Advertisement

Biological Cybernetics

, Volume 113, Issue 1–2, pp 93–104 | Cite as

Locomotion: exploiting noise for state estimation

  • John GuckenheimerEmail author
  • Aurya Javeed
Original Article

Abstract

Running, walking, flying and swimming are all processes in which animals produce propulsion by executing rhythmic motions of their bodies. Dynamical stability of the locomotion is hardly automatic: millions of older people are injured by falling each year. Stability frequently requires sensory feedback. We investigate how organisms obtain the information they use in maintaining their stability. Assessing stability of a periodic orbit of a dynamical system requires information about the dynamics of the system off the orbit. For locomotion driven by a periodic orbit, perturbations that “kick” the trajectory off the orbit must occur in order to observe convergence rates toward the orbit. We propose that organisms generate excitations in order to set the gains for stabilizing feedback. We hypothesize further that these excitations are stochastic but have heavy-tailed, non-Gaussian probability distributions. Compared to Gaussian distributions, we argue that these are more effective for estimating stability characteristics of the orbit. Finally, we propose experiments to test the efficacy of these ideas.

Keywords

Locomotion Stochastic dynamical system Floquet multiplier 

References

  1. Blickhan R, Full RJ (1993) Similarity in multilegged locomotion: bouncing like a monopode. J Comp Physiol A 173(5):509–517.  https://doi.org/10.1007/BF00197760 CrossRefGoogle Scholar
  2. Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New York.  https://doi.org/10.1007/978-1-4419-0320-4 CrossRefGoogle Scholar
  3. Cabrera JL, Milton JG (2004a) Human stick balancing: tuning Lévy flights to improve balance control. Chaos 14(3):691–698.  https://doi.org/10.1063/1.1785453 CrossRefGoogle Scholar
  4. Cabrera JL, Milton JG (2004b) Stick balancing: on-off intermittency and survival times. Nonlinear Stud 11(3):305–318Google Scholar
  5. Collins JJ, De Luca CJ (1993) Open-loop and closed-loop control of posture: a random-walk analysis of center-of-pressure trajectories. Exp Brain Res 95(2):308–318.  https://doi.org/10.1007/BF00229788 CrossRefGoogle Scholar
  6. Collins JJ, De Luca CJ (1994) Random walking during quiet standing. Phys Rev Lett 73(5):764–767.  https://doi.org/10.1103/PhysRevLett.73.764 CrossRefGoogle Scholar
  7. Grassberger P, Procaccia I (1983) Characterization of strange attractors. Phys Rev Lett 50(5):346–349.  https://doi.org/10.1103/PhysRevLett.50.346 CrossRefGoogle Scholar
  8. Guckenheimer J (1995) A robust hybrid stabilization strategy for equilibria. IEEE Trans Automat Control 40(2):321–326.  https://doi.org/10.1109/9.341802 CrossRefGoogle Scholar
  9. Guckenheimer J (2014) From data to dynamical systems. Nonlinearity 27(7):R41.  https://doi.org/10.1088/0951-7715/27/7/R41 CrossRefGoogle Scholar
  10. Hamilton JD (1994) Time series analysis. Princeton University Press, PrincetonGoogle Scholar
  11. Hannan EJ, Kanter M (1977) Autoregressive processes with infinite variance. J Appl Probab 14(2):411–415.  https://doi.org/10.2307/3213015 CrossRefGoogle Scholar
  12. Hartman P (2002) Ordinary differential equations, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, corrected reprint of the second (1982) edition.  https://doi.org/10.1137/1.9780898719222
  13. Holmes P, Full RJ, Koditschek D, Guckenheimer J (2006) The dynamics of legged locomotion: models, analyses, and challenges. SIAM Rev 48(2):207–304.  https://doi.org/10.1137/S0036144504445133 CrossRefGoogle Scholar
  14. Javeed A (2017) An uncertainty principle for estimates of Floquet multipliers. arXiv:1711.10992
  15. Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New York.  https://doi.org/10.1007/b98854 Google Scholar
  16. Maus HM, Revzen S, Guckenheimer J, Ludwig C, Reger J, Seyfarth A (2015) Constructing predictive models of human running. J R Soc Interface.  https://doi.org/10.1098/rsif.2014.0899 Google Scholar
  17. Peterka RJ (2000) Postural control model interpretation of stabilogram diffusion analysis. Biol Cybern 82(4):335–343.  https://doi.org/10.1007/s004220050587 CrossRefGoogle Scholar
  18. Revzen S, Guckenheimer JM (2008) Estimating the phase of synchronized oscillators. Phys Rev E 78:051907.  https://doi.org/10.1103/PhysRevE.78.051907 CrossRefGoogle Scholar
  19. Ristroph L, Bergou AJ, Ristroph G, Coumes K, Berman GJ, Guckenheimer J, Wang ZJ, Cohen I (2010) Discovering the flight autostabilizer of fruit flies by inducing aerial stumbles. Proc Natl Acad Sci USA 107(11):4820–4824.  https://doi.org/10.1073/pnas.1000615107 CrossRefGoogle Scholar
  20. Schaal S, Sternad D, Atkeson CG (1996) One-handed juggling: a dynamical approach to a rhythmic movement task. J Mot Behav 28(2):165–183.  https://doi.org/10.1080/00222895.1996.9941743 CrossRefGoogle Scholar
  21. Sussmann HJ (1978) On the gap between deterministic and stochastic ordinary differential equations. Ann Probab 6(1):19–41.  https://doi.org/10.1214/aop/1176995608 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentCornell UniversityIthacaUSA
  2. 2.Center for Applied MathematicsCornell UniversityIthacaUSA

Personalised recommendations