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Spatiotemporal pattern of periodic rhythms in delayed Van der Pol oscillators for the CPG-based locomotion of snake-like robot

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Abstract

This is the further research on the delayed half-center oscillator (DHCO) neural system presented in our previous paper (Song and Xu in Nonlinear Dyn 108:2595–2609, 2022. https://doi.org/10.1007/s11071-022-07222-y). The DHCO is used to construct a CPG (central pattern generator) neural system to control locomotion of a snake-like robot with pitch-yaw connecting configuration. To this end, we firstly give an improved model of the VDP (Van der Pol) oscillator. Employing mutually coupled delay, a pair of VDP oscillator is connected to produce an half-center oscillator (HCO) module with time delay that is called as a DHCO (delayed HCO) model. Based on the analysis of the Hopf bifurcation, periodic rhythm and their spatiotemporal patterns of the DHCO are illustrated in the different regions of parameters. The DHCO presents periodic rhythms with synchronous and anti-synchronous patterns, which is to control joint actuators combined in snake-like robot with pitch-yaw connecting configuration. To realize a backward propulsive wave to promote snake-like robotic locomotion, based on the DHCOs, we construct a chain type of the CPG neural system combined with a new unidirectional delay in which phase difference can be regulated. Numerical simulations are illustrated that the CPG neural system can control snake-like robot to move with serpentine, rectilinear, and side-winding patterns in the forward and backward directions. The results show that the snake-like robot can be controlled in expected locomotion patterns for a region but not a fixed value of the controlling parameters. Further, the corresponding regions of the parameters are obtained by using theoretical dynamical analysis but not a trial-and-error method. The snake-like robot gets smooth and stable gait transition with parameter changing.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Wang, J.K., Chen, W.N., Xiao, X., Xu, Y.X., Li, C., Jia, X., Meng, M.Q.-H.: A survey of the development of biomimetic intelligence and robotics. Bio. Intell. Robot. 1(100001), 1–12 (2021)

    Google Scholar 

  2. Ryczko, D., Simon, A., Ijspeert, A.J.: Walking with salamanders: From molecules to biorobotics. Trend. Neurosci. 43(11), 916–930 (2020)

    Article  Google Scholar 

  3. Grillner, S., Manira, A.E.: Current principles of motor control, with special reference to vertebrate locomotion. Physiol. Rev. 100(1), 271–320 (2019)

    Article  Google Scholar 

  4. Flood, T.F., Iguchi, S., Gorczyca, M., White, B., Ito, K., Yoshihara, M.: A single pair of interneurons commands the Drosophila feeding motor program. Nature 499(9), 83–87 (2013)

    Article  Google Scholar 

  5. Gupta, V., Mittal, M., Mittal, V., Saxena, N.K.: A critical review of feature extraction techniques for ECG signal analysis. J. Inst. Eng. India Ser. B 102, 1049–1060 (2021)

    Article  Google Scholar 

  6. Yu, Y., Han, F., Wang, Q.S., Wang, Q.Y.: Model-based optogenetic stimulation to regulate beta oscillations in Parkinsonian neural networks. Cogn. Neurodyn. 16, 667–681 (2022)

    Article  Google Scholar 

  7. Aminzare, Z., Srivastava, V., Holmes, P.: Gait transitions in a phase oscillator model of an insect central pattern generator. SIAM J. Appl. Dynamic. Syst. 17(1), 626–671 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta, V., Mittal, M.: A novel method of cardiac arrhythmia detection in electrocardiogram signal. Int. J. Med. Eng. Informatics 12(5), 489–499 (2020)

    Article  Google Scholar 

  9. Ijspeert, A.J.: Central pattern generators for locomotion control in animals and robots: a review. Neural. Netw. 24(4), 642–653 (2008)

    Article  Google Scholar 

  10. Lobato-Rios, V., Ramalingasetty, S.T., Özdil, P.G., Arreguit, J., Ijspeert, A.J., Ramdya, P.: NeuroMechFly, a neuromechanical model of adult Drosophila melanogaster. Nat. Methods 19, 620–627 (2022)

    Article  Google Scholar 

  11. Liu, J.D., Tong, Y., Liu, J.G.: Review of snake robots in constrained environments. Robot. Auto. Syst. 141, 103785 (2021)

