Design of Bacteria Bottle Clamping Elements Based on Regression Models

The fixture design of the bacteria bottle plays a vital role in designing a bottled fungus picking robot to save labor cost in the picking process of bottled fungus. This paper proposed a kind of clamping element design method based on regression models. Several sets of clamping elements were designed according to the appearance data of bacteria bottle. A single-factor test was conducted by using these clamping elements, and three levels of every factor were selected to obtain the desired values of clamping elements. Then, a Box–Behnken test was performed by selected levels. The established regression models described a numeric relationship between the variation of vital measurement points and all variation sources under a precise clamping element layout. To solve the problems in obtaining the direct parameters, a response surface method was presented based on the regression models. Finally, a test was used to demonstrate the effect of the optimized clamping elements when clamping a bacteria bottle. Through the related analysis and optimization, it was demonstrated that the holding effect of the clamping elements was the best under these conditions: the inner arc area was 4948 mm2; the downward displacement was 1.48 mm; and the rubber thickness was 3.69 mm. It showed that the proposed method was feasible, and the assembly quality after optimizing had been greatly improved. It can provide a reference for designing the bottle fixture of a picking machine.


Introduction
It is a fact that some mushrooms are cultivated by bacteria bottles in most edible fungus factories. Besides, Flammulina velutipes (F. velutipes) is one of the six most cultivated mushroom species in the world, and over 300,000 t are produced per year [1]. As for the production process of bottled F. velutipes, it is still picked by manual drawing. As China's population aging progresses, labor cost also increases significantly. e contribution of labor to overall production costs for growers is generally at least 40%. Automation has the potential of improving the quality of fresh produce, lowering production costs and reducing the requirement of labor [2]. As a result, designing a kind of picking robot, such as F. velutipes harvester, contributes to increasing factories' profits. During the process of production, amounts of bacteria bottles are applied. erefore, it is necessary to pay more attention to designing clamping elements that can be used in picking robots.
A fixture is a work-holding device that locates and supports the workpiece [3]. Moreover, clamping elements are essential parts of a fixture. Fixtures should have appropriate structures and supply reasonable force to hold objects. Some methods were proposed to avoid the workpiece deformations, which can be caused by the clamping system itself. Gonzalo et al. [4] presented an analysis to identify the causes of the static deformations during clamping and a method to correct the geometrical distortion and deformation of a clamped workpiece by the evaluation of the reaction forces in the selected relevant clamping points. Vukelic et al. [5] proposed a method that was based on preprocessing of contact interfaces and indenting the clamping elements into the workpiece. It realized to use lower clamping forces to achieve deeper indenting and higher tangential load capacity. Sun et al. [6] introduced a measurement prototype for the real-time dynamic clamping force monitoring, which can be employed to realize the realtime vibration monitoring in thin-walled components milling, as well as to preliminarily identify the machining quality of the thin-walled components. Huamin et al. [7] suggested a linear programming technology to verify the force existence and the force feasibility at the corresponding clamping point. Because the proposed method transforms the continuous design problem of clamping forces into the discrete analysis problem of workpiece stability, it can apply to the complex workpiece and benefit the development of automated fixture design. Wu and Xi [8] designed a synchronous plane of the clamping mechanism and proposed a design criterion for the locking mechanism's clamping force.
rough those, they obtained the relevant motion curves, which could be served as a new idea for designing clamping elements of the robot. Li et al. [9] designed a new type of clamping end-effector's structure for the robot to realize the pickup of the box goods. e finite element analysis of key components improved the feasibility of structural design, which verified the feasibility of the structure design and provided references for subsequent optimization. Din et al. [10] designed a new type of Arcan fixture by a mechanical test to obtain applicable design standards. Papastathis et al. [11] presented an adaptive fixture able to reconfigure the clamping elements; it also uses position and clamping force feedback. Besides, modeling of the workpiece-fixture system allows the implementation of optimization strategies for attaining a suitable fixture structure. Wu et al. [12] proposed a computerized framework for computer-aided fixture design and evaluation for near-net shaped jet engine blades based on the mechanical model and mathematical model. It evaluated and optimized the fixture of the same type blade to obtain the positioning and mechanical behavior of the fixture and can realize high-precision manufacturing of near-net-shaped jet engine blades. Yu [13] designed a more applicable robust fixture by using a response surface method to establish the vector regression mode, which had improved the assembly quality. Liu et al. [14] improved the fixture quality by optimizing the clamping force plan and the fixture layout design in a finite element model. Nategh and Parvaz [15] developed a mathematical foundation of clamping system design and an algorithm by using the concept of screw theory and the minimum norm principle. It relegalized the minimum number of clamps, maximum contrariety of clamping force components with respect to the twists tending to break contact between the workpiece and the locators, and maintenance of workpiece's static stability under different wrenches. Zhang et al. [16] proposed a design method to improve the clamping apparatus for replacement of insulator strings in the ultrahigh-voltage (UHV) transmission line. e mechanical properties of the clamping apparatus are analyzed to determine the structure with the fixtures of carbon fiber composites by the finite element method. It could improve the working efficiency and reduced the labor intensity. Zhu [17] designed the pneumatic clamping device and arc-shaped parts and optimized them by orthogonal test and regression analysis. Wang et al. [18] designed and optimized holding devices of the banana picking manipulator by using Pro/E and ANSYS software. e results having a dedicated application value met the stiffness and strength requirement for harvesting bunches of bananas. Medellín et al. [19] presented a biaxial tensile testing fixture using finite element analysis, which can be used along with a universal testing machine for the characterization of sheet metals. Calabrese et al. [20] optimized the performance of the fixtures used in thin-walled workpiece machining depending on the local rigidity characteristics of the component to be machined by designing topology optimized.
At present, objects clamped are relatively dense in agriculture or industry. However, there is no research on the fixture for holding bacteria bottles with fresh materials used to design clamping elements for the mushroom harvester. erefore, we combined experimental research with mathematics analysis, which was first applied to design fixtures of bacteria bottles with culture. And we ensured the minimum size of clamps to keep contact between the bottle and locators. It contributed to the recyclable lifetime of the bottles.
Pull-out is a universal method when labors pick up the bottled F. velutipes. In designing a bottled mushroom picking robot, it is necessary to ensure that the bacteria bottles are clamped appropriately by the holding parts during the pull-out process of picking F. velutipes. And the proper holding capacity can increase the recyclable lifetime of the bottles. erefore, studying the mechanical properties of bacteria bottles has become necessary. Several scholars [21][22][23] have conducted many mechanical tests using plastic bottles to obtain related data. erefore, we presented a set of clamping elements and conducted a compression test, aiming to design a fixture used in exploiting a robot that can harvest the bottled F. velutipes by pull-out. e levels of Box-Behnken test were selected by the single-factor test. According to the obtained data, the variation models were established to describe accurately the relationship between the clamping quality and variation sources under specific clamping elements. And the response surface modeling method was employed to obtain the interaction between factors that significantly influence clamping quality. Finally, optimized parameters were corroborated by a confirmation test, which supplied a foundation for the bacteria bottle fixture design.

