Optimization of Design Parameters Using SQP for an Agricultural Pipe Extraction Device

Abstract

Removal of agricultural pipes used in crop support and greenhouse agriculture is manpower-intensive. However, most agricultural workers are elderly. Therefore, auxiliary devices should be used to allow pipe removal with as little force as possible. In this study, the design parameters of an extraction device were optimized within constraints to minimize the force required to remove agricultural pipes. The optimization parameters are the length of each link and the initial link angle of the device. The driving force, according to the design parameters, was calculated by applying the theory of kinematics. An optimal design plan was derived using an optimization algorithm to minimize the force driving the device within the desired constraint. As a result of the optimization, it was confirmed that the force required to remove the pipe was reduced by 87.1% compared with the initial design value and was designed to suit the user’s convenience.

1. Introduction

When a crop grows and produces fruit, it cannot bear its weight and falls down. The fruit touches the soil and becomes soft or rotten. In particular, during the summer and rainy seasons, crops are frequently exposed to typhoons or rain and wind, which increases the probability of crop damage. Therefore, agricultural pipes are widely used for the healthy growth of crops. The use of pipes in agriculture is a major way to prevent crops from being bent or overturned in the wind and rain and is directly related to improving yields. However, in winter, when farming is over, the agricultural pipe must be removed from the hardened bottom of the farmland. It is difficult to remove the pipe without using a tool because a large force is continuously required. Therefore, methods of removing agricultural pipes using various tools have been proposed. Representative examples include a crop-holder recovery device and a pepper-support extraction device. These existing devices allow the removal of an agricultural pipe by pulling the lever or applying force by hand.

In this study, an agricultural pipe extraction device (hereafter referred to as an extraction device) was proposed and optimized. This extraction device is operated by foot, unlike conventional hand-operated instruments. Compared to existing devices that require pulling the lever by hand, the stepping method has the advantage that the agricultural pipe can be easily removed with a small force using an appropriate link. In particular, considering that the work is performed continuously, the effect of reducing the workload experienced by users is expected to be significant, as compared to the existing method. Optimization of the agricultural pipe extraction device was performed in the direction of minimizing the force required during operation. To this end, ergonomic design variable constraints were applied to prevent using excessive force when operating the extraction device.

Various studies have been conducted on parameter optimization to minimize the force targeting the link structure. Cho and Kim (2010) [1] conducted a study on the optimal design of a tractor clutch-link system to minimize the pedal force generated when the clutch pedal was depressed. The link system was formulated, and a feasible direction method was used as an optimization algorithm. The operating force was reduced by 14% by optimizing the clutch system of commercial tractors. Kim et al. (2013) [2] performed a link structure optimization study for a four-section link manipulator for starfish collection and presented a mechanism that improved the collection distance set as an objective function by 56%. Kim et al. (2011) [3] optimized a link structure with a maximum walking stride for a walking robot. The four-section link theory and the Janssen mechanism were applied for optimization, and through this, it was possible to propose the position and length of each link with a 214% improved walking stride, as compared with the existing model. Koushkaki et al. (2022) [4] conducted a DEM simulation to simulate the draft force, and it was concluded that the draft force increased with an increase in working depth and forward speed. Im et al. (2022) [5] conducted research to optimize the bucket size of a metering device for each group of garlic cloves investigated was determined using MBD simulation. Using the developed garlic clove 3D models, a simulation was performed by increasing the angle of the bucket size control guide, and the angle immediately before the descent of the garlic clove was regarded as the optimal angle. Cha et al. (2014) [6] set and optimized the position and length of latches as design variables to shorten the opening time of circuit breakers. For optimal design, an interlocking model of ADAMS (Automated Dynamic Analysis of Mechanical Systems) and VisualDOC was constructed, and optimization was performed using GA as a nonlinear optimization algorithm. Through this, the opening time was reduced by 2.7 ms, which showed an improvement effect of 22.5%.

Although many studies have been conducted on link structures, to the best of our knowledge, there have been no studies on the design of hand tools for agriculture. Due to the aging of agricultural workers, the development of agricultural machinery should promote comfort and convenience. However, studies to improve the convenience of using agricultural machinery in consideration of the body of the elderly are lacking. Moreover, there is no research on hand tools such as the pipe extraction device selected as the subject of study in this study. Minimizing the force required to operate agricultural machinery is important because it can prevent user injury and increase comfort.

