Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Kimanzi, Stephen M.
Doctor of Philosophy - PhD
University of British Columbia
Tremors represent one of the most prevalent types of movement disorders, often leading to significant functional impairments. With the expanding landscape of research and development in addressing neurodegenerative movement disorders like Parkinson’s Disease, there is a growing focus on devising effective solutions. While existing medical interventions, including surgical procedures like Deep Brain Stimulation and medications such as Levodopa, have shown varying degrees of success, there is increasing interest in exploring alternative approaches. One such approach involves mitigating tremors by suppressing their manifestation in specific areas of the body, particularly the palm. The aim of this research is to conceptualize, design, and assess a tremor suppression device capable of attenuating tremors across multiple degrees of freedom (DOFs). This is achieved by framing tremors as vibrations and employing vibration absorber design techniques to develop a passive, non-linear multi-degree-of-freedom (MDOF) attenuator. Through dynamic modeling of the human arm complex at the musculoskeletal level, coupled with iterative refinement of the device through rigorous experimental evaluation, the final prototype demonstrates significant efficacy in reducing tremors across multiple DOFs. Specifically targeting the most troublesome tremor DOFs of pronation-supination and flexion-extension, the device achieves reductions exceeding 70%. Patient outcomes corroborate the findings of the experimental evaluation, with a notable two-thirds retention rate observed among nearly 1000 units deployed. These results underscore the promising potential of the developed device in mitigating the debilitating impacts of neurological movement disorders. Furthermore, they lay the groundwork for the advancement of assistive devices aimed at improving the quality of life for individuals affected by such conditions.
Modeling, Design, and Evaluation of aNovel Passive Hand Tremor Attenuatorfor Broadband SuppressionbyStephen M. KimanziB.A.Sc., The University of British Columbia, 2016M.A.Sc., The University of British Columbia, 2018A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2024© Stephen M. Kimanzi, 2024The following individuals certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, a dissertation entitled:Modeling, Design, and Evaluation of a Novel Passive HandTremor Attenuator for Broadband Suppressionsubmitted by Stephen M. Kimanzi in partial fulfilment of the requirementsof the degree of Doctor of Philosophy.Dr. Hadi Mohammadi, School of EngineeringSupervisorDr. Zheng Liu, School of EngineeringSupervisory Committee MemberDr. Ray Taheri, School of EngineeringSupervisory Committee MemberDr. Sumi Siddiqua, School of EngineeringUniversity ExaminerDr. Shahram Payandeh, Simon Fraser UniversityExternal ExamineriiAbstractTremors represent one of the most prevalent types of movement disorders,often leading to significant functional impairments. With the expandinglandscape of research and development in addressing neurodegenerativemovement disorders like Parkinson’s Disease, there is a growing focus ondevising effective solutions. While existing medical interventions, includingsurgical procedures like Deep Brain Stimulation and medications such asLevodopa, have shown varying degrees of success, there is increasing interestin exploring alternative approaches. One such approach involves mitigatingtremors by suppressing their manifestation in specific areas of the body,particularly the palm. The aim of this research is to conceptualize, design,and assess a tremor suppression device capable of attenuating tremors acrossmultiple degrees of freedom (DOFs). This is achieved by framing tremors asvibrations and employing vibration absorber design techniques to develop apassive, non-linear multi-degree-of-freedom (MDOF) attenuator. Throughdynamic modeling of the human arm complex at the musculoskeletal level,coupled with iterative refinement of the device through rigorous experimentalevaluation, the final prototype demonstrates significant efficacy in reducingtremors across multiple DOFs. Specifically targeting the most troublesometremor DOFs of pronation-supination and flexion-extension, the deviceachieves reductions exceeding 70%. Patient outcomes corroborate the findingsof the experimental evaluation, with a notable two-thirds retention rateobserved among nearly 1000 units deployed. These results underscore thepromising potential of the developed device in mitigating the debilitatingimpacts of neurological movement disorders. Furthermore, they lay thegroundwork for the advancement of assistive devices aimed at improving thequality of life for individuals affected by such conditions.iiiLay SummaryThe research presented here investigates into the details of designing andoptimizing a novel solution aimed at alleviating the challenges encounteredby individuals affected by tremor-related conditions, notably Parkinson’sand Essential tremor diseases. Adopting a perspective rooted in vibrationengineering, the methodologies and techniques employed follow establishedprinciples in mechanical vibration to effectively absorb and mitigate thesetremors. The conclusion of these efforts yields a ground-breaking devicecapable of significantly reducing tremors, achieving success rates exceeding70% in certain cases. Beyond addressing the immediate needs of tremorpatients, this work holds broader implications for the field of vibrationabsorption. By systematically introducing a novel approach to tacklinglow-frequency, high-amplitude vibrations, it contributes to the advancementof knowledge and techniques in this area. In essence, this research not onlyoffers tangible solutions to improve the quality of life for individuals grapplingwith tremor-related conditions but also expands the horizons of vibrationengineering and neurological tremor mitigation, paving the way for futureinnovations and advancements in the field.ivPrefaceThis thesis represents my research accomplishments at The University ofBritish Columbia, completed under the guidance of my supervisor, Dr. HadiMohammadi, and the valued input of the supervisory committee members,Dr.Zheng Liu and Dr. Ray Taheri. Herein, I detail the contributions of allcollaborators and co-authors to the publications and material submitted forpublication.∗ I was responsible for the identification and design of the researchprogram, conceptualizing the main research questions and objectivesin close consultation with my supervisor and committee members.∗ I am the author of the manuscripts and some of the work presentedin this thesis has been submitted for publication under the followingtitles:1. Parametric Evaluation and Optimization of a Novel See-SawActuator for Tremor Attenuation, published in Journal of MedicalEngineering and Technology, related to Chapter 5.1.2. Dynamic Modeling of Human Arms for the Attenuation of HandTremor, submitted to Journal of Medical Engineering and Technology,related to Chapter 9.1.∗ My industry partner Steadiwear Inc was responsible for all ethicsapprovals for research involving human participants.∗ Generative Artificial Intelligence was not used for any aspects of thisresearch.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxivChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background on Tremors . . . . . . . . . . . . . . . . . . . . . 41.2 Problem Statement and Research Objectives . . . . . . . . . 71.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . 71.2.2 Research Objectives . . . . . . . . . . . . . . . . . . . 71.3 Scope and Thesis Outline . . . . . . . . . . . . . . . . . . . . 81.3.1 Project Scope . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Interdisciplinary Context and Scope of Author’s Expertise 91.3.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . 9Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . 112.1 Parkinson’s Disease . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Pathophisiology of Parkinson’s Disease . . . . . . . . . 122.1.2 Clinical Diagnosis and Manifestations . . . . . . . . . 14viTABLE OF CONTENTS2.1.3 Medical Therapies for Parkinson’s Disease . . . . . . . 152.2 Essential Tremor Disease . . . . . . . . . . . . . . . . . . . . . 172.3 Alternative Therapies for Tremor Mitigation . . . . . . . . . . 172.3.1 Wearable Tremor Suppression Technologies . . . . . . 182.3.2 Brief overview on anatomy of the upper limb . . . . . 192.3.3 Exoskeleton Devices . . . . . . . . . . . . . . . . . . . 242.3.4 Functional Electrical Stimulation . . . . . . . . . . . . 262.3.5 Afferent Stimulation . . . . . . . . . . . . . . . . . . . 292.4 Quantification and Assessment of Tremor . . . . . . . . . . . 312.5 Tremors as Vibrations . . . . . . . . . . . . . . . . . . . . . . 322.6 Effective Exoskeleton WTSD Design . . . . . . . . . . . . . . 33Chapter 3: Theoretical Framework . . . . . . . . . . . . . . . . 363.1 Simple Arm Models . . . . . . . . . . . . . . . . . . . . . . . 373.1.1 Flexion-Extension Lumped Parameter Model . . . . . 393.1.2 Forearm Pronation-Suppination Model . . . . . . . . . 463.2 Inertial Vibration Absorber . . . . . . . . . . . . . . . . . . . 49Chapter 4: Conceptual Designs . . . . . . . . . . . . . . . . . . 554.1 Design A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Design B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Design C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Chapter 5: Detailed Mathematical Modeling . . . . . . . . . . 705.1 Design B: The T-Beam Absorber . . . . . . . . . . . . . . . . 715.1.1 Performance Analysis . . . . . . . . . . . . . . . . . . 785.1.2 Parametric Evaluation . . . . . . . . . . . . . . . . . . 785.1.3 Optimizing the T-Beam for a broader bandwidth . . . 865.1.4 Evaluating the T-Beam in FPS . . . . . . . . . . . . . 915.1.5 Exploring the T-Beam for MDOF Tremors . . . . . . 955.2 Non-Linear Vibration Absorption . . . . . . . . . . . . . . . . 1075.2.1 A simple Non-Linear Absorber . . . . . . . . . . . . . 1095.3 Non-Linear T-Beam . . . . . . . . . . . . . . . . . . . . . . . 1175.3.1 Modeling the T-Beam Magnetic Springs . . . . . . . . 1175.3.2 Modeling the Non-Linear T-Beam . . . . . . . . . . . 1295.3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . 1315.4 Non-Linear MDOF Device . . . . . . . . . . . . . . . . . . . . 1425.4.1 Selecting the Design Iteration . . . . . . . . . . . . . . 1435.4.2 Mathematical Model of Design C . . . . . . . . . . . . 1445.4.3 Numerical Simulation of the MDOF Design . . . . . . 145viiTABLE OF CONTENTSChapter 6: Prototyping and Implementation . . . . . . . . . . 1516.1 Iterative Prototyping Phase . . . . . . . . . . . . . . . . . . . 1526.1.1 Outer Cover . . . . . . . . . . . . . . . . . . . . . . . . 1536.1.2 Height Adjustment Mechanism . . . . . . . . . . . . . 1546.2 Polymagnetic Springs . . . . . . . . . . . . . . . . . . . . . . 156Chapter 7: Model Validation and Experimental Evaluation . 1607.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 1617.1.1 Multi-DOF Variable Parameter Tremor Inducer . . . . 1617.1.2 FPS Tremor Test Platform . . . . . . . . . . . . . . . 1657.1.3 Instrumentation and Data Processing . . . . . . . . . 1687.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . 1757.2.1 Validation Process . . . . . . . . . . . . . . . . . . . . 1757.2.2 Tremor Response Analysis . . . . . . . . . . . . . . . . 1767.2.3 Frequency Domain Comparison . . . . . . . . . . . . . 1827.2.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . 1847.2.5 Model Refinements . . . . . . . . . . . . . . . . . . . . 1857.3 In-Vitro Evaluation of Absorber Prototypes . . . . . . . . . . 1867.3.1 T-Beam Experimental Evaluation . . . . . . . . . . . . 1867.3.2 Steadi-Two Experimental Evaluation . . . . . . . . . . 190Chapter 8: Patient Tests and Real-World Applications . . . . 2188.1 Patient On-boarding and Data Collection . . . . . . . . . . . 2198.2 Phase I Evaluations . . . . . . . . . . . . . . . . . . . . . . . 2228.2.1 Patient Characterization and Evaluation . . . . . . . . 2228.2.2 Overall Analysis for Phase I . . . . . . . . . . . . . . . 2318.3 Phase II Evaluations . . . . . . . . . . . . . . . . . . . . . . . 2358.3.1 Clinical Investigation Protocol . . . . . . . . . . . . . 2358.3.2 Data Collection Procedure . . . . . . . . . . . . . . . . 2398.3.3 Phase II Data Analysis . . . . . . . . . . . . . . . . . 2408.4 Quality of Life Evaluations . . . . . . . . . . . . . . . . . . . 256Chapter 9: Discussion . . . . . . . . . . . . . . . . . . . . . . . . 2599.1 Biofidelic Models of the human arm . . . . . . . . . . . . . . 2609.1.1 Flexion-Extension . . . . . . . . . . . . . . . . . . . . 2609.1.2 Pronation-Supination . . . . . . . . . . . . . . . . . . 2709.1.3 Combined DOF Analysis . . . . . . . . . . . . . . . . 2729.2 Simulation of Neurological Tremors . . . . . . . . . . . . . . . 2749.3 Discussion of the Final Attenuator Performance . . . . . . . . 2899.3.1 Performance Discussion of the T-Beam . . . . . . . . . 289viiiTABLE OF CONTENTS9.3.2 Steadi-Two: Design, Functionality, and Impact . . . . 2919.4 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . 2949.4.1 Model Assumptions and Simplifications . . . . . . . . 2949.4.2 Experimental Constraints . . . . . . . . . . . . . . . . 2959.4.3 Posture and DOF Interactions . . . . . . . . . . . . . 2959.4.4 Scaling Effects and Prototyping Challenges . . . . . . 2959.4.5 Rigid Body Motion vs. Vibration Analysis . . . . . . . 2969.4.6 Fatigue Life and Long-Term Performance . . . . . . . 296Chapter 10:Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 29710.1 Core Contributions . . . . . . . . . . . . . . . . . . . . . . . . 29810.2 Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 29910.2.1 Vibrations Engineering . . . . . . . . . . . . . . . . . . 29910.2.2 Neurological Tremors . . . . . . . . . . . . . . . . . . 301Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Appendix A: Derivations . . . . . . . . . . . . . . . . . . . . . . . . 316A.1 Theory of Design and Optimization of a Vibration Absorber . 316A.2 T-Beam Pivot Change . . . . . . . . . . . . . . . . . . . . . . 339Appendix B: Non-Linear Work . . . . . . . . . . . . . . . . . . . . 343B.1 Krylov-Bogoliubov Method . . . . . . . . . . . . . . . . . . . 343B.2 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . 346B.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 346B.2.2 Regular Perturbation . . . . . . . . . . . . . . . . . . 347B.2.3 Poincare-Lindstedt Method . . . . . . . . . . . . . . . 352B.3 The Duffing Oscillator . . . . . . . . . . . . . . . . . . . . . . 361B.3.1 Determining the Frequency . . . . . . . . . . . . . . . 361Appendix C: Modeling Code . . . . . . . . . . . . . . . . . . . . . 366C.1 Simplified Arm Model . . . . . . . . . . . . . . . . . . . . . . 366C.2 T-Beam PSO Algorithm . . . . . . . . . . . . . . . . . . . . . 369C.3 Fourier Transform and Butterworth Filter Code . . . . . . . . 371C.3.1 Butterworth Filter . . . . . . . . . . . . . . . . . . . . 371Appendix D: Steadi-Two Evaluation . . . . . . . . . . . . . . . . . 372D.1 DoE and In-Vitro Results . . . . . . . . . . . . . . . . . . . . 372D.2 Participant Information Form . . . . . . . . . . . . . . . . . . 374D.3 Participant Feedback Form . . . . . . . . . . . . . . . . . . . 375ixList of TablesTable 1.1 Hyperkinetic Movement Disorders . . . . . . . . . . . 2Table 1.2 Hypokinetic Movement Disorders . . . . . . . . . . . . 3Table 1.3 Classification of Tremors . . . . . . . . . . . . . . . . . 5Table 2.1 Key aspects addressed in this chapter. . . . . . . . . . 12Table 2.2 Coupled limb properties for tremor propagation . . . . 22Table 2.3 Mean weight and reduction percentages for Exoskeleton,Functional Electrical Stimulation, and Afferent StimulationDevices. . . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 2.4 Distribution of devices targeting single and multipledegrees of freedom (DOFs) for Exoskeleton, FunctionalElectrical Stimulation, and Afferent Stimulation devices. 33Table 3.1 Summary of human-arm physical parameters . . . . . 37Table 5.1 Goodness of fit comparison for Exponential vs. Polynomialfits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Table 5.2 Parameter description for magnetic spring model . . . 121Table 5.3 Possible Mass and Height Values for Parametric Evaluation131Table 5.4 Experimental Factors and Treatment Levels . . . . . . 137Table 5.5 Performance Results for T-Beam height h = 0.5 in . . 139Table 5.6 Frequency Sweep Results with the MDOF Device . . . 147Table 7.1 Voltage-Frequency Mapping for Induced Tremors . . . 164Table 7.2 Sensors Implemented in Experimental Phase . . . . . . 168Table 7.3 Error Analysis of Predicted and Measured Results . . 184Table 7.4 T-Beam Performance for FPS, WFE, and WRUD . . . 187Table 7.5 Performance Ranking for the Steadi-Two Device . . . 191Table 7.6 Qualitative performance for the 12mm mode of theSteadi-Two device. . . . . . . . . . . . . . . . . . . . . 193Table 7.7 Qualitative performance for the 11mm mode of theSteadi-Two device. . . . . . . . . . . . . . . . . . . . . 193xLIST OF TABLESTable 7.8 Qualitative performance for the 9mm mode of theSteadi-Two device. . . . . . . . . . . . . . . . . . . . . 194Table 7.9 Steadi-Two In-Vitro Results . . . . . . . . . . . . . . . 198Table 7.10 Percentage Reduction for Straight Arm Posture . . . . 202Table 7.11 Reductions for Elbow Bend Posture . . . . . . . . . . 203Table 7.12 Reductions for Picking Posture . . . . . . . . . . . . . 205Table 7.13 Reductions for Drinking Posture . . . . . . . . . . . . 208Table 7.14 Straight Arm Posture Reductions for the Patient Device211Table 7.15 Drinking Posture Reductions for the Patient Device . 212Table 7.16 Picking Posture Reductions for the Patient Device . . 214Table 7.17 Percentage Reduction on the RIG for the Steadi-Two . 216Table 8.1 Patient Demographics and Tremor Classification . . . 239Table 8.2 Phase II Experimental Naming Convention . . . . . . 240Table 9.1 Parameters for Flexion-Extension . . . . . . . . . . . . 261Table 9.2 Biomechanical dynamic parameters for the FPS DOF 271Table 9.3 Predicted vs. Measured Results for the FE DOF forthe T-Beam . . . . . . . . . . . . . . . . . . . . . . . . 290Table 9.4 FPS and FE Performance across Stead-Two Iterations 292Table 9.5 Patient Reductions for the FPS DOF for each Posture 293Table A.1 Comparison of DVA System performance for amplitudereduction . . . . . . . . . . . . . . . . . . . . . . . . . 329Table A.2 DVA Optimization techniques for a SDOF . . . . . . . 335Table A.3 Optimal DVA Parameters for Various Techniques . . . 336Table D.1 Steadi-Two DoE Factors and Treatment Levels . . . . 372Table D.2 Amalgamated In-Vitro Results for the Steadi-Two . . 373xiList of FiguresFigure 1.1 Breakdown of Movement Disorders . . . . . . . . . . 6Figure 2.1 Parkinsonian neurological process . . . . . . . . . . . 16Figure 2.2 The human arm degrees of freedom . . . . . . . . . . 20Figure 2.3 Device group performance-to-weight distribution . . . 34Figure 3.1 A Comprehensive Flexion-Extension Model . . . . . . 