Design of the Clamping Elements.
e bacteria bottle is a kind of plastic bottle. It has three types, including small, medium, and large. Medium bottles have the most commonly used in the process of cultivating F. velutipes in a factory. Figure 1 shows the medium bacteria bottle with external dimensions. e fixture should hold a plastic bottle tightly. And the rubber is attached to the interior to reduce the impact of the metal fixture. e main body of the bottle, length of 140 mm, is an excellent position to hold. us, the clamping elements were devised based on the parameters of the central bottle part. A V-block has broad applications for clamping in terms of holding plastic bottles. It does not have large contact areas with the bacteria bottle, which could decrease the lifetime of the bottle due to high load. erefore, circular surface contact is more conductive to protecting the bottle. It has smaller contact pressure that exists between the bottle and the clamping elements. Besides, clamping the whole bottle is not a scientific method. It should have an appropriate area, including the central angle and thickness of it. e thickness of the rubber is ignored because it cannot significantly influence the area of the clamping elements. To make errors under 5.6%, the radius value "54 mm" is applied. As a result, the structure of the clamping elements is shown in Figure 2(a). en, where B is the internal arc area of the clamping elements and β ′ (rad) is the central angle corresponding to the clamping arc of the clamping elements. e value of it can be{π/2, 2π/3, 5π/6}. h (mm) is the thickness of the clamping elements. e value of it can be{25, 35, 45}. According to the combination, the areas can be 2121 mm 2 (π/2 × 25), 2969 mm 2 (π/2 × 35), 3817 mm 2 (π/2 × 45), 2827 mm 2 (2π/ 3 × 25), 3958 mm 2 (2π/3 × 35), 5089 mm 2 (2π/3 × 45), 3534 mm 2 (5π/6 × 25), 4948 mm 2 (5π/6 × 35), and 6362 mm 2 (5π/6 × 45). Areas selected should have an appropriate range and interval. Besides, we considered errors. e areas of the clamping elements, 2121 mm 2 , 2969 mm 2 , 3958 mm 2 , 4948 mm 2 , and 6362 mm 2 , were used to make the corresponding parts, as shown in Figure 2(b).

