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Optimizing basket transplanting parameters using the mechanical properties of tomato plug seedlings

The collision between tomato plug seedlings and the planting device during basket transplanter function causes the seedlings’ root lump to lose some mass. To resolve this problem, we establish a contact mechanics model using Hertzian contact theoretical and Newtonian methods, enabling us to formally derive the maximum deformation of the root lump. The equation of motion describing the collision between the tomato plug seedling and planting device is constructed using the principles of impulse and momentum. Then, we identify the optimization function of contact angle θ—which consists of the maximum deformation of the root lump, the angular velocity of the tomato plug seedling, and the absolute velocity of the root lump—via multi-objective methods. When the angular velocity of the planting device is 1.05 rad/s or 1.15 rad/s, the optimal contact angle is 67°. Further, we examine the velocity of the tomato plug seedling after collision. A small plant height reduces the root lump’s mass loss; we found that 12-cm-tall tomato plug seedlings lose less root lump mass than taller seedlings. Finally, for angular velocities of 1.05 rad/s and 1.15 rad/s, we carry out of tomato plug seedling crush tests with the optimized parameters, achieving minimum root lump mass losses of 8.76% and 7.43%, respectively.


Introduction
The basket transplanters can form holes and cast seedlings at one time and thus are applied into the transplanting of tomato plug seedlings [1]. To improve their efficiency, basket transplanters generally have auxiliary seedling transplanting machines [2]. Workers place tomato plug seedlings into the cast seedling cup of the feed device. When a tomato plug seedling falls freely from the cast seedling cup, its root lump collides with the inner wall of the planting device. This motion results in loss of the root lump's mass and decreases the plug's survival rate.
Under a theoretical framework, Song et al [3] revealed the relationship between rice seedlings' movement and distance when the plugs are turned into a guide. Zhang et al [4] adopted a high-speed video camera to observe the collision between the rice plug seedling and the guide device; concluding that the rice plug seedling slides along the guide device. Wang et al [5] studied the breakage rate of broccoli seedlings during mechanized planting. Orthogonal experiments based on agronomic requirements have been carried out to obtain optimal parameters. Chen et al [6] developed a transplanting mechanism with planetary deformed elliptic gears for plug seedling transplant and built  10.1088/1757-899X/657/1/012046 2 differential equations for the collision of broccoli plug seedlings and the planting derive. In all the studies discussed above, the collision between the root lump and planting device is plastic, the plug seedling slides along the wall of the planting device, and the velocity of the plug seedlings was neglected during the collision between the stem and planting device In this paper, we focus on the collision between tomato plug seedlings and the planting device of a basket transplanter. The contact model and collision model of tomato plug seedlings are established, the relationships of the working parameters of transplanters with the deformation of root lumps and the movement speed of tomato plug seedlings are studied, and the working parameters are optimized to reduce the quality loss of root lumps.

Collision analysis
We assume uniform rotation of the planting device about a fixed axis in a plane perpendicular to this axis. When cast seedlings cup put tomato seedling plug, the attitude of the seedling (its verticality or slope) is a stochastic process. Two common sets of contacts are collisions between the root lump and planting device, and between the stem and the edge of the planting device.

Mechanics model
We divide our analysis into two branches: a contact mechanics analysis and a collision analysis. Contact deals with the instant impact between a root lump and the planting device-that is, those that deform the surface-whereas collision addresses the velocity from just before to just after the collision.

