Citation
Locomotion Pattern Prediction Based on Whole-Body Angular and Linear Momentum Variations

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Title:
Locomotion Pattern Prediction Based on Whole-Body Angular and Linear Momentum Variations
Creator:
Jackson, Jennifer N
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (96 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Biomedical Engineering
Committee Chair:
FREGLY,BENJAMIN J
Committee Co-Chair:
BANKS,SCOTT ARTHUR
Committee Members:
CONRAD,BRYAN
HASS,CHRISTOPHER J
DE WITT,JOHN
Graduation Date:
12/13/2013

Subjects

Subjects / Keywords:
Calibration ( jstor )
Coordinate systems ( jstor )
Feet ( jstor )
Gait ( jstor )
Kinematics ( jstor )
Modeling ( jstor )
Momentum ( jstor )
Parametric models ( jstor )
Toes ( jstor )
Walking ( jstor )
Biomedical Engineering -- Dissertations, Academic -- UF
biomechanics -- contact -- dynamics -- ground -- modeling -- momentum
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Biomedical Engineering thesis, Ph.D.

Notes

Abstract:
800x600 Computational models are valuable for developing andtesting hypotheses that otherwise would be infeasible to exploreexperimentally. They can also provide a theoretical framework to explainexperimental observations. In the case of pathological gait, despite manystudies reporting that the central nervous system (CNS) regulates angularmomentum during walking, no simple control law currently exists to explain howthe CNS makes walking efficient or even possible. Fewer studies have looked athow linear momentum is regulated during human locomotion, although recentfindings indicate regulation occurs for various locomotion tasks. Acomputational model that uses basic momentum considerations to predictachievable, improved gait patterns for individuals with pathological gait couldbe a valuable tool to aid clinicians in making objective, highly effectivetreatment decisions. To create a framework for predictive gait optimization, themain objectives are to: 1) Eliminate the pelvis residual loads and improve footmarker tracking by enhancing the residual elimination algorithm through markerweight, tracked acceleration curve, feedback gain, and select model joint andinertial parameter adjustments; 2) Develop a foot-ground contact model thatmatches all three force components, center of pressure location, and freemoment for both feet using physics to model the foot-ground interactions; 3) Demonstratethat whole-body momentum variations for gait tasks cluster differently from oneanother and that these clusters can be viewed as “momentum signatures” fordifferent gait patterns; and 4) Develop an optimization methodology to predictdifferent subject-specific gait patterns using a subject-specific computationalmodel that matches a specified momentum signature. Eliminating the residualloads makes the resulting motions dynamically consistent while closely trackingthe foot markers. This is essential for use with the foot-ground contact modelthat frees up the motion of the foot, which was constrained using previousinverse dynamics methods. Although this method did not yield desired results,other methods utilizing implicit numerical integration may have success usingthe developed foot-ground contact model to predict new motions based onexperimental data and whole-body angular and linear momentum principles, whichmay help identify where to focus rehabilitation efforts that are likely toproduce the largest functional improvement for a particular patient. Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-parent:"";mso-padding-alt:0in 5.4pt 0in 5.4pt;mso-para-margin:0in;mso-para-margin-bottom:.0001pt;mso-pagination:widow-orphan;font-size:10.0pt;font-family:"Times New Roman","serif";} ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: FREGLY,BENJAMIN J.
Local:
Co-adviser: BANKS,SCOTT ARTHUR.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31
Statement of Responsibility:
by Jennifer N Jackson.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Embargo Date:
12/31/2014
Resource Identifier:
907780689 ( OCLC )
Classification:
LD1780 2013 ( lcc )

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1 LOCOMOTION PATTERN PREDICTION BASED ON WHOLE BODY ANGULAR AND LINEAR MOMENTUM VARIATIONS By JENNIFER NOELLE JACKSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF T HE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013

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2 2013 Jennifer Noelle Jackson

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3 To my extraordinary Mom whose light still shines bright ly

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4 ACKNOWLEDGMENTS d support of my extraordinary mom, who passed away in 2009. She taught me what true strength really is, and she always had the utmost confidence in me no matter how much I doubted myself. She never stopped learning or striving to meet her goals and no mat ter how difficult things got, she always had a smile on her face and believed that things would work out somehow She taught me to fight for the things I want. I carry that with me. She is the light that still guides me, and I have always done (and still d o) my best to make her proud. This is as much an achievement for me as it is for her. Through out life, we never get anywhere on our own, and I owe so much to my PhD advisor, Dr. B.J. Fregly. I will always be eternally grateful for his immeasurable patienc e, guidance and support He never gave up on me, and I hope he knows how much respect, admiration, and gratitude I have for him. He is an amazing role model, person, and mentor. Even though h Dr. Ted Conway has also be en a source of support and guidance over the years. We reconnected at a conference in 2010, and I am so grateful for his encouragement and confidence in me, especi ally at a low point in my life. I have no doubt that our paths will continue to cross and tha t his encouragement and support will not end when I graduate. I would like to thank my committee members for their guidance and time along the course of my PhD: Dr. Scott Banks, Dr. Chris Hass, Dr. John De Witt, and Dr. Bryan Conrad They are great resear chers and have been wonderful role models for me. As the ones who often go unnoticed, I would also like t o thank April Lane Derfinyak, Tifiny McDonald and Andrea Fabic for all of their help with dep artment paperwork and issues

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5 They truly care about their students and made me feel like I mattered as a part of the BME department. Also, a special thank you to my encouraging labmates and peers who have enriched this experience in great ways: Jonathan Walter, Laura (Lingmin) Li, Alli son Hall and Ira Hill I a m proud to call them my friends, in addition to colleagues. Before I was able to finish my research, I took a job with the National Institutes of Health. Everyone warned me that it would be hard to finish my PhD once I had a full time job, and although it took a lot longer than I thought (and was extremely difficult) I finally completed my work. My department chair Leighton Chan and new boss Fran Gavelli have been so supportive and encouraging since I joined their team at NIH M y new NIH family is all roo ful environment to work in ever y day I cannot express how much I love going to work every day, feeling like my work is valued and that I make a difference, and of course the people I work with. One of my many bless ings has been my new mentor, Fran. We think so much wonderful honest, and trusting work gether as I build my career at NIH with her group I have been very fortunate over these years to have and make the most amazing frien ds whose support means so much to me. I owe all of my friends a world of thanks for their support and encouragement, as w ell as forgiveness and understanding. Specifically I would like to thank Kerri Morton, Jessica Cobb, and Allison Hall Despite not always knowing what to say, their support through the darkest times was what helped keep me going. Kerri has been my family a been through everything together, and she has been a comforting presence in my life.

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6 She has been the sister I always wished I had. Jessica has been a model for patience and creativity Her supp ort and thoughtfulness has bee n invaluable and I am thankful every day that she is in my life Allison has been a wonderful mentor and a great friend. Her friendship is one of the treasures that the PhD process has brought to my life. My cousins Larry and Sue Conyers have been suppor tive an d encouraging and randomly remind me how special my mom was. Their love and support mean more to me than they are probably aware. I really enjoy the late night conversations I get to have during holidays with my cousin s Jake and Jesse (s he is going to be a great doctor one day !) I will always be indebted to Jackie Davis and Ellen Buhl w ho opened their hearts to me so we could deal with loss together There are not enough wonderful things I can say about Jackie special bond. She has been inspirational for me in moving forward and helping me celebrate milestones that my mom cannot physically be a part of. She has a very special place in my heart, an d her love and support is my light through dark times. In working through loss, Jim Probert and Tamera Martin have been instrumental in helping me process and move forward. Words cannot express the depth of my gratitude to Jim for his support through the t oughest situations I have ever gone through. I did not do that alone, and I hope he knows the impact he has had I have to thank my Gainesville girl s, Dawn Olmstead, Keri Patrick, and Erin Patrick who brought me into their family circle. They are always so supportive and encouraging been months between phone calls, we never skip a

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7 beat. Despite our crazy schedules, my group of cra fty ladies has helped me stay sane: Jessica Cobb, Lori Pirzer, Allison Hall, and Morgan Hughes. Many of my favorite memories are from projects and events we did together. Lori Pirzer has been especially understanding and supportive regarding loss, and I ho pe she knows how much her friendship and compassion mean to me. She understands when others cannot, which One nice thing about living in one place for so long is slowly building relationships with people who care about how your day is going and encourage you through obstacles. I was lucky to find that in David Laird and Jim Pierce They were not just my neighbors; they became dear friends. Love and support from Divya Agrawal, Jor dan Awerbach, and JJ Buchholz has also helped me make it through this process. These amazing people always make me smile. I am grateful to all of my friends, and i f I named every person who has had an i mpact on my life over the last 8 years, then I would n ever end this acknowledgment. We are shaped by our experiences and the people we meet. I am thankful for all of the people in my life. If you are reading this now, thank you for your time and interest in my work.

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8 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ .......... 10 LIST OF FIGURES ................................ ................................ ................................ ........ 11 LIST OF ABBREVIATIONS ................................ ................................ ........................... 13 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 16 2 ENHANCED RESIDUAL ELIMINATION ALGORITHM ................................ ........... 19 Background ................................ ................................ ................................ ............. 19 Methods ................................ ................................ ................................ .................. 21 Enhanced REA Gait Model ................................ ................................ ............... 21 Enhanced REA Development ................................ ................................ ........... 22 Enhanced REA Evaluation ................................ ................................ ............... 25 Results ................................ ................................ ................................ .................... 27 Discussion ................................ ................................ ................................ .............. 28 3 DEVELOPMENT OF A SUBJECT SPECIFIC GROUND CONTACT MODEL ........ 38 Background ................................ ................................ ................................ ............. 38 Methods ................................ ................................ ................................ .................. 41 Experimental Data ................................ ................................ ............................ 41 Foot Ground Contact Model Development ................................ ....................... 42 Foot Ground Contact Model Calibration ................................ ........................... 44 Results ................................ ................................ ................................ .................... 45 Discussion ................................ ................................ ................................ .............. 47 4 PREDICTIVE GAIT MODEL USING MOMENTUM VARIATIONS .......................... 60 Motivation ................................ ................................ ................................ ............... 60 Methods ................................ ................................ ................................ .................. 61 Experimental Data ................................ ................................ ............................ 62 Gait Model ................................ ................................ ................................ ........ 62 Residual Elimination Algorithm (REA) ................................ .............................. 63 Foot Ground Contact Model ................................ ................................ ............. 64 Momentum Signatures ................................ ................................ ..................... 64 Results ................................ ................................ ................................ .................... 66

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9 Predictive Gait Optimization ................................ ................................ ............. 66 Momentum Signatures ................................ ................................ ..................... 66 Discussion ................................ ................................ ................................ .............. 67 5 CONCLUSIONS ................................ ................................ ................................ ..... 78 APPENDIX: COST FUNCTION DETAILS FOR FOOT GROUND CONTACT MODEL CALIBRATION AND TESTING ................................ ................................ ........ 81 LIST OF REFERENCES ................................ ................................ ............................... 91 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 96

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10 LIST OF TABLES Table page 2 1 Mean root mean square (RMS) marker distance error comparison between the 29 DOF one back joint model using the ori ginal residual elimination algorithm (REA), 29 DOF one back joint model using the enhanced REA (1BJM), 29 DOF two back joint model using the enhanced REA (2BJM D), and 32 DOF two back joint model using the enhanced REA (2BJM I). All results are for the cal ibration walking trial. Units are in mm. ............................... 36 2 2 Mean root mean square (RMS) marker distance error comparison for individual and combinations of modifications for the enhanced REA. Adjustments are defined as: Mod 1 = marker weights, Mod 2 = tracked acceleration curves, Mod 3 = feedback gains, and Mod 4 = model parameters. All results are for the calibration walking trial. Units are in mm. Bold text indicates the two best combinations of modifica tions. ......................... 37 3 1 Calibrated parameters for the right and left feet. The maximum, mean, minimum, and standard deviations are reported for the spring stiffness ( ) and damping ( ) values, in units of N/m and m/s, respectively, as well as the coefficients of friction (unitless). ................................ ................................ .......... 58 3 2 RMS errors for ground reaction components and kinematics for both feet, where CoP is center of pressure The RMS mean and standard deviations are calculated for five testing trials. The anterior posterior (AP) direction is forward, normal or superior inferior (SI) direction is up, and medial lateral (ML) direction is to the right. The RMS errors for the tra nslational (Trans) and rotational (Rot) kinematics for the hindfoot (HF) and toes are also reported. ..... 59 4 1 RMS variations about the mean of normalized central angular and linear momentum for each of five trials from the three tasks studied. .......................... 77

