Fifteen observations on the structure of energy-minimizing gaits in many simple biped models

Journal of the Royal Society Interface

Volumn 8, Issue 54, 2011, Pages 74-98

  Srinivasan, Manoj ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Gaits; Legged locomotion; Metabolic cost; Minimize energy; Optimization and optimal control; Walking and running

Indexed keywords

ANIMALS; BEHAVIORAL RESEARCH; CONSTRAINT THEORY; CONTROL; COST FUNCTIONS; COSTS; METABOLISM; OPTIMIZATION;

GAITS; LEGGED LOCOMOTION; METABOLIC COST; MINIMIZE ENERGY; OPTIMIZATION AND OPTIMAL CONTROL; WALKING AND RUNNING;

MUSCLE;

ADENOSINE TRIPHOSPHATASE;

ANIMAL ANATOMY; ARTICLE; ENERGY CONSERVATION; ENZYME ACTIVITY; GAIT; HEAT LOSS; HUMAN; MATHEMATICAL MODEL; METABOLIC RATE; MUSCLE FORCE; MUSCLE METABOLISM; NONHUMAN; RUNNING; STRUCTURE ANALYSIS; VELOCITY;

ANIMALIA;

EID: 77956865663     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2009.0544     Document Type: Article

References (76)

1. Daffertshofer, A., Lamoth, C. J. C., Meijer, O. G. & Beek, P. J. 2004 PCA in studying coordination and variability: a tutorial. Clin. Biomech. 19, 415-428. (doi:10.1016/j.clinbiomech.2004.01.005)
2. Alexander, R. McN. 1976 Mechanics of bipedal locomotion. Perspectives in experimental biology, vol. 1, pp. 493-504. New York, NY: Pergamon Press.
3. Kuo, A. D., Donelan, J. M. & Ruina, A. 2005 Energetic consequences of walking like an inverted pendulum: step-to-step transitions. Exer. Sport Sci. Rev. 33, 88-97. (doi:10.1097/00003677-200504000-00006)
4. Ruina, A., Bertram, J. E. & Srinivasan, M. 2005 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. 14, 170-192.
5. Srinivasan, M. & Ruina, A. 2006 Computer optimization of a minimal biped model discovers walking and running. Nature 439, 72-75. (doi:10.1038/nature04113) (Pubitemid 43053630)
6. Rashevsky, N. 1948 On the locomotion of mammals. Bull. Math. Biophys. 10, 11-23. (doi:10.1007/BF02478326)
7. McMahon, T. A. & Cheng, G. C. 1990 The mechanics of running: how does stiffness couple with speed? J. Biomech. 23(Suppl. 1), 65-78. (doi:10.1016/0021-9290(90)90042-2)
8. Farley, C. T., Blickhan, R., Saito, J. & Taylor, C. R. 1991 Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits. J. Appl. Physiol. 71, 2127-2132.
9. Blickhan, R. & Full, R. J. 1993 Similarity in multilegged locomotion: bouncing like a monopode. J. Comp. Physiol. A. 173, 509-517.
10. Geyer, H., Seyfarth, A. & Blickhan, R. 2006 Compliant leg behaviour explains basic dynamics of walking and running. Proc. R. Soc. B 273, 2861-2867. (doi:10.1098/rspb.2006.3637)
11. O'Connor, S. M. & Kuo, A. D. 2006 Dynanic biopedal locomotion on compliant legs. J. Biomech. 39(Suppl. 1), S359.
12. Srinivasan, M. & Holmes, P. 2008 How well can springmass-like telescoping leg models fit multi-pedal sagittal plane locomotion? J. Theor. Biol. 255, 1-7. (doi:10.1016/j.jtbi.2008.06.034)
13. Borelli, J. A. 1680 On the movement of animals (De Motu Animalium, Pars prima) (Maquet, P. transl. 1989), p. 152. Berlin, Germany: Springer-Verlag.
14. Alexander, R. McN. 1989 Optimization and gaits in the locomotion of vertebrates. Physiol. Rev. 69, 1199-1227.
15. Alexander, R. McN. 1996 Optima for animals. Princeton, NJ: Princeton University Press.
16. Srinivasan, M. 2009 Optimal speeds for walking and running, and walking on a moving walkway. Chaos 19, 026112.
17. Chow, C. K. & Jacobson, D. H. 1971 Studies of human locomotion via optimal programming. Math. Biosci. 10, 239-306. (doi:10.1016/0025-5564(71)90062- 9)
18. Hatze, H. 1976 The complete optimization of the human motion. Math. Biosci. 28, 99-135. (doi:10.1016/0025-5564(76)90098-5)
19. Beletskii, V. V. & Chudinov, P. S. 1977 Parametric optimization in the problem of bipedla locomotion. Izv. An SSSR Mekhanika Tverdogo (Mech. Solids), pp. 25-35.
20. Alexander, R. McN. 1980 Optimum walking techniques for quadrupeds and bipeds. J. Zool. Lond. 192, 97-117. (doi:10.1111/j.1469-7998.1980.tb04222.x)
21. Alexander, R. McN. 1992 A model of bipedal locomotion on compliant legs. Phil. Trans. R. Soc. Lond. B 338, 189-198. (doi:10.1098/rstb.1992.0138)
