Determining the minimal mulch film damage caused by the up-film transplanter

Abstract

To satisfy the increasing demand for improving productivity while reducing pollution, this article proposes a method to determine the minimal mulch film damage in mechanical transplantation operations, which can be used to guide the design of all the up-film transplanters. Specifically, the whole working process of digging the hole for a typical dibble-type transplanter working in Xinjiang province of China is analyzed, in which the dibble motion serves to illustrate the effectiveness of the proposed method. Both the ideal condition and the realistic environmental force-involved condition are taken into consideration during analytically modeling the process of the mulch film damage. Based on the presented two-layer soil-layered model and basic motion principles, a series of equations is derived to describe the typical dibble-type transplanter motion involved gravity and soil counter-force. Furthermore, the derived relations between working parameters and structural parameters of the transplanter guide the search of a minimum value of the mulch film damage. Finally, simulation results verify the effectiveness of the presented theoretical calculation. More remarkably, the dibble motion in the preliminary field trial is similar to that in the simulations. The obtained results provide a new way to deal with the optimal design and operations of novel up-film transplanters.

Introduction

In recent years, advanced mechatronic systems and robotic technologies are increasingly being used in primary industries such as agriculture, forestry, and fisheries. Practical applications of advanced technologies are contributing to the enhancement of the quality of primary industries, typically involving improved productivity, cost-effective management, and reduced pollution. In the agriculture sector, the seedling transplanting is an efficient means to increase the yield for it to offer a shorter growing season, higher survival rate, and stronger lodging resistance. ¹ However, the seedling transplanting is a heavy physical labor, which hinders the promotion. A transplanter is an agricultural machine used for transplanting seedlings into the field, which greatly facilitates farming with increased productivity. Many countries have shown a keen interest in the development of a variety of mechanical transplanters,^2,3 ranging from semi-automatic ⁴ to automatic.^5–10 At present, mechanical transplants are quite qualified for seeding tomatoes, strawberries, nursery stock, and so on.

It is generally thought that no universal transplanter exists. The mechanical structures of the transplanters differ greatly since there are many means to achieve the goal. For example, a planetary gear train with elliptical gears and an incomplete non-circular gear are used for rice plug seedlings, ¹¹ while a planetary five-bar structure of transplanting mechanism is designed to transplant seedlings on mulch film. ¹² Besides, there are many differences between the agriculture machine and the in-door machine, such as the working condition, working time, and manufacturing cost. The transplanter is usually used intensively in the working season and stored in the farm slack season. It implies that the transplanter should be reliable, easy-maintenance, and low-cost. In addition, another important reason for this non-generality is that the design of a transplanter tightly depends on various field conditions. For instance, the rice transplanter is extensively used in paddy fields,^13–15 but not suitable for dry-land transplanting.

Although there are many kinds of transplanter which can be used for dry-land transplanting,^16,17 what attract us most to the up-film transplanter is the ability to save water resources. For instance, in Xinjiang, China, like other dry areas, the plastic mulch film is wildly used for water saving, which makes the up-film transplanter the best choice for transplanting. However, there is a problem with the transplanting operations, in which the up-film transplanter may cause undesirable damage to plastic mulch film. In reality, large plastic mulch film damage generally influences the effect of the mulch film in certain aspects such as soil insulation, precise management of soil moisture, and weed control. ¹⁸ Although a parameter combination for the particular transplanter achieving satisfied degree of the mulch film damage may be figured out through experimental research, a universal method is still missing to determine the structural parameter of a transplanter with the minimal mulch film damage.

In this article, the objective is to numerically model and analyze the mulch film damage of a typical dibble-type up-film transplanter. With this model, the relationship between the structural features of the transplanter and the mulch film damage can be established via quantitative analysis with different conditions. Both simulations and tests are offered to show the effectiveness of the proposed method to determine the minimal mulch film damage. The novelty and contribution of this article lies in three aspects: (1) a strategy for dealing with the mulch film damage is proposed, which can be adapted to all kinds of up-film transplanters; (2) a two-layer soil-layered model is developed to show the effect of environmental force quantitatively; and (3) the quantitative relations among working parameters, mechanical parameters, and the value of the minimal mulch film damage are determined. The kinematic model describing the dibble’s motions is derived and verified. In particular, both the ideal condition and the environmental force-involved condition are comparatively discussed. The obtained results shed light on the optimal design and operations of novel up-film transplanters.

The remainder of the article is organized as follows. The proposed strategy is briefly introduced in section “Strategy for dealing with the mulch film damage.” A quantitative relationship between the machine features of a typical dibble-type transplanter and the minimal mulch film damage under ideal condition is built in section “Minimal mulch film damage in ideal condition.” The analysis of dibble motion in conjunction with the influence of environmental force is detailed in section “Minimal mulch film damage considering the environmental force.” Results from simulations and field tests are provided in section “Results and analysis.” Finally, section “Conclusion” concludes the article with an outline of future work.