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11 LIST OF FIGURES Figure page 2 1 Schematics of the 29 DOF one back joint (a) and 29 or 32 DOF t wo back joint (b) full body gait models used to evaluate the enhanced residual elimination algorithm (REA). ................................ ................................ ............... 32 2 2 Schematic of the original residual elimination algorithm [16]. ............................. 33 2 3 Comparison between experimental markers (blue) and model markers (red) for walking, marching, running, and bounding at 0%, 25%, 50%, 75%, and 100% of the locomotion cycle. Units are in meters. ................................ ............ 34 2 4 Mean root mean square (RMS) marker distance error comparison for each segment for all locomotion tasks: walking, marching, running, and bounding over all 5 trials. The mean and standard devia tions are over five data trials for all segments markers for each motion except marching, which only had four useable trials. Units are in mm. ................................ ................................ ... 35 3 1 Viscoelastic element placement for the rig ht foot. The heel, toe, and medial and lateral toe markers were used to define a uniform rectangular grid (5 x markers. The elements are separated into active hindfoot, active toes, an d inactive elements. The black line is the shoe outline and the gray line is the toes axis. ................................ ................................ ................................ ............ 52 3 2 Ground reaction components for both right (solid) and left (dashed) feet. The top row shows t he anterior posterior (AP), superior inferior (Normal), and medial lateral (ML) force curve comparisons, respectively, and the bottom row shows the center of pressure (CoP) location in the AP and ML directions and the free moment curve comparisons for the experimental data (blue solid and shaded) and model optimization (orange solid and shaded). ...................... 53 3 3 Comparison of right foot kinematic curves over time for one gait cycle excluded from calibrat ion. The top row shows the hindfoot (HF) translational kinematics in units of meters. Since the rotation sequence was 3 1 2, the hindfoot rotational kinematic curves in units of degrees are presented in this order in row 2. The third row shows toe flexion in units of degrees. ................... 54 3 4 Comparison of left foot kinematic curves over time for one gait cycle excluded from calibration. The top row shows the hindfoot (HF) translational kinematics in uni ts of meters. Since the rotation sequence was 3 1 2, the hindfoot rotational kinematic curves in units of degrees are presented in this order in row 2. The third row shows toe flexion in units of degrees. ................... 55 3 5 Center of pressure paths for three force plates shown in blue for all five walking trials. The subject hit the first and last force plates with his left foot

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12 and the middle force plate with his right foot. The shape of the foot is super imposed (in green for the first force plate, black for the second force plates, and red for the third force plate) under this path to verify that the center of pressure is underneath the foot during movement. The hooking pattern observed for the middle for ce plate is highlighted in the orange circles. ................................ ................................ ................................ ................ 56 3 6 Differences between parabolic toes surfaces for (A) Adidas Samba sneakers used in this study and (B) other athletic shoes. The vertical he ights of the elements in the toes segment were fitted along a parabolic surface determined from shoe measurements. ................................ ............................... 57 4 1 Ground reaction curves without REA. Ground reaction curves resulting f rom a forward dynamics simulation on a right foot only model that includes damping but does not include REA. Experimental curves are in blue (solid), and calculated curves are in orange (dashed). The top row is forces and the bottom row is moments. ................................ ................................ ..................... 72 4 2 Ground reaction curves with REA. Ground reaction curves resulting from a forward dynamics simulation on a right foot only model that includes damping and REA. Experimental curves are in blue ( solid), and calculated curves are in orange (dashed). The top row is forces and the bottom row is moments. ...... 73 4 3 Momentum curves for gait tasks. Whole body central angular (left) and linear ( right) momentum for walking (blue), running (red), and marching (black). The x direction points anteriorly, the y direction points superiorly, and the z direction points to the right. Momentum values are normalized using body mass M, average speed V, and b ody height H. ................................ .................. 74 4 4 body central angular (left) and linear (right) momentum for walking, running, and marching. The x direction points anteriorly, the y dire ction points superiorly, and the z direction points to the right. Momentum values are normalized using body mass, average speed, and body height. ................................ ................................ ................................ 75 4 5 Vertical ground reaction force curves f or both feet resulting from a static optimization of a full body model that only allowed the vertical translation of the pelvis to vary while minimizing the vertical acceleration. Experimental curves are in blue (solid), and calculated model curves are in orange (dashed). ................................ ................................ ................................ ............ 76

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13 LIST OF ABBREVIATION S CNS Central Nervous System REA Residual Elimination Algorithm DOF Degrees of Freedom IRB Internal Review Board 1BJM One Back Joint Model 2BJM D Two Back Joint Model Dependent de grees of freedom 2BJM I Two Back Joint Model Independent degrees of freedom RMS Root mean square Mod Modification vGRF Vertical Ground Reaction Force AP Anterior Posterior ML Medial Lateral SI Superior Inferior Co P Center of Pressure HF Hindfoot N Newton s mm millimeters Nm Newton meters deg degrees CoM Center of Mass

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14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy L OCOMOTION PATTERN PREDICTION BASED ON WHOLE BODY ANGULAR AND LINEAR MOMENTUM VARIATIONS By Jennifer Noelle Jackson December 2013 Chair: Benjamin J. Fregly Major: Biomedical Engineering Computational models are valuable for developing and testing hypoth eses th at otherwise would be infeasible to explore experimentally. They can also provide a theoretical framework to explain experimental observations. In the case of pathological gait, despite many studies reporting that the central nervous system (CNS) re gulates angular momentum during walking, no simple control law currently exists to explain how the CNS makes walking efficient or even possible. Fewer studies have looked a t how linear momentum is regulat ed during human locomotion, although recent findings indicate regul ation occurs for various locomotion tasks. A computational model that uses basic momentum considerations to predict achievable, improved gait patterns for individuals with pathological gait could be a valuable tool to aid clinicians in makin g objective, highly effective treatment decisions. To create a framework for predictive gait optimization, the main objectives are to : 1) E liminate the pelvis residual loads and improve foot marker tracking by enhancing the residual elimination algorithm through marker weight tracked a cceleration curve feedback gain and select model joi nt and inertial parameter adjustments ; 2) Develop a foot ground contact mode l that matches all three force components center of pressure

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15 location, and free moment for bo th feet using physics to model the foot ground interactions; 3) Demonstrate that whole body momentum variations for gait tasks cluster and 4 ) Develop an optimization methodology to predict different subject specific gait patterns using a subject specific computational model that matches a specified momentum signature Eliminating the residual loads makes the resulting motions dynamically consistent while closely tracking the foot markers. This is essential for use with the foot ground contact model that frees up the motion of the foot, which was constrained using previous inverse dynamics methods Alt hough this method did not yield desired results other methods utiliz ing implicit numerical integration may have success using the developed foot ground contact model to predict new motions based on experimental data and whole body angular and li near momentum principles which may help identify where to focus rehabilitation efforts that are likely to produce the largest functional improvement for a particular patient.

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16 CHAPTER 1 INTRODUCTION Computational models are valuable for developing and testing hypotheses th at otherwise would be infeasib le to explore experimentally. They can also provide a theoretical framework to explain experimental observations. In the case of pathological gait, despite many studies reporting that the central nervous system (CNS) regulates angular momentum during walki ng, no simple control law currently exists to explain how the CNS makes walking efficient or even possible. Fewer studies have looked a t how linear momentum is regulat ed during human locomotion, although recent findings indicate regul ation occurs for vario us locomotion tasks. A computational model that uses basic momentum considerations to predict achievable, improved gait patterns for individuals with pathological gait could be a valuable tool to aid clinicians in making objective, highly effective treatme nt decisions. Patient specific gait optimizations capable of predicting post treatment changes in joint motions and loads could improve treatment design for gait related disorders. To maximize potential clinical utility, such optimizations should utilize t hree dimensional patient specific musculoskeletal models, generate dynamically consistent gait motions that reproduce pre treatment marker measurements closely, and achieve accurate foot motions to permit deformable foot ground contact modeling. Results ob tained from computational models are dependent upon how the foot ground interaction is modeled. Subject specific foot ground contact models capable of calculating all six ground reaction components (i.e., three forces and either three moments or the center of pressure and free moment) as outputs are needed for forward dynamic gait optimizations to predict new motions for individual subjects. The foot ground contact

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17 models reported in the literature are valuable for answering generic questions, but there is a need for deformable subject specific foot ground contact models, especially since a ot ground contact pattern may change after treatment. To create a framework for predictive gait optimization, the main objectives of this work are to: 1) Elim inate the pelvis residual loads and improve foot marker tracking by enhancing the residual elimination algorithm through marker weight, tracked acceleration curve, feedback gain, and select model joint and inertial parameter adjustments; 2) Develop a foot ground contact model that matches all three forces, center of pressure location, and free moment for both feet using physics to model the foot ground interactions; 3) Demonstrate that whole body momentum variations for gait tasks cluster differently from o ne another and that these clusters can be viewed as methodology to predict different subject specific gait patterns using a subject specific computational model that matches a specified momentum signature. First, we will enhance an existing residual elimination algorithm (REA) to produce dynamically consistent motion with improved foot marker tracking within a single gait optimization framework. We will investigate four primar y enhancements to the original REA: 1) manual modification of tracked marker weights, 2) automatic modification of tracked joint acceleration curves, 3) automatic modification of algorithm feedback gains, and 4) automatic calibration of model joint and ine rtial parameter values. We will evaluate the enhanced REA using a three dimensional full body skeletal model and movement data collected from a subject who performed four distinct gait patterns: walking, marching, running, and bounding.

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18 Second, we will dev elop a complete foot ground contact model for both feet. This will be accomplished by using a) a two segment foot model with seven degrees of freedom (three translations and three rotations at the heel and one degree of freedom for toe flexion) and b) 38 v iscoelastic elements with locations dependent on shoe size and shape to calculate the ground reaction components. Next we will develop a three step calibration approach for determining subject specific parameters that match 1) the vertical force component, 2) all three force components, and 3) all six ground reaction components (three forces, the anterior posterior and medial lateral center of pressure location, and the free moment). The resulting parameters will then be tested on the same trial, as well as four additional trials excluded from calibration. Last, we will show that whole body momentum variations for gait tasks cluster differently from one another, thus creating momentum signatures. Then we will combine the enhanced REA framework and the foot ground contact model to seek to predict different subject specific gait patterns using a subject specific computational model that matches a specified momentum signature. This method will be tested using a foot only model possessing 2 segments and 7 degree s of freedom and a full body gait model possessing 16 segments and 31 degrees of freedom.