    Article  Google Scholar 

  12. Wang, Z., Gao, Q., Zhao, H.: CPG-inspired locomotion control for a snake robot basing on nonlinear oscillator. J. Intell. Robot. Syst. 85(2), 209–227 (2017)

    Article  Google Scholar 

  13. Conradt, J., Varshavskaya, P.: Distributed central pattern generator control for a serpentine robot. In: Proceedings of Artificial Neural Networks and Neural Information, Istanbul, Turkey, pp. 338–341 (2003)

  14. Inoue, K., Ma, S., Cheng, J.: Neural oscillator network-based controller for meandering locomotion of snake-like robot. In: Proceedings of IEEE International Conference on Robotics and Automation, New Orleans, USA, pp. 5064–5069 (2004)

  15. Lu, Z.L., Ma, S.G., Li, B., Wang, Y.: Gaits-transferable CPG controller for a snake-like robot. Sci. China F 51(3), 293–305 (2008)

    MATH  Google Scholar 

  16. Wu, X., Ma, S.: Adaptive creeping locomotion of a CPG-controlled snake-like robot to environment change. Auton. Robots 28(3), 283–294 (2010)

    Article  Google Scholar 

  17. Wu, X., Ma, S.: Neurally controlled steering for collision-free behavior of a snake robot. IEEE Trans. Control Syst. Technol. 21(6), 2443–2449 (2013)

    Article  Google Scholar 

  18. Lu, Q., Wang, X.Y., Tian, J.: A new biological central pattern generator model and its relationship with the motor units. Cogn. Neurodyn. 16(1), 135–147 (2022)

    Article  Google Scholar 

  19. Ijspeert, A.J., Crespi, A., Ryczko, D., Cabelguen, J.M.: From swimming to walking with a salamander robot driven by a spinal cord model. Science 315(9), 1416–1420 (2007)

    Article  Google Scholar 

  20. Tang, C.Q., Ma, S.G., Wang, Y.C.: A cubic CPG model for snake-like robot to adapt to environment. In: IEEE International Conference on Information and Automation, pp. 24–29 (2010)

  21. Nor, N.M., Ma, S.: Smooth transition for CPG-based body shape control of a snake-like robot. Bioinspir. Biomim. 9(1), 1–11 (2013)

    Article  Google Scholar 

  22. Bing, Z.S., Cheng, L., Chen, G., Röhrbein, F., Huang, K., Knoll, A.: Towards autonomous locomotion: CPG-based control of smooth 3D slithering gait transition of a snakelike robot. Bioinspir. Biomim. 12(3), 035001 (2017)

    Article  Google Scholar 

  23. Qiao, G.F., Zhang, Y., Wen, X.L., Wei, Z., Cui, J.Y.: Triple-layered central pattern generator-based controller for 3D locomotion control of snake-like robots. Int. J. Adv. Robot. Syst. 14(6), 1–13 (2017)

    Article  Google Scholar 

  24. Manzoor, S., Cho, Y.G., Choi, Y.: Neural oscillator based CPG for various rhythmic motions of modular snake robot with active joints. J. Intell. Robot Syst. 94(3), 641–654 (2019)

    Article  Google Scholar 

  25. Manzoor, S., Khan, U., Ullah, I.: Serpentine and rectilinear motion generation in snake robot using central pattern generator with gait transition. Iran. J. Sci. Technol. Trans. Electric. Eng. 44(1), 1093–1103 (2019)

    Google Scholar 

  26. Zhu, F.Y., Wang, R.B., Aihara, K., Pan, X.C.: Energy-efficient firing patterns with sparse bursts in the Chay neuron model. Nonlinear Dyn. 100(3), 2657–2672 (2020)

    Article  Google Scholar 

  27. Zhong, H.X., Wang, R.B.: A new discovery on visual information dynamic changes from V1 to V2: corner encoding. Nonlinear Dyn. 105(4), 3551–3570 (2021)

    Article  Google Scholar 

  28. Du, M.-M., Li, J.-J., Yuan, Z.-X., Fan, Y.-C., Wu, Y.: Astrocyte and ions metabolism during epileptogenesis: a review for modeling studies. Chin. Phys. B 29(3), 038701 (2020)

