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A data-driven reconstruction method for dynamic systems with multistable property

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Abstract

In this work, Sparse Identification of Nonlinear Dynamics (SINDy) algorithm is generalized to nonlinear dynamic systems with multistable property. This study comes from the application of a bistable bionic joint. In its dynamical model reconstruction, we find that the classical SINDy fails due to the lacking ergodicity with tiny amount of time history signals. Aiming at the accuracy and precision of model reconstruction of multistable nonlinear dynamic systems, a generalized data-driven reconstruction method is proposed based on data assembly principle and sparsification parameter determination technique. The datasets are expanded by different initial conditions for the dynamics ergodic. Besides, to resolve the difficulty on the determination of sparsification parameter, a novel bi-objective optimization scheme considering both sparsity and accuracy of the reconstruction model is constructed. Based on the Pareto front and Knee point theory of the bi-objective optimization scheme, it provides the precisely determination technique for the sparsification parameter range. Two numerical examples are conducted on bistable and quad-stable nonlinear systems to verify the effectiveness of the generalized data-driven reconstruction method. Then, the experimental prototype of the bionic joint with bistable property is carried out, which possesses both geometric and constitutive nonlinearity. Assembling the experimental signals with measurement noise according to the proposed data assembly principle, the generalized reconstruction method proposed in this paper can capture the position of the stable equilibriums and accurately reconstruct the dynamic model of the bistable bionic joint. Since the ergodic dataset can be further extended, proposed reconstruction method for dynamic systems with multistable property has effectiveness and robustness. The generalized data-driven reconstruction method successfully solves the problems of non-ergodic data and undetermined sparsification parameter in the model reconstruction of multistable systems, which has potential applications of multistable nonlinear systems in the fields of robotics, deployable structures, etc.

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All data generated or analyzed during this study are included in this published article.

References

  1. Wang, X., Zhou, H.Y., Kang, H.W., Au, W., Chen, C.: Bio-inspired soft bistable actuator with dual actuations. Smart Mater. Struct. 30, 125001 (2021)

    Google Scholar 

  2. Li, S.Y., Wang, K.W.: Fluidic origami with embedded pressure dependent multi-stability: a plant inspired innovation. J. R. Soc. Interface 12, 20150639 (2015)

    Google Scholar 

  3. Wu, S., Ze, Q.J., Dai, J.Z., Udipi, N., Paulino, G.H., Zhao, R.K.: Proc. stretchable origami robotic arm with omnidirectional bending and twisting, Multistable inflatable origami structures at the metre scale. Natl. Acad. Sci. USA 118, e2110023118 (2021)

    Google Scholar 

  4. Rothemund, P., Ainla, A., Belding, L., Preston, D.J., Kurihara, S., Suo, Z.G., Whitesides, G.M.: A soft, bistable valve for autonomous control of soft actuators. Sci. Robot. 3, eaar7986 (2018)

    Google Scholar 

  5. Wang, Y.Z., Gupta, U., Parulekar, N., Zhu, J.: A soft gripper of fast speed and low energy consumption. Sci. China-Technol. Sci. 62, 31 (2019)

    Google Scholar 

  6. Jin, T., Li, L., Wang, T.H., Wang, G.P., Cai, J.G., Tian, Y.Z., Zhang, Q.: Origami-inspired soft actuators for stimulus perception and crawling robot applications. IEEE Trans. Robot. 38, 748 (2022)

    Google Scholar 

  7. Jin, L., Khajehtourian, R., Mueller, J., Rafsanjani, A., Tournat, V., Bertoldi, K., Kochmann, D.M.: Guided transition waves in multistable mechanical metamaterials. Proc. Natl. Acad. Sci. USA 117, 2319 (2020)

    MATH  Google Scholar 

  8. Singh, N., van Hecke, M.: Design of pseudo-mechanisms and multistable units for mechanical metamaterials. Phys. Rev. Lett. 126, 248002 (2021)

    Google Scholar 

  9. Brunck, V., Lechenault, F., Reid, A., Adda-Bedia, M.: Elastic theory of origami-based metamaterials. Phys. Rev. E 93, 033005 (2016)

