The collection of Camellia oleifera flower buds, a critical process for improving the pollination and fruit set rates of this important oil-bearing tree, has not yet been mechanized. To address the inefficiency of manual collection, this research focuses on the design and dynamic analysis of a specialized robotic end effector. Based on measured mechanical property parameters of the flower buds, a rotary-twist type end effector was conceived. This study details the structural modeling of the end effector, establishes its dynamic equations coupled with the flower bud, performs a dynamic response analysis using the fourth-order Runge-Kutta method, and validates the design through field trials. The results confirm the feasibility of the proposed end effector, providing essential data and a theoretical foundation for the development and motion planning of automated camellia flower bud harvesting robots.
Introduction and Background
Camellia oleifera is a vital woody oil crop. However, its pollination faces significant challenges due to a short blooming period coinciding with unfavorable conditions for pollinating insects, leading to notoriously low fruit set. Mechanizing the collection of pollen-rich flower buds from selected high-yield cultivars, such as the ‘Sanhua’ series, is a promising strategy to overcome this bottleneck. The key to automation lies in a reliable and delicate end effector attached to a robotic manipulator. This end effector must successfully detach the bud without damage, requiring a harvesting method informed by the bud’s biomechanics. Mechanical tests on ‘Sanhua’ buds at three optimal developmental stages (bud-cracking, initial-opening, and petal-standing) revealed that the torsional moment required for detachment (average 0.0178 N·m) is significantly lower than the tensile (8.968 N) and shear forces (13.94 N). This fundamental insight directly inspired the design of a rotary-twist harvesting mechanism as the core of the proposed end effector.
Mechanical Design of the Rotary-Twist End Effector
The primary design objective for the end effector was to apply a controlled torsional force to the flower bud’s pedicel. The overall design is a hollow conical structure, as shown in the accompanying figure. The wider front end serves as the collection aperture, while the narrower rear connects to a pneumatic transport tube for conveying harvested buds to a storage unit.
The internal mechanism is the core of the end effector‘s function. It consists of six identical friction rollers arranged in a circular pattern within the conical housing. These rollers create a central passage aligned with the inlet and outlet. The conical shape of the rollers allows different sections of their surface to contact buds of varying diameters, ensuring effective gripping. To minimize damage while providing sufficient grip, the roller surface features fine spiral ribs. The kinematic design ensures all six rollers rotate synchronously in the same direction. This is achieved through a gear train driven by a DC motor. The motor shaft is connected to a central drive gear (Gear A), which meshes with six idler gears. These idler gears, in turn, mesh with six follower gears (Gears B1-B6) attached to the roller shafts. This arrangement guarantees the required uniform rotational direction for all rollers.
Modeling and Dynamic Analysis
1. Mechanical Model of the Flower Bud
To analyze the interaction dynamics, the camellia bud is modeled as an elastic sphere. Although bud geometry varies, under the clamping force of the rollers, it deforms to a near-spherical contact profile. For dynamic calculations, an equivalent sphere radius \( r \) is derived from the average length of buds across the three harvest stages. The mass \( m_f \) is the average measured mass. The moments of inertia about any axis through the center are assumed equal for the idealized sphere:
$$ J_x = J_y = J_z = \frac{2}{5} m_f r^2 $$
The values used in the model are summarized below:
| Parameter | Symbol | Value |
|---|---|---|
| Average Bud Radius | \( r \) | 41.89 mm |
| Average Bud Mass | \( m_f \) | 1.93 g |
2. Force Analysis During Harvesting
At the moment of detachment, the flower bud is in a state of force and moment equilibrium. Analyzing a pair of opposing rollers, the equilibrium equations are established. The normal forces \( F_{N1} \) and \( F_{N4} \) from the rollers have components in the x and z directions. The bud’s weight \( G \) and the suction force from the transport tube \( F_{suct} \) act in the z-direction. The friction forces arising from the rollers’ twist (\( F_{S1}, F_{S4} \)) counter the detachment resistance.
$$ \begin{cases}
F_{Nx} = F_{N1}\sin\theta + F_{N4}\sin\theta = F_{S1}\cos\alpha_1 + F_{S4}\cos\alpha_4 \\[6pt]
F_{Nz} = 6F_{N1}\cos\theta = G + F_{suct}
\end{cases} $$
Where \( \theta \) is the cone angle of the roller and \( \alpha \) is the angle of the resultant friction force. From the moment equilibrium about the bud’s center, the resisting torque \( M_f \) from the pedicel is balanced by the sum of the rollers’ inertial torques and the torque from friction:
$$ M_f = 6 \left( M_1 + r F_{S1} \sin\alpha_1 \right) $$
Where \( M_1 = J_r \beta \) is the inertial torque of a single roller, \( J_r \) is its moment of inertia, and \( \beta \) is its angular acceleration. Combining and simplifying these equations for all six rollers, the total resisting torque is derived as:
$$ M_f = 6J_r \frac{d\omega}{dt} + \frac{r}{6} (m_f g + F_{suct}) \tan\theta \tan\alpha_1 $$
Assuming the roller is a hollow cone of mass \( m_r \) with base radius \( r_r \), its moment of inertia is approximated by \( J_r = \frac{1}{2} m_r r_r^2 \).
