Application of Multiple Deep Neural Networks to MultiSolution Synthesis of Linkage Mechanisms
Abstract
:1. Introduction
References  Year  Mechanism  Curve Descriptor  NN Model  Additional Features (Selected Portions Only) 

Hoskins & Kramer [16]  1993  CrankRocker  Power spectrum  Radial basis NN  Hybridizing a gradientbased numerical method 
Yannou & Vasiliu [17]  2001  CrankRocker  Fourier series  MLFFNN  Developing an integrated predesign platform RealisMe 
Xie & Chen [23]  2007  CrankRocker  Fourier series  MLFFNN  Extending FD to the image space of kinematic mapping 
Erkaya & Uzmay [24]  2009  SliderCrank  Cartesian positions  MLFFNN  Modelling joint clearance as a massless link 
GalánMarín et al. [13]  2009  CrankRocker  Wavelet  MLFFNN  Sampling precise points at a nonconstant time interval 
Khan et al. [18]  2015  CrankRocker  Fourier series  MLFFNN  Hybridizing a local optimization procedure 
Ahmadi et al. [25]  2016  General fourbar  Cartesian positions  GMDHtype NNs  Integrating game theory and multiobjective optimization 
Li & Chen [19]  2017  General fourbar  Fourier series  MLFFNN  Proposing arc length normalization 
Deshpande & Purwar [21]  2018  General fourbar  Signature method  AutoEncoder  Integrating machine learning and computational kinematics for defectfree and parttowhole synthesis 
Mo et al. [26]  2019  CrankRocker  Fourier series  MLFFNN  Obtaining a high precision linkager mechanism 
Yim et al. [27]  2021  General fourbar  Fourier series  Deep MLFFNN  Determining mechanism topology and endeffector location simultaneously based on big data 
Kapsalyamov et al. [28]  2022  Sixlinkagebar  Cartesian positions  Deep MLFFNN  Integrating computational kinematics and machine learning to syntherize two joint trajectories (ankle and knee) 
Yim et al. [29]  2023  Spatial linkage  Fourier series  Deep MLFFNN  Making the NN handle multiclass classification to improve the previous planar linkage synthesis approach 
2. Problem Definition and Formulation
2.1. Fourier Descriptor Formulation
2.2. Fourier Coefficient Normalizing and Learning
2.3. OnetoMany Mapping Issues
 (1)
 Cognate linkages
 (2)
 Factors of normalization
 (3)
 Incomplete coupling at precise points
2.4. Learning OnetoMany Mapping by Neural Networks
3. Synthesis Using Multiple DNNs
3.1. Dataset Partition and Generation for DNN Training Flow
3.1.1. MultiSolution Distribution Evaluation by Random Restart Local Searches (MDERRLS)
Algorithm 1: MDERRLS 
Input:

Begin

3.1.2. Dataset Generation & Partition
3.1.3. Training DNNs by Partitioned Datasets
3.2. Predicting Flow to Obtain One or Multiple Solutions
3.2.1. MultiFacet Query
3.2.2. Voting Method
4. Experiments and Discussions
4.1. MDERRLS Evaluation and Selection of Subspace Partitions
4.2. Training Parameter Selection
4.3. Comparison with Literature Cases
5. Application to Design an Industrial SixBar Ladle Mechanism
5.1. Design Precise Points and Partition Schemes
5.2. MultiDNNs Training Results
5.3. Design Refinement in a ShortTime Response
6. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Data Set Amount  Mean (Standard Deviation) of Prediction Errors  (Unit: rad)  

$$\mathit{\theta}:\left[\frac{1}{2}\mathit{\pi},\frac{1}{2}\mathit{\pi}\right]$$

$$\mathit{\theta}:\left[\frac{1}{4}\mathit{\pi},\frac{1}{2}\mathit{\pi}\right]$$

$$\mathit{\theta}:\left[0,\frac{1}{2}\mathit{\pi}\right]$$

$$\mathit{\theta}:\left[\frac{1}{2}\mathit{\pi},0\right]$$
 
5000  0.7309 (0.0132)  0.3411 (*0.004)  0.0329 (0.0079)  0.0372 (0.0149) 
10,000  0.7404 (0.0039)  0.3005 (0.0114)  0.0282 (0.0159)  0.0547 (0.0213) 
50,000  0.7655 (0.0001)  0.2918 (0.0095)  0.0137 (0.0062)  0.0232 (0.0106) 
2 Regions  MAX (SUM) 