Strategy for dealing with the mulch film damage

To obtain the minimal mulch film damage, the whole working process of digging the hole is studied to figure out how the mulch film damage is formed. The process of manual transplanting work is illustrated in Figure 1. As for a typical working process, the farmer first digs a hole, transplants the seedling, moves ahead, and repeats the previous actions. In such a way, the seedling suffers no impact and there is no extra mulch film damage. The whole transplanting process, in general, can be divided into two basic motions: one is the intermittent motion in the vertical direction, drilling holes; the other is the horizontal movement, moving forward. The trajectory of the dibble motion is shown in Figure 2, where h denotes the planting depth, and S indicates the planting distance. Ideally, the up-film transplanter should exactly execute human-like actions without extra mulch film damage. However, the working performance would be inefficient for the frequent starting and braking during the operations. To overcome this problem, the trajectory needs some change. Because the dibble makes no damage to the mulch film while they are not contacted, the transplanting process can be described below. When the transplanter moves forward uniformly, the dibble drills intermittently and has a trajectory where the beneath-film part is approximate to a straight line perpendicular to the ground. By this way, both the efficiency and the zero extra mulch film damage can be satisfied. The trajectory of the dibble motion can be plotted as Figure 3.

Figure 1.

Process of manual transplanting work: (a) non-work status of soil, (b) dig a hole, (c) put the seedling into the hole, and (d) cover soil on the seedling.

Figure 2.

The ideal dibble motion trajectory causing zero extra mulch film damage.

Figure 3.

A possible trajectory of the dibble motion.

Although the portrayed motion seems easier to achieve, it is unrealistic in the context of mechanical engineering. The trajectory in Figure 3 indicates that the dibble should perform a round-trip rectilinear motion in a very short time, corresponding to very high acceleration. The acceleration of the dibble not only imposes a burden to the transplanter but also possibly impacts the seedlings. To reduce the acceleration, the restraints on the dibble motion are relaxed. Thus, the whole motion can be simplified as below: when the transplanter moves forward uniformly, the dibble performs the intermittent drilling motion and intends to obtain the minimal plastic mulch film damage. The possible trajectories shown in Figure 4 are more feasible. Intuitively, different damage degrees are relevant to different trajectories. To verify this hypothesis, the minimal mulch film damage will be defined and the direct relationship will be developed in the following section.

Figure 4.

Possible trajectories of the dibble motion.

Minimal mulch film damage in ideal condition

Introduction of a typical dibble-type transplanter

Before going into the relation between structural parameters of the transplanter and the mulch film damage, a specific object should be chosen to begin the study. To date, a variety of mechanical structures of up-film transplanter have been developed.^12,19,20 For the convenience of the real-world applications, a kind of up-film transplanter which is widely used in Xinjiang province of China is selected, whose structure diagram is depicted in Figure 5. More specifically, the core mechanism of the dibble-type transplanter is the parallel four-bar mechanism. The eccentric disk, the driving disk and the square shaft act as the bars. They ensure that the dibble is vertical to the ground so as to avoid impact of the seedling during the transplanting. When the transplanter moves forward uniformly, the rotated driving disk and the eccentric disk drive the dibble to implement the desired intermittent digging.

Figure 5.

Illustration of mechanical structure of a dibble-type transplanter.

Mathematic model of the dibble motion

To model the dibble motion mathematically, the Cartesian coordinate system is established with the forward direction as the positive direction of x axis and the vertical direction upward to the planting ground as the positive direction of y axis. In particular, one of the lowest points of the dibble trajectory is set as the original point. Further let the lowest point of a dibble be the reference point, the dibble motion can then be mathematically described as follows

$$\{\begin{array}{c}x=v_{0}t-R\sin \omega t\hfill \\ \multicolumn{1}{c}{y=R-R\cos \omega t}\end{array}$$

(1)

Mulch film damage determined by the dibble motion

To simplify the problem of determination of mulch film damage, the dibble is considered as a shapeless line always perpendicular to the ground, so that the mulch film damage degree can be calculated by the distance of the dibble movement. Figure 6 shows trajectories of the dibble with different transplanting speed ratios $(r)$ . As can be observed, when the value of r increases, the mulch film damage first decreases to the minimum and then increases. Hence, the reasonably minimal damage will only occur in the case of $r<1$ .

Figure 6.

Trajectories of the dibble under different conditions.

Specifically, when to discuss the condition of $r<1$ , the whole plastic mulch film damage can be divided into two parts. One part is caused by the dibble movement on the film, called on-film damage, represented by $w_1$ in Figure 7. The other part is caused by the dibble movement beneath the film, called beneath-film damage, represented by $w_2$ in Figure 7. Both parts contribute to the whole damage, but the larger one does more. Based on this fact, it can be inferred that the whole damage is equal to the larger one. In particular, the whole damage will reach its minimum when both parts have equal sizes (Figure 8).

Figure 7.

The mulch film damage relevant to the dibble motion: (a) $w_1<w_2$ , (b) $w_1=w_2$ , and (c) $w_1>w_2$ .

Figure 8.

Calculation of the mulch film damage.

