To enhance the spraying efficiency of agricultural drones, a cooperative operation method utilizing a formation of multiple agricultural UAVs is proposed. This method integrates an improved artificial potential field (APF) path planning algorithm with a sliding mode path tracking control strategy. This comprehensive approach addresses the critical challenges of obstacle avoidance in complex agricultural environments and precise formation control during spraying operations, significantly advancing intelligent agriculture practices.
The core of the methodology involves establishing accurate mathematical models for individual agricultural drones and the coordinated formation. An improved APF algorithm, enhanced with a loop force mechanism to overcome local minima, efficiently generates obstacle-avoidance spraying paths. Subsequently, dedicated sliding mode control laws are designed for both the translational motion and rotational attitude dynamics of each agricultural UAV, ensuring accurate path tracking by the entire formation.
1. System Modeling
1.1 Agricultural UAV Dynamics: The dynamic behavior of a single agricultural drone, essential for spraying operations, is modeled considering translational motion and rotational attitude. The translational dynamics under disturbances are given by:
$$
\begin{align}
\dot{\mathbf{p}} &= \mathbf{v} \\
\dot{\mathbf{v}} &= -\mathbf{G} + \mathbf{h}\mathbf{T} + \mathbf{d}_v
\end{align}
$$
where $\mathbf{p} = [x, y, z]^T$ represents the position vector (m), $\mathbf{v} = [v_x, v_y, v_z]^T$ is the velocity vector (m/s), $\mathbf{G} = [g_x, g_y, g_z]^T$ is the gravitational acceleration vector (m/s²), $\mathbf{T} = [T_x, T_y, T_z]^T$ is the thrust vector (N), $\mathbf{d}_v$ represents disturbances in the motion model, and $\mathbf{h}$ is the thrust coefficient matrix defined as:
$$
\mathbf{h} = \frac{1}{m}
\begin{bmatrix}
0 & 0 & \sin\theta\cos\psi\cos\phi + \sin\psi\sin\phi \\
0 & 0 & \sin\theta\sin\psi\cos\phi – \cos\psi\sin\phi \\
0 & 0 & \cos\theta\cos\phi
\end{bmatrix}
$$
Here, $m$ is the mass of the agricultural UAV (kg), and $\phi$, $\theta$, $\psi$ are the roll, pitch, and yaw angles (rad), respectively. The attitude dynamics under disturbances are modeled as:
$$
\begin{align}
\dot{\mathbf{\Theta}} &= \mathbf{W}\boldsymbol{\omega} \\
\dot{\boldsymbol{\omega}} &= \mathbf{f} + \boldsymbol{\Gamma} + \mathbf{d}_{\omega}
\end{align}
$$
where $\mathbf{\Theta} = [\phi, \theta, \psi]^T$ is the attitude angle vector (rad), $\boldsymbol{\omega} = [p, q, r]^T$ is the angular velocity vector (rad/s), $\mathbf{J} = \text{diag}\{J_x, J_y, J_z\}$ is the inertia matrix (kg·m²), $\boldsymbol{\Gamma} = \mathbf{J}^{-1}\boldsymbol{\Gamma}_0 = [\Gamma_x, \Gamma_y, \Gamma_z]^T$ is the control moment vector (N·m), $\mathbf{d}_{\omega}$ represents disturbances in the attitude model, $\mathbf{f} = \mathbf{J}^{-1}(-\boldsymbol{\omega} \times \mathbf{J}\boldsymbol{\omega})$, and $\mathbf{W}$ is the transformation matrix:
$$
\mathbf{W} =
\begin{bmatrix}
1 & \sin\phi\tan\theta & \cos\phi\tan\theta \\
0 & \cos\phi & -\sin\phi \\
0 & \sin\phi/\cos\theta & \cos\phi/\cos\theta
\end{bmatrix}
$$
1.2 Agricultural UAV Formation Model: For cooperative spraying, a leader-follower formation structure is employed. The leader agricultural drone’s path is planned based on the field geometry and obstacles. Followers maintain a predefined relative position to the leader. The formation geometry is defined by:
$$
\begin{bmatrix}
x_f \\ y_f \\ z_f
\end{bmatrix}
=
\begin{bmatrix}
x_l – 2x_c \\ y_l – 2y_c \\ z_l – 2z_c
\end{bmatrix}
$$
where $(x_l, y_l, z_l)$ is the leader’s position, $(x_f, y_f, z_f)$ is a follower’s position, and $(x_c, y_c, z_c)$ represents the maximum effective spraying radius components relative to the leader at operating height $z_c$. This ensures complete coverage of the swath width by the agricultural drone formation.
