This paper applies a robotics-inspired approach to derive a low-dimensional forward-dynamic hybrid model of human walking in the sagittal plane. The low-dimensional model is derived as a subdynamic of a higher-dimensional anthropomorphic hybrid model. The hybrid model is composed of models for single support (SS) and double support (DS), with the transition from SS to DS modeled by a rigid impact to account for the impact at heel-contact. The transition from DS to SS occurs in a continuous manner. Existing gait data are used to specify, via parametrization, the low-dimensional model that is developed. The primary result is a one-degree-of-freedom model that is an exact subdynamic of the higher-dimensional anthropomorphic model and describes the dynamics of walking. The stability properties of the model are evaluated using the method of Poincaré. The low-dimensional model is validated using the measured human gait data. The validation demonstrates the observed stability of the measured gait.
Issue Section:
Research Papers
1.
2007, CGA, Clinical Gait Analysis Home Page, http://guardian.curtin.edu.au/cga/http://guardian.curtin.edu.au/cga/.
2.
Westervelt
, E. R.
, Grizzle
, J. W.
, and Koditschek
, D. E.
, 2003, “Hybrid Zero Dynamics of Planar Biped Walkers
,” IEEE Trans. Autom. Control
0018-9286, 48
(1
), pp. 42
–56
.3.
Westervelt
, E. R.
, Buche
, G.
, and Grizzle
, J. W.
, 2004, “Experimental Validation of a Framework for the Design of Controllers That Induce Stable Walking in Planar Bipeds
,” Int. J. Robot. Res.
, 23
(6
), pp. 559
–582
. 0278-36494.
Pandy
, M. G.
, 2001, “Computer Modeling and Simulation of Human Movement
,” Annu. Rev. Biomed. Eng.
1523-9829, 3
, pp. 245
–273
.5.
Peng
, C. Y.
, and Ono
, K.
, 2003, “Numerical Analysis of Energy-Efficient Walking Gait With Flexed Knee for Four-DOF Planar Biped Model
,” JSME Int. J., Ser. C
, 46
(4
), pp. 1346
–1355
. 1344-76536.
Pop
, C.
, Khajepour
, A.
, Huissoon
, J. P.
, and Patla
, A. E.
, 2003, “Experimental/Analytical Analysis of Human Locomotion Using Bondgraphs
,” ASME J. Biomech. Eng.
0148-0731, 125
(4
), pp. 490
–498
.7.
van der Kooij
, H.
, Jacobs
, R.
, Koopman
, B.
, and van der Helm
, F.
, 2003, “An Alternative Approach to Synthesizing Bipedal Walking
,” Biol. Cybern.
, 88
(1
), pp. 46
–59
. 0340-12008.
Zajac
, F. E.
, Neptune
, R. R.
, and Kautz
, S. A.
, 2003, “Biomechanics and Muscle Coordination of Human Walking Part II: Lessons From Dynamical Simulations, Clinical Implications, and Concluding Remarks
,” Gait and Posture
0966-6362, 17
(1
), pp. 1
–17
.9.
Cavagna
, G. A.
, Heglund
, N. C.
, and Taylor
, C. R.
, 1977, “Mechanical Work in Terrestrial Locomotion: Two Basic Mechanisms for Minimizing Energy Expenditure
,” Am. J. Physiol.
, 235
, pp. R243
–R61
. 0002-951310.
Lee
, C. R.
, and Farley
, C. T.
, 1998, “Determinants of the Center of Mass Trajectory in Human Walking and Running
,” J. Exp. Biol.
, 201
(21
), pp. 2935
–2944
. 0022-094911.
Garcia
, M.
, Chatterjee
, A.
, Ruina
, A.
, and Coleman
, M.
, 1998, “The Simplest Walking Model: Stability, Complexity, and Scaling
,” ASME J. Biomech. Eng.
0148-0731, 120
(2
), pp. 281
–288
.12.
Kuo
, A. D.
, 2002, “Energetics of Actively Powered Locomotion Using the Simplest Walking Model
,” ASME J. Biomech. Eng.
0148-0731, 124
(1
), pp. 113
–120
.13.
McGeer
, T.
, 1993, “Dynamics and Control of Bipedal Locomotion
,” J. Theor. Biol.
, 163
(3
), pp. 277
–314
. 0022-519314.
Kuo
, A. D.
, 2001, “A Simple Model of Bipedal Walking Predicts the Preferred Speed-Step Length Relationship
,” ASME J. Biomech. Eng.