Given the definitions of $w_1$ , $w_2$ , and the symmetry of the dibble trajectory, both values can be estimated. Set $y=0$ , and substitute it into equation (1), equation (2) is yielded as follows

$$t=\frac{1}{\omega }{\cos }^{-1}\frac{R-h}{R}$$

(2)

The half damage on the horizontal line of h can be calculated as

$$x_1=R(r{\cos }^{-1}k-\sqrt{1-k^2})$$

(3)

where

$$k=\frac{R-h}{R}$$

(4)

$$r=\frac{v_0}{\omega R}$$

(5)

Taking account of the symmetry of motion trajectory, we have $w_1=2\left|x_1\right|$ . However, it is hard to calculate the value of the beneath-film damage, which cannot be obtained at a fixed level. Considering both the trajectory described by the function $x=f(y)$ and the definition of the beneath-film damage, the problem seems much easier. Because of $t\ge 0$ , $x_2\le 0$ . The value of y making x reach its extremum can be calculated by taking a derivative with respect to y. It follows

$$\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{v_0-\omega R\cos \omega t}{\omega R\sin \omega t}$$

(6)

Set equation (6) equal to 0, the satisfied point is obtained

$$v_0=\omega R\cos \omega t$$

(7)

Substituting equation (7) into equation (1), it yields that when $y=R-v_0/\omega$ , $x_2$ has an extreme value

$$x_2=R(r{\cos }^{-1}r-\sqrt{1-r^2})$$

(8)

Furthermore, a combination of equations (3) and (8) leads to the estimation of the minimal mulch film damage. By observing the sign difference, we have

$$\begin{array}{c}{x}_1=-x_2\hfill \\ \mathrm{or}\hfill \\ r{\cos }^{-1}k-\sqrt{1-k^2}=\sqrt{1-r^2}-r{\cos }^{-1}r\hfill \end{array}$$

(9)

which means that the value of the minimal mulch film damage is equal to $2R|r{\cos }^{-1}r-\sqrt{1-r^2}|$ when equation (9) is established.

Because r has a corresponding value for every given value of k, equation (9) can further be simplified. A curve fitting of k and r is made using the fzero function and cftool command provided by MATLAB, giving an approximately quantitative relation between r and k as equation (10). The curve fitting result is shown in Figure 9.

$$r=0.2744k+0.73$$

(10)

Figure 9.

Illustration of the curve fitting between r and k.

Substituting equations (4) and (5) into equation (10), the following relation can be derived

$$v_0=\omega (1.0044R-0.2744h)$$

(11)

In reality, equation (11) offers a numerical relationship between $v_0$ , R, h, and $\omega$ , which can be used to calculate the minimal mulch film damage equal to $2R|r{\cos }^{-1}r-\sqrt{1-r^2}|$ .

Minimal mulch film damage considering the environmental force

Introduction of the environmental force

All the above discussions are based on a simple fact ignoring the gravity and other forces, failing to accurately reflect the real situation. As a matter of fact, digging deeper needs larger force. Therefore, if the gravity of the transplanter remains constant, the machine would always balance at a certain depth level.

Based on the above analysis, the soil model in the simulation can be divided into two parts: one part, called soft layer, is between the mulch film and the balance plane; the other part, called solid layer, is beneath the balance plane. The soft layer has no impact on the dibble’s motion trajectory, only reflecting the transplanting depth. However, the solid layer is so hard to penetrate that the dibble can only move on the balance plane. According to the simulation result shown in Figure 10, the trajectory of the dibble has changed dramatically with the environmental force involved. Hence, it is necessary that some adjustments should be made to assess the minimal mulch film damage theory. Specifically, the mathematic model of the new dibble motion law is first derived. Second, the mulch film damage is combined with the model. At last, the minimal mulch film damage is defined and calculated.

Figure 10.

A simulation result involved with the environmental force.

Analysis of the component of the dibble motion

To highlight the quantitative relationship between the structural parameters and working parameters, Figure 11 gives a simplified schematic diagram of transplanting mechanism. Likewise, the whole mulch film damage is divided into two parts. Because of the existing force, there must be always one dibble or two dibbles contacting with the balance plane to counterbalance the gravity. This implies that the on-film damage caused by the middle dibble has begun with the left dibble contacting with the solid layer, then finished with the right dibble balancing the force. The beneath-film damage is calculated by the distance that the chosen dibble moving on the balance plane. A similar strategy of decomposing the whole dibble motion into horizontal and circular movements is still working. The dibble circular movement involved with the force will be analyzed below, for the reason that the force merely influences the circular movement.

Figure 11.

The simplified kinematic model of the transplanting mechanism: (a) an arbitrary position of the model and (b) a special position of the model.