2. Improved Artificial Potential Field Path Planning
Traditional APF methods suffer from local minima, particularly dangerous near obstacles like trees or poles in fields. The improved APF introduces a loop force to escape these minima.
2.1 Repulsive Potential Field: The repulsive field $U_{po}$ from an obstacle located at $\mathbf{X}_o$ acting on the agricultural UAV formation at $\mathbf{X}_p$ is:
$$
U_{po}(\mathbf{X}_p, \mathbf{X}_o) =
\begin{cases}
+\infty & 0 \leq l(\mathbf{X}_p, \mathbf{X}_o) < R_0 \\
b \left[ \left( \frac{l(\mathbf{X}_p, \mathbf{X}_o)}{\rho_0} \right)^2 – 2\ln\left( \frac{l(\mathbf{X}_p, \mathbf{X}_o)}{\rho_0} \right) – 1 \right] & R_0 \leq l(\mathbf{X}_p, \mathbf{X}_o) \leq R_0 + \rho_0 \\
0 & l(\mathbf{X}_p, \mathbf{X}_o) > R_0 + \rho_0
\end{cases}
$$
where $l(\mathbf{X}_p, \mathbf{X}_o)$ is the distance (m), $R_0$ is the obstacle radius (m), $\rho_0$ is the repulsive field radius (m), and $b$ is a gain coefficient. The corresponding repulsive force $\mathbf{F}_{po}$ is the negative gradient:
$$
\mathbf{F}_{po}(\mathbf{X}_p, \mathbf{X}_o) = -\nabla U_{po} =
\begin{cases}
+\infty & 0 \leq l < R_0 \\
\frac{b}{\rho_0} \left( \frac{l}{\rho_0} – \frac{1}{l} \right) \frac{\partial l}{\partial \mathbf{X}_p} & R_0 \leq l \leq R_0 + \rho_0 \\
\mathbf{0} & l > R_0 + \rho_0
\end{cases}
$$
2.2 Attractive Potential Field: The attractive field $U_{pt}$ pulling the agricultural drone formation towards the target $\mathbf{X}_t$ is:
$$
U_{pt}(\mathbf{X}_p, \mathbf{X}_t) =
\begin{cases}
-a_1 l^2(\mathbf{X}_p, \mathbf{X}_t) & 0 \leq l(\mathbf{X}_p, \mathbf{X}_t) < R_p \\
-a_2 l^2(\mathbf{X}_p, \mathbf{X}_t) & l(\mathbf{X}_p, \mathbf{X}_t) \geq R_p
\end{cases}
$$
where $l(\mathbf{X}_p, \mathbf{X}_t)$ is the distance to target (m), $R_p$ is the formation radius (m), and $a_1$, $a_2$ are gain coefficients ($a_2 > a_1$). The attractive force $\mathbf{F}_{pt}$ is:
$$
\mathbf{F}_{pt}(\mathbf{X}_p, \mathbf{X}_t) = -\nabla U_{pt} =
\begin{cases}
2a_1 l(\mathbf{X}_p, \mathbf{X}_t) \frac{\partial l}{\partial \mathbf{X}_p} & 0 \leq l < R_p \\
2a_2 l(\mathbf{X}_p, \mathbf{X}_t) \frac{\partial l}{\partial \mathbf{X}_p} & l \geq R_p
\end{cases}
$$
2.3 Loop Force for Local Minima Escape: When the total force $\mathbf{F}_p = \mathbf{F}_{pt} + \mathbf{F}_{po}$ approaches zero near an obstacle ($l < R_0 + \rho_0$), a loop force $\mathbf{F}_{pot}$ is activated:
$$
\mathbf{F}_{pot}(\mathbf{X}_p, \mathbf{X}_o, \mathbf{X}_t) =
\mu \frac{ \| \mathbf{F}_p \cdot \mathbf{F}_{pt} \| }{ \| \mathbf{F}_{pt} \| } \cdot \frac{ \mathbf{F}_{po} \cdot \mathbf{F}_{pt} \cdot \mathbf{F}_p }{ \| \mathbf{F}_{po} \cdot \mathbf{F}_{pt} \cdot \mathbf{F}_p \| }
$$
where $\mu$ is a scaling factor. This force introduces a perturbation perpendicular to the resultant force direction, enabling the agricultural UAV formation to escape stagnation points.
