Agricultural drones, particularly multi-rotor unmanned aerial vehicles (agricultural UAVs), have become indispensable tools for modern precision agriculture, enabling efficient and targeted pesticide and fertilizer application. However, these agricultural drones face significant challenges during spraying operations due to disturbances like motor dynamic characteristics, airframe vibrations, and unpredictable mechanical friction caused by turbulent airflow. These factors induce flight deviations, leading to inaccurate spraying, potential crop damage from over-application, and safety risks. To address this critical issue, we propose a novel robust control strategy integrating an Extended State Observer (ESO) with Backstepping control.
Mathematical Modeling of the Agricultural UAV with Disturbances
The core dynamics of a multi-rotor agricultural UAV, focusing on attitude control (pitch $\phi$, roll $\theta$, yaw $\psi$), are governed by the rotational motion driven by brushless DC motors:
$$A\dot{\omega} + B\omega = K_T i$$
where $\omega = [\dot{\phi}, \dot{\theta}, \dot{\psi}]^T$ is the angular velocity vector, $i = [i_1, i_2, i_3]^T$ is the motor current vector, $K_T$ is the motor torque constant matrix, and $A$, $B$ are inertia and damping matrices respectively.
The motor electrical dynamics are:
$$C\dot{i} + Di = u$$
where $u$ is the motor input voltage vector, $C$ is the inductance matrix, and $D$ is the resistance matrix.
Defining the state variables $x_1 = \eta = [\phi, \theta, \psi]^T$ (attitude angles), $x_2 = \dot{\eta} = \omega$, and $x_3 = i$, the nominal state-space model is:
$$\begin{cases}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= -A^{-1}B x_2 + A^{-1}K_T x_3 \\
\dot{x}_3 &= -C^{-1}D x_3 + C^{-1}u
\end{cases}$$
To accurately represent real-world spraying conditions, we incorporate lumped disturbances affecting the agricultural drone:
$$\begin{cases}
\dot{x}_1 &= x_2 + d_1 \\
\dot{x}_2 &= -A^{-1}B x_2 + A^{-1}K_T x_3 + d_2 \\
\dot{x}_3 &= -C^{-1}D x_3 + C^{-1}u + d_3
\end{cases}$$
where $d_1$ (deg/s) represents disturbances on angular position (e.g., wind gusts, payload shift), $d_2$ (deg/s²) represents disturbances on angular acceleration (e.g., vibration-induced torques, aerodynamic imbalances), and $d_3$ (A/s) represents disturbances on motor current dynamics (e.g., electrical noise, varying back-EMF).
Anti-Disturbance Backstepping Robust (ADBSR) Control Design
The control objective is to ensure the agricultural UAV accurately tracks desired attitude trajectories ($x_{1c} = \eta_{d}$) despite $d_1$, $d_2$, $d_3$. The ADBSR control structure employs cascaded ESOs for disturbance estimation and rejection within a Backstepping framework.
Extended State Observer (ESO) Design
Three ESOs are designed to estimate the states and disturbances in each channel:
- Attitude Angle ($x_1$) ESO:
$$\begin{cases}
\dot{\hat{x}}_1 &= \hat{x}_2 + \hat{d}_1 – 2r\tilde{x}_1 \\
\dot{\hat{d}}_1 &= -r^2 \tilde{x}_1 + (x_1 – x_{1c})
\end{cases}$$where $\hat{x}_1$ is the estimated attitude, $\hat{d}_1$ is the estimated disturbance on $x_1$, $\tilde{x}_1 = \hat{x}_1 – x_1$ is the estimation error, $x_{1c}$ is the desired attitude trajectory, and $r > 0$ is the observer gain.
- Angular Velocity ($x_2$) ESO:
$$\begin{cases}
\dot{\hat{x}}_2 &= -A^{-1}B \hat{x}_2 + A^{-1}K_T \hat{x}_3 + \hat{d}_2 – 2p\tilde{x}_2 \\
\dot{\hat{d}}_2 &= -p^2 \tilde{x}_2 + (x_2 – x_{2c})
\end{cases}$$where $\hat{x}_2$ is the estimated angular velocity, $\hat{d}_2$ is the estimated disturbance on $x_2$, $\tilde{x}_2 = \hat{x}_2 – x_2$ is the estimation error, $x_{2c}$ is the desired angular velocity (virtual control input), and $p > 0$ is the observer gain.