3. Dynamic Model of the End Effector
The end effector (housing and rollers combined) is modeled as a rigid hollow conical body of mass \( m_2 \). Its interaction with the flower bud during the instantaneous detachment event is modeled via a spring-damper system in the horizontal direction, representing the structural elasticity and damping of the system. The kinetic energy \( T \), potential energy \( U \), and dissipated energy \( W \) of the end effector are:
$$ T = \frac{1}{2}m_2 \dot{x}^2 + \frac{1}{2}J_2 \dot{\theta}^2, \quad U = \frac{1}{2}k_2 x^2, \quad W = b_2 \dot{x} x $$
Here, \( x \) and \( \theta \) are the horizontal displacement and angular displacement of the end effector, \( k_2 \) is the equivalent stiffness, \( b_2 \) is the damping coefficient, and \( J_2 \) is the end effector‘s moment of inertia. The work done by the interaction force \( F_2(t) \) (derived from \( M_f \)) is \( F_2(t) \cdot x \).
Applying the principle of conservation of energy, \( T + U + W = F_2(t) \cdot x \), and differentiating with respect to time, leads to the equation of motion for the end effector. Substituting \( x = R\theta \) where \( R \) is the effective radius, and simplifying with the assumption \( R \approx r_{hole} \) (inner radius), the final dynamic equation is obtained:
$$ \frac{3}{2} m_2 \ddot{x} + b_2 \dot{x} + k_2 x = F_2(t) $$
The interaction force \( F_2(t) \) is:
$$ F_2(t) = \frac{3 m_r r_r^2}{R} \frac{d\omega}{dt} + \frac{r}{6R}(m_f g + F_{suct}) \tan\theta \tan\alpha_1 $$
The parameters for the dynamic model are listed in the following table:
| Parameter | Symbol | Value / Range |
|---|---|---|
| End-Effector Mass | \( m_2 \) | 1 kg |
| Damping Coefficient | \( b_2 \) | 0.2 – 0.4 N·s/m |
| Stiffness Coefficient | \( k_2 \) | 2000 – 4000 N/m |
4. Dynamic Response Simulation via Runge-Kutta Method
The second-order ordinary differential equation (ODE) governing the end effector‘s motion is solved numerically to analyze its dynamic response to the harvesting impulse. The fourth-order Runge-Kutta method, a powerful and standard numerical technique for solving ODEs, is employed for this purpose. The method approximates the solution at the next time step using a weighted average of four increments, providing excellent accuracy and stability for dynamic systems.
The general formulation for a first-order ODE \( \frac{dy}{dt} = f(t, y) \) is:
$$ y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$
where \( h \) is the step size, and
$$ \begin{aligned}
k_1 &= f(t_n, y_n), \\
k_2 &= f(t_n + \frac{h}{2}, y_n + \frac{h}{2}k_1), \\
k_3 &= f(t_n + \frac{h}{2}, y_n + \frac{h}{2}k_2), \\
k_4 &= f(t_n + h, y_n + h k_3).
\end{aligned} $$
To apply this to the second-order equation of motion, it is first decomposed into two coupled first-order equations in terms of displacement \( x \) and velocity \( v \):
$$ \begin{cases}
\dot{x} = v \\[6pt]
\dot{v} = \displaystyle \frac{F_2(t) – b_2 v – k_2 x}{1.5 m_2}
\end{cases} $$
These equations are then integrated simultaneously using the Runge-Kutta routine. The maximum expected displacement \( x_{max} \) is estimated from the peak interaction force: \( x_{max} = F_2(t)_{peak} / k_2 \approx 7 \text{ mm} \). This value and the parameters from the table are used as inputs for the simulation.
Simulation Results and Field Validation
The Runge-Kutta simulation provided the time-domain response of the end effector—displacement, velocity, and acceleration. The results showed that the displacement amplitude was extremely small, on the order of \( 10^{-5} \) mm, and rapidly attenuated. The velocity and acceleration profiles followed a similar, highly damped trend. This indicates that the end effector exhibits minimal vibration or jerk in response to the detachment force impulse. The dynamic stability is crucial for maintaining positional accuracy if the end effector is integrated with vision-based targeting systems, as it implies the harvesting action does not significantly perturb the system’s pose.
To validate the design and simulation findings, a prototype of the rotary-twist end effector was constructed and tested in a camellia orchard. The field trials involved multiple harvesting cycles on different trees. The end effector successfully performed both single-bud and cluster (2-3 buds) harvesting operations. Key performance metrics were recorded: the average time for single-bud collection was 1.57 seconds per bud, and for cluster collection was 6.08 seconds per cluster. The harvesting success rate (clean removal without significant damage) was approximately 90% for both modes. Critically, no significant shaking or instability of the end effector was observed during operation, corroborating the dynamic stability predicted by the Runge-Kutta simulation.
Conclusion
This study presented a comprehensive design and analysis of a novel rotary-twist end effector for automated camellia flower bud harvesting. The mechanical design was directly driven by the measured biomechanical properties of the bud, specifically its low torsional detachment moment. A dynamic model coupling the end effector and the flower bud was established, leading to a second-order differential equation of motion. The fourth-order Runge-Kutta method was effectively applied to solve this equation and simulate the dynamic response. The simulation predicted minimal displacement and high damping, indicating inherent stability. These theoretical findings were strongly supported by field trials, where the prototype end effector demonstrated effective and stable harvesting performance. This work provides a validated theoretical framework and a practical design for a camellia harvesting end effector, forming a critical component for the future development of fully automated pollination systems for Camellia oleifera. The methodology combining biomechanics-informed design, dynamic modeling, numerical simulation, and field validation serves as a robust template for the development of other delicate agricultural robotic end effectors.