2.4 Resultant Force and Velocity Command: The total force acting on the formation is:
$$
\mathbf{F}(\mathbf{X}_p, \mathbf{X}_o, \mathbf{X}_t) = \mathbf{F}_{pt}(\mathbf{X}_p, \mathbf{X}_t) + \mathbf{F}_{po}(\mathbf{X}_p, \mathbf{X}_o) + \mathbf{F}_{pot}(\mathbf{X}_p, \mathbf{X}_o, \mathbf{X}_t)
$$
This force determines the formation’s commanded velocity $\mathbf{v}_p$ and orientation ($\psi_p$, $\theta_p$):
$$
\mathbf{v}_p = \frac{\mathbf{F}}{m_p} \delta, \quad
\psi_p = \arctan\left(\frac{F_y}{F_x}\right), \quad
\theta_p = \arctan\left(\frac{F_z}{\sqrt{F_x^2 + F_y^2}}\right)
$$
where $m_p$ is the formation mass (kg), and $\delta$ is the computation step size. Integrating $\mathbf{v}_p$ provides the leader agricultural drone’s path $\mathbf{p}_l = [x_l, y_l, z_l]^T$:
$$
x_l = \int (v_p \cos\theta_p \cos\psi_p) dt, \quad y_l = \int (v_p \cos\theta_p \sin\psi_p) dt, \quad z_l = \int (v_p \sin\theta_p) dt
$$
Follower agricultural UAV paths are derived using the formation model. This method efficiently generates smooth, obstacle-free spraying paths within 1.2 seconds.
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3. Sliding Mode Path Tracking Control
Precise tracking of the planned path $\mathbf{p}_d$ by each agricultural UAV is achieved via hierarchical sliding mode control (SMC), robust against mass variation (spray payload depletion) and disturbances.
3.1 Translational Motion Control: Define the position tracking error $\mathbf{e}_1 = \mathbf{p} – \mathbf{p}_d$. Introduce an auxiliary error $\mathbf{e}_2 = \mathbf{v} + c_1 \mathbf{e}_1 – \dot{\mathbf{p}}_d$, where $c_1 > 0$. The sliding surface is:
$$
\mathbf{S}_1 = k_1 \mathbf{e}_1 + \mathbf{e}_2 = (k_1 + c_1)\mathbf{e}_1 + \dot{\mathbf{e}}_1, \quad k_1 > 0
$$
Consider the Lyapunov function $V_1 = \frac{1}{2} \mathbf{e}_1^T \mathbf{e}_1 + \frac{1}{2} \mathbf{S}_1^T \mathbf{S}_1$. Its derivative, using the motion dynamics, is:
$$
\dot{V}_1 = -\mathbf{e}_1^T c_1 \mathbf{e}_1 + \mathbf{e}_1^T \mathbf{e}_2 + \mathbf{S}_1^T \left[ (k_1 + c_1) \dot{\mathbf{e}}_1 – \mathbf{G} + \mathbf{h}\mathbf{T} + \mathbf{d}_v – \ddot{\mathbf{p}}_d \right]
$$
To ensure $\dot{V}_1 \leq 0$, the thrust control law $\mathbf{T}$ is designed as:
$$