38Figure 3.2 Idealization of the arm in flexion-extension as threepoint masses located at the centroid of the upper arm,forearm, palm. . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.3 Lumped mass model of the arm. . . . . . . . . . . . . 40Figure 3.4 Displacement response of the arm to 2.5Hz tremor.The system experiences a near resonance response. . . 44Figure 3.5 FRC for Flexion-Extension Arm Model . . . . . . . . 45Figure 3.6 Pronation-Suppination DOF . . . . . . . . . . . . . . 46Figure 3.7 Free Body Diagram of the FPS DOF . . . . . . . . . 46Figure 3.8 FPS Tremor Response . . . . . . . . . . . . . . . . . . 48Figure 3.9 Inertial Absorber Placed on the Arm in FE DOF . . 49Figure 3.10 Den-Hartog Optimized Inertial Absorber in FE DOFat Resonance . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.11 Frequency Performance of a Den-Hartog OptimizedInertial Absorber . . . . . . . . . . . . . . . . . . . . 53Figure 3.12 Comparison of Optimization Techniques for LinearAbsorbers . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.1 Design Idea A: Inertial Absorber . . . . . . . . . . . . 56Figure 4.2 Design Idea B: Rotational System . . . . . . . . . . . 57Figure 4.3 Rotating Attenuator with Magnetic Springs . . . . . 58Figure 4.4 Magnetic Rotating System with Base Magnets at anAngle: Maximum Stiffness at Extreme Positions . . . 60Figure 4.5 Rotating System Magnetic Overlap Area . . . . . . . 61xiiLIST OF FIGURESFigure 4.6 Offset Magnetic Centers for Rotating System . . . . . 62Figure 4.7 Overlap of Offset Magnetic Faces for Rotating SystemIteration 2 . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.8 Design Idea C: Combined Design for MDOF Tremors 65Figure 4.9 Modified MDOF Device with Multiple Masses andMagnets . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.10 Final Iteration of the MDOF Device . . . . . . . . . . 69Figure 5.1 T-Beam Design Parametric Definition . . . . . . . . . 71Figure 5.2 T-Beam in Flexion-Extension DOF . . . . . . . . . . 71Figure 5.3 T-Beam Free Body Diagram . . . . . . . . . . . . . . 72Figure 5.4 T-Beam Performance in Flexion-Extension DOF atResonance . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 5.5 T-Beam Movement at Resonance in Flexion-ExtensionDOF . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.6 Comparing T-Beam and Inertial Absorber FrequencyResponse . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.7 RMS Reduction Curves for T-Beam and Inertial Absorber 77Figure 5.8 Varying T-Beam Dynamic Mass Response . . . . . . 80Figure 5.9 Varying T-Beam Stiffness Response . . . . . . . . . . 81Figure 5.10 Varying T-Beam Damping Response . . . . . . . . . . 82Figure 5.11 Varying T-Beam Arm Length a . . . . . . . . . . . . 84Figure 5.12 Varying T-Beam Height q . . . . . . . . . . . . . . . 85Figure 5.13 PSO Optimized T-Beam Performance at Resonance . 88Figure 5.14 FRC for PSO Optimized T-Beam vs. Den HartogInertial Absorber . . . . . . . . . . . . . . . . . . . . 89Figure 5.15 RMS Reduction Percentages for Optimized T-Beamand Inertial Absorber . . . . . . . . . . . . . . . . . . 90Figure 5.16 Free Body Diagram of the T-Beam in the FPS DOF . 91Figure 5.17 FPS T-Beam Performance . . . . . . . . . . . . . . . 92Figure 5.18 Performance of the FPS Optimized T-Beam . . . . . 94Figure 5.19 Dual T-Beam setup for the FE DOF . . . . . . . . . 95Figure 5.20 Dual T-Beam response FE DOF . . . . . . . . . . . . 97Figure 5.21 Dual T-Beam response FPS DOF . . . . . . . . . . . 99Figure 5.22 Dual Half T-Beam response FE DOF . . . . . . . . . 101Figure 5.23 Dual Half T-Beam response FPS DOF . . . . . . . . 102Figure 5.24 Exploring T-Beam Orientation Changes . . . . . . . . 103Figure 5.25 Performance of dual deviated T-Beam in FE . . . . . 105Figure 5.26 Sketches Depicting the Position of the Arm whenDrinking . . . . . . . . . . . . . . . . . . . . . . . . . 106xiiiLIST OF FIGURESFigure 5.27 FRCs for Varying Non-Linearity of a Duffing Oscillator108Figure 5.28 Non-Linear Dynamic Vibration Absorber . . . . . . . 109Figure 5.29 Response of Non-Linear Dynamic Vibration Absorber 114Figure 5.30 Vibration Reduction with Non-Linear Dynamic VibrationAbsorbers . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 5.31 Measuring the Axial F − d Relationship for the T-Beam118Figure 5.32 Curve fit to generate a F − d relationship . . . . . . . 119Figure 5.33 N42 Neodymium magnetic F − d relationship . . . . 120Figure 5.34 Parameters for calculating the magnetic spring stiffness122Figure 5.35 Overlap area of top magnet with bottom magnet . . . 123Figure 5.36 Parameters for calculating are of a segment . . . . . . 124Figure 5.37 Discretizing the magnetic overlap area . . . . . . . . . 126Figure 5.38 F − θ profile for the N42 magnets over T-Beam rangeof motion . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 5.39 Final Modified T-Beam for HFLA and LFHA Tremors 128Figure 5.40 Non-Linear T-Beam Free body digram . . . . . . . . 129Figure 5.41 Infinite Growth in Non-Linear Model . . . . . . . . . 133Figure 5.42 Resonance Response of the Non-Linear T-Beam . . . 134Figure 5.43 Resonance Response of the Non-Linear T-Beam forthe FE DOF . . . . . . . . . . . . . . . . . . . . . . . 135Figure 5.44 Resonance Response of the Non-Linear T-Beam in FPS136Figure 5.45 Performance trend for the non-linear T-Beam at h =0.5 in in FE and FPS. . . . . . . . . . . . . . . . . . . 141Figure 5.46 MDOF Design Iteration 3 . . . . . . . . . . . . . . . . 143Figure 5.47 MDOF Design Iteration 3 Response . . . . . . . . . . 145Figure 5.48 MDOF Device FE and FPS Resonance SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . 146Figure 5.49 MDOF Device FE and FPS Bandwidth Reduction . . 149Figure 5.50 Comparing Bandwidth Performance for T-Beam andMDOF Device . . . . . . . . . . . . . . . . . . . . . . 150Figure 6.1 Final Steadi-Two Device Render . . . . . . . . . . . . 152Figure 6.2 Steadi-Two Outer Cover Notch . . . . . . . . . . . . . 153Figure 6.3 Steadi-Two Height Adjustment Knob . . . . . . . . . 154Figure 6.4 Steadi-Two Height Adjustment Screw . . . . . . . . . 155Figure 6.5 Sample Polymagnet Face . . . . . . . . . . . . . . . . 156Figure 6.6 Force-Displacement Relation for Normal vs. Polymagnets158Figure 6.7 Steadi-Two Design . . . . . . . . . . . . . . . . . . . . 159Figure 7.1 Experimental Tremor Inducer . . . . . . . . . . . . . 162xivLIST OF FIGURESFigure 7.2 Experimental Tremor Setup . . . . . . . . . . . . . . 163Figure 7.3 Rig Displacement Seismographs . . . . . . . . . . . . 166Figure 7.4 FPS Tremor Test Platform ”RIG” . . . . . . . . . . . 167Figure 7.5 Fast Fourier Transform of Raw Experimental Data . . 170Figure 7.6 Comparison of Raw and 4th Order Butterworth FilteredData . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Figure 7.7 Displacement of the Palm center of mass for X, Y ,and Z axes under induced tremor . . . . . . . . . . . 173Figure 7.8 Comparison of the Trapezoidal Method and Simpson’sRule for One-Dimensional Numerical Integration . . . 174Figure 7.9 Validation of the FPS Model and RIG Response comparedto Induced Tremor . . . . . . . . . . . . . . . . . . . . 177Figure 7.10 Windowed RMS Comparison of the FPS Model, RIGResponse compared to Induced Tremor . . . . . . . . 178Figure 7.11 Windowed RMS Comparison of the FPS Model, RIGResponse compared to Induced Tremor . . . . . . . . 181Figure 7.12 FFT and PSD Response of Predicted and MeasuredResults . . . . . . . . . . . . . . . . . . . . . . . . . . 183Figure 7.13 T-Beam Response for a 2Hz induced tremor for FPS,FE, and WRUD . . . . . . . . . . . . . . . . . . . . . 188Figure 7.14 Steadi-Two Qualitative Test Setup . . . . . . . . . . . 192Figure 7.15 Performance of Steadi-Two iteration two using WeSUGyroscopic Sensor at 10V . . . . . . . . . . . . . . . . 197Figure 7.16 FPS Frequency Performance of Steadi-Two iterationone using WeSU Gyroscopic Sensor . . . . . . . . . . 199Figure 7.17 Performance of the Steadi-Two iteration two deviceat 6.4Hz for a straight arm posture . . . . . . . . . . 201Figure 7.18 Performance of the Steadi-Two iteration two deviceat 6.4Hz for a 90o elbow bend. . . . . . . . . . . . . . 202Figure 7.19 FPS Frequency Performance of Steadi-Two iterationthree using WeSU Gyroscopic Sensor for the ElbowBend Posture . . . . . . . . . . . . . . . . . . . . . . . 203Figure 7.20 Performance of the Steadi-Two iteration two deviceat 6.4Hz for the picking posture. . . . . . . . . . . . . 204Figure 7.21 FPS Frequency Response of the Steadi-Two iterationtwo using WeSU Gyroscopic Sensor for the PickingPosture . . . . . . . . . . . . . . . . . . . . . . . . . . 206Figure 7.22 Performance of the Steadi-Two iteration two deviceat 6.4Hz for the drinking posture. . . . . . . . . . . . 207xvLIST OF FIGURESFigure 7.23 FPS Frequency Response of the Steadi-Two iterationtwo using WeSU Gyroscopic Sensor for the DrinkingPosture . . . . . . . . . . . . . . . . . . . . . . . . . . 208Figure 7.24 Performance of the Steadi-Two patient device at 6.4Hzfor the straight arm posture. . . . . . . . . . . . . . . 210Figure 7.25 Performance of the Steadi-Two patient device at 6.4Hzfor the drinking posture. . . . . . . . . . . . . . . . . 211Figure 7.26 FPS Frequency Response of the Steadi-Two PatientDevice using WeSU Gyroscopic Sensor for the DrinkingPosture . . . . . . . . . . . . . . . . . . . . . . . . . . 212Figure 7.27 Performance of the Steadi-Two patient device at 6.4Hzfor the picking posture. . . . . . . . . . . . . . . . . . 213Figure 7.28 FPS Frequency Response of the Steadi-Two PatientDevice using WeSU Gyroscopic Sensor for the PickingPosture . . . . . . . . . . . . . . . . . . . . . . . . . . 214Figure 7.29 Performance of the Steadi-Two patient device at 5.5Hzfor the picking posture. . . . . . . . . . . . . . . . . . 215Figure 7.30 FPS Frequency Response of the Steadi-Two PatientDevice on the RIG . . . . . . . . . . . . . . . . . . . . 217Figure 8.1 Posture Definitions for Patient Evaluations . . . . . . 219Figure 8.2 Process Flow Diagram for Patient Evaluations . . . . 221Figure 8.3 Characterizing Patient Tremor . . . . . . . . . . . . . 222Figure 8.4 Characterizing Patient Tremor in FPS DOF . . . . . 223Figure 8.5 Patient Tremor Reduction in 3 DOFs . . . . . . . . . 225Figure 8.6 Frequency Domain Analysis for Absorber Response . 226Figure 8.7 Tremor Reductions for All Postures in MDOF . . . . 228Figure 8.8 Failure of the Steadi-Two in FPS DOF Straight ArmPosture . . . . . . . . . . . . . . . . . . . . . . . . . . 230Figure 8.9 Multi-Patient Tremor Reductions for All Postures inthe FPS DOF . . . . . . . . . . . . . . . . . . . . . . 232Figure 8.10 Performance of the Steadi-Two for another Patient . 233Figure 8.11 FFT Analysis for Second Patient Results . . . . . . . 234Figure 8.12 Example of Excluded Tremor Pattern . . . . . . . . . 238Figure 8.13 Patient with Arm in Posture 1 . . . . . . . . . . . . . 241Figure 8.14 Phase II Moderate Severity Tremor Posture 1 . . . . 242Figure 8.15 Heatmap of Tremor Reduction in Posture 1 . . . . . . 244Figure 8.16 Patient with Arm in Posture 2 . . . . . . . . . . . . . 245Figure 8.17 Phase II Moderate Severity Tremor Posture 2 . . . . 246Figure 8.18 Heatmap of Tremor Reduction in Posture 2 . . . . . . 248xviLIST OF FIGURESFigure 8.19 Patient with Arm in Drinking Posture . . . . . . . . . 249Figure 8.20 Phase II Moderate Severity Tremor Drinking Posture 250Figure 8.21 Heatmap of Tremor Reduction in Drinking Posture . 252Figure 8.22 Pouring Posture Demonstration . . . . . . . . . . . . 253Figure 8.23 Archimedes Spiral Test . . . . . . . . . . . . . . . . . 254Figure 8.24 Freehand Archimedes Spiral Test . . . . . . . . . . . . 254Figure 8.25 Real Archimedes Spiral Test . . . . . . . . . . . . . . 255Figure 8.26 QUEST Survey Response for the Steadi-Two . . . . . 257Figure 9.1 Musculoskeletal representation of the human arm . . 262Figure 9.2 FRC for 3 DOF Flexion-Extension Model . . . . . . . 266Figure 9.3 FRCs for Reduced DOF Flexion-Extension Model . . 269Figure 9.4 Musculoskeletal Model of the FPS DOF . . . . . . . . 270Figure 9.5 Dynamic model of the forearm pronation-supinationdegree of freedom. . . . . . . . . . . . . . . . . . . . . 271Figure 9.6 FPS Arm Model Frequency Response . . . . . . . . . 272Figure 9.7 Varying Attenuator Positions for MDOF Tremors . . 273Figure 9.8 Parkinsonian FPS Rest Tremor . . . . . . . . . . . . 275Figure 9.9 Parkinsonian WFE Rest Tremor . . . . . . . . . . . . 276Figure 9.10 Parkinsonian WRUD Rest Tremor . . . . . . . . . . . 278Figure 9.11 Postural tremor for Parkinsonsian patient 12 for theirleft hand. We can infer that accely (middle plot)represents the FPS DOF due to harmonic nature oftremor and the high amplitude. . . . . . . . . . . . . 280Figure 9.12 Postural tremor for Parkinsonsian patient 12 for theirright hand. Their tremor significantly differs both inamplitude and frequency compared to the left hand.This highlights the variation in tremor even for thesame patient. . . . . . . . . . . . . . . . . . . . . . . . 282Figure 9.13 Kinetic tremor for Parkinsonsian patient 12 for theirleft hand. This tremor differs from the postural tremorfor the same hand showing that tremor is posturedependent. . . . . . . . . . . . . . . . . . . . . . . . . 283Figure 9.14 Kinetic tremor for Parkinsonsian patient 12 for theirright hand. This highlights the variation in tremor forthe same patient, same posture but different hands. . 284Figure 9.15 Postural tremor for Parkinsonsian patient 25 for theirleft hand. This patient’s tremor for the left hand issimilar to patient 12’s tremor in all three DOFs. . . . 285xviiLIST OF FIGURESFigure 9.16 Postural tremor for Parkinsonsian patient 25 for theirright hand. The FPS DOF (middle figure) shows asimilar RMS response to the left hand postural tremorbut the other two DOFs a significantly different. Thisvariation is the main challenge for broadband tremorsuppression. . . . . . . . . . . . . . . . . . . . . . . . 286Figure 9.17 Suggested Models for Mild, Moderate, and SevereTremors . . . . . . . . . . . . . . . . . . . . . . . . . . 288Figure A.1 Dynamic Vibration Absorber . . . . . . . . . . . . . . 317Figure A.2 Effect of a Tuned Mass Damper . . . . . . . . . . . . 322Figure A.3 Relationship between the mass ratio and effectiveabsorber frequency bandwidth . . . . . . . . . . . . . 324Figure A.4 Simulated response of a vibrating system with variousabsorber configurations . . . . . . . . . . . . . . . . . 326Figure A.5 Frequency analysis of an optimized TMD withoutprimary damping . . . . . . . . . . . . . . . . . . . . 327Figure A.6 Accounting for the main system inherent damping . . 328Figure A.7 Comparing primary system to absorber response . . . 330Figure A.8 Response of absorbers optimized with various methods337Figure A.9 T-Beam Pivot Change . . . . . . . . . . . . . . . . . . 339Figure A.10 T-Beam Pivot Change Equilibrium Displacement . . 340Figure A.11 New T-Beam FBD . . . . . . . . . . . . . . . . . . . . 341Figure A.12 New vs. Old T-Beam Comparison . . . . . . . . . . . 342Figure B.1 Comparing Solutions for Regular Perturbation. . . . . 351Figure B.2 Failure of Regular Perturbation . . . . . . . . . . . . 355Figure B.3 Comparison of Poincare-Lindstedt, Regular Perturbation,and Numerical Solutions. . . . . . . . . . . . . . . . . 360Figure B.4 Frequency Response of the Duffing Oscillator for aHardening and Softening Spring. . . . . . . . . . . . . 365xviiiList of AbbreviationsADL Activities of Daily LivingCAD Computer Aided DesignCNS Central Nervous SystemDoE Design of ExperimentsDOF Degree of FreedomDVA Dynamic Vibration AbsorberDW Dead Weight ”Sham”EFE Elbow Flexion-ExtensionEMG ElectromyographyEOM Equation of MotionET Essential TremorFBD Free Body DiagramFES Functional Electrical StimulationFFT Fast Fourier TransformxixLIST OF ABBREVIATIONSFPS Forearm Pronation-SupinationFRC Frequency Response CurveFTM Fahn-Tolosa-MarinH12 Higher Height ModeHFLA High Frequency Low AmplitudeHVPL Heart Valve Performance LabIMU Inertial Measurement UnitsKBM Krylov-Bogoliubov MethodL8 Lower Height ModeLFHA Low Frequency High AmplitudeMDOF Multi-Degree of FreedomMDs Movement DisordersPD Parkinsons DiseasePSD Power Spectral DensityPSO Particle Swarm OptimizationQoL Quality of LifeQUEST Quality of life Essential Tremor QuestionnaireRMS Root Mean SquareSAA Shoulder Abduction-AductionSDOF Single Degree of FreedomSEIR Shoulder Internal-External RotationSFE Shoulder Flexion-ExtensionSNR Signal to Noise RatioTETRAS The Essential Tremor Rating Assessment ScalexxLIST OF ABBREVIATIONSTSD Tremor Suppression DeviceUPDRS Unified Parkinsons Disease Rating ScaleWFE Wrist Flexion-ExtensionWRUD Wrist Radial-Ulnar DeviationWTSD Weighted Tremor Suppression DevicexxiAcknowledgementsI would like to acknowledge the following for their contributions and supportduring my studies:∗ My supervisor, Dr. Hadi Mohammadi, and my committee membersDr. Ray Taheri and Dr. Zhang Liu; whose guidance, mentorship,and unwavering support have been pivotal throughout my academicjourney. Their emphasis on understanding the fundamentals andmultidisciplinary components of my work deepened my knowledge andinspired me to explore new perspectives. Through their insightfulfeedback and thought-provoking discussions, my research has beensignificantly enriched.∗ My lab manager, Dylan, for his invaluable support in sourcing researchmaterials and fostering a collaborative and intellectually stimulatingenvironment. His enthusiasm for discussing research ideas and providingconstructive feedback has been instrumental in refining my work.∗ MITACS, for their generous financial support, which enabled me tofocus on my research. Their funding was critical in providing theresources necessary to achieve my academic and professional goals.xxii∗ SteadiWear Inc., for their financial support and for offering me theopportunity to grow as a professional. Collaborating with their teamallowed me to contribute to meaningful and impactful work to improvethe lives of those ailed by Parkinson’s Disease and Essential Tremors.Their belief in my capabilities was a profound motivator throughoutmy journey.∗ My mother, Christine, and my brothers, Simon and David, whosesacrifices and patience allowed me to pursue my academic dreams.Their unwavering love, unselfish support, and constant encouragementwere the foundation upon which I built my success. Their belief in myjourney will forever inspire my gratitude and perseverance.∗ My partner and best friend, Nicole, whose unwavering encouragementand emotional support have been my anchor during challenging times.Her enduring patience and remarkable sacrifices to accommodate mydemanding schedule ensured that I remained focused and organized.Her belief in my potential, even during moments of self-doubt, providedme with the confidence and clarity needed to navigate this journeysuccessfully.xxiiiDedicationDedicated to my father who instilled in me the passion for knowledge, and tomy mother for teaching me to be tenacious.xxivCHAPTER 1IntroductionMovement Disorders (MDs) are often categorized into two distinct subgroupsnamely Hyperkinetic MDs (referred to as dyskinesias) and HypokineticMDs which are further subdivided to akinesia, hypokenisia, bradykenisia,and rigidity [1].Mechanically, Hyperkinetic MDs are characterized by often repetitive andexcessive involuntary movements which interrupt regular motor activity.Disorders such as chorea, dystonia, clonus, myoclonus, stereotypes, tics andtremor fall under Hyperkinetic MDs [1–4]. Table 1.1 summarizes Hyperkineticdisorders and their cardinal features.On the other hand, Hypokinetic MDs comprise of akinesia referring to alack of movement, hypokenisea implying a reduced amplitude of movement,bradykenisia which is best described as slow movement and rigidity implyingimpeded movement. Table 1.2 summarizes Hypokinetic MDs.1Table 1.1: Characteristics of Hyperkinetic movement disorders [2]Hyperkinetic Disorder Key FeaturesTremor Rhythmic, involuntary oscillations of a bodypartClonus Rhythmic oscillations around a joint inresponse to sudden passive contractionStereotypy Repeated and continuous motor actsMyoclonus Very rapid, jerky, shock-like movementsDystonia Sustained muscular contractions frequentlycausing twisting and abnormal posturing ofa body partChorea Nonrhythmic, nonstereotyped, semipurposive,relatively slow movements that flow from onepart of the body to anotherTic Brief, nonrhythmic, rapid movements that areusually orofacial but may involve the voiceor any part of the body; presence of urge toperform; can be suppressed voluntarily for ashort period, with rebound increase after theperiod of suppression2Table 1.2: Characteristics of Hypokinetic movement disordersHypokinetic Disorder Key FeaturesRigidity Characterized by increased muscle tone thatproduces abnormal resistance to passivemovement[5]Bradykinesia Slow movementAkinesia Absence/lack of movementHypokinesia Diminished amplitude of movementThere is a variation in the classification and analysis of MDs due to theirnon-elucidated causes and the type of study. Thus, a given description andunderstanding of MDs depends on purpose of the body of work they arecaptured. Generally, classifications of MDs are based on the way they manifesti.e. their observable characteristics. Thus, analysis of the neurologicalprocess tends to focus on cause/manifestation type of classification whereasstatistical approaches to quantify the severity of MDs opt for movement-basedphenomenology.The study of movement disorders dates as far back as 180AD according toGerpen [6]. The Greek physician Galen is credited with the original effortto classify tremors independently from other MDs such as palpitations andseizures. A more robust classification of tremors emerged in 1680 [6] whenSylvius De la Boe recognized that tremors could also occur at rest. Thisclassification has been fine tuned with the expansion of research into MDs,specifically tremors. Currently, there are various ways MDs are classified(neurological causality or movement based) as well as various approaches(both medical and mechanical) to combat them. The nature and reliabilityof these mitigation approaches varies with each movement disorder. Thepurpose of this body of work is to introduce these common movementdisorders; which are Parkinson’s Disease and Essential Tremor, discuss theirpathophisiology and medical approaches to treatment followed by a broadreview of alternative therapies for mitigation. The aim of this approach is topropose an iteratively designed and evaluated novel approach to combatingthese common MDs.31.1. BACKGROUND ON TREMORS1.1 Background on TremorsTremor; defined as an involuntary, approximately rhythmic, and roughlysinusoidal movement [2, 3, 5–9] is the most common type of movementdisorder [3]. It is said to affect upto 10 million people a day in the UnitedStates [7]. It is distinctive from other MDs given its periodic movements withdiscernible properties. This distinction is important in addressing tremorsbecause misdiagnosis can occur due to similarities of various MDs at onset[7]. With the prevalence and debilitating impacts of tremor, research intounderstanding and tackling the challenges faced by patients has grown overthe years.There are two broad classifications of tremor: action and rest tremor [3].Action tremor is the most prevalent form of tremor and it occurs aftersustained extension of the arm during voluntary movements such as writingor eating. It is further subdivided into postural, isometric and kinetic tremor.On the other hand, rest tremor, as the name suggests, occurs when the patientis at rest with arms supported. Table 1.3 provides a summary of action andrest tremors and their definitions. There are other categories of tremorssuch as physiological action tremor that can be present in every personunder specific conditions[10]. These are not considered here as this review isconcerned with neurological tremors resulting from chemical imbalances anddeficiency. Additionally, physiological tremors can sometimes be mitigatedonce the exacerbating conditions are removed. For instance, trembling handsafter holding onto a weight at an elevated height for a period of time is aform of physiological tremor that can present even in everyone.41.1. BACKGROUND ON TREMORSTable 1.3: Definition of various tremors[2, 3] based on manifestation.Tremor Definition and Fundamental FeaturesRest Tremor Tremor that occurs in a body part withoutvoluntary activation and fully supportedagainst gravity. Observing a patient whosearm is supported on a rest can give insight onresting tremor[2]Action Tremor Any tremor produced by voluntary musclecontraction. Subdivided to postural, isometricand kinetic tremor.Postural Tremor Tremor that presents when voluntarily holdinga position/posture against gravity.Isometric Tremor Tremor occurring as a result of musclecontraction against rigid/stationary objects.Kinetic Tremor Tremor that occurs during any kind ofmovement. If tremor occurs duringtargeted/directed movement, it is calledintention tremor (Task Specific Tremor)The pathophisiology of action and rest tremors identifies Parkinson’s Diseaseas the most common cause of rest tremor and Essential Tremor as themost common cause of action tremor [3]. Therefore, understanding thesetwo movement disorders in terms of their causes, diagnoses and treatmentoptions is fundamental understanding present approaches and devisingpotential alternative therapies to mitigate challenges faced by patients.Figure 1.1 recaps the classification of movement disorders leading to thesub-classifications for both PD and ET.51.1.BACKGROUNDONTREMORSFigure 1.1: Classification of movement disorders, divided into hyperkinetic and hypokinetic types. The subgroupsfor Parkinson’s disease and essential tremor are highlighted within this framework.61.2. PROBLEM STATEMENT AND RESEARCH OBJECTIVES1.2 Problem Statement and Research Objectives1.2.1 Problem StatementWe have seen that tremors are among the most prevalent movement disorders,resulting insignificant functional impairments and diminished quality of life.Currently existing medical solutions, as covered in Chapter 2 are insufficientin addressing patient needs and are primarily focused on reducing tremorsymptoms but do not directly address the mechanical manifestation of tremorsin affected body parts. This creates a gap and a need for the developmentof non-invasive devices that can consistent attenuation for a variety oftremors and impacted activities. The need for alternative approaches, suchas tremor suppression through mechanical vibration attenuation, remains anunder-explored yet critical area of research.1.2.2 Research ObjectivesTo address this gap, this thesis, and the work done to complete it definesthe following research objectives to guide the development of a novel devicefor tremor suppression.1. Dynamic Modeling: Develop dynamic models of the human armcomplex at the musculoskeletal level to study tremor propagation andits mechanical manifestation on different degrees of freedom (DOFs).2. Device Conceptualization: Ideate non-invasive, passive tremorsuppression devices capable of multi-degree-of-freedom (MDOF) tremorattenuation.3. Simulation and Validation: Assess the performance of these devicethrough rigorous mathematical model simulation and physical experimentalevaluations. The goal is to understand what how design affectsperformance and functionality in critical DOFs.4. Patient-Centric Evaluation: Evaluate the final device’s efficacyand performance on real patient tremors. Additionally, by assessingnon-tangible factors such as its rate of adoption and ease of usabilityby patients, we aim to iteratively improve the device to assist a widepopulation of patients.71.3. SCOPE AND THESIS OUTLINE1.3 Scope and Thesis OutlineBuilding on these objectives, the scope of this work focuses on addressingthe identified challenges through a targeted and practical approach. Theproject scope below outlines the primary areas of focus, ensuring that eachobjective is achieved systematically.1.3.1 Project Scope1. Common Movement Disorders: The primary focus of this work ison neurological movement disorders characterized by tremors, specificallyParkinson’s Disease and Essential Tremor disease. Moreover, this workconcerns itself with the manifestation of these tremors rather than theneurological processes that result in these disorders.2. Multi-Degree of Freedom Devices: The modeling and design workcompleted is focused on realizing a device capable of tremor attenuationin multiple degrees of freedom. Devices that fail to meet this criteriawill be limited to conceptual analysis.3. Mathematical Modeling and Analysis: By implementing mathematicalmodels, we can evaluate and understand different devices and theirperformance in various scenarios. However, for the scope of this work,the degree of detail provided for some mathematical models may belimited due to pre-existing copyright ownership agreements.4. Experimental Testing: This research involves in-vitro tests forvarious iterations of devices. The results provided from this section ofthe research are to provide a contextual understanding of the iterativenature of the work. The data provided showcases various aspects of thedevelopment process but does not cover the entirety of experimentalwork conducted.5. Patient Evaluation: The data from various patient evaluationsconducted throughout the course of this research is used to showcasethe congruency between modeling, experimental evaluation and realworld application.81.3. SCOPE AND THESIS OUTLINE1.3.2 Interdisciplinary Context and Scope of Author’sExpertiseThis work focuses on addressing neurological tremors from an interdisciplinaryperspective, leveraging principles of engineering to tackle a complex biomedicalchallenge. The author, as an engineer by training, possesses expertise inmechanical systems, dynamic modeling, and vibration analysis. These skillshave facilitated the conceptualization and design of solutions targeting themechanical manifestations of tremors, framing the problem within the domainof vibration engineering.Thus, the author is limited in their biological understanding, particularlyconcerning the physiological and neurological origins of tremors. Theselimitations have been addressed through extensive review of existing research,collaboration with domain experts, and a focused examination of the mechanicalinteractions central to this study. This interdisciplinary approach allows theauthor to contribute a unique perspective that complements conventionalbiological and medical strategies.1.3.3 Thesis OutlineThis thesis is organized into a series of interconnected chapters, each addressinga specific aspect of the research objectives to realize a novel tremor suppressiondevice (TSD). The structure is as detailed below:∗ Chapter 2 provides a comprehensive review of existing research relevantto the study of neurological tremors and their mitigation. It exploresprior work in mathematical modeling, design methodologies, prototypingtechniques, and experimental evaluations, identifying key gaps thatthis research seeks to address.∗ Chapter 3 develops the theoretical foundation for the research, focusingon dynamic modeling of the human arm complex at the musculoskeletallevel. It presents the human arm dynamic models used to frame tremorsas vibrations and describes the assumptions and analysis techniquesemployed.∗ Chapter 4 outlines the conceptual design process for various tremorsuppression devices using vibration absorber principles.91.3. SCOPE AND THESIS OUTLINE∗ Chapter 5 covers extensive mathematical evaluation for the conceptualdesign and includes discussion on the criteria for selection as well asjustification for the chosen design.∗ Chapter 6 details the fabrication and iterative refinement of the tremorsuppression device. It discusses the materials, tools, and techniquesused, along with the challenges encountered and the modificationsmade to optimize the prototype’s performance.∗ Chapter 7 presents the experimental methods and results used to assessthe performance of the tremor suppression device. It focuses on thedevice’s efficacy in attenuating tremors in key degrees of freedom andprovides a statistical analysis of the collected data. It also describesthe validation setup, analyzes discrepancies, and highlights how themodels informed the device’s design and development.∗ Chapter 8 evaluates the device in a real-world context through patienttesting and deployment. It discusses the testing protocol, user feedback,and the device’s adoption and usability, emphasizing its impact onimproving patient quality of life.∗ Chapter 9 discusses the findings from the modeling, validation, andexperimental work. It explores the implications of the results, discussesthe limitations and constraints of the study, and proposes directionsfor future research.∗ Chapter 10 summarizes the key findings of the research and theirsignificance. It reflects on the contributions of the study to the field ofneurological tremor mitigation and reiterates its potential to advanceassistive device development.∗ In the Appendices, derivations, code scripts, and background knowledgenecessary to complete this research have been provided.10CHAPTER 2Literature ReviewChapter 1 provided a concise overview of tremor classification, highlightingParkinson’s disease (PD) and essential tremor (ET) as the most prevalentforms of rest and action tremors. This chapter will explore the literatureon Parkinsons and Essential Tremor and their existing solutions for tremormitigation. The key areas of discussion in this chapter are summarized inTable 2.1.2.1 Parkinson’s DiseaseIn 1817, the British Physician Dr. James Parkinson described PD, thennamed Shaking Palsy (Paralysis Agitans) as an ”involuntary tremulousmotion” leading to diminished muscular power both in action and rest [11].While the cause of PD was not well understood at the time, the hypothesiswas that it was dependent on the diseased state of the medulla spinalis[11]. Over time, as the study of PD expanded, a more concise definitiondescribes Parkinson’s Disease as a ”neurological disorder characterized by thedegeneration of dopaminergic neurons in the Substantia Nigra pars compacta(SNpc)” [12].112.1. PARKINSON’S DISEASETable 2.1: Key aspects addressed in this chapter.Key Aspect DescriptionPathophysiologicalProcessesAn exploration of the clinical mechanismsunderlying the development of both PD and ET.Clinical Diagnosis A discussion of diagnostic procedures andthe factors influencing diagnostic accuracy andreliability.PhysicalManifestationsExamination of the common physical symptomsand their effects on patients’ quality of life.Medical TreatmentOptionsA review of available medical treatments,emphasizing their limitations and associated risks.These shortcomings underscore the need foralternative tremor mitigation strategies, providingthe impetus for this research.Tremors as Vibration Re-framing tremors as vibration phenomena toenable the application of vibration absorption andengineering design principles.Current AlternativeTherapiesAn evaluation of current tremor suppressiontechnologies, including their methodologies,performance metrics, and limitations.Criteria for TremorSuppression DevicesDevelopment of a general framework for designingeffective tremor suppression devices that addressthe gaps identified in existing solutions.2.1.1 Pathophisiology of Parkinson’s DiseaseThe human brain regulates and coordinates voluntary movement in collaborationwith the spinal cord. Anatomically, the brain has four main structures: thelarge domed Cerebrum; the Cerebellum; the Thalamus and nearby structuresforming the inner Diencephalon [13] and finally the brain stem at its base.Functionally, the brain consists of the Fornix which is an important partof the limbic system [14–16], the Hippocampus which is responsible formemory, the Amygdala which is responsible for emotions and the Thalamuswhich forms the connection to between the Cortex and the brain stem andultimately connects to the spinal cord. The Thalamus is medial to the BasalGanglia [4, 17, 18].122.1. PARKINSON’S DISEASEThe Basal Ganglia, sometimes referred to as the Basal Nucleus, is responsiblefor muscle tone and initiation of smooth and learned movement patterns.The Basal Ganglia comprises of caudate nucleus (CN) and caudate putamen(CP) which together form the Striatum, the internal and external GlubusPallidus (GPi, GPe), the nucleus accumbens, the Subthalamic Nucleus (STN)and the Substantia Nigra (SN) comprised of the pars reticulata (SNpr) andpars compacta (SNpc).In normal movement when Cortex intends to initiate motion, neurologicalsignals relayed to the Basal Ganglia, specifically the Striatum are sent backto the motor Cortex via the Thalamus from the GPi. The Cortex relaysthese signals to the spinal cord thus smooth and coordinated movement isinitiated. In this process, the input to- and output from the Basal Gangliaare key since it is known that the output from the Basal Ganglia is disruptedin the case of Parkinsons disease [4, 19].In the Basal Ganglia, there are interconnections between the GluterminergicNeurons (GLUT) which are excitatory neurons and the GabaminergicNeurons (GABA) which are inhibitory [19, 20]. However, it is the DoperminergicNeurons (DN) in the SNpc that release dopamine into the Striatum. Dopaminecan bind onto either D1 receptors in which it is excitatory and stimulates theGLUT neuron in the Striatum or D2 receptor in which it is inhibitory andinhibits the GABA neuron. Therefore, if Dopamine binds to a D1 receptor,it stimulates the GLUT neurons and inhibits the GABA neurons in theStriatum and results in the Thalamus relaying signals to the Cortex thusinitiating motion to the musculoskeletal system.However, in the case of Parkinson’s disease, the death of the DN in the SNpcleads to reduced Dopamine production. It is theorized that this death canbe caused by protein misfolding, mitochondrial dysfunction among otherpossibilities but there is no certainty as to the cause of this degeneration [20].With reduced Dopamine, the inhibitory receptor D2 is unbound and unableto inhibit GABA neurons in the Striatum. This inhibits the GABA neuronin the GPe which is now unable to inhibit the GLUT neuron in the STN[20]. The excitatory GLUT neuron secretes glutamate which stimulates theGABA neuron in the GPi and in the SNpr. Moreover, the lack of Dopamineto bind with the D1 receptor fails to stimulate the GABA neuron that wouldnormally inhibit the GPi which results in an overly active