Journal of Mechanical Science and Technology

, Volume 28, Issue 4, pp 1393–1401 | Cite as

A modeling study of mechanical energetic optimality in incline walking

  • Keonyoung Oh
  • Jae-Kwan Ryu
  • Sukyung Park


To maintain steady level walking, collision loss is predominantly compensated for with push-off propulsion, and negligible additional work is performed during the single support phase. The observed energy balance during the double support phase is energetically optimal. However, unlike level walking, significant work proportional to the incline slope was observed during the single support phase, which raises the question of whether energetic optimality applies to incline walking. In this study, we examined the energetic optimality of incline walking using a simple work-energy relationship. Work performed by the leading and trailing leg over a gait cycle was estimated for various incline slopes, and the optimal push-off impulse that minimized the total work performed was calculated. The model prediction for least costly gait occurred when push-off propulsion provided all of the necessary work for raising or lowering the body center of mass (CoM) and collision compensation. When we assumed that the generation of optimal propulsion is gradually scaled to obey a feasible push-off constraint, which was estimated based on the allowable plantar flexor torque and the weight support of the trailing leg, the predicted slope-proportional increase in mechanical work done by the ground reaction force (GRF) during the single support phase was consistent with the empirical data. This result implies that the energetic optimality of incline walking can be described from a mechanical perspective and is subject to a feasible push-off propulsion constraint. However, the implication of the mechanical perspective of energetic optimality on the metabolic cost should be further examined and compared using empirical data.


