Quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their agility, efficiency, and versatility in applications such as agriculture, surveillance, and disaster management. However, achieving precise and rapid trajectory tracking for a quadrotor remains challenging due to external disturbances like wind gusts and model uncertainties. Traditional control methods, such as PID, often fall short in providing robust performance under such conditions. This paper addresses these challenges by proposing a composite control strategy that integrates an Adaptive Sliding Mode Disturbance Observer (ASMDO) and a Non-singular Terminal Sliding Mode Controller (NTSMC). The ASMDO enables finite-time estimation of disturbances without prior knowledge of their derivative bounds, while the NTSMC ensures accurate and fast trajectory tracking. Theoretical analysis and simulation results validate the effectiveness of the proposed approach.
The dynamics of a quadrotor are inherently nonlinear and underactuated, making them susceptible to disturbances. The attitude dynamics can be represented as:
$$ \ddot{\phi} = -\frac{l H_1}{I_{xx}} \dot{\phi} + u_1 + d_1 $$
$$ \ddot{\theta} = -\frac{l H_2}{I_{yy}} \dot{\theta} + u_2 + d_2 $$
$$ \ddot{\psi} = -\frac{l H_3}{I_{zz}} \dot{\psi} + u_3 + d_3 $$
Here, $\phi$, $\theta$, and $\psi$ denote the roll, pitch, and yaw angles, respectively; $l$ is the arm length; $I_{xx}$, $I_{yy}$, and $I_{zz}$ are moments of inertia; $H_i$ are drag coefficients; $u_i$ are control inputs; and $d_i$ represent external disturbances. The primary objective is to design a controller that compensates for $d_i$ and ensures finite-time convergence of tracking errors.
The proposed ASMDO is designed to estimate disturbances in finite time, even when the bounds of disturbance derivatives are unknown. For the roll angle $\phi$, the observer is formulated as follows. Define an auxiliary variable $s_1 = \dot{\phi} – \eta$, where $\eta$ satisfies:
$$ \dot{\eta} – a \dot{\phi} = u_1 + \hat{d}_1 – a s_1 + v $$
with $a = \frac{l H_1}{I_{xx}}$, $\hat{d}_1$ as the estimated disturbance, and $v = \lambda_1 \text{sgn}(s_1)$ as the sliding mode term. The disturbance estimate $\hat{d}_1$ is given by:
$$ \hat{d}_1 = \xi + \lambda_2 \dot{\phi} $$
$$ \dot{\xi} = \lambda_2 (-a \dot{\phi} – u_1 – \hat{d}_1) + (\hat{\gamma} + \lambda_3) \text{sgn}(v) $$
where $\xi$ is an auxiliary state, $\hat{\gamma}$ is the estimated bound of $\dot{d}_1$, and $\lambda_1$, $\lambda_2$, $\lambda_3$ are gains. The adaptive law for $\hat{\gamma}$ is:
$$ \dot{\hat{\gamma}} = -\delta_1 \hat{\gamma} + |v| $$
with $\delta_1 > 0$. The estimation error $\tilde{d}_1 = d_1 – \hat{d}_1$ converges to a bounded set in finite time, as proven using Lyapunov stability theory.
The NTSMC is designed to achieve finite-time tracking of desired trajectories. For the roll angle, define the tracking error $e = \phi_d – \phi$. The non-singular terminal sliding surface is:
$$ s_2 = \dot{e} + k_1 e + k_2 |e|^{\zeta_1 / \zeta_2} \text{sgn}(e) $$
where $k_1$, $k_2 > 0$, and $\zeta_1$, $\zeta_2$ are positive odd integers with $\zeta_1 / \zeta_2 > 1$. The control law is:
$$ u = u_e + u_{s2} – \hat{d}_1 $$
with equivalent control $u_e$ and robust control $u_{s2}$ given by:
$$ u_e = \ddot{\phi}_d – a \dot{\phi} + k_1 \dot{e} + k_2 \frac{\zeta_1}{\zeta_2} |e|^{\zeta_1 / \zeta_2 – 1} \dot{e} $$
$$ u_{s2} = \rho_1 s_2 + \rho_2 \tanh\left(\frac{s_2}{\psi}\right) $$
Here, $\rho_1$, $\rho_2 > 0$, and $\psi$ is a boundary layer thickness to reduce chattering. The compensation term $-\hat{d}_1$ cancels out the estimated disturbance, enhancing robustness.
