Abstract
The inherent dynamics of bipedal, passive mechanisms are studiedto investigate the relation between motions constrained to two-dimensional (2D)planes and those free to move in a three-dimensional (3D) environment. Inparticular, we develop numerical and analytical techniques usingdynamical-systems methodology to address the persistence and stabilitychanges of periodic, gait-like motions due to the relaxation ofconfiguration constraints and the breaking of problem symmetries. Theresults indicate the limitations of a 2D analysis to predictthe dynamics in the 3D environment. For example, it is shownhow the loss of constraints may introduce characteristically non-2Dinstability mechanisms, and how small symmetry-breaking terms may result inthe termination of solution branches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pratt, J.E., ‘Virtual model control of a biped walking robot’, Master's Thesis, MIT, Department of Electrical Engineering and Computer Science, 1995.
Pratt, J.E. and Pratt, G.A., ‘Exploiting natural dynamics in the control of a 3D bipedal walking simulation’, in International Conference on Climbing and Walking Robots (CLAWAR99), Portsmouth, U.K., 1999.
Formal'sky, A.M., Ballistic Locomotion of a Biped. Design and Control of Two Biped Machines, CISM Advanced School on Modelling and Simulation of Human and Walking Robots Locomotion, Udine, 1996.
Howell, G.W. and Baillieul, J., ‘Simple controllable walking mechanism which exhibit bifurcations’, in Proceedings of the 37th IEEE Conference on Decision and Control, IEEE, New York, 1998, 3027–3032.
Berbuyk, V.E. and Boström, A.E. and Lytwyn, B.A. and Peterson, B., ‘Optimization of control laws of the bipedal locomotion systems’, Advances in Computational Multibody Dynamics 37 1999, 703–728.
McGeer, T., ‘Passive dynamic walking’, International Journal of Robotics Research 9, 1990, 62–82.
Garcia, M., Chatterjee, A. and Ruina, A., ‘Efficiency, speed, and scaling of passive dynamical bipedal walking’, Dynamics and Stability of Systems 15(2), 2000, 75–99.
Adolfsson, J., Dankowicz, H. and Nordmark, A., ‘3-D stable gait in passive bipedal mechanisms’, in Proceedings of European Mechanics Colloquium, Euromech 372, J.A.C. Ambrósio and W.O. Schiehlen (eds), 1999, 253–259.
Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., ‘Passive-dynamic models of human gait’, in Proceedings of the Conference on Biomechanics and Neural Control of Human Movement, 1998, 32–33.
Dankowicz, H.J., Adolfsson, J. and Nordmark, A.B., ‘Repetetive gait of passive bipedal mechanisms in a three-dimensional environment’, Journal of Biomechanical Engineering 123(1), 2001, 40–46.
Adolfsson, J., Dankowicz, H. and Nordmark, A., ‘3D passive walkers: Finding periodic gaits in the presence of discontinuities’, Nonlinear Dynamics 24, 2001, 205–229.
Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., ‘The simplest walking model: Stability, complexity, and scaling’, ASME Journal of Biomechanical Engineering 120, 1998, 281–288.
McGeer, T., ‘Passive walking with knees’, in Proceedings of the IEEE Conference on Robotics and Automation, Vol. 2, IEEE, New York, 1990, 1640–1645.
Piiroinen, P.T., Dankowicz, H.J. and Nordmark, A.B., ‘On a normal-formal analysis for a class of passive bipedal walkers’, International Journal of Bifurcation and Chaos 11(9), 2001, 2411–2425.
Collins, S., Wisse, M. and Ruina, A., ‘A 3-D passive-dynamic walking robot with two legs and knees’, International Journal of Robotics Research 20(7), 2001, 607–615.
Coleman, M. and Ruina, A., ‘An uncontrolled toy that can walk but cannot stand still’, Physical Review Letters 80(16), 1998, 3658–3661.
Adolfsson, J., ‘Passive control of mechanical systems — Bipedal walking and autobalancing’, Ph.D. Thesis, Royal Institute of Technology, Department of Mechanics, Sweden, 2001.
Parker, T.S. and Chua, L.O., Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.
Leine, R., ‘Bifurcations in discontinuous mechanical systems of Filippov-type’, Ph.D. Thesis, Technische Universiteit Eindhoven, The Netherlands, 2000.
Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin, 1983.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Piiroinen, P.T., Dankowicz, H.J. & Nordmark, A.B. Breaking Symmetries and Constraints: Transitions from 2D to 3D in Passive Walkers. Multibody System Dynamics 10, 147–176 (2003). https://doi.org/10.1023/A:1025540401249
Issue Date:
DOI: https://doi.org/10.1023/A:1025540401249