Nonlinear Dynamics

, Volume 93, Issue 3, pp 1183–1199 | Cite as

The simple chaotic model of passive dynamic walking

  • Saeed Montazeri Moghadam
  • Maryam Sadeghi Talarposhti
  • Ali Niaty
  • Farzad Towhidkhah
  • Sajad Jafari
Original Paper


Recent findings on the dynamical analysis of human locomotion characteristics such as stride length signal have shown that this process is intrinsically a chaotic behavior. The passive walking has been defined as walking down a shallow slope without using any muscular contraction as an active controller. Based on this definition, some knee-less models have been proposed to present the simplest possible models of human gait. To maintain stability, these simple passive models are compelled to show a wide range of different dynamics from order to chaos. Unfortunately, based on simplifications, for many years the cyclic period-one behavior of these models has been considered as the only stable response. This assumption is not in line with the findings about the nature of walking. Thus, this paper proposes a novel model to demonstrate that the knee-less passive dynamic models also have the ability to model the chaotic behavior of human locomotion with some modifications. The presented novel model can show chaotic behavior as a stable and acceptable answer using a chaotic function in heel-strike condition. The represented chaotic model is also able to simulate different types of motor deficits such as Parkinson’s disease only by manipulating the value of chaotic parameter. Our model has extensively examined in complexity and chaotic behavior using different analytical methods such as fractal dimension, bifurcation and largest Lyapunov exponent, and it was compared with conventional passive models and the stride signal of healthy subjects and Parkinson patients.


Chaotic-passive-walking Dynamic-locomotion Stride signal Largest Lyapunov exponent Fractal dimension 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Elftman, H.: Studies of gait. J. Bone. Jt. Surg. Am. 48, 363–377 (1966)CrossRefGoogle Scholar
  2. 2.
    Ijspeert, A.J.: Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21(4), 642–653 (2008)CrossRefGoogle Scholar
  3. 3.
    Iqbal, S., et al.: Bifurcations and chaos in passive dynamic walking: a review. Robot. Auton. Syst. 62(6), 889–909 (2014)CrossRefGoogle Scholar
  4. 4.
    Xiang, Y., Arora, J.S., Abdel-Malek, K.: Physics-based modeling and simulation of human walking: a review of optimization-based and other approaches. Struct. Multidiscip. Optim. 42(1), 1–23 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Craik, R.L.: Gait Analysis: Theory and Application, pp. 143–158 (1995)Google Scholar
  6. 6.
    Fish, D.J., Nielsen, J.-P.: Clinical assessment of human gait. J. Prosthet. Orthot. 5(2), 39 (1993)CrossRefGoogle Scholar
  7. 7.
    Galley, P., Forster, A.: Human Movement: An Introductory Text for Physiotherapy Students. Churchill Livingstone, London (1987)Google Scholar
  8. 8.
    Kuo, A.D., Donelan, J.M.: Dynamic principles of gait and their clinical implications. Phys. Ther. 90(2), 15 (2010)CrossRefGoogle Scholar
  9. 9.
    Ferris, D.P., Sawicki, G.S., Daley, M.A.: A physiologist’s perspective on robotic exoskeletons for human locomotion. Int. J. Humanoid Robot. 4(03), 507–528 (2007)CrossRefGoogle Scholar
  10. 10.
    Hase, K., et al.: Human gait simulation with a neuromusculoskeletal model and evolutionary computation. J. Vis. Comput. Animat. 14(2), 73–92 (2003)CrossRefGoogle Scholar
  11. 11.
    Whittle, M.W.: Gait Analysis: An Introduction. Butterworth-Heinemann, London (2014)Google Scholar
  12. 12.
    Doyle, T.L., et al.: Discriminating between elderly and young using a fractal dimension analysis of centre of pressure. Int. J. Med. Sci. 1(1), 11–20 (2004)CrossRefGoogle Scholar
  13. 13.
    Kay, B.A.: The dimensionality of movement trajectories and the degrees of freedom problem: a tutorial. Hum. Mov. Sci. 7(2), 343–364 (1988)CrossRefGoogle Scholar
  14. 14.
    Kelso, J., et al.: Phase-locked modes, phase transitions and component oscillators in biological motion. Phys. Scr. 35(1), 79 (1987)CrossRefGoogle Scholar
  15. 15.