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19 CHAPTER 2 ENHANCED RESIDUAL EL IMINATION ALGORITHM Background Common clinical conditions that limit walking function, specifically osteoarthritis [1] stroke [2] [3] are the main causes of disability among U.S. adults, resulting in billions of dollars in medical care and lost productivity annually [4] As walking ability diminishes, quality of life decreases and the risk of death increases [5 7] making this problem a critical one for public health. Traditional treatment design methods for these conditions rely heavily on subjective clinical ju dgment and have been ineffective at restoring normal walking ability [8] suggesting that new treatment design methods are necessary to address the needs of these clinical populations. Computational models could provide an objective alternative to tradi tional treatment walking function treatment walking data, a computational model should ideally be three dimensional, patient specific, and dynamically consistent. Further more, sinc e treatment in many cases may ground contact pattern such models should also incorporate patient specific deformable foot ground contact models. A number of research groups have made important strides in this area, ac hieving some but not all of these features. Our lab successfully used a three dimensional patient specific dynamic walking model to predict walking pattern changes to reduce the peak knee adduction moment in a patient with bilateral medial knee osteoarthri tis [9] However, the model was dynamically inconsistent since it used inverse rather than forward dynamics, leaving fictitious residual loads acting on the pelvis. Furthermore, no

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20 physical foot ground contact model was included, making it difficult to predict new walking motions with altered foot paths [10] In comparison, other groups have successfully generated forward dynamic simulations of walking [11 15] While Piazza and his colleagues restricted their mo dels to the sagittal plane [12, 13] they did incorporate a somewhat subject specific foot ground contact model by scanning the other groups used generic foot ground contact models in their two [15] or three dimensional gait models [11, 14] None of these models were subject specific in the sense of using joint and inertial paramete r values that were calibrated to subject movement data. In addition, Remy and Thelen [16] presented a residual elimination algorithm that generates three dimensional dy namic simulations of walking that are dynamically consistent and reproduce experimental surface marker measurements closely. The main drawbacks of their method were that no deformable foot ground contact model was used (ground reactions were applied direct ly to each foot) and the simulated foot path did not closely match the experimentally measured foot marker motions, making it difficult to incorporate a deformable foot ground contact model into the algorithm for reproducing an experimentally measured walk ing motion as a starting point. These existing gait optimization methods have limited ability to simulate patient specific changes in foot ground contact patterns, since either the foot path must be prescribed [9] or else generic foot ground contact models are used [11, 13 15] This study extends the residual el imination algorithm (REA) framework developed by Remy and Thelen [16] While the kinematics are dynamically consistent, the inability of the REA to produce accurate foo t placement hinders the prediction of post treatment

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21 gait motions with altered foot ground contact patterns. To achieve better matching of foot marker motion with the REA, we investigate four potential enhancements that adjust: 1) marker weights, 2) tracke d acceleration curves, 3) feedback gains, and 4) model parameter values. While the marker weights are manually adjusted, each of the remaining changes are automated within the enhanced REA optimization methodology. Each modification is explored individuall y and in combinations to determine which ones yield the lowest hindfoot and overall segment marker errors. In addition, greater back flexibility may help the model achieve better matching of all marker motion. Thus, the potential benefit of using a two joi nt back model with and without additional degrees of freedom is also investigated. Methods Enhanced REA Gait Model We used a modified version of an existing three dimensional full body dynamic skeletal model to develop and evaluate our enhanced REA. The e xisting skeletal model possesses 14 segments and 27 degrees of freedom (DOFs) [9] The ankles were modeled as two non intersecting pin joints, the knees as pin joints, the hips as ball and socket joints, the back as a ball and socket joint at approximately the L4 L5 level, the shoulders as universal joints, and the elbows as pin joints, with the pelvis being connected to ground via a six DOF joint. We made two modifications to this model to explore how increased model complexity affects marker tracking errors. First, to accommodate non sagittal arm motion, we added internal/external rotational DOFs to both shoulders, thereby increasing the number of DOFs to 29 (Fig. 2 1a). Second, we split the back into two segments by adding a second ball and socket joint at the T8 T9 level, further increasing

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22 the number of DOFs to 32 (Fig. 2 1b). The additional should er DOFs were used in all evaluations, while the additional back DOFs were treated three ways to explore how back flexibility affected the results: 1) upper back joint locked so that all motion occurred in the lower back joint, 2) upper back joint free to m ove independently from lower back joint, and 3) upper back joint prescribed to move identically to lower back joint. We derived the equations of motion for each model using Autolev symbolic manipulation software (Motion Genesis, Palo Alto, CA). Enhanced RE A Development The original REA [16] (Fig. 2 2) finds new initial conditions and generalized accelerations that produce a dynamically consistent walking motion (i.e., pelvis residual loads eliminated) while tracking exp erimentally measured marker positions. To accomplish this, small variations are introduced to account for the differences between the model and experimental ( accelerations: ( 2 1) At the initial time frame, the pseudoinverse method is used to calculate the minimum changes in all generalized accelerations required to balance an underdetermined system of six whole body dynamics equations. Acceleration changes are calculated relative to desir ed values defined by a critically damped feedback strategy: (2 2 ) In this equation, is a desired generalized acceleration, is a feedback ga in on joint position error, is a feedback gain on joint velocity error, and are the model generalized coordinate and velocity values, respectively, , and are the experiment al generalized coordinate, velocity, and acceleration values, respectively,

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23 derived from an initial inverse kinematics analysis. The updated model generalized accelerations are numerically integrated, and the process is repeated for each subsequent time frame. Using the initial conditions as design variables, an optimization repeats the entire numerical integration process until errors between experimental and model marker positions are minimized. To improve the ability of the o riginal REA to match measured marker positions accurately, we investigated four modifications to the algorithm. First, to improve foot marker tracking, we adjusted the marker tracking weights in the original REA cost function. The original weights were 5 f or the dynamic lower body markers and 1 for the remaining markers [16] Higher marker weights are used to emphasize errors on markers that we want to track more closely. We increased the weight on the hindfoot markers to 100 to track the feet closely and the weight on the shoulder markers to 10 to prevent the trunk from falling over. We chose large foot marker weights to encourage achievement of hindfoot marker errors within 1 mm of the best case solution produced when each hindfoot was treated as isolated free body [17] All other markers weights were less than 10 (pel vis: 8, thigh: 2.8, shank: 1.2, toe: 0, arms: 3, trunk: 0.2, and xiphoid: 1.5). Since the foot is modeled as a single rigid body, there is nothing to account for toe rotation. Hence, we are only concerned with matching the hindfoot markers so the toe marke r weight is zero. These were the only manual modifications made to the algorithm. The remaining modifications were automated within the enhanced REA optimization methodology. Second, to account for errors in the experimental , and curves being tracked in Eq. ( 2 2), we allowed our REA to adjust these curves. The experimental

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24 curves have errors since they are obtained from an initial inverse kinematics analysis using a model with uncalibrated joint para meters, and the experimental and curves have errors since they are produced by differentiating noisy and inaccurate curves [18] We parameterized the initial curves using a combination of polynomial and Fourier coefficients [19] thereby allowing us to calculate the associated and cu rves analytically. We then treated these coefficients as design variables to be adjusted by the optimization. We used twenty coefficients to parameterize each of the 29 (single back joint model) or 32 (two back joint model) curves. This modif ication added 580 or 640 design variables (depending on the back joint model) to the optimization, which significantly increased computation time. Third, to improve the performance of the feedback control system, we changed the single gain in the original formulation into nine gains adjusted by the optimization, with each value controlling a different set of generalized coordinates. In the original REA, the position gain was 100 and the velocity gain was 20 for all DOFs. In our m odified formulation, we used an initial guess of 100 for each of our nine position gains: three pelvis translations (1 gain), three pelvis rotations (1 gain), all back rotations (1 gain), all arm DOFs (1 gain), both ankle DOFs (4 gains ; each DOF had its ow n gain ), and the remaining lower body DOFs (1 gain). We defined the associated gains by ( 2 3) which drives the feedback error terms to zero in a critically damped manner [20] Fourth, to improve marker tracking while maintaining zero pelvis residual loads, we included select joint and inertial parameter values as addi tional design variables in the optimizations. We allowed the masses and inertias of each body segment to be

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25 adjusted by up to 10% from their initial values calculated from the literature [21] We also allowed joint parameter values for the lower body segments, as well as center of mass locations for all segments, to be adjusted by up to 5 mm or 5 degrees from their initial values calcula ted from the literature [21] None of these limits were reached by any of our optimizations. The optimization design variables and i nputs to the model were not the same. The enhanced REA optimization framework had 86 6 design variables: 32 initial position values, 32 initial velocity values, 9 position gain values, 640 Bspline coefficients (20 coefficients for each of 32 DOFs), 39 cente r of mass values, 13 mass values, 39 inertia values, and 62 lower body joint parameter values. The inputs to the numerical integrator are the curr ent time point and the generalized coordinate and velocity values at that time point. Model inputs were the jo int locations and joint attachment positions in the segment reference frames; the position, velocity, and ground reaction force and moment values at the current time point; the electrical center positions of each force plate; and the mass, inertia, and cen ter of mass values of each segment. The outputs from the model are the A matrix and b vector that satisfy A*x = b, where x is a vector of the 32 generalized acceleration values The outputs from the numerical integrator were the velocity and acceleration v alues at the current time point. Enhanced REA Evaluation We evaluated our enhanced REA using experimental gait data collected from a single healthy subject (male, age 46, height 1.7 m, weight 69 kg) who performed four gait tasks: walking, marching, runnin g, and bounding. The study was IRB approved, and the subject gave informed consent prior to testing. Surface marker positions were measured using a 14 camera Vicon/Peak motion capture system (Vicon Motion

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26 Systems, Inc., Lake Forest, CA), while ground react ion forces and moments were measured using three six axis Bertec force plates (Type 4060 08, Bertec Corp., Columbus, OH). The data collection protocol and marker set were identical to a previous study [9] Four markers were placed on each foot (including a toe marker), three on each shank and thigh, three on the pelvis, three on the torso, and one on e ach elbow and wrist. This marker set permitted tight tracking of lower body segment motions and less stringent tracking of upper body segment motions. To facilitate REA evaluation, we collected five trials of each gait pattern with three clean foot force p late strikes. Using these data and the three variations of our full body gait model, we followed a four step process to evaluate our enhanced REA. Out of the five recorded walking trials, the trial with the velocity closest to the mean of all five trials w as chosen to calibrate the model. First, a calibration optimization step used data from the calibration walking trial to determine the model joint and inertial parameters for the subject, as well as a set of marker weights (determined manually) that tracke d hindfoot markers as closely as possible without degrading the tracking of other segment markers. The cost function minimized marker coordinate errors, changes in initial joint and inertial parameter values, and differences between the initial and final s tate (i.e., near periodic motion). Second, the resulting optimized joint and inertial parameter values were held constant and used with the walking calibration data set in a testing optimization step to determine the optimal marker weights that best repro duced marker errors from the calibration optimization step. To ensure that the enhanced REA can be reliably used for

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27 various locomotion tasks, we also performed these same two steps on the bounding calibration trial (our most dynamic task). Using optimized marker weights from both the walking and bounding calibration trials as a guide, marker weights were evaluated using each of our three gait models (29 DOF one back joint model (1BJM), 29 DOF two back joint model with dependent DOFs (2BJM D), and 32 DOF tw o back joint model with independent DOFs (2BJM I)). Marker weights were manually chosen that yielded the lowest root mean square (RMS) marker distance errors for the bounding calibration trial without significantly degrading the RMS marker distance errors for the walking calibration trial. Therefore, there was a trade off in selecting marker weights, as well as a model, suitable for various location tasks. The new cost function minimized marker coordinate errors and enforced periodicity. Third, the optimal marker weights from the testing optimization step were used to test the enhanced REA on four walking trials excluded from the calibration optimization step. In addition to walking, we also investigated how well our enhanced REA performed for all running, m arching, and bounding trials. As a final step, the impact of each enhancement was evaluated individually and in all possible combinations using the same calibration walking trial with the 29 DOF one back joint model (chosen based on results from the previo us step) to determine which enhancement(s) yielded the lowest RMS marker distance errors compared with those from the original REA. Results The enhanced REA significantly reduced foot marker errors while slightly reducing overall leg marker errors for wal king (Table 2 1). Compared to the original REA, all segment marker errors were reduced for each of the three gait models, except for a slight increase in the trunk marker errors for the 32 DOF two back joint model. The

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28 single back joint model yielded the l owest foot marker errors compared to the other models and second lowest overall marker error. While the 32 DOF two back joint model had the highest pelvis marker errors, it produced some of the smallest marker errors for other segments. Despite having the lowest pelvis marker errors, the 29 DOF two back joint model did not improve upper body marker errors and produced the highest overall marker errors of the three models. Foot markers were tracked closely for all three models with RMS errors approximately w ithin 1 mm of the best case solution produced when each hindfoot was treated as an isolated free body. Pelvis residual forces and torques remained below 1e 12 N and 1e 12 Nm, respectively, for all cases. The enhanced REA was also able to eliminate residua ls for the three other gait tasks while minimizing segment marker errors (Fig. 2 3). The mean RMS marker distance errors were generally higher for the more dynamic motions of running and bounding compared to walking and marching (Fig. 2 4). Foot markers we re tracked closely for all locomotion tasks with considerably lower mean RMS errors compared to all other segments. The pelvis residual forces and torques remained below 1e 11 N and 1e 11 Nm, respectively, for all tasks. Compared to results from the origi nal REA, all enhancements individually and in combinations reduced RMS foot marker distance errors, though some enhancements increased RMS marker distance errors for other segments (Table 2 2). The lowest mean RMS foot marker error was achieved by leaving out the feedback gain adjustments from the enhanced REA, though the reduction was trivial. Discussion This study evaluated improvements to the original REA developed by Remy and Thelen [16] with the goal of tracking fo ot marker trajectories more closely while still

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29 maintaining dynamic consistency. Use of all four modifications together resulted in significantly better foot marker tracking as well as somewhat improved leg marker tracking. The omission of feedback gain ad justments yielded slightly better results overall, though the improvements were small, suggesting that feedback gain adjustment was not worth the effort. Compared to results from the original REA, all four modifications resulted in some marker error impro vements and slightly altered kinematics (Table 2 1). To account for inaccuracies in the experimental kinematic data, we allowed the tracked position, velocity, and acceleration curves to vary, resulting in small but reasonable changes in the marker traject ories produced by our enhanced REA. The optimized kinematics showed slightly more rotation of the pelvis, back, and shoulders to reduce overall marker errors (Fig. 2 3). Surprisingly, splitting the trunk into two segments did not significantly improve mark er errors compared to a single segment trunk (Table 2 1). Splitting the back into two segments at the T8 T9 level may not have been the best way to test the benefits of a second back joint for the model because there is not much flexibility in the rib cage Alternatively, choosing the second back joint at the T12 L1 level could potentially be more useful because the back has more flexibility in the lumbar region. However, th e marker errors for all segments were comparable between the three models with the h ighest pelvis marker errors for the 32 DOF two back joint model, which also had the lowest arm marker errors. However, the free rotational motion of the lower trunk segment in this model resulted in a non unique solution. Overall, the reduced complexity of the one back joint model was sufficient for improving both foot marker tracking, as well as tracking of markers on other segments.

