Selection Procedures for the Largest Lyapunov Exponent in Gait Biomechanics

  • Peter C. Raffalt
  • Jenny A. Kent
  • Shane R. Wurdeman
  • Nicholas StergiouEmail author


The present study was aimed at investigating the effectiveness of the Wolf et al. (LyE_W) and Rosenstein et al. largest Lyapunov Exponent (LyE_R) algorithms to differentiate data sets with distinctly different temporal structures. The three-dimensional displacement of the sacrum was recorded from healthy subjects during walking and running at two speeds; one low speed close to the preferred walking speed and one high speed close to the preferred running speed. LyE_R and LyE_W were calculated using four different time series normalization procedures. The performance of the algorithms were evaluated based on their ability to return relative low values for slow walking and fast running and relative high values for fast walking and slow running. Neither of the two algorithms outperformed the other; however, the effectiveness of the two algorithms was highly dependent on the applied time series normalization procedure. Future studies using the LyE_R should normalize the time series to a fixed number of strides and a fixed number of data points per stride or data points per time series while the LyE_W should be applied to time series normalized to a fixed number of data points or a fixed number of strides.


Locomotion Dynamics Walking Variability Nonlinear analysis 



This work was supported by the Center for Research in Human Movement Variability and the National Institutes of Health (P20GM109090 and R15HD08682).

Supplementary material

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Supplementary material 1 (PDF 301 kb)


  1. 1.
    Alkjaer, T., P. C. Raffalt, H. Dalsgaard, E. B. Simonsen, N. C. Petersen, H. Bliddal, and M. Henriksen. Gait variability and motor control in people with knee osteoarthritis. Gait Posture 42:479–484, 2015.CrossRefGoogle Scholar
  2. 2.
    Bruijn, S. M., O. G. Meijer, P. J. Beek, and J. H. van Dieen. Assessing the stability of human locomotion: a review of current measures. J. R. Soc. Interface 10:20120999, 2013.CrossRefGoogle Scholar
  3. 3.
    Bruijn, S. M., O. G. Meijer, S. M. Rispens, A. Daffertshofer, and J. H. van Dieen. Letter to the editor: “Sensitivity of the Wolf’s and Rosenstein’s algorithms to evaluate local dynamic stability from small gait data sets”. Ann. Biomed. Eng. 40:2505–2506, 2012; (author reply 2507-2509).CrossRefGoogle Scholar
  4. 4.
    Bruijn, S. M., J. H. van Dieen, O. G. Meijer, and P. J. Beek. Is slow walking more stable? J. Biomech. 42:1506–1512, 2009.CrossRefGoogle Scholar
  5. 5.
    Buzzi, U. H., N. Stergiou, M. J. Kurz, P. A. Hageman, and J. Heidel. Nonlinear dynamics indicates aging affects variability during gait. Clin. Biomech. 18:435–443, 2003.CrossRefGoogle Scholar
  6. 6.
    Cignetti, F., L. M. Decker, and N. Stergiou. Sensitivity of the Wolf’s and Rosenstein’s algorithms to evaluate local dynamic stability from small gait data sets. Ann. Biomed. Eng. 40:1122–1130, 2012.CrossRefGoogle Scholar
  7. 7.
    Cignetti, F., L. M. Decker, N. Stergiou, et al. Sensitivity of the Wolf’s and Rosenstein’s algorithms to evaluate local dynamic stability from small gait data sets: response to commentaries by Bruijn. Ann. Biomed. Eng. 40:2507–2509, 2012.CrossRefGoogle Scholar
  8. 8.
    Cohen, J. Statistical Power Analysis for the Behavioural Sciences. Hillsdale, NJ: Lawrence Erlbaum, 1988.Google Scholar
  9. 9.
    Diedrich, F. J., and W. H. Warren, Jr. Why change gaits? Dynamics of the walk-run transition. J. Exp. Psychol. Hum. Percept. Perform. 21:183–202, 1995.CrossRefGoogle Scholar
  10. 10.
    Diedrich, F. J., and W. H. Warren. The dynamics of gait transitions: effects of grade and load. J. Mot. Behav. 30:60–78, 1998.CrossRefGoogle Scholar
  11. 11.
    Dingwell, J. B., and J. P. Cusumano. Nonlinear time series analysis of normal and pathological human walking. Chaos 10:848–863, 2000.CrossRefGoogle Scholar
  12. 12.
    Dingwell, J. B., J. P. Cusumano, P. R. Cavanagh, and D. Sternad. Local dynamic stability versus kinematic variability of continuous overground and treadmill walking. J. Biomech. Eng. 123:27–32, 2001.CrossRefGoogle Scholar
  13. 13.
    Dingwell, J. B., and L. C. Marin. Kinematic variability and local dynamic stability of upper body motions when walking at different speeds. J. Biomech. 39:444–452, 2006.CrossRefGoogle Scholar
  14. 14.
    England, S. A., and K. P. Granata. The influence of gait speed on local dynamic stability of walking. Gait Posture 25:172–178, 2007.CrossRefGoogle Scholar
  15. 15.
    Hattie, J., and R. W. Cooksey. Procedures for assessing the validities of tests using the “Known-Groups” method. Appl. Psychol. Meas. 8:295–305, 1984.CrossRefGoogle Scholar
  16. 16.
    Hausdorff, J. M., S. L. Mitchell, R. Firtion, C. K. Peng, M. E. Cudkowicz, J. Y. Wei, and A. L. Goldberger. Altered fractal dynamics of gait: reduced stride-interval correlations with aging and Huntington’s disease. J. Appl. Physiol. 82(262–269):1997, 1985.Google Scholar
  17. 17.
    Ihlen, E. A. F., K. S. van Schooten, S. M. Bruijn, M. Pijnappels, and J. H. van Dieen. Fractional stability of trunk acceleration dynamics of daily-life walking: toward a unified concept of gait stability. Front. Physiol. 8:516, 2017.CrossRefGoogle Scholar
  18. 18.
    Jordan, K., J. H. Challis, J. P. Cusumano, and K. M. Newell. Stability and the time-dependent structure of gait variability in walking and running. Hum. Mov. Sci. 28:113–128, 2009.CrossRefGoogle Scholar
  19. 19.
    Myers, S. A., J. M. Johanning, N. Stergiou, R. I. Celis, L. Robinson, and Pipinos, II. Gait variability is altered in patients with peripheral arterial disease. J. Vasc. Surg. 49:924-931.e921, 2009.Google Scholar
  20. 20.
    Myers, S. A., Pipinos, II, J. M. Johanning, and N. Stergiou. Gait variability of patients with intermittent claudication is similar before and after the onset of claudication pain. Clin. Biomech. 26:729–734, 2011.CrossRefGoogle Scholar
  21. 21.
