Application of Product Design Concepts and Hybrid System Dynamics to Demonstrate Zeno Behavior and Zeno Periodic Orbits in a Physical Double Pendulum Setup

This thesis aims to explain how the concepts of functional modeling are implemented in the development and validation of real-world hybrid dynamic systems. I also discuss how control theory is integrated with the design process in order to understand the significance of periodic orbits on a simple dynamic system.
Two hybrid system applications with different levels of complexity will be considered in this thesis ? an anthropomorphic Bipedal walking robot and a Double Pendulum with a mechanical stop. The primary objectives of this project are to demonstrate the phenomena of Zeno and zeno periodic orbits in hybrid dynamic systems involving impacts. Initially, I describe the salient features of the product design procedure and then explain the significance of functional modeling as a part of this process. We then discuss hybrid dynamic systems and the occurrence of Zeno behavior in their mathematical form. Also, the necessary conditions for existence of Zeno and zeno equilibrium points are provided. Then the theory of completed Lagrangian hybrid systems is explained in detail.
We then examine the two hybrid dynamic systems being considered for this project. Prior research undertaken on bipedal walking is explored to understand their design and achievement of stable walking gaits with appropriate actuation mechanisms. Based on this insight, a suitable design procedure is employed to develop the bipedal robot model. The desired actuation mechanisms for all the configurations considered for this model as well as the challenges faced in employing optimal actuation will be discussed. However, due to the high level of complexity of the bipedal robot model, a simpler hybrid dynamic system is considered to simplify fabrication and control of the model. This is the motivation behind designing and building the Double Pendulum model with a mechanical stop in an attempt to observe zeno behavior in this system.
We begin by formally demonstrating that the ?constrained? double pendulum model displays Zeno behavior and complete this Zeno hybrid system to allow for solutions to be carried past the Zeno point. The end result is periods of unconstrained and constrained motions in the pendulum, with transitions to the constrained motion occurring at the Zeno point. We then consider the development of a real physical pendulum with a mechanical stop and introduce non-plastic impacts. Later, we verify through experimentation that Zeno behavior provides an accurate description of the behavior of the physical system. This provides evidence to substantiate the claim that Zeno behavior, while it does not technically occur in reality, provides an accurate method for predicting the behavior of systems undergoing impacts and that the theory developed to understand Zeno behavior can be applied to better understand these systems.