Doctoral defense
Recurrent Dynamics of Nonsmooth Systems with Application to Human Gait
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| Defendant |
Main Advisor |
Extra Advisor |
Date |
| Petri Piiroinen |
Harry Dankowicz |
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2002-11-29 |
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| Opponent |
| Gabor Stepan, Department of Applied Mechanics, Budapest University of Technology and Economics, Hungary
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| Evaluation committee |
Michael Benedicks, Institutionen för Matematik, KTH
Viktor Berbyuk, Dept of machine and Vehicle Systems, Chalmers
Tom Wadden, Objecta Systems AB, Stockholm, Sweden
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AbstractThe focus of this thesis is on the development of methods of analysis and control of recurrent motions in nonsmooth dynamical systems, with particular emphasis on low-dimensional models of two-legged walking. In particular, passive walkers---bipedal rigid-body mechanisms that achieve sustained gait down incined planes with gravity as the only source of energy---are analyzed to demonstrate the existence of a variety of gait-like recurrent motions, such as periodic, quasiperiodic, and chaotic gait; and to establish the sensitivity of such motions to changes in system parameters and to small perturbations. As an example, the present study contains a careful analysis of the transition between gait motions prevented from exhibiting lateral dynamics and those of fully three-dimensional walkers. It is shown that instability mechanisms appear for the latter mechanisms that cannot be anticipated by a study of the constrained models, suggesting only limited applicability of a two-dimensional analysis to understanding actual human gait. To suggest ways to apply the present study to the clinical context, an idea on how to expand the three-dimensional passive walkers to include muscles is also discussed. A control algorithm is presented that relies on the presence of discontinuities for controlling the local stability of periodic and other recurrent motions. The method allows one to predict the effects of the control strategy entirely from information about the uncontrolled system. This method is applied to the passive walkers to stabilize highly unstable periodic gait and to switch between different walking patterns. Finally, a method based on the discontinuity-mapping approach is derived to predict the characteristic changes in system behavior that occur following a grazing intersection of a quasiperiodic attractor with a state-space discontinuity. The method is applied to a simple model example representing a two-frequency, quasiperiodic oscillation of a forced van-der-Pol oscillator with a two-dimensional impact surface in a three-dimensional state space.
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