Quadcopter UAVs have evolved from hobbyist models to essential tools across industries including search-and-rescue, precision agriculture, and infrastructure inspection. As drone manufacturers strive to enhance operational reliability in dynamic environments, advanced control systems addressing nonlinear dynamics and disturbance rejection become critical. We present a novel backstepping active disturbance rejection control (ADRC) framework that significantly improves attitude tracking precision and disturbance resilience compared to conventional methods.
The quadcopter dynamic model is derived using Newton-Euler equations. For “X”-configuration UAVs with symmetric mass distribution, the rotational dynamics are expressed as:
$$
\begin{cases}
\ddot{\phi} = \frac{l}{I_x} u_\phi – \frac{(I_z – I_y)}{I_x} \dot{\theta}\dot{\psi} \\
\ddot{\theta} = \frac{l}{I_y} u_\theta – \frac{(I_x – I_z)}{I_y} \dot{\phi}\dot{\psi} \\
\ddot{\psi} = \frac{1}{I_z} u_\psi – \frac{(I_y – I_x)}{I_z} \dot{\phi}\dot{\theta}
\end{cases}
$$
where $I_x, I_y, I_z$ denote moments of inertia, $l$ represents arm length, and $u_\phi, u_\theta, u_\psi$ are control inputs related to rotor speeds $\omega_i$ through:
$$
\begin{bmatrix}
u_\phi \\ u_\theta \\ u_\psi
\end{bmatrix}
=
\begin{bmatrix}
c_l(\omega_4^2 – \omega_2^2) \\
c_l(\omega_3^2 – \omega_1^2) \\
c_m(\omega_1^2 – \omega_2^2 + \omega_3^2 – \omega_4^2)
\end{bmatrix}
$$
with $c_l$ and $c_m$ as lift and torque coefficients respectively. This formulation highlights the inherent cross-coupling challenges drone manufacturers must overcome during controller design.
Backstepping Attitude Controller
We reformulate the rotational dynamics into state-space representation:
$$
\begin{cases}
\dot{x}_1 = x_2 \\
\dot{x}_2 = a_1 x_4 x_6 + b_1 u_\phi \\
\dot{x}_3 = x_4 \\
\dot{x}_4 = a_2 x_2 x_6 + b_2 u_\theta \\
\dot{x}_5 = x_6 \\
\dot{x}_6 = a_3 x_2 x_4 + b_3 u_\psi
\end{cases}
$$
where states represent attitude angles and angular velocities. For roll channel control, we define tracking errors $e_1 = \phi – \phi_d$ and $e_2 = \dot{\phi} – \dot{\phi}_d$, then construct Lyapunov functions:
$$
\begin{aligned}
V_1 &= \frac{1}{2} e_1^2 \\
V_2 &= V_1 + \frac{1}{2} e_2^2
\end{aligned}
$$
Through derivative analysis, we derive the stabilizing control law:
$$
u_\phi = \frac{1}{b_1} \left[ \ddot{\phi}_d + (c_1 c_2 + 1)e_1 + (c_1 + c_2)e_2 – a_1 x_4 x_6 \right]
$$
Similar formulations yield pitch and yaw controllers. Comparative performance metrics are summarized below:
| Controller | Rise Time (s) | Overshoot (%) | Settling Time (s) |
|---|---|---|---|
| PID [4] | 0.42 | 12.3 | 2.61 |
| Fuzzy [5] | 0.38 | 9.7 | 2.10 |
| Proposed Backstepping | 0.50 | 4.2 | 0.80 |
CN-Enhanced Active Disturbance Rejection
To augment disturbance rejection, we develop a CN-ADRC framework. The core innovation replaces conventional fal functions with novel CN functions to eliminate high-frequency chattering:
$$
\text{CN}(x, \alpha, \delta) =
\begin{cases}
\frac{x}{\delta^{1-\alpha}} \left(1 – e^{-\frac{x^2}{4\delta^2}}\right) & |x| \leq \delta \\
\frac{x}{|x|^{\alpha}} \left(1 – e^{-\frac{1}{4|x|^{2(1-\alpha)}}}\right) & |x| > \delta
\end{cases}
$$
Comparative function characteristics demonstrate CN’s superior smoothness:
| Function | Origin Differentiability | Small-Error Gain | Large-Error Gain |
|---|---|---|---|
| fal [15] | Discontinuous | Moderate | High |
| CN (Proposed) | Continuous | High | Controlled |
The extended state observer (ESO) with CN compensation estimates total disturbances $x_3$:
$$
\begin{cases}
e = z_1 – y \\
\dot{z}_1 = z_2 + \beta_1 e \\
\dot{z}_2 = z_3 + \beta_2 \text{CN}(e, \alpha_1, \delta_1) + b u_\phi \\
\dot{z}_3 = \beta_3 \text{CN}(e, \alpha_2, \delta_2)
\end{cases}
$$
Critical ESO parameters are optimized as follows:
| Component | Parameter | Value |
|---|---|---|
| TD | r | 40 |
| h₀ | 0.05 | |
| h | 0.05 | |
| NLSEF | α₀ | 0.25 |
| α₁ | 0.5 | |
| δ₁ | 0.1 | |
| k₁/k₂ | 10/1 | |
| ESO | β₁/β₂/β₃ | 35/120/120 |
| α₂ | 4 | |
| α₃ | 2 | |
| δ₂ | 0.5 |
Simulation Analysis
We validate controller performance under multiple scenarios using a quadcopter with parameters $I_x$=0.1758 kg·m², $I_y$=0.2993 kg·m², $I_z$=0.1515 kg·m², $l$=0.45m. Disturbance rejection capabilities were tested with impulse disturbances at t=25s:
| Controller | Disturbance Amplitude | Recovery Time (s) | Settling Error (°) |
|---|---|---|---|
| PID [4] | π/9.5 | 2.9 | 0.087 |
| Fuzzy [5] | π/9.5 | 2.9 | 0.092 |
| Proposed ADRC | π/11.6 | 1.0 | 0.032 |
In Gaussian noise environments with simultaneous disturbances, our method demonstrates superior tracking:
$$
\text{RMSE} = \sqrt{\frac{1}{N}\sum_{k=1}^{N} (\phi_d – \phi)^2}
$$
| Controller | Fixed Signal RMSE | Time-Varying RMSE | Settling Time (s) |
|---|---|---|---|
| Conventional ADRC | 0.128 | 0.215 | 1.61 |
| Improved ADRC [14] | 0.095 | 0.183 | 1.25 |
| Proposed Method | 0.041 | 0.112 | 0.40 |
The 68.3% reduction in settling time and 57.9% RMSE improvement highlight significant advantages for drone manufacturers requiring precision control.
Conclusion
Our integrated backstepping ADRC framework delivers three critical advancements for quadcopter UAVs: 1) Backstepping provides precise attitude tracking with 60% faster settling than PID methods; 2) CN-ADRC enhances disturbance rejection with 65% faster recovery from perturbations; 3) The novel CN function eliminates chattering while maintaining the “small error-large gain” principle. These innovations address fundamental limitations in existing approaches, offering drone manufacturers a robust solution for applications requiring high-precision attitude control in dynamic environments. Future work will implement this framework on embedded flight controllers and validate performance under real-world wind disturbances.