Contact mechanics analysis
When the root lump collides with the planting device, deformation of the root lump occur. These deformations are classified as contact mechanics due to the instantaneous impact [7]. The colliding parts of the root lump can be compared to the face of sphere, and those of the planting device can be considered to be a surface. Thus, the instantaneous collision model can be simplified as describing contact between a flexible sphere and a rigid plane.
According to Hertzian contact theoretical analysis [7], the normal force F can be described as When the equation of motion is applied to each body of the system (the root lump and planting device), similar equations are derived by Newtonian methods.
Then, equation (3) can be solved: Substituting equation (4) into equation (1) gives Thus, integrating the resulting equation (5) When the slope of deformation of the root lump is zero, the maximum compression of the root lump max is  In general, there are two types of impact. The motion in figure 2A is caused by the collision between the root lump and the sloped wall of the planting device. The motion in figure 2B occurs due to the collision between the root lump and the vertical wall of the planting device. Global inertial coordinates XOY are considered. The symbols in figure 2 are listed in table 1.   Just before the root lump strikes the planting device, its angular velocity 1 and velocity uX in the X direction is zero. Thus, the Y direction with a velocity of uY is Neglecting air resistance, the planting device is subjected to a constant angular velocity about the axis of rotation. Then, the velocity uE of the planting device is Also neglecting the lump-device frictional force in the collision and applying the relative velocity equation to the collision points [8], the relative velocity uN1 of the root lump to the planting device in the collision direction is If the root lump crashes into the vertical wall of the planting device, angle  between the inclined wall of planting device and the vertical plane is set to 0. Thus, equation (10) is reduced to The relative velocity uN1 and uN2 are shown as functions of the planting device's angular velocity  and contact angle . The structural parameters of the planting device and tomato plug seedling are given in table 2 [9]. Then, three-dimensional representations of the solutions given by both equations (10) and (11) can be obtained. As shown in figure 3, velocity uN1 is higher than velocity uN2. Thus, attention is focused on the collision between the root lump and the sloped wall of the planting device.
From equation (7), the maximum deformation max of the root lump depends on the relative velocity before collision. The influence of the interaction between angular velocity  of the planting device and contact angle  on the maximum deformation max of the root lump is plotted in figure 4. From the curved surface diagram, when angular velocity  is fixed at a low speed, maximum deformation max gradually decreases with increasing contact angle . When angular velocity  is fixed at a high velocity, the trend in maximum deformation max has the opposite effect on contact angle . When contact angle  is fixed at a low value, the change in maximum deformation max with angular velocity  is at approximately one level. However, when the contact angle  was fixed at a large angle, the maximum deformation max gradually increased with the angular velocity . Therefore, contact angle  is the major factor influencing the deformation of the root lump.

Collision motion analysis
2.4.1. Collision between the root lump and planting device. To describe the characteristics of this collision, we assume a dumbbell shape for the distribution of the tomato seedling plug's structure [10]. The link mass between balls A and B is negligible. Figure 5 shows an example of such a body. The origin of the X and Y coordinates is located at the ground O. The origin of the relative X and Y coordinates has been placed at collision point O. The symbols in figure 5 are listed in table 1. Using impulse and momentum [11], it is possible to write three equations that define the particle's motion: The first two of these equations represent the linear impulse in the X and Y directions, and the third equation represents the impulse momentum about point C of the center of mass. Neglecting fractional terms, impulse IX in the X directions is zero. The X and Y components of the velocity are indicated in figure 5; then, Since the tomato plug seedling is in translation just before collision, angular velocity 1 = 0.
equation (14) can be written as   To determine the relation between the velocities of points A and C, the absolute velocity can be determined: The ratio of the restitution impulse to the deformation impulse is called the coefficient of restitution k1 [11]. From the above equations, this value for ball A can be expressed in terms of the particles' initial and final velocities as The absolute velocity A u of ball A has a magnitude that is found from In the studies discussed above, the planar movement equation of the tomato plug seedling was obtained. The relative parameters in the above equations are given in tables 2 and 3 [6]. Given equation (27), the effects of the angular velocity  of the planting device and contact angle  on the angular velocity 2 of the tomato plug seedling can be examined in figure 6. When the angular velocity  of the planting device is fixed, the angular velocity 2 of the tomato plug seedling decreases as contact angle  increases. When contact angle  is fixed at a low value, angular velocity 2 increases with the rotation velocity ; however, when the contact angle  is fixed at a high value, there is a single decreasing curve. Thus, the angular velocity 2 of the tomato seedling decreases from -5.78 rad/s to -2.04 rad/s or from -5.89 rad/s to -1.93rad/s when the angular velocity  of the planting device is 1.05 rad/s and 1.15 rad/s, respectively.     10 the angular velocity  of the planting device is fixed, the absolute velocity uʹA decreases as contact angle  increases. If the contact angle  is a small constant, absolute velocity uʹA is on the order of angular velocity . However, when contact angle  is fixed at a high value, absolute velocity uʹA increases asymptotically with angular velocity . When the angular velocity  is 1.05 rad/s, the absolute velocity uʹA decreases from 2.87 m/s to 2.19 m/s, and when the angular velocity  is 1.15 rad/s, the absolute velocity uʹA decreases from 2.88 m/s to 2.18 m/s.