For convenience, the balance plane maintains consistent with the no-force condition, as shown in Figure 11. Then, trigonometric equations can be derived as follows

$$l_1\sin {\theta }_0+l_2+l_3=\mathrm{OE}=l_1\sin \theta +l_2\sin \beta +l_3\sin \alpha$$

(12)

$$\angle \mathrm{CAB}=\frac{\pi }{2}-\alpha +\beta$$

(13)

Assuming that the rod AE clockwisely rotates an angle of $\mathrm{\Delta }\theta$ (in degree) around the point E over a time interval $\mathrm{\Delta }t$ . It means $\theta$ has a decrease of $\mathrm{\Delta }\theta$ . Meanwhile, both rods AB and BE clockwisely rotate $\mathrm{\Delta }\theta$ around the point E, thus the angle between the rod AB and the horizontal direction is reduced by $\mathrm{\Delta }\theta$ . Over the same time interval, the rod AC clockwisely rotates $\omega \mathrm{\Delta }t$ around the point A. Because the rod AB keeps parallel to the rod CD, while the rod CD keeps perpendicular to the rod CG, the angle between the rod CG and the horizontal direction increases by $\mathrm{\Delta }\theta$ . The following equations can be easily obtained

$$\mathrm{\Delta }\angle \mathrm{CAB}=\omega \mathrm{\Delta }t$$

(14)

$$\mathrm{\Delta }\alpha =-\mathrm{\Delta }\theta$$

(15)

$$\omega \mathrm{\Delta }t=\mathrm{\Delta }\beta -\mathrm{\Delta }\alpha =\mathrm{\Delta }\beta +\mathrm{\Delta }\theta$$

(16)

The relations between the angle velocities of these rods can further be expressed as equations (17) and (18)

$$\frac{d\angle \mathrm{CAB}}{\mathrm{dt}}=\frac{d\beta }{\mathrm{dt}}-\frac{d\alpha }{\mathrm{dt}}$$

(17)

$$\frac{d\alpha }{\mathrm{dt}}=-\frac{d\theta }{\mathrm{dt}}$$

(18)

According to equations (17) and (18), the final solution is given by

$$\omega =\frac{d\beta }{\mathrm{dt}}-\frac{d\alpha }{\mathrm{dt}}=\frac{d\beta }{\mathrm{dt}}+\frac{d\theta }{\mathrm{dt}}$$

(19)

By taking a derivative with respect to t, equation (20) can be deduced from equation (12)

$$l_1\frac{d\theta }{\mathrm{dt}}\cos \theta +l_2\frac{d\beta }{\mathrm{dt}}\cos \beta +l_3\frac{d\alpha }{\mathrm{dt}}\cos \alpha =0$$

(20)

Substituting equation (19) into equation (20), we obtain the following equation

$$\frac{d\theta }{\mathrm{dt}}=\frac{\omega l_2\cos \beta }{-l_1\cos \theta +l_2\cos \beta +l_3\cos \alpha }$$

(21)

In practice, equation (21) is hard to use for the reason that $\theta$ , $\alpha$ , and $\beta$ are time-varying. To solve this problem, we need to get an initial state of the simplified transplanting mechanism. We focus on two special balance positions determining the range of $\theta$ depicted in Figure 12. Specifically, Figure 12(a) gives the position relative to the minimal $\theta$ , whereas Figure 12(b) shows the position relative to the maximal $\theta$ . When the rods AC and CG are both perpendicular to the horizontal plane, $\theta$ will reach its minimal value ${\theta }_0$ with $\alpha =\pi /2$ , $\beta =\pi /2$ , and zero angular velocity. When the point A reaches its lowest place, $\theta$ attains a maximum value. At this moment, two dibbles contact with the balance plane coincidentally, indicating that the line $C\prime C$ is parallel to the x axis. Thereby, we have

$$\{\begin{array}{c}\beta =\frac{\pi }{4}\hfill \\ \alpha =\frac{\pi }{2}-\theta +{\theta }_0=\frac{\pi }{2}-{\theta }_{max}+{\theta }_0\hfill \end{array}$$

(22)

$$A=B\sin \theta +C\cos \theta =\sqrt{B^2+C^2}\sin (\theta +\phi )$$

(23)

Figure 12.

Two special positions of the transplanting mechanism: (a) $\theta$ reaches its minimum ${\theta }_0$ and (b) $\theta$ reaches its maximum ${\theta }_{\max }$ .

Since equation (23) is well known, ²¹ substituting equation (22) into equation (12) yields

$${\theta }_{max}={\sin }^{-1}\frac{A}{\sqrt{B^2+C^2}}-\phi$$

(24)

where

$$\{\begin{array}{c}A=l_1\sin {\theta }_0+(1-\frac{\sqrt{2}}{2})l_2+l_3\hfill \\ B=l_1+l_3\sin {\theta }_0\hfill \\ C=l_3\cos {\theta }_0\hfill \\ \tan \phi =\frac{C}{B}\hfill \end{array}$$

(25)

