Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-03T01:40:21.271Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy-optimal relative timing of stance-leg push-off and swing-leg retraction in walking

Published online by Cambridge University Press:  17 September 2015

S. Javad Hasaneini ,
John E. A. Bertram  and
Chris J. B. Macnab
S. Javad Hasaneini*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, USA Department of Electrical and Computer Engineering, University of Calgary, Canada Department of Cell Biology and Anatomy, University of Calgary, Canada
John E. A. Bertram
Affiliation:
Department of Cell Biology and Anatomy, University of Calgary, Canada
Chris J. B. Macnab
Affiliation:
Department of Electrical and Computer Engineering, University of Calgary, Canada
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Summary

Swing-leg retraction in walking is the slowing or reversal of the forward rotation of the swing leg at the end of the swing phase prior to ground contact. For retraction, a hip torque is often applied to the swing leg at about the same time as stance-leg push-off. Due to mechanical coupling, the push-off force affects leg swing, and hip torque affects the stance-leg extension. This coupling makes the energetic costs of retraction and push-off depend on their relative timing. Here, we find the energy-optimal relative timing of these actions. We first use a simplified walking model with non-regenerative actuators, a work-based energetic-cost, and impulsive actuations. Depending on whether the late-swing hip torque is retracting or extending (pushing the leg forward), we find that the optimum is obtained by applying the impulsive hip torque either following or prior to the impulsive push-off force, respectively. These trends extend to other bipedal models and to aperiodic gaits, and are independent of step lengths and walking speeds. In one simulation, the cost of a walking step is increased by 17.6% if retraction torque comes before push-off. To consider non-impulsive actuation and the cost of force production, we add a force-squared (F2) term to the work cost. We show that this cost promotes simultaneous push-off force and retracting torque, but does not change the result that any extending torque should come prior to push-off. A high-fidelity optimization of the Cornell Ranger robot is consistent with the swing-retraction trends from the models above.

Keywords

legged locomotionbipedhip torqueoptimizationenergetic cost minimizationsimultaneous impulses
Type
Articles
Information
Robotica , Volume 35 , Issue 3 , March 2017 , pp. 654 - 686
Copyright
Copyright © Cambridge University Press 2015 

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

1. Inman, V. T., Ralston, H. J. and Todd, F., Human Walking (Williams & Wilkins, Baltimore, 1981).Google Scholar
2. Poggensee, K. L., Sharbafi, M. A. and Seyfarth, A., “Characterizing Swing-Leg Retraction in Human Locomotion,” Proceedings of 17th International Conference on Climbing and Walking Robots, Poznan, Poland (Jul. 2014).Google Scholar
3. Muybridge, E., Animal Locomotion (University of Pennsylvania, Philadelphia, P.A., 1887).Google Scholar
4. Rose, J. and Gamble, J. G., Human Walking, 2nd ed. (Williams & Wilkins, Baltimore, 1994).Google Scholar
5. Hasaneini, S. J., Energy Efficient Bipedal Locomotion Ph.D. dissertation (University of Calgary, Calgary, Alberta, Canada, Jan. 2014).Google Scholar
6. Wisse, M., Atkeson, C. G. and Kloimwieder, D. K., “Swing Leg Retraction Helps Biped Walking Stability,” Proceedings 5th IEEE-RAS International Conference on Humanoid Robots, Tsukuba, Japan (Dec. 2005) pp. 295–300.Google Scholar
7. Seyfarth, A., Geyer, H. and Herr, H., “Swing-leg retraction: A simple control model for stable running,” J. Exp. Biol. 206, 2547–2555 (2003).Google Scholar
8. Hobbelen, D. G. E. and Wisse, M., “Swing-leg retraction for limit cycle walkers improves disturbance rejection,” IEEE Trans. Robot. 24 (2), 377–389 (2008).Google Scholar
9. Karssen, J. G. D., Haberland, M., Wisse, M. and Kim, S., “The optimal swing-leg retraction rate for running,” Proceedings 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China (May 2011) pp. 4000–4006.Google Scholar
10. Hasaneini, S. J., Macnab, C. J., Bertram, J. E. and Ruina, A., “Seven Reasons to Brake Leg swing Just Before Heel Strike,” Online Proceedings of Dynamic Walking Conference (Pittsburgh, P.A., 2013).Google Scholar
11. Bhounsule, P. A., Cortell, J., Grewal, A., Hendriksen, B., Karssen, J. D., Paul, C. and Ruina, A., “Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge,” Int. J. Robot. Res. 33 (10), 1305–1321 (2014).Google Scholar
12. Ruina, A., Bertram, J. E. A. and Srinivasan, M., “A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition,” J. Theor. Biol. 237 (2), 170–192 (Nov. 2005).Google Scholar
13. Kuo, A. D., “Energetics of actively powered locomotion using the simplest walking model,” J. Biomed. Eng. 124, 113–120 (2002).Google Scholar
14. Bertram, J. E. A. and Hasaneini, S. J., “Neglected losses and key costs: tracking the energetics of walking and running,” J. Exp. Biol. 216 (6), 933–938 (Mar. 2013).CrossRefGoogle ScholarPubMed
15. Hasaneini, S. J., Macnab, C. J., Bertram, J. E. and Leung, H., “Swing-Leg Retraction Efficiency in Bipedal Walking,” Intelligent Robots and Systems (IROS 2014), 2014 IEEE/RSJ International Conference on, IEEE (Chicago: IL, 2014) pp. 2515–2522.Google Scholar
16. McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 62–82 (1990).CrossRefGoogle Scholar
17. Hasaneini, S. J., Macnab, C. J., Bertram, J. E. and Leung, H., “Optimal Relative Timing of Stance Push-Off and swing leg retraction,” Intelligent Robots and Systems (IROS), 2013 IEEE/RSJ International Conference on. IEEE (Tokyo: Japan, 2013) pp. 3616–3623.Google Scholar
18. Srinivasan, M. and Ruina, A., “Computer optimization of a minimal biped model discovers walking and running,” Nature Mag. 439 (7072), 72–75 (Jan. 2006).CrossRefGoogle ScholarPubMed
19. Hasaneini, S. J., Macnab, C. J. B., Bertram, J. E. A. and Leug, H., “The dynamic optimization approach to locomotion dynamics: human-like gaits from a minimally-constrained biped model,” J. Adv. Robot. 27 (11), 845–859 (Jul. 2013).Google Scholar
20. Srinivasan, M., “Fifteen observations on the structure of energy-minimizing gaits in many simple biped models,” J. R. Soc. Interface 8 (54), 74–98 (2011).Google Scholar
21. Rebula, J. R. and Kuo, A. D., “The cost of leg forces in bipedal locomotion: A simple optimization study,” PloS one 10 (2), e0117384, (2015).Google Scholar
22. Donelan, J. M., Kram, R. and Kuo, A. D., “Simultaneous positive and negative external mechanical work in human walking,” J. biomech. 35 (1), 117–124 (2002).Google Scholar
23. Donelan, J. M., Kram, R. and Kuo, A. D., “Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking,” J. Exp. Biol. 205 (23), 3717–3727 (2002).Google Scholar
24. Mochon, S. and McMahon, T. A., “Ballistic walking: An improved model,” Math. Biosci. 52 (1), 241–260 (1980).CrossRefGoogle Scholar
25. Basmajian, J., “The human bicycle,” Biomechanics VA, 5, pp. 297–302 (1976).Google Scholar
26. McMahon, T. A., Muscles, Reflexes, and Locomotion (Princeton University Press, Princeton, NJ, 1984).Google Scholar
27. Kuo, A. D., Donelan, J. M. and Ruina, A., “Energetic consequences of walking like an inverted pendulum: step-to-step transitions,” Exercise Sport Sci. Rev. 33 (2), 88–97 (2005).CrossRefGoogle Scholar
28. Margaria, R., Biomechanics and Energetics of Muscular Exercise (Clarendon Press, UK, 1976).Google Scholar
29. Kuo, A. D., “A mechanical analysis of force distribution between redundant, multiple degree-of-freedom actuators in the human: Implications for the central nervous system,” Human Mov. Sci. 13 (5), 635–663 (1994).CrossRefGoogle Scholar
30. Bhounsolue, P. A., A controller design framework for bipedal robots: Trajectory optimization and event-based stabilization Ph.D. dissertation (Cornell University, Ithaca, NY, USA, May 2012).Google Scholar
31. Patterson, M. A. and Rao, A. V., “Gpops-ii: A matlab software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming,” ACM Trans. Math. Softw. (TOMS) 41 (1), 1 (2014).Google Scholar
32. Gill, P. E. and Murray, W., “Snopt: An sqp algorithm for large-scale constrained optimization,” SIAM J. Opt. 12 (4), 979–1006 (2002).Google Scholar
33. Winter, D. A., Biomechanics and Motor Control of Human Movement (John Wiley & Sons Inc., Hoboken, N.J., 2005).Google Scholar
34. Bertram, J. E. A. and Ruina, A., “Multiple walking speed-frequency relations are predicted by constrained optimization,” J. Theor. Biol. 209, 445–453 (2001).Google Scholar
35. Bertram, J. E. A., “Constrained optimization in human walking: cost minimization and gait plasticity,” J. Exp. Biol. 208, 979–991 (2005).Google Scholar
36. Spong, M. W., Hutchinson, S. and Vidyasagar, M., Robot Modeling and Control (John Wiley & Sons Inc., New York, 2006).Google Scholar
37. Formal'skii, A. M., “Ballistic walking design via impulsive control,” J. Aerospace Eng. 23 (2), 129–138 (Apr. 2010).Google Scholar