In modern agriculture, the use of crop spraying drones has revolutionized pesticide application by enabling precise and efficient spraying operations. However, traditional fixed or limited-movement sprayers often lead to uneven coverage, missed spots, or over-spraying, reducing effectiveness and increasing environmental impact. To address these challenges, we designed a three-degree-of-freedom (3-DOF) spray manipulator for integration with spraying UAVs. This system allows for dynamic adjustment of spray parameters, such as spray width, angle, and accompanying blade speed, enhancing droplet deposition and coverage on target crops like cotton, wheat, and rice. In this paper, we present the comprehensive design, kinematic and dynamic analysis, workspace evaluation, trajectory planning, and simulation of the manipulator, demonstrating its potential to improve precision in agricultural spraying operations.
The 3-DOF spray manipulator is configured as a PRR type, where P denotes a prismatic joint and R represents revolute joints. It consists of a base sliding joint, a shoulder rotation joint, an elbow pitch joint, accompanying blades, a power box, and a nozzle. The base joint employs a servo motor driving a ball screw for linear motion, assisted by linear guides and sliders to enhance stability and precision. The shoulder and elbow joints are connected via support rods to transmit torque from servos, increasing overall rigidity. To mitigate the adverse effects of UAV rotor downwash on droplet drift, especially during vertical spraying, accompanying blades are attached to the spray rod. These blades generate a wind field aligned with the spraying direction, reinforcing droplet deposition and penetration when the nozzle is coaxial with the UAV rotors. The manipulator is symmetrically mounted on the drone, and for analysis, we focus on a single unit. During operation, the servo motor controls the base movement, while two servos manage the shoulder rotation and elbow pitch, enabling 3-DOF spraying. Combined with the UAV’s flight direction and vertical movement, the nozzle achieves five-degree-of-freedom adjustments, significantly improving flexibility. The accompanying blades adjust their rotational speed based on the nozzle’s orientation, ensuring optimal wind field support. This design allows for targeted spraying with minimal drift, as the nozzle can be positioned directly above crops or at controlled angles, typically not exceeding 45 degrees for side spraying.
To analyze the manipulator’s motion, we employed the modified Denavit-Hartenberg (D-H) method to establish a kinematic model. The coordinate frames are assigned as follows: Frame {0} at the base, Frame {1} at the sliding joint, Frame {2} at the shoulder joint, Frame {3} at the elbow joint (virtual joint for transformation), and Frame {4} at the nozzle tip. The modified D-H parameters are summarized in Table 1, where $d_i$ represents link offset, $\theta_i$ denotes joint angle, $\beta_{i-1}$ is the link twist, and $a_{i-1}$ is the link length. For instance, $d_1$ is the variable for the prismatic joint, ranging from -200 mm to 200 mm, while $\theta_2$ and $\theta_4$ are variables for the revolute joints, with ranges of 0° to 180° and 0° to 60°, respectively. The virtual joint (Joint 3) facilitates coordinate transformation without actual movement.