$$\frac{d\theta }{\mathrm{dt}}=\{\begin{array}{c}\begin{array}{cc}\multicolumn{1}{c}{\frac{-\sqrt{2}\omega l_2}{2l_1\cos {\theta }_{max}-\sqrt{2}{l}_2-2l_3\sin ({\theta }_{max}-{\theta }_0)}}& t=\mathrm{nT}=\frac{n\pi }{2\omega }\hfill \end{array}\hfill \\ \frac{\omega l_2\cos (\omega t_1+{\theta }_{max}-\theta +\frac{3\pi }{4})}{-l_1\cos \theta +l_2\cos (\omega t_1+{\theta }_{max}-\theta +\frac{3\pi }{4})+l_3\sin (\theta -{\theta }_0)}{t}_1=t-\mathrm{nT},t\in (\frac{n\pi }{2\omega },\frac{(2n+1)\pi +4({\theta }_0-{\theta }_{max})}{4\omega })\hfill \\ \frac{\omega l_2\sin (\theta -\omega t_2-{\theta }_0)}{-l_1\cos \theta +l_2\sin (\theta -\omega t_2-{\theta }_0)+l_3\sin (\theta -{\theta }_0)}{t}_2=t-\frac{(2n+1)T+4({\theta }_0-{\theta }_{max})}{4\omega },t\in (\frac{(2n+1)\pi +4({\theta }_0-{\theta }_{max})}{4\omega },\frac{(n+1)\pi }{2\omega })\hfill \\ \begin{array}{cc}\multicolumn{1}{c}{\frac{\sqrt{2}\omega l_2}{2l_1\cos {\theta }_{max}+\sqrt{2}{l}_2-2l_3\sin ({\theta }_{max}-{\theta }_0)}}& t=(n+1)T=\frac{(n+1)\pi }{2\omega }\hfill \end{array}\hfill \end{array}$$

(26)

$$\{\begin{array}{c}{l}_1\sin \theta +l_2\sin (\omega t_1+{\theta }_{max}-\theta +\frac{\pi }{4})+l_3\cos (\theta -{\theta }_0)=l_1\sin {\theta }_0+l_2+l_3{t}_1=t-\mathrm{nT},t\in (\frac{n\pi }{2\omega },\frac{(2n+1)\pi +4({\theta }_0-{\theta }_{max})}{4\omega })\hfill \\ \multicolumn{1}{c}{l_1\sin \theta +l_2\cos (\theta -\omega t_2-{\theta }_0)+l_3\cos (\theta -{\theta }_0)=l_1\sin {\theta }_0+l_2+l_3{t}_2=t-\frac{(2n+1)T+4({\theta }_0-{\theta }_{max})}{4\omega },t\in (\frac{(2n+1)\pi +4({\theta }_0-{\theta }_{max})}{4\omega },\frac{(n+1)\pi }{2\omega })}\end{array}$$

(27)

For the sake of simplicity, the condition of $\theta ={\theta }_{\max }$ is regarded as the initial state of the transplanting mechanism. the new law of the dibble motion can be described as follows. With the rod AC rotating clockwise around the point A, the rod CG holds contacted with the balance plane, whereas the rod $C\prime G\prime$ maintains in the no-load state. The angle $\theta$ decreases from ${\theta }_{\max }$ to ${\theta }_0$ . During this process, the angle $\beta$ alters from $\pi /4$ to $\pi /2$ . When the angle $\theta$ reaches its minimal value, ${\theta }_0$ , we get $\beta =\pi /2$ . After that, the angle $\theta$ increases from ${\theta }_0$ to ${\theta }_{\max }$ , making a complete cycle. At the beginning of a new cycle, the angular velocity of $\theta$ suffers an abrupt change, since the state of the rod CG changes from load to no-load. According to equations (19), (21), (22), and all the above analysis, the angular velocity of $\theta$ is a periodic function with a period of $T=t_1+t_2$ , whose exact value can be determined by equation (26). The function between $\theta$ and t is expressed in equation (27).

Analysis of the mulch film damage caused by the dibble motion

Building the relationship between the mulch film damage and the dibble motion is not an easy task under the environmental force condition, since there are several time-varying parameters involved. To tackle this problem, we divide the whole forming process of the mulch film damage into several parts and then give detailed discussions about them separately.

Forming time of the mulch film damage

As a first step to calculate the mulch film damage, we should know the time interval during which it is forming. According to the definition of the beneath-film damage, it is just the distance that the rod CG moves on the balance plane, meaning that the time interval is the minimal positive period of the function. Specifically, two special positions are examined in determining the on-film damage. As schematically shown in Figure 13, the CG rod rotates from one special position to the other, with the same condition $\theta =\theta \prime$ . There are two different values of $\beta$ with respect to the positions. The whole on-film damage forming process consists of three phases: the first phase is that the rod CG moves from the position where $\theta =\theta \prime$ and $\beta ={\beta }_a$ to the balance position where the rod CG is at the right side of the point A with $\theta ={\theta }_{\max }$ and $\beta =\pi /4$ ; the second phase is that the rod CG moves from the right balance position to the left balance position with $\theta ={\theta }_{\max }$ and $\beta =3\pi /4$ ; the third phase is that the rod CG moves from the balance position to the position where $\theta =\theta \prime$ and $\beta ={\beta }_b$ . Combining with the previous equations and analysis, the two specified positions can be mathematically described as follows

$$\{\begin{array}{c}{l}_1\sin \theta \prime +l_2\sin {\beta }_a+l_3\sin \alpha +h\hfill \\ =l_1\sin \theta \prime +l_2\sin ({\beta }_a+\frac{\pi }{2})+l_3\sin \alpha \hfill \\ l_1\sin \theta \prime +l_2\sin {\beta }_b+l_3\sin \alpha +h\hfill \\ =l_1\sin \theta \prime +l_2\sin ({\beta }_b-\frac{\pi }{2})+l_3\sin \alpha \hfill \end{array}$$

(28)

Figure 13.