- Motor Current ($x_3$) ESO:
$$\begin{cases}
\dot{\hat{x}}_3 &= -C^{-1}D \hat{x}_3 + C^{-1}u + \hat{d}_3 – 2q\tilde{x}_3 \\
\dot{\hat{d}}_3 &= -q^2 \tilde{x}_3 + (x_3 – x_{3c})
\end{cases}$$where $\hat{x}_3$ is the estimated motor current, $\hat{d}_3$ is the estimated disturbance on $x_3$, $\tilde{x}_3 = \hat{x}_3 – x_3$ is the estimation error, $x_{3c}$ is the desired motor current (virtual control input), and $q > 0$ is the observer gain.
Backstepping Robust Control Law
The control law is derived recursively using the Backstepping technique, integrating the disturbance estimates ($\hat{d}_1, \hat{d}_2, \hat{d}_3$) for robustness.
- Step 1 (Attitude Tracking): Define the tracking error $e_1 = x_1 – x_{1c}$. Using the Lyapunov function $V_1 = \frac{1}{2}e_1^T e_1 + \frac{1}{2}\tilde{d}_1^T \tilde{d}_1 + \frac{r^2}{2}\tilde{x}_1^T \tilde{x}_1$ ($\tilde{d}_1 = \hat{d}_1 – d_1$), the virtual control input (desired angular velocity) is derived to stabilize $e_1$:
$$x_{2c} = -K_1 e_1 – \hat{d}_1 + \dot{x}_{1c}$$
where $K_1 = K_1^T > 0$ is a positive definite gain matrix. This ensures $\dot{V}_1 \leq -e_1^T K_1 e_1 – 2r^3 \tilde{x}_1^T \tilde{x}_1 \leq 0$.
- Step 2 (Angular Velocity Tracking): Define the velocity error $e_2 = x_2 – x_{2c}$. Using the Lyapunov function $V_2 = \frac{1}{2}e_2^T e_2 + \frac{1}{2}\tilde{d}_2^T \tilde{d}_2 + \frac{p^2}{2}\tilde{x}_2^T \tilde{x}_2$ ($\tilde{d}_2 = \hat{d}_2 – d_2$), the virtual control input (desired motor current) is derived:
$$x_{3c} = K_T^{-1}A \left( -K_2 e_2 + A^{-1}B x_2 – \hat{d}_2 + \dot{x}_{2c} \right)$$
where $K_2 = K_2^T > 0$. This ensures $\dot{V}_2 \leq -e_2^T K_2 e_2 – 2p^3 \tilde{x}_2^T \tilde{x}_2 \leq 0$.
- Step 3 (Motor Current Tracking): Define the current error $e_3 = x_3 – x_{3c}$. Using the Lyapunov function $V_3 = \frac{1}{2}e_3^T e_3 + \frac{1}{2}\tilde{d}_3^T \tilde{d}_3 + \frac{q^2}{2}\tilde{x}_3^T \tilde{x}_3$ ($\tilde{d}_3 = \hat{d}_3 – d_3$), the actual control input (motor voltage) is derived:
$$u = C \left( -K_3 e_3 + C^{-1}D x_3 – \hat{d}_3 + \dot{x}_{3c} \right)$$
where $K_3 = K_3^T > 0$. This ensures $\dot{V}_3 \leq -e_3^T K_3 e_3 – 2q^3 \tilde{x}_3^T \tilde{x}_3 \leq 0$.
Stability Analysis
The overall stability of the closed-loop agricultural drone system is guaranteed by the composite Lyapunov function $W = V_1 + V_2 + V_3$. Its derivative is:
$$\dot{W} = \dot{V}_1 + \dot{V}_2 + \dot{V}_3 \leq -e_1^T K_1 e_1 – e_2^T K_2 e_2 – e_3^T K_3 e_3 – 2r^3 \tilde{x}_1^T \tilde{x}_1 – 2p^3 \tilde{x}_2^T \tilde{x}_2 – 2q^3 \tilde{x}_3^T \tilde{x}_3 \leq 0$$
Since $\dot{W}$ is negative semi-definite, by Lyapunov’s stability theorem, the tracking errors $e_1, e_2, e_3$ and the observer estimation errors $\tilde{x}_1, \tilde{x}_2, \tilde{x}_3$ are globally uniformly bounded. Furthermore, Barbalat’s lemma ensures that $e_1, e_2, e_3 \to 0$ as $t \to \infty$, proving asymptotic tracking for the agricultural UAV despite disturbances.