Experimental
Procedure. Several bacteria bottles with fresh culture were applied rather than empty bottles due to the great differences in mechanical properties between them. To obtain the desired values of the clamping elements by the compression test, the single-factor and Box-Behnken tests were designed. First, the clamping elements attached rubber were fixed to the testing machine with pins. e position between the bottle and the upper one until they contacted tightly with each other without any interaction. According to the two designed tests, the compression test was conducted using the electronic universal testing machine (UTM6503) under a sliding speed of 5 mm/min. When the testing machine stopped, the long axis and short axis of the bottle mouth were measured by an electronic vernier caliper (SR44). After that, a pointer push-pull force meter (NK-50) was used to measure the friction coefficient between the bottle and the rubber. e data obtained were recorded for further analysis.
Furthermore, the stress analysis of the bottle in the test is shown in Figure 3. It was not analyzed and drawn in Figure 3 because the gravity of the bottle is equal to the support produced by the inferior clamping element [24]. en, where F is the pull force measured by the dynamometer, N; F f1 and F f2 are the sliding friction force generated by the positive pressure in the pull direction of the push-pull dynamometer, N; F N1 and F N2 are the positive pressure of the clamping element on the bottle, N; and μ is the equivalent friction coefficient between the clamping element and the bottle.

Design of Experiments.
According to the designed bacteria bottle clamping elements, the main factors affecting the holding quality are the inner arc surface area, the downward displacement, and the rubber thickness of the clamping elements. And consequences are the pressure on the bacteria bottle, the ratio of the long and short axis, and the pull on the bacteria bottle.

Single-Factor Test.
Factors and levels of the singlefactor test were selected by the designed clamping elements and the final effect, as shown in Table 1. Some details were as follows.

Journal of Mathematics
Areas selected should have an appropriate range and interval. Based on the self-made clamping elements, the inner arc surface area's levels are "2121 mm 2 , 2969 mm 2 , 3958 mm 2 , 4948 mm 2 , and 6362 mm 2 ." It can include different angles (β′) and thickness (h) of the clamping elements. e thickness of the rubber, "1 mm, 2 mm, 3 mm, 4 mm, and 5 mm," is chosen as the single-factor test levels according to the existing ones in the market. e compression test of the whole bottle was conducted with the area 2121 mm 2 and 6362 mm 2 , as shown in Figure 4. And the curves were obtained, as shown in Figure 5. It illustrated that the curves' trends were increasing quickly when the displacement was beyond 2 mm. rough the pull-out test of the bottled F. velutipes, the average pull-out force required for picking is 30 N. When the safety factor is 1.1, the maximum safe pull-out load is about 32.94 N [25]. According to the model of pull-out (Figure 3), it is calculated that the pressure was at least 35 N to ensure to pick bottled F. velutipes under the coefficient of friction of 0.47. erefore, the force above 35 N is necessary, and the displacement is 1.4 mm at this time according to Figure 5. If the downward displacement were too large, the pressure on the bottle would be too tremendous. It would affect the subsequent cycle usage and lifetime of bottles. erefore, we took 0.1 mm as a value interval. And the singlefactor test levels of downward displacements were 1.4 mm, 1.5 mm, 1.6 mm, 1.7 mm, and 1.8 mm.

Box-Behnken Test.
e Box-Behnken technique is a technique that makes an efficient design space, and it requires only the fraction of full factorial points to estimate the   second-order effects of the response surface [26]. Based on the single-factor test results, the Box-Behnken test was performed with a combination of A, B, and C factors each at three levels (high, +1, medium, 0, and low, −1). e results are shown in Table 2.

e Influence of the Internal Arc Surface Areas on the Detection Indicator.
ree physical quantities were used as detection indicators to evaluate the clamping effect. ey are pressure value, long and short axis ratio of bottle mouth, and pull value. ey can reflect the clamping elements' holding effect and the degree of deformation of the bacteria bottle.
When the downward displacement of the clamping was 1.6 mm and the rubber thickness was 3 mm, the bacteria bottle pressure test was carried out by using different clamping elements which had different arc surface areas. Figure 6 shows the results of the pressure, pull, and the ratio with different areas changing. e range of tensile force measured by push-pull dynamometer is from 0 to 50 N. When the value exceeded the range, it was calculated according to the equivalent friction coefficient between the bacteria bottle and rubber obtained from the test. en, the corresponding value was obtained. e average of the equivalent friction coefficient between the bottle and the rubber was 0.458. As shown in Figure 6, it demonstrated that the pressure of the bacteria bottle and the tensile force produced by the dynamometer had similar trends, and the ratio fluctuated with the increase of the inner arc surface area of the clamping elements. e clamping elements need to meet the requirements for picking bottled F. velutipes, and it will not exert a significant force on the bacteria bottle, so as not to affect subsequent use. erefore,   Journal of Mathematics it is necessary to have a small deformation of the bacteria bottle. It should have more appropriate pressure and tension. Besides, the selected inner arc surface areas should conclude three kinds of central angles. Based on the results, the three levels of the arc surface area were selected. ey were 2121 mm 2 , 3958 mm 2 , and 4948 mm 2 .

e Influence of Downward Displacement on Detection
Indicator.
e bacteria bottle pressure test was carried out while the arc surface area was 3958 mm 2 and the rubber thickness was 3 mm. Figure 7 shows the pressure, pull, and the ratio with different downward displacements.
All three detection indexes have increased with downward displacement increasing. To meet the requirements, it should cause small damage and deformation to the bacteria bottle. ree downward displacements were selected. ey were 1.4 mm, 1.5 mm, and 1.6 mm.

3.1.3.
e Influence of Rubber ickness on Detection Indicator. When the arc surface area was 3958 mm 2 and the rubber thickness was 3 mm, the bacteria bottle pressure test was carried out using different clamping elements with different thicknesses of rubber. Figure 8 presents the pressure, pull, and the ratio with the different thickness of rubber changing. e three detection index values showed a downward trend when the thickness of the rubber was more exceptional than 3 mm. To meet the picking requirements, the tensile force was not less than 35 N. When the rubber thickness was 5 mm, it was not sufficient for the test. As for the "1 mm" thick rubber, the pressure of the bacteria bottle was immense, and the thickness of 1 mm was relatively thinner than 3 mm. It was not conducive to reducing the buffer of the force from the clamping component. erefore, the thicknesses, "2 mm, 3 mm, and 4 mm," were chosen as the comprehensive consideration factors.

e Results of Box-Behnken Test.
To explore the optimal value of the clamping elements' relevant parameters, the three values, the pressure on the bacteria bottle, the lengthto-short axis ratio of the bottle mouth, and the value of the tensile force, were used as the response values. e experimental design scheme and results are shown in Table 3.
A mathematical model was exploited. And the relevant summary results of quadratic and linear models were specified in ANOVA, as shown in Table 4 [27]. e response surface is constructed as a surrogate model based on the most important effects. To find the relationship between the objective function and factor values, the following second-order polynomial model is used, in which {x1, ..., xk} are the factors and βi s are the coefficients and e is the approximation error [26]. en, Comparing the values between the predicted at the design point and the average prediction error, the value of adequate precision (AP) in this model showed that it was well above (4). e models can be used to navigate the design space. Moreover, according to equation (        Journal of Mathematics where Y 1 is the code value of the pressure value; Y 2 is the code value of the ratio; Y 3 is the code value of pull value; A represents the code values of the inner surface area, B represents the code values of the down pressure displacement, and C represents the code values of the rubber thickness. e parameters whose p value is less than 0.05 are statistically significant. It can be seen in Table 4. It has been found that the models are fairly well fitted with the experimental values [28].

Effect of Experimental Parameters on Detection
Indicators. e response surface method [29][30][31][32] is a common way to analyze the effect of factors on the detection indicators. erefore, according to the established regression equation, we obtained the relationships between the factors and indicators by establishing the response surfaces. Figure 9 shows the effect of experimental parameters on pressure value. e range of pressure value was from 33.843 N to 97.608 N. With the value of displacement increasing, the pressure value increased first and decreased later. e same trend was demonstrated in the relationship between the rubber thickness and the pressure value. However, the pressure value was increased with the areas increasing. Several designed points are out of the predicted value. Moreover, according to the contours of the response surface, it showed that the ellipse's curvature in Figure 9(b) was large. erefore, there was an interaction between the arc surface area of the clamping elements and the rubber thickness, and the interaction had a significant effect on the pressure of the bacteria bottle.  Journal of Mathematics e effect of experimental parameters on the long and short axis ratio is shown in Figure 10. e range of the ratio was from 1.0074 to 1.0341. ree factors have different effects on the ratio. When the displacement increased, the ratio increased first and decreased later. e same phenomenon of change can be found in the effect of thickness. e ratio decreased with the area decreasing, respectively. It also has several designed points beyond the predicted value. e contours of the response surface showed that the elliptic curvature in Figure 10(b) was the largest.
ere was an interaction between the arc surface area of the clamping element and the rubber thickness. Figure 11 shows the influence of the three factors on the pull value of the bacteria bottle. e pull value was in the range of 31.055 N to 89.54 N. e more significant the area is, the higher the pull value is. Moreover, the pull value increased first and decreased later due to the increase of the displacement or the thickness. It can be found that some designed points are out of range. e contour of the response surface showed that the elliptic curvature in Figure 10(b) was the largest. It had an interaction between the arc surface area of the clamping elements and the rubber thickness.

Experimental Verification.
According to the Design-Expert software, the multiobjective and nonlinear optimization method was used to solve the regression model under the target, and the optimal parameter combination was finally obtained. e optimization constraints were min Y 1 (A, B, C), min Y 2 (A, B, C), and Y 3 (A, B, C) Finally, the optimized parameters were obtained. e inner arc area of the clamping element was 4948 mm 2 . e downward pressure displacement was 1.48 mm. e rubber thickness was 3.69 mm. e pressure of the bacteria bottle was 73.84 N. e long and short axis ratio of the bacteria bottle mouth was 1.02, and the tensile force of the bacteria bottle was 67.75 N.  Journal of Mathematics To verify the feasibility of the optimization results, experimental verification of the optimization parameters was performed. e inner arc surface area of the clamping element was 4948 mm 2 , the downward displacement was 1.48 mm, and the rubber thickness was 3.7 mm. Under these conditions, the bacteria bottle pressure test was repeated three times, and an average was obtained. As shown in Table 5, the results were obtained and were close to the predicted values. erefore, it appears that the relevant test research meets the clamping component's performance and technical requirements.