22. Davy, D. T. & Audu, M. L. 1987 A dynamic optimization technique for predicting muscle forces in the swing phase of gait. J. Biomech. 20, 187-201. (doi:10.1016/0021-9290(87)90310-1)
23. Minetti, A. E. & Alexander, R. McN. 1997 A theory of metabolic costs for bipedal gaits. J. Theor. Biol 186, 467-476. (doi:10.1006/jtbi.1997.0407)
24. Anderson, F. C. & Pandy, M. G. 2001 Dynamic optimization of human walking. J. Biomech. Eng. 123, 381-390. (doi:10.1115/1.1392310)
25. Safonova, A., Pollard, N. & Hodgins, J. K. 2003 Optimizing human motion for the control of a humanoid robot. In Proc. Int. Conf. on Robotics and Automation, pp. 1390-1397.
26. Liu, C. K., Hertzmann, A. & Popović, Z. 2005 Learning physics-based motion style with nonlinear inverse optimization. ACM Trans. Graph. 24, 1071-1081. (doi:10.1145/1073204.1073314)
27. Ren, L., Jones, R. & Howard, D. 2007 Predictive modelling of human walking over a complete gait cycle. J. Biomech. 40, 1567-1574. (doi:10.1016/j.jbiomech.2006.07.017)
28. Doke, J., Donelan, M. J. & Kuo, A. D. 2005 Mechanics and energetics of swinging the human leg. J. Exp. Biol. 208, 439-445. (doi:10.1242/jeb.01408)
29. Srinivasan, M. & Ruina, A. 2007 Idealized walking and running gaits minimize work. Proc. R. Soc. A 463, 2429-2446. (doi:10.1098/rspa.2007.0006)
30. Srinivasan, M. 2006 Why walk and run: energetic costs and energetic optimality in simple mechanics-based models of a bipedal animal. PhD thesis, Cornell University, Ithaca.
31. Cavagna, G. A., Saibene, F. P. & Margaria, R. 1964 Mechanical work in running. J. Appl. Physiol 19, 249-256.
32. Alexander, R. McN. 1988 Elastic mechanisms in animal movement. Cambridge, UK: Cambridge University Press.
33. Alexander, R. McN. 2003 Principles of animal locomotion. Princeton, NJ: Princeton University Press.
34. Ralston, H. J. 1958 Energy-speed relation and optimal speed during level walking. Eur. J. Appl. Physiol. (Int. Z. Angew. Physiol.) 17, 277-283.
35. Hoyt, D. F. & Taylor, C. R. 1981 Gait and the energetics of locomotion in horses. Nature 292, 239-240. (doi:10.1038/292239a0)
36. Kuo, A. D. 2001 A simple model predicts the step length-speed relationship in human walking. J. Biomech. Eng. 123, 264-269. (doi:10.1115/1.1372322)
37. Tucker, V. A. 1975 The energetic cost of moving about. Am. Sci. 63, 413-419.
38. Radhakrishnan, V. 1998 Locomotion: dealing with friction. Proc. Natl Acad. Sci. USA 95, 5448-5455. (doi:10.1073/pnas.95.10.5448)
39. Margaria, R. 1976 Biomechanics and energetics of muscular exercise. Oxford, UK: Clarendon Press.
40. Alexander, R. McN. 2002 Work or force minimization as a criterion for the evolution of locomotion. In Fourth World Congress of Biomechanics, Calgary, Canada.
41. Alexander, R. McN. 1997 A minimum energy cost hypothesis for human arm trajectories. Biol. Cybern. 76, 97-105. (doi:10.1007/s004220050324)
42. Alexander, R. McN. 1997 Optimum muscle design for oscillatory movements. J. Theor. Biol. 184, 253-259. (doi:10.1006/jtbi.1996.0271)
43. Woledge, R. C., Curtin, N. A. & Homsher, E. 1985 Energetic aspects of muscle contraction. London, UK: Academic Press.
44. Ma, S. P. & Zahalak, G. I. 1991 A distribution-moment model of energetics in skeletal muscle. J. Biomech. 24, 21-35. (doi:10.1016/0021-9290(91) 90323-F)
45. Crowninshield, R. D. 1978 Use of optimization techniques to predict muscle forces. J. Biomech. Eng. 100, 88-92.
46. Crowninshield, R. D. & Brand, R. A. 1981 A physiologically based criterion of muscle force prediction in locomotion. J. Biomech. 14, 793-801. (doi:10.1016/0021-9290(81)90035-X)
47. Fenn, W. O. 1924 The relation between the work performed and the energy liberated in muscular contraction. J. Physiol. 58, 373-395.
48. Roussel, L., Canudas-De-Wit, C. & Goswami, A. 1998 Generation of energy optimal complete gait cycles for biped robots. In Proc. 1998, IEEE International Conference on Robotics and Automation, vol. 3, pp. 2036-2041.