Two special positions for calculation of the on-film damage: (a) beginning position of the on-film damage and (b) ending position of the on-film damage.

The values of $\beta$ can be calculated by equation (29)

$$\{\begin{array}{c}{\beta }_a=\frac{\pi }{4}-{\sin }^{-1}\frac{h}{\sqrt{2}{l}_2}\hfill \\ {\beta }_b={\sin }^{-1}\frac{h}{\sqrt{2}{l}_2}+\frac{3\pi }{4}\hfill \end{array}$$

(29)

Combining equation (28) with equations (12) and (29) yields

$$l_1\sin \theta \prime +l_2\sin {\beta }_a+l_3\cos (\theta \prime -{\theta }_0)+h=l_1\sin {\theta }_0+l_2+l_3$$

(30)

$$A=B\sin \theta \prime +C\cos \theta \prime =\sqrt{B^2+C^2}\sin (\theta \prime +\phi )$$

(31)

$$\theta \prime ={\sin }^{-1}\frac{A}{\sqrt{B^2+C^2}}-\phi$$

(32)

where

$$\{\begin{array}{c}\begin{array}{c}A=l_1\sin {\theta }_0+l_2+l_3-h-l_2\sin {\beta }_a\hfill \\ \multicolumn{1}{c}{=l_1\sin {\theta }_0+l_2+l_3-\frac{1}{2}(\sqrt{2l_2^2-h^2}+h)}\end{array}\hfill \\ B=l_1+l_3\sin {\theta }_0\hfill \\ C=l_3\cos {\theta }_0\hfill \\ \tan \phi =\frac{C}{B}\hfill \end{array}$$

(33)

According to equation (22), the time spent on the process where the CG rod moves from the position where $\theta =\theta \prime$ and $\beta ={\beta }_a$ to the special position with $\theta ={\theta }_{\max }$ and $\beta =\pi /4$ can be calculated as

$$t_a=\frac{{\theta }_{max}-\theta \prime +(\frac{\pi }{4}-{\beta }_a)}{\omega }$$

(34)

Similarly, the time spent on the process where the rod CG moves from the position with $\theta ={\theta }_{\max }$ and $\beta =3\pi /4$ to the special position with $\theta =\theta \prime$ and $\beta ={\beta }_b$ can be calculated as

$$t_b=\frac{\theta \prime -{\theta }_{max}+({\beta }_b-\frac{3\pi }{4})}{\omega }$$

(35)

Thus, the total forming time of the on-film damage becomes

$$t_{\mathrm{on}-\mathrm{film}}=t_a+t_b+T=\frac{\pi +4\gamma }{2\omega }$$

(36)

where

$$\gamma ={\sin }^{-1}\frac{h}{\sqrt{2}{l}_2}$$

(37)

According to the prior analysis, $t_{\mathrm{beneath}-\mathrm{film}}=T$ . The process of forming the on-film damage includes that of the beneath-film, so the analysis of the on-film damage is sufficient. Note that the effect of angle $\alpha$ is ignored in the interest of simplicity. Hence, the horizontal distance traveled by the point G can be served to describe the mulch film damage. In this sense, the point G can be viewed as the reference point of a dibble.

Forming velocity of the mulch film damage

It is a tough task to calculate the exact distance traveled by the point G in that the velocity of the point G is not regular. Figure 14 illustrates the velocity components of the point G. According to the theory of composition of velocities, the following relation holds

$$l_1\sin \theta \frac{d\theta }{\mathrm{dt}}-l_2\sin \beta \frac{d\beta }{\mathrm{dt}}-l_3\sin \alpha \frac{d\alpha }{\mathrm{dt}}+v_0=v_{\mathrm{Gx}}$$

(38)

Here, $v_{\mathrm{Gx}}$ denotes the x axis component of the velocity of the point G. In the above discussion, the angular velocity of $\theta$ suffers an abrupt change at the point where the angle $\theta$ reaches its maximum, ${\theta }_{\max }$ . The time begins at the point where $\theta =\theta \prime$ and $\beta ={\beta }_a$ , and the value of $v_{\mathrm{Gx}}$ is calculated by equation (39)

$$\{\begin{array}{c}(l_1\sin \theta +l_2\sin \beta +l_3\sin \alpha )\frac{d\theta }{\mathrm{dt}}-\omega l_2\sin \beta +v_0=v_{\mathrm{Gx}}\hfill \\ \frac{\sqrt{2}\omega l_2(N-M)}{2M+\sqrt{2}{l}_2}+v_0=v_{\mathrm{Gx}}t\to t_a^{-}\mathrm{or}t\to (T+t_a)-\hfill \\ \frac{-\sqrt{2}\omega l_2(N+M)}{2M-\sqrt{2}{l}_2}+v_0=v_{\mathrm{Gx}}t\to t_a^{+}\mathrm{or}t\to (T+t_a)+\hfill \end{array}$$

(39)

where

$$\{\begin{array}{c}M=l_1\cos {\theta }_{max}-l_3\sin ({\theta }_{max}-{\theta }_0)\hfill \\ \multicolumn{1}{c}{N=l_1\sin {\theta }_{max}+l_3\cos ({\theta }_{max}-{\theta }_0)}\end{array}$$

(40)

Figure 14.