    Buzzi, U.H., et al.: Nonlinear dynamics indicates aging affects variability during gait. Clin. Biomech. 18(5), 435–443 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yamada, N.: Chaotic swaying of the upright posture. Hum. Mov. Sci. 14(6), 711–726 (1995)CrossRefGoogle Scholar
  17. 17.
    Yamada, N.: Nature of variability in rhythmical movement. Hum. Mov. Sci. 14(3), 371–384 (1995)CrossRefGoogle Scholar
  18. 18.
    Blin, O., Ferrandez, A.-M., Serratrice, G.: Quantitative analysis of gait in Parkinson patients: increased variability of stride length. J. Neurol. Sci. 98(1), 91–97 (1990)CrossRefGoogle Scholar
  19. 19.
    Factor, S.A., Weiner, W.: Parkinson’s Disease: Diagnosis & Clinical Management. Demos Medical Publishing, New York (2007)Google Scholar
  20. 20.
    Morris, M.E., et al.: Stride length regulation in Parkinson’s disease. Brain 119(2), 551–568 (1996)CrossRefGoogle Scholar
  21. 21.
    Shumway-Cook, A., Woollacott, M.H.: Motor Control: Translating Research into Clinical Practice. Lippincott Williams & Wilkins, Philadelphia (2007)Google Scholar
  22. 22.
    Kellert, S.H.: In the Wake of Chaos: Unpredictable Order in Dynamical Systems. University of Chicago Press, Chicago (1994)zbMATHGoogle Scholar
  23. 23.
    Nijmeijer, H., Schumacher, J.: Four decades of mathematical system theory. In: Polderman, J.W., Trentelman, H.L. (eds.) The Mathematics of Systems and Control: From Intelligent Control to Behavioral Systems, pp. 73–83. University of Groningen Press, Groningen (1999)Google Scholar
  24. 24.
    Walleczek, J.: Self-Organized Biological Dynamics and Nonlinear Control: Toward Understanding Complexity, Chaos and Emergent Function in Living Systems. Cambridge University Press, Cambridge (2006)Google Scholar
  25. 25.
    Fradkov, A.L., Evans, R.J.: Control of chaos: methods and applications in engineering. Annu. Rev. Control 29(1), 33–56 (2005)CrossRefGoogle Scholar
  26. 26.
    Hilborn, R.C.: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press, New York (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D 9(1–2), 189–208 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hausdorff, J.M., et al.: Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J. Appl. Physiol. 78(1), 349–358 (1995)CrossRefGoogle Scholar
  29. 29.
    Hausdorff, J.M., et al.: Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. J. Appl. Physiol. 80(5), 1448–1457 (1996)CrossRefGoogle Scholar
  30. 30.
    Harbourne, R.T., Stergiou, N.: Movement variability and the use of nonlinear tools: principles to guide physical therapist practice. Phys. Ther. 89(3), 267 (2009)CrossRefGoogle Scholar
  31. 31.
    Shim, Y., Husbands, P.: Chaotic exploration and learning of locomotion behaviors. Neural Comput. 24(8), 2185–2222 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Collins, S.H.: Dynamic Walking Principles Applied to Human Gait. ProQuest, Ann Arbor (2008)Google Scholar
  33. 33.
    Hirata, K., Kokame, H.: Stability analysis of linear systems with state jump-motivated by periodic motion control of passive walker. In: IEEE Conference on Proceedings of Control Applications, CCA 2003 (2003)Google Scholar
  34. 34.
    Suzuki, S., Furuta, K.: Enhancement of stabilization for passive walking by chaos control approach. IFAC Proc. Vol. 35(1), 133–138 (2002)CrossRefGoogle Scholar
  35. 35.
    Katoh, R., Mori, M.: Control method of biped locomotion giving asymptotic stability of trajectory. Automatica 20(4), 405–414 (1984)CrossRefzbMATHGoogle Scholar
  36. 36.
    Gritli, H., Khraief, N., Belghith, S.: Chaos control in passive walking dynamics of a compass-gait model. Commun. Nonlinear Sci. Numer. Simul. 18(8), 2048–2065 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mitra, S., Riley, M.A., Turvey, M.T.: Chaos in human rhythmic movement. J. Mot. Behav. 29(3), 195–198 (1997)CrossRefGoogle Scholar
  38. 38.
