In my research, I address the critical challenge of position and attitude control for a quadcopter carrying LiDAR during point cloud data collection along transmission lines. The quadcopter operates in complex environments where external disturbances, such as wind gusts, electromagnetic interference, and structural obstacles, can severely degrade control performance. To overcome these issues, I propose a novel active disturbance rejection control (ADRC) strategy based on complementary sliding mode control (CSMC). This approach leverages a finite-time convergent extended state observer (ESO) to estimate both the system states and lumped disturbances, enabling real-time compensation and enhanced robustness. The quadcopter’s dynamics are highly nonlinear and coupled, making traditional control methods inadequate. My method ensures asymptotic convergence of tracking errors while maintaining control input continuity, which is vital for smooth quadcopter operation in noisy environments.
The quadcopter dynamics are derived using the Euler-Lagrange formulation, considering the quadcopter as a rigid body with symmetric structure and uniform mass distribution. The model incorporates external disturbances and uncertainties, which are common in real-world scenarios. The equations of motion for the quadcopter are expressed as follows:
$$ \begin{align*}
\ddot{x} &= \frac{(\cos \phi \sin \theta \cos \psi + \sin \phi \sin \psi) u_1}{m} + d_x, \\
\ddot{y} &= \frac{(\cos \phi \sin \theta \sin \psi – \sin \phi \cos \psi) u_1}{m} + d_y, \\
\ddot{z} &= \frac{(\cos \phi \cos \theta) u_1}{m} – g + d_z, \\
\ddot{\phi} &= \frac{l}{I_x} u_2 + q r \frac{I_y – I_z}{I_x} – \frac{J_r}{I_x} q \Omega_r + d_\phi, \\
\ddot{\theta} &= \frac{l}{I_y} u_3 + p r \frac{I_z – I_x}{I_y} – \frac{J_r}{I_y} p \Omega_r + d_\theta, \\
\ddot{\psi} &= \frac{u_4}{I_z} + p q \frac{I_x – I_y}{I_z} + d_\psi,
\end{align*} $$
where \(x, y, z\) represent the position coordinates, \(\phi, \theta, \psi\) are the roll, pitch, and yaw angles, \(m\) is the mass, \(l\) is the distance from the propeller center to the center of gravity, \(g\) is gravitational acceleration, \(I_x, I_y, I_z\) are moments of inertia, \(d_i\) (for \(i = x, y, z, \phi, \theta, \psi\)) denote lumped disturbances including external forces and model uncertainties, \(u_1, u_2, u_3, u_4\) are control inputs, and \(\Omega_r = -\Omega_1 + \Omega_2 – \Omega_3 + \Omega_4\) with \(\Omega_i\) being propeller speeds. To simplify the underactuated nature of the quadcopter, I introduce virtual control inputs as:
$$ \begin{align*}
v_x &= \frac{(\cos \phi \sin \theta \cos \psi + \sin \phi \sin \psi) u_1}{m}, \\
v_y &= \frac{(\cos \phi \sin \theta \sin \psi – \sin \phi \cos \psi) u_1}{m}, \\
v_z &= \frac{(\cos \phi \cos \theta) u_1}{m} – g, \\
v_\phi &= \frac{l u_2}{I_x}, \quad v_\theta = \frac{l u_3}{I_y}, \quad v_\psi = \frac{u_4}{I_z}.
\end{align*} $$
This transformation decouples the system into six independent channels, each controlled separately. The actual control inputs \(u_1, \phi_d, \theta_d\) are derived through decoupling algorithms to handle coupling effects. For the quadcopter, the dynamics can be generalized as a second-order system:
$$ \begin{align*}
\dot{x}_1 &= x_2, \\
\dot{x}_2 &= u + f(x_1, x_2),
\end{align*} $$
where \(x_1 = [x, y, z, \phi, \theta, \psi]^T\), \(u = [v_x, v_y, v_z, v_\phi, v_\theta, v_\psi]^T\), and \(f(x_1, x_2)\) represents the lumped disturbance vector. This formulation allows me to design an ESO for state and disturbance estimation.
I develop a finite-time convergent ESO to accurately estimate the quadcopter states and disturbances. By extending the lumped disturbance \(f(x_1, x_2)\) as an additional state variable \(x_3\), the system becomes:
$$ \begin{align*}
\dot{x}_1 &= x_2, \\
\dot{x}_2 &= x_3 + u, \\
\dot{x}_3 &= \dot{f}(x_1, x_2).
\end{align*} $$
The ESO is designed as:
$$ \begin{align*}
\dot{\hat{x}}_1 &= \hat{x}_2 + \beta_1 \text{sig}^{\alpha_1}(e_1), \\
\dot{\hat{x}}_2 &= \hat{x}_3 + \beta_2 \text{sig}^{\alpha_2}(e_1) + u, \\
\dot{\hat{x}}_3 &= \beta_3 \text{sig}^{\alpha_3}(e_1),
\end{align*} $$
where \(\hat{x}_i\) are the estimated states, \(e_1 = x_1 – \hat{x}_1\) is the estimation error, \(\beta_i > 0\) are gains, \(\alpha_i \in (0,1)\) are tuning parameters, and \(\text{sig}^\alpha(\cdot) = |\cdot|^\alpha \text{sgn}(\cdot)\). The observer error dynamics converge in finite time, ensuring rapid and accurate estimation. For the quadcopter, this ESO provides robust disturbance rejection by compensating for uncertainties in real-time.
