Skip to main content
SpringerLink
Log in

A cardioid oscillator with asymmetric time ratio for establishing CPG models

  • Original Article
  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

Nonlinear oscillators are usually utilized by bionic scientists for establishing central pattern generator models for imitating rhythmic motions by bionic scientists. In the natural word, many rhythmic motions possess asymmetric time ratios, which means that the forward and the backward motions of an oscillating process sustain different times within one period. In order to model rhythmic motions with asymmetric time ratios, nonlinear oscillators with asymmetric forward and backward trajectories within one period should be studied. In this paper, based on the property of the invariant set, a method to design the closed curve in the phase plane of a dynamic system as its limit cycle is proposed. Utilizing the proposed method and considering that a cardioid curve is a kind of asymmetrical closed curves, a cardioid oscillator with asymmetric time ratios is proposed and realized. Through making the derivation of the closed curve in the phase plane of a dynamic system equal to zero, the closed curve is designed as its limit cycle. Utilizing the proposed limit cycle design method and according to the global invariant set theory, a cardioid oscillator applying a cardioid curve as its limit cycle is achieved. On these bases, the numerical simulations are conducted for analyzing the behaviors of the cardioid oscillator. The example utilizing the established cardioid oscillator to simulate rhythmic motions of the hip joint of a human body in the sagittal plane is presented. The results of the numerical simulations indicate that, whatever the initial condition is and without any outside input, the proposed cardioid oscillator possesses the following properties: (1) The proposed cardioid oscillator is able to generate a series of periodic and anti-interference self-exciting trajectories, (2) the generated trajectories possess an asymmetric time ratio, and (3) the time ratio can be regulated by adjusting the oscillator’s parameters. Furthermore, the comparison between the simulated trajectories by the established cardioid oscillator and the measured angle trajectories of the hip angle of a human body show that the proposed cardioid oscillator is fit for imitating the rhythmic motions of the hip of a human body with asymmetric time ratios.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Grillner S (1985) Neurobiological bases of rhythmic motor acts in vertebrates. Science 228(4696):143–149

    Article  PubMed  CAS  Google Scholar 

  2. Hodgkin AL, Huxley AF (1952) Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J Physiol 116(4):449–472

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  3. Pinto CMA, Santos AP (2011) Modelling gait transition in two-legged animals. Commun Nonlinear Sci Numer Simul 16(12):4625–4631

    Article  Google Scholar 

  4. Matsuoka K (1987) Mechanisms of frequency and pattern control in the neural rhythm generators. Biol Cybern 56(5–6):345–353

    Article  PubMed  CAS  Google Scholar 

  5. Fukuoka Y, Kimura H, Cohen AH (2003) Adaptive dynamic walking of a quadruped robot on irregular terrain based on biological concepts. Int J Robot Res 22(3–4):187–202

    Article  Google Scholar 

  6. Awrejcewicz J, Pyryev Y (2005) Chaos prediction in the doffing-type system with friction using Melnikov’s function. Nonlinear Anal Real World Appl 2(1):12–24

    Article  Google Scholar 

  7. Bhuiyan MSH, Choudhury IA, Dahari M (2015) Development of a control system for artificially rehabilitated limbs: a review. Biol Cybern 109(2):141–162

    Article  PubMed  CAS  Google Scholar 

  8. Buchli J, Righetti L, Ijspeert AK (2005) A dynamical systems approach to learning: a frequency-adaptive Hopper Robot, 8th European Conference on Artificial Life, Canterbury, England

  9. Righetti L, Buchli J, Ijspeert A (2006) Dynamic hebbian learning for adaptive frequency oscillators. Phys D Nonlinear Phenom 216(2):269–281

    Article  CAS  Google Scholar 

  10. de Pina Filho AC, Dutra MS, Raptopoulos LSC (2005) Modeling of a bipedal robot using mutually coupled Rayleigh oscillators. Biol Cybern 92(1):1–7

    Article  Google Scholar 

  11. Nandi GC, Ijspeert AJ, Chakraborty P, Nandi A (2009) Development of Adaptive Modular Active Leg (AMAL) using bipedal robotics technology. Robot Auton Syst 57(6–7):603–616

    Article  Google Scholar 

  12. Wieczorek S (2009) Stochastic bifurcation in noise-driven lasers and Hopf oscillators. Phys Rev E 79(3):036209

    Article  CAS  Google Scholar 

  13. Hu YH, Liang JH, Wang TM (2014) Parameter synthesis of coupled nonlinear oscillators for CPG-based robotic locomotion. IEEE Trans Ind Electron 61(11):6183–6191

