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Fractional central pattern generators for bipedal locomotion

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Abstract

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots.

In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative D α f(x), with α non-integer, is a generalization of the concept of an integer derivative, where α=1. The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators which has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs’ rhythms in bipeds, occur, for 0<α≤1. We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to unity.

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References

  1. Baleanu, D.: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520–2523 (2009)

    Article  Google Scholar 

  2. Bayden, A.G.: Evolution of central pattern generators for the control of a five-link planar bipedal walking mechanism (2008). arXiv:0801.0830v3

  3. Buono, P.L., Golubitsky, M.: Models of central pattern generators for quadruped locomotion I. Primary gaits. J. Math. Biol. 42, 291–326 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  5. Collins, J.J., Stewart, I.: Coupled nonlinear oscillators and the symmetries of animal gaits. J. Nonlinear Sci. 3, 349–392 (1993). ISSN 1432-1467

    Article  MATH  MathSciNet  Google Scholar 

  6. Collins, J.J., Stewart, I.: Hexapodal gaits and coupled nonlinear oscillator models. Biol. Cybern. 68, 287–298 (1993)

    Article  MATH  Google Scholar 

  7. Collins, J.J., Stewart, I.: A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern. 71, 95–103 (1994)

    Article  MATH  Google Scholar 

  8. Degallier, S., Santos, C., Righetti, L., Ijspeert, A.: Movement generation using dynamical systems: A humanoid robot performing a drumming task. In: IEEE-RAS International Conference on Humanoid Robots (2006)

  9. Golubitsky, M., Stewart, I.: In: The Symmetry Perspective. Progress in Mathematics, vol. 200. Birkhauser, Basel (2002)

    Google Scholar 

  10. Golubitsky, M., Stewart, I.: Nonlinear dynamics of network: The groupoid formalism. Bull. Am. Math. Soc. 43, 305–364 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: A modular network for legged locomotion. Physica D 115, 56–72 (1998). ISSN 0167-2789

    Article  MATH  MathSciNet  Google Scholar 

  12. Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: Symmetry in locomotor central pattern generators and animal gaits. Nature 401, 693–695 (1999). ISSN 0028-0836

    Article  Google Scholar 

  13. Grillner, S.: Locomotion in vertebrates: central mechanisms and reflex interaction. Physiol. Rev. 55, 247–304 (1975)

    Article  Google Scholar 

  14. Kopell, N., Ermentrout, J.: Coupled oscillators and the design of central pattern generators. Math. Biosci. 90, 87–109 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lewis, M.A., Etienne-Cummings, R., Hartmann, M.J., Xu, Z.R., Cohen, A.H.: An in silico central pattern generator: Silicon oscillator, coupling, entrainment, and physical computation. Biol. Cybern. 88, 137–151 (2003). ISSN 0340-1200

    Article  MATH  Google Scholar 

  16. Liu, G.L., Habib, M.K., Watanabe, K., Izumi, K.: Central pattern generators based on Matsuoka oscillators for the locomotion of biped robots. Artif. Life Robot. 12, 264–269 (2008). ISSN 1433-5298

    Article  Google Scholar 

  17. Machado, J.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal. Model. Simul. 27, 107–122 (1997)

    MATH  Google Scholar 

  18. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mann, R.A.: Biomechanics. In: Jahss, M.H. (ed.) Disorders of the Foot, pp. 37–67. Saunders, Philadelphia (1982)

    Google Scholar 

  20. Mann, R.A., Moran, G.T., Dougherty, S.E.: Comparative electromyography of the lower extremity in jogging, running and sprinting. Am. J. Sports Med. 14, 501–510 (1986)

    Article  Google Scholar 

  21. Matos, V., Santos, C.P., Pinto, C.M.A.: A brainstem-like modulation approach for Gait transition in a quadruped robot. In: Proceedings of The 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2009, St. Louis, MO, October 2009

  22. Marder, E., Bucher, D.: Central pattern generators and the control of rhythmic movements. Curr. Biol. 11(23), 986–996 (2001). ISSN 0960-9822

    Article  Google Scholar 

  23. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  24. Morris, C., Lecar, H.: Voltage oscillations in the Barnacle giant muscle fiber. Biophys. J. 35, 193–213 (1981)

    Article  Google Scholar 

  25. Nigmatullin, R.: The statistics of the fractional moments: Is there any chance to “read quantitatively” any randomness? Signal Process. 86(10), 2529–2547 (2006)

    Article  MATH  Google Scholar 

  26. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974)

    Google Scholar 

  27. Oustaloup, A.: La Commande CRONE: Commande Robuste d’Ordre non Entier. Hermes, London (1991)

    MATH  Google Scholar 

  28. Pinto, C.M.A., Golubitsky, M.: Central pattern generators for bipedal locomotion. J. Math. Biol. 53(3), 474–489 (2006). ISSN 0303-6812

    Article  MATH  MathSciNet  Google Scholar 

  29. Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Righetti, L., Ijspeert, A.: Design methodologies for central pattern generators: An application to crawling humanoids. In: Proceedings of Robotics: Science and Systems, pp. 191–198 (2006)

  31. Righetti, L., Buchli, J., Ijspeert, A.J.: Dynamic Hebbian learning in adaptive frequency oscillators. Physica D 216, 269–281 (2006). ISSN 0167-2789

    Article  MATH  MathSciNet  Google Scholar 

  32. Righetti, L., Ijspeert, A.J.: Pattern generators with sensory feedback for the control of quadruped locomotion. In: IEEE International Conference on Robotics and Automation. ICRA 2008, No. 19–23, pp. 819–824 (2008). ISBN 1050-4729

  33. Rinzel, J., Ermentrout, G.B.: Analysis of neural excitability and oscillations. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling: From Synapses to Networks. MIT Press, Cambridge (1989). ISBN 0-262-11231-0

    Google Scholar 

  34. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  35. Taga, G., Yamaguchi, Y., Shimizu, H.: Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol. Cybern. 65, 147–169 (1991)

    Article  MATH  Google Scholar 

  36. Tenore, F., Etienne-Cummings, R., Lewis, M.A.: Entrainment of silicon central pattern generators for legged locomotory control. In: Neural Information Processing Systems, NIPS 2003, Vancouver and Whistler, British Columbia, Canada, 8–13 December 2003. MIT Press, Cambridge (2004). ISBN 0-262-20152-6

    Google Scholar 

  37. Tenreiro Machado, J.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001)

    MATH  MathSciNet  Google Scholar 

  38. Tenreiro Machado, J.A.: Fractional derivatives: Probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 14(9–10), 3492–3497 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Centro de Matemática da Universidade do Porto and Department of Mathematics, Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072, Porto, Portugal

    Carla M. A. Pinto

  2. Department of Electrical Engineering, Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072, Porto, Portugal

    J. A. Tenreiro Machado

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  1. Carla M. A. Pinto
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  2. J. A. Tenreiro Machado
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Correspondence to Carla M. A. Pinto.

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Pinto, C.M.A., Tenreiro Machado, J.A. Fractional central pattern generators for bipedal locomotion. Nonlinear Dyn 62, 27–37 (2010). https://doi.org/10.1007/s11071-010-9696-4

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  • DOI: https://doi.org/10.1007/s11071-010-9696-4

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