2.4.2.
Collision between the stem and planting device. The free-body diagram of the inclined tomato plug seedling shown in figure 8 indicates that during the brief time from just before to just after the collision, the weights of the stem and resistance force have been ignored. The symbols in figure 8 are listed in table 1. According to both impulse and momentum theories, impulse IXʹ in the X direction is zero because fractional terms are ignored. Then

Substitution of equation (30) into equation (13) leads to
Because the tomato plug seedling is not rotating just before collision, angular velocity 1 = 0.
Equation (14) can be rewritten as To study the relation between the velocities of points D and C, the absolute velocity after the collision can be determined: From absolute motion analysis, the Y projections of absolute velocity vʹDYʹ can be written as From the definition of the coefficient of restitution, k2 for particle D can be expressed in terms of the particles' initial and final velocities as Equations (31), (32), (34), and (35) can be solved simultaneously to obtain the relative velocity vʹCYʹ of the center of mass and angular velocity 3 of the tomato plug seedling just after collision: According to planar kinematics analysis, the X and Y components of the velocity of ball A are The absolute velocity A v of ball A has a magnitude that is found from We now discuss the collision of the stem, whose physical structure is defined in tables 2 and 3. From equation (37), we can plot the data as shown in figure 9, allowing us to examine the effects of the angular velocity  of the planting device and contact angle  on the rotation velocity 3 of the tomato seedling plug. When angular velocity  is fixed, angular velocity 3 decreases as contact angle  increases. When contact angle  is fixed at a low value, rotation velocity 3 increases with angular velocity ; however, when contact angle  is fixed at a high value, the reverse occurs. Thus, the angular velocity of a tomato seedling 3 decreases from -8.27 rad/s to -1.96 rad/s or from -8.56 rad/s to -1.89 rad/s when the angular velocity  of the planting device is 1.05 rad/s or 1.15 rad/s, respectively.
Using equation (40), we can show the influence of the interaction between angular velocity  of the planting device and contact angle  on the absolute velocity vʹA of the root lump in figure 10. When angular velocity  of the planting device is fixed, absolute velocity vʹA decreases as contact angle  increases; when contact angle  is fixed, absolute velocity vʹA decreases in roughly inverse proportion with angular velocity  of the planting device. These results reveal that when angular velocity  is 1.05 rad/s, absolute velocity vʹA decreases from 2.67 m/s to 2.17 m/s, and when angular velocity  is 1.15 rad/s, absolute velocity vʹA decreases from 2.65 m/s to 2.19 m/s.    We next discuss the effects of distance b from collision point D to the center A of the root lump mass and incline angle  on the absolute velocity vʹA of ball A in figure 12. When b is fixed, absolute velocity vʹA decreases as incline angle  increases. When incline angle  is fixed at a low value, absolute velocity vʹA remains the same. However, if incline angle  is fixed at a large angle, absolute velocity vʹA is concave. Thus, when distance b is equal to d, absolute velocity vʹA is effectively at its minimum.

Optimization of working parameters
Decreasing the velocity of the tomato plug seedling during the collision and the deformation of the root lump can reduce the root lump mass loss [12]. Thus, the maximum deformation max, angular velocities 2 and 3 of the tomato seedling plug, and absolute velocities uʹA and vʹA of the root lump should be decreased. However, when the angular velocity  of the planting device equal 1.05 rad/s or 1.15 rad/s in previous research of contact and collision analysis, the contact angle  is decreased to  (7), (27), (28), (37), and (40)-are combined to construct objective function f(). Thus, the optimization function is:   The curve in figure 13 is established by solving equation (41), specific parameters of which are listed in tables 2 and 3. Figure 13 reveals the different optimizing function curves of contact angle . When the angular velocity  of the planting device is 1.05 rad/s or 1.15 rad/s, f() is minimized and the contact angles are 68.1° and 66.5°, respectively. Hence, the optimal contact angle is 67° because the manual planting frequency was changed from 50 to 55 plants/minute. According to figures 11 and 12, as the distance between collision point D and the center of mass C decrease, the angular velocity 3 of the tomato plug seedling and the velocity vʹA of the root lump also decreases. If collision point D is located below center of mass C, angular velocity 3 is positive and is identical to the direction of 3 in figure 8. The results indicated that the root lump failed to enter the planting device due to its rotation. Reducing the plant height of tomato plug seedlings decreases the distance between the center of mass and the root lump, which can reduce the occurrence of collision phenomena between the planting device and the bottom of the center of mass of the tomato plug seedling. According to agronomic requirement, the general plant height range is 12-19 cm [14]. Then, based on the studies discussed above, 12-cm-tall tomato plug seedlings have an advantage.

Experimental verification test
The experiments described here were designed to confirm the optimized working parameters in crush resistance experiments. In this experiment, the plug seedling tray has 128 cells, the tomato seedlings were 45 days old and 12-19 cm in height. The transplant mechanism experimental bed is shown in figure 14, and includes the planting device, feeding device, cast seedling cup. Figure 14. The transplanting mechanism test bench. 1) Planting device; 2) Cast seedling cup; 3) Feeding device. Table 5 provides a set of direct test parameters. Conditions 1 and 5 is the optimum work parameters through optimizing analysis. The conditions were chosen by varying the contact angle and plant height. In this experiment, 200 tomato plug seedlings were fed in succession, and each condition was repeated three times.