\mathbf{T} = \mathbf{h}^{-1} \left[ \mathbf{G} – (k_1 + c_1) \dot{\mathbf{e}}_1 + \ddot{\mathbf{p}}_d – k_2 \mathbf{S}_1 – \eta_1 \text{sgn}(\mathbf{S}_1) – (\mathbf{S}_1^T)^{-1} \mathbf{e}_1^T \mathbf{e}_2 – D_v \text{sgn}(\mathbf{S}_1) \right]
$$
where $k_2 > 0$, $\eta_1 > 0$, $D_v$ bounds $\|\mathbf{d}_v\|$, and $\text{sgn}$ is the signum function. Substitution proves $\dot{V}_1 \leq -c_1 \|\mathbf{e}_1\|^2 – k_2 \|\mathbf{S}_1\|^2 – \eta_1 \|\mathbf{S}_1\| \leq 0$, guaranteeing stability. From $\mathbf{T}$, the attitude commands $\phi_d$, $\theta_d$, $\psi_d$ are computed:
$$
\psi_d = \arctan\left(\frac{T_y}{T_x}\right), \quad \phi_d = \arcsin\left(\frac{T_x \sin\psi_d – T_y \cos\psi_d}{\sqrt{T_x^2 + T_y^2 + T_z^2}}\right), \quad \theta_d = \arctan\left(\frac{T_x \cos\psi_d + T_y \sin\psi_d}{T_z + mg}\right)
$$
3.2 Rotational Attitude Control: Define the attitude tracking error $\mathbf{e}_3 = \mathbf{\Theta} – \mathbf{\Theta}_d$. Introduce $\mathbf{e}_4 = \mathbf{W}\boldsymbol{\omega} + c_2 \mathbf{e}_3 – \dot{\mathbf{\Theta}}_d$, $c_2 > 0$. The attitude sliding surface is:
$$
\mathbf{S}_2 = k_3 \mathbf{e}_3 + \mathbf{e}_4 = (k_3 + c_2)\mathbf{e}_3 + \dot{\mathbf{e}}_3, \quad k_3 > 0
$$
Consider $V_2 = \frac{1}{2} \mathbf{e}_3^T \mathbf{e}_3 + \frac{1}{2} \mathbf{S}_2^T \mathbf{S}_2$. Its derivative is:
$$
\dot{V}_2 = -\mathbf{e}_3^T c_2 \mathbf{e}_3 + \mathbf{e}_3^T \mathbf{e}_4 + \mathbf{S}_2^T \left[ (k_3 + c_2) \dot{\mathbf{e}}_3 + \mathbf{W}(\mathbf{f} + \boldsymbol{\Gamma} + \mathbf{d}_{\omega}) – \ddot{\mathbf{\Theta}}_d \right]
$$
The control moment law $\boldsymbol{\Gamma}$ ensuring $\dot{V}_2 \leq 0$ is:
$$
\boldsymbol{\Gamma} = \mathbf{W}^{-1} \left[ -\mathbf{W}\mathbf{f} – (k_3 + c_2) \dot{\mathbf{e}}_3 + \ddot{\mathbf{\Theta}}_d – k_4 \mathbf{S}_2 – \eta_2 \text{sgn}(\mathbf{S}_2) – (\mathbf{S}_2^T)^{-1} \mathbf{e}_3^T \mathbf{e}_4 – \mathbf{W} D_{\omega} \text{sgn}(\mathbf{S}_2) \right]
$$
where $k_4 > 0$, $\eta_2 > 0$, and $D_{\omega}$ bounds $\|\mathbf{d}_{\omega}\|$. Stability is confirmed as $\dot{V}_2 \leq -c_2 \|\mathbf{e}_3\|^2 – k_4 \|\mathbf{S}_2\|^2 – \eta_2 \|\mathbf{S}_2\| \leq 0$. This dual-loop SMC structure ensures precise path tracking by each agricultural drone in the formation despite disturbances.