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30 Our improvements to the original REA came at a significant computational cost. The original REA formulation simultaneously a dapts only the initial positions and velocities of the 29 generalized coordinates (total design variables = 58). The enhanced REA formulation has 866 design variables that also account for parameterization of joint kinematics, feedback gains, and joint and inertial parameters. For the original REA, the time required to process one walking trial using the 29 DOF one back joint model was 9 minutes. In contrast, for the 29 DOF one back joint, 29 DOF two back joint, and 32 DOF models, the enhanced REA using all four modifications required 1.9 hours, 3.4 hours, and 2.9 hours, respectively, to process the same walking trial. Not only were results for the one back joint model sufficient in terms of accuracy, but the computation time was much lower compared to both of the two back joint models. Our analysis of each REA enhancement performed separately and in combinations (Table 2 2) revealed that adjustment of the tracked kinematic curves had the most impact on reducing foot marker errors. Use of two modifications together led to improvements compared to the original REA but did not achieve the desired level of accuracy for the foot markers. Use of groups of three modifications, specifically all but the feedback gain adjustments, resulted in the best overall marker tracking errors. Therefore, the best approach appears to be a combination of manual adjustments to marker weights, automatic adjustments to tracked kinematic curves, and automatic adjustments to model parameter values if the goal is to reduce foot marker e rrors without increasing other segment marker errors. With these modifications, simulations generated with the enhanced REA can be used for applications that utilize foot ground contact models, where foot position and

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31 orientation are important for modelin g foot floor interaction. This increased accuracy should allow the use of a deformable foot ground contact model with REA generated simulations to predict new gait motions for investigating novel rehabilitation strategies. These findings also demonstrate t hat greater back flexibility does not improve marker tracking significantly in computational gait models, suggesting that a single back joint model is sufficient for simulating various gait patterns.

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32 Figure 2 1. Schematics of the 29 DOF one back joint (a) and 29 or 32 DOF two back joint (b) full body gait models used to evaluate the enhanced residual elimination algorithm (REA).

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33 Figure 2 2 Schematic of the original residual elimination algorithm [16]

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34 Figure 2 3. Comparison between experimental markers (blue) and model markers (red) for walking, marching, running, and bounding at 0%, 25%, 50%, 75%, and 100% of the locomotion cycle. Units are in meters.

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35 Figure 2 4. Mean root mean square (RMS) marke r distance error comparison for each segment for all locomotion tasks: walking, marching, running, and bounding over all 5 trials. The mean and standard deviations are over five data trials for all segments markers for each motion except marching, which on ly had four useable trials. Units are in mm.

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36 Table 2 1. Mean root mean square (RMS) marker distance error comparison between the 29 DOF one back joint model using the original residual elimination algorithm (REA), 29 DOF one back joint model using the enh anced REA (1BJM), 29 DOF two back joint model using the enhanced REA (2BJM D), and 32 DOF two back joint model using the enhanced REA (2BJM I). All results are for the calibration walking trial. Units are in mm. Segment Original REA 1BJM 2BJM D 2BJM I Pe lvis 16.7 14.5 14.3 16.5 Thigh 20.8 19.2 19.0 18.9 Shank 17.9 13.0 12.6 12.5 Foot 16.0 3.1 3.4 3.3 Arm 34.9 19.1 19.3 17.6 Trunk 25.9 21.7 26.1 21.2 Leg 18.0 12.1 12.0 12.3 All 22.3 14.8 15.3 14.5

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37 Table 2 2. Mean root mean square (RMS) marker di stance error comparison for individual and combinations of modifications for the enhanced REA. Adjustments are defined as: Mod 1 = marker weights, Mod 2 = tracked acceleration curves, Mod 3 = feedback gains, and Mod 4 = model parameters. All results are fo r the calibration walking trial. Units are in mm. Bold text indicates the two best combinations of modifications. Segment Pelvis Thigh Shank Foot Arm Trunk Leg All Original REA 16.29 19.96 18.28 18.37 36.65 26.23 18.50 23.17 Mod 1 19.64 21.00 17.66 16.2 6 40.61 29.64 18.50 24.30 Mod 2 12.68 19.60 15.41 7.83 24.67 25.22 14.05 17.11 Mod 3 14.38 16.67 15.04 12.87 37.58 26.01 14.79 20.58 Mod 4 14.56 16.08 15.39 16.00 28.14 23.65 15.64 19.41 Mods 1&2 18.10 23.29 16.95 4.09 24.54 27.64 15.25 18.46 Mods 1& 3 18.82 16.73 13.99 12.57 33.92 30.73 15.06 20.85 Mods 1&4 15.88 14.34 9.98 7.79 23.43 23.49 11.44 15.55 Mods 2&3 12.82 19.46 15.45 8.21 24.81 25.37 14.15 17.43 Mods 2&4 10.96 17.88 13.98 6.75 21.32 22.05 12.60 15.35 Mods 3&4 11.89 14.05 12.29 10.15 32 .49 23.49 12.12 17.57 Mods 1 3 19.04 21.73 15.04 4.17 27.43 28.97 14.42 18.63 Mods 1 2,4 14.27 19.29 13.19 3.07 17.76 21.60 12.20 14.71 Mods 1,3 4 17.25 14.07 10.31 8.26 22.70 25.63 11.79 15.94 Mods 2 4 11.02 17.89 13.99 6.73 21.69 22.02 12.60 15.42 M ods 1 4 14.46 19.15 12.99 3.14 19.10 21.67 12.14 14.75

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38 CHAPTER 3 DEVELOPMENT OF A SUB JECT SPECIFIC GROUND CONT ACT MODEL Background Musculoskeletal models have been used to investigate kinematics and muscle function for normal tasks like walking and run ning [13, 22 24] as well as a variety of clinical conditions such as osteoarthritis [25, 26] post stroke hemiparesis [27, 28] and cerebral palsy [29 31] Musculoskeletal models have also been used to design novel treatments for patients with knee osteoarthritis [9] Gait optimizations utilize these models to predict clinical outcomes. Many clinical populations are heterogeneous, and the development of patient specific musculoskeletal model s will be a critical step towards designing new motions for patient specific treatment. An important factor for predicting new motions with patient specific musculoskeletal models is accurate modeling of the interaction between the foot and the ground bec ause model estimates are dependent on how foot ground interaction is modeled [24] The foot ground contact models reported in the literature are valuable for answering generic questions, but there is a need for deformable subject specific foot ground contact mode ls [32] especially ot ground contact pattern may change after treatment Previous studies have shown promising results for foot ground contact modeling using various approaches for tasks, such as walking [13, 14, 23, 24, 33 39] running [11, 23, 24, 39 41] and jumping [42] The most notable differences between methods a re the number and location of contact points and the number of foot segments. Many models assume contact at a handful of discrete points, but in reality, contact is continuous over the surface of the foot/shoe. Researchers have reported application of cont act elements at a single point [41] along the midline of [33, 34] or center of

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39 pressure [38] under the foot, at a sma ll number of specific points in each foot segment [14, 24, 35, 36, 40, 42] and 30 or more locations under the foot [11, 13, 23] However, the number of segments seemed to ha ve a greater effect on the reproducibility of experimental ground reaction force curves compared to the number/placement of contact points. This could be attributed to the foot ground modeling techniques, as well as the unrealistic assumption that contact only occurred at a small number of points under the foot. Most of these studies used either a single segment foot [23, 36, 39 42] or two foot segments (hindfoot and toes) [13, 14, 24, 33 35, 38] The single segment foot models exhibited more discontinuities in force curves, especially at heel contact, probably due to their simplicity and inability to account for rolling motion under a single rigid body. New constraint based co ntact models that account for rolling motion have produced ground reaction curves without discontinuities using a single segment foot model [39] However, the rolling constraint cannot be used in predictive optimizations beca use of its dependence on measured center of pressure data. Both two and three segment foot models produced groun d reaction force curves with discontinuities, but it was reported that viscoelas tic elements in particular reduce discontinuities when modeling foot ground interaction [33] While progress has been made toward reproducing experimental ground reaction force curves, the discontinuities in the reported model ground reaction force curves, especially the vertical ground reaction force will be problematic for predictive gait models that need smoother curves to numerically integrate new motions over time. Despite the many ways that contact has been modeled, none of those studies reported moment or center of pressure and free moment cur ves for walking, which is a

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40 major limitation for predictive gait modeling. While a few research groups have reported free moment curves, they used force data obtained from force plates, and thus did not require a foot model [37, 43, 44] which would be needed for predictive gait models. Regardless of how well some of the planar models reproduced the vertical ground reaction force, they lack the ability to determine forces, and thus moments, in other directions. In re ality, motion is not constrained to a single plane so in order to better dimensional model is needed. However, a three dimensional model with two foot segments is not enough to avert problems when modeling contact events especially at heel strike [11, 14, 24, 33, 34, 38] which leads to the need for subject specific parameters and viscoelastic elements. Optimizations were used to determine contact element parameters i n some cases [35, 36, 38, 42] but in most of those studies, the parameters, and consequently the foot model, were not subject specific. In addition, none of the models reported data for the contralateral foot whose stance phase is discontinuous over the gait cycle. Thus, a three dimensional deformable subject specific foot ground contact model capable of reproducing experimental forces, center of pressure, and free moment curves for both feet is needed for predictiv e gait modeling. In this study, we develop a deformable subject specific foot ground contact model that calculates the center of pressure and free moment, in addition to the ground reaction forces using a two segment foot model. We present a three step ca libration process for determining subject specific parameters that reproduce experimental data curves by matching 1) the vertical ground reaction force, 2) all ground reaction forces, and 3) all six ground reaction components (three forces, the anterior po sterior and

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41 medial lateral center of pressure location, and the free moment) The optimized parameters are then tested using the same trial and four additional trials excluded from calibration. This process is repeated for the other foot. Methods Experimen tal Data We collected motion capture and ground reaction force and moment data from one healthy subject (male, age 46, height 1.7 m, weight 69 kg) performing overground gait. The gait cycle starts at right foot contact and ends at subsequent right foot co ntact. The study was IRB approved and the subject gave informed consent. A 14 camera Vicon/Peak motion capture system (Vicon Motion Systems, Inc., Lake Forest, CA) measured surface marker positions, and three six axis Bertec force plates (Type 4060 08, Ber tec Corp., Columbus, OH) calculated ground reaction forces and moments. The marker set and protocol for data collection were identical to a previous study [9] Six markers (four dynamic and two static) were placed on each foot for tracking and to determine foot segments (hindfoot and toes). Adidas Samba sneakers were specifically used for this study b ecause the bottom is essentially flat (i.e., in a static pose, the vertical distance between the toe tip and ground was less than 5 mm) an d possesses no built in cushi oning. We collected five trials of walking data with three clean foot force plate strikes The trial with the velocity closest to the mean of all five recorded walking trials was chosen to calibrate the model. The optimized parameters were then tested on the calibration trial, as well as the four other trials excluded from model calibration.