GABA neuronfrom the GPi to the Thalamus [19]. This results in excessive inhibition132.1. PARKINSON’S DISEASEof the Thalamus (since no excitatory neurons were active) and Thalamusinhibition leads to a suppression of the Thalamus-Cortex-Spine pathway andthe resulting motion is not smooth but uncontrolled and uncoordinated. Thismotion can manifest as the clinical symptoms of Parkinson’s Disease. Figure2.1 (adapted from [21]) summarizes the Parkinsonian neurological processeswith arrows thickened to indicate the excessive inhibition of the Thalamus.2.1.2 Clinical Diagnosis and ManifestationsThe cardinal features of Parkinson’s disease include: i) rigidity (stiffness)which is an increase in muscle that opposes motion irrespective of velocity ofmotion; ii) tremors ranging between 4 - 6 Hz [3, 12, 22, 23]; iii) a pill-rollingtremor typically originating in the hand that may progress and involve thetongue, lips,legs; iv) bradykinesia and hypokinesia which are the main cuasesof disability; v) postural instability which develops late in the disease course[4, 5, 12, 13, 24, 25]. The main symptoms of bradykinesia and hypokinesiainclude hypomimia which is a decrease in facial expressions, hypophomiawhich is a reduction in voice tone, reduction of manual dexterity for fine motorskills, shuffling of gait, festinating gait and freezing. Non-motor features ofPD include behavioural changes and abnormalities, the most common andfrequently encountered psychological problem being depression [5, 12, 25–27].Others include autonomic dysfunction which is loss of autonomy, sleepdisturbances, disorders of sensation which may affect primary senses, anxiety,and gastrointestinal and impulse-control disorder to name a few [4, 26, 28].The diagnosis for PD involves physical examination. It is challenging toproperly diagnose since PD manifests differently in every patient a la ”Ifyou have met one person with Parkinson’s, you have met one person withParkinson’s” [20]. Therefore, examination requires ruling out other conditionswith similar manifestations which can be difficult at onset [4, 14]. Nonetheless,the presence of certain conditions, either individually or together can be astrong indicator for PD. These include [12, 23]:∗ Unilateral onset∗ Clinical course for 10 or moreyears∗ Presence of rest tremor∗ Progressive disorder∗ Excellent response toLevodopa (⩾ 70%)∗ Response to Levodopatreatment for 5+ years142.1. PARKINSON’S DISEASEWhen it comes to the treatment of Parkinson’s disease, the underlyingchallenge remains the lack of certainty regarding the cause of death of thedopaminergic neurons in the SNpc. Moreover, no treatment yet has beencapable of slowing or curing the progression of the disease [25]. Therefore,available therapies for PD seek to mitigate the manifestations of the disease[12]. For instance, the Michael J. Fox foundation, one of the largestorganizations at the forefronts in the search for a cure for Parkinson’s disease[17] with their motto ”Here. Until Parkinson’s isn’t,” insist on the importanceof alleviating the burden on the patients caused by the manifestations whilethe search for a cure continues [29, 30].2.1.3 Medical Therapies for Parkinson’s DiseaseFor treatment, pharmacotherapy remains the mainstay in the treatmentof PD. Levodopa, which is a precursor to dopamine is administered incombination with carbidopa which is an inhibitor that blocks peripheralconversion of Levodopa to dopamine before crossing the blood-brain barrier[4, 18, 31]. However, there is a myriad of complications related to theadministration of Levodopa. These include [18]:∗ Dyskinesia∗ Motor Fluctuations∗ Psychosis∗ Hallucinations∗ Cognitive impairment∗ Insomnia and NauseaDeep brain stimulation (DBS), which involves the insertion of electrical probesto either the STN or GPi has been found to be effective in moderate-to-severePD [12, 32]. However, DBS remains a late stage option with surgicaltreatment being 10-13 years after diagnosis. In addition to this time delay, thecost of DBS is significantly high such that it may be outside of many patientsaffordability range. Serious complications from DBS include intracerebralhaemorrhage, worsened cognition, hardware complications [4] to name a few.Therefore, while DBS is a potential therapy, it faces multiple barriers thathinder its potency.152.1.PARKINSON’SDISEASEFigure 2.1: The basal ganglia circuitry involved in movement modulation and Parkinson’s disease symptommanifestation is illustrated here. Both direct and indirect pathways are shown, with excitatory (green) andinhibitory (red) neurotransmitter actions, focusing on the roles of glutamate and GABA across regions like thecortex, striatum, thalamus, and substantia nigra pars compacta (SNpc). Dopaminergic input from SNpc to thestriatum modulates these pathways, and disruptions in this balance, particularly in Parkinson’s disease, result inincreased inhibitory output to the thalamus, leading to motor impairments162.2. ESSENTIAL TREMOR DISEASE2.2 Essential Tremor DiseaseEssential tremor is consists of bilateral, visible, persistent and mostly symmetricaltremor [3, 6, 7] ranging from 4-12Hz [22, 23, 33]. The hallmark differencebetween ET and PD is that ET is most pronounced during active movementswhile PD manifestations are pronounced during rest. ET can affect up to 10million people a day in the U.S. [7] but despite its widespread nature andthe impact it has on those it affects, the pathophisiology of ET is currentlyunknown [3, 6, 7, 23, 33, 34]. Approximately 50% of ET patients have afamily history of ET which points to a possible genetic mutation [7]. Inaddition to its undetermined root cause, there is a limited number of tools toquantify ET but The Essential Tremor Rating Assessment Scale (TETRAS)is one such tool [3, 7].The diagnosis for ET involves a differential diagnosis where the patient isobserved for rest tremor and action tremors before declaring it present. Thisprocess is further complicated by the different ways in which tremor canpresent in patients and makes proper diagnosis and ongoing challenge [2, 3].Pharmacotherapy is the main therapy available for ET with beta-blockerslike Propranolol [9] making up the major interventions available. Like PD,these seek to minimize the impact on the patients quality of life by mitigatingthe effects of the manifestations. However, just like the medication used inPD treatments, complications such as dizziness and drowsiness, bradycardiaamong others continue to plague the treatment of ET [6, 23, 33].2.3 Alternative Therapies for Tremor MitigationThe continuum of care for PD and ET patients begins with diagnosis followedby the prescription of appropriate medication be it Levodopa or Propranololrespectively. However, given a 50−60% effectiveness rate for such medicationcombined with diminishing potency over long term use and other side effects[9], some patients find themselves in need of alternative therapies. Thesetake the form of task specific tools such as the liftware spoon [35, 36] whichis an active system that aids patients in consuming their meals. Thisspoon has a reported tremor reduction of 76% [35]. While task specifictools like the Liftware spoon increase self reliability for patients as theyperform the intended task, they are limited on the improvement they canprovide to their overall quality of life. Thus, there is another sub-categoryof alternative therapies comprised of devices broadly classified as wearable172.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONtremor suppression devices. As the name suggest, these devices are worn onthe portion of the limb experiencing tremors and achieve tremor suppressionthrough different mechanisms based on their design.2.3.1 Wearable Tremor Suppression TechnologiesWearable tremor technology is not a recently novel concept. In fact, Aisenat al. [37] published their work in 1992 that explained the use of a form ofwearable tremor suppression device to attenuate tremor of chronic alcoholism.This device relied on inertial loading which is a technology that has sincebecome one of the main approaches behind the design and fabrication ofwearable orthoses. Consequently, reviewing alternative therapies requires anintroduction of the various working principles of wearable tremor suppressiondevices. With each principle, we review devices produced through implementingthese techniques.The working principles of wearable tremor suppression devices (WTSD)can be classified into three categories: i) exoskeleton devices, ii) functionalelectrical stimulation (FES) devices, iii) afferent stimulation devices [9].Exoskeletons include orthoses whose functionality depends on biomechanicalloading. They combat tremor either by altering the limb impedance due tothe added inertia or through application of opposition forces to counteractinduced tremors. They can be rigid frames or flexible cables or straps.On the other hand, FES utilizes electrical stimulation to produce musclecontraction for biomechanical loading. The stimulation is above the motorthreshold hence the muscular response. Finally, afferent stimulation utilizessensory stimulation that generates a response in the central nervous system(CNS) . The stimulation is below the motor threshold thus no muscularresponse occurs and as result, there is no biomechanical loading. Thesedesign principles have been employed to create devices with varying degreesof success in tremor reduction. We review each principle in depth andhighlight some of the devices created. With variability in research for tremormitigation, there is a plethora of devices created some of which are notcaptured in this review. For different reviews of currently available devicesand their performance, comprehensive device reviews have been presentedby Lora-Millan at al. [9] and Fromme at al. [34].182.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONIt is important to introduce a key aspect of tremor suppression design herewhich is the consideration for the human arm biomechanics. Alternativetherapies, specifically those relying on biomechanical loading must notinterfere with general mobility of the human arm in their functioning. Thus,we briefly introduce the human arm biomechanics before reviewing the devicesavailable for tremor suppression.2.3.2 Brief overview on anatomy of the upper limbThe first consideration in the design of the WTSD, Fromme [34] argues thatthe degrees of freedom (DOF) of the chosen anatomical representation ofthe upper limb should match the mobility of the limb to avoid inhibition ofuser movement. By definition, the arm has 7 DOF [34, 38] beginning fromthe wrist heading towards the shoulder: in the wrist; flexion and extension(WFE) , radial and ulnar deviation (WURD) , pronation and supinationin the forearm (FPS) , flexion and extension in the elbow (EFE) , and inthe shoulder; flexion-extension (SFE) , abduction-adduction (SAA) , andinternal-external rotation (SIER) . Moreover, consideration for the location ofligaments, tendons, nerves and blood vessels is important to avoid unforeseenharm [34] especially in devices relying on electrical stimulation. Figure 2.2summarizes the aforementioned DOF of the upper limb.With the multiple DOF of the upper limb, it is important to consider theirinteraction during tremor propagation. These interactions influence thekinematics of the arm and thus tremor propagation. Therefore, propertieslike inertia, joint stiffness and damping must be considered depending onthe target DOF combination. Furthermore, the addition of the WTSD maychange these properties and can lead to a shift in tremor from the suppresseddistal joint to a proximal joint as observed by Manto at al. [39]. Manto atal. argues for caution not only design but placement of the WTSD.For optimal tremor suppressing WTSD design, discerning the propagation oftremors is key. According to Davidson and Charles [38], most mechanicalloading WTSDs have investigated tremors in singular DOF. In fact, majorityhave considered the manifestation of the tremor at the endpoint of anoutstretched arm thus targeting the suppression of WFE tremors. However,this treats tremors in the human arm independent from each other (in this caseit is as though they only originate in the wrist) as opposed to multiple tremors192.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONFigure 2.2: Human arm degrees of freedom. The shoulderhas three DOF (flexion-extension:SFE, external-internalrotation:SEIR, abduction-adduction:SAA), the elbow has1 DOF (flexion-extension:EFE),the forearm has one DOF(pronation-suppination:FPS), and the wrist as two DOF (flexionextension:WFE,radial-ulnar deviation:WRUD).202.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONpropagating and manifesting in various parts of the limb. To overcome this,they suggest considering the propagation of tremor throughout the limb byidentifying the source, distribution and key manifestations.Fundamentally, the propagation of tremors is aided by the mechanics of thelimb. That is to say, regardless of the neurological sensations or physiologicalsources that lead to tremor, they can be presented as joint torques that areattenuated by the limb mechanics whose output is manifested as the tremor[38]. The amplitude of the tremor is then dependent on the input torque tothe joint and the limb parameters of inertia, stiffness, and damping. Whileit is possible to design a WTSD by using the resultant tremors withoutconsidering the biomechanics of the arm, this approach fails to capture theinteraction between the WTSD and the limb parameters. This may lead totremor amplification in the interaction between limb and WTSD.In their work, Davidson and Charles built and simulated a 7-DOF modelof the human arm to investigate tremor propagation. They produced aset of parameters based on body segment parameters of a human male inthe 50th percentile [38] and the result was a full 7-by-7 inertia, stiffness,damping matrix summarized in Table 2.2. Notably, the parameters in thistable were determined to be independent of tremor frequency between 4-12Hz after a sensitivity analysis [38] which is ideal for modeling and studyingParkinsonian and Essential tremors.In Table 2.2, each ij (row-column) pairing represents the change in torquein the i-DOF that is associated with a change in acceleration, velocity, orposition (respective to inertia, damping, or stiffness) in the j-DOF [38]. Forinstance, if we are studying the effects of a tremor in the FPS DOF, we caninclude the effect of an induced added tremor in the WFE DOF by usingthe coupling matrix and determine its effect on the inertia, damping, andstiffness in the FPS DOF. This is a significant result as it allows analysis ofthe tremor response without having to induce tremors throughout the entirelimb.212.3.ALTERNATIVETHERAPIESFORTREMORMITIGATIONTable 2.2: Joint inertia, damping, and stiffness matrices [38] required for analysis of MDOF tremorsInteria (kg m2)WFE WRUD FPS EFE SAA SFE SIERWFE 0.003 0 0 0 0.009 0 0.011WRUD 0 0.003 0 -0.012 0 -0.014 0FPS 0 0 0.002 0 -0.002 0 0EFE 0 -0.012 0 0.076 0 0.076 0SAA 0.009 0 -0.002 0 0.196 0 0.083SFE 0 -0.014 0 0.076 0 0.269 0SIER 0.011 0 0 0 0.083 0 0.079Damping (Nms/rad)WFE WRUD FPS EFE SAA SFE SIERWFE 0.028 -0.003 0.001 0 0 0 0WRUD -0.003 0.082 0.008 0 0 0 0FPS 0.001 0.008 0.021 0 0 0 0EFE 0 0 0 0.607 0 0.187 0SAA 0 0 0 0 0.383 0.184 0.267SFE 0 0 0 0.187 0.184 0.756 0.02SIER 0 0 0 0 0.267 0.02 0.524Stiffness(Nm/rad)WFE WRUD FPS EFE SAA SFE SIERWFE 0.992 -0.099 0.018 0 0 0 0WRUD -0.099 2.92 0.291 0 0 0 0FPS 0.018 0.291 0.756 0 0 0 0EFE 0 0 0 8.67 0 2.67 0SAA 0 0 0 0 5.468 2.626 3.821SFE 0 0 0 2.67 2.626 10.8 0.279SIER 0 0 0 0 3.821 0.279 7.486222.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONThere are some important observations from the work by Davidson andCharles:1. Tremor appears to spread to a narrow set of DOF which implies thatmultiple input tremors do not necessarily mean a more complicatedanalysis. This observation is noted and further explained in thediscussion by Davidson at al.. [38].2. Tremor appears to have a proximal-distal increase in magnitude ratio[38]. This is to say that an inputs in a proximal DOF (shoulder andelbow) affect distal DOF (forearm and wrist) more than a proximalDOF. However, despite this propagation pattern, a distal input createsa larger distal response than a proximal input of equal magnitude andalso creates a bigger proximal response than a proximal input of equalmagnitude [38].3. Inertial loading produces varying results of tremor reduction dependingon the frequency; it decreases the natural frequency but also decreasesthe damping ratio which raises resonance peaks [38]. However, ittypically leads to a decrease of the magnitude ratio within the studiedfrequencies of 4-12 Hz [38].These observations imply that the design of alternative tremor mitigationsdevices can be done sequentially beginning with a single DOF torque inputand building the WTSD to attenuate the effect of the tremor on multiple DOFusing the coupling matrices. Moreover, since the tremor propagation is foundto have a larger response when it is distally induced, an appropriate vibrationinducer/simulator can be created to function at a distal location to aid islaboratory experiments. Therefore, the designed WTSD can be evaluated onits efficacy without having to go through patient testing in the prototypingphase. It is important to note that Davidson and Charles generated thecoupling matrices presented in Table 2.2 for a specific posture of the arm.This means the device must be evaluated in various postures for completeness.With the brief overview of the human arm, we continue with the review ofvarious technologies and subsequent devices attenuate tremor due to theaforementioned neurological disorders.232.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATION2.3.3 Exoskeleton DevicesBiomechanical loading has extensively been studied as a viable approach totremor suppression. In their 1974 publication, Joyce and Rack [40] foundthat the principal tremor frequency could be adjusted between 2− 12 Hz byapplying different mass and spring combinations. They noted that the limbmuscles would act as a frequency filter. Their experiments also identifiedthat certain loading scenarios of the limb could lead to either amplificationor reduction of tremor amplitudes. The work of Joyce and Rack [40] hadfocused on generic tremor but in 1999, Bilodeau at al. [41] conducted astudy investigating the impact of strength training in improving steadinessfor persons with essential tremor. In a three group study, one group trainedwith heavy weights, one with light weights, and one with no weights at all.The study subjects that trained with heavy weights showed an increasedsteadiness while the other two groups did not. However, the increasedsteadiness did not translate into improved functionality for the subjectstrained with heavy weights.Homberg at al. [42] also studied the effect of inertial loading to limb propertiesat the natural frequency. By using a combination of dorsally located wristaccelerometer and demodulating the surface Electromyography (EMG)signals, they noted a gradual decrease in peak frequency with increasinginertia for physiological tremor, similar to the behaviour vibration absorption.Additionally, they reported clearer peaks of EMG signals at higher inertias forphysiological tremors. On the contrary, the response to inertia in the case ofpathological tremors PD and ET showed relatively constant peak frequencyleading to clear distinction between tremor types. This study did not howeverreport on the changes to amplitude of tremor through inertial loading.Another study by Burne at al. [43] showed that muscle loading throughtargeted torque delivery identified the underlying tremor characteristics betterthan postural analysis used in diagnosis showing potential for better tremorunderstanding and mitigation through biomechanical loading. From the workin these studies, various exoskeleton devices have been commercialized toassist PD and ET patients.One of the first exoskeleton devices to implement biomechanical loadingas a way to combat tremor was designed by Rocon at al. [44]. TheWearable Orthosis for Tremor Assessment and Suppression (WOTAS)is an active orthosis that applies forces to mitigate tremor. It targets 3242.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONDOF in the arm; elbow flexion-extension, forearm pronation-supination,and wrist flexion-extension. WOTAS was designed with two tremor controlstrategies that would be implemented once the motion had been decipheredas voluntary or tremulous. This was achieved using a two stage phaselag measurement [9]. A passive passive control mode combated tremor byadjusting the biomechanical properties of the upper limb and an activecontrol mode through the application of opposing forces to the tremor basedon an estimate of volitional movement [44]. Patient tests were conducted on10 participants with varying degrees of ET severity and other mixed tremorsand using power spectral density (PSD) analysis, tremor was reduced by81.2% for the active control mode and 70% for the passive mode.Estimating tremulous movement when designing an active exoskeleton orthoticwas also implemented by Taheri at al. [45]. They created an adaptivesuppression algorithm capable of estimating the fundamental frequency of atremor and determining the appropriate force required to suppress it. Usinga pneumatic cylinder as part of the orthotic, the required force would begenerated and applied [45]. They tested their algorithm using data generatedfrom ten patients with severe tremor and were able to track the fundamentalfrequency with less than 3s response time [45]. Additionally, they wereable to suppress tremor movement on average by 98.1% and 74.3% at thefundamental and second-harmonic frequency with the average positionalerror on volitional movement being 2.08%.With a similar idea of estimating volitional movements, Matsumoto at al. [46]improved on their initial design to develop an exoskeleton aimed at assistingET patients when eating. The exoskeleton included a motor, connectedby gears to a rotating arm. Tremor would be suppressed by mechanicallyconstraining the elbow joint through torque generated by the motor; themotor essentially acting as a brake to the rotation of the joint. However,fully restricting elbow joint mobility during eating is not ideal and thus theyimplemented real time EMG analysis with a short-time Fourier transformalgorithm to discern volitional and tremulous movement [46]. It is reportedthat the real time EMG analysis was not integrated into their experimentalphase exoskeleton [9] but in its place, a switch triggered by an ET patient[46] was used to account for voluntary movement. Their experimental workshowed tremor reduction ranging from 50− 80% in yet another successfulshowcase of biomechanical loading as a means of combating tremor.252.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONDespite their excellent tremor suppression, the exoskeletons presented aboveare limited by their bulky form and rigid structure [9, 44]. This limits boththeir wearability as and usability in day to day activities which has beennoted to deter users [9]. This limitation has paved way for the advent of asub-group of exoskeletons often referred to as soft-exoskeletons [9]. They areless restrictive to user movement as they consist of cables and flexible wraps.For instance, the weighted bracelet for fine motor activity by Brack [47]under patent number US 2008/0072622 A1 is worn by a looping stringcontaining magnetic beads in-between non-magnetic beads. It resemblesa fashionable jewellery piece and as Brack [47] argues, users wearing thebracelet proximal to the wrist have enhanced fine-motor coordination.Zahedi at al. [48] designed a soft exoskeleton for tremor suppression (SETS)equipped with a magneto-rheological fluid as the controllable flexible actuator.This device weighed 255g and could yield upto 11N of damping force[48]. The advantage of using magneto-rheological fluid is that this devicecould proportionally respond to different tremors with varying degrees ofintensity without sacrificing user mobility by having to further restrict motion.Simulated tests indicated that the device could combat tremor in three degreesof freedom (forearm pronation-suppination, elbow flexion-extension, wristflexion-extension). Collected data showed a reduction in the magnitude ofangular acceleration by 61.39% and angular velocity by 56.22%. The authorsplan to commercialize the SETS given that it is lightweight and with asignificant damping force.The adoption of exoskeletons devices that biomechanically load the armto reduce tremor is hindered by their often bulky and in the case of softexoskeletons, lack of patient experimental verification. Thus, there is anincreasing body of research on different ways to achieve biomechanical loadingof the arm muscles without need for bulky orthoses.2.3.4 Functional Electrical StimulationThe effect of biomechanical loading can also be achieved through electricalstimulation over the motor threshold as studied by Jitkritsadakul at al. [49].By delivering constant electrical muscle stimulation (EMS) at 50 Hz to 34 PDpatients for ten seconds, they compared the response before, during and afterthe stimulation. Results from this test showed a peak amplitude reduction of262.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATION49.6±38.89% and average reduction in RMS tremor of 43.8±33.2%. Notably,when using peak amplitude 70.6% of the patients showed tremor reductiongreater than 30% while 60.1% showed greater than 30% tremor reductionwhen the RMS metric was considered. It is important to acknowledge thatthe performance of functional electrical stimulation (FES) in this instancepales compared to the exoskeleton devices previously presented. However,noting the drawbacks of bulky devices and the chance of users not respondingwell to a device, FES as a means of combating tremulous movement remainsa worthwhile approach.In a separate publication, Jitkritsadakul at al. [50] documented the fabricationand validation of a tremor glove built on the principle presented in [49]. Atotal of 30 PD patients participated in the test with 15 patients wearingthe glove and the other 15 wearing a sham device [50]. Results indicated areduction in both RMS and peak angular velocity by 56.86± 37.97% in theX-axis and 49.64± 71.48% in the Y-axis [50]. Notice that there are instanceswhere the glove caused amplification in tremor which is not unique to FEStremor mitigation approach. In fact, amplification has been noted in bothmedication [9] and even some exoskeleton orthoses [34].In practice, there are two main strategies for FES that result in tremorsuppression. The first is low current out-of-phase stimulation of musclesantagonist to that responsible for the tremor [51]. Stimulating the antagonistmuscle to that causing tremulous motion above the motor threshold causesthe muscle to produce a force sufficient to counteract that of the tremor [9].The second strategy is co-contraction where a joint impedance is increasedby stimulating the pair of antagonist muscle at the affected joint [51]. Bothof these strategies require precise timing and control of the EMS in order towork as intended.Prochazka at al. [52, 53] designed an out-of-phase based FES device by firstdetermining the relevant frequency response parameters in an open loopconfiguration. These parameters were used in a closed-loop FES systemwhere the neural activation due to tremorgenic muscles was treated as adisturbance that the closed-loop system rejected in order to achieve theparameters determined during the open loop frequency response. To avoidhaving volitional movements be rejected by the closed loop system, theybuilt an analog feedback filter such that tremor movements (2− 5 Hz) were272.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONattenuated but voluntary movements (0− 1 Hz) remained unaffected [53].Although their system showed promising results with average suppressionranging between 36− 91%, limitations like gain selection for the closed loopsystem and muscle fatigue due to constant electrical stimulation as well asthe complicated process of estimating neuromuscular response [52] meanthat the success of the device greatly varies from individual to individual.Notably, Gillard at al. [54] conducted a study comparing the use of analogand digital filters in the closed-loop FES system and found that the meanreduction for the digital filter was 84% while that of the analog filter was65% [54]. This adds to the complexity of properly designing a closed-loopFES system.Dosen at al. implemented the out-of-phase approach through a two phaseprocess that involved an initial stage of recording and analyzing EMG signalsfrom the tremogenic bursts [55]. Then in the second phase, using theiterative Hilbert Transform (IHT), the timing of tremor burst was predictedand stimulation of muscles antagonist to tremor muscles was initiated. Totest their approach, they had six patients (four with PD and two withET) presenting primarily wrist flexion-extension tremor [55]. One of theET patients did not experience any attenuation but in the other five, theattenuation ranged from 46 − 81%. One challenge with FES approach totremor mitigation is muscle fatigue due to constant electrical stimulation.Dosen at al. noted this in their work and proposed stimulation below themotor threshold as a means to overcome fatigue. This approach is coveredin the afferent stimulation section.The muscle cocontraction strategy to increase joint impedance was applied byBo at al. implementing an intermittent stimulation while patients performeda static task [56]. The validation involved 10 ET patients with 8 of thepatients experiencing average tremor amplitude reduction of 66.9± 21.7%.Of the two remaining patients, one had no change in amplitude and the otherexperienced amplification [56]. They also noted the challenge of muscle fatiguedue to constant stimulation and pointed out that since different musclesare involved during the motion of the forearm, the FES electrodes needto activate different muscles to achieve cocontraction successfully duringmotion. This may lead to errors since the timing of tremogenic muscleoutburst varies from muscle to muscle making consistent success challenging.To overcome some challenges posed by FES, it has been hypothesized thatlow-level stimulation below the motor threshold be implemented to interfere282.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONwith the cerebello-thalamo-cortical circuit [50]. This is achievable throughafferent stimulation.2.3.5 Afferent StimulationThe presence of sensory activity during tremulous movement has been studiedand documented by various authors [9, 49, 55]. Through procedures suchas EMG, the response of the central nervous system and resulting sensorystimulation during tremor can be studied and the pathways leading to eitherPD or ET can be deciphered. In fact, studies such as the one conducted byDosen at al. [55] implemented EMG to predict the timing of tremogenic burstto muscles based on the activation of motor neurons. Even some exoskeletondevices such as the one by Matsumoto at al. [46] implement real-time EMGsignal analysis to identify and target the appropriate muscles for tremorsuppression.Heo at al. investigated the effect of continuous sensory electrical stimulation(SES) for the suppression of tremors [57]. SES differs from FES in that nomuscle activation occurs during SES. In this study, Heo at al. concludedfrom tests conducted on 14 PD patients that continuous SES was effectivein decreasing angular velocity of tremor during and post the SES sessions.Additionally, they found that tremor frequency was significantly decreasedduring the session making low-level electrical stimulation, colloquially referredto as afferent stimulation, a valid approach for tremor suppression. Intheir other studies [58–60], they showed that SES was capable of reducingtremors in ET patients and even in patients with scans without evidence ofdopaminergic deficit (SWEDDs) who are often misdiagnosed with PD [60].On a different note, the team at Cala Health in a collaborative effort acrossmultiple publications [61–63] documented the effectiveness of their devicewhich worked to reduce tremor by first calibrating the tremor frequency onthe patient hand [61]. Once calibrated, the device would then stimulate themedian and radial nerves of the wrist at bursts matching the tremor frequency.Their hypothesis for this stimulation was that it would neuromodulate theventral intermediate nucleus of the thalamus [62] resulting in tremor reduction.The process of stimulation lasted 40 minutes. In total, three separate studieswere conducted and in one of the studies with 77 patients, 40 received thetreatment and 37 received a sham treatment. The results of this study were292.3. ALTERNATIVE THERAPIES FOR TREMOR MITIGATIONpresented using The Essential Tremor Rating Assessment Scale (TETRAS)Archimedes spiral task. There authors reported not significant improvementfor patients receiving the treatment compared to those who received thesham treatment [61]. However, in another study with 23 ET patients where10 received the treatment while 13 received the sham treatment [62], theassessment for tremor reduction with the Archimedes spiral from TETRASand results showed 60± 80.4% reduction [9]. This highlights the challenge inafferent stimulation and in the greater sense, combating neurological tremorsince each patient responds differently to treatment.Even with these seemingly successful devices, the challenge with afferentstimulation remains the variability in functionality of each device’s approachwhich could mean the effectiveness of each device varies from patient topatient. Additionally, since each device works based on different principles,performance comparison is challenging.The review of the various alternative approaches to design devices for tremorsuppression highlights a few noteworthy points:∗ Every approach, including medical, is aimed at mitigating the manifestationsof tremors as there is currently no cure. This implies lessons learnedfrom using a particular approach can be applied to other approaches.∗ In most cases, a situation arises where a patient experiences amplificationwhen using a device. Thus, understanding the limitations of eachapproach and the better understanding the patient remains a criticalstep in assisting patients.∗ Inertial loading devices appear to have a long history of repeated successcompared to FES and Afferent Stimulation. This can be attributed tothe application of mechanical vibration absorption techniques whichoffer optimization techniques for better performance.∗ Tremor reduction reporting is not standardized. Studies use variousquantification and reporting for device performance. This makescomparison of different devices and technologies challenging, especiallyin reviews and meta-analyses. For completeness of this review, we alsoreview the different ways in which both tremors and device performancesare reported.302.4. QUANTIFICATION AND ASSESSMENT OF TREMOR2.4 Quantification and Assessment of TremorQuantifying the severity of tremors is an important step in both identifyingthe type of tremor and selection of the appropriate therapy steps to relievethe patients from its debilitating effects. Clinically, there are three commonscales used namely the Unified Parkinson’s Disease Rating Scale (UPDRS),The Essential Tremor Rating and Assessment Scale (TETRAS) and theFahn-Tolosa-Marin scale [9]. The TETRAS scale has been widely adoptedin studies dealing with the attenuation of ET. TETRAS has two sections; aperformance section with 9 items including handwriting and voice tremorwhich are rated on a scale of 0− 4 and an activities of daily living section(ADL) with 12 items such as occupational impairment and social impactamong other which are also rated on a scale of 0− 4 [64].While these clinical scales offer a readily and easily implemented avenue forrating tremors, issues such as bias and ”white coat syndrome” where patientsexert more effort in presence of medical personnel may plague the reliabilityof the results [9]. However, tremors can also be discerned if their frequencyand amplitude are known. Given the ubiquity of inertial measurement units(IMUs) , assessing the severity of tremors can be achieved with both easeand accuracy. Moreover, the performance of any wearable tremor attenuatorcan be readily quantified by measuring the frequency and amplitude of thetremor with and without the device.The measurements of amplitude and frequency can be manipulated usingwell know metrics and tools of data assessment to study reduction valuesbeyond just averages and percentages. One such metric is the root meansquare (RMS) of the amplitude which takes into account all the recordedamplitude data (both positive and negative) and presents a figure that canbe compared between two tests. Another metric is the power spectral density(PSD) which provides different peaks based on the power at each frequency.This means the most dominant frequency appears as the highest peak on thespectrum with several peaks of other frequencies depending on the recordedspectrum. The RMS and PSD are widely adopted and specifically, the PSDmetric is often used for oscillatory motion which, by definition, tremors arerhythmic and oscillatory.312.5. TREMORS AS VIBRATIONS2.5 Tremors as VibrationsA growing body of work is the treatment of tremors as vibrations [65–69].The quantities frequency and amplitude are terms also used to quantify avibration. A vibration is described as any motion that repeats itself over aperiod of time [70] and vibrating systems are characterized by a means ofstoring potential energy (spring element), a means of storing kinetic energy(mass or inertia element) and a means by which energy is dissipated overtime (damping element). Vibrations differ from rigid body motion in thateach element of a vibrating system moves with respect to a fixed coordinatesystem. Tremors, similar to vibrations, occur with regular amplitudes andfrequencies and the human arm has inertial, stiffness and damping elementsso without loss of generality, they can be represented as vibrations. Sincetremors are unique and properties of the human arm change with orientation,this definition of tremors as vibrations must be cautiously applied.The