    Gritli, H., Belghith, S., Khraief, N.: Cyclic-fold bifurcation and boundary crisis in dynamic walking of biped robots. Int. J. Bifurc. Chaos 22(10), 1250257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Gritli, H., Belghith, S.: Bifurcations and chaos in the semi-passive bipedal dynamic walking model under a modified OGY-based control approach. Nonlinear Dyn. 83(4), 1955–1973 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gritli, H., Belghith, S., Khraief, N.: OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot. Nonlinear Dyn. 79(2), 1363–1384 (2015)CrossRefzbMATHGoogle Scholar
  41. 41.
    Gritli, H., Belghith, S.: Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: analysis of local bifurcations via the hybrid Poincaré map. Chaos Solitons Fractals 98, 72–87 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Gritli, H., Belghith, S.: Walking dynamics of the passive compass-gait model under OGY-based control: emergence of bifurcations and chaos. Commun. Nonlinear Sci. Numer. Simul. 47, 308–327 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Gritli, H., Belghith, S.: Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map. Chaos Solitons Fractals 81, 172–183 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Garcia, M., et al.: The simplest walking model: stability, complexity, and scaling. J. Biomech. Eng. Trans. ASME 120(2), 281–288 (1998)CrossRefGoogle Scholar
  45. 45.
    Garcia, M., Chatterjee, A., Ruina, A.: Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dyn. Stab. Syst. 15(2), 75–99 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Goswami, A., Thuilot, B., Espiau, B.: Compass-like biped robot part I: stability and bifurcation of passive gaits. INRIA (1996)Google Scholar
  47. 47.
    McGeer, T.: Passive dynamic walking. Int. J. Robot. Res. 9(2), 62–82 (1990)CrossRefGoogle Scholar
  48. 48.
    McGeer, T.: Passive walking with knees. In: IEEE International Conference on Robotics and Automation, Proceedings. IEEE (1990)Google Scholar
  49. 49.
    McGeer, T.: Principles of walking and running. Adv. Comp. Environ. Physiol. 11, 113–139 (1992)CrossRefGoogle Scholar
  50. 50.
    Gritli, H., Khraeif, N., Belghith, V.: Falling of a passive compass-gait biped robot caused by a boundary crisis. In: Proceedings of the 4th Chaotic Modeling and Simulation International Conference, Crete, Greece (2011)Google Scholar
  51. 51.
    Kinsella-Shaw, J.M., Shaw, B., Turvey, M.T.: Perceiving ’walk-on-able’ slopes. Ecol. Psychol. 4(4), 223–239 (1992)CrossRefGoogle Scholar
  52. 52.
    Gritli, H., Belghith, S., Khraeif, N.: Intermittency and interior crisis as route to chaos in dynamic walking of two biped robots. Int. J. Bifurc. Chaos 22(03), 1250056 (2012)CrossRefzbMATHGoogle Scholar
  53. 53.
    Goswami, A., Espiau, B., Keramane, A.: Limit cycles and their stability in a passive bipedal gait. In: 1996 IEEE International Conference on Robotics and Automation. Proceedings. IEEE (1996)Google Scholar
  54. 54.
    Goswami, A., Espiau, B., Keramane, A.: Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Auton. Robots 4(3), 273–286 (1997)CrossRefGoogle Scholar
  55. 55.
    Goswami, A., Thuilot, B., Espiau, B.: A study of the passive gait of a compass-like biped robot: symmetry and chaos. Int. J. Robot. Res. 17(12), 1282–1301 (1998)CrossRefGoogle Scholar
  56. 56.
    McGeer, T.: Passive dynamic biped catalogue, 1991. In: Experimental Robotics II. Springer, Berlin, pp. 463–490 (1993)Google Scholar
  57. 57.
    Sabaapour, M.R., Hairi Yazdi, M., Beigzadeh, B.: Passive dynamic turning in 3D biped locomotion: an extension to passive dynamic walking. Adv. Robot. 30(3), 218–231 (2016)CrossRefGoogle Scholar
  58. 58.
    Safa, A.T., Naraghi, M.: The role of walking surface in enhancing the stability of the simplest passive dynamic biped. Robotica 33(1), 195 (2015)CrossRefGoogle Scholar
  59. 59.
    Spong, M.W.: The passivity paradigm in the control of bipedal robots. In: Climbing and Walking Robots, pp. 775–786. Springer (2005)Google Scholar
  60. 60.