| Link i | $\theta_i$ (°) | $d_i$ (mm) | $\beta_{i-1}$ (°) | $a_{i-1}$ (mm) |
|---|---|---|---|---|
| 1 | 0 | $d_1$ | -90 | 0 |
| 2 | $\theta_2$ | 0 | 90 | 0 |
| 3 | 90 | 0 | -90 | 0 |
| 4 | $\theta_4$ | 0 | -90 | $a_4$ |
The homogeneous transformation matrix between consecutive frames is given by:
$$^{i-1}_i T = R_x(\beta_{i-1}) D_x(a_{i-1}) R_z(\theta_i) D_z(d_i) = \begin{bmatrix}
\cos \theta_i & -\sin \theta_i & 0 & a_{i-1} \\
\sin \theta_i \cos \beta_{i-1} & \cos \theta_i \cos \beta_{i-1} & -\sin \beta_{i-1} & -\sin \beta_{i-1} d_i \\
\sin \theta_i \sin \beta_{i-1} & \cos \theta_i \sin \beta_{i-1} & \cos \beta_{i-1} & \cos \beta_{i-1} d_i \\
0 & 0 & 0 & 1
\end{bmatrix}$$
For simplicity, we denote sine and cosine functions as ‘s’ and ‘c’, respectively. Substituting the parameters from Table 1, we derive the transformation matrices:
$$^0_1 T = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & d_1 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}, \quad ^1_2 T = \begin{bmatrix}
c\theta_2 & -s\theta_2 & 0 & 0 \\
0 & 0 & -1 & 0 \\
s\theta_2 & c\theta_2 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}, \quad ^2_3 T = \begin{bmatrix}
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}, \quad ^3_4 T = \begin{bmatrix}
c\theta_4 & -s\theta_4 & 0 & a_4 \\
0 & 0 & 1 & 0 \\
-s\theta_4 & -c\theta_4 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}$$
The forward kinematics model is obtained by multiplying these matrices:
$$^0_4 T = ^0_1 T \cdot ^1_2 T \cdot ^2_3 T \cdot ^3_4 T = \begin{bmatrix}
s\theta_2 s\theta_4 & c\theta_4 s\theta_2 & -c\theta_2 & 0 \\
-c\theta_2 s\theta_4 & -c\theta_2 c\theta_4 & -s\theta_2 & d_1 \\
-c\theta_4 & s\theta_4 & 0 & -a_4 \\
0 & 0 & 0 & 1
\end{bmatrix}$$
This matrix defines the position and orientation of the nozzle tip relative to the base frame. For inverse kinematics, we solve for joint variables given the end-effector pose. Pre-multiplying both sides by $^0_1 T^{-1}$ yields:
$$\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & d_1 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
n_x & o_x & a_x & p_x \\
n_y & o_y & a_y & p_y \\
n_z & o_z & a_z & p_z \\
0 & 0 & 0 & 1
\end{bmatrix} = ^1_4 T$$
From element (3,4), we get $p_y – d_1 = 0$, so $d_1 = p_y$. Comparing elements (1,1) and (2,2), we derive $s\theta_2 s\theta_4 = n_x$ and $-s\theta_4 = -o_z$, leading to $\theta_2 = \arccos(n_x / o_z)$ and $\theta_4 = \arccos(o_y / a_x)$. These equations enable real-time control of joint parameters to adjust spraying operations.
The workspace of the spray manipulator defines all reachable positions of the nozzle, crucial for assessing performance in crop spraying drones. Using Monte Carlo simulation with joint variable ranges (Table 2), we generated 10,000 random configurations to map the workspace. The results show that the nozzle moves within ranges of 0–86 mm along the x-axis, -203–202 mm along the y-axis, and -250––200 mm along the z-axis, relative to the base. The workspace is continuous without voids, ensuring comprehensive coverage for targeted spraying. To compute the workspace volume, we applied convex hull algorithms in MATLAB to extract boundary points, forming a 3D envelope. Triangulation of this envelope yielded a volume of 1.54 dm³. Further analysis examined the impact of structural parameters on volume, as summarized in Table 3. The link offset $d_1$ and end spray rod length $l$ positively correlate with volume, with $l$ having the most significant effect, while the shoulder link length $a_4$ shows negligible influence. Additionally, joint variable ranges affect volume; for instance, expanding the pitch joint range from -60° to 60° increases volume from 0 to 3.08 dm³, highlighting its importance in spraying UAV applications.