    Google Scholar 

  10. Karpov, E.G., Danso, L.A., Klein, J.T.: Negative extensibility metamaterials: occurrence and design-space topology. Phys. Rev. E 96, 023002 (2017)

    Google Scholar 

  11. Ji, J.C., Luo, Q., Ye, K.: Vibration control based metamaterials and origami structures: a state-of-the-art review. Mech. Syst. Signal Proc. 161, 107945 (2021)

    Google Scholar 

  12. Melancon, D., Gorissen, B., Garcia-Mora, C.J., Hoberman, C., Bertoldi, K.: Multistable inflatable origami structures at the metre scale. Nature 592, 545 (2021)

    Google Scholar 

  13. Arnouts, L.I.W., Massart, T.J., De Temmerman, N., Berke, P.Z.: Multi-objective optimisation of deployable bistable scissor structures. Autom. Constr. 114, 103154 (2020)

    Google Scholar 

  14. Zirbel, S.A., et al.: Accommodating thickness in origami-based deployable arrays. J. Mech. Des. 135, 111005 (2013)

    Google Scholar 

  15. Johnson, D.R., Harne, R.L., Wang, K.W.: a disturbance cancellation perspective on vibration control using a bistable snap-through attachment. J. Vib. Acoust.-Trans. ASME 136, 031006 (2014)

    Google Scholar 

  16. Yang, K., Harne, R.L., Wang, K.W., Huang, H.: Investigation of a bistable dual-stage vibration isolator under harmonic excitation. Smart Mater. Struct. 23, 045033 (2014)

    Google Scholar 

  17. Yan, B., Ma, H.Y., Jian, B., Wang, K., Wu, C.Y.: Nonlinear dynamics analysis of a bi-state nonlinear vibration isolator with symmetric permanent magnets. Nonlinear Dyn. 97, 2499 (2019)

    MATH  Google Scholar 

  18. Ye, K., Ji, J.C.: An origami inspired quasi-zero stiffness vibration isolator using a novel truss-spring based stack Miura-ori structure. Mech. Syst. Signal Proc. 165, 108383 (2022)

    Google Scholar 

  19. Yang, T., Cao, Q.J., Hao, Z.F.: A novel nonlinear mechanical oscillator and its application in vibration isolation and energy harvesting. Mech. Syst. Signal Proc. 155, 107636 (2021)

    Google Scholar 

  20. Filipov, E.T., Tachi, T., Paulino, G.H.: Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proc. Natl. Acad. Sci. USA 112, 12321 (2015)

    Google Scholar 

  21. Li, S.Y., Wang, K.W.: Fluidic origami: a plant-inspired adaptive structure with shape morphing and stiffness tuning. Smart Mater. Struct. 24, 105031 (2015)

    Google Scholar 

  22. Zhang, Q.W., Fang, H.B., Xu, J.: Programmable stopbands and supratransmission effects in a stacked Miura-origami metastructure. Phys. Rev. E 101, 042206 (2020)

    Google Scholar 

  23. Zareei, A., Deng, B.L., Bertoldi, K.: Harnessing transition waves to realize deployable structures. Proc. Natl. Acad. Sci. USA 117, 4015 (2020)

    Google Scholar 

  24. Liu, Y.D., Liu, B.H., Yin, T.H., Xiang, Y.H., Zhou, H.F., Qu, S.X.: Bistable rotating mechanism based on dielectric elastomer actuator. Smart Mater. Struct. 29, 015008 (2020)

    Google Scholar 

  25. Arnouts, L.I.W., De Temmerman, N., Massart, T.J., Berke, P.Z.: Geometric design of triangulated bistable scissor structures taking into account finite hub size. Int. J. Solids Struct. 206, 84 (2020)

    Google Scholar 

  26. García-Mora, C.J., Sánchez-Sánchez, J.: Geometric strategies to design a bistable deployable structure with straight scissors using stiff and flexible rods. Int. J. Solids Struct. 238, 111381 (2022)

    Google Scholar 

  27. Dai, W., Yang, J., Wiercigroch, M.: Vibration energy flow transmission in systems with Coulomb friction. Int. J. Mech. Sci. 214, 106932 (2022)