Based on the ESO estimates, I design an ADRC scheme using complementary sliding mode control. The tracking errors are defined as \(e = x_{1d} – x_1\) and \(\dot{\hat{e}} = x_{2d} – \hat{x}_2\), where \(x_{1d}\) and \(x_{2d}\) are desired states. To avoid velocity sensor noise, I use the estimated error derivative. The generalized sliding manifold is constructed as:
$$ s_1 = \dot{\hat{e}} + 2c e + c^2 \int_0^t e \, d\tau, $$
where \(c = \text{diag}(c_x, c_y, c_z, c_\phi, c_\theta, c_\psi)\) is a positive diagonal matrix. The complementary sliding manifold is defined as:
$$ s_2 = \dot{\hat{e}} – c^2 \int_0^t e \, d\tau. $$
The relationship between \(s_1\) and \(s_2\) is given by \(\dot{s}_1 = \dot{s}_2 + c(s_1 + s_2)\). To ensure control input continuity, I integrate the sign function and propose the ADRC law:
$$ u = \dot{x}_{1d} – \hat{x}_3 + c(2\dot{\hat{e}} + c e + s_1) + k_1 \text{sig}^{1/2}(s_1 + s_2) + \int k_2 \text{sgn}(s_1 + s_2) \, dt, $$
where \(k_1\) and \(k_2\) are positive diagonal gain matrices. This controller compensates for disturbances estimated by the ESO and guarantees smooth operation of the quadcopter.
I analyze the stability of the closed-loop system using Lyapunov theory. Consider the Lyapunov function \(V = \frac{1}{2} s_1^T s_1 + \frac{1}{2} s_2^T s_2\). Differentiating \(V\) and substituting the control law yields:
$$ \dot{V} = -(s_1 + s_2)^T c (s_1 + s_2) – k_1 \|s_1 + s_2\|^{3/2} – \|s_1 + s_2\| \int k_2 \, dt. $$
Since \(c, k_1, k_2 > 0\), \(\dot{V} \leq 0\), ensuring that \(s_1\) and \(s_2\) converge to zero asymptotically. Consequently, the tracking error \(e\) approaches zero, proving the system’s stability for the quadcopter.
To validate the proposed method, I conduct simulations comparing it with non-complementary sliding mode ADRC and cascade PID control. The quadcopter parameters are set as \(m = 1.1 \, \text{kg}\), \(l = 0.21 \, \text{m}\), \(I_x = I_y = 1.22 \, \text{kg} \cdot \text{m}^2\), \(I_z = 2.2 \, \text{kg} \cdot \text{m}^2\), \(J_r = 0.2 \, \text{Ns}^2/\text{rad}\), and \(g = 9.81 \, \text{m/s}^2\). The initial conditions are zero for position, attitude, and angular velocity. The ESO parameters are \(\beta_1 = 30\), \(\beta_2 = 300\), \(\beta_3 = 500\), \(\alpha_1 = \alpha_2 = \alpha_3 = 0.675\), and the controller gains are:
$$ c = \text{diag}(2, 1, 8, 10, 10, 2), \quad k_1 = \text{diag}(7, 7, 15, 40, 80, 7), \quad k_2 = \text{diag}(0.5, 0.4, 0.2, 0.2, 0.4, 0.2). $$
The desired trajectory for the quadcopter is \(x_d = \sin(0.1t)\), \(y_d = \cos(0.1t)\), \(z_d = 0.1t\), and \(\psi_d = 0\), with \(\phi_d\) and \(\theta_d\) computed from the virtual controls. Model uncertainties are introduced by reducing \(m, I_x, I_y, I_z\) by 10%, and lumped disturbances are set as:
$$ \begin{align*}
d_x &= \cos(t), \quad d_y = \sin(t), \quad d_z = \sin(t)\cos(t), \\
d_\phi &= \sin(0.5t), \quad d_\theta = \cos(0.5t), \quad d_\psi = \sin(0.5t)\cos(0.5t).
\end{align*} $$
The simulation results demonstrate that the proposed ADRC with CSMC achieves higher tracking accuracy and smoother control inputs compared to other methods. For instance, position and attitude errors converge rapidly with minimal oscillation. The control inputs remain continuous, avoiding chattering effects common in traditional sliding mode control. To quantify performance, I compute the Integral Absolute Derivative Control (IADC) signal, defined as \(\int_0^{t_f} |\dot{u}| \, dt\), for each control input. The values are summarized in the table below:
| Control Method | \(u_1\) IADC | \(u_2\) IADC | \(u_3\) IADC | \(u_4\) IADC |
|---|---|---|---|---|
| Proposed ADRC-CSMC | 502.00 | 307.98 | 1466.42 | 7.48 |
| Non-Complementary ADRC | 884.97 | 352.29 | 4889.51 | 26.55 |
The lower IADC values for the proposed method indicate smoother control actions, which is crucial for the quadcopter’s stability during point cloud data collection. Additionally, the quadcopter’s response to time-varying disturbances is effectively suppressed, highlighting the robustness of the approach.
In conclusion, my proposed ADRC strategy based on complementary sliding mode control significantly enhances the position and attitude control of a quadcopter in challenging environments. The finite-time ESO provides accurate disturbance estimation, while the CSMC ensures asymptotic error convergence and control continuity. This method is particularly beneficial for applications like transmission line inspection, where precision and reliability are paramount. Future work could focus on adapting this controller for quadcopter swarms or integrating it with machine learning techniques for improved autonomy.