    Article  Google Scholar 

  14. Rayleigh JWS, Lindsay RB (1976) The theory of sound. Dover Publications, Mineola

    Google Scholar 

  15. Wu YH, Han MA, Chen XF (2004) On the study of limit cycles of the generalized Rayleigh Lienard oscillator. Int J Bifurc Chaos 14(8):2905–2914

    Article  Google Scholar 

  16. Acebron JA, Bonilla LL, Vicente CJP (2005) The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 77(1):137–185

    Article  Google Scholar 

  17. Tsimring LS, Rulkov NF, Larsen ML (2005) Repulsive synchronization in an array of phase oscillators. Phys Rev Lett 95(1):1–4

    Article  CAS  Google Scholar 

  18. Van der Pol B, Van der Mark J (1928) The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Philos Mag 6(38):763–775

    Article  Google Scholar 

  19. Bi QS (2004) Dynamical analysis of two coupled parametrically excited van der Pol oscillators. Int J Non-linear Mech 39(1):33–54

    Article  Google Scholar 

  20. Ijspeert AJ, Crespi A, Ryczko D (2007) From swimming to walking with a salamander robot driven by a spinal cord model. Science 315(5817):1416–1420

    Article  PubMed  CAS  Google Scholar 

  21. Zielinska T (1996) Coupled oscillators utilized as gait rhythm generators of a two-legged walking machine. Biol Cybern 74(3):263–274

    Article  PubMed  CAS  Google Scholar 

  22. Buchli J, Righetti L, Ijspeert AJ (2006) Engineering entrainment and adaptation in limit cycle systems—from biological inspiration to applications in robotics. Biol Cybern 95(6):645–664

    Article  PubMed  Google Scholar 

  23. Chatterjee S, Dey S (2013) Nonlinear dynamics of two harmonic oscillators coupled by Rayleigh type self-exciting force. Nonlinear Dyn 72(1–2):113–128

    Article  Google Scholar 

  24. Suchorsky M, Rand R (2009) Three oscillator model of the heartbeat generator. Commun Nonlinear Sci Numer Simul 14(5):2434–3449

    Article  Google Scholar 

  25. Murray MP, Drought AB, Kory RC (1964) Walking patterns of normal men. J Bone Jt Surg 46(2):335–360

    Article  CAS  Google Scholar 

  26. Wu QD, Liu CJ, Zhang JQ, Chen QJ (2009) Survey of locomotion control of legged robots inspired by biological concept. Sci China Ser F Inf Sci 52(10):1715–1729

    Article  Google Scholar 

  27. Righetti L, Ijspeert AJ (2008) Pattern generators with sensory feedback for the control of quadruped locomotion. In: Proceedings of the IEEE international conference on robotics and automation, Washington, pp 819–824

  28. Xu L, Wang DH, Fu Q, Yuan G, Hu LZ (2016) A novel four-bar linkage prosthetic knee based on magnetorheological effect: principle, structure, simulation and control. Smart Mater Struct 25(11):115007

    Article  Google Scholar 

  29. Fu Q, Wang DH, Xu L, Yuan G (2017) A magnetorheological damper based prosthetic knee (MRPK) and the sliding mode tracking control method for the MRPK based lower-limb prosthesis. Smart Mater Struct 26(4):045030

    Article  Google Scholar 

Download references

Acknowledgements

This work has been partialy supported by the LPMT, CAEP (Grant No. 2015-01-001), the National Natural Science Foundation of China (Grant No. 51675070), and the Fundamental Research Funds for the Central Universities (Project No. CDJZR12 12 00 05)

Author information

Authors and Affiliations

  1. Key Laboratory of Optoelectronic Technology and Systems of the Ministry of Education of China, Chongqing University, Chongqing, 400044, China

    D. H. Wang & G. Yuan

  2. Precision and Intelligence Laboratory, Department of Optoelectronic Engineering, Chongqing University, Chongqing, 400044, China

    Q. Fu, D. H. Wang, L. Xu & G. Yuan

Authors
  1. Q. Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. H. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. H. Wang.

Additional information

Communicated by J. Leo van Hemmen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, Q., Wang, D.H., Xu, L. et al. A cardioid oscillator with asymmetric time ratio for establishing CPG models. Biol Cybern 112, 227–235 (2018). https://doi.org/10.1007/s00422-018-0746-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-018-0746-1

Keywords

Search

Navigation