Modeling Walking with an Inverted Pendulum Not Constrained to the Sagittal Plane. Numerical Simulations and Asymptotic Expansions

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DOI: 10.4236/am.2017.81006    238 Downloads   323 Views  


Inverted pendulum models are commonly used to study the bio-mechanics of biped walkers. In its simplest form, the inverted pendulum consists of a point mass attached to two straight mass-less legs. Most works constrain the motion of the mass to the sagittal plane, i.e. the plane perpendicular to the ground that contains the direction toward the biped is walking. In this article, we remove this constrain to study the oscillations, the mass experiences in the direction perpendicular to the sagittal plane as the biped walks. While small, these lateral oscillations are unavoidable and of importance in the understanding of balance and stability of walkers, as well as walkers induced oscillations in pedestrian bridges.

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Goldsztein, G. (2017) Modeling Walking with an Inverted Pendulum Not Constrained to the Sagittal Plane. Numerical Simulations and Asymptotic Expansions. Applied Mathematics, 8, 57-76. doi: 10.4236/am.2017.81006.


[1] Abdulrehem, M.M. and Ott, E. (2009) Low Dimensional Description of Pedestrian-Induced Oscillation of the Millennium Bridge. Chaos: An Interdisciplinary Journal of Nonlinear Science, 19, Article ID: 013129.
[2] Bachmann, H. (1992) Case Studies of Structures with Man-Induced Vibrations. Journal of Structural Engineering, 118, 631-647.
[3] Bocian, M., Macdonald, J.H.G. and Burn, J.F. (2012) Biomechanically Inspired Modelling of Pedestrian-Induced Forces on Laterally Oscillating Structures. Journal of Sound and Vibration, 331, 3914-3929.
[4] Bodgi, J., Erlicher, S. and Argoul, P. (2007) Lateral Vibration of Footbridges under Crowd-Loading: Continuous Crowd Modeling Approach. Key Engineering Materials, 347, 685-690.
[5] Bruno, L., Venuti, F. and Nascé, V. (2012) Pedestrian-Induced Torsional Vibrations of Suspended Footbridges: Proposal and Evaluation of Vibration Countermeasures. Engineering Structures, 36, 228-238.
[6] Dallard, P., Fitzpatrick, T., Flint, A., Low, A., Smith, R.R., Willford, M. and Roche, M. (2001) London Millennium Bridge: Pedestrian-Induced Lateral Vibration. Journal of Bridge Engineering, 6, 412-417.
[7] Eckhardt, B., Ott, E., Strogatz, S.H., Abrams, D.M. and McRobie, A. (2007) Modeling Walker Synchronization on the Millennium Bridge. Physical Review-Series E, 75, Article ID: 021110.
[8] Ingólfsson, E.T., Georgakis, C.T., Ricciardelli, F. and Jonsson, J. (2011) Experimental Identification of Pedestrian-Induced Lateral Forces on Footbridges. Journal of Sound and Vibration, 330, 1265-1284.
[9] Ingólfsson, E.T., Georgakis, C.T. and Jonsson, J. (2012) Pedestrian-Induced Lateral Vibrations of Footbridges: A Literature Review. Engineering Structures, 45, 21-52.
[10] Macdonald, J.H.G. (2008) Lateral Excitation of Bridges by Balancing Pedestrians. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465, 1055-1073.
[11] Venuti, F. and Bruno, L. (2009) Crowd-Structure Interaction in Lively Footbridges under Synchronous Lateral Excitation: A Literature Review. Physics of Life Reviews, 6, 176-206.
[12] Adamczyk, P.G., Collins, S.H. and Kuo, A.D. (2006) The Advantages of a Rolling Foot in Human Walking. Journal of Experimental Biology, 209, 3953-3963.
[13] Alexander, R.M. (1991) Energy-Saving Mechanisms in Walking and Running. Journal of Experimental Biology, 160, 55-69.
[14] Alexander, R.M. (1992) A Model of Bipedal Locomotion on Compliant Legs. Philosophical Transactions of the Royal Society B: Biological Sciences, 338, 189-198.
[15] Coleman, M.J., Garcia, M., Mombaur, K. and Ruina, A. (2001) Prediction of Stable Walking for a Toy That Cannot Stand. Physical Review E, 64, Article ID: 022901.
[16] Coleman, M.J. and Ruina, A. (1998) An Uncontrolled Walking Toy That Cannot Stand Still. Physical Review Letters, 80, 3658.
[17] Donelan, J.M., Kram, R. and Kuo, A.D. (2002) Simultaneous Positive and Negative External Mechanical Work in Human Walking. Journal of Biomechanics, 35, 117-124.
[18] Garcia, M., Chatterjee, A. and Ruina, A. (2000) Efficiency, Speed, and Scaling of Two-Dimensional Passive-Dynamic Walking. Dynamics and Stability of Systems, 15, 75-99.
[19] Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M. (1998) The Simplest Walking Model: Stability, Complexity, and Scaling. Journal of Biomechanical Engineering, 120, 281-288.
[20] Geyer, H., Seyfarth, A. and Blickhan, R. (2006) Compliant Leg Behaviour Explains Basic Dynamics of Walking and Running. Proceedings of the Royal Society of London B: Biological Sciences, 273, 2861-2867.
[21] Goswami, A., Espiau, B. and Keramane, A. (1997) Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws. Autonomous Robots, 4, 273-286.
[22] Kuo, A.D. (2001) A Simple Model of Bipedal Walking Predicts the Preferred Speed-Step Length Relationship. Journal of Biomechanical Engineering, 123, 264-269.
[23] Kuo, A.D. (2002) Energetics of Actively Powered Locomotion Using the Simplest Walking Model. Journal of Biomechanical Engineering, 124, 113-120.
[24] Kuo, A.D. (2007) The Six Determinants of Gait and the Inverted Pendulum Analogy: A Dynamic Walking Perspective. Human Movement Science, 26, 617-656.
[25] McGeer, T. (1993) Dynamics and Control of Bipedal Locomotion. Journal of Theoretical Biology, 163, 277-314.
[26] Xiang, Y., Arora, J.S. and Abdel-Malek, K. (2010) Physics-Based Modeling and Simulation of Human Walking: A Review of Optimization-Based and Other Approaches. Structural and Multidisciplinary Optimization, 42, 1-23.
[27] McGeer, T. (1990) Passive Dynamic Walking. The International Journal of Robotics Research, 9, 62-82.
[28] Alexander, R.M. (1995) Simple Models of Human Movement. Applied Mechanics Reviews, 48, 461-470.
[29] Katoh, R. and Mori, M. (1984) Control Method of Biped Locomotion Giving Asymptotic Stability of Trajectory. Automatica, 20, 405-414.
[30] Roos, P.E. and Dingwell, J.B. (2011) Influence of Simulated Neuromuscular Noise on the Dynamic Stability and Fall Risk of a 3d Dynamic Walking Model. Journal of Biomechanics, 44, 1514-1520.
[31] Pai, Y.-C. and Patton, J. (1997) Center of Mass Velocity-Position Predictions for Balance Control. Journal of Biomechanics, 30, 347-354.
[32] Dickinson, M.H., Farley, C.T., Full, R.J., Koehl, M.A.R., Kram, R. and Lehman, S. (2000) How Animals Move: An Integrative View. Science, 288, 100-106.
[33] Holmes, P., Full, R.J., Koditschek, D. and Guckenheimer, J. (2006) The Dynamics of Legged Locomotion: Models, Analyses, and Challenges. Siam Review, 48, 207-304.
[34] Ghigliazza, R.M., Altendorfer, R., Holmes, P. and Koditschek, D. (2005) A Simply Stabilized Running Model. SIAM Review, 47, 519-549.
[35] Peuker, F., Maufroy, C. and Seyfarth, A. (2012) Leg-Adjustment Strategies for Stable Running in Three Dimensions. Bioinspiration & Biomimetics, 7, Article ID: 036002.
[36] Seipel, J.E. and Holmes, P. (2005) Running in Three Dimensions: Analysis of a Point-Mass Sprung-Leg Model. The International Journal of Robotics Research, 24, 657-674.
[37] Salazar, H.R.M. and Carbajal, J.P. (2011) Exploiting the Passive Dynamics of a Compliant Leg to Develop Gait Transitions. Physical Review E, 83, Article ID: 066707.
[38] Srinivasan, M. and Ruina, A. (2007) Idealized Walking and Running Gaits Minimize Work. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 463, 2429-2446.
[39] Srinivasan, M. and Ruina, A. (2006) Computer Optimization of a Minimal Biped Model Discovers Walking and Running. Nature, 439, 72-75.
[40] Grizzle, J.W., Chevallereau, C., Sinnet, R.W. and Ames, A.D. (2014) Models, Feedback Control, and Open Problems of 3d Bipedal Robotic Walking. Automatica, 50, 1955-1988.
[41] Hof, A.L., Vermerris, S.M. and Gjaltema, W.A. (2010) Balance Responses to Lateral Perturbations in Human Treadmill Walking. The Journal of Experimental Biology, 213, 2655-2664.
[42] Bauby, C.E. and Kuo, A.D. (2000) Active Control of Lateral Balance in Human Walking. Journal of Biomechanics, 33, 1433-1440.
[43] Donelan, J.M., Shipman, D.W., Kram, R. and Kuo, A.D. (2004) Mechanical and Metabolic Requirements for Active Lateral Stabilization in Human Walking. Journal of Biomechanics, 37, 827-835.
[44] Kuo, A.D. (1999) Stabilization of Lateral Motion in Passive Dynamic Walking. The International Journal of Robotics Research, 18, 917-930.
[45] Lyon, I.N. and Day, B.L. (1997) Control of Frontal Plane Body Motion in Human Stepping. Experimental Brain Research, 115, 345-356.
[46] MacKinnon, C.D. and Winter, D.A. (1993) Control of Whole Body Balance in the Frontal Plane during Human Walking. Journal of Biomechanics, 26, 633-644.
[47] Donelan, J.M., Kram, R., et al. (2001) Mechanical and Metabolic Determinants of the Preferred Step Width in Human Walking. Proceedings of the Royal Society of London B: Biological Sciences, 268, 1985-1992.

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