0148-0731, 123
, pp. 264
–269
.15.
Kuo
, A. D.
, Donelan
, J. M.
, and Ruina
, A.
, 2005, “Energetic Consequences of Walking Like an Inverted Pendulum: Step-to-Step Transitions
,” Exerc Sport Sci. Rev.
, 33
, pp. 88
–97
. 0091-633116.
Glitsch
, U.
, and Baumann
, W.
, 1997, “The Three-Dimensional Determination of Internal Loads in the Lower Extremity
,” ASME J. Biomech. Eng.
, 30
(11-12
), pp. 1123
–1131
. 0021-929017.
Siegler
, S.
, and Liu
, W.
, 1997, “Inverse Dynamics in Human Locomotion
,” Three-Dimensional Analysis of Human Locomotion
, P.
Allard
, A.
Cappozzo
, A.
Lundberg
, and C. L.
Vaughan
, eds., Wiley
, New York
, pp. 191
–209
.18.
Anderson
, F. C.
, and Pandy
, M. G.
, 2001, “Dynamic Optimization of Human Walking
,” ASME J. Biomech. Eng.
0148-0731, 123
(5
), pp. 381
–390
.19.
Neptune
, R. R.
, Kautz
, S. A.
, and Zajac
, F. E.
, 2001, “Contributions of the Individual Ankle Plantar Flexors to Support, Forward Progression and Swing Initiation During Walking
,” J. Biomech.
0021-9290, 34
(11
), pp. 1387
–1398
.20.
Pandy
, M. G.
, and Berme
, N.
, 1988, “A Numerical Method for Simulating the Dynamics of Human Walking
,” J. Biomech.
0021-9290, 21
(12
), pp. 1043
–1051
.21.
Ogihara
, N.
, and Yamazaki
, N.
, 2001, “Generation of Human Bipedal Locomotion by a Bio-Mimetic Neuro-Musculo-Skeletal Model
,” Biol. Cybern.
, 84
(1
), pp. 1
–11
. 0340-120022.
Taga
, G.
, 1995, “A Model of the Neuro-Musculo-Skeletal System for Human Locomotion I: Emergence of Basic Gait
,” Biol. Cybern.
0340-1200, 73
(2
), pp. 97
–111
.23.
Winter
, D. A.
, 2005, Biomechanics and Motor Control of Human Movement
, Wiley
, New York
.24.
Delp
, S. L.
, and Loan
, J. P.
, 2000, “A Computational Framework for Simulating and Analyzing Human and Animal Movement
,” Comput. Sci. Eng.
1521-9615, 2
(5
), pp. 46
–55
.25.
Thelen
, D. G.
, Anderson
, F. C.
, and Delp
, S. L.
, 2003, “Generating Dynamic Simulations of Movement Using Computed Muscle Control
,” J. Biomech.
0021-9290, 36
(3
), pp. 321
–328
.26.
Dankowicz
, H.
, Adolfsson
, J.
, and Nordmark
, A. B.
, 2001, “Repetitive Gait of Passive Bipedal Mechanisms in a Three-Dimensional Environment
,” ASME J. Biomech. Eng.
0148-0731, 123
(1
), pp. 40
–46
.27.
Full
, R.
, and Koditschek
, D. E.
, 1999, “Templates and Anchors: Neuromechanical Hypotheses of Legged Locomotion on Land
,” J. Exp. Biol.
, 202
, pp. 3325
–3332
. 0022-094928.
Ghigliazza
, R. M.
, Altendorfer
, R.
, Holmes
, P.
, and Koditschek
, D. E.
, 2005, “A Simply Stabilized Running Model
,” SIAM Rev.
0036-1445, 47
(3
), pp. 519
–549
.29.
Holmes
, P.
, Full
, R. J.
, Koditschek
, D. E.
, and Guckenheimer
, J.
, 2006, “The Dynamics of Legged Locomotion: Models, Analyses, and Challenges
,” SIAM Rev.
0036-1445, 48
, pp. 207
–304
.30.
Westervelt
, E. R.
, Grizzle
, J. W.
, Chevallereau
, C.
, Choi
, J. H.
, and Morris
, B.
, 2007, Feedback Control of Dynamic Bipedal Robot Locomotion
, Taylor & Francis
, London
/CRC
, Boca Raton, FL
.31.
Hansen
, A. H.
, Childress
, D. S.
, and Knox
, E. H.