    Spong, M.W., Holm, J.K., Lee, D.: Passivity-based control of bipedal locomotion. IEEE Robot. Autom. Mag. 14(2), 30–40 (2007)CrossRefGoogle Scholar
  61. 61.
    Taga, G., Yamaguchi, Y., Shimizu, H.: Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol. Cybern. 65(3), 147–159 (1991)CrossRefzbMATHGoogle Scholar
  62. 62.
    Spong, M.W., Bullo, F.: Controlled symmetries and passive walking. IEEE Trans. Autom. Control 50(7), 1025–1031 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Iida, F., Tedrake, R.: Minimalistic control of a compass gait robot in rough terrain. In: IEEE International Conference on Robotics and Automation. ICRA’09 (2009)Google Scholar
  64. 64.
    Gritli, H., Belghith, S.: Displayed phenomena in the semi-passive torso-driven biped model under OGY-based control method: birth of a torus bifurcation. Appl. Math. Model. 40(4), 2946–2967 (2016)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Alexander, R.: Simple models of human movement. Appl. Mech. Rev. 48(8), 461–469 (1995)CrossRefGoogle Scholar
  66. 66.
    May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261(5560), 459–467 (1976)CrossRefzbMATHGoogle Scholar
  67. 67.
    Baumol, W.J., Benhabib, J.: Chaos: significance, mechanism, and economic applications. J. Econ. Perspect. 3(1), 77–105 (1989)CrossRefGoogle Scholar
  68. 68.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Ni, X., Chen, W., Liu, J.: The effects of parameters on the stability of passive dynamic walking. In: 3rd International Conference on. 2009 Bioinformatics and Biomedical Engineering. ICBBE 2009. IEEE (2009)Google Scholar
  70. 70.
    Dingwell, J.B., John, J., Cusumano, J.P.: Do humans optimally exploit redundancy to control step variability in walking? PLoS Comput. Biol. 6(7), e1000856 (2010)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefzbMATHGoogle Scholar
  72. 72.
    Sarbaz, Y., et al.: Gait spectral analysis: an easy fast quantitative method for diagnosing Parkinson’s disease. J. Mech. Med. Biol. 12(03), 1250041 (2012)CrossRefGoogle Scholar
  73. 73.
    Giladi, N., Nieuwboer, A.: Understanding and treating freezing of gait in parkinsonism, proposed working definition, and setting the stage. Mov. Disord. 23(S2), S423–S425 (2008)CrossRefGoogle Scholar
  74. 74.
    Yogev, G., et al.: Dual tasking, gait rhythmicity, and Parkinson’s disease: which aspects of gait are attention demanding? Eur. J. Neurosci. 22(5), 1248–1256 (2005)CrossRefGoogle Scholar
  75. 75.
    Bond, J.M., Morris, M.: Goal-directed secondary motor tasks: their effects on gait in subjects with Parkinson disease. Arch. Phys. Med. Rehabil. 81(1), 110–116 (2000)CrossRefGoogle Scholar
  76. 76.
    Camicioli, R., et al.: Verbal fluency task affects gait in Parkinson’s disease with motor freezing. J. Geriatr. Psychiatry Neurol. 11(4), 181–185 (1998)CrossRefGoogle Scholar
  77. 77.
    Theiler, J.: Estimating fractal dimension. JOSA A 7(6), 1055–1073 (1990)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Hans, S.: Space-Filling Curves. Springer, Berlin (2012)Google Scholar
  79. 79.
    Accardo, A., et al.: Use of the fractal dimension for the analysis of electroencephalographic time series. Biol. Cybern. 77(5), 339–350 (1997)CrossRefzbMATHGoogle Scholar
  80. 80.
    Higuchi, T.: Approach to an irregular time series on the basis of the fractal theory. Physica D 31(2), 277–283 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Acharya, R., et al.: Analysis of cardiac health using fractal dimension and wavelet transformation. ITBM-RBM 26(2), 133–139 (2005)CrossRefGoogle Scholar
  82. 82.
    Manabe, Y., et al.: Fractal dimension analysis of static stabilometry in Parkinson’s disease and spinocerebellar ataxia. Neurol. Res. 23(4), 397–404 (2001)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Biomedical EngineeringAmirkabir University of TechnologyTehranIran

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