In this study, sequential quadratic programming (SQP) was used as an optimization algorithm. SQP is an algorithm that quickly and accurately converges the objective function and is widely used as an objective function optimization algorithm for various optimal designs. Kang et al. (2018) [7] conducted a study on the optimization of the cross-sectional design of the main wing of a human-powered aircraft. Variational asymptotic beam section analysis (VABS) and geometric precision beam theory were introduced for optimization. The objective function obtained through this process was optimized through the SQP algorithm, and a mass reduction of 7.88% was obtained while maintaining the bending and torsional deformation. Bark (1998) [8] used the SQP algorithm for the optimization of structures under multiple loading conditions. It was confirmed that the formula is simple and easy to converge to the optimal solution when the SQP algorithm is used for optimization. As a result, although the state variables become an implicit function of the sizing variables, the dimension of the optimization problem can be greatly reduced. In addition, Shin et al. (2022) [9] conducted a study to optimize the efficiency of a 3 K-type planetary reducer using the SQP technique and confirmed that the highest efficiency gear parameters could be obtained under constraints. As such, the SQP algorithm is judged to be suitable for optimizing mechanical systems. There are not many cases in which an algorithm such as SQP is used in the process of optimizing design variables, such as the link length of agricultural equipment. For agricultural hand tools such as the pipe extraction device, which is the subject of this study, no research has been conducted on machine improvement through link length optimization. If hand tools can be improved through the process of setting the objective function, design, and operating conditions, it is expected that the agricultural environment can be dramatically improved.

Therefore, in this study, each design variable was optimized to minimize the objective function value using the SQP algorithm. In setting the objective function, we tried to obtain design parameters so that the user could conveniently operate the instrument with little force and without the risk of injury.

2. Materials and Methods

2.1. Extraction Device Modeling

The initial design of the extraction device is shown in Figure 1. The left side of the device consists of a gripping part for holding the pipe, and the right side is composed of an active part to which the user applies force. The action part is depicted in detail in an enlarged view. In this study, it was assumed that the tongs gripped the pipe with sufficient force and that the subject of the study was limited to the action part to which the user applied a force.

The operating mechanism of the extraction device presented in this study is shown in Figure 2. When the user exerts pedal effort on this device, the force converted into vertical force by the link structure is transmitted to the gripper holding the agricultural pipe. It is a device that uses the principle that the pipe embedded in the farmland is lifted and removed by the force. This allows users to quickly and easily remove agricultural pipes by simply pedaling with their feet.

In order to facilitate the analysis, Figure 1 was schematized as a two-dimensional plane, as shown in Figure 3. Because point A is fixed and does not move when the extraction device is used, point A is regarded as the reference point (origin of the coordinate system).

Link3 consists of a sliding joint capable of only up-and-down translational movement, and point A is a fixed-rotation joint. Points B and C are rotational joints that are not fixed and are configured to move in connection with each link. The coordinates of each point are shown in Figure 3, where each link is assumed to be a rigid body. The mass and moment of inertia of each link are not considered because their influence is negligible compared with the force factor considered in the analysis. The variables to be considered in the design are the length of Link1 ($L_1$), length of the left side of Link2 from point B ($L_2$), length of Link2 from the right side of point B ($L_3$), the horizontal distance between point C and point A (k), angle (α) made by the extension line of Link3 with Link2, and angle (β) made by Link1 with the ground. Among the six variables, β can be expressed as a function of the remaining variables. Therefore, this final link system can be modeled with the five variables. β can be represented by Equation (1).

$$\beta =\mathrm{co}{\mathrm{s}}^{-1}(\frac{L_2\cos \alpha -k}{L_1})$$

(1)

2.2. Objective Function for Optimization

In the analysis of this study, a static equilibrium state was assumed. The vertical force required to remove the pipe is set to P, and the force that Link1 exerts on Link2 is set to $F_{12}$. In addition, the vertical force of the acting part applied to generate the reaction force against P is set to $F_r$. To express $F_r$ as an equation, a force analysis on Link2 was performed, as shown in Figure 4, and Equation (2) was derived based on the analysis results.

$$F_r=\frac{P\cdot L_2\cdot \sin \left(\alpha +\beta \right)}{\left(L_2+L_3\right)\sin \beta -L_2\cdot \sin \left(\alpha +\beta \right)\cdot \cos \alpha }$$

(2)

By substituting Equation (1) into Equation (2), β can be eliminated from the design variables. Finally, the equation of $F_r$ consisting of the five variables was selected as the objective function.

2.3. Optimal Design Method

The optimal design involves finding the parameters in the direction of minimizing the value of the objective function of the system, which takes into account the constraints proposed by the designer. In this study, the SQP algorithm was used as an optimization algorithm. The SQP algorithm is a nonlinear optimization design algorithm that has the advantage of increasing the convergence speed and improving the accuracy through quadratic modeling of the objective function (Kang et al., 2018) [7].

The algorithm for the SQP method is represented by Equations (3)–(5).

Minimize

$$\frac{1}{2}{d}^T{H}^{k}d+\nabla f{\left(x_k\right)}^{T}d$$

(3)

Subject to

$$\nabla g_i{\left(x_k\right)}^{T}d+g_i\left(x_k\right)=0\left(i=1,\dots ,m\right)$$

(4)

$$\nabla g_i{\left(x_k\right)}^{T}d+g_i\left(x_k\right) \le 0\left(i=m_e+1,\dots ,m\right)$$

(5)

Here, d is the search direction vector, k is the number of iterations, H is the Hessian matrix, $f\left(x_k\right)$ is the objective function, and $g_i\left(x_k\right)$ is the constraint expression. The initial value $x_0$ of the design variable and the initial matrix $H_0$ of the Hessian matrix are obtained. After obtaining the search direction vector, a new $x_k$ is selected through Equation (6). This process is repeated, and when the optimal design variable is found, the algorithm is terminated (Cha et al., 2013) [10].

2.4. Constraints

If there is a contradiction in the range of parameters according to the objective function during optimization using the optimal design method, an error may occur. In this case, the initial height from the ground to the operating position of the extraction device is designed unrealistically. That is, if the constraints of the objective function are not considered during optimization, the height from the ground to the operating position of the extraction device may increase or decrease more than necessary because of the lengths of Link1 and Link2 and the angles α and β. Considering the action of actually stepping on the extraction device by lifting one’s leg, if the operating distance range between the footrest and the road surface is significantly reduced or increased, the extraction device operation is difficult. Consequently, work efficiency is lowered. In addition, owing to excessive motion, fatigue of the lower extremity muscles and pain in the knee joint may occur.

In order to solve this problem, constraints were set using an ergonomic approach. The main design objective applied to ergonomics is to design objects, equipment, machinery, etc., for efficient use by humans. Its importance is emerging in all fields where interaction with humans occurs (Kim and Chung, 2016) [11].

In this study, the values required for the constraints were defined based on the Korean body size data of Size Korea and the EMG (Electromyography) measurement values according to the knee joint. Considering the average age of farmers engaged in agriculture in Korea (KOSIS, Daejeon, Republic of Korea, 2022) [12], anthropometric data from 40-to-69-year-old individuals were used. The EMG values measured by Hwang et al. (2015) [13] were used to assess changes in the lower extremity muscle activation and fatigue according to the knee joint angle.

According to the 8th Korean anthropometric survey (Size Korea, 2020) [14], the height of the anterior superior iliac spine (ASIS) was 914 mm, and the height of the tibiale was 447 mm among the body sizes of men aged 40–69 years. For women, they were 837 mm and 409 mm, respectively.

Table 1. ASIS, tibiale height measurement standard (source: Size Korea).

Name	Anterior Superior Iliac Spine (ASIS)	Tibiale
Definition	The most anterior palpable part of the iliac crest of the hip	At the top of the upper medial condyle of the tibia
Measuring instrument	Vertical ruler

and Figure 5 show the definitions and measurement tools for the heights of the anterior superior iliac spine (ASIS) and tibiale.

The height of the footrest was designed based on the body size of an elderly woman, and there are several reasons for selecting the body size standard. If it is based on the body size of a male, the height of the footrest can be designed higher. In this design, the angle of the knee that needs to be lifted increases to more than 60° when used by an elderly woman with relatively small body size. As a result, muscle fatigue greatly increases, causing the user to feel uncomfortable during the operation, which may eventually lead to injury.

On the other hand, if the device is designed based on the body size of an elderly woman, most people can use the device without difficulty. In addition, rural society is rapidly aging. It is appropriate to design mechanical systems to meet these trends. Therefore, for more universal use, it was designed based on the body size of an elderly woman. (Kim et al., 2020 [15]).

EMG values measured by Hwang et al. (2015) [13] were used to determine the activity and fatigue of the lower body muscles according to the angle of the initial knee joint when the extraction device was operated. Electromyography was used to record the change in the current generated by the muscle movement by attaching electrodes to the skin surface. This method is mainly used to analyze muscle activity and muscle fatigue in many studies.

According to Hwang et al. (2015) [13], muscle activity according to the knee joint angle is highest when the knee joint angle is 60°, among 30°, 60°, and 90°. Muscle activity tends to increase as the knee joint angle increases. However, when the knee joint is bent more than 60°, the pressure on the knee bone increases, and pain occurs, resulting in lower muscle activity and increased muscle fatigue. Therefore, in this study, the initial knee joint angle was selected as 60°, considering the muscles activated to operate the extraction device and the burden on the body.

Assuming that the extraction device is operated, Figure 6 shows the height of the anterior iliac spine (A), height of the tibiale (B), and height from the ground to the sole of the foot (H). The figure corresponds to the case when the leg is lifted, and the knee is bent at a certain angle (θ). The constraints of the objective function are selected based on the average anthropometric data of Koreans aged 40 to 69 years and the value of the initial knee joint angle. To this end, the height (H) from the ground to the sole of the foot is calculated, as shown in Equation (6).

H = A − [(A − B) cos (θ) + B]

(6)

The values of the heights of the anterior iliac spine (A) and tibiale (B) and the initial knee angle (θ) of women aged 40–69 years presented by Size Korea (2020) [14] were substituted into Equation (6).

As a result, the height from the ground to the sole of the foot was calculated as 222 mm. Because this result was obtained with the average body size data of women aged 40 to 69 years, the footrest height was set to a maximum of 200 mm or less in consideration of users with a smaller body size than average. However, if the design height of the scaffolding is too low, the operating range of the mechanism is limited, and there is a concern that the pipe cannot be pulled out. Therefore, the height of the scaffolding was set to a minimum of 150 mm or more. In addition, constraint conditions were set such that the overall height and width of the equipment were each less than 300 mm, considering the distance between crops and convenience when moving the equipment.

2.5. Optimization Model Construction

Table 2. Initial values and range of the design variables.

Design Variable	min	max	Initial
Angle between the extension line of Link3 and Link2 ($\alpha$) [°]	0	60	21.37
Length of Link1 ($L_1$) [mm]	30	500	218.19
Length of Link2 from Point B to Point C ($L_2$) [mm]	30	500	119.39
Length of Link2 from Point B to Point D $L_3$ [mm]	30	500	203.21
Horizontal distance between Points A and C (k) [mm]	0	50	27.84

shows the initial values of the design variables and ranges of each parameter set, reflecting ergonomic factors. The objective function for design variable optimization is expressed as Equation (7) by substituting Equations (1) and (2). In order to compare the size of the operating force for any P, the ratio of the vertical force (P) and the operating force ($F_r$) was expressed as R. This is a non-dimensional and is represented by Equation (8).

Equation (9) limits the maximum value of the width of the equipment as a constraint, and Equation (10) represents the range of the overall height of the device. Equation (11) is a constraint on the height of the scaffolding of the device defined in the constraints section.

Find $\alpha ,L_1,L_2,L_3, \mathrm{and} k$

To minimize

$$F_r\left(\alpha ,L_1,L_2,L_3,k\right)=\frac{P\cdot L_2\cdot \sin \left(\alpha +{\cos }^{-1}\left(\frac{L_2\cos \alpha -k}{L_1}\right)\right)}{\left(L_2+L_3\right)\sin \left({\cos }^{-1}\left(\frac{L_2\cos \alpha -k}{L_1}\right)\right)-L_2\cdot \sin \left(\alpha +{\cos }^{-1}\left(\frac{L_2\cos \alpha -k}{L_1}\right)\right)\cdot \cos \alpha }$$

(7)

Subject to

$$R=\frac{F_r}{P}$$

(8)

$$\left(L_2+L_3\right)\cdot \cos \alpha \le 300$$

(9)

$$L_1\cdot \sin ({\cos }^{-1}(\frac{L_2\cos \alpha -k}{L_1}))+L_2\cdot \sin \alpha +k\le 300$$

(10)

$$150\le L_1\cdot \sin ({\cos }^{-1}(\frac{L_2\cos \alpha -k}{L_1}))-L_3\cdot \sin \alpha \le 200$$

(11)

The optimization process of this study is briefly shown in Figure 7. The initial design variables were determined in the model before optimization, and the objective function was derived based on the statics theory. In order to increase user convenience, the overall length and height of the footrest were set as the constraints. After setting the objective function, constraints, and initial design variables, the optimization process of the SQP algorithm was performed using the FMINCON module in MATLAB (Version R2022a) [16].

3. Results and Discussion

Optimal Design Result

During the optimization, the value of $F_r$ for P converged after 21 iterations. The process of optimizing the ratio R using the SQP algorithm is shown in Figure 8, and the process of optimizing each design variable is shown in Figure 9. From Figure 8, it can be seen that the ratio R has an extremely large value in the first three iterations but converges stably thereafter. However, as shown in Figure 9, all parameters selected for the optimization converged after the 17th iteration. Thus, the optimization process was interrupted after the 21st iteration.

The final derived optimal design variable values are listed in

Table 3. Optimization results.

Optimization Results
Angle between the extension line of Link3 and Link2 ($\alpha$) [°]	26.56
Length of Link1 ($L_1$) [mm]	287.52
Length of Link2 from Point B to Point C ($L_2$) [mm]	30
Length of Link2 from Point B to Point D ($L_3$) [mm]	305.41
Horizontal distance between points A and C (k) [mm]	50

. In Table 3, it can be seen that the optimized lengths of Link1 and Link3 are increased compared to the initial design length, while the optimized length of Link2 is considerably decreased. In addition, the angle α between Link2 and Link3 increased slightly, and the horizontal distance k between points A and C also increased. Each design variable satisfies all the previously defined inequality constraint Equations (9)–(11).

If the ratio R of the operating force $F_r$ to the vertical force P is calculated using the initial design variable values, it is 0.683, and the ratio R calculated through the optimized design variables in Table 3 is about 0.082. It can be confirmed that the optimized operating force is reduced by about 87.1% from the initial operating force.

In addition, the height of the initial scaffold (127.31 mm) is out of the range of the set constraints. This value is judged to be unsuitable for pulling out the pipe because it reduces the operating range of the device. On the other hand, the height of the optimized footrest is 150 mm, which has a sufficient operating range. It is sufficient to apply force with the foot without the risk of injury based on the body of an elderly woman. In addition, the overall length and height are 300 mm, which is suitable for removing pipes while moving between crops.

4. Conclusions

In this study, optimization was performed by targeting the following design parameters of an agricultural pipe-extraction device: link length and initial angle. For easy interpretation, the structure through which the force is transmitted was modeled in two dimensions, and the objective function was formulated through kinematic theory. Constraints were set using the body size and muscle activity data, and the SQP algorithm was used as the optimization method with the FMINCON module in MATLAB (Version R2022a) [16]. Through the optimization, it was possible to secure parameter values such that the force required to drive the extraction device was reduced by 87.1% compared with the initial design value. In addition, the extraction device with the optimized parameters can be moved and used by elderly farmers without the risk of injury.

The entire device mechanism includes the process of holding the pipe through tongs. However, in this study, force analysis and optimal design were conducted only from the footing of the extraction device to lifting the link. The optimal design of the extraction device can be more accurate if a design plan is derived for the entire mechanism of the extraction device, considering the process of picking up the pipe through tongs.

In addition, if an actual extraction device is manufactured using parameters obtained through optimal design, the force for driving the extraction device can be actually measured through a force-measuring instrument such as a load cell. Through this, the validity of the optimal design model presented in this study can be verified, and it will be possible to confirm how much the force required to actually drive the extraction device is reduced.

In this study, the design was performed to minimize the force required for agricultural operations using agricultural hand tools. In addition, it was possible to select the optimal design variables by selecting and applying the objective function, design condition, and operating conditions in the parameter selection of the hand tool. These studies can help operators’ health by improving the agricultural operating environment and improving the convenience of using agricultural equipment.