, 2004, “Roll-Over Shapes of Human Locomotor Systems: Effects of Walking Speed
,” Clin. Biomech. (Bristol, Avon)
, 19
(4
), pp. 407
–414
. 0268-003332.
Furusho
, J.
, and Sano
, A.
, 1990, “Sensor-Based Control of a Nine-Link Biped
,” Int. J. Robot. Res.
, 9
(2
), pp. 83
–98
. 0278-364933.
Goswami
, A.
, 1999, “Postural Stability of Biped Robots and the Foot-Rotation Indicator (FRI) Point
,” Int. J. Robot. Res.
0278-3649, 18
(6
), pp. 523
–533
.34.
Spong
, M. W.
, Hutchinson
, S.
, and Vidyasagar
, M.
, 2005, Robot Modeling and Control
, Wiley
, New York
.35.
Amirouche
, F. M. L.
, 1992, Computational Methods in Multibody Dynamics
, Prentice-Hall
, Englewood Cliffs, NJ
.36.
Hürmüzlü
, Y.
, and Marghitu
, D. B.
, 1994, “Rigid Body Collisions of Planar Kinematic Chains With Multiple Contact Points
,” Int. J. Robot. Res.
, 13
(1
), pp. 82
–92
. 0278-364937.
Isidori
, A.
, 1995, Nonlinear Control Systems
, 3rd ed., Springer-Verlag
, Berlin
.38.
Choi
, J. H.
, 2005, “Model-Based Control and Analysis of Anthropomorphic Walking
,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.39.
Purser
, J. L.
, Weinberger
, M.
, Cohen
, H. J.
, Pieper
, C. F.
, Morey
, M. C.
, Li
, T.
, Williams
, G. R.
, and Lapuerta
, P.
, 2005, “Walking Speed Predicts Health Status and Hospital Costs for Frail Elderly Male Veterans
,” J. Rehabil. Res. Dev.
, 42
, pp. 535
–545
. 0748-771140.
Dingwell
, J. B.
, and Kang
, H. G.
, 2007, “Differences Between Local and Orbital Dynamic Stability During Human Walking
,” ASME J. Biomech. Eng.
, 129
(4
), pp. 586
–593
. 0148-073141.
Hürmüzlü
, Y.
, and Basdogan
, C.
, 1994, “On the Measurement of Dynamic Stability of Human Locomotion
,” ASME J. Biomech. Eng.
0148-0731, 116
(1
), pp. 30
–36
.42.
Hürmüzlü
, Y.
, Basdogan
, C.
, and Stoianovici
, D.
, 1996, “Kinematics and Dynamic Stability of the Locomotion of Post-Polio Patients
,” ASME J. Biomech. Eng.
0148-0731, 118
(3
), pp. 405
–411
.43.
Srinivasan
, S.
, Westervelt
, E. R.
, and Hansen
, A. H.
, 2007, “A Low-Dimensional Sagittal Plane Forward-Dynamic Model for Asymmetric Gait and Its Application to Study the Gait of Transtibial Prosthesis Users
,” ASME J. Biomech. Eng.
, accepted.44.
Srinivasan
, S.
, 2007, “Low-Dimensional Modeling and Analysis of Human Gait With Application to the Gait of Transtibial Prosthesis Users
,” Ph.D. thesis, The Ohio State University, Columbus, OH.45.
Hansen
, A.
, and Childress
, D.
, 2004, “Effects of Shoe Heel Height on Biologic Roll-Over Characteristics During Walking
,” J. Rehabil. Res. Dev.
, 41
(4
), pp. 547
–554
. 0748-771146.
Hansen
, A. H.
, and Childress
, D. S.
, 2005, “Effects of Adding Weight to the Torso on Roll-Over Characteristics of Walking
,” J. Rehabil. Res. Dev.
, 42
(3
), pp. 381
–390
. 0748-771147.
Choi
, J. H.
, and Grizzle
, J. W.
, 2005, “Feedback Control of an Underactuated Planar Bipedal Robot With Impulsive Foot Action
,” Robotica
, 23
, pp. 567
–580
. 0263-574748.
Srinivasan
, S.
, and Westervelt
, E. R.
, 2005, “An Analytically Tractable Model for a Complete Gait Cycle
,” Proceedings of the 20th Congress of the International Society of Biomechanics
, Cleveland, OH, July 31–August 5, 2005.49.
Perry
, J.
, 1982, Atlas of Limb Prosthetics: Surgical, Prosthetic, and Rehabilitation Principles
, 2nd ed., Mosby-Year Book
, Mosby
, pp. 359
–369
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