Velocity components of the point G in the forming process of on-film damage: (a) velocities at the beginning of the process and (b) velocities at the end of the process.

During the whole time $t_{\mathrm{on}-\mathrm{film}}$ , the sign of $v_{\mathrm{Gx}}$ depends on the angular velocity of $\theta$ . The whole movement of the point G can be divided into four phases: the first phase is the movement from the special position where $\theta =\theta \prime$ and $\beta ={\beta }_a$ to the position where $\theta ={\theta }_{\max }$ and $\beta =\pi /4$ ; the second phase is the movement from the position where $\theta ={\theta }_{\max }$ and $\beta =\pi /4$ to the position where $\theta ={\theta }_0$ and $\beta =\pi /2$ ; the third phase is the movement from the position where $\theta ={\theta }_0$ and $\beta =\pi /2$ to the position where $\theta ={\theta }_{\max }$ and $\beta =3\pi /4$ ; the fourth phase is the movement from the position where $\theta ={\theta }_{\max }$ and $\beta =3\pi /4$ to the position where $\theta =\theta \prime$ and $\beta ={\beta }_b$ . In the first and fourth phases, the angles $\theta$ , $\beta$ , and $\alpha$ share a constraint as

$$l_1\sin \theta +l_2\sin (\beta \pm \frac{\pi }{2})+l_3\sin \alpha =l_1\sin {\theta }_0+l_2+l_3$$

(41)

In the second and third phases, the angles $\theta ,\beta ,\mathrm{and}\alpha$ have a constraint as

$$l_1\sin \theta +l_2\sin \beta +l_3\sin \alpha =l_1\sin {\theta }_0+l_2+l_3$$

(42)

According to equations (39), (41), (42), and the definition of time t, the calculation of $v_{\mathrm{Gx}}$ can be simplified as equation (43)

$$\{\begin{array}{c}\frac{\omega l_2\sin \beta (l_1(\sin \theta -\cos \theta )+l_3(\cos (\theta -{\theta }_0)+\sin (\theta -{\theta }_0)))}{l_1\cos \theta +l_2\sin \beta -l_3\sin (\theta -{\theta }_0)}+v_0=v_{\mathrm{Gx}},0\le t<t_a\hfill \\ \frac{\omega l_2(l_1\sin (\theta +\beta )+l_3\cos (\theta +\beta -{\theta }_0))}{-l_1\cos \theta +l_2\cos \beta +l_3\sin (\theta -{\theta }_0)}+v_0=v_{\mathrm{Gx}},t_a<t<t_a+T\hfill \\ \frac{\omega l_2\sin \beta (l_1(\sin \theta +\cos \theta )+l_3(\cos (\theta -{\theta }_0)-\sin (\theta -{\theta }_0)))}{-l_1\cos \theta +l_2\sin \beta +l_3\sin (\theta -{\theta }_0)}+v_0=v_{\mathrm{Gx}},t_a+T<t<t_a+t_b+T\hfill \end{array}$$

(43)

$$\{\begin{array}{c}{l}_1(\cos \theta \prime -\cos {\theta }_{max})+l_2(\frac{\sqrt{2}}{2}-\cos {\beta }_a)+l_3(\sin ({\theta }_{max}-{\theta }_0)-\sin (\theta \prime -{\theta }_0))+v_0{t}_a=D_{x1}\hfill \\ -\sqrt{2}{l}_2+v_{0}T=D_{x2}\hfill \\ l_1(\cos {\theta }_{max}-\cos \theta \prime )+l_2(\frac{\sqrt{2}}{2}+\cos {\beta }_b)+l_3(\sin (\theta \prime -{\theta }_0)-\sin ({\theta }_{max}-{\theta }_0))+v_0{t}_b=D_{x3}\hfill \end{array}$$

(44)

Definition and calculation of the minimal mulch film damage

The horizontal displacement of the point G in different phases can be obtained via equation (44), given by the integration of equation (43). The on-film damage can then be calculated by $w_2=|D_{x1}+D_{x2}+D_{x3}|$ , but the beneath-film damage cannot be simply calculated by $\left|D_{x3}\right|$ . The major reason is that the dibble movement beneath the film is a round-trip rectilinear motion, implicating $w_1\ne 0$ when $D_{x2}=0$ . The exact value of the beneath-film damage can be calculated by equation (45), where $t_z$ means some value of t making $v_{\mathrm{Gx}}=0$

$$w_2=max(\left|\underset{t_a}{\overset{t_z}{\int }}{v}_{\mathrm{Gx}}\mathrm{dt}\right|,\left|\underset{t_z}{\overset{T+t_a}{\int }}{v}_{\mathrm{Gx}}\mathrm{dt}\right|)$$

(45)