advantage of this definition is that there are numerous vibration analysistools and techniques that can aid in developing tremor attenuation devices.One such technique is the concept of vibration absorption. In vibrationabsorption, an absorber system is designed and tuned to attenuate thevibration of a primary system [70]. In the context of tremor attenuation, theprimary system is the human arm and thus the absorber system becomes thewearable tremor suppression device. It is important to study the mechanics ofthe primary system before designing an absorber as we intend to minimize itsvibrations. Thus, understanding vibration absorption is critical in the designof a successful WTSD, especially those relying on biomechanical loading.322.6. EFFECTIVE EXOSKELETON WTSD DESIGN2.6 Effective Exoskeleton WTSD DesignIn the review of alternative therapies, exoskeleton devices appear to edgeout the FES and Afferent Stimulation devices. This is evident in Table 2.3which is an amalgamation of 20+ devices reviewed during the exploratoryphase of this research. Exoskeleton devices have the lowest mean weight1.42kg and the highest RMS reduction of 63.02%. In Figure 2.3, we can seethe distribution of reduction compared to device weight. We observe thatexoskeleton devices have a better reduction-to-weight ratio i.e. they are inthe top left corner of the graph.Table 2.3: Mean weight and reduction percentages for Exoskeleton, FunctionalElectrical Stimulation, and Afferent Stimulation Devices.Device Group Mean Weight (kg) Mean Reduction (%)Exoskeleton 1.42 63.02FES 1.51 59.03AF 1.56 56.86Additionally, exoskeleton devices target more DOFs compared to FES andAF devices. This makes them better suited to combat MDOF tremorswhen well designed compared to their counterparts. In Table 2.4, sevenExoskeleton devices target two or more DOFs compared to three fromFunctional Electrical Stimulation, and five from Afferent Stimulation.Table 2.4: Distribution of devices targeting single and multiple degrees offreedom (DOFs) for Exoskeleton, Functional Electrical Stimulation, andAfferent Stimulation devices.Device Group No. Target 1 DOF No. Target 2+ DOFExoskeleton 6 7FES 5 3AF 2 5332.6.EFFECTIVEEXOSKELETONWTSDDESIGNFigure 2.3: Scatter plot showing the reduction percentage (%) against the weight (kg) for different device groups.Exoskeletons are represented by red circles, FES by blue squares, and AF by green triangles, highlighting theperformance distribution across various devices.342.6. EFFECTIVE EXOSKELETON WTSD DESIGNNow that we have identified exoskeleton devices as superior compared toFES and AF devices, we must consider common design considerations andshortcomings that may hinder their efficacy to combat tremors. The primaryconsideration is the total device mass which must be minimized to reducestrain loading on the arm. Additionally, the burden of proper operationmust be alleviated from the patient so positioning of the device for excellentperformance must be predetermined and easily achievable by the patient. Thismeans performance optimization is key for the device to perform effectivelywithout requiring extensive training on its use. Finally, while designing taskspecific devices offer assistance to patients when performing the intendedtask, the overall quality of life is not necessarily always improved. Therefore,as a general guideline, a Weighted Tremor Suppression Device should aim toachieve the following:∗ The device should target at least 3 of the 7 identified DOF. It shouldaid those whose daily lives are affected by hand tremors by attenuatingtremors in WFE, WRUD, FPS.∗ It must avoid restricting the regular movements of the hand, wrist andfingers.∗ The design must be simple and sufficient so that little to no maintenancefrom the patient is required for optimal functionality.∗ The device must be lightweight. The weight of the WTSD should bekept under 300 grams to avoid loading of the hand which may causeother complications.∗ The device must not interfere with voluntary motion.Therefore, itshould be virtually insensitive to voluntary motion (< 3 Hz) whileattenuating involuntary motion (> 3Hz).∗ The device must be compact and conform to the aspect ratio of thehuman limb to avoid looking bulky and out of place.This list is a general guideline since every approach for WTSD design varies.For instance, when considering vibration absorber devices, optimization isoften needed for effective performance within the specified tremor frequencyrange. Whether optimization is through experimental iteration as done by[7, 71, 72], mathematical optimization such as H2 [73] or by through solvingthe differential equations as done by Gebai at al.. [68, 69], it remains amainstay in achieving the aspirations presented in these guidelines.35CHAPTER 3Theoretical FrameworkThis chapter establishes the theoretical foundation for addressing tremorattenuation by presenting generalized mathematical models of the humanarm complex. These models characterize tremors as mechanical vibrationsand analyze their dynamic behaviour across key degrees of freedom. Thepurpose of this modeling is to provide a framework for understanding themechanical challenges posed by tremors and to identify the parameters thatwill inform the subsequent design process. Detailed analyses specific to thedesign concepts are discussed in the following chapter.There is a significant departure from the neurological nature of tremors inthis chapter as we begin to frame tremors as vibrations. This enables theapplication of vibration engineering principles to address the problem. Byanalyzing the dynamics of the human arm complex at the musculoskeletallevel, this chapter provides the necessary groundwork for the subsequentdesign and development of the tremor suppression device. The insightsderived from these models guide the conceptualization phase discussed inthe next chapter.363.1. SIMPLE ARM MODELS3.1 Simple Arm ModelsWith 7 DOF for the human arm, there are multiple parameters of inertia,length, stiffness, and damping that need to be identified. Contini at al. [74]conducted a detailed study for the determination of body segment parameters.From their work, for a human male weighing 165lbs, the mass, length andcentroid locations for the arm are summarized in Table 3.1.Table 3.1: Summary of human-arm physical parametersRight Hand Length(m)Mass (kg) Desity(kg/m3) CentroidUpper Arm l1 0.364 2.07 1088 0.427l1Forearm l2 0.299 1.16 1108.6 0.417l2Palm l3 0.203 0.54 1112.6 0.361l3The kinematic parameters identified in Table 3.1 are universal to the humanarm regardless of the DOF. However, dynamic parameters such as stiffnessand damping coefficients are specific to a given DOF. Additionally, interactionsbetween DOF identified in Table 2.2 result in different dynamic propertiesthus the design, modeling and optimization of the TSD will begin with asingle DOF and add complexity when necessary to incorporate multiple DOF.Throughout this process, iterations and optimization will will be implementedto identify the optimal properties of the tuned TSD and provide insight onthe biofidelity of the biomechanical models. Figure 3.1 identifies the physicallocation of the kinematic parameters.373.1.SIMPLEARMMODELSFigure 3.1: Flexion-extension model illustrating the segments of the upper arm, forearm, and palm, with theirrespective lengths denoted as li. The centroids of each segment are represented by di, providing key parameters fordynamic modeling and analysis.383.1. SIMPLE ARM MODELS3.1.1 Flexion-Extension Lumped Parameter ModelIn Chapter 2, the Flexion-Extension (FE) and Forearm Pronation-Suppination(FPS) DOF are the most problematic DOF for neurological tremor patients.We begin our analysis with a simple arm model that combines the upperarm, forearm, and wrist and considers FE at the shoulder. The arm model isnow idealized as three point masses located at the centroid of the upper arm,forearm, and palm. The free body diagram (FBD) in Figure 3.2 representsthis idealization.Figure 3.2: Idealization of the arm in flexion-extension as three point masseslocated at the centroid of the upper arm, forearm, palm.A lumped mass model of the arm to represent a single point mass M locatedat a distance ci from the shoulder where the mass and distance are determinedby:lc =∑3i=1mi ci∑3i=1mi=m1 c1 +m2 c2 +m3 c3m1 +m2 +m3M =3∑i=1mi393.1. SIMPLE ARM MODELSThe resulting lumped mass model of the arm is shown in Figure 3.3. Usingthe properties of the arm defined in Table 3.1, the parameters for this armmodel are:M = 3.77 kg, lc = 0.3502m,L = 0.8660mDynamic Parameters: The stiffness k and damping c for the arm mustbe determined. This is done by modeling the tremor as a harmonic momentMt =Mo cos(ωt) acting on the arm. The deviation of the arm due to tremoris described by θ. The equation of motion (EOM) for the arm model isobtained using Newton’s second law for rotating systems which translates tothe sum of moments taken at the shoulder joints such that ΣM = Iθ¨. Thus:I θ¨ + c l2c θ˙ + k l2c θ =M (t) (3.1)Figure 3.3: Lumped mass model of the arm.The inertia, equivalent damping and stiffness are given by:I =M l2cceq = cl2ckeq = kl2c403.1. SIMPLE ARM MODELSThe equation is normalized by inertia to become Equation 3.2θ¨ + 2 ζ ωn θ˙ + ω2n θ =M(t) (3.2)Where: ωn =√keqIζ =ceqcc= c l22√I keqWe concern ourselves with the steady-state behaviour and make the assumptionthat the motion in steady-state is harmonic. We intend to characterize thesystem to obtain the appropriate stiffness k and damping c for the armsuch that we can observe the response over time for a given input frequency.Note that since the system is under harmonic excitation, the response is alsoharmonic and can be described by Equation 3.3.θ(t) = Θ cos(ωt− ϕ) (3.3)We now characterize the system to obtain the appropriate stiffness anddamping for the arm such that we can observe the response over time for agiven input frequency. We rewrite the response of the arm as follows:θ(t) = A cos(ωt) +B sin(ωt)Where:Θ =√A2 +B2 & ϕ = arctan(BA)The velocity and acceleration of the response are:{θ˙(t) = −ω (A sin(ωt)−B cos(ωt))θ¨(t) = −ω2 (A cos(ωt) +B sin(ωt))413.1. SIMPLE ARM MODELSSubstitute these derivations into Equation 3.2 resulting in the followingequation:−ω2(A cos(ωt) +B sin(ωt))+ 2ζωωn(−A sin(ωt) +B cos(ωt))+ ω2n(A cos(ωt) +B sin(ωt))=M(t)This equation should hold for all values of t and when evaluated at t = 0and t = π2ω (over a period), it gives the following equations:{−ω2A+ 2ζωωnB + ω2nA =Mo−ω2B − 2ζωωnA+ ω2n = 0This is a system of linear equations which can be solved algebraically for Aand B as follows:[ω2n − ω2 2ζωωn−2ζωωn ω2n − ω2][AB]=[Mo0]Solving algebraically, we obtain the constants A and B as:A =Mo(ω2n − ω2)ω4n +(4ζ2 − 2)ω2ω2n + ω4nB =2ζωωn Moω4n +(4ζ2 − 2)ω2ω2n + ω4nRecall that Θ =√A2 +B2 and thus we obtain an expression for theamplitude of the steady state response given by:Θ =Mo√(ω2n − ω2)+ (2ζωωn)2423.1. SIMPLE ARM MODELSFor the shoulder joint, the stiffness and damping have the following relationshipc = 0.002 k [68]. Making this substitution in the above equation, we obtainan expression for the maximum amplitude Θ given the mass, stiffness, tremoramplitude and frequency as follows:Θ =Mo√(kM − ω2)+(2 0.002 k2√k Mω√kM)2The tremor is modeled to have a maximum amplitude of Mo = 0.22Nmat a frequency of 2.5Hz. The maximum amplitude of the arm in SFE isapproximately Θ = 2o [75]. Substituting all the known parameters, wecalculate the stiffness k = 876.25N/m and the resulting natural frequencyof the arm is ωn = 2.43Hz with c = 1.75 Ns/m. Simulating the responseof the arm for this tremor using adaptive-step Runge-Kutta(RK) for 20s,the response is shown in Figure 3.4. Notice that the maximum amplitude isapproximately 2o and since the tremor frequency here is close to resonance,the amplitude is at its maximum here.Parkinsons and Essential tremors vary in frequency from 4Hz to 12Hz[22, 23, 33]. This variation is dependent on DOF, arm parameters amongother factors. Therefore, it is important to understand the behaviour of thearm under tremors of varying frequencies. This is achieved by determiningthe maximum amplitude in each frequency and the result is in Figure 3.5.433.1.SIMPLEARMMODELSFigure 3.4: Displacement response of the arm to 2.5Hz tremor. The system experiences a near resonance response.443.1.SIMPLEARMMODELSFigure 3.5: Normalized amplitude response of the arm over the frequency ratio r = ω/ωn. Maximum amplitudeoccurs around r = 1 when the system is at resonance.453.1. SIMPLE ARM MODELS3.1.2 Forearm Pronation-Suppination ModelThe pronation-suppination DOF is prominent on the forearm and presents achallenge to patients daily lives by impacting the movement of the forearmand wrist. It is noted that a combination of FE, specifically WFE and FPSpresent the most debilitating tremor manifestation in patients. Therefore,designing a T-Beam capable of mitigating tremors in both these DOFs isthe ultimate goal. Note that WFE is modeled and tackled similar to the FEtremors presented earlier only at a much lower inertia and tremor amplitude.The manifestation of tremor in FPS is depicted by the Figure 3.6. Furtheranalysis of this model and its biomechanical fidelity is covered in Chapter 9.Figure 3.6: Human arm model showing the pronation-suppination DOFThe dynamic model of the arm in this DOF is represented in Figure 3.7.Figure 3.7: Dynamic representation of the human arm in the FPS DOF463.1. SIMPLE ARM MODELSThe equation of motion for this system is given by Equation 3.4.J5θ¨ + c5θ˙ + k5θ = T (t) (3.4)The inertia, damping, and stiffness of the forearm can be obtained from thework done by Davidson and Charles [38]. However, we will first create amodel of our own using a similar approach to that applied in the FE DOF.We take the mass of the forearm and palm from Table 9.1 and the averageradius of the arm in this section is obtained from Contini’s work. There isconsiderable debate about the portion of the forearm to be included in theFPS inertia model since longer arm lengths result in higher inertia which canlead to inaccurate modeling and analysis. For the purposes of this study, thetwo-thirds of the forearm is used calculate an inertia of J5 = 0.00326kgm2and subsequently determine stiffness and damping. Additionally, we use thesame tremogenic torque function definition for FPS as we did for FE however,due to the reduced inertia of the arm in this DOF, we introduce a reductionfactor for the torque amplitude such that MFPSo = 0.006 ×MFEo . Usingthe equation below as for the FE model and maintaining the relationshipc = 0.002k, we can determine the stiffness for a maximum tremor amplitudeof 2o at a frequency of 2Hz.Θ =Mo√(kJ − ω2)+(2 0.002 k2√k Jω√kJ)2This results in a stiffness k5 = 0.4718Nm/rad and damping c = 9.4369 ×10−4Nms/rad. Notice how low (almost negligible) damping is for the FPSDOF compared to the FE model. The natural frequency of this DOF isωFPSn = 1.92Hz. The tremor response for this model is provided in Figure3.8.473.1. SIMPLE ARM MODELSFigure 3.8: Response of the FPS DOF to tremor.The tremor response for the FPS DOF shows a longer transient statecompared to the FE model which is attributed to the extremely low dampingas well as the low inertia. This makes the FPS DOF extremely problematicto tremor patients.With these two simplified models of the most problematic DOFs, we canestablish the theoretical foundation for evaluating tremor suppression designs.This is best achieved by introducing a simple inertial absorber model. Thismodel allows for an initial analysis of absorber performance at the resonantfrequency, serving as a baseline for understanding vibration attenuation. Theanalysis is extended to assess the absorber’s efficacy over a broader range oftremor frequencies. We seek to evaluate absorbers designs that demonstratepotential for significant tremor reduction across a wide frequency ranges sothat approach is also demonstrated here.483.2. INERTIAL VIBRATION ABSORBER3.2 Inertial Vibration AbsorberThe simplest form of a tremor attenuator is an inertial absorber consistingof a mass, spring, and damper.When placed at a distance d from the end ofthe arm in FE as depicted in Figure 3.9, the equation of motion (EOM) forthe arm-absorber complex is obtained using Lagrange’s equations as follows:T =12I1 θ˙1 +12ma x˙aHere I1 =M l2cV =12k1 l2c θ21+12ka[xa − θ1 (l − d)]2R =12c1 l2c θ˙1 +12ca(x˙a − θ˙1)The general form of Lagrange’sequations:ddt(∂L∂q˙i)− ∂L∂qi+∂ℜ∂q˙i= QiWhere L = T − VFigure 3.9: Inertial absorber with mass ma, stiffness ka and damping calocated a distance d from the end of the arm. The motion of the absorber isdescribed by xa.493.2. INERTIAL VIBRATION ABSORBERThe EOMs for this system are given by Equation 3.5 and Equation 3.6 whichare represented in matrix form by Equation 3.7.I1 θ¨1 +(c1 l2c + ca)θ˙1 − ca x˙a +((d− l)2ka + k1 l2c)θ1+ ka (d− l) xa =Mo cos (ωt) (3.5)ma x¨a − ca θ˙1 + ca x˙a + ka (d− l) θ1+ ka xa = 0 (3.6)[I1 00 ma] [θ¨1x¨a]+[c1 l2c + ca −ca−ca ca] [θ˙1x˙a]+[(d− l)2ka + k1 l2c ka (d− l)ka (d− l) ka] [θ1xa]=[M(t)0]From the theory of vibration absorption, covered in detail in Appendix A.1,a perfectly tuned inertial absorber at resonance completely minimizes theamplitude. The optimization of an inertial absorber is extensively studied invibrations [67–69, 76, 77] and Appendix A provides a detailed introductionand analysis on the topic of vibration absorption covering several optimizationtechniques. The most commonly applied optimization technique for aninertial absorber is the Den Hartog [76] approach which is given in Equation3.7 where µ = mam . For the current tremor problem, we implement an inertialabsorber with a dynamic mass of with µ = 0.102 which results in an optimalabsorber stiffness of ka = 13.5N/m and ca = 0.49Ns/m. We simulate theresponse of the arm with the inertial absorber using Matlab® and obtain thetime response at resonance shown in Figure 3.10. The RMS tremor reductionat resonance with the Den-Hartog optimized inertial absorber is 75%.