* r_{2}:2  55 (100) 
r_{3}:2  57 (72) 
r_{4}:2  56 (71) 
Ψ:2  13 (22) 
r_{5}:2  55 (76) 
CFG:2  15 (25) 
4 Regions  MAX (SUM) 
Ψ:4  6 (11) 
Ψ:2, CFG:2  3 (6) 
8 Regions  MAX (SUM) 
Ψ:4, CFG:2  2 (6) 
Ψ:8  2 (6) 
Data Amount  RMSE of Predictions in Different Partition Methods  Mean (std)  

Ψ:2  CFG:2  Ψ:4  Ψ:2, CFG:2  Ψ:8  Ψ:4, CFG:2  No Partition  
80,000  0.5597 (0.0423)  0.596 (0.0302)  0.5349 (0.0202)  *0.5138 (0.0131)  0.5552 (0.0222)  0.5162 (0.0223)  1.4377 (0.0496) 
400,000  0.3298 (0.023)  0.4262 (0.0301)  0.2866 (0.0261)  0.2758 (0.0171)  0.3212 (0.013)  0.2922 (0.014)  1.3279 (0.0367) 
800,000  0.277 (0.0167)  0.3286 (0.0296)  0.2296 (0.0125)  0.232 (0.0244)  0.2488 (0.011)  0.2368 (0.0132)  1.2264 (0.0382) 
1,600,000  0.2452 (0.0181)  0.2725 (0.0244)  0.2197 (0.0182)  0.2139 (0.0227)  0.2114 (0.0075)  0.2061 (0.0148)  1.1914 (0.0262) 
Study Cases  ${\mathit{\theta}}_{1}$  r_{1}  r_{2}  r_{3}  r_{4}  $\mathsf{\phi}$  r_{5}  CFG  Projection 

#1  −0.52  234.23  64.62  249.61  360.60  4.22  163.67  0  None 
#2  −0.27  135.82  35.68  246.41  265.25  0.02  358.53  0  Horizontal 
#3  −0.45  299.31  76.90  361.66  402.66  4.71  298.47  0  None 
#4  −0.19  234.23  85.09  260.69  141.45  0.36  230.54  0  Horizontal 
2 Regions  MAX (SUM) 

C_{y}:2  28 (32) 
r_{1}:2  16 (28) 
r_{2}:2  23 (26) 
r_{3}:2  17 (23) 
r_{4}:2  20 (25) 
r_{5}:2  19 (22) 
r_{6}:2  13 (18) 
r_{7}:2  *9 (14) 
4 Regions  MAX (SUM) 
r_{7}:4  3 (8) 
r_{7}:2, r_{6}:2  3 (6) 
8 Regions  MAX (SUM) 
r_{7}:4, r_{6}:2  1 (2) 
r_{7}:8  2 (6) 
DNN No  o_{x}  o_{y}  c_{x}  c_{y}  r_{1}  r_{2}  r_{3}  r_{4}  r_{5}  r_{6}  r_{7} 

1  13.92  92.89  186.72  615.26  219.84  498.5  323.27  1427.38  452.83  1058.51  964.14 
2  62.56  186.41  215.45  710.56  207.65  467.71  301.6  1367.77  451.54  910.34  1088.36 
3  95.67  115.96  186.42  618.8  203.89  438.73  259.53  1851.51  550.87  1393.61  1597.08 
4  114.26  176.75  210.67  690.03  200.5  426.18  297.89  1505.03  450.45  1045.28  1530.64 
5  4.78  246.23  217.5  709.54  194.58  469.81  305.44  1352.53  347.01  1081.34  725.82 
6  18.19  234.81  212.45  714.23  197.21  475.72  321.05  1486.01  391.81  1094.67  1129.86 
7  93.63  358.67  263.03  862  188.17  447.75  296.24  1470.82  371.21  1051.77  1210.44 
8  37.82  133.99  213.72  684.79  222.4  503.03  331  1319.76  334.43  791.37  1208.21 
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Chen, C.H. Application of Multiple Deep Neural Networks to MultiSolution Synthesis of Linkage Mechanisms. Machines 2023, 11, 1018. https://doi.org/10.3390/machines11111018
Chen CH. Application of Multiple Deep Neural Networks to MultiSolution Synthesis of Linkage Mechanisms. Machines. 2023; 11(11):1018. https://doi.org/10.3390/machines11111018
Chicago/Turabian StyleChen, ChiuHung. 2023. "Application of Multiple Deep Neural Networks to MultiSolution Synthesis of Linkage Mechanisms" Machines 11, no. 11: 1018. https://doi.org/10.3390/machines11111018