4. Simulation Analysis
The performance of the proposed agricultural UAV formation control system was rigorously evaluated in simulation using MATLAB.
4.1 Simulation Setup: A rectangular field (90m x 150m) containing three obstacles (Tree: (20m, 3m), R0=3m; Water Tower: (60m, 57m), R0=7.5m; Pylon: (120m, 85m), R0=5m) was defined. A formation of 5 agricultural drones, flying at 5m altitude, each with a spraying radius of 3m, was controlled. Key parameters are listed below.
| Parameter | Description | Value |
|---|---|---|
| $R_p$ | Formation Radius | 15 m |
| $\rho_0$ | Repulsive Field Radius | 0.5 m |
| $b$ | Repulsive Gain | 2.5 |
| $a_1$ | Attractive Gain 1 | 1.5 |
| $a_2$ | Attractive Gain 2 | 2.5 |
| $m_p$ | Formation Mass | 25 kg |
| $\delta$ | APF Step Size | 0.1 m |
| $c_1, k_1, k_2$ | Translational SMC Gains | 8, 6, 10 |
| $c_2, k_3, k_4$ | Attitude SMC Gains | 9, 5, 12 |
4.2 Path Planning Performance: The Improved APF generated smooth, obstacle-avoiding spraying paths within 1.2 seconds. Comparatively, an A* algorithm (Li et al., 2021) produced a jagged path unsuitable for precise agricultural drone tracking and took 2.1 seconds.
4.3 Path Tracking Performance: The proposed SMC was compared against an Adaptive Trajectory Tracking (ADAT) method (Si et al., 2023). While ADAT managed straight paths, it exhibited significant oscillation near obstacles (max error: 1.89m), risking collision. The SMC achieved precise tracking throughout, including curves, with a maximum deviation of only 0.12m.
5. Field Test Results
The method was validated using 5 HaoYing X9 Plus agricultural drones spraying liquid nitrogen fertilizer on a 90m x 150m wheat field containing three poles (P1: (22m, 40m), P2: (68m, 61m), P3: (95m, 82m), R0=1.5m). Performance was compared against a baseline strategy combining Improved A* (IA) path planning and ADAT control (IA-ADAT). Key metrics were tracking error, over-spray area (heavy spraying), under-spray area (missed spraying), and operation time.
| Method | Max Tracking Error (m) | Over-Spray Area (m² / %) | Under-Spray Area (m² / %) | Operation Time (s) |
|---|---|---|---|---|
| IA-ADAT | 1.55 | 236.3 / 1.75% | 252.4 / 1.87% | 251 |
| Proposed Method | 0.21 | 41.6 / 0.31% | 45.7 / 0.34% | 239 |
The results demonstrate the superiority of the proposed approach. The low tracking error (0.21m) directly translates to minimal over-spray (0.31% of total area) and under-spray (0.34%), optimizing chemical usage and ensuring uniform crop protection. The reduced operation time (239s vs 251s) highlights the efficiency gain from precise, coordinated path following by the agricultural UAV formation.
6. Conclusion
This work presents a comprehensive solution for efficient and precise cooperative spraying using agricultural drone formations. The key contributions are:
- An Improved APF path planner incorporating a loop force effectively solves the local minima problem, generating smooth, obstacle-avoiding spraying paths for the leader agricultural UAV in 1.2 seconds.
- A hierarchical sliding mode control structure provides robust tracking for both translational motion and rotational attitude of each agricultural drone within the formation, achieving a maximum path deviation of only 0.12m despite disturbances like payload variation.
- Field validation confirms significant practical benefits: drastically reduced over-spray (0.31%) and under-spray (0.34%) areas compared to baseline methods, alongside shorter operation times, leading to substantial efficiency gains and resource savings.
The method significantly enhances the operational capability of agricultural UAV swarms, enabling efficient, high-coverage, low-waste spraying in complex fields with obstacles. Future work will focus on integrating real-time perception for dynamic obstacles and refining the formation model for variable-rate spraying applications, further advancing intelligent agricultural drone operations.