ν =11 + µ; ζopta =√3 µ8 (1 + µ)(3.7)503.2.INERTIALVIBRATIONABSORBERFigure 3.10: Time response of the arm without (in blue) and with (in red) the Den-Hartog optimized inertialabsorber at 2.5Hz.513.2. INERTIAL VIBRATION ABSORBERThis is an acceptable performance for the absorber at resonance. However,neurological tremors may occur over a range of frequencies and amplitudes.For simplicity, we presume a similar tremor amplitude for all frequenciesand evaluate the multi-frequency performance for the inertial absorber. Thefrequency response curve (FRC) is shown in Figure 3.11. Observe that theamplitude at resonance is greatly reduced but the performance of the inertialabsorber is bound to a region between two frequencies near the resonantpeak. This region is the performance bandwidth [70] and signifies the regionwithin which the absorber effectiveness is at its maximum. Additionally,tremor reduction outside the performance bandwidth is effectively zero whichis seen on the purple graph with reference on the left y-axis.The bandwidth limitation of the inertial absorber is not a consequence of theDen Hartog optimization technique. In Figure 3.12, the FRCs from otheroptimization techniques for inertial absorbers are presented. We can observethat they are also incapable of increasing the effective bandwidth of theinertial absorber in addition to varying performances at resonance.This short exercise shows that the inertial absorber is not viable in tremorreduction for the frequency ranges of neurological tremors. Thus, we do notlabour to evaluate it in the FPS DOF since it does not fulfil the FE DOFcriteria. This challenge of increasing absorber bandwidth is a mainstay invibration absorber design and analysis. A potential solution is increasing theoverall mass of the absorber which is well explained in Appendix A.1 whenlooking the FRC in Figure A.3. While this approach is can be effective, it isunsuitable when tackling tremor attenuation since an increase in device massmay cause user discomfort and other associated complications (especiallywith long term use).This short foray into the fundamentals of vibration absorber evaluation usingthe inertial absorber as our example, shows that the simplified models ofthe human arm in FPS and FE DOFs are viable vibration systems. In thechapter that follows, we will see how the proposed designs perform whenevaluated on these models.523.2.INERTIALVIBRATIONABSORBERFigure 3.11: Frequency response of the arm without (in blue) and with (in red) an inertial absorber. Tremorreduction is maximum at resonance with a limited performance bandwidth.533.2.INERTIALVIBRATIONABSORBERFigure 3.12: Frequency response curves comparing optimization techniques for inertial absorber. From Top Leftgoing right: a) No absorber b) Den-Hartog [76] c)Krenk [77] d)Zilleti [67] e)Iwata & Warburton f)Miller [78] &Yamaguchi54CHAPTER 4Conceptual DesignsIn the presentation of alternative tremor therapies in Chapter 2, exoskeletonweighted tremor suppression devices stands out in their performance althoughtheir weight is a limitation. In this chapter, we focus on the conceptualizationof the weighted tremor suppression device using the theoretical frameworkin Chapter 3. We detail the process of translating the mathematical insightsinto design concepts using vibration absorber principles. Various designoptions are explored, mathematically evaluated, and refined based on thecriteria of feasibility, effectiveness, and practicality. The final device mustadhere to the guidelines in Chapter 2.6 and forms the foundation for theprototyping and implementation phase that follows.On the use of Engineering Concepts: The concepts of vibrations engineeringand review of current tremor suppression devices presented in Chapter 2 indicatethat designing a tremor attenuator involves theoretical and experimental components.In this research, the design and evaluation of the aforementioned device will bepresented in a combination of theoretical and experimental work. This is to saythat they will be used interchangeably in order to leverage their strengths insteadof using a single approach to exhaustion before employing the other.554.1. DESIGN A4.1 Design AConceptually, the most fundamental design for a vibration absorber is aninertial absorber with a mass that vertically oscillates attached to a springand damper. This design is depicted in Figure 4.1. From the theory ofvibration absorption, we can hypothesize that this design will be effectivefor tremor reduction at a single frequency for a single DOF. This is becausethe inertial absorber best performs when it is perfectly tuned to the primaryand induced frequency tremor. Therefore, we can expect its performance tohave a small performance bandwidth.Figure 4.1: Simple inertial absorber with a mass, spring, and damper.4.2 Design BThe limitation of the inertial absorber’s frequency bandwidth stems from itsreliance on the vertical oscillation of the mass for vibration absorption. Thismeans the design effectively achieves a performance at a single frequencyratio ν = ωa/ωp. This limitation can be overcome by changing the movementof the mass from vertical translation to rotational about a pivot with themass at an arm. This design concept is shown in Figure 4.2. In addition tothe rotating orientation of this system, there are two masses and two springson the opposite end of the arm which increases the effective inertia. Thehypothesis is that by parametrically selecting the appropriate combination ofmass, arm span, and the spring stiffness, the bandwidth of the absorber canbe increased leading to better performance compared to the inertial absorberin Design A.564.2. DESIGN BFigure 4.2: Absorber concept with two masses fixed on the end of an arm oflength l rotating about a pivot. Moving the masses away from the rotationcenter increases the effective inertia.The topic of bandwidth increase in absorber performance is important,especially in the context of tremors where different DOFs experience tremorsin different frequency ranges. A challenge of increasing the bandwidth is thatit does not necessarily mean equal performance for all DOFs. This is becausetremors in MDOF scenarios may have varying amplitude. This amplitudevariation may result in sub-optimal performance in one DOF while otherDOFs show ideal performance.In the pursuit of designing an attenuator with a wide performance frequencybandwidth, we can incorporate non-linear springs into the design. Theconcept of non-linearity and its implications is discussed in detail both inChapter 5 and in Appendix B. For now, we focus on the various ways inwhich non-linearity can be incorporated into design. One approach is usingmulti-stiffness springs with varying pitch or wire width to alter the stiffnesswith deflection. This approach is limited by the amount of deflection thespring experiences and the frequency of oscillation for the system. Thus, avibrating system experiencing varying frequencies and amplitudes, which canbe the case with tremors in MDOF scenarios, may not perform excellentlywith a non-liner stiffness spring that depends on deflection to change stiffnessdomains. Another approach is the use of the interaction between magneticfaces to produce magnetic springs. In fact, several publications from theHVPL [65, 66], explore the concept of magnetic springs for the rotating systemproposed above. In Figure 4.3, the prototype from the work conducted inthese studies is shown.574.2. DESIGN BFigure 4.3: Rotating actuator with magnetic springs for non-linearity. Smallmagnets are fitted either side and act as both the inertia due to their weightand springs due to their interactions.The concept of magnetic springs is not new and is quite ubiquitous inengineering applications [79–81]. Primarily, the proposed magnets for useare Neodymium magnets and some of the major benefits of incorporatingthem for springs include:∗ Neodymium magnets do not easily demagnetize thus resulting magneticsprings perform as expected for extended periods.∗ Magnetic springs can be tuned to perform in exact ranges by combiningmagnets that attract and repel.∗ Magnetic springs can be adjusted to provide high attraction or repulsiveforces even when small in size making them ideal for size reductionthus improving aspect ratio.The hypothesis behind the use of magnetic springs in this design is as follows:magnetic springs will provide the restoring force when a tremor causes theT-shaped arm to rotate from its equilibrium. This means that high amplitudedisplacement of the arm is possible provided the magnetic field (read magneticsprings) is strong enough to restore to equilibrium.584.2. DESIGN BThe prototype design of the rotating system in Figure 4.3 incorporates highstrength Neodymium magnets (grade N52) which results in a high stiffnesswhen the system is at equilibrium. With a high baseline stiffness and relativelylow dynamic mass of the magnets, we can anticipate this design to primarilytarget High Frequency Low Amplitude (HFLA) tremors with great success.However, as discussed earlier, the targeted Parkinsonian tremors range from4 − 6Hz and Essential Tremors range from 4 − 12Hz. Additionally, wenote that generally, low frequency tremors are often accompanied with hightremor amplitude. This means the current proposed prototype may havea smaller bandwidth even with the non-linear springs incorporated. Tomitigate this, two design modifications are conceptualized to alter the highbaseline stiffness at equilibrium with the intent of increasing the performancebandwidth.These proposed designs can achieve increased bandwidth either by: a)lowering the stiffness at equilibrium and increasing the stiffness at maximumdeflection. This would allow response to low frequency; b) by controlling theoverlap between the two magnetic faces as the system oscillates. This wouldallow a smoother stiffness profile throughout the dynamic range of the devicethus targeting both low and high amplitude tremors. In addition to thesedesign changes for stiffness, another proposed modification is the addition ofdynamic masses instead of relying on the weight of the magnets. This couldpotentially improve performance by changing the inertia ratio µ = Ia/Iarmbetween the absorber and the arm for a given DOF. Finally, we include abearing to reduce the friction at the pivot and introduce a viscous damperin the system. These two design concepts are discussed in detail below.594.2. DESIGN BIteration 1: Maximum Stiffness at Maximum DeflectionThe original prototype is ineffective at low frequencies due to the highbaseline stiffness but with high stiffness, a greater restoring force is achievedwhen displaced from equilibrium. The proposed solution to this challengeis to design the absorber such that the bottom magnets are at angle to thehorizontal plane as shown in Figure 4.4.During the operation for this design concept, when the device is at equilibrium,the minimal overlap between the two magnetic faces would result in a lowequilibrium stiffness. As the system oscillates towards the extreme positions,the two magnetic faces would meet axially with complete overlap (shown inred hatch). This would increase the stiffness and subsequently the repulsiveforce and restore the system to equilibrium. The hypothesis here is thatthe device can respond to low and high frequency tremors since the baselinestiffness is reduced compared to the previous prototype.Figure 4.4: Magnetic Rotating System with Base Magnets at an Angle:Maximum Stiffness at Extreme Positions. Angled bottom magnet to ensureaxial magnetic interaction (red hatch) at extreme position to maximize therestoring force.604.2. DESIGN BIteration 2: Controlling Overlap AreaThe first iteration would greatly reduce the stiffness at baseline given theorientation of the bottom magnet. A glaring concern is that this mayextremely lower the baseline stiffness and result in an overall system stiffnessthat is completely outside the limits of the targeted tremor frequency range.This second iteration aims to pre-emptively mitigate this by proposing adesign that achieves low equilibrium stiffness by setting the two opposingmagnetic faces off-center. In the initial prototype shown in Figure 4.3, thetop and bottom magnets axially constrained at rest. This is what results ina high initial stiffness. Consequently, when maximum displacement occurs,the overlap area is at its minimum and the restoring force is greatly reduced.This potentially makes the system ineffective for combating low frequencyhigh amplitude (LFHA) tremors. Figure 4.5 shows a sketch of the currentprototype overlap area at rest (in red hatch) and in blue (for maximumdisplacement).Figure 4.5: Magnetic face overlap for the current prototype at rest (inred) resulting in a high equilibrium stiffness and reduced overlap (in blue)resulting in a low stiffness and consequently low restoring force at maximumdisplacement.614.2. DESIGN BWe surmise from this depiction that reduction in overlap decreases theeffective magnetic field interaction between the two magnetic faces. Since thefringe magnetic fields are negligible compared to the core field, the effectivespring stiffness is significantly decreased as the deflection angle decreases i.e.magnetic stiffness decreases from equilibrium to maximum deflection. Thesecond iteration solves this by offsetting the centers of the top and bottommagnets such that they are no longer equidistant from the center as shownin Figure 4.6.Figure 4.6: Offset magnetic centers to minimize initial baseline stiffness.In this design, the overlap area is reduced at rest which translates to lowerequilibrium stiffness compared to the current prototype. As the system isdisturbed from equilibrium, the overlap area grows consequently increasingthe stiffness and the restoring force which is maximum the extreme position.We can geometrically optimize this design such that the two magnetic facescompletely overlap at the maximum deflection. Figure 4.7 shows the overlaparea between the two magnetic faces.624.2.DESIGNBFigure 4.7: Conceptual Iteration 2 to increase the overlap area over the range of motion from equilibrium tomaximum displacement. The overlap area shown in red hatch is maximized at maximum deflection. This reducesthe baseline stiffness at equilibrium allowing the system to respond to low amplitude tremors. At maximumdeflection, the magnetic faces have maximum interaction resulting in high stiffness allowing the system to respondto high amplitude tremors.634.3. DESIGN C4.3 Design CThe potential merits of Design B and its iterations are yet to be evaluatedbut a glaring challenge is its likely inability to combat multi-degree offreedom tremors since it is a single DOF system. We can anticipate this byconsidering that the necessary combination of mass, stiffness, and arm spanfor one DOF may not necessarily imply similar performance in another DOF.This is because, as discussed and presented in Table 2.2, each DOF has anassociated inertia, stiffness, and damping thus tremors present differentlyand expectedly, the combating mechanisms must be different.A possible solution to this predicament could be the design and optimizationof multiple devices based on Design B and provide each device as anindependent solution for each DOF. However, this is not the goal of thisresearch and additionally, such solutions already exist as activity specificdevices as presented in Chapter 2. Moreover, requiring patients to wearmultiple devices at a time for tremor mitigation is both cumbersome andprone to failure when the incorrect device is used.We can overcome both these problems by designing a device that combinesthe two devices in one as shown in Figure 4.8. We would need to incorporatea swivel bearing to the top of the arm to allow pivoting during tremogenicmovement but lock the rotational movement of the device to avoid magneticmisalignment. Placing the swivel bearing at the top is advantageous sincebottom placement of the swivel may become problematic when large amplitudetremors occur. This is because of the increased moment arm would require ahigher restoring force to restore equilibrium.644.3.DESIGNCFigure 4.8: Conceptual design for combined single DOF systems to combat MDOF tremors. Each individual SDOFdesign is optimized for a specific DOF allowing for optimal performance in MDOF.654.3. DESIGN CBy combining two independently optimized absorbers from design B intoone device, we would eliminate the need for patients to wear to devices andreduce the risk of wrong device-DOF combination.An obvious challenge for this proposed design is instability and imbalancesince two differently optimized devices will likely have differing massesand magnetic spring strengths. Thus, blindly combining two differentlyoptimized devices to target MDOF tremors may prove ineffective in practice.Additionally, the problem of incorrect DOF association remains a potentialfailure since this design must be worn in the orientation of optimized DOFs toeffectively reduce tremors. Another potential challenge on the topic of DOFsis that this design may also face challenges when the user is transitioningbetween DOFs (for instance during an activity like drinking). This is becauseits optimal performance depends on the correct DOF association.These potential shortfalls can be mitigated by designing a device that operatesequally in all orientations and all DOFs. Admittedly, this may result inperformance trade-off in certain DOFs. However, through iterative analysis,we can sufficiently improve performance to be within acceptable margins.One potential design alteration for equi-performance for all DOFs andorientations is shown by the conceptual sketch in Figure 4.9. By placingmultiple smaller masses and magnets in a circular shape instead of twocrossing rotating systems based on Design B, this design would effectivelymimic multiple rotating systems at once. The advantage here would beequal performance for all DOFs regardless of device orientation and thiswould require no prior information to the user. Moreover, the use of smallermasses on the periphery would increase the device inertia leading to a betterinertia ratio and potentially reduce tremors more effectively. These potentialbenefits would make the device more robust and likely result in a morereliable performance spectrum.664.3.DESIGNCFigure 4.9: Modified MDOF device with multiple equal masses and magnets for stability and even magnetic field.This device is potentially more stable than the previous concept and offers a uniform performance regardless oforientation.674.3. DESIGN CFrom a manufacturing standpoint, this improved concept device is quitechallenging to assemble given the number of magnets and masses required perdevice. Given that these are high strength Neodymium magnets, assemblingthis many small magnets is a difficult process because the attractive forcesbetween the magnets may dislodge them from their sockets. In addition tothis, since there are magnets on the underside of the top disk, misalignmentmay still occur as the magnets seek opposite poles to attract. This mayresult in a device locked in a single position rendering it ineffective. Aproposed design solution to this problem is having creating a device that usesa magnetic pair from two ring magnets. These ring magnets would producea uniform field eliminating any potential attraction between magnets thatmay result in dislodging. In the manufacturing phase, the ring magnets canbe made thin enough that their weight is negligible both for assembly andthe overall dynamic inertia of the system. Finally, instead of using smallmasses in the periphery of the device, a single ring mass made from a densenon-magnetic material can be implemented. This would simplify assemblyand have a uniform inertia regardless of tremor orientation.Another challenge with this current design concept is having a single heightbetween the magnetic faces. This results in a single equilibrium stiffnessand thus a limited bandwidth for optimal operation. This means that if theheight is designed to have an equilibrium stiffness tuned for lower frequencytremors, higher frequency tremors would be less attenuated and vice versa.To combat the potential problem of having a single equilibrium stiffness,the design is modified to include height changing capabilities which wouldallow the equilibrium distance between the top and bottom magnets to bechanged. This would result in a lower or higher equilibrium stiffness basedon the selected height option. The hypothesis is that a higher stiffnessat equilibrium would target high frequency tremors and conversely, lowerequilibrium stiffness would attenuate lower frequencies since the inertiaremains constant. The new conceptual design that incorporates these changesis shown in Figure 4.10. As a final note, the swivel bearing should be lowfriction to allow for maximum displacement when tremors occur.684.3.DESIGNCFigure 4.10: Updated MDOF device using ring magnets to provide an even field and doughnut mass for higherinertia with low weight. Variable height allows changing equilibrium stiffness to target different tremor frequencybandwidth. The ”effective springs” are highlighted to show magnetic field uniformity and do not representmechanical springs.69CHAPTER 5Detailed Mathematical ModelingThe conceptual designs introduced in the previous chapter are potentialsolutions to addressing MDOF tremors. In this chapter,exhaustive mathematicalanalysis of the proposed tremor suppression designs is provided. The analysisfocuses on theoretical modeling and the system dynamics of each device fromthe perspective of vibration absorption. By applying advanced mathematicaltechniques and some key assumptions, we evaluate the feasibility, performance,and optimization of the designs under various conditions. The insights gainedfrom this analysis serve to validate the conceptual designs and identify themost effective configurations for tremor attenuation.Modeling Assumptions: In this chapter, some key assumptions are madewhen mathematically modeling the systems:∗ The masses on they T-Beam are treated as point masses and the weight ofthe connecting links in negligible.