Discussion
e clamping elements' design is similar to banana fixtures [17,18] in terms of research methods, though they have different mechanical tests subjects. Besides, some models and methods using the design of Wu and Xi [8] are also evaluated in our study. It can improve the property of the clamping elements due to using these reasonable methods. e pressure test of a whole bacteria bottle was conducted for selecting levels of the single-factor test. e results are shown in Table 1. Compared with other studies [23], the test machine is used to produce a uniform force on the bottle's surface instead of using different weights. It is more direct, accurate, and convenient.
A Box-Behnken test was designed and used to obtain related data. e results are shown in Tables 2 and 3. Moreover, the mathematical models were established by referring the analysis of variance and response surface methods [27,30]. e relationships among three factors were obtained after analyzing formulas (4) to (6). Figures 9  to 11 show the response surface exhibiting the effects of three factors on the detection indicators. e response surfaces reflect that they have a good agreement of the variance with mathematics analysis, and results are observed. In detail, the estimated value and measured value are presented in Table 5.
ey have a good fit in an acceptable range of errors. Finally, the optimized parameters are validated by its great agreement with well-established analytical models and verification experiments.  In the present study, we investigated the fact that labors picked the bottled mushroom due to lack of the harvester. And a set of clamping elements was devised to effectively hold the bacteria bottles, which was an essential part of designing a picking robot. We conducted the Box-Behnken test to obtain data that were used to establish the regression models. e response surface method was applied to obtain the relationship between the variation and variation resources.
e results demonstrated that the mathematical models had a good fit to the designed experiment. e verification test confirmed that the optimized clamping elements had a good clamping effect. is finding is significant because it provides the possibility of using a robot to pick the bottled mushroom, especially the bottled F. velutipes. To our knowledge, this was the first time a mechanical test was conducted using the bacteria bottles with fresh culture material, aiming to design and optimize a fixture for bottled mushroom picking robots. However, the bacteria bottle's mechanical properties have a few differences from that of the plastic bottle because of the different contents in bottles. And the effect of the response surface method is verified again. Compared to other studies, we connected the mechanical test with the response surface method to design a fixture. It is more direct and concise in optimizing parameters of clamping elements.  To improve the clamping elements' holding effect, the method of mechanical test needs to be investigated systematically. e range of the pointer push-pull force meter is limited. A further study should be performed to verify whether using a force transducer can make a difference in this study or not. Besides, the method of choosing the area of the clamping elements should be improved further. Finding a more accurate project becomes necessary in the future. In this study, we demonstrated that the clamping elements were designed and optimized by the mechanical test and response surface method. e desired values have a good agreement with the experiment.
is suggests the feasibility of designing a fixture of a thin-walled pipe by using the method of our study process. e optimized clamping elements would be a valuable part of the application in designing a bacteria bottle fixture of the picking robot.

Conclusion
In this paper, we proposed a kind of clamping element for designing a fixture of the bottled F. velutipes picking machine. After the experimental research, the influences of process parameters on pressure, long and short axis ratio, and rubber thickness were investigated. To obtain the desired values of the clamping elements' parameters, the adequacy of the proposed model was examined and the confirmation test was carried out to study the accuracy of the optimized parameters. e differences between the actual value and the predicted value were seen as within the acceptable limit. For these experimental investigations, the following conclusion has been drawn: (a) e clamping elements were designed according to the medium bacteria bottles that were used universally in the industrial production of edible fungi. e pressure test of the bacteria bottle was carried out using the clamping elements. e significant levels were obtained by conducting a single-factor test. (b) A Box-Behnken test was used to establish a mathematical regression model between the performance indicators of the bacteria bottle pressure test and three test factors (including the inner arc surface area, the displacement of the pressure, and the rubber thickness). e influence and interaction of each factor on the test index were obtained by establishing variance models. Among these experimental factors, the inner arc surface areas significantly influence the pressure and pull value of the bacteria bottle. e ratio of the long and short axis can be influenced dramatically by changing the rubber thickness. (c) Mathematical models were established for analyzing the test data, and the response surface method was used to optimize the regression equation. e optimal parameters were as follows: the area of the clamping element's internal arc surface was 4948 mm 2 , the downward displacement was 1.48 mm, and the rubber thickness was 3.69 mm.
Meanwhile, the bacteria bottle's pressure value was 73.84 N, the long and short axis ratio of the bacteria bottle was 1.02, and the strain value of the bacteria bottle was 67.75 N. e same test method was used to verify the optimization results. e actual values were as follows: the pressure value of the bacteria bottle was 75.69 N. e ratio of the long and short axis of the bacteria bottle mouth was 1.03, and the strain value of the bacteria bottle was 69.45 N. (d) e method of the mechanical test needs to be investigated systematically to improve the clamping elements' holding effect. e designed clamping elements are used to hold the bottled F. velutipes. It is important to study whether different cultures in bottles have different effects on the results or not. It is also necessary to develop a kind of fixture that can hold several bacteria bottles to improve the efficiency of harvesting in the future.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.