    Google Scholar 

  28. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 113, 3932 (2016)

    MATH  Google Scholar 

  29. Kaheman, K., Kutz, J.N., Brunton, S.L.: SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proc. R. Soc. A-Math. Phys. Eng. Sci. 476, 20200279 (2020)

    MATH  Google Scholar 

  30. Li, S., Kaiser, E., Laima, S., Li, H., Brunton, S.L., Kutz, J.N.: Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems. Phys. Rev. E 100, 022220 (2019)

    Google Scholar 

  31. Chu, H.K., Hayashibe, M.: Discovering interpretable dynamics by sparsity promotion on energy and the Lagrangian. IEEE Robot. Autom. Lett. 5, 8977323 (2020)

    Google Scholar 

  32. Kaiser, E., Kutz, J.N., Brunton, S.L.: Data-driven discovery of Koopman eigenfunctions for control. Mach. Learn.-Sci. Technol. 2, 035023 (2021)

    Google Scholar 

  33. Bhattacharya, D., Chen, L.K., Xu, W.L.: Sparse machine learning discovery of dynamic differential equation of an esophageal swallowing robot. IEEE Trans. Ind. Electron. 67, 8765629 (2020)

    Google Scholar 

  34. Rudy, S.H., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3, e1602614 (2017)

    Google Scholar 

  35. Guan, Y.F., Brunton, S.L., Novosselov, I.: Sparse nonlinear models of chaotic electroconvection. R. Soc. Open Sci. 8, 202367 (2021)

    Google Scholar 

  36. Kaptanoglu, A.A., Callaham, J.L., Aravkin, A., Hansen, C.J., Brunton, S.L.: Promoting global stability in data-driven models of quadratic nonlinear dynamics. Phys. Rev. Fluids 6, 094401 (2021)

    Google Scholar 

  37. Mendible, A., Koch, J., Lange, H., Brunton, S.L., Kutz, J.N.: Data-driven modeling of rotating detonation waves. Phys. Rev. Fluids 6, 050507 (2021)

    Google Scholar 

  38. Qian, J., Sun, X., Xu, J., Fang, H.: Design and dynamic analysis of a novel bio-inspired erecting structure. Chin. J. Theor. Appl. Mech. 53, 2023–2036 (2021). (in Chinese)

    Google Scholar 

  39. Tusar, T., Filipic, B.: Visualization of Pareto front approximations in evolutionary multiobjective optimization: a critical review and the prosection method. IEEE Trans. Evol. Comput. 19, 6777535 (2015)

    Google Scholar 

  40. Hua, Y.C., Liu, Q.Q., Hao, K.R., Jin, Y.C.: A survey of evolutionary algorithms for multi-objective optimization problems with irregular Pareto fronts. IEEE-CAA J. Automatica Sin. 8, 9321268 (2021)

    Google Scholar 

  41. Deb, K., Gupta, S.: Understanding knee points in bicriteria problems and their implications as preferred solution principles. Eng. Optim. 43, 1175 (2011)

    Google Scholar 

  42. Das, I.: On characterizing the “knee” of the Pareto curve based on Normal-Boundary Intersection. Struct. Optim. 18, 107 (1999)

    Google Scholar 

  43. Branke, E., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization, vol 3242, p 722. Berlin (2004)

  44. Chartrand, R.: Numerical differentiation of noisy, nonsmooth data. ISRN Appl. Math. 2011, 164564 (2011)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to gratefully acknowledge the support from the National Natural Science Foundation of China under Grants 12122208, 11972254, 11932015.

Funding

The funding was provided by National Natural Science Foundation of China (Grant Nos. 12122208, 11972254, 11932015).

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Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, People’s Republic of China

    Jiawei Qian, Xiuting Sun & Jian Xu

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  1. Jiawei Qian
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  2. Xiuting Sun
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Correspondence to Xiuting Sun.

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Qian, J., Sun, X. & Xu, J. A data-driven reconstruction method for dynamic systems with multistable property. Nonlinear Dyn 111, 4517–4541 (2023). https://doi.org/10.1007/s11071-022-08082-2

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