∗ The magnetic fields are uniform and fringe magnetic effects are negligible.∗ The dynamic parameters of the human arm remain constant over multiplefrequencies and coulomb damping is negligible.∗ Geometric and boundary non-linearity effects are negligible and the onlymaterial non-linearity comes from the magnetic field interactions.705.1. DESIGN B: THE T-BEAM ABSORBER5.1 Design B: The T-Beam AbsorberDesign B is based on the one proposed by the Heart Valve Performance Lab(HVPL). It is a see-saw like actuator with a T-Shaped arm connected toa base with a pivot in the centre [66] [66] (hereinafter T-Beam). Figure5.1 shows the T-Beam design with the dimensions and components labelled.The design consists of two masses mT at a distance a from the center of theT with a height q. The pivot is a bearing with damping cb and two springsof equal stiffness k1 = k2 = kb connect the masses on the T to the base.Figure 5.1: See-saw like absorber (T-Beam) design with two masses, twosprings, and a bearing in the pivot acting as a damper.Consider the T-Beam placed on the arm as depicted in Figure 5.2 at adistance d from the end. For initial analysis, the T-Beam is placed furthestaway from the shoulder such that d ≈ 0.Figure 5.2: T-Beam and arm complex for the flexion-extension DOF.715.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.3 shows the FDB for the T-Beam when displaced from equilibriumdue to a tremor in the arm. The displacement of the T-Beam is described bythe angle θ the T makes with the horizontal base. The vertical componentof the force due to the springs creates a moment about the pivot.Figure 5.3: T-Beam FBD displaced from equilibrium. θb describes the motionof the T-Beam.The EOMs for this system are derived by taking the sum of moments aboutthe shoulder joint and the pivot point for the T-Beam. The resulting EOMsare given by Equation 5.1 and Equation 5.2. Notice that the T-Beam ismodeled with linear springs instead of the non-linear springs proposed byMasoumi at al. [65]. This is to identify the point at which non-linearitybecomes a necessity.725.1. DESIGN B: THE T-BEAM ABSORBERI1 θ¨1 + k1 l2c θ1 + c1 l2c θ˙1 + kb1 (a sin α− x1) (l − d+ a) cos θ1 + · · ·+kb2 (a sin α− x2) (l − d− a) cos θ1+cb(θ˙1 − θ˙b)(l − d) cos θ1 =M (t)(5.1)Ib θ¨b + cb(θ˙b − θ˙1)+ kb1 (a sin α− x1) (q cos θb + a cos α) + · · ·+ kb2 (a sin α− x2) (a cos α− q cos θb) = 0 (5.2)Where :x1 = (l − d+ a) θ1x2 = (l − d− a) θ1α = π2 − θbI1 =M l2cIb = 2 mT(a2 + q2)The performance of the T-Beam at resonance, using the same kinematicand dynamic parameters as the inertial absorber shown in Chapter 3.2 witha = 2cm and q = 2cm, results in Figure 5.4. The T-Beam has an RMSreduction of 75%but also shows a transient response which was absent fromthe inertial absorber response. While this performance meets the minimumexpectation of 50% reduction, it is sub-par since the T-Beam has doublethe weight of the inertial absorber thus it should theoretically perform better.If we observe the movement of the T-Beam in Figure 5.5, we see that itspeak to peak amplitude is 1.6o in the transient state and 0.8o in steady-state.Theoretically, the T-Beam in its present geometric configuration can achievea 45o peak to peak amplitude i.e. 22.5o maximum displacement fromequilibrium. Therefore, analyzing the T-Beam at multiple frequencies isimportant to investigate if higher displacements from equilibrium lead resultin better performance.735.1.DESIGNB:THET-BEAMABSORBERFigure 5.4: Time response of primary system without absorbers (blue), with inertial absorber (red), with T-Beamabsorber (yellow). The T-Beam has lower reduction in this configuration with a slight beat phenomenon.745.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.5: Time response of the T-Beam when the primary system experiencenear resonance tremorThe performance of the T-Beam compared to the absorberless and inertialabsorber responses is provided in Figure 5.6. This is a fundamental result asit shows the unoptimized T-Beam reducing tremors over a wider bandwidthcompared to the optimized inertial absorber. Notably, the inertial absorberstill outperforms the T-Beam at resonance but its bandwidth is limited. Thismay be initially attributed to the overall increased mass and stiffness of theT-Beam but inspecting the FRC also indicates a shift in the amplitude ofthe response which is not necessarily a consequence of increased mass. Thisresult encourages further analysis for better understanding.755.1.DESIGNB:THET-BEAMABSORBERFigure 5.6: Frequency response of absorberless(blue), with inertial absorber(green), and with T-Beam absorber(red)for the primary system.765.1. DESIGN B: THE T-BEAM ABSORBERIf we compare the RMS tremor reduction over the range of frequencies resultsin Figure 5.7, we notice that the both the optimized inertial absorber andthe unoptimized T-Beam cause an amplification just before resonance. Theirperformance then improves and peaks at resonance but for the inertialabsorber, the performance rapidly plummets past resonance while theperformance of the T-Beam gradually decreases. This establishes thatthe unoptimized T-Beam is better performing over the range of frequencies.With these results, we further investigate the influence of individual T-Beamparameters on its performance to fundamentally determine if there is a betterconfiguration to achieve acceptable tremor reduction across the frequencybandwidth.Figure 5.7: Tremor reduction frequency response for the inertial (blue) andT-Beam (purple) absorbers.775.1. DESIGN B: THE T-BEAM ABSORBER5.1.1 Performance AnalysisWe have observed that the T-Beam does not outperform the inertial absorberat resonance even with a higher dynamic mass and stiffness. However, whenobserving the frequency response for a range of tremor frequencies, theviability of the T-Beam becomes clear. Its performance bandwidth andperformance fall off are both better compared to the inertial absorber. Thereare some important points to be made from this:∗ The inertial absorber is capable of performance tuning to target theresonance frequency but becomes limited due to its bandwidth. Thisis known limitation with inertial vibration absorbers [70].∗ The influence of T-Beam mass, stiffness, length, height, and dampingcoefficient must be studied to fundamentally understand how theyimpact the T-Beam performance.We now explore the influence of each T-Beam parameter to its performance.Once we have established the impacts, we can select the optimal combinationof parameters for maximized performance either at resonance or over afrequency range. The selection of ideal parameter combination can beachieved either by blind variation of parameters or through optimizationwith the appropriate parameter ranges determined from the parametricevaluation.5.1.2 Parametric EvaluationIn order to determine the influence of each parameter, we take a simplifiedapproach of varying one parameter within a specified range while keeping allother parameters fixed from the base performance case from the previousresults. This avoids confounding parameter influence. We can then compareeach variation to the base absorberless system to understand the fundamentalimpact to performance.785.1. DESIGN B: THE T-BEAM ABSORBERThe range of evaluation for each parameter is provided below as well as theprediction of how changing each parameter influences in the performance.These ranges are selected such that the resulting T-Beam would still adhereto the WTSD guidelines provided at the end of Chapter 2. The hypothesisof the impact of each parameter change provided here serves as a basis ofevaluating the T-Beam’s adherence to known vibration absorber principles.This is important in both design analysis and future performance predictions.∗ mT = [40, 80, 120]g. The limitation here is such that the total T-Beammass must not exceed 300g so that it is not heavy on the arm thusminimizing strain loading [82]. The dynamic mass is one componentthe total mass limitation thus at the minimum, 60g is available for theconstruction components of the T-Beam. This target can be achievedusing high strength plastic or 3D printed components.∗ kb1 = kb2 = kb = [5, 15, 30]N/m. Note that stiffness changes thenatural frequency ratio of the T-Beam to the Arm and affects the”tuning” of the T-Beam. Thus, stiffness is expected to affect thefrequency at which the T-Beam has its peak amplitude response. Thereare other implications to changing the stiffness but this effect shouldbe readily observable.∗ cb = [0.01, 0.1, 0.35]Ns/m. The damping coefficient is expected toprimarily affect the peak of the amplitude and the phase shift introducedwith a damping coefficient.∗ a = [5, 20, 40]mm. Changing the length of the T changes the inertiaof the T-Beam which is expected to impact the tremor reduction. Itis important to observe how significant this change is. Notice thatthe range here is quite small. This is done deliberately to ensure theparameter remains within an acceptable margin so the final device fitswith the proportions of the human arm.∗ q = [5, 20, 80]mm. Similar to the length of the T and its impacts, theheight of the T is constrained within these specified limits.795.1. DESIGN B: THE T-BEAM ABSORBERDynamic Parameters: Mass, Stiffness, DampingMass: By varying the T-Beam mass in the given range, we obtain theresponse shown in Figure 5.8. The results indicate that the change in dynamicmass alone does not significantly affect the performance. From vibrationabsorber design, it is logical that a higher mass reduces the maximum tremoramplitude. However, for the above analysis, increasing the mass from 40g to120g increases the new peak resonance reduction by 0.9%. From the inertiadefinition for the T-Beam, this mass increase results in a 33% increase ininertia which should result in better reduction but that is not the case. Thus,it is likely the parametric combination may result in a T-Beam outside ofthe performance range for the specified tremor.Figure 5.8: Response of primary system with varying mass of T-Beam forthree masses. Performance is largely unchanged for a 3X mass increase.805.1. DESIGN B: THE T-BEAM ABSORBERStiffness: Conducting a similar analysis for the stiffness yields the resultsdisplayed in Figure 5.9. For lower stiffness values, the response of the absorbersystem resembles that of the non absorber system. As stiffness increases,the resonance point shifts towards the left mitigating the initial primaryresonance. This leads to tremor reduction at and after resonance.Further increase in stiffness introduces a new resonance peak. This isnoteworthy since a sub-resonance tremor may lead to amplification. Thisalso signifies that infinitely increasing the stiffness of the system with fixedparameters will not result in better performance implying the existence ofan optimal value of stiffness for a given combination of parameters. Finally,notice that increasing stiffness increases then decreases the effectivenessbandwidth but has no impact on the amplitude of tremor at resonance. Thus,other parameters must be adjusted to minimize the peak response.Figure 5.9: Response of primary system for varying T-Beam stiffness.Stiffness changes affect resonance performance, introduce new resonancepeaks, and affect the performance bandwidth.815.1. DESIGN B: THE T-BEAM ABSORBERDamping: Varying the damping coefficient yields the results displayed inFigure 5.10. For very low damping (c = 0.01Ns/m), tremor reductionis confined to the region around resonance with amplification over therest of the frequency range. As the damping increases, the maximumamplitude is decreased while the resonance peak shifts to the left. However,continued increase past a certain value causes a decrease in performance i.ethe maximum amplitude at resonance increases beyond the original maximumamplitude. This suggests the existence of an optimal damping.Figure 5.10: Response of primary system for varying T-Beam damping.Performance indicates existence of an optimal damping value for eachparametric combination.825.1. DESIGN B: THE T-BEAM ABSORBERGeometric Parameters: T-Beam Length and HeightIt is also essential to understand the influence of the geometric parametersto the T-Beam’s performance. This is important as we want to minimize thesize of the T-Beam to conform its aspect ratio to that the human arm. Webegin with varying the T-Beam arm length parameter a and the response isshown in Figure 5.12. We make the following observations:∗ Very low values of a lead to increased amplitude at resonance whilealso shifting the resonance frequency. This is to be expected as theparameter influences the inertia and stiffness elements in the equationof motion.∗ As the T length increases, the resonant amplitude as well as theeffectiveness range are improved positively upto a certain point whereperformance is optimal. However, further increase leads to amplification.This implies that the T length is vital in maximizing the effective bandwidthof the absorber and reducing peak amplitude at the new resonant point.For the variation of the T height q , the result is shown in Figure 5.12.Notice that varying the parameter does not have significant impact onthe performance apart from shifting the resonance peak. However, sinceconfounding effects are not studied here, it would be incorrect to concludethis parameter has minimal influence in performance.835.1.DESIGNB:THET-BEAMABSORBERFigure 5.11: Performance of the T-Beam when varying T length a. Increasing length is effective but has limitedimprovement range for a given frequency.845.1.DESIGNB:THET-BEAMABSORBERFigure 5.12: Performance of the T-Beam when varying T height q. The T-Beam height change is not effective onits own.855.1. DESIGN B: THE T-BEAM ABSORBER5.1.3 Optimizing the T-Beam for a broader bandwidthThe analysis of individual parametric influences to the performance of theT-Beam has revealed that there exists optimal combinations for performance.We can determine these optimal combinations that maximize tremor reductionmanually by changing the parameters based on the findings of this analysis.However, we have only analyzed the influence of each parameter but havenot observed the confounding effect of multi-parameter adjustment. Thiscan be achieved using either full or partial factorial Design of Experiment(DoE) approach which for five factors at a minimum of three levels is 243experiments.This approach is both computationally intensive and not viable at thisstage of research. Thus, we employ optimization algorithms that studythe tremor reduction either at the resonant frequency or over a range offrequencies by manipulating the parameters using a defined objective functionfor minimization. For example, Borges [83] implemented a genetic algorithmin the study of a non-linear absorber to optimize its performance over awide frequency range with great success. Therefore, we implement a readilyavailable algorithm called Particle Swarm Optimization (PSO) algorithmwhich is built into Matlab® and the implementation script is provided inAppendix C.2. We optimize for performance at resonance with the followingparameter ranges:∗ mT = 10→ 120g. The mass of the T-Beam must be low to avoid strainloading the arm.∗ kb1 = kb2 = kb = 20 → 200N/m. The stiffness was observed to shiftthe resonant peak further to the left and increasing the reductionbandwidth hence a wider range for this parameter.∗ cb = 0.01→ 0.5Ns/m. Damping influence depends on the combinationsof other parameters hence a wider range.∗ a = 10→ 30mm and q = 10→ 30mm. The desire is to maintain theaspect ratio of the T-Beam on the arm so the geometric parametersare equally constrained.865.1. DESIGN B: THE T-BEAM ABSORBERThe objective function for optimal performance is the reduction percentageat a tremor frequency of 2.5Hz which is near the resonance of the primarysystem. We calculate tremor reduction as the Root Mean Square (RMS)difference between the absorberless and absorber system. Executing the PSOalgorithm, we obtain the following optimal parameters:∗ m = 120g∗ kb1 = 64.4N/m∗ kb2 = kb1N/m∗ cb = 0.1Ns/m∗ a = 30mm∗ q = 0.5mmSimulating the system with these parameters at 2.5Hz, we obtain the timeresponse in Figure 5.5. Observe that the optimized T-Beam now performsmuch better than the inertial absorber at this frequency. The RMS tremorreduction is calculated to be 99.34%. This is a remarkable result as theT-Beam not only reduces the steady-state but also the transienttremor amplitude.Evaluating the optimal T-Beam over the frequency range, the results aredisplayed in Figure 5.14 and the RMS reduction is given in Figure 5.15.We can see that the T-Beam outperforms the inertial absorber for most ofthe relevant frequency spectrum. However, the T-Beam introduces two newresonant peaks at much lower frequencies but they are negligible since tremors,as noted earlier, typically occur in the 4− 12Hz range. This theoreticallyestablishes the T-Beam as a viable tremor suppression device.875.1.DESIGNB:THET-BEAMABSORBERFigure 5.13: Performance of optimized Inertial and T-Beam absorbers at 2.5Hz.885.1.DESIGNB:THET-BEAMABSORBERFigure 5.14: Frequency response without(blue), with optimized inertial (green), and optimized T-Beam (red)absorbers.895.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.15: Tremor reduction for the optimized inertial (blue) and T-Beam(purple) absorbers.With this satisfactory performance in the FE DOF, both at resonance andover the entire relevant tremor frequency range, we can analyse the T-Beam’sperformance in the FPS DOF which is the other problematic DOF discussedearlier.NOTE: We modify the T-Beam by moving the pