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14 - Modeling human walking as an inverted pendulum of varying length

Published online by Cambridge University Press:  10 August 2009

By
Jack T. Stern ,
Brigitte Demes  and
D. Casey Kerrigan
Edited by
Fred Anapol ,
Rebecca Z. German  and
Nina G. Jablonski
Jack T. Stern
Affiliation:
Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA
Brigitte Demes
Affiliation:
Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA
D. Casey Kerrigan
Affiliation:
Department of Physical Medicine and Rehabilitation, School of Medicine, University of Virginia, Charlottesville, VA 22908-1007 USA
Fred Anapol
Affiliation:
University of Wisconsin, Milwaukee
Rebecca Z. German
Affiliation:
University of Cincinnati
Nina G. Jablonski
Affiliation:
California Academy of Sciences, San Francisco
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Summary

Symbols and abbreviations

  1. Ac axial (centripetal or centrifugal) acceleration of the mass

  2. ΔC forward translation of the substrate contact point

  3. COM center of mass

  4. D instantaneous horizontal distance traveled by the mass since t = 0

  5. Dh the total horizontal distance traveled by the mass during one swing of the inverted pendulum, i.e., the step length

  6. f stride frequency

  7. Fc centripetal force required to keep the mass on a circular path

  8. GRFv vertical component of the ground reaction force

  9. H instantaneous value of the height of the point mass (or COM)

  10. H height of the point mass (or COM) at the middle of the first double-support phase

  11. H height of the point mass (or COM) at the middle of the second double-support phase

  12. H height of the point mass (or COM) at the middle of the single-support phase

  13. ΔH maximum vertical excursion of the point mass (or COM) from MDS to MSS

  14. I moment of inertia of the mass about the pivot point of the inverted pendulum

  15. Lc the axially directed force exerted on the virtual stance limb (VSL) by the mass

  16. -Lc the axially directed force exerted on the mass by the virtual stance limb (VSL)

  17. M mass

  18. MDS middle of double-support phase

  19. MSS middle of single-support phase

  20. R the length of the virtual stance limb (from substrate contact-point to the point mass or its surrogate) at any moment in time

  21. R length of the virtual stance limb at MDS

  22. R length of the virtual stance limb at MSS

  23. S stature

  24. t time

  25. t half the duration of the swing of the inverted pendulum

  26. V velocity of the mass

  27. Vc axial (centripetal or centrifugal) velocity of the mass

  28. Vo orbital (tangential) velocity of the mass

  29. […]

Type
Chapter
Information
Shaping Primate Evolution
Form, Function, and Behavior
 , pp. 297 - 333
Publisher: Cambridge University Press
Print publication year: 2004

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References

Alexander, R. McN. (1976). Mechanics of bipedal locomotion. In: Perspectives in Experimental Biology, vol. 1, ed. P. S. Davis. Oxford: Pergamon. pp. 493–504CrossRef
Alexander, R. McN. (1984). Walking and running. Amer. Sci., 72, 348–354Google Scholar
Alexander, R. McN. (1991). Energy saving mechanisms in walking and running. J. Exp. Biol., 160, 55–69Google ScholarPubMed
Alexander, R. McN. (1992a). The Human Machine. New York, NY: Columbia University Press
Alexander, R. McN. (1992b). A model of bipedal locomotion on compliant legs. Phil. Trans. Roy. Soc. Lond., 338, 189–198CrossRefGoogle Scholar
Alexander, R. McN. and Jayes, A. S. (1978). Vertical movements in walking and running. J. Zool. Lond., 185, 27–40CrossRefGoogle Scholar
Andriacchi, T. P., Ogle, J. A., and Galante, J. O. (1977). Walking speed as a basis for normal and abnormal gait measurements. J. Biomech., 10, 261–268CrossRefGoogle ScholarPubMed
Beuter, A. and Lefebvre, R. (1988). Un modèle théorique de transition de phase dans la locomotion humaine. Can. J. Sport Sci., 13, 247–253Google Scholar
Braune, W. and Fischer, O. (1889). In: On the Centre of Gravity of the Human Body, translated by P. G. J. Maquet and R. Furlong. Berlin: Springer, 1985
Cappozzo, A. (1981). Analysis of the linear displacement of the head and trunk during walking at different speeds. J. Biomech., 14, 411–425CrossRefGoogle ScholarPubMed
Cavagna, G. A., Thys, H., and Zamboni, A. (1976). The sources of external work in level walking and running. J. Physiol., 262, 639–657CrossRefGoogle ScholarPubMed
Croskey, M. I., Dawson, P. M., Luessen, A. C., Marohn, I. E., and Wright, H. E. (1922). The height of the center of gravity in man. Amer. J. Physiol., 61, 171–185Google Scholar
Crowe, A., Schiereck, P., Boer, R. W., and Keessen, W. (1995). Characterization of human gait by means of body center of mass oscillations derived from ground reaction forces. IEEE Trans. Biomed. Eng., 42, 293–303CrossRefGoogle ScholarPubMed
Duff-Raffaele, M., Kerrigan, D. C., Corcoran, P. J., and Saini, M. (1996). The proportional work of lifting the center of mass during walking. Amer. J. Phys. Med. Rehabil., 75, 375–379CrossRefGoogle ScholarPubMed
Engsberg, J. R., Tedford, K. G., and Harder, J. A. (1992). Center of mass location and segment angular orientation of below-knee amputee and able-bodied children during walking. Arch. Phys. Med. Rehabil., 73, 1163–1168Google ScholarPubMed
Fischer, O. (1899). Der Gang des Menschen II. Theil: die Bewegung des Gesammtschwerpunktes und die äusseren Kräfte. Abhandl. K. Sächsischen Gesellsch. Wissensch., 43, 1–130Google Scholar
Fischer, O. (1903). Kinematics of the swing of the leg. In: The Human Gait, translated by P. Maquet and R. Furlong. Berlin: Springer, 1987. pp. 315–384
Gard, S. A. and Childress, D. S. (1997). The effect of pelvic list on the vertical displacement of the trunk during normal walking. Gait and Posture, 5, 233–238CrossRefGoogle Scholar
Gard, S. A. and Childress, D. S. (1999). The influence of stance-phase knee flexion on the vertical displacement of the trunk during normal walking. Arch. Phys. Med. Rehabil., 80, 26–32CrossRef
Grieve, D. W. and Gear, R. J. (1966). The relationship between length of stride, step frequency, time of swing and speed of walking for children and adults. Ergonomics, 5, 379–399CrossRefGoogle Scholar
Helene, O. (1984). On “waddling” and race walking. Amer. J. Phys., 52, 656CrossRefGoogle Scholar
Hreljac, A. (1993). Preferred and energetically optimal gait transition speeds in human locomotion. Med. Sci. Sports and Exerc., 25, 1158–1162CrossRefGoogle ScholarPubMed
Hreljac, A. (1995). Determinants of the gait transition speed during locomotion: kinematic factors. J. Biomech., 28, 699–677CrossRefGoogle ScholarPubMed
Iida, H. and Yamamuro, T. (1987). Kinetic analysis of the center of gravity of the human body in normal and pathological gaits. J. Biomech., 20, 987–995CrossRefGoogle ScholarPubMed
Inman, V. T., Ralston, H. J., and Todd, F. (1981). Human Walking. Baltimore, MD: Williams and Wilkins
Kerrigan, D. C., Viramontes, B. E., Corcoran, P. J., and LaRaia, P. J. (1995). Measured versus predicted vertical displacement of the sacrum during gait as a tool to measure biomechanical gait performance. Amer. J. Phys. Med. Rehabil., 74, 3–8CrossRefGoogle ScholarPubMed
Kerrigan, D. C., Della Croce, U., Marciello, M., and Riley, P. O. (2000). A refined view of the determinants of gait: significance of heel rise. Arch. Phys. Med. Rehabil., 81, 1077–1080CrossRefGoogle ScholarPubMed
Kerrigan, D. C., Riley, P. O., Lelas, J. L., and Della Croce, U. (2001) Quantification of pelvic rotation as a determinant of gait. Arch. Phys. Med. Rehabil., 82, 217–220CrossRefGoogle Scholar
Kram, R., Domingo, A., and Ferris, D. P. (1997). Effect of reduced gravity on the preferred walk–run transition speed. J. Exp. Biol., 200, 821–826Google ScholarPubMed
Lamoreux, L. W. (1971). Kinematic measurements in the study of human walking. Bull. Prosthet. Res., 10–15, 3–84Google Scholar
Lee, C. R. and Farley, C. T. (1998). Determinants of the center of mass trajectory in human walking and running. J. Exp. Biol., 201, 2935–2944Google ScholarPubMed
Martin, R. and Saller, K. (1959). Lehrbuch der Anthropologie. Band II Stuttgart: Gustav Fischer
Minetti, A. E., Capelli, C., Zamparo, P., di Prampero, P. E., and Saibene, F. (1995). Effects of stride frequency on mechanical power and energy expenditure of walking. Med. Sci. Sports and Exerc., 27, 1194–1202CrossRefGoogle ScholarPubMed
Minetti, A. E. and Saibene, F. (1992). Mechanical work rate minimization and freely chosen stride frequency of human walking: a mathematical model. J. Exp. Biol., 170, 19–34Google ScholarPubMed
Mochon, S. and McMahon, T. A. (1980a). Ballistic walking. J. Biomech., 13, 49–57CrossRefGoogle Scholar
Mochon, S. and McMahon, T. A. (1980b). Ballistic walking: an improved model. Math. Biosci., 52, 241–260CrossRefGoogle Scholar
Murray, M. P., Kory, R. C., Clarkson, B. H., and Sepic, S. B. (1966). Comparison of free and fast speed walking patterns of normal men. Amer. J. Phys. Med., 45, 8–24CrossRefGoogle ScholarPubMed
Noble, B., Metz, K., Pandolf, K. B., Bell, C. W., Cafarelli, E., and Sime, W. E. (1973). Perceived exertion during walking and running. II. Med. Sci. Sports, 5, 116–120Google ScholarPubMed
Pandy, M. G. and Berme, N. (1988a). A numerical method for simulating the dynamics of human walking. J. Biomech., 21, 1043–1051CrossRefGoogle Scholar
Pandy, M. G. and Berme, N. (1988b). Synthesis of human walking: a planar model for single support. J. Biomech., 21, 1053–1060CrossRefGoogle Scholar
Reynolds, T. R. (1987). Stride length and its determinants in humans, early hominids, primates and mammals. Amer. J. Phys. Anthropol., 72, 101–115CrossRefGoogle ScholarPubMed
Saini, M., Kerrigan, D. C., Thirunarayan, M. A., and Duff-Raffaele, M. (1998). The vertical displacement of the center of mass during walking: a comparison of four measurement methods. J. Biomech. Eng., 120, 133–139CrossRefGoogle ScholarPubMed
Saunders, J. B. deC., Inman, V. T., and Eberhart, H. D. (1953). The major determinants in normal and pathological gait. J. Bone Joint Surg., 38A, 543–558CrossRefGoogle Scholar
Shimba, T. (1984). An estimation of center of gravity from force platform data. J. Biomech., 17, 53–60CrossRefGoogle ScholarPubMed
Siegler, S., Seliktar, R., and Hyman, W. (1982). Simulation of human gait with the aid of a simple mechanical model. J. Biomech., 15, 415–425CrossRefGoogle ScholarPubMed
Simon, S. R., Knirk, J. K., Mansour, J. M., and Koskinen, M. F. (1977). The dynamics of the center of mass during walking and its clinical applicability. Bull. Hosp. Joint Dis., 38, 112–116Google ScholarPubMed
Thirunarayan, M. A., Kerrigan, D. C., Rabuffetti, M., Della Croce, U., and Saini, M. (1996). Comparison of three methods for estimating vertical displacement of center of mass during level walking in patients. Gait and Posture, 4, 306–314CrossRefGoogle Scholar
Thorstensson, A. and Robertson, H. (1987). Adaptations to changing speed in human locomotion: speed of transition between walking and running. Acta Physiol. Scand., 131, 211–214CrossRefGoogle ScholarPubMed
Thorstensson, A., Nilsson, J., Carlson, H., and Zomlefer, R. (1984). Trunk movements in human locomotion. Acta Physiol. Scand., 121, 9–22CrossRefGoogle ScholarPubMed
Waters, R. L., Morris, J., and Perry, J. (1973). Translational motion of the head and trunk during normal walking. J. Biomech., 6, 167–172CrossRefGoogle ScholarPubMed
Whittle, M. W. (1997). Three-dimensional motion of the center of gravity of the body during walking. Hum. Move. Sci., 16, 347–355CrossRefGoogle Scholar