    Raffalt, P. C., M. K. Guul, A. N. Nielsen, S. Puthusserypady, and T. Alkjaer. Economy, movement dynamics, and muscle activity of human walking at different speeds. Sci. Rep. 7:43986, 2017.CrossRefGoogle Scholar
  22. 22.
    Rosenstein, M. T., J. J. Collins, and C. J. De Luca. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65:117–134, 1993.CrossRefGoogle Scholar
  23. 23.
    Sauer, T., and J. A. Yorke. How many delay coordinates do you need? Int. J. Bifurc. Chaos 3:737–744, 1993.CrossRefGoogle Scholar
  24. 24.
    Sauer, T., J. A. Yorke, and M. Casdagli. Embedology. J. Stat. Phys. 65:579–616, 1991.CrossRefGoogle Scholar
  25. 25.
    Scafetta, N., D. Marchi, and B. J. West. Understanding the complexity of human gait dynamics. Chaos 19:026108, 2009.CrossRefGoogle Scholar
  26. 26.
    Stenum, J., S. M. Bruijn, and B. R. Jensen. The effect of walking speed on local dynamic stability is sensitive to calculation methods. J. Biomech. 47:3776–3779, 2014.CrossRefGoogle Scholar
  27. 27.
    Stergiou, N. Innovative Analyses of Human Movement. Champaign, Illinois: Human Kinetics, 2004.Google Scholar
  28. 28.
    Stergiou, N. Nonlinear Analysis for Human Movement Variability. Boca Raton, Florida: Taylor & Francis Group, 2016.CrossRefGoogle Scholar
  29. 29.
    Takens, F. Detecting strange attractors in turbulence. Dyn. Syst. Turbul. Lect. Notes Math. 898:366–381, 1981.CrossRefGoogle Scholar
  30. 30.
    Terrier, P., and O. Deriaz. Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking. J. Neuroeng. Rehabil. 8:12, 2011.CrossRefGoogle Scholar
  31. 31.
    Terrier, P., and O. Deriaz. Non-linear dynamics of human locomotion: effects of rhythmic auditory cueing on local dynamic stability. Front. Physiol. 4:230, 2013.CrossRefGoogle Scholar
  32. 32.
    Terrier, P., V. Turner, and Y. Schutz. GPS analysis of human locomotion: further evidence for long-range correlations in stride-to-stride fluctuations of gait parameters. Hum. Mov. Sci. 24:97–115, 2005.CrossRefGoogle Scholar
  33. 33.
    van Schooten, K. S., S. M. Rispens, M. Pijnappels, A. Daffertshofer, and J. H. van Dieen. Assessing gait stability: the influence of state space reconstruction on inter- and intra-day reliability of local dynamic stability during over-ground walking. J. Biomech. 46:137–141, 2013.CrossRefGoogle Scholar
  34. 34.
    Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining lyapunov exponents from a time-series. Physica D 16:285–317, 1985.CrossRefGoogle Scholar
  35. 35.
    Wurdeman, S. R. State-space reconstruction. In: Nonlinear Analysis for Human Movement Variability, edited by N. Stergiou. Boca Raton, Florida: Taylor & Francis Group, 2016.Google Scholar
  36. 36.
    Wurdeman, S. R., S. A. Myers, A. L. Jacobsen, and N. Stergiou. Prosthesis preference is related to stride-to-stride fluctuations at the prosthetic ankle. J. Rehabil. Res. Dev. 50:671–686, 2013.CrossRefGoogle Scholar
  37. 37.
    Wurdeman, S. R., S. A. Myers, and N. Stergiou. Transtibial amputee joint motion has increased attractor divergence during walking compared to non-amputee gait. Ann. Biomed. Eng. 41:806–813, 2013.CrossRefGoogle Scholar
  38. 38.
    Wurdeman, S. R., S. A. Myers, and N. Stergiou. Amputation effects on the underlying complexity within transtibial amputee ankle motion. Chaos 24:013140, 2014.CrossRefGoogle Scholar
  39. 39.
    Zampeli, F., C. O. Moraiti, S. Xergia, V. A. Tsiaras, N. Stergiou, and A. D. Georgoulis. Stride-to-stride variability is altered during backward walking in anterior cruciate ligament deficient patients. Clin. Biomech. 25:1037–1041, 2010.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2019

Authors and Affiliations

  • Peter C. Raffalt
    • 1
    • 2
    • 3
  • Jenny A. Kent
    • 3
  • Shane R. Wurdeman
    • 3
    • 4
  • Nicholas Stergiou
    • 3
    • 5
    Email author
  1. 1.Julius Wolff Institute for Biomechanics and Musculoskeletal RegenerationCharité – Universitätsmedizin BerlinBerlinGermany
  2. 2.Department of Biomedical SciencesUniversity of CopenhagenCopenhagen NDenmark
  3. 3.Department of Biomechanics and Center for Research in Human Movement VariabilityUniversity of Nebraska at OmahaOmahaUSA
  4. 4.Department of Clinical and Scientific AffairsHanger ClinicHoustonUSA
  5. 5.College of Public Health984355 University of Nebraska Medical CenterOmahaUSA

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