Path synthesis of spatial revolute–spherical–cylindrical–revolute mechanisms using deep learning

Abstract

The design of single-degree-of-freedom spatial mechanisms tracing a given path is challenging due to the highly non-linear relationships between coupler curves and mechanism parameters. This work introduces an innovative application of deep learning to the spatial path synthesis of one-degree-of-freedom spatial revolute–spherical–cylindrical–revolute (RSCR) mechanisms, aiming to find the non-linear mapping between coupler curve and mechanism parameters and generate diverse solutions to the path synthesis problem. Several deep learning models are explored, including multi-layer perceptron (MLP), variational autoencoder (VAE) plus MLP, and a novel model using conditional-\beta -\text{VAE} (\text{c}-\beta -\text{VAE}). We found that the \text{c}- \beta -\text{VAE} model with β = 10 achieves superior performance by predicting multiple mechanisms capable of generating paths that closely approximate the desired input path. This study also builds a publicly available database of over 5 million paths and their corresponding RSCR mechanisms. The database provides a solid foundation for training deep learning models. An application in the design of human upper-limb rehabilitation mechanism is presented. Several RSCR mechanisms closely matching the wrist and elbow path collected from human movements are found using our deep learning models. This application underscores the potential of RSCR mechanisms and the effectiveness of our model in addressing complex, real-world spatial mechanism design problems.

Graphical abstract

Keywords

spatial mechanism / neural networks / path synthesis / machine learning / deep learning / generative models

References

[1] Zhao P, Zhang Y T, Guan H W, Deng X T, Chen H D. Design of a single-degree-of-freedom immersive rehabilitation device for clustered upper-limb motion. Journal of Mechanisms and Robotics, 2021, 13(3): 031006

[2] Song W, Zhao P, Li X, Deng X T, Zi B. Data-driven design of a six-bar lower-limb rehabilitation mechanism based on gait trajectory prediction. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2023, 31: 109–118

[3] ZhangYDeng X TZhouBZhaoP. Design and optimization of a multi-mode single-DOF watt-I six-bar mechanism with one adjustable parameter. In: Proceeding of Advances in Mechanism, Machine Science and Engineering in China. Singapore: Springer, 2022

[4] Deng X T, Purwar A. A matrix-based approach to unified synthesis of planar four-bar mechanisms for motion generation with position, velocity, and acceleration constraints. ASME Journal of Computing and Information Science in Engineering, 2024, 24(12): 121003

[5] Watanabe K, Sekine T, Nango J. Kinematic analysis and branch identification of RSCR spatial four link mechanisms. JSME International Journal Series C Mechanical Systems, Machine Elements and Manufacturing, 1998, 41(3): 450–459

[6] ThompsonJ M. Computer aided design and synthesis of the RSCR spatial mechanism. Thesis for the Master’s Degree. Blacksburg: Virginia Polytechnic Institute and State University, 1987

[7] BagciC. The RSRC space mechanism – analysis by 3 × 3 screw matrix, synthesis for screw generation by variational methods. Dissertation for the Doctoral Degree. Oklahoma State University, 1969

[8] Chiang C H, Chieng W H, Hoeltzel D A. Synthesis of the RSCR mechanism for four precision positions with relaxed specifications. Mechanism and Machine Theory, 1992, 27(2): 157–167

[9] Ananthasuresh G K, Kramer S N. Analysis and optimal synthesis of the RSCR spatial mechanisms. Journal of Mechanical Design, 1994, 116(1): 174–181

[10] Shrivastava A K, Hunt K H. Dwell motion from spatial linkages. Journal of Engineering for Industry, 1973, 95(2): 511–518

[11] Wang X, Guo W Z. The design of looped-synchronous mechanism with duplicated spatial assur-groups. Journal of Mechanisms and Robotics, 2019, 11(4): 041014

[12] Osman M O M, Segev D. Kinematic analysis of spatial mechanisms by means of constant distance equations. Transactions of the Canadian Society for Mechanical Engineering, 1972, 1(3): 129–134

[13] Osman M O M, Bahgat B M, Dukkipati R V. Kinematic analysis of spatial mechanisms using train components. Journal of Mechanical Design, 1981, 103(4): 823–830

[14] Huang T C, Youm Y. Exact displacement analysis of four-link spatial mechanisms by the direction cosine matrix method. Journal of Applied Mechanics, 1984, 51(4): 921–928

[15] Regenwetter L, Nobari A H, Ahmed F. Deep generative models in engineering design: A review. Journal of Mechanical Design, 2022, 144(7): 071704

[16] Vasiliu A, Yannou B. Dimensional synthesis of planar mechanisms using neural networks: application to path generator linkages. Mechanism and Machine Theory, 2001, 36(2): 299–310

[17] Galán-Marín G, Alonso F J, Del Castillo J M. Shape optimization for path synthesis of crank-rocker mechanisms using a wavelet-based neural network. Mechanism and Machine Theory, 2009, 44(6): 1132–1143

[18] ChuiC K. An Introduction to Wavelets. San Diego: Academic Press, 1992

[19] Deshpande S, Purwar A. A machine learning approach to kinematic synthesis of defect-free planar four-bar linkages. Journal of Computing and Information Science in Engineering, 2019, 19(2): 021004

[20] Deshpande S, Purwar A. Computational creativity via assisted variational synthesis of mechanisms using deep generative models. Journal of Mechanical Design, 2019, 141(12): 121402

[21] Sharma S, Purwar A. A machine learning approach to solve the alt–burmester problem for synthesis of defect-free spatial mechanisms. Journal of Computing and Information Science in Engineering, 2022, 22(2): 021003

