Date of Award

Spring 1-1-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Rodger Kram

Second Advisor

James D. Meiss

Third Advisor

David M. Bortz

Abstract

Mathematical modeling and analysis have been an integral part of legged locomotion research for many years. While models from the very simple inverted pendulum model of walking and the spring-mass model of running to multi-segmental models with numerous muscles across each joint have been used to explore the process of legged locomotion, they are not always sufficient to explain how human locomotion adapts to a changing environment. Using techniques from dynamical and control systems, I : 1) identify the time scales involved in metabolic minimization in running, 2) explore how stability differs between walking and running, 3) develop an algorithm for optimal control in discrete physical systems, and 4) examine the changes in leg mechanics involved in uphill and downhill running. For the first project, I use ideas from control systems to identify the processes involved in metabolic minimization in running. For the second project, I use ideas of orbital and local stability, measured using Floquet multipliers and finite time Lyapunov exponents, to try to quantify dynamic stability in walking and running at preferred and transition speeds. For the third project, I define a new method for the constrained optimization problem underlying Discrete Mechanics in Optimal Control. For the fourth project, I use a control systems approach to examine the changes in leg dynamics from level to uphill and downhill running. By exploring the adaptations that occur with changing environment, I hope to reveal the mechanisms, and possibly some of the strategies, that lead to stable locomotion.

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