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A mathematical framework to study fast walking of human

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Abstract

The body of a walking human is an elaborated dynamic system that operates adaptively in various conditions such as fast walking. Due to dynamic redundancies, the individual motor control strategies for speeding up the walking can be different among normal subjects. However, in reality, we see that the pattern of motion is quite similar among people and it is only the profile of hip joint motion along its path which determines the speed. The objective of the current paper is to develop a mathematical framework to investigate time optimal motion of a human during walking. To this end, a nine-link planar biped model is used. The motion is considered to take place in sagittal plane and to follow a normal pattern of motion. The solution is obtained using a phase plane method to solve minimum time problem which is subjected to inequality constraints of variable maximum joint torques and stability conditions. The solution method can be used to find the maximum possible speed of a human with specific body characteristics and to obtain a hip joint trajectory which could produce that speed. The proposed method can be utilized to study quantitative effect of different parameters such as joint strength in fast walking.

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References

  1. Perry, J.: Gait Analysis: Normal and Pathological Function. SLACK Incorporated, Thorofare (2010)

    Google Scholar 

  2. Murray, M.P., Mollinger, L.A., Gardner, G.M., Sepic, S.B.: Kinematic and EMG patterns during slow, free, and fast walking. J. Orthop. Res. 2, 272–280 (1984)

    Article  Google Scholar 

  3. Winter, D.A.: Kinematic and kinetic patterns in human gait: variability and compensating effects. Hum. Mov. Sci. 3, 51–76 (1984)

    Article  Google Scholar 

  4. Andriacchi, T.P., Ogle, J.A., Galante, J.O.: Walking speed as a basis for normal and abnormal gait measurements. J. Biomech. 10, 261–268 (1977)

    Article  Google Scholar 

  5. Lelas, J.L., Merriman, G.J., Riley, P.O., Kerrigan, D.C.: Predicting peak kinematic and kinetic parameters from gait speed. Gait Posture 17, 106–112 (2003)

    Article  Google Scholar 

  6. Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model: stability, complexity, and scaling. J. Biomech. Eng. 120, 281–288 (1998)

    Article  Google Scholar 

  7. Chevallereau, C., Aoustin, Y.: Optimal reference trajectories for walking and running of a biped robot. Robotica 19, 557–569 (2001)

    Article  Google Scholar 

  8. Miossec, S., Aoustin, Y.: A simplified stability study for a biped walk with underactuated and overactuated phases. Int. J. Robot. Res. 24, 537–551 (2005)

    Article  Google Scholar 

  9. Kirk, D.E.: Optimal Control Theory: An Introduction. Dover, New York (2004)

    Google Scholar 

  10. Bobrow, J.E., Dubowsky, S., Gibson, J.S.: Time-optimal control of robotic manipulators along specified paths. Int. J. Robot. Res. 4, 3–17 (1985)

    Article  Google Scholar 

  11. Sadigh, M.J., Mansouri, S.: Application of phase-plane method in generating minimum time solution for stable walking of biped robot with specified pattern of motion. Robotica 31, 837–851 (2013)

    Article  Google Scholar 

  12. Anderson, D.E., Madigan, M.L., Nussbaum, M.A.: Maximum voluntary joint torque as a function of joint angle and angular velocity: model development and application to the lower limb. J. Biomech. 40, 3105–3113 (2007)

    Article  Google Scholar 

  13. Vukobratovic, M., Potkonjak, V., Tzafestas, S.: Human and humanoid dynamics. J. Intell. Robot. Syst. 41, 65–84 (2004)

    Article  Google Scholar 

  14. Vukobratovic, M., Borovac, B.: Zero-moment point—thirty five years of its life. Int. J. Humanoid Robot. 01, 157–173 (2004)

    Article  Google Scholar 

  15. Gross, M.M., Stevenson, P.J., Charette, S.L., Pyka, G., Marcus, R.: Effect of muscle strength and movement speed on the biomechanics of rising from a chair in healthy elderly and young women. Gait Posture 8, 175–185 (1998)

    Article  Google Scholar 

  16. Pavol, M.J., Owings, T.M., Foley, K.T., Grabiner, M.D.: Influence of lower extremity strength of healthy older adults on the outcome of an induced trip. J. Am. Geriatr. Soc. 50, 256–262 (2002)

    Article  Google Scholar 

  17. Grabiner, M.D., Owings, T.M., Pavol, M.J.: Lower extremity strength plays only a small role in determining the maximum recoverable lean angle in older adults. J. Gerontol., Ser. A, Biol. Sci. Med. Sci. 60, 1447–1450 (2005)

    Article  Google Scholar 

  18. Wojcik, L., Thelen, D.G., Schultz, A.B., Ashton-Miller, J., Alexander, N.B.: Age and gender differences in peak lower extremity joint torques and ranges of motion used during single-step balance recovery from a forward fall. J. Biomech. 34, 67–73 (2001)

    Article  Google Scholar 

  19. Pfeiffer, F., Johanni, R.: A concept for manipulator trajectory planning. In: Proceedings of 1986 IEEE International Conference on Robotics and Automation, pp. 1399–1405 (1986)

    Chapter  Google Scholar 

  20. Zlajpah, L.: On time optimal path control of manipulators with bounded joint velocities and torques. In: Proceedings of IEEE ICRA, Minneapolis, MN, pp. 1572–1577 (1996)

    Google Scholar 

  21. Moon, S.B., Ahmad, S.: Time-optimal trajectories for cooperative multi-manipulator systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 27, 343–353 (1997)

    Article  Google Scholar 

  22. Ghasemi, M.H., Sadigh, M.J.: A direct algorithm to compute the switching curve for time-optimal motion of cooperative multi-manipulators moving on a specified path. Adv. Robot. 22, 493–506 (2008)

    Google Scholar 

  23. Bobrow, J.E., McCarthy, J.M., Chu, V.K.: Minimum-time trajectories for two robots holding the same workpiece. In: 29th IEEE Conference on Decision and Control, vol. 6, pp. 3102–3107 (1990)

    Chapter  Google Scholar 

  24. Winter, D.A.: Biomechanics and Motor Control of Human Movement. Wiley, Hoboken (2009)

    Book  Google Scholar 

  25. Farizeh, T., Mansouri, S., Sadigh, M.J.: Effect of increase in single support phase on walking speed of a biped robot. In: 2011 IEEE International Conference on Robotics and Biomimetics, pp. 287–292. IEEE Press, New York (2011)

    Chapter  Google Scholar 

  26. Farizeh, T., Mansouri, S., Sadigh, M.J.: A Parameter Study on Effect of Pattern of Motion on Walking Speed of Biped. Volume 4: Dynamics, Control and Uncertainty, Parts A and B p. 381. ASME, Montreal (2012)

    Google Scholar 

  27. Farizeh, T., Sadigh, M.J.: Effect of Heel to Toe Walking on Time Optimal Walking of a Biped During Single Support Phase. Volume 3: Biomedical and Biotechnology Engineering p. V003T03A054. ASME, Montreal (2014)

    Google Scholar 

  28. England, S.A., Granata, K.P.: The influence of gait speed on local dynamic stability of walking. Gait Posture 25, 172–178 (2007)

    Article  Google Scholar 

  29. Webpage: The importance of shin and ankle strength for racewalkers. http://www.racewalking.org/shins.htm

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Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, Daneshgah Blvd., Isfahan, 84156-83111, Iran

    Tara Farizeh

  2. College of Engineering, Department of Mechanical Engineering, University of Tehran, Shahid Farshi Moghaddam Ave., North Karghar St., North Amirabad Campus, Tehran, Iran

    Mohammad Jafar Sadigh

Authors
  1. Tara Farizeh
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  2. Mohammad Jafar Sadigh
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Correspondence to Mohammad Jafar Sadigh.

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Farizeh, T., Sadigh, M.J. A mathematical framework to study fast walking of human. Multibody Syst Dyn 40, 99–122 (2017). https://doi.org/10.1007/s11044-015-9494-3

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  • DOI: https://doi.org/10.1007/s11044-015-9494-3

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