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42 F oot Ground Contact Model Development The foot ground contact model was developed using a dynamic foot model and five trials of motion capture and ground reaction force and moment data. A two segment (hindfoot and toes) dynamic foot model was constructed po ssessing seven degrees of freedom (DOFs). The equations of motion were derived using Autolev symbolic manipulation software (OnLine Dynamics, Sunnyvale, CA). The translations and rotations of the foot in the lab frame and toe flexion with respect to the hi ndfoot during locomotion were determined from the marker motion capture data using an inverse kinematics analysis. The foot model was incorporated into a Matlab (The Mathworks, ective segment (hindfoot or toes). Foot model parameter values were determined from static trial data. During the static trial, each foot was outlined using a marker wand to determine foot shape and size (Fig. 3 1). The lateral and medial toe markers were used to define the toes axis. The heel, toe, and medial and lateral toe markers were used to define a uniform rectangular grid (5 x 11 elements) whose long axis was aligned with the heel to toe marker direction. A linear spring damper element was placed at the center of each grid element. The location of each spring damper element was defined relative to its respective foot segment, which was based on the anterior posterior position of the s of th e hindf oot and toes segments were used to determine the vertical deformation and deformation rate of each viscoelastic element throughout the gait cycle.

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43 To determine the forces generated by our foot ground contact model, we first calculated the vertical contact force. Vertical contact force (vGRF) generated by each viscoelastic element, was modeled using nonlinear damping: ( 3 1) In this equation, is the spring s tiffness, and are the spring deformation and deformation rate, respectively, is the damping coefficient, is the change in vertical position within the foot segment, is the s pring number, and is the time frame. The nonlinear damping term produces physically realistic hysteresis during spring compression [45] as opposed to linear damping terms that exhibit force discontinuities at the transitions into and out of contact. The contact forces from each e lement were added together to obtain the total vGRF at each time frame. Using the vertical contact force, we calculated the horizontal friction forces generated by each element. The friction force exerted by each viscoelastic element, was ca lculated as a vector in the slip direction and then separated into components using the following model proposed by Hollars [46] : ( 3 2) In this equation, is the normal force at the contact point, is the slip velocity between the shoe (foot) and ground at the contact point, is the transition velocity, and , and are the coefficients of sta tic, dynamic, and viscous friction, respectively.

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44 The transition velocity acts as an upper limit on the drift velocity, which can be arbitrarily small by setting to a sufficiently small value. For this model, we chose 1e 2 based on sensitivity studies. Foot Ground Contact Model Calibration The foot ground contact model was calibrated in three stages using a calibration gait trial. First, a non linear least squares optimization algorithm modified the kine matics (5 DOFs; horizontal translations were excluded), the height of the spring damper elements in the vertical direction in their respective foot segments, and viscoelastic parameter values (38 stiffness and 38 damping values) to best match the experimental vertical ground reaction force curve. The cost function minimized changes in spring stiffness and damping values about their calculated mean values, subject to constraints on marker position errors and vGRF errors. The cost function also included terms to minimize errors in the kinematic, marker, and vGRF derivative curves to help the optimizer find a solution with smoother curves and fewer oscillations. The , and vertical element height values wer e bounded to be greater than zero through the use of virtual springs that ramp up error when a value falls outside of its range. In the second stage, we used a non linear least squares optimization algorithm to best match all three ground reaction force c urves. The cost function modified the kinematics (all 7 DOFs), the coefficients of friction (static, dynamic, and viscous), and the viscoelastic parameter values to best match all three ground reaction force curves. The height of the spring damper elements in the vertical direction was defined as the value determined from the first stage. The cost function was the same as that in the previous stage, with additional terms for the anterior posterior (AP) and medial lateral

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45 (ML) forces and derivative errors, a s well as additional virtual springs for the marker errors and coefficients of friction. A virtual spring was also placed on the difference between the static and dynamic coefficients of friction to maintain realistic values (i.e., > ). The third stage built upon the previous stage to match the center of pressure location and the free moment, in addition to the ground reaction forces. This problem formulation had the same design variables as the previous calibration stage. In addition to the constraints on the marker position errors and the ground reaction force curves and their first derivatives, constraints on the center of pressu re locations and the free moment curve were introduced, as well as their fi rst derivatives. Results from the previous optimization were used as initial guesses in this final stage of calibration. The optimized stiffness and damping values for each viscoelastic element, as well as the coefficients of fr iction were then tested on four additional gait trials excluded from model calibration. The calibration and testing process was repeated for the second foot. Cost function details for the calibration and testing process can be found in Appendix A. Results Our subject specific foot ground contact model closely reproduced experimental force, center of pressure location, and free moment curves (Fig. 3 2 ) with minor kinematic changes for both right (Fig. 3 3) and left (Fig. 3 4) feet The calibrated coefficien ts used to reproduce those curves were similar for both feet, but the left foot had a larger range of stiffness and damping values compared to the right foot (Table 3 1). The calibrated viscoelastic parameter values for all elements were greater than zero, with mean values of 4112 2193 N/m and 6057 6532 N/m and mean values of

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46 1.09e 4 2.82e 5 s/m and 7.60e 3 7.70e 3 s/m for the right and left foot, respectively. The optimized coefficients of friction for the right foot wer e 0.133, 0.090, and 0.009 for the static, dynamic, and viscous coefficients of friction, respectively, while the same coefficients of friction for the left foot were 0.166, 0.063, and 0.012. Like the viscoelastic parameter values, there was a larger differ ence between the dynamic and static coefficients of friction for the left compared to the right foot. Our model achieved close agreement between the experimental curves and model curves for all five testing trials In terms of root mean square (RMS) error s (Table 3 2), the force curves had the largest changes compared to the experimental curves with RMS errors that range from 2.5 N to 18.8 N for the right foot and 3.0 N to 11.8 N for the left foot (Fig. 3 2). The center of pressure locations deviated less from the experimental locations with RMS errors that range from 3.6 mm to 10.9 mm for the right foot and 3.5 mm to 6 mm for the left foot. Differences between the experimental and model free moment curves were the most consistent of all ground reaction cur ves with RMS errors between 1.0 and 1.4 N/m for the right foot and 1.2 and 2.0 N/m for the left foot. Minimal changes were needed to the kinematic curves for both the right (Fig. 3 3) and left (Fig. 3 4) feet to find a good solution. All RMS errors were un der 4 m m for the translational DOFs and under 2 degrees for all rotational DOFs (Table 3 2) The left foot required slightly larger kinematic changes compared to the right foot with the largest mean RMS errors of 3.8 mm for the right foot and 4.3 mm for th e left foot translational kinematics and 0.6 deg for the right foot and 2.0 deg for the left foot rotational kinematics. The average hindfoot marker distance RMS errors were ap proximately 5 mm for both feet.

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47 Discussion The goal of this study was to develo p a three dimensional subject specific deformable foot ground contact model that reproduces experimental ground reaction curves for both feet. Our subject specific foot ground contact model generated force, center of pressure, and free moment curves that c losely matched experimental curves for both feet. Only small variations to the translational and rotational kinematics were needed to achieve the desired results. With accurate modeling of foot ground interaction during gait, patient specific musculoskelet al models can predict new and It was challenging to develop a foot ground contact model that matched all six experimental ground reaction curves. Compared to curves reported in the literature [14, 33, 34, 36, 38] all three forces produced by our model for both feet did not exhibit spikes. The forces were smooth, even if some parts of the curves do not match the experimental da ta as well as others. However, it was particularly difficult to match the center of pressure and free moment curves. Because small vertical ground reaction forces yield large changes in the center of pressure location values it was problematic for the opt imizer to closely match center of pressure locations at the start and end of stance phase. Therefore, values were excluded from the calibration that corresponded to when the vertical ground reaction force was less than 100 N. This also helped to explain so me of the differences between the free moment curves. While the small scale makes the differences appear large, the same trends can be seen in the experimental and model free moment curves. This finding is important because research has shown that the free moment significantly affects arm swing during walking [39] and thus affects model estimates.

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48 The right foot ground reaction curves more closely matched the experimental data compared to the left foot curves. To achieve the se results for both feet, we had to tweak the cost function weights slightly when working with contralateral (left) foot data. While the shoe should have uniform properties, the foot does not. Therefore, each element had its own stiffness and damping value so that soft tissues could also be accounted for with this model. It is possible that higher ranges of values for those parameters led to greater deviations from the experimental curves, but it is important to note that they were not much higher than thos e for the right foot. This suggests that other combinations of viscoelastic parameter values may also produce promising results. In addition, it may be important to adjust damping values for numerical reasons when implementing this contact model into a pre dictive gait model. There were differences between calibrated parameters for each foot, particularly with the stiffness and damping parameters. This could be due to mechanical differences b etween force plates, as well as the discontinuous nature of the gr ound reaction curves for the left foot or even possible slight asymmetrie s on the part of the subject. We investigated mechanical differences in the force plates and found differences in the center of pressure path for all five walking trials (Fig. 3 5 ). In particular, there is a hooking pattern we see for the middle force plate that we do not see for the other force plates. This hooking pattern was also observed in data for other healthy subjects collected from the same lab. This could indicate a mechanic al difference between the experimental data we are trying to match for each foot, which could possibly lead to differences between the calibrated foot ground contact models of both feet. In addition, since the viscoelastic elements do not distinguish betwe en cushioning from the shoe

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49 and the soft tissues of the foot, the parameter differences could be accounting for The order of the optimization stages was important. We tried optimizing all six ground reaction compone nts starting from initial guesses, but the optimi zer was unable to find a solution that closely matched any of the components. Since the friction forces depend on the vertical ground reaction force and the center of pressure and free moment depend on all o f the forces, it follows that a good vertical ground reaction force curve helps the optimizer find a good solution. We were able to obtain a good solution by breaking the problem into stages that used the optimized parameters from the previous step. In ad dition, we tried using different optimization algorithms before choosing the non investigated during the calibration process development. A constrained non linear optimization al gorithm worked well for the vertical ground reaction force. However, that algorithm took much longer to find a less optimal solution, compared to the non linear least squares algorithm, when used for all three forces and was unable to find a feasible solut ion when the moments were included in the cost function. Some parameters were more important than others to find an optimal solution. To generate force, the spring damper elements need to penetrate the floor. This could be accomplished by pushing the who le body into the floor, leading to larger marker errors, or by creating an offset for each element within its respective foot segment frame that pushed the element into the floor. For uniformity, the same offset value was used for all elements. Similarly, we tried adjusting the height of the elements in the toes segment so

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50 that they varied along a parabolic surface to mimic the bottom of many shoes (Fig. 3 6 ) However, the slope adjustment of that parabola competed with the vertical offset value during the optimization and often led to a slope near zero. Therefore, we removed this parameter from the problem formulation and assumed the bottom of the shoe to be flat. However, this parameter could be more valuable for studies using shoes with a vertical distanc e between toe tip and ground that is greater than 1.5 cm. In addition to those parameters, we investigated methods for parameterizing kinematic curves. Instead of parameterizing those curves directly, we reduced the kinematic errors by parameterizing the a djustment curves, which were added to the original kinematic curves, with 25 Bspline coefficients. Bspline coefficients are determined for curves based on the degree of that curve and then are linearly combined to produce that curve. The number of coeffici ents, as well as the parameterization method, were determined from a sensitivity analysis that used a range of 20 to 30 coefficients and either Bspline or polynomial plus Fourier coefficients. While our model generated a good solution, our study was not w ithout limitations. Our method was tested on one subject with one pair of shoes, but we showed that the model could also predict accurate results for multiple trials excluded from calibration. In addition, our model does not explicitly account for soft tis sues, ligaments, or muscles that cushion the foot and aid in the coordinated effort of walking. The viscoelastic elements do not differentiate between soft tissues and shoe cushioning. We also simplified the model to two segments even though there are many bones in the foot that articulate and transmit forces during movement. The toes rotation axis was moved to the floor to eliminate any unrealistic gaps that could form when the toes flex during the

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51 transition between the flat foot and toe off portions of s tance phase. While our results were good for the chosen 5 by 11 grid of spring damper elements, further studies should investigate whether other grid densities could lead to improvements in the results and/or computation time. In addition, future work sho uld look at the sensitivity of optimized ground contact forces and kinematics to large changes in model parameter values, especially stiffness values, since average values and variations were quite different between the two feet, which is not expected sinc e physica lly they should be very similar The goal of this study was to develop a deformable subject specific foot ground contact model that reproduces experimental ground reaction curves. Our model successfully reproduced experimental curves for four tria ls excluded from model calibration. To use foot ground contact models for predictive optimizations, all six ground reaction components are required. Predictive gait optimization can be an important tool for developing rehabilitation strategies by predictin treatment outcome. This work lays the foundation for subject specific predictive gait optimizations that may lead to improvements in rehabilitation medicine.

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52 Figure 3 1. Viscoelastic element placement for the right foot. The heel, toe, and medial and lateral toe markers were used to define a uniform rectangular grid (5 x 11 markers. The elements are separated into active hindfoot, active toes, and inactive element s. The black line is the shoe outline and the gray line is the toes axis.

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53 Figure 3 2. Ground reaction components for both right (solid) and left (dashed) feet. The top row shows the anterior posterior (AP), superior inferior (Normal), and medial lateral (ML) force curve comparisons, respectively, and the bottom row shows the center of pressure (CoP) location in the AP and ML directions and the free moment curve comparisons for the experimental data (blue solid and shaded ) and model optimization (orange s olid and shaded).

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54 Figure 3 3 Comparison of right foot kinematic curves over time for one gait cycle excluded from calibration. The top row shows the hindfoot (HF) translational kinematics in units of meters. Since the rotation sequence was 3 1 2, the h indfoot rotational kinematic curves in units of degrees are presented in this order in row 2. The third row shows toe flexion in units of degrees.

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55 Figure 3 4. Comparison of left foot kinematic curves over time for one gait cycle excluded from calibrati on. The top row shows the hindfoot (HF) translational kinematics in units of meters. Since the rotation sequence was 3 1 2, the hindfoot rotational kinematic curves in units of degrees are presented in this order in row 2. The third row shows toe flexion i n units of degrees.

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56 Figure 3 5 Center of pressure paths for three force plates shown in blue for all five walking trials. The subject hit the first and last force plates with his left foot and the middle force plate with his right foot. The shape of the foot is superimposed (in green for the first force plate, black for the second force plates, and red for the third force plate) under this path to verify that the center of pressure is underneath the foot during movement. The hooking pattern observed for the middle force plate is highlighted in the orange circles.

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57 A B Figure 3 6. Differences between parabolic toes surfaces for (A) Adidas Samba sneakers used in this study and (B) other athletic shoes. The ve rtical heights of the elements in the toes segment were fitted along a parabolic surface determined from shoe measurements.

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58 Table 3 1. Calibrated parameters for the right and left feet. The maximum, mean, minimum, and standard deviations are reported fo r the spring stiffness ( ) and damping ( ) values, in units of N/m and m/s, respectively, as well as the coefficients of friction (unitless). Right Foot Left Foot min 796 590 mean 4112 6057 max 9993 37563 stdev 2193 6532 min 2.13E 06 2.10E 03 mean 1.09E 04 7.60E 03 max 2.24E 04 2.63E 02 stdev 2.82E 05 7.70E 03 static mu 8.96E 02 6.27E 02 dynamic mu 1.33E 01 1. 69E 01 viscous mu 9.30E 03 1.17E 02

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59 Table 3 2. RMS errors for ground reaction components and kinematics for both feet, where CoP is center of pressure. The RMS mean and standard deviations are calculated for five testing trials. The anterior posterior (AP) direction is forward, normal or superior inferior (SI) direction is up, and medial lateral (ML) direction is to the right. The RMS errors for the translational (Trans) and rotational (Rot) kinematics for the hindfoot (HF) and toes are also reported. Foot Trial AP Force (N) Normal Force (N) ML Force (N) AP CoP (mm) ML CoP (mm) Free Moment (Nm) HF AP Trans (mm) HF SI Trans (mm) HF ML Trans (mm) HF AP Rot (deg) HF SI Rot (deg) HF ML Rot (deg) Toe Flexio n (deg) Calibration 6.58 5.65 2.57 9.00 3 .40 1.25 3.60 2.70 1.50 0.74 0.46 0.44 0.24 Right Test Mean 8.86 7.23 6.61 7.96 5.14 1.24 3.52 2.66 1.70 0.21 0.35 0.50 0.30 Test St Dev 5.38 2.86 6.89 2.01 2.55 0.14 0.18 0.09 0.17 0.14 0.15 0.13 0.20 Calibration 4.93 7.40 3.51 5.40 3.70 1.55 3.80 2.70 1.80 1.74 0.79 0.63 1.96 Left Test Mean 7.78 9.82 4.43 5.22 3.96 1.66 3.94 2.74 1.94 1.27 0.74 0.60 1.44 Test St Dev 2.70 0.92 0.97 0.54 0.61 0.29 0.29 0.19 0.51 0.48 0.16 0.08 0.23

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60 CHAPTER 4 PREDICTIVE GAIT MODEL USING MOMENTUM VARIA TIONS Motiva tion No simple control law currently exists to explain how the central nervous system (CNS) makes walking efficient or even possible. Researchers in robotics have developed control strategies for bipedal gait through the regulation of whole body central an gular momentum, but there is little data to support the use of whole body linear momentum for similar purposes. Furthermore, it is currently unknown whether normal and pathological gait patterns can be characterized by variations in central angular and lin ear momentum. It is possible that new compensation strategies for pattern into a pattern closer to normal gait. Momentum variations may help explain pathological gait patter ns. Many researchers have reported that whole body angular momentum is highly regulated (i.e., close to zero) about all three spatial directions in walking [47 50] Since the active generation of angular momentum ma kes bipedal maneuverability and stability possible, some have hypothesized that angular momentum could serve as a simple c ontrol parameter for walking [49 ] Silverman and Neptune showed that angular momentum is regulated differently for amputees vs. non amputees [51] Simoneau and Krebs [52] looked at both angular and linear momentum components of elderly subjects and concluded that compared to non fallers, the momentum control in fallers appears to be impaired, thereby contributing to instability. If other patient populations also experience instability during locomotion (i.e. cerebral palsy or stroke), rehabilitation strategi es that lead to mo re highly regulat ed momentum values may be the most effective treatment

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61 options. If so, momentum may provide a simple means for identifying where to focus rehabilitation efforts. Preliminary results from our lab indicate that different locomotion tasks exh ibit clusters of momentum variation values. This finding suggests that an optimization approach that targets specified values while accounting for patient specific limitations may predict which rehabilitation strategies would be the most beneficial. Althou gh numerous studies show that angular momentum is regulated about the center of mass during walking no study has shown how regul ation of momentum (both central angular and linear) can be used to characterize different locomotion tasks. This novel approach will be valuable for people with disabilities resulting from either neurological disorders (i.e., cerebral palsy or stroke) or mobility impairments (i.e., dependence on a cane or crutches) by helping them to ach ieve more normal gait patterns. Given six va lues defining central angular and linear momentum variations over a gait cycle, the pro posed optimization approach seeks to predict a physically realistic gait motion for one subject Methods To develop predictive gait optimizations that target momentum va riation values, we combined the enhanced residual elimination algorithm (REA) framework and the foot ground contact model from previous chapters. The clusters of momentum variation values, determined from our preliminary work, for different gait tasks are referred to as momentum signatures. These values will be defined as design variables in our predictive gait optimizations that seek to produce dynamically consistent motion with adaptable foot ground contact.

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62 Experimental Data We collected experimental g ait data from a single healthy subject (male, age 46, height 1.7 m, weight 69 kg) who performed three gait tasks: walking marching, and running The IRB approved this study, and the subject gave informed consent prior to data collection A 14 camera Vicon /Peak motion capture system (Vicon Motion Systems, Inc., Lake Forest, CA) was used to measure s urfac e marker positions while three six axis Bertec force plates (Type 4060 08, Bertec Corp., Columbus, OH) were used to measure ground reaction forces and mome nts The marker set and data collection protocol were nearly identical to a previous study [9] Fo ur markers were placed on each foot (including a toe marker), three on each shank and thigh, three on the pelvis, four on the torso, and one on each elbow and wrist. Two static markers were used to determine where to separate the hindfoot and toes segments of each foot. For each gait task, w e collected five trials with three clean foot force plate strikes. Gait Model The gait model used in this study was developed by modifying an existing three dimensional full body dynamic skeletal model that possesses 14 segments and 27 degrees of freedom (DOFs) [9] In the existing model, the pelvis was connected to the ground through a six DOF joint while the hips were modeled as ball and socket joints, the knees as pin joints, the ankles as two non intersecting pin joints, the back as a ball and socket joint at approximately the L4 L5 level, the shoulders as univer sal joints, and the elbo ws as pin joints. The existing model was modified as follows: internal/external rotation of both shoulders was added to account for non sagittal arm motion, the back was split into two segments at the T8 T9 level to increase flexibi lity in the model, and each foot was split into two segments (hindfoot and toes) with the toes axes modeled as

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63 pin joints. The gait model now possesses 16 segments and 34 total DOFs. However, since we determined that the increased back flexibility did not lead to better results in Chapter 2, the second back joint was locked into place so the model was reduced to 31 DOFs. We incorporated the foot model used to calculate the deformation and deformation rates of each viscoelastic element into the three dimensi onal full body gait model so all calculations could be done seamlessly with one model. We derived the equ ations of motion for this model using Autolev symbolic manipulation software (Motion Genesis, Palo Alto, CA). In addition, we also investigated the REA methodology using the foot only model. Residual Elimination Algorithm ( REA ) The enhanced REA (see Chapter 2 for more details) was used to produce dynamically consistent motion in our predictive gait optimizations and will be briefly described here. The o riginal REA [16] produces a dynamically consistent walking motion (i.e., pelvis residual loads eliminated) by finding new initial conditions and generalized acceleratio ns while tracking experimentally measured marker positions. The pseudoinverse method is used to calculate small variations between model and experimental accelerations that are required to balance an underdetermined system of six whole body dynamics equati ons. These variations are used to update the model generalized accelerations, which are then numerically integrated. The entire numerical integration process is repeated using an optimization with the initial conditions as design variables until the experi mental and model marker position errors are minimized. To weights on the markers and specifically put a higher weight on the hindfoot markers, 2) allowed the coefficients th at parameterize the acceleration curves being tracked to vary

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64 as design variables in the optimization, 3) allowed DOFs to have different feedback gains, and 4) included certain lower body joint and inertial parameters to change as design variables in the o ptimization. For this work, only the initial conditions and acceleration curve parameterization coefficients were included as design variables. All other values were defined as those determined from the REA calibration trial. Foot Ground Contact Model The foot ground contact model (see Chapter 3 for more details) was used to calculate ground reactions for the predictive gait optimization and will be described here briefly. Our foot ground contact model was developed using a two segment (hindfoot and toes) dynamic foot model posse ssing 7 DOFs. Autolev symbolic manipulation software (OnLine Dynamics, Sunnyvale, CA) was used to derive the dynamics equations An inverse kinematics analysis was used to determine t he translations and rotations of the foot in the lab frame and toe flexion with respect t o the hindfoot during walking. The foot model was incorporated into a Matlab (The Mathworks, Natick, MA) (hindfoot or toes) which was based on the anterior center relative to the toes axis. The motion of the hindfoot and toes segments was used to determine the vertical deformation and deformation rate of each viscoelastic el ement throughout the gait cy cle to calculate the forces and moments in each direction. Momentum Signatures An existing three dimensional, 14 segment, 27 degree of freedom full body dynamical model [53 57] was modified to calculate central an gular and linear momentum quantities during various gait tasks. The equations of motion for the original model and the equations for whole body angular and linear momentum about the body

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65 symbolic manipulation software (OnLine Dynamics, Sunnyvale, CA). These momentum equations were incorporated into a Matlab program (The Mathworks, Natick, MA) to calculate whole body central angular and linear momentum for the three orthogonal directions of the lab oratory reference frame. The x direction pointed anteriorly, the y direction pointed superiorly, and the z direction pointed to the right. The calculations were performed for 5 trials per gait task (total of 15 trials), and results from the same tasks were averaged together. For each trial, the central angular momentum for each segment was calculated using ( 4 1) where is the angular momentum of the body due to the motion of the center of mass ( CoM) and is the angular momentum of the body due to its rotation about the CoM. is the angular momentum of the i th segment, is the inertia tensor of the segment about its CoM is the angular velocity, and are the position and velocity of the segment CoM relative to the whole body CoM, and is the mass of the segment. Simi larly, the linear momentum for each segment was calculated using ( 4 2) where is the linear momentum of the i th segment, is the mass of the segment, and is the velocity of the segment CoM. The whole body central angular momentum and the whole body linear momentum were calculated as a summation of the segmental angular and linear momenta, respectively.

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66 Results Predictive Gait Optimizatio n The REA gait model with adaptable foot ground contact was unable to reproduce experimental ground reaction data. To determine if this method is viable, the REA methodology was investigated using a 7 DOF right foot only model, which is a much simpler mode l compared to the 31 DOF full body model. The same outcome was observed. Without REA, a forward dynamics simulation that used damping to overcome any failures during repeated numerical integration could mostly reproduce reasonable ground reactions (Fig. 4 1). However, the same model using REA to find dynamically consistent motion that reproduces ground reactions was either unable to find a solution or was unable to reproduce the ground reactions (Fig. 4 2). Momentum Signatures Based on our preliminary work on momentum regula tion, w e found that all three normalized central angular and linear momentum components varied about a constant value and these values were small (Table 4 1) While five normalized momentum components varied about a constant value of zero the normalized anterior component of linear momentum varied about a constant value of one (Fig. 4 3) Thus, all six momentum components may be beneficial for characterizing CNS control of human locomotion from a high level perspective. When root mean squ are (RMS) variations in the six momentum components about their mean values were plotted for the three different tasks, we found that the variations for each task clustered toge ther (Fig. 4 4 ). e it possible to delineate between walking, marching, and running based solely on whole body momentum variations.

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67 Two important observations can be made from these preliminary results. First, they provide a basis for characterizing gait patterns based on variations in central angular and linear momentum. If these values are clustered for various locomotion tasks, then appropriate ranges of these values can be determined that will define normal walking patterns. Second, if these momentum signatures cluster differently for different pathological gait patterns, then these quantities could be used to determine how changes in gait patterns affect these values and vice versa. It is possible that due to variability in the measurements, the concept of momentum sign atures may not hold for additional subjects so further analysis is needed to validate this concept. Discussion The purpose of this study was to predict new motion using REA with a foot ground contact model to optimize whole body linear and central angular momentum variation values about a constant value. This purpose was dependent upon the hypothesis that REA with a foot ground contact model is a viable option for predicting new motion. After much investigation, this method is not supported REA with foot g round contact was prone to numerical integration failures because REA adjusts the kinematics for dynamic consistency and small changes in kinematics lead to large changes in ground reactions. These kinematic tweaks necessary for dynamic consistency prevent ed the optimizer from finding kinematic curve coefficients and initial conditions that produced accurate ground reactions, which often led to numerical integration failures or results that did not match the experimental data. The steps we took to eliminate numerical integration issues were: 1) redesigning our foot ground contact model so that experimental ground reactions were reproduced without much change in the foot kinematics; 2) eliminating interfacing issues between the full body gait and foot only mo dels; 3) calculating initial

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68 conditions for kinematic curves that closely reproduced the experimental vertical ground reaction forces (vGRFs); and 4) testing this method using both the full body gait and foot only models with and without additional damping terms to eliminate numerical integration issues. Therefore, while momentum signatures remain a plausible variable to use for predicting new motions, this method for testing that idea did allow it to come to fruition. In the early stages of our analysis, we thought the numerical integration failures were caused by an inaccurate foot ground contact model. The complete foot ground contact model developed and used in this study has evolved from earlier versions of the foot ground contact model that were test ed solely on the right foot and only over stance phase. Since we did not verify that the model would produce zero force and moment values during swing phase (i.e., the foot was lifted off the ground), it was possible that the kinematics used to reproduce e xperimental ground reactions would not be realistic during swing phase. In addition, trying to splice together the calculated kinematics from stance phase with the swing phase portions of experimental kinematic curves left discontinuities that could lead t o numerical integration failures. The model was then updated to calculate ground reactions and optimized kinematics over the entire gait cycle. However, the kinematics required to reproduce the experimental ground reaction data deviated too much from the e xperimental kinematics. That model left doubt as to its viability within an REA framework when the kinematics needed more than slight tweaking to closely reproduce experimental ground reactions. The foot ground contact model described in Chapter 3 does not require much change to the kinematics to closely reproduce ground reactions so this source of failure was eliminated.

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69 Problems resulting from the use of separate full body gait and foot only Autolev models were eventually eliminated. During our investig ation, we realized that the origin of the hindfoot segment was different from the point we thought we chose meaning that calculated marker parameters defined relative to the hindfoot segment origin were being applied to a different point This finding was discovered when trying to match hindfoot markers using both models. To eliminate any doubts about interfacing two separate models, we were able to combine them to be one seamless model that can derive dynamics equations as well as calculate ground reactio ns using the two non orthogonal ankle DOFs instead of trying to convert those kinematics into kinematics in orthogonal laboratory reference frame directions for an additional set of calculations. This eliminated an extra step during the optimization proces s, and thus sped up calculations although it did not eliminate numerical integration problems. We investigated initial conditions as possible sources of error. We modified the model by eliminating gravity (i.e., setting it to zero) and replacing the vert ical ground reaction force from both feet to the origin of the pelvis (i.e., the midpoint between the two static ASIS markers). Then a static solver was used to settle the model onto the ground by allowing the vertical translation of the pelvis to move wit h the condition that the acceleration in the vertical direction fall below 1e 3 m/s 2 This was done for each time frame. However, this did not guarantee that the distribution between left and right components was correct (Fig. 4 5). To distribute the force s properly, we first tried to also allow the rotation about the anterior posterior direction to change for double support time frames, but this gave the model too much freedom and it repeatedly fell over. To remedy this, we performed an optimization that a llowed the ankle kinematics to change

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70 while holding the vertical translation determined from the static model constant to match the experimental vGRF curve. A separate optimization was performed for each foot. With that, we were able to start out with extr emely good initial conditions that reproduced the vGRF, which is what all of the other forces, and subsequently moments, are based on. However, even with good initial conditions, we still had numerical integration failures. We investigated REA with foot ground contact using both full body gait and foot only models. The foot only model has a small mass and low inertia so we thought that it might be too stiff. We tried running an optimization that had a high mass and inertia for the foot to see if that fixe d numerical integration problems. While the optimization was able to complete one iteration, a second iteration never completed but the optimization did not fail. In case the foot only model was too stiff, we also investigated the full body gait model that 1) allowed all 31 DOFs to change and 2) allowed only the 6 pelvis DOFs to change. Remy and Thelen [16] found that results were similar for REA without a foot ground co ntact model that allowed 29 DOFs to change compared to 6 pelvis DOFs. We thought we would have similar success. Spreading the total error over multiple DOFs seemed promising, but we continued to have numerical integrator issues. Finally when our efforts we re exhausted for getting the full body model to work within an REA framework, whether it was because of numerical integrat ion issues or because the optim i z er either got stuck or could not find a solution, we tried using a right foot only model wit h 7 DOFs. The idea was that if we could not get a simpler 7 DOF model to wo rk, then it was un likely that we could get a more complicated m odel working. To circumvent numerical integration issues, additional damping was included in the foot

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71 only model. While these a dditional damping terms eliminated numerical integrator issues, the method of REA with a foot ground contact model was unable to produce any solution that was close to realistic (Fig. 4 2). In addition, using damping negates the purpose of REA, which is dy namic consistency. However, it did not work. Despite all of our attempts and approaches to this problem, we were unable to predict realistic motion using REA with a foot ground contact model. The positive outcomes from this study were the concept of mome ntum signatures to quantify motion using only six values, as well as a foot ground contact model that closely reproduces experimental ground reactions for both feet without much modification to the hindfoot and toe flexion kinematics. We still believe that predictive gait models using momentum signatures as targets in an optimization may lead to personalized rehabilitation strategies, but REA with a foot ground contact model is not a viable method to carry out that goal. Other methods that perform numerical integration implicitly may yield better results with this foot ground contact model.

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72 Figure 4 1. Ground reaction curves without REA. Ground reaction curves resulting from a forward dynamics simulation on a right foot only model that includes damping but does not include REA. Experimental curves are in blue (solid), and calculated curves are in orange (dashed). The top row is forces and the bottom row is moments.

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73 Figure 4 2. Ground reaction curves with REA. Ground reaction curves resulting from a forward dynamics simulation on a right foot only model that includes damping and REA. Experimental curves are in blue (solid), and calculated curves are in orange (dashed). The top row is forces and the bottom row is moments.

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74 Figure 4 3. Momentum curv es for gait tasks. Whole body central angular (left) and linear (right) momentum for walking (blue), running (red), and marching (black). The x direction points anteriorly, the y direction points superiorly, and the z direction points to the right. Momentu m values are normalized using body mass M, average speed V, and body height H.

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75 Figure 4 4. body central angular (left) and linear (right) momentum for walking, running, and marching. The x direction points anteriorly, the y direction points superiorly, and the z direction points to the right. Momentum values are normalized using body mass, average speed, and bod y height

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76 Figure 4 5. Vertical ground reaction force curves for both feet resulting from a static optimi zation of a full body model that only allowed the vertical translation of the pelvis to vary while minimizing the vertical acceleration. Experimental curves are in blue (solid), and calculated model curves are in orange (dashed).

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77 Table 4 1. RMS variation s about the mean of normalized central angular and linear momentum for each of five trials from the three tasks studied. Angular Momentum Linear Momentum Task x y z x y z Gait 0.0096 0.0064 0.0076 0.0519 0.1288 0.0293 Running 0.0074 0.0025 0.0075 0.0154 0.1684 0.0167 Marching 0.0142 0.0054 0.0184 0.0566 0.2604 0.0336

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78 CHAPTER 5 CONCLUSIONS The main objectives of this work were to: 1) Eliminate the pelvis residual loads and improve foot marker tracking by enhancing the residual elimination algor ithm through marker weight, tracked acceleration curve, feedback gain, and select model joint and inertial parameter adjustments; 2) Develop a foot g round contact model that matches all three forces, center of pressure location, and free moment for both fe et using physics to model the foot ground interactions; 3) Demonstrate that whole body momentum variations for gait tasks cluster differently from one another and that these clusters can ) Develop an optimization methodology to predict different subject specific gait patterns using a subject specific computational model that matches a specified momentum signature. We evaluated the following enhancements to the original REA: 1) manual modif ication of tracked marker weights, 2) automatic modification of tracked joint acceleration curves, 3) automatic modification of algorithm feedback gains, and 4) automatic calibration of model joint and inertial parameter values. The evaluation was performe d using a three dimensional full body skeletal model and movement data collected from a subject who performed four distinct gait patterns: walking, marching, running, and bounding. When all four enhancements were implemented together, the enhanced REA achi eved dynamic consistency with lower marker tracking errors for all segments, especially the feet, compared to the original REA. When each enhancement was implemented separately, the most important one was automatic modification of tracked joint acceleratio n curves, while the least important enhancement was automatic modification of algorithm feedback gains. The pelvis residual forces and torques were

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7 9 all below 1e 12 N and 1e 12 Nm, respectively. Thus, our enhanced REA framework successfully eliminated pelvi s residual loads and improved foot marker tracking. Our deformable foot ground contact model was able to closely reproduce a ll six experimental ground reaction curves for both feet and for all trials. However, the center of pressure location in the forwar d direction exhibited the largest changes at the beginning and end of stance phase when the normal forces were small. Only minor deviations to the kinematic curves were needed to achieve these results. The spring damper element parameters (stiffness and da mping) were greater than zero and the coefficients of friction were within realistic ranges. The overall goal of this work was to predict new motion using REA with a deformable foot ground contact model to optimize whole body linear and central angular mo mentum variation values using specified momentum signatures. This purpose was dependent upon the hypothesis that REA with a foot ground contact model is a viable option for predicting new motion. After much investigation, this method is not supported Whil e REA and the foot ground contact model worked well separately, the combination of REA with foot ground contact was prone to numerical integration failures because REA adjusts the kinematics for dynamic consistency and small changes in kinematics lead to l arge changes in ground reactions. Therefore, while momentum signatures remain a plausible target to use in predicting new motions, this method for testing that idea did allow it to come to fruition. Sometimes how not to do something can be valuable for ot hers trying to achieve the same goal but unfortunately most journals will not publish negative results We are trying to help get models to the next step of predicting new motions, and the REA model

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80 with adaptable foot ground contact is not the way to acc omplish that. However, this work provides ma ny contributions to the biomechanics modeling field: 1. I showed that momentum variations are task dependent. Because of this, predictive gait models using momentum signatures as targets in an optimization may st ill lead to personalized rehabilitation strategies. 2. While it works without a deformable foot ground contact model, the enhanced REA is subject specific so it can still be used to investigate an not change. 3 And last, I developed a foot ground contact model that closely reproduces all six components of experimental ground reactions for both feet without much modification to the hindfoot and toe flexion kinematics. This can be used in other mode ls that require foot ground biomechanics modeling community, which is to accurately predict new motions for any subject.

