Modeling human walking as an inverted pendulum of varying length

doi:10.1017/CBO9780511542336.019

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

Published online by Cambridge University Press: 10 August 2009

Jack T. Stern ,

Brigitte Demes and

D. Casey Kerrigan

Edited by

Fred Anapol ,

Rebecca Z. German and

Nina G. Jablonski

Show author details

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

Book contents

Get access

Summary

Symbols and abbreviations

Ac axial (centripetal or centrifugal) acceleration of the mass
ΔC forward translation of the substrate contact point
COM center of mass
D instantaneous horizontal distance traveled by the mass since t = 0
Dh the total horizontal distance traveled by the mass during one swing of the inverted pendulum, i.e., the step length
f stride frequency
Fc centripetal force required to keep the mass on a circular path
GRFv vertical component of the ground reaction force
H instantaneous value of the height of the point mass (or COM)
H height of the point mass (or COM) at the middle of the first double-support phase
H height of the point mass (or COM) at the middle of the second double-support phase
H height of the point mass (or COM) at the middle of the single-support phase
ΔH maximum vertical excursion of the point mass (or COM) from MDS to MSS
I moment of inertia of the mass about the pivot point of the inverted pendulum
Lc the axially directed force exerted on the virtual stance limb (VSL) by the mass
-Lc the axially directed force exerted on the mass by the virtual stance limb (VSL)
M mass
MDS middle of double-support phase
MSS middle of single-support phase
R the length of the virtual stance limb (from substrate contact-point to the point mass or its surrogate) at any moment in time
R length of the virtual stance limb at MDS
R length of the virtual stance limb at MSS
S stature
t time
t half the duration of the swing of the inverted pendulum
V velocity of the mass
Vc axial (centripetal or centrifugal) velocity of the mass
Vo orbital (tangential) velocity of the mass
[…]

Type: Chapter
Information: Shaping Primate Evolution
Form, Function, and Behavior
, pp. 297 - 333

DOI: https://doi.org/10.1017/CBO9780511542336.019 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 Scholar PubMed

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, 189–198CrossRef Google Scholar

Alexander, R. McN. and Jayes, A. S. (1978). Vertical movements in walking and running. J. Zool. Lond., 185, 27–40CrossRef Google 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, 261–268CrossRef Google Scholar PubMed

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, 411–425CrossRef Google Scholar PubMed

Cavagna, G. A., Thys, H., and Zamboni, A. (1976). The sources of external work in level walking and running. J. Physiol., 262, 639–657CrossRef Google Scholar PubMed

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, 293–303CrossRef Google Scholar PubMed

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, 375–379CrossRef Google Scholar PubMed

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 Scholar PubMed

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, 233–238CrossRef Google 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, 379–399CrossRef Google Scholar

Helene, O. (1984). On “waddling” and race walking. Amer. J. Phys., 52, 656CrossRef Google Scholar

Hreljac, A. (1993). Preferred and energetically optimal gait transition speeds in human locomotion. Med. Sci. Sports and Exerc., 25, 1158–1162CrossRef Google Scholar PubMed

Hreljac, A. (1995). Determinants of the gait transition speed during locomotion: kinematic factors. J. Biomech., 28, 699–677CrossRef Google Scholar PubMed

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, 987–995CrossRef Google Scholar PubMed

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, 3–8CrossRef Google Scholar PubMed

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, 1077–1080CrossRef Google Scholar PubMed

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, 217–220CrossRef Google 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 Scholar PubMed

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 Scholar PubMed

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, 1194–1202CrossRef Google Scholar PubMed

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 Scholar PubMed

Mochon, S. and McMahon, T. A. (1980a). Ballistic walking. J. Biomech., 13, 49–57CrossRef Google Scholar

Mochon, S. and McMahon, T. A. (1980b). Ballistic walking: an improved model. Math. Biosci., 52, 241–260CrossRef Google 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, 8–24CrossRef Google Scholar PubMed

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 Scholar PubMed

Pandy, M. G. and Berme, N. (1988a). A numerical method for simulating the dynamics of human walking. J. Biomech., 21, 1043–1051CrossRef Google Scholar

Pandy, M. G. and Berme, N. (1988b). Synthesis of human walking: a planar model for single support. J. Biomech., 21, 1053–1060CrossRef Google Scholar

Reynolds, T. R. (1987). Stride length and its determinants in humans, early hominids, primates and mammals. Amer. J. Phys. Anthropol., 72, 101–115CrossRef Google Scholar PubMed

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, 133–139CrossRef Google Scholar PubMed

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, 543–558CrossRef Google Scholar

Shimba, T. (1984). An estimation of center of gravity from force platform data. J. Biomech., 17, 53–60CrossRef Google Scholar PubMed

Siegler, S., Seliktar, R., and Hyman, W. (1982). Simulation of human gait with the aid of a simple mechanical model. J. Biomech., 15, 415–425CrossRef Google Scholar PubMed

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 Scholar PubMed

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, 306–314CrossRef Google 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, 211–214CrossRef Google Scholar PubMed

Thorstensson, A., Nilsson, J., Carlson, H., and Zomlefer, R. (1984). Trunk movements in human locomotion. Acta Physiol. Scand., 121, 9–22CrossRef Google Scholar PubMed

Waters, R. L., Morris, J., and Perry, J. (1973). Translational motion of the head and trunk during normal walking. J. Biomech., 6, 167–172CrossRef Google Scholar PubMed

Whittle, M. W. (1997). Three-dimensional motion of the center of gravity of the body during walking. Hum. Move. Sci., 16, 347–355CrossRef Google Scholar

Book contents

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

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive