ANALYSIS OF THE MAXIMUM FINITE TIME LYAPUNOV EXPONENT IN .

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ANALYSIS OF THE MAXIMUM FINITE TIMELYAPUNOV EXPONENT IN TIME DOMAIN USINGDATA FROM TORSO STABILITY TESTSA thesis presented to the faculty of the Graduate School ofWestern Carolina University in partial fulfillment of therequirements for the degree of Master of Science in TechnologyByChaoke DongAdvisor: Dr. Martin L. TanakaDepartment of Engineering & TechnologyCommittee Members:Dr. Michael S. June, Department of Engineering & TechnologyDr. Yanjun Yan, Department of Engineering & TechnologyNovember 2015 2015 by Chaoke Dong

To my families, for their endless supportii

ACKNOWLEDGEMENTSI would like to give special thanks to my thesis director, Dr. Martin L. Tanaka, whose guidanceand support made all my research possible. I am really grateful that he introduced the fantastictopic – torso stability to me, so that I can have this valuable opportunity to work on it. I havereally good experience to work with Dr. Tanaka. His passion for research has always been anencouragement for me, and he is always willing to help me. I learned a lot from our discussions.My thanks also go to my committee members, Dr. Michael S. June and Dr. Yanjun Yan.Although they are very busy, they never rejected me whenever I needed to talk to them about myresearch. Especially Dr. Yanjun Yan, she gave me some suggestions on how to organize Matlabcodes and reduce the runtime of codes.I would like to extend my thanks to all the other professors who taught me during my graduateyears, Dr. Sudhir Kaul, Dr. Aaron Ball, Dr. Guanghsu Chang, Prof. Michael Clare, and Dr.Robert Adams. I really enjoyed their class and learned a lot from their classes. I would also liketo thank all my classmates in the MST program for the beneficial discussions we had together.In addition, I would like to express my sincere gratitude to the Western Carolina UniversityGraduate School for awarding research Grants to Dr. Tanaka. Funds from these grants were usedto purchase the materials used to improve the Basin of Stability chair in the research. I wouldalso like to thank Mr. Monty Graham for providing assistance and help purchasing the materialsneeded to complete this research.Finally, I would like to thank my husband, and my other family members for their endlesssupport.iii

TABLE OF CONTENTSList of Tables . viList of Figures . viiAbstract . ixCHAPTER 1: Introduction . 1CHAPTER 2: Literature Review . 42.1 Low Back Pain (LBP) . 42.2 Core Stability . 42.3 Current Method . 62.3.1 Torso Stability Experiments and Experimental Devices. 62.3.2 Mathematical Model and Simulation of Torso Stability . 102.3.3 Techniques Used to Study The Dynamical System . 132.4 Basin of Stability (BoS) . 16CHAPTER 3: Methods . 173.1 Research Plan . 173.2 Establishing The Testing Environment . 173.3 Conducting The Test . 283.3.1 Objective of Torso Stability Experiment . 283.3.2 Test Subjects . 283.3.3 Preliminary Test . 293.3.4 Full Scale Test. 343.4 Data Collection . 363.5 Data Analysis . 37CHAPTER 4: Results and Disscussion . 394.1 Analysis and Results of Preliminary Data . 394. 2 Analysis and Results of Full Scale Study . 444.2.1 Different Stages of Stability Test . 454.2.2 Effect of Gender on The Lyapunov Exponent . 524.2.3 Effect of Different Filters on The Lyapunov Exponent . 554.2.4 Effect of Different Evolution Times on The Lyapunov Exponent. 57CHAPTER 5: Conclusion and Future Work . 60Bibliography . 62Appendices. 67Appendix A: Main Code for How to Calculate The Lyapunov Exponent in Time Domain . 67iv

Appendix B: Main Code for How to Align The Time Plots with The Lypunov Exponent Plots atAssigned Time Zero . 70Appendix C: Main Code for How to Average the Lypunov Exponent. 73v

LIST OF TABLES3.14.1The basic information of test subjects . 29Lists of cutoff frequencies for different filters . 56vi

LIST OF 4.24.34.44.54.64.74.84.94.10Basins of stability found from mathematical models of seated torso stability. (a) The finitetime Lyapunov exponent field shows Lagrangian coherent structures (LCS); (b) The LCSviewed from the top . 3The anatomy of core . 5A subject positioned on an unstable seat in a test apparatus . 7Wobble chair (above) and spring layout (below) . 8The integrated configuration of the BoS chair . 9Main parts of the BoS chair . 10(a) Wobble chair with the movement of lumbar spine and (b) Simplified mathematicalmodel. 11(a) Time plot for simulated experimental trials. (b) The trajectories in state space . 12How two initially close trajectories diverge or separate . 13Image of the wood frames arranged around the chair. 18Locking up three safety frames . 18Using clamp to open and close the safety frame easily . 19(a) Adjust tool; (b) Step stool. 20Example of Xsens software showing data readouts (left) and a 3D representation of thesensor heading (right). . 21Gyroscopic sensor is fixed behind the seat . 21(a) The beam frame (b) The support rope hanging on the beam and (c) Support feet . 23(a) Using eyelets to hold the foam; (b) Using small wood piece to hold the foam . 24Grip tape. 25(a) Foam tape for sharp edge; (b) Foam tape for sharp tip . 25Front panel of custom LabView program . 26Block diagram of custom LabView program. 27(a) The subject held the rope to keep balance. (b) The subject kept balance with armcrossed. (c)The subject fell . 30Basin of stability test protocol . 32Basin of stability test protocol (continue) . 33Testing sheet . 35Raw data collected from LabView . 36Flow chart depicting how the FTLE is calculated and plotted . 38Time plots for 5 trials . 40Plots in state space . 40Plot of angle vs. time and Lyapunov exponent at different evolution time . 41Time plots and FTLE plots in the same period . 42Average FTLE vs. time with evolution time 50 milliseconds . 44Time plot for 55 trials and lining up at t 0 . 45Centered time plot for 55 trials and lining up at t 0 . 46State space plot . 47Different stages of stability test . 49FTLE field with reverse time flow. 50vii

4.114.124.134.144.154.164.17State-space-average Lyapunov exponent from mathematical model . 51Average FTLE at evolution time 50 milliseconds for male subject . 53Average FTLE at evolution time 50 milliseconds for female subject . 53Average FTLE vs time at evolution time 50 ms for different gender subject . 54Average Lyapunov exponent plots with different filters (male, evolution time 50 ms). 57Average FTLE plots with different evolution times . 58Normalized average FTLE vs time with different evolution time . 59viii

ABSTRACTANALYSIS OF THE MAXIMUM FINITE TIME LYAPUNOVEXPONENT IN TIME DOMAIN USING DATA FROM TORSOSTABILITY TESTChaoke Dong, M.S.T.Western Carolina University (November 2015)Director: Dr. Martin L. TanakaLow back pain (LBP) is a common health problem that affects most people during their life. Thegoal of this research is to track the dynamics of seated stability through the falling region anddetermine how the maximum finite time Lyapunov exponent (FTLE) changes over time. TheFTLE describes how quickly two initially close points in state space diverge. Human torsostability tests were conducted using an unstable sitting apparatus capable of attaining largedeflection angles. Angle data was collected using a gyroscopic sensor and Matlab was used tocalculate the FTLE in the time domain. The analysis results for the Lyapunov exponent in thetime domain were consistent with the results found in state space. Deterministic behavior of thedynamical system was also detected. A suite of parameters were investigated in the data analysis.The Lyapunov exponent was found to be sensitive to changes in evolution time but not sensitiveto the cutoff frequency of the low pass filter when it was above 3.2 Hz. A key point of thisresearch was to understand how the Lyapunov exponent changed with time as it approached acritical event. Tracking the Lyapunov exponent in the time domain may be a useful indicator topredict a future event. Moreover, this approach may be generalizable to other dynamic systemsthat have critical transitions. This research helps to better understand torso stability and buildonto the foundation of knowledge to diagnose and prevent LBP.ix

CHAPTER 1: INTRODUCTIONLow back pain (LBP) is a very common problem in modern health care. About 40% of peopleworldwide have reported LBP at some point of their lives [1]. This number is ever greater indeveloped countries with four out of five people experiencing LBP [2]. LBP affects people of allages, from adolescent to the elderly. The prevalence of LBP on adolescents is lower than adults,but has been increasing in recent years [3]. It is difficult to estimate the incidence of LBPbecause LBP may recur over time after the LBP patient has alleviated pain using medication,physical therapy, or surgery. LBP is one of top 10 reasons that people consult a physician [4].Furthermore, LBP is also a major cause of activity limitations and work absence, which leads toa high economic burden on individuals, families, and government [3, 5]. In the United States, ithas been estimated that about 100 to 200 billion dollars is lost each year because of LBP [6, 7, 8].Often the cause of LBP is obscure and difficult to distinguish, since many differentfactors may lead to the same or similar symptoms. Torso instability has been associated withLBP and is the topic of interest for researchers. Although there seems to be a link between LBPand torso instability, the relationship has not been clearly understood [9]. Therefore, a series ofstudies have been performed to better understand this relationship [10, 11, 28, 30]. Usingvarious unstable sitting apparatus, the torso stability was quantified. Test subjects maintainedbalance on the chair while their movement was recorded.Movement was analyzed usingkinematic variability parameters and the maximum finite time Lyapunov exponent.1

From the studies above, it seems that the maximum finite time Lyapunov exponent is acommon and useful tool to quantify torso stability from time series data. The Lyapunov exponentmeasures the divergence rate between two points which are initially close in the state space.Generally, previous researchers have calculated the maximum Lyapunov exponent as a singlescalar value [10, 11, 31, 39, 40] with the primary goal of identifying the presence ofdeterministic chaos. This was accomplished by calculating the average divergence rate over theentire time series. However, this approach does not show how the dynamic system changes withtime. It assumes a constant divergence rate through the entire time series that may be simplyrepresented by the average value.In addition, mathematical models have been developed [12]. Simulations were carriedout to generate time series data, from which the maximum finite time Lyapunov exponents werecalculated. The magnitude of the finite time Lyapunov exponent was found to change withrespect to its location in state space [13 - 16]. When plotted, the finite time Lyapunov exponent(FTLE) field shows structural features that aligned with the dynamical behavior of the system(Figure 1.1). Although the state space representation of a system shows more information aboutthe system dynamics than a single scalar value, it still does not take into account the componentof time.2

(a)(b)Figure 1.1: Basins of stability found from mathematical models of seated torso stability. (a) Thefinite time Lyapunov exponent field shows Lagrangian coherent structures (LCS) [13]; (b) TheLCS viewed from the top [13, 45]The purpose of this research is to track the dynamics of seated stability through thefalling region and to determine how the maximum finite time Lyapunov exponent changes overtime. Specifically we are interested in understanding how the FTLE changes prior to an unstableevent. In our case, this event is the loss of stability that results in falling during a torso stabilitytest. If we are able to identify features of the FTLE that occur prior to a loss of stability, thismay be useful in predicting and possibly preventing spinal instability injuries and the resultingLBP.3

CHAPTER 2: LITERATURE REVIEW2.1 Low Back Pain (LBP)The low back includes the lumbar region of spine, surrounding muscles and connective tissue.Its main functions are to provide structural support, enable torso movement, and protect thespinal cord [18]. LBP refers to the disorder of muscles, tissue and bones in low back whichresults in pain [19]. Symptoms usually are back stiffness, numbness of the lower extremities anddifficulty moving or standing straight. It is commonly diagnosed by X-ray test, CT scanning orMRI scanning. One common cause of LBP is back muscle strain or ligament strain which isdeveloped by some movements, like lifting a heavy object, twisting, or forward-bending [20]. Asearly as 1995, Adam and Dolan proposed that mechanical fatigue damage may be the underlyingreason for LBP [21]. The current treatment for LBP is medication, physical therapy, and surgery,which may alleviate the pain of patient, but these treatments do not always eradicate the LBPbased on the patients' experiences. Often the cause of LBP is

time. It assumes a constant divergence rate through the entire time series that may be simply represented by the average value. In addition, mathematical models have been developed [12]. Simulations were carried out to generate time series data, from which the maximum finite time Lyapunov exponents were calculated.

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