Liye et al 2023a - Revision history

Lvliye at 21:40, 29 October 2023

2023-10-29T21:40:00Z

Rimni at 09:00, 3 October 2023

2023-10-03T09:00:12Z

Rimni at 08:59, 3 October 2023

2023-10-03T08:59:34Z

Rimni at 08:55, 3 October 2023

2023-10-03T08:55:18Z

Rimni at 13:48, 2 October 2023

2023-10-02T13:48:44Z

Rimni: /* References */

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‎References

Rimni: /* References */

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‎References

Rimni at 11:55, 2 October 2023

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Rimni at 11:52, 2 October 2023

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 ==Acknowledgment==
-The research was supported by the Zhejiang Province Key Research and Development Project (2021C01070).
+The research was supported by the Zhejiang Province Key Research and Development Project (2021C01070) and Zhejiang Province Science and Technology Plan Project (2023C01174).
 ====Data availability statement====

@@ Line 335: / Line 335: @@
 ====4.2.1 Problem definition====
-The surrogate models built in Section 3 need to be used instead of the simulation model for optimization design and analysis. In the whole gantry optimization process, total deformation and gantry weight are the two optimization objectives. In fact, these two optimization goals are contradictory to each other. Structures with less deformation have better rigidity and corresponding weight will be larger. Therefore, this problem is a multi-objective optimization design problem. There are two main constraints. The first one is the maximum stress. The material of the gantry is 25MnV alloy steel with yield strength of 590.0 MPa, and the safety factor is taken as 1.8, then the allowable stress of the gantry is <math display="inline">\left[ \sigma \right] =</math><math>\frac{590}{1.8}=327.8</math> MPa, and the maximum stress of the gantry <math display="inline">{\sigma}_{\max}\leq \left[ \sigma \right]</math>. The other one is the gantry size. Considering the specific applications and working conditions, load, and local stability, it is necessary to constrain geometry size of the non-standard I-beam. To obtain the optimal design of the forklift gantry, multi-objective generic algorithm (MOGA) is used herein. The entire gantry optimization problem is then described as Eq. (8). To obtain the optimal design of the forklift gantry, generic algorithm (GA) is used herein
+The surrogate models built in Section 3 need to be used instead of the simulation model for optimization design and analysis. In the whole gantry optimization process, total deformation and gantry weight are the two optimization objectives. In fact, these two optimization goals are contradictory to each other. Structures with less deformation have better rigidity and corresponding weight will be larger. Therefore, this problem is a multi-objective optimization design problem. There are two main constraints. The first one is the maximum stress. The material of the gantry is 25 MnV alloy steel with yield strength of 590.0 MPa, and the safety factor is taken as 1.8, then the allowable stress of the gantry is <math display="inline">\left[ \sigma \right] =</math><math>\frac{590}{1.8}=327.8</math> MPa, and the maximum stress of the gantry <math display="inline">{\sigma}_{\max}\leq \left[ \sigma \right]</math>. The other one is the gantry size. Considering the specific applications and working conditions, load, and local stability, it is necessary to constrain geometry size of the non-standard I-beam. To obtain the optimal design of the forklift gantry, multi-objective generic algorithm (MOGA) is used herein. The entire gantry optimization problem is then described as Eq. (8). To obtain the optimal design of the forklift gantry, generic algorithm (GA) is used herein
 {| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;"

@@ Line 244: / Line 244: @@
 ===4.1 Accuracy analysis of surrogate models===
-In order to explore the performance of different individual surrogate models intuitively, both global criteria (i.e., <math display="inline">{R}^{2}</math> and <math display="inline">NRMSE</math>) and local criterion (i.e., <math display="inline">NMAE</math>) are employed which are displayed by bar charts in [[#img-6|Figure 6]] and listed in [[#tab-1|Table 1]]. It is obvious that the four surrogate models can better approximate the total deformation and maximum stress. For the gantry weight, the formula can usually be established with the given four design variables, so surrogate models can fit the gantry weight well. However, the fitting accuracy of the RBF model for the gantry weight is poor, probably due to the overfitting during the fitting process, which leads to a sharp decrease in the global fitting accuracy. In addition, it is found that surrogate models with better global accuracy do not necessarily have the best local accuracy, as in[[#img-6|Figure 6]](a), the global accuracy of PRS, KRG, and SVR are better than that of RBF, but the local accuracy of RBF model is better than that of the other three surrogate models. Considering the global and local performance together, for total deformation, maximum stress, and weight of gantry, the KRG model performs best, followed by the PRS model and the SVR model, and finally by the RBF model.
+In order to explore the performance of different individual surrogate models intuitively, both global criteria (i.e., <math display="inline">{R}^{2}</math> and <math display="inline">NRMSE</math>) and local criterion (i.e., <math display="inline">NMAE</math>) are employed which are displayed by bar charts in [[#img-6|Figure 6]] and listed in [[#tab-1|Table 1]]. It is obvious that the four surrogate models can better approximate the total deformation and maximum stress. For the gantry weight, the formula can usually be established with the given four design variables, so surrogate models can fit the gantry weight well. However, the fitting accuracy of the RBF model for the gantry weight is poor, probably due to the overfitting during the fitting process, which leads to a sharp decrease in the global fitting accuracy. In addition, it is found that surrogate models with better global accuracy do not necessarily have the best local accuracy, as in [[#img-6|Figure 6]](a), the global accuracy of PRS, KRG, and SVR are better than that of RBF, but the local accuracy of RBF model is better than that of the other three surrogate models. Considering the global and local performance together, for total deformation, maximum stress, and weight of gantry, the KRG model performs best, followed by the PRS model and the SVR model, and finally by the RBF model.
 <div id='img-6'></div>

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 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Lv_119229664-image1.jpeg|462px]]
+|style="text-align: center;padding:10px;"| [[Image:Draft_Lv_119229664-image1.jpeg|400px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 1'''. Structural diagram of AGV (1-forklift body, 2 -lifting system, 3-lifting platform, 4- docking mechanism)

@@ Line 776: / Line 776: @@
 The research was supported by the Zhejiang Province Key Research and Development Project (2021C01070).
-==Data availability statement==
+====Data availability statement====
 The data that support the findings of this study are available from the corresponding author upon reasonable request.

← Older revision		Revision as of 13:43, 2 October 2023
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	==References==		==References==
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 When designing the gantry, great attention should be paid to the influence of the height of I-beam. Compared with the simulation-based GSA, the surrogate-based GSA method can save a lot of time. In this problem, using 10000 Monte Carlo sample points, it needs to build a model 20000 times, and it takes about 48 hours to run the Sobol’s GSA method once, while it takes about 30 minutes to run the gantry model once, and it takes about 10000 hours to run 20000 times, which increases the speed by about 2083 times. The surrogate model technique has proven to be more effective for GSA method and more suitable for GSA of mechanical equipment and systems.
-==5 Conclusions==
+==5. Conclusions==
 This paper reviewed the previous gantry optimization design methods and surrogate models in different fields, as well as the development history of forklift industrial robots. Due to the gap between the domestic forklift research in terms of scientific application and product production, this paper compares and studies the application of four surrogate models, namely PRS, KRG, RBF and SVR, in the multi-objective optimization design of forklift gantry. The main conclusions are as follows.
-) The 3-D model of the gantry was established through statistical analysis and numerical settings, and four parameters, namely height ''h'', width ''b'', waist thickness ''d'', and average thickness ''t'' of the I-beam were selected as the design variables. Using the LHS sampling method, 40 training points and 10 testing points with good projection and space-filling property were selected.
+) The 3-D model of the gantry was established through statistical analysis and numerical settings, and four parameters, namely height <math display="inline"> h </math>, width <math display="inline"> b </math>, waist thickness <math display="inline"> d </math>, and average thickness <math display="inline"> t </math> of the I-beam were selected as the design variables. Using the LHS sampling method, 40 training points and 10 testing points with good projection and space-filling property were selected.
-) Global accuracy criteria (''R''<sup>2</sup> and ''NRMSE'') and local criterion (NMAE) were used to evaluate the prediction accuracy of each surrogate model. Comparing the total deformation, maximum stress, and I-beam weight, the KRG model performed the best, followed by the PRS model and the SVR model, and finally by the RBF model.
+) Global accuracy criteria (<math display="inline">R^2</math> and ''NRMSE'') and local criterion (NMAE) were used to evaluate the prediction accuracy of each surrogate model. Comparing the total deformation, maximum stress, and I-beam weight, the KRG model performed the best, followed by the PRS model and the SVR model, and finally by the RBF model.
-) Multi-objective optimization design for maximum total deformation and I-beam weight was conducted using I-beam geometry and maximum stress as constraints. Results of before and after optimization show that, for the total deformation, PRS, RBF, KRG, and SVR model decreases 16.2%, 17.2%, 23.2%, and 25.3%, respectively; For the maximum stress, PRS, RBF, KRG, and SVR model decreases 17.1%, 18.6%, 20.4%, and 20.8%, respectively; For the I-beam weight, PRS, RBF, KRG, and SVR model decreases 14.9%, 12.8%, 11.9%, and 8.1%, respectively. The optimal design based on KRG model has the best performance.
+) Multi-objective optimization design for maximum total deformation and I-beam weight was conducted using I-beam geometry and maximum stress as constraints. Results of before and after optimization show that, for the total deformation, PRS, RBF, KRG, and SVR model decreases 16.2%, 17.2%, 23.2%, and 25.3%, respectively. For the maximum stress, PRS, RBF, KRG, and SVR model decreases 17.1%, 18.6%, 20.4%, and 20.8%, respectively. For the I-beam weight, PRS, RBF, KRG, and SVR model decreases 14.9%, 12.8%, 11.9%, and 8.1%, respectively. The optimal design based on KRG model has the best performance.
 ) Finally, using the KRG model and Sobel’s GSA method, the GSA of the gantry was investigated. The height of the I-beam has the greatest influence on the performance of the gantry, and the average thickness also has a certain influence on the maximum stress of the gantry, so it is necessary to pay more attention to the height of the I-beam and the average thickness of the I-beam when designing forklift gantries in the future.

@@ Line 344: / Line 344: @@
 | <math>\mathrm{min}\,\quad F\left( b,h,d,t\right) ={\omega }_{1}{f}_{1}+{\omega }_{2}{f}_{2}</math>
-<math>S.t.\quad {\sigma }_{max}\leq 327.8\quad \quad \quad</math>
+<math>S.t.\quad {\sigma }_{\max}\leq 327.8\quad \quad \quad</math>
 <math>68.0\leq b\leq 94.0\quad</math>