It is worth noting that the value of ${\int }_{t_a}^{t_z}{v}_{\mathrm{Gx}}\mathrm{dt}$ is definitely negative as the result of the negative value of $v_{\mathrm{Gx}}$ . There would be a much larger mulch film damage if the value of $v_{\mathrm{Gx}}$ is positive. Actually, the previous definition of the minimal mulch film damage is still valid, but the calculation method simply making $w_1=w_2$ is helpless. Notice that $v_{\mathrm{Gx}}$ suffers sudden changes at the end of the first phase and the beginning of the fourth phase, and that $T=\pi /2\omega$ is much larger than the value of $t_a+t_b=2\gamma /\omega$ . In such a context, it can be inferred that the whole mulch film damage is mainly determined by the beneath-film damage. In practice, it is still hard to directly use equation (45) to calculate the beneath-film damage. By inspecting the components of the beneath-film damage, the difference between the two parts can be expressed by $D_{x2}$ . Sequentially, the condition of $D_{x2}=0$ is conjectured to be helpful in estimating the minimal damage. It follows that

$$v_0=\frac{\sqrt{2}{l}_2}{T}=\frac{2\sqrt{2}\omega l_2}{\pi }$$

(46)

In addition, the exact value of $t_z$ can be calculated by equations (43) and (46). Substituting it into equation (45), as a consequence, we can numerically estimate the incurred beneath-film damage.

Results and analysis

To evaluate the proposed method to determine the minimal mulch film damage caused by the up-film transplanter, simulations and field tests have been performed. Specifically, we have created a three-dimensional (3D) model of the up-film transplanter in automatic dynamic analysis of mechanical systems (ADAMS), allowing extensive analysis capabilities of the dynamics of moving parts. The basic feature parameters for the up-film transplanter are set as follows: $l_1=622.27\mathrm{mm}$ , $l_2=200\mathrm{mm}$ , $l_3=154.44\mathrm{mm}$ , $h=60\mathrm{mm}$ , and ${\theta }_0=15.{296}^{\circ }$ . More detailed parametric setting and numerical results are shown in Table 1 and Figure 15. Besides, preliminary field tests have been made, in which a typical test scenario is shown in Figure 16.

Table 1.

Parametric setting and results of the simulation.

Trajectories of the point G	Simulation settings		The on-film damage (mm)	The beneath-film damage (mm)
	$v_0(\mathrm{mm}/\mathrm{s})$	$\omega (\circ /\mathrm{s})$
Trajectory 1	245	80	13.8	57.7
Trajectory 2	251	80	23.0	53.9
Trajectory 3	257	80	31.0	57.1
Trajectory 4	260	80	35.3	58.5

Note: Trajectory 2 has the minimal mulch damage among all the trajectories.

Figure 15.

Trajectories of the point G under different settings.

Figure 16.

A preliminary field trial: (a) prototype of the developed dibble-type transplanter and (b) scenario of transplanting operation.

The main simulation and test results are summarized as follows: first, the presumption that the whole mulch film damage is decided by the beneath-film damage is confirmed, because the on-film damage is much smaller than the beneath-film damage (see Table 1). Second, the comparisons between Trajectory 2 and other trajectories reveal the effectiveness of equation (46). Furthermore, the comparison between Trajectory 2 and Trajectory 3 verifies the necessity of the adjustment, for the value of $v_0$ in Trajectory 3 is calculated by equation (11). At last, due to the contractile capability of the mulch film, real mulch film damage occurred in the field tests is smaller than the calculation results, so very small differences between different trajectories could hardly be distinguished. However, the dibble motion displayed in the real-world transplanting operations is similar to that in the simulation, demonstrating the effectiveness of the soil-layered theory. It should be remarked that only the perpendicular component of the soil counter-force is considered in this study. In reality, the horizontal component of the soil counter-force may affect the mulch film damage, which is another question worthy of more in-depth investigation.

Conclusion

In this article, we have examined influencing factors of the mulch film damage caused by a typical dibble-type transplanter. The proposed strategy for solving the minimal mulch film damage problem is as follows: first, the dibble’s motion is expressed in the form of mathematical model and then the relation between the motion and the mulch film damage is figured out, and finally, the minimal value of mulch film damage is estimated. The minimal value can be easily calculated in the ideal condition, whereas the situation turns complex in the realistic environmental force-involved condition. Hence, a two-layer soil-layered theory is proposed to simplify the problem so that the quantitative relations among working parameters, mechanical parameters, and the value of the minimal mulch can be determined. Simulations and preliminary field tests show the effectiveness of the presented theoretical analyses. From the simulations, it can be concluded that it is possible to model the mulch damage process effectively and obtain good results. This suggests that there is potential in reducing the mulch film damage using mechanical and control measures during transplanting operations.

The future work will focus on several aspects. In this article, the main motivation is to figure out the relationship between the mulch film damage and the dibble motion, so some unrelated factors are simplified or omitted. For instance, the dibble shape is simplified as a shapeless line always perpendicular to the ground, but the dibble shape actually has some connection with the mulch film damage. Besides, to what extent does the tilt angle of the dibble during transplanting influences the mulch film damage still remains unknown. Also, the soil model merely consisting of two layers can be extended to more layers with different characteristics. Of course, more thorough experimental investigation on the newly developed up-film transplanter is demanded for better support.