    Article  Google Scholar 

  29. Guo, Y., Zhou, P., Yao, Z., Ma, J.: Biophysical mechanism of signal encoding in an auditory neuron. Nonlinear Dyn. 105, 3603–3614 (2021)

    Article  Google Scholar 

  30. Yuan, Y., Pan, X.C., Wang, R.B.: Biophysical mechanism of the interaction between default mode network and working memory network. Cogn. Neurodyn. 15(6), 1101–1124 (2021)

    Article  Google Scholar 

  31. De Sarkar, S.S., Sharma, A.K., Chakraborty, S.: Chaos, antimonotonicity and coexisting attractors in Van der Pol oscillator based electronic circuit. Analog. Integr. Circ. Sig. Process 110, 211–229 (2022)

    Article  Google Scholar 

  32. Tuna, M.: A novel secure chaos-based pseudo random number generator based on ANN-based chaotic and ring oscillator: design and its FPGA implementation. Analog. Integr. Circ. Sig. Process 105, 167–181 (2020)

    Article  Google Scholar 

  33. Bay, J.S., Hemami, H.: Modeling of a neural pattern generator with coupled nonlinear oscillator. IEEE Trans. Biomed. Eng. 34(4), 297–306 (1987)

    Article  Google Scholar 

  34. Zielinska, T.: Coupled oscillators utilized as gait rhythm generators of a two-legged walking machine. Biol. Cybern. 74(3), 263–273 (1996)

    Article  MATH  Google Scholar 

  35. Dutra, M.S., Filho, A.C., Romano, V.F.: Modeling of a bipedal locomotor using coupled nonlinear oscillators of Van der Pol. Biol. Cybern. 88, 286–292 (2003)

    Article  MATH  Google Scholar 

  36. Jasni, F., Shafie, A.A.: Van Der Pol central pattern generator (VDP-CPG) model for quadruped robot. In: Trends in Intelligent Robotics, Automation, and Manufacturing. Springer, Berlin, pp. 167–175 (2012)

  37. Yu, H., Guo, W., Deng, J., Li, M., Cai, H.: A CPG-based locomotion control architecture for hexapod robot. In: IEEE International Conference on Intelligent Robots and Systems, pp. 5615–5621 (2013)

  38. Yu, H.T., Gao, H.B., Ding, L., Li, M.T., Deng, Z.Q., Liu, G.J.: Gait generation with smooth transition using CPG-Based locomotion control for hexapod walking robot. IEEE Trans. Ind. Electron. 63(9), 5488–5500 (2016)

    Article  Google Scholar 

  39. Yao, S., Ding, L., Song, Z., Xu, J.: Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay. Nonlinear Dyn. 95, 1549–1563 (2019)

    Article  MATH  Google Scholar 

  40. Korotkov, A.G., Levanova, T.A., Zaks, M.A., Maksimov, A.G., Osipov, G.V.: Dynamics in a phase model of half-center oscillator: two neurons with excitatory coupling. Commun. Nonlinear Sci. Numer. Simulat. 104, 106045 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  41. Song, Z.G., Xu, J.: Self-/mutual-symmetric rhythms and their coexistence in a delayed half-center oscillator of the CPG neural system. Nonlinear Dyn. 108, 2595–2609 (2022)

    Article  Google Scholar 

  42. Song, Z., Zhen, B., Hu, D.: Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays. Cogn. Neurodyn. 14, 359–374 (2020)

    Article  Google Scholar 

  43. Song, Z., Xu, J., Zhen, B.: Multitype activity coexistence in an inertial two-neuron system with multiple delays. Int. J. Bifurcation and Chaos 25(13), 1530040 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This research is supported by the National Natural Science of China under Grant Nos. 12172212 and 11932015.

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Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Zigen Song & Jian Xu

  2. Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, China

    Xiaojun Huang

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  1. Zigen Song
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Correspondence to Jian Xu.

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Song, Z., Huang, X. & Xu, J. Spatiotemporal pattern of periodic rhythms in delayed Van der Pol oscillators for the CPG-based locomotion of snake-like robot. Nonlinear Dyn 110, 3377–3393 (2022). https://doi.org/10.1007/s11071-022-07783-y

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