Constraint Heel strike Incline walking Mechanical work Push-off 


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  1. [1]
    J. M. Donelan, R. Kram and A. D. Kuo, Simultaneous positive and negative external mechanical work in human walking, Journal of Biomechanics, 35 (2002) 117–124.CrossRefGoogle Scholar
  2. [2]
    J. M. Donelan, R. Kram and A. D. Kuo, Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking, The Journal of Experimental Biology, 205(23) (2002) 3717–3727.Google Scholar
  3. [3]
    A. Hernandez, A. Silder, B. C. Heiderscheit and D. G. Thelen, Effect of age on center of mass motion during human walking, Gait and Posture, 30 (2009) 217–222.CrossRefGoogle Scholar
  4. [4]
    J. Yeom and S. Park, A gravitational impulse model predicts collision impulse and mechanical work during a step-to-step transition, Journal of Biomechanics, 44(1) (2011) 59–67.CrossRefGoogle Scholar
  5. [5]
    S. Kim and S. Park, The oscillatory behavior of the CoM facilitates mechanical energy balance between push-off and heel strike, Journal of Biomechanics, 45 (2012) 326–333.CrossRefGoogle Scholar
  6. [6]
    A. D. Kuo, Energetics of actively powered locomotion using the simplest walking model, Journal of Biomechanical Engineering, 124(1) (2002) 113–120.CrossRefGoogle Scholar
  7. [7]
    A. Ruina, J. E. Bertram and M. Srinivasan, A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudoelastic leg behavior in running and the walk-to-run transition, J. Theor. Biol., 237(2) (2005) 170–192.CrossRefMathSciNetGoogle Scholar
  8. [8]
    A. E. Minetti, L. P. Ardigo and F. Saibene, Mechanical determinants of gradient walking energetics in man, Journal of Physiology, 471 (1993) 725–735.Google Scholar
  9. [9]
    J. S. Gottschall and R. Kram, Mechanical energy fluctuations during hill walking: the effects of slope on inverted pendulum exchange, The Journal of Experimental Biology, 209 (2006) 4895–4900.CrossRefGoogle Scholar
  10. [10]
    J. R. Franz, N. E. Lyddon and R. Kram, Mechanical work performed by the individual legs during uphill and downhill walking, Journal of Biomechanics, 45 (2012) 257–262.CrossRefGoogle Scholar
  11. [11]
    M. Kuster, S. Sakurai and G. A. Wood, Kinematic and kinetic comparison of downhill and level walking, Clinical Biomechanics, 10(2) (1995) 79–84.CrossRefGoogle Scholar
  12. [12]
    A. S. McIntosh, K. T. Beatty, L. N. Dwan and D. R. Vickers, Gait dynamics on an inclined walkway, Journal of Biomechanics, 39 (2006) 2491–2502.CrossRefGoogle Scholar
  13. [13]
    P. G. Adamczyk and A. D. Kuo, Redirection of center-of-mass velocity during the step-to-step transition of human walking, The Journal of Experimental Biology, 212(16) (2009) 2668–2678.CrossRefGoogle Scholar
  14. [14]
    K. Oh, J. Baek and S. Park, Gait strategy changes with acceleration to accommodate the biomechanical constraint on push-off propulsion, Journal of Biomechanics, 45 (2012) 2920–2926.CrossRefGoogle Scholar
  15. [15]
    R. R. Neptune and K. Sasaki, Ankle plantar flexor force production is an important determinant of the preferred walk-to-run transition speed, The Journal of Experimental Biology, 208(5) (2005) 799–808.CrossRefGoogle Scholar
  16. [16]
    D. A. Cunningham, D. Morrison, C. L. Rice and C. Cooke, Ageing and isokinetic plantar flexion, European Journal of Applied Physiology, 56(1) (1987) 24–29.CrossRefGoogle Scholar
  17. [17]
    A. R. Fugl-Meyer, L. Gustafsson and Y. Burstedt, Isokinetic and static plantar flexion characteristics, European Journal of Applied Physiology, 45(2) (1980) 221–234.CrossRefGoogle Scholar
  18. [18]
    G. Thelen, A. B. Schultz, N. B. Alexander and J. A. Ashton-Miller, Effects of age on rapid ankle torque development, Journals of Gerontology Series A: Biological and Medical Sciences, 51(5) (1996) 226–232.CrossRefGoogle Scholar
  19. [19]
    S. Park, F. B. Horak and A. D. Kuo, Postural feedback responses scale with biomechanical constraints in human standing, Experimental Brain Research, 154 (2004) 417–427.CrossRefGoogle Scholar
  20. [20]
    S. Kim, F. B. Horak, P. Carlson-Kuhta and S. Park, Postural Feedback Scaling Deficits in Parkinson’s Disease, Journal of Neurophysiology, 102 (2009) 2910–2920.CrossRefGoogle Scholar
  21. [21]
    S. Kim, C. G. Atkeson and S. Park, Perturbation-dependent selection of postural feedback gain and its scaling, Journal of Biomechanics, 45 (2012) 1379–1386.CrossRefGoogle Scholar
  22. [22]
    H. M. Maus, S. W. Lipfert, M. Gross, J. Rummel and A. Seyfarth, Upright human gait did not provide a major mechanical challenge for our ancestors, Nature Communications, 1 (2010) doi: 10.1038/ncomms1073.Google Scholar
  23. [23]
    J. Doke, J. M. Donelan and A. D. Kuo, Mechanics and energetics of swinging the human leg, The Journal of Experimental Biology, 208 (2005) 439–445.CrossRefGoogle Scholar
  24. [24]
    B. R. Umberger, Stance and swing phase costs in human walking, Journal of the Royal Society Interface, 7 (2010) 1329–1340.CrossRefGoogle Scholar
  25. [25]
    R. Neptune, S. A. Kautz and F. E. Zajac, Contributions of the individual ankle plantar flexors to support, forward progression and swing initiation during walking, Journal of Biomechanics, 34(11) (2001) 1387–1398.CrossRefGoogle Scholar
  26. [26]
    G. S. Sawicki and D. P. Ferris, Mechanics and energetics of level walking with powered ankle exoskeletons, The Journal of Experimental Biology, 211(9) (2008) 1402.CrossRefGoogle Scholar
  27. [27]
    S. Sawicki and D. P. Ferris, Powered ankle exoskeletons reveal the metabolic cost of plantar flexor mechanical work during walking with longer steps at constant step frequency, The Journal of Experimental Biology, 212(1) (2009) 21–31.CrossRefGoogle Scholar
  28. [28]
    P. Kao, C. L. Lewis and D. P. Ferris, Invariant ankle moment patterns when walking with and without a robotic ankle exoskeleton, Journal of Biomechanics, 43 (2010) 203–209.CrossRefGoogle Scholar
  29. [29]
    Park, H. Choi, K. Ryu, S. Kim and Y. Kim, Kinematics, kinetics and muscle activities of the lower extremity during the first four steps from gait initiation to the steady-state walking, Journal of Mechanical Science and Technology, 23 (2009) 204–211.CrossRefGoogle Scholar
  30. [30]
    J. H. Park and S. Chung, Optimal locomotion trajectory for biped robot’ D2’ with knees stretched, heel-contact landings, and toe-off liftoffs, Journal of Mechanical Science and Technology, 25(12) (2011) 3231–3241.CrossRefGoogle Scholar
  31. [31]
    T. L. Heiden, D. G. Lloyd and T. R. Ackland, Knee joint kinematics, kinetics and muscle co-contraction in knee osteoarthritis patient gait, Clinical Biomechanics, 24 (2009) 833–841.CrossRefGoogle Scholar
  32. [32]
    P. Mahaudens, X. Banse, M. Mousny and C. Detrembleur, Gait in adolescent idiopathic scoliosis: kinematics and electromyographic analysis, European Spine Journal, 18 (2009) 512–521.CrossRefGoogle Scholar
  33. [33]
    L. Peterson, S. A. Kautz and R. R. Neptune, Muscle work is increased in pre-swing during hemiparetic walking, Clinical Biomechanics, 26 (2011) 859–866.CrossRefGoogle Scholar
  34. [34]
    M. Seyedali, J. M. Czerniecki, D. C. Morgenroth and M. E. Hahn, Co-contraction patterns of trans-tibial amputee ankle and knee musculature during gait, Journal of NeuroEngineering and Rehabilitation, 9 (2012) doi:10.1186/1743-0003-1189-1129.Google Scholar
  35. [35]
    R. R. Neptune, K. Sasaki and S. A. Kautz, The effect of walking speed on muscle function and mechanical energetics, Gait and Posture, 28(1) (2008) 135–143.CrossRefGoogle Scholar
  36. [36]
    G. S. Sawicki, C. L. Lewis and D. P. Ferris, It pays to have a spring in your step, Exercise and Sport Sciences Reviews, 37(3) (2009) 130–138.CrossRefGoogle Scholar
  37. [37]
    D. Chang, J. Kim, D. Choi, K. Cho, T. Seo and J. Kim, Design of a slider-crank leg mechanism for mobile hopping robotic platforms, Journal of Mechanical Science and Technology, 27(1) (2013) 207–214.CrossRefGoogle Scholar
  38. [38]
    D. M. Wanta, F. J. Nagle and P. Webb, Metabolic response to graded downhill walking, Medicine and Science in Sports and Exercise, 25 (1993) 159–162.CrossRefGoogle Scholar
  39. [39]
    L. C. Hunter, E. C. Hendrix and J. C. Dean, The cost of walking downhill: Is the preferred gait energetically optimal?, Journal of Biomechanics, 43 (2010) 1910–1915.CrossRefGoogle Scholar
  40. [40]
    H. C. Doets, D. Vergouw, H. E. Veeger and H. Houdijk, Metabolic cost and mechanical work for the step-to-step transition in walking after successful total ankle arthroplasty, Human Movement Science, 28(6) (2009) 786–797.CrossRefGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringKorea Advanced Institute of Science and Technology (KAIST)DaejeonKorea
  2. 2.LIG Nex1 Pangyo R&D CenterGyeonggi-doKorea

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