49. Chevallereau, C. & Aoustin, Y. 2001 Optimal reference trajectories for walking and running of a biped robot. Robotica 19, 557-569.
50. Bhounsule, P. 2008 Cornell ranger: computer simulation and experimental fits. Dynamic walking. Delft, The Netherlands: Holland.
51. Bhounsule, P. & Ruina, A. 2009 Cornell ranger: energyoptimal control. In Dynamic walking. Vancouver, Canada.
52. Rudin, W. 1987 Real and complex analysis. New York, NY: McGraw-Hill.
53. Alexander, R. McN. 1997 Invited editorial on 'interaction of leg stiffness and surface stiffness during human hopping'. J. Appl. Physiol. 82, 13-14.
54. Blickhan, R. 1989 The spring-mass model for running and hopping. J. Biomech. 22, 1217-1227. (doi:10.1016/0021-9290(89)90224-8)
55. Ghigliazza, R. M., Altendorfer, R., Holmes, P. & Koditschek, D. 2003 A simply stabilized running model. SIAM J. Appl. Dyn. Syst. 2, 187-218. (doi:10.1137/S1111111102408311)
56. Chatterjee, A. & Garcia, M. 2000 Small slope implies low speed for Mcgeer's passive walking machines. Dyn. Stab. Syst. 15, 139-157.
57. Garcia, M., Chatterjee, A. & Ruina, A. 2000 Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dyn. Stab. Syst. 15, 75-99.
58. Donelan, J. M., Kram, R. & Kuo, A. D. 2002 Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking. J. Exp. Biol. 205, 3717-3727.
59. Collins, S. H., Ruina, A., Tedrake, R. & Wisse, M. 2005 Efficient bipedal robots based on passive dynamic walkers. Science 307, 1082-1085. (doi:10.1126/science.1107799)
60. Gomes, M. & Ruina, A. In preparation. A walking model with no energy cost.
61. McGeer, T. 1989 Powered flight, child's play, silly wheels and walking machines. In Proc. IEEE Int. Conf. Robotics and Automation, vol. 3, pp. 1592-1597. (doi:10.1109/ROBOT.1989.100205)
62. Saunders, J. B., Inman, V. T. & Eberhart, H. D. 1953 The major determinants in normal and pathological gait. J. Bone Joint Surg. 35, 543-558.
63. Athans, M. & Falb, P. L. 1969 Optimal control. New York, NY: McGraw Hill.
64. Usherwood, J. R. 2005 Why not walk faster? Biol. Lett. 1, 338-341. (doi:10.1098/rsbl.2005.0312)
65. Ortega, J. D. & Farley, C. T. 2005 Minimizing center of mass vertical movement increases metabolic cost in walking. J. Appl. Physiol. 99, 2099-2107. (doi:10.1152/japplphysiol.00103.2005)
66. Gordon, K. E., Ferris, D. P. & Kuo, A. D. 2009 Metabolic and mechanical energy costs of reducing vertical center of mass movement during gait. Arch. Phys. Med. Rehabil. 90, 136-144. (doi:10.1016/j.apmr.2008.07.014)
67. Alexander, R. McN. 1991 Energy-saving mechanisms in walking and running. J. Exp. Biol. 160, 55-69.
68. Roberts, T. J., Marsh, R. L., Weyand, P. G. & Taylor, C. R. 1997 Muscular force in running turkeys: the economy of minimizing work. Science 275, 1113-1115. (doi:10.1126/science.275.5303.1113)
69. Fukunaga, T., Kubo, K., Kawakami, Y., Fukashiro, S., Kanehisa, H. & Maganaris, C. N. 2001 In vivo behavior of human muscle tendon during walking. Proc. R. Soc. Lond. B 268, 229-233. (doi:10.1098/rspb.2000.1361)
70. Cavagna, G. A., Heglund, N. C. & Taylor, C. R. 1977 Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure. Am. J. Physiol. 233, 243-261.
71. Alexander, R. McN. 1984 Elastic energy stores in running vertebrates. Am. Zool. 24, 85-94.
72. Biewener, A. A. 1998 Muscle-tendon stresses and elastic energy storage during locomotion in the horse. Comp. Biochem. Physiol. B 120, 73-87. (doi:10.1016/S0305-0491(98)00024-8)
73. Beletskii, V. V., Berbyuk, V. E. & Samsonov, V. A. 1982 Parametric optimization of motions of a bipedal walking robot. Izv. An SSSR Mekhanika Tverdogo (Mech. Solids) 17, 28-40.
74. Kram, R. & Taylor, C. R. 1990 Energetics of running: a new perspective. Nature 346, 265-267. (doi:10.1038/346265a0)
75. Pontzer, H. 2005 A new model predicting locomotor cost from limb length via force production. J. Exp. Biol. 208, 1513-1524. (doi:10.1242/jeb.01549)
76. Marsh, R. L., Ellerby, D. J., Carr, J. A., Henry, H. T. & Buchanan, C. I. 2004 Partitioning the energetics of walking and running: swinging the legs is expensive. Science 303, 80-83. (doi:10.1126/science.1090704)