \) | 0.01 |
| \( k_1 \) | 1.49e-3 N·s²/rad |
| \( k_2 \) | 1.35e-2 N·s²/rad |
| \( k_3 \) | 1.6e-4 N·s²/rad |
| \( g \) | 9.81 m/s² |
The desired trajectory is set as \( x_d(t) = 3\cos(t) \), \( y_d(t) = 3\sin(t) \), \( z_d(t) = 2 + 0.5t \), and \( \psi_d(t) = 0.5 \) rad. External disturbances and actuator faults are introduced at specific times: at 5 s, disturbances \( \sin(3\pi t) \) for position and \( \cos(3\pi t) \) for attitude; at 10 s, additional faults \( 2\sin(3\pi t) + \sin(3\pi t) + 0.12 \) for position and \( 2\sin(3\pi t) + \cos(3\pi t) + 0.5 \) for attitude. The controller parameters are tuned for optimal performance, as shown in Table 2.
| Parameter | Value |
|---|---|
| \( \eta_x, \eta_y, \eta_z \) | 1.5 |
| \( \mu_x, \mu_y, \mu_z \) | 2 |
| \( p_x, p_y, p_z \) | 1.5 |
| \( n_x, n_y, n_z \) | 1.45 |
| \( \gamma_{x1}, \gamma_{y1}, \gamma_{z1} \) | 0.4, 2, 0.5 |
| \( \lambda_x, \lambda_y, \lambda_z \) | 1.2, 1.2, 1.5 |
| \( b_x, b_y, b_z \) | 0.02 |
| \( h_x, h_y, h_z \) | 5 |
| \( \eta_\phi, \eta_\theta, \eta_\psi \) | 3 |
| \( \mu_\phi, \mu_\theta, \mu_\psi \) | 2.5 |
| \( p_\phi, p_\theta, p_\psi \) | 1.6 |
| \( n_\phi, n_\theta, n_\psi \) | 1.52 |
| \( \gamma_{\phi 1}, \gamma_{\theta 1}, \gamma_{\psi 1} \) | 0.05, 0.01, 0.01 |
| \( \lambda_{\phi 1}, \lambda_{\theta 1}, \lambda_{\psi 1} \) | 0.01 |
| \( b_\phi, b_\theta, b_\psi \) | 0.02 |
| \( h_\phi, h_\theta, h_\psi \) | 7 |
Simulation results demonstrate that the ANFITSMC achieves superior tracking performance compared to FTSMC and TSMC. For instance, the 3D trajectory tracking shows that ANFITSMC has minimal overshoot and smooth curves, while FTSMC and TSMC exhibit significant oscillations, especially under disturbances and faults. The roll and pitch angle tracking errors converge rapidly with ANFITSMC, whereas the other methods struggle to maintain accuracy. The adaptive laws effectively estimate parameters and disturbances, as seen in the convergence of \( \hat{k}_z \) and \( \hat{\sigma}_z \) to constant values near the true bounds, enhancing robustness.
We also conducted experimental validation using a quadcopter platform equipped with a Pixhawk 2.4.8 flight controller and Raspberry Pi for external control. The experiment involved flying the quadcopter for 100 seconds at a desired height of 2 meters, with a fan simulating disturbances and assumed actuator faults at 20 seconds. The results, as shown in the trajectory data, confirm that the quadcopter maintains stable tracking despite the faults, with only minor deviations. This real-world test underscores the practicality of the ANFITSMC for quadcopter applications.
In conclusion, the proposed adaptive non-singular fast integral terminal sliding mode control effectively addresses the challenges of parameter uncertainties, external disturbances, and actuator faults in quadcopters. The integration of adaptive laws ensures accurate estimation and compensation, leading to robust trajectory tracking. Stability analysis via Lyapunov theory guarantees asymptotic convergence of errors. Simulations and experiments validate the method’s superiority over conventional approaches. Future work will focus on fault observer-based fault-tolerant control to further enhance robustness and expand the applicability of quadcopters in dynamic environments. The continuous development of such adaptive strategies is crucial for advancing autonomous quadcopter operations in real-world scenarios.