To validate the proposed method, both Software-in-the-Loop (SITL) and Hardware-in-the-Loop (HITL) simulations were conducted. The quadrotor parameters used in simulations are summarized in Table 1.
| Parameter | Value |
|---|---|
| Mass $M$ (kg) | 1.400 |
| Arm Length $l$ (m) | 0.225 |
| Moment of Inertia $I_{xx}$ (kg·m²) | 0.0211 |
| Moment of Inertia $I_{yy}$ (kg·m²) | 0.0219 |
| Moment of Inertia $I_{zz}$ (kg·m²) | 0.0366 |
| Drag Coefficient $H$ (N·(m·s⁻¹)⁻²) | 0.0730 |
The ASMDO and NTSMC parameters are listed in Table 2. These values were tuned to optimize performance in both SITL and HITL environments.
| Component | Parameter | Value |
|---|---|---|
| ASMDO | $\lambda_1$ | 1.2 |
| $\lambda_2$ | 2.0 | |
| $\lambda_3$ | 3.0 | |
| NTSMC | $k_1$ | 7.0 |
| $k_2$ | 8.0 | |
| $\rho_1$ | 13.0 | |
| $\rho_2$ | 15.0 | |
| $\zeta_1 / \zeta_2$ | 3 |
In SITL simulations, the control algorithm was implemented in MATLAB/Simulink and integrated with FlightGear for visualization. The quadrotor was subjected to a step disturbance at $t = 2$ s. Figure 1 shows the disturbance estimation performance of ASMDO for the roll angle. The observer accurately estimates $d_1$ within finite time, demonstrating its effectiveness.
The tracking performance of the proposed method is compared with ADRC and PID controllers in Figures 2-4. For the roll angle $\phi$, the proposed controller maintains precise tracking even after disturbance injection, while ADRC exhibits a recovery delay and PID fails to converge. Similar results are observed for pitch $\theta$ and yaw $\psi$ angles, highlighting the robustness of the composite controller.
In HITL simulations, the algorithm was deployed on a Pixhawk 4 flight controller using RflySim. The quadrotor was commanded to follow a circular trajectory. Figure 5 illustrates the path tracking results, where the proposed controller achieves minimal deviation. The attitude tracking errors for roll and pitch are shown in Figures 6 and 7, respectively. Although small errors persist due to model inaccuracies, the overall performance validates the practical applicability of the method.
The finite-time stability of the closed-loop system is analyzed using Lyapunov theory. Consider the Lyapunov function $V = \frac{1}{2} s_2^2$ for the tracking error dynamics. Its derivative satisfies:
$$ \dot{V} = s_2 \dot{s}_2 \leq -\rho_1 s_2^2 – (\rho_2 + \tilde{d}_1) |s_2| $$
Since $\tilde{d}_1$ is bounded, selecting $\rho_2 > \max |\tilde{d}_1|$ ensures $\dot{V} < 0$, guaranteeing finite-time convergence.
For the ASMDO, the estimation error dynamics are:
$$ \dot{\tilde{d}}_1 = -\lambda_2 \tilde{d}_1 – (\hat{\gamma} + \lambda_3) \text{sgn}(v) + \dot{d}_1 $$
Using the Lyapunov function $V_2 = \tilde{d}_1^2 + \frac{1}{2} \tilde{\gamma}^2$, where $\tilde{\gamma} = \gamma – \hat{\gamma}$, it can be shown that:
$$ \dot{V}_2 \leq – \min\left(2\lambda_3, \frac{\delta_1}{2}\right) V_2^{1/2} + \sigma_0 $$
Thus, $\tilde{d}_1$ converges to a bounded set in finite time.
In conclusion, this paper presents a novel composite control strategy for quadrotor attitude control. The ASMDO provides finite-time disturbance estimation without requiring prior knowledge of disturbance bounds, while the NTSMC ensures rapid and accurate trajectory tracking. The proposed method is validated through extensive simulations, demonstrating superior performance compared to existing techniques. Future work will focus on real-world flight tests to further assess the algorithm’s robustness.
The versatility of quadrotor systems necessitates advanced control strategies to handle diverse operational conditions. The integration of adaptive observers and sliding mode controllers offers a promising framework for enhancing quadrotor autonomy and reliability. Future research could explore applications in swarm robotics and autonomous navigation, where disturbance rejection is critical.