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  • Modeling human walking as an inverted pendulum of varying length
    • By Jack T. Stern, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, Brigitte Demes, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, D. Casey Kerrigan, Department of Physical Medicine and Rehabilitation, School of Medicine, University of Virginia, Charlottesville, VA 22908-1007 USA
  • Edited by Fred Anapol, University of Wisconsin, Milwaukee, Rebecca Z. German, University of Cincinnati, Nina G. Jablonski, California Academy of Sciences, San Francisco
  • Book: Shaping Primate Evolution
  • Online publication: 10 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542336.019
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  • Modeling human walking as an inverted pendulum of varying length
    • By Jack T. Stern, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, Brigitte Demes, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, D. Casey Kerrigan, Department of Physical Medicine and Rehabilitation, School of Medicine, University of Virginia, Charlottesville, VA 22908-1007 USA
  • Edited by Fred Anapol, University of Wisconsin, Milwaukee, Rebecca Z. German, University of Cincinnati, Nina G. Jablonski, California Academy of Sciences, San Francisco
  • Book: Shaping Primate Evolution
  • Online publication: 10 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542336.019
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Modeling human walking as an inverted pendulum of varying length
    • By Jack T. Stern, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, Brigitte Demes, Department of Anatomical Sciences, School of Medicine, Health Sciences Center, Stony Brook University, Stony Brook, NY 11794-8081, USA, D. Casey Kerrigan, Department of Physical Medicine and Rehabilitation, School of Medicine, University of Virginia, Charlottesville, VA 22908-1007 USA
  • Edited by Fred Anapol, University of Wisconsin, Milwaukee, Rebecca Z. German, University of Cincinnati, Nina G. Jablonski, California Academy of Sciences, San Francisco
  • Book: Shaping Primate Evolution
  • Online publication: 10 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542336.019
Available formats
×