| Joint Name | Joint Variable | Min Value | Max Value |
|---|---|---|---|
| Prismatic Joint | $d_1$ (mm) | -200 | 200 |
| Shoulder Joint | $\theta_2$ (°) | 0 | 180 |
| Elbow Joint | $\theta_4$ (°) | 0 | 60 |
| Parameter | Effect on Workspace Volume |
|---|---|
| Link Offset $d_1$ | Positive correlation; volume increases linearly with range expansion. |
| Shoulder Link Length $a_4$ | Negligible effect; volume remains nearly constant. |
| End Spray Rod Length $l$ | Strong positive correlation; largest impact on volume. |
| Prismatic Joint Range | Steady increase in volume with expanded range. |
| Shoulder Joint Range | Rapid initial increase, then gradual; volume peaks at full range. |
| Elbow Joint Range | Significant effect; volume grows exponentially with range expansion. |
Trajectory planning ensures smooth and continuous motion of the spray manipulator during operations, minimizing vibrations and shocks. We used a seventh-order polynomial for interpolation between key points in the workspace: start point A, waypoints B, C, D, and end point E. The joint variables for these points are: $q_A = [0.2, 0, 0, 0]$, $q_B = [0.4, 0, 0, -\pi/3]$, $q_C = [0.2, -\pi/2, 0, -\pi/3]$, $q_D = [0, -\pi, 0, -\pi/3]$, $q_E = [0.2, -\pi, 0, 0]$. The polynomial is defined as:
$$\theta(t) = u_0 + u_1 t + u_2 t^2 + u_3 t^3 + u_4 t^4 + u_5 t^5 + u_6 t^6 + u_7 t^7$$
With boundary conditions for position, velocity, acceleration, and jerk set to zero at start and end times $t_0$ and $t_d$:
$$\theta(t_0) = \theta_0, \quad \dot{\theta}(t_0) = 0, \quad \ddot{\theta}(t_0) = 0, \quad \dddot{\theta}(t_0) = 0$$
$$\theta(t_d) = \theta_d, \quad \dot{\theta}(t_d) = 0, \quad \ddot{\theta}(t_d) = 0, \quad \dddot{\theta}(t_d) = 0$$
Differentiating the polynomial gives velocity, acceleration, and jerk functions. Solving these constraints, we obtain the trajectory equation:
$$\theta(t) = \theta_0 + \frac{35(\theta_d – \theta_0)}{t_d^4} t^4 – \frac{84(\theta_d – \theta_0)}{t_d^5} t^5 + \frac{70(\theta_d – \theta_0)}{t_d^6} t^6 – \frac{20(\theta_d – \theta_0)}{t_d^7} t^7$$
Simulating over 0 to 10 seconds, we plotted joint displacement, velocity, and acceleration curves. The results show smooth, continuous changes without discontinuities, confirming stable motion for precise spraying in crop spraying drones. The virtual joint (Joint 3) remains static, as expected.
For dynamic analysis, we built a virtual prototype in ADAMS. The model includes mass and material properties, with fixed constraints between the manipulator and UAV base. Joints are assigned appropriate motion pairs, and drive functions are defined using STEP5 functions for segmented motion. The prismatic joint drive is: step5(time, 0, 0, 2, 200) + step5(time, 2, 200, 4, 0) + step5(time, 4, 0, 6, -200) + step5(time, 6, -200, 8, 0). The shoulder joint drive is: step5(time, 2, 0d, 6, -180d). The elbow joint drive is: step5(time, 0, 0d, 2, 45d) + step5(time, 6, 0d, 8, -45d). A 210 N load is applied to the accompanying blades to simulate reaction forces. Simulation over 8 seconds with 1000 steps yielded dynamics data. The nozzle trajectory aligns with planned paths, and joint angle curves match drive functions. Torque analysis reveals maximum torques of 353 N·mm for the prismatic joint, 11 N·mm for the shoulder joint, and 49 N·mm for the elbow joint. The blade load most affects the prismatic joint, but all values are within acceptable limits for spraying UAV operations.
In conclusion, our design and analysis of a 3-DOF spray manipulator for crop spraying drones demonstrate significant improvements in spraying precision and coverage. The kinematic model enables accurate control of spray parameters, while workspace analysis ensures comprehensive reach. Trajectory planning with seventh-order polynomials guarantees smooth motion, and dynamics simulations validate structural integrity. This system enhances the capabilities of spraying UAVs, reducing chemical waste and environmental impact. Future work will focus on field testing and integration with real-time control systems for adaptive spraying. For additional details, refer to the following resource: nan.