[22] Nurizada A, Purwar A. An invariant representation of coupler curves using a variational autoencoder: Application to path synthesis of four-bar mechanisms. Journal of Computing and Information Science in Engineering, 2024, 24(1): 011008

[23] LeeSKimJ KangN. Deep generative model-based synthesis of four-bar linkage mechanisms considering both kinematic and dynamic conditions. In: Proceedings of the ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Boston: ASME, 2023, V03AT03A016

[24] Purwar A, Chakraborty N. Deep learning-driven design of robot mechanisms. Journal of Computing and Information Science in Engineering, 2023, 23(6): 060811

[25] NobariA HSrivastava AGutfreundDXuKAhmedF. LInK: learning joint representations of design and performance spaces through contrastive learning for mechanism synthesis. 2024, arXiv preprint arXiv:2405.20592

[26] YimN HRyu JKimY Y. Big data approach for synthesizing a spatial linkage mechanism. In: Proceedings of IEEE International Conference on Robotics and Automation. London: IEEE, 2023, 7433–7439

[27] KingmaD PWelling M. Auto-encoding variational Bayes. 2013, arXiv preprint arXiv:1312.6114

[28] Nurizada A, Dhaipule Z, R A. A dataset of 3M single-DOF planar 4-, 6-, and 8-bar linkage mechanisms with open and closed coupler curves for machine learning-driven path synthesis. ASME Journal of Mechanical Design, 2025, 147(4): 041702

[29] Nurizada A, Lyu Z, Purwar A. Path generative model based on conditional β-variational auto encoder for four-bar mechanism design. Journal of Mechanisms and Robotics, 2025, 17(6): 061004

[30] DengXNurizada APurwarA. Synthesizing spatial RSCR mechanisms for path generation using a deep neural network. In: Proceedings of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASME, 2024

[31] Javier G dJEduardo B. Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. New York: Springer, 2011, 72–74

[32] VirtanenPGommers ROliphantT EHaberlandMReddyT CournapeauDBurovski EPetersonPWeckesserWBrightJ vander Walt S JBrettMWilsonJ MillmanK JMayorov NNelsonA R JJonesEKernR LarsonECarey C JPolatİFengYMooreE W VanderPlasJLaxalde DPerktoldJCimrmanRHenriksen IQuinteroE AHarrisC RArchibald A MRibeiroA HPedregosaFvanMulbregt PSciPy1.0 Contributors. SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 2020, 17: 261–272

[33] Chase T R, Mirth J A. Circuits and branches of single-degree-of-freedom planar linkages. Journal of Mechanical Design, 1993, 115(2): 223–230

[34] PieglLTiller W. The NURBS Book. 2nd ed. Berlin: Springer, 1997

[35] LyuZ JPurwar A. Design and development of a sit-to-stand device using a variational autoencoder-based deep neural network. In: Proceedings of ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. St. Louis: ASME, 2022, V007T07A027

[36] JolliffeI T. Principal Component Analysis. 2nd ed. New York: Springer, 2002

[37] Yu S C, Chang Y, Lee J J. A generative model for path synthesis of four-bar linkages via uniform sampling dataset. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2023, 237(4): 811–829

[38] SenerSUnel M. Geometric invariant curve and surface normalization. In: Campilho A, Kamel M, eds. Image Analysis and Recognition. Berlin: Springer, 2006, 445–456

[39] Purwarlab. RSCR Mechanisms. 2025, available at www.kaggle.com/datasets/purwarlab/rscr-mechanisms website

[40] HaykinS. Neural Networks: A Comprehensive Foundation. 2nd ed. Upper Saddle River: Prentice Hall, 1998

[41] AgarapA F. Deep learning using rectified linear units (ReLU). 2019, arXiv preprint arXiv:1803.08375

[42] BaJ LKiros J RHintonG E. Layer normalization. 2016, arXiv preprint arXiv:1607.06450

[43] Srivastava N, Hinton G, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 2014, 15(1): 1929–1958

[44] VaswaniAShazeer NParmarNUszkoreitJJonesL GomezA NKaiser ŁPolosukhinI. Attention is all you need. In: Proceedings of the 31st International Conference on Neural Information Processing Systems (NIPS 2017). Red Hook, NY: Curran Associates Inc., 2017, 6000–6010

[45] Han K, Wang Y H, Chen H T, Chen X H, Guo J Y, Liu Z H, Tang Y H, Xiao A, Xu C J, Xu Y X, Yang Z H, Zhang Y M, Tao D C. A survey on vision transformer. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023, 45(1): 87–110

[46] SuttonR SBarto A G. Reinforcement Learning: An Introduction. 2nd ed. Cambridge: MIT Press, 2018

[47] SohnKLee HYanX C. Learning structured output representation using deep conditional generative models. In: Proceedings of the 28th International Conference on Neural Information Processing Systems. Cambridge: MIT Press, 2015, 3483–3491

[48] HigginsIMatthey LPalABurgessCGlorotX BotvinickMMohamed SLerchnerA. β-VAE: Learning basic visual concepts with a constrained variational framework. In: Proceedings of International Conference on Learning Representations (ICLR 2017). 2017

[49] Rote G. Computing the minimum Hausdorff distance between two point sets on a line under translation. Information Processing Letters, 1991, 38(3): 123–127

[50] McNemar Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 1947, 12(2): 153–157

[51] RaschkaS. Model evaluation, model selection, and algorithm selection in machine learning. 2020, arXiv preprint arXiv:1811.12808

RIGHTS & PERMISSIONS

The Author(s). This article is published with open access at link.springer.com and journal.hep.com.cn