Simulation Validation
We evaluated the ADBSR control performance on the agricultural UAV model in MATLAB under significant disturbances:
Desired Trajectory:
$$x_{1c} = \begin{bmatrix} 3t – 6\cos t \\ e^{0.25t} – 10\cos t \\ 5\cos t + t + e^{0.25t} \end{bmatrix} \text{(deg)}$$
Disturbances:
$$\begin{aligned}
d_1 &= [0.3t + 0.2\cos t]_{3\times1} \text{ (deg/s)} \\
d_2 &= [e^{0.05t} + 0.2\sin t]_{3\times1} \text{ (deg/s²)} \\
d_3 &= [0.1t + e^{0.03t} + 0.1\sin t]_{3\times1} \text{ (A/s)}
\end{aligned}$$
Control Parameters: $r=5, p=3, q=6, K_1=\text{diag}\{3, 5, 7\}, K_2=\text{diag}\{6, 4, 3\}, K_3=\text{diag}\{8, 4, 6\}$.
Disturbance Estimation Accuracy
The designed ESOs demonstrated high-fidelity disturbance estimation, crucial for effective rejection in the agricultural UAV control loop. Maximum estimation errors were remarkably low:
| Disturbance Signal | Maximum Estimation Error |
|---|---|
| $d_1$ (Angular Position) | 0.03 deg/s |
| $d_2$ (Angular Acceleration) | 0.03 deg/s² |
| $d_3$ (Motor Current) | 0.04 A/s |
Attitude Tracking Performance
We compared ADBSR against Adaptive Fuzzy Active Disturbance Rejection (AFADR) control and Joint Observer with FeedForward Compensation (JOFFC) control under identical disturbances. The ADBSR controller showed superior robustness for the agricultural drone:
| Control Method | Max Pitch Error (deg) | Max Roll Error (deg) | Max Yaw Error (deg) |
|---|---|---|---|
| AFADR | 6.2 | 8.3 | 9.2 |
| JOFFC | 3.5 | 3.8 | 4.4 |
| ADBSR (Proposed) | 0.4 | 0.6 | 0.5 |
The significantly smaller errors with ADBSR highlight its effectiveness in maintaining precise attitude control for the agricultural UAV, essential for accurate spraying paths.
Field Testing Results
Practical validation was conducted using a C50 agricultural UAV (50 kg payload) spraying nitrogen fertilizer on a 100m x 18m rice field. Flight parameters were set to 2.5m altitude, 1.5m/s speed, and 1.8m effective spray swath. Controller performance was assessed based on overlapping (re-spray) and missed (under-spray) areas:
| Control Method | Re-Spray Area (m² / %) | Under-Spray Area (m² / %) | Flight Time (s) |
|---|---|---|---|
| AFADR | 64.08 / 3.56% | 67.14 / 3.73% | 695 |
| JOFFC | 33.12 / 1.84% | 35.10 / 1.95% | 684 |
| ADBSR (Proposed) | 4.14 / 0.23% | 4.86 / 0.27% | 676 |
The ADBSR controller drastically reduced re-spray and under-spray areas by effectively rejecting disturbances, leading to more precise flight paths and chemical application. The reduced flight time further indicates less need for corrective maneuvers, enhancing operational efficiency for the agricultural UAV.
Conclusion
This work successfully developed and validated an Anti-Disturbance Backstepping Robust (ADBSR) control strategy for agricultural drones performing spraying operations. The key contributions are:
- High-Fidelity Disturbance Estimation: The multi-channel Extended State Observer (ESO) accurately estimated critical disturbances ($d_1, d_2, d_3$) affecting the agricultural UAV, with maximum errors below 0.04 deg/s, 0.03 deg/s², and 0.04 A/s.
- Robust Trajectory Tracking: Integrating ESO estimates within the Backstepping control law yielded exceptional attitude tracking performance for the agricultural drone. Maximum pitch, roll, and yaw errors were minimized to 0.4°, 0.6°, and 0.5° respectively under significant simulated disturbances, outperforming AFADR and JOFFC methods.
- Enhanced Spraying Precision: Field tests on a 50kg payload agricultural UAV confirmed the controller’s practical efficacy. Re-spray and under-spray areas were reduced to just 0.23% and 0.27% of the total field area, significantly improving chemical application efficiency and minimizing environmental impact while also reducing operational time.
The ADBSR control law effectively mitigates the detrimental effects of motor dynamics, airframe vibration, and aerodynamic friction on agricultural UAV stability during spraying. This ensures precise flight path adherence, crucial for avoiding crop damage and optimizing resource use. Future work will focus on extending this robust control framework to coordinated multi-agent systems for swarm-based agricultural drone operations, further scaling the efficiency of precision spraying.