Research On Chaos In Passive Dynamic Walking Using Nonlinear Time-series Analysis

During the past decades, bipedal walking has been very popular in robotics research as it highly resembles human locomotion, and, thus, we can get a better understanding about human walking. The enhancement of prosthetic limbs also gave new impetus to bipedal robots. In recent years, the discovery of chaos in robotics has generated much enthusiasm amongst researchers. Studying and analyzing the walking gait time-series with dynamical systems theory can reveal the dynamic walking mechanism. The researchers have carried out numerous studies on nonlinear dynamic systems. Deterministic chaos has usually been regarded as an alternative to random phenomena for explaining erratic dynamics. Despite the fact that many efforts to find evidence for chaotic dynamics in robotics, useful applications of chaos in robotics have seldom been examined. Consequently, chaos has a little influence on most of the robotics community. We attempted to fill this gap by taking up chaos as the main subject of this work. This thesis is focused on the application of the nonlinear time-series analysis to human walking and passive dynamic walking(PDW) and reveals the association between chaos and bipedal walking.The nonlinear time-series analysis has been applied to human gait and passive dynamic walking(PDW) with particular emphasis on developing understanding about human walking so as to address the needs of patients with Parkinson’s disease(PD) and persons with disabilities. In this thesis, the results of chaos theory(in robotics) have been elaborated and the chaos phenomenon in passive dynamic walking has been discussed.Since dynamic walking is identical to human walking and it consumes less energy than static walking, so we focused only on PDW. The human and robot walking is a nonlinear dynamical phenomenon since it exhibits chaotic quantifiers like correlation dimensions and Lyapunov exponents. We found that the walking stability is different when the value of correlation dimensions and Lyapunov exponents are different, and experimental results based on a compass-gait robot prototype proved this idea.Chapter one introduced PDW, chaos theory, nonlinear time-series analysis and presented a detailed literature review of chaotic dynamics in robotics. Major applications of chaos theory in robotics have been reviewed. This chapter shows that deterministic chaos is a tremendous idea in science and an omnipresent phenomenon in various robotic domains. Chaos theory has deepened the understanding of humans to the physical world and the nonlinear time-series analysis method is an efficient method to reveal the complicated mechanism in biped walking. Chapter one also presents a panoramic view of efforts for uncovering chaotic dynamics in passive bipedal robots. The last two decades of the chaos research in bipedal walking has resulted in a major shift in the way PDW phenomena is perceived now. This chapter describes the present status and future of nonlinear dynamics study in passive walking. By analyzing the chaos phenomenon in different PDW robots,we can extend the understanding of the chaos detection and controlling methods. Though the present research in chaos and bifurcation has furthered the understanding about passive biped robot but still, a lot of questions needs to be answered. The association between chaos theory and the real world is the time-series analysis in terms of nonlinear dynamics.Chapter two of this thesis also presented the general framework of nonlinear timeseries analysis, which was used for discovering nonlinearities in human and passive biped robots. Nonlinear time-series analysis has been widely used in problems, which either lack a mathematical model or need access to multiple variables. Based on the reconstructed states, the time-series can then be examined from the standpoint of nonlinear dynamics. Finally, the fundamental dynamics can be modeled, predicted and described in terms of chaotic invariants such as Lyapunov exponents and fractal dimensions.Chapter three focused on the application of chaotic time-series analysis on different normal and pathological human locomotion gaits. Low embedding dimension estimates for the walking data meant low dimensional chaos. All the normal and pathological walking gaits exhibited chaotic dynamics as they showed positive Lyapunov exponents and fractal dimensions. The chaotic measures for patients with PD indicated that the gait dynamics are chaotic but less correlated and extremely sensitive than the gaits for young and healthy subjects. Such quantitative analysis may aid physicians in diagnostic and therapeutic strategies of movement disorders like PD. The understanding of various factors that administer a healthy walking gait may help in gait rehabilitation procedures for the injured and ill people.In chapters, four and five we performed nonlinear time-series analysis of the passive compass gait-biped model. We investigated the dynamics of a simple PDW, 2D model that loosely look like human legs using time-series analysis based on nonlinear dynamics. The walking model was developed using Lagrange equation and the time-series was generated by its numerical simulation in MATLAB. We analyzed the walking of passive compass-gait biped(non-linear model) by means of chaotic quantifiers(Lyapunov exponents and correlation dimensions). After comparing the simulation results with the nonlinear time-series analysis results, we found that the walking stability is getting higher with the diminution of the value of correlation dimensions. As a result, we can optimize the robot parameters based on the changing value of the relative exponents.Finally, to validate the simulation results, we performed experiments using passive prototype biped. After measuring the mechanical parameters and slope angle of the walking surface, a series of experiments was performed. We recorded the number of successful walking attempts and drew the comparison to find out the dynamical parameters for which the walking performance of the biped is superior. The biped exhibited steady gaits only when the applied initial conditions were within the basin of attraction of this dynamical system. The experiments were done by changing different slope angles and centers of mass. It was found that with the increase in the slope angle, the walking step length becomes smaller and the gait becomes aperiodic. With the increment in position of the center of mass(COM) of the robot leg, the values of correlation dimension were getting smaller, which implied that the walking stability was getting higher. In other words,lesser values of correlation dimension imply a better walking performance, which proved the conclusion that the values of the chaotic quantifiers like Lyapunov exponents and correlation dimensions could predict the walking stability for the passive walking robot.