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81 APPENDIX A COST FUNCTION DETAIL S FOR FOOT GROUND CO NTACT MODEL CALIBRAT ION AND TESTING

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91 LIST OF REFERENCES [1] Murphy, L., and Helmick, C. G., 2012, "The impact of osteoarthritis in the United States: a population health perspective: A population based re view of the fourth most common cause of hospitalization in U.S. adults," Orthop Nurs 31(2), pp. 85 91. [2] Perry, J., Garrett, M., Gronley, J. K., and Mulroy, S. J., 1995, "Classification of walking handicap in the stroke population," Stroke 26(6), pp. 982 989. [3] Davie, C., 2008, "A review of Parkinson's disease," British Medical Bulletin 86(1), pp. 109 127. [4] Center for Disease Control and Prevention 2009, "Prevalence and Most Common Causes of Disability Among Adults -United States, 2005," pp. 421 426. [5] Ostir, G. V., Berges, I. M., Kuo, Y. F., Goodwin, J. S., Fisher, S. R., and Guralnik, J. M., 2013, "Mobility activity and its value as a prognostic indicator of survival in hospitalized older adults," J Am Geriatr Soc 61(4), pp. 551 557. [ 6] Mutikainen, S., Rantanen, T., Aln, M., Kauppinen, M., Karjalainen, J., Kaprio, J., and Kujala, U. M., 2011, "Walking ability and all cause mortality in older women," Int J Sports Med 32(3), pp. 216 222. [7] Blair, S. N., Kohl, H. W., Paffenbarger, R. S., Clark, D. G., Cooper, K. H., and Gibbons, L. W., 1989, "Physical fitness and all cause mortality. A prospective study of healthy men and women," JAMA 262(17), pp. 2395 2401. [8] Bogey, R., and Hornby, G. T., 2007, "Gait training strategies utilized in poststroke rehabilitation: are we really making a difference?," Top Stroke Rehabil 14(6), pp. 1 8. [9] Fregly, B. J., Reinbolt, J. A., Rooney, K. L., Mitchell, K. H., and Chmielewski, T. L., 2007, "Design of Patient Specific Gait Modifications for Kne e Osteoarthritis Rehabilitation," IEEE Trans Biomed Eng 54 (9), pp. 1687 1695. [10] Fregly, B. J., Reinbolt, J. A., and Chmielewski, T. L., 2008, "Evaluation of a patient specific cost function to predict the influence of foot path on the knee adduction t orque during gait," Com p Methods Biomech Biomed Eng 11(1), pp. 63 71. [11] Neptune, R. R., Wright, I. C., and Van Den Bogert, A. J., 2000, "A Method for Numerical Simulation of Single Limb Ground Contact Events: Applicat ion to Heel Toe Running," Comp Met hods Biomech Biomed Eng 3(4), pp. 321 334. [12] Piazza, S. J., and Delp, S. L., 1996, "The influence of muscles on knee flexion during the swing phase of gait," J Biomech 29(6), pp. 723 733.

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92 [13] Mahboobin, A., Cham, R., and Piazza, S. J., 2010, "The i mpact of a systematic reduction in shoe floor friction on heel contact walking kinematics -A gait simulation approach," J Biomech 43(8), pp. 1532 1539. [14] Anderson, F. C., and Pandy, M. G., 2001, "Dynamic optimization of human walking," J Biomech Eng 123(5), pp. 381 390. [15] Miller, R. H., Brandon, S. C., and Deluzio, K. J., 2013, "Predicting sagittal plane biomechanics that minimize the axial knee joint contact force during walking," J Biomech Eng 135(1), p. 011007. [16] Remy, C. D., and Thelen, D. G., 2009, "Optimal Estimation of Dynamically Consistent Kinematics and Kinetics for Forward Dynamic Simulation of Gait," J Biomech Eng 131 (3), p. 031005. [17] Sderkvist, I., and Wedin, P. A., 1993, "Determining the Movements of the Skeleton Us ing Wel l Configured Markers," J Biomech 26 (12), pp. 1473 1477. [18] Cahout, V., Luc, M., and David, A., 2002, "Static Optimal Estimation of Joint Accelerations for Inverse Dynamic s Problem Solution," J Biomech 35 (11), pp. 1507 1513. [19] Nagurka, M. L., and Yen, V., 1990, "Fourier Based Optimal Control o f Nonlinear Dynamic Systems," J Dyn Syst Meas Control 112 (1), pp. 17 26. [20] Greenwood, D. T., 1988, Principles of Dynamics (2nd ed.), Prentice Hall, Inc., Upper Saddle River, NJ. [21] de Leva, P., 1996, Adjustments to Zatsiorsky Seluyanov's Segment Inertia Parameters," J Biomech 29 (9), pp. 1223 1230. [22] Kidder, S. M., Abuzzahab, F. S., Harris, G. F., and Johnson, J. E., 1996, "A system for the analysis of foot and ankle kinematics during gait," IEEE T rans Rehabil Eng 4(1), pp. 25 32. [23] Sasaki, K., and Neptune, R. R., 2006, "Differences in muscle function during walking and running at the same speed," J Biomech 39(11), pp. 2005 2013. [24] Dorn, T. W., Lin, Y. C., and Pandy, M. G., 2012, "Estimate s of muscle function in human gait depend on how foot gro und contact is modelled," Comp Methods Biomech Biomed Eng 15(6), pp. 657 668. [25] Gerus, P., Sartori, M., Besier, T. F., Fregly, B. J., Delp, S. L., Banks, S. A., Pandy, M. G., D'Lima, D. D., and Lloyd, D. G., 2013, "Subject specific knee joint geometry improves predictions of medial tibiofemoral contact forces," J Biomech 46(16), pp. 2778 86.

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94 [38] Remy, C. D., 2006, "Integration of an Adaptive Ground Contact Model Into the Dynamic Simulation of Gait," Master's Thesis, University of Wisconsin Madison USA. [39] Hamner, S. R., Seth, A., Steele, K. M., and Delp, S. L., 2013, "A rolling constraint reproduces ground reaction forces and mom ents in dynamic simulations of walking, running, and crouch gait," J Biomech 46(10), pp. 1772 1776. [40] Cole, G. K., Nigg, B. M., van Den Bogert, A. J., and Gerritsen, K. G., 1996, "The clinical biomechanics award paper 1995 Lower extremity joint loadin g during impact in running," Clin Biomech (Bristol, Avon), 11(4), pp. 181 193. [41] Gerritsen, K. G., van den Bogert, A. J., and Nigg, B. M., 1995, "Direct dynamics simulation of the impact phase in heel toe running," J Biomech 28(6), pp. 661 668. [42] Wilson, C., King, M. A., and Yeadon, M. R., 2006, "Determination of subject specific model parameters for visco elastic elements," J Biomech 39(10), pp. 1883 1890. [43] Holden, J. P., and Cavanagh, P. R., 1991, "The free moment of ground reaction in dist ance running and its changes with pronation," J Biomech 24(10), pp. 887 897. [44] Almosnino, S., Kajaks, T., and Costigan, P. A., 2009, "The free moment in walking and its change with foot rotation angle," Sports Med Arthrosc Rehabil Ther Technol 1(1), p. 19. [45] Hunt, K. H., and Crossley, F.R.E., 1975, "Coefficient of restitution interpreted as dam ping in vibroimpact," J App Mech 42, pp. 440 445. [46] Sherman, M. A., Seth, A., and Delp, S. L., 2011, "Simbody: multibody dynamics for biomedical resear ch," Procedia IUTAM 2, pp. pp. 241 261. [47] Gu, J. W., 2003, "The regulation of angular momentum during human walking," Undergraduate thesis, MIT, USA. [48] Popovic, M., Gofmann, A., and Herr, H., 2004, "Angular momentum regulation during human walking : biomechanics and control," In Proceedings of the IEEE International Conference on Robotics and Automation New Orleans, LA, USA, pp. 2405 2411. [49] Herr, H., and Popovic, M., 2008, "Angular momentum in human walking.," J Exp Biol 211(Pt 4), pp. 467 48 1. [50] Bennett, B. C., Russell, S. D., Sheth, P., and Abel, M. F., 2010, "Angular momentum of walking at different speeds.," Hum Mov Sci 29(1), pp. 114 124.

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95 [51] Silverman, A. K., and Neptune, R. R., 2011, "Differences in whole body angular momentum bet ween below knee amputees and non amputees across walking speeds," J Biomech 44(3), pp. 379 385. [52] Simoneau, G. a. K., D., 2000, "Whole body momentum during gait: apreliminary study of non fallers and frequent fallers," J App Biomech 16, pp. 1 13. [ 53] Sawyer, W. G., Hamilton, M.A., Fregly, B.J., and Banks, S.A., 2003, "Temperature modeling in a total knee joint replacement using patient specific kinematics," Tribology Letters 15, pp. 343 351. [54] Blankevoort, L., Kuiper, J. H., Huiskes, R., and G rootenboer, H. J., 1991, "Articular contact in a three dimensional model of the knee.," J Biomech 24(11), pp. 1019 1031. [55] Fregly, B. J., Rahman, H. A., and Banks, S. A., 2005, "Theoretical accuracy of model based shape matching for measuring natural knee kinematics with single plane fluoroscopy.," J Biomech Eng 127(4), pp. 692 699. [56] Moro oka, T. A., Hamai, S., Miura, H., Shimoto, T., Higaki, H., Fregly, B. J., Iwamoto, Y., and Banks, S. A., 2007, "Can magnetic resonance imaging derived bone mode ls be used for accurate motion measurement with single plane three dimensional shape registration?," J Orthop Res 25(7), pp. 867 872. [57] Moro oka, T. A., Hamai, S., Miura, H., Shimoto, T., Higaki, H., Fregly, B. J., Iwamoto, Y., and Banks, S. A., 2008, "Dynamic activity dependence of in vivo normal knee kinematics.," J Orthop Res 26(4), pp. 428 434.

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96 BIOGRAPHICAL SKETCH Jennifer Jackson graduated from the University of Central Florida with a bachelor's degree in aerospace engineering in 2003 with ho nors. She earned her master's degree in mechanical engineering in 2005 under the advisement of Dr. Ted Conway. Her estimator of post mortem interval was featured on the Disc overy Channel in 2003. In 2005, Ms. Jackson was awarded a graduate research fellowship (GRFP) from the National Science Foundation, which brought her to the Computational Biomechanics Lab under the direction of Dr. B.J. Fregly at the University of Florida For her dissertation, she develop ed a deformable foot ground contact model and a (i.e., clustering of whole body momentum variations) to facilitate identification of rehabi litation st rategies for pathological gait. In 2013, Ms. Jackson was hired by the Rehabilitation Medicine Department at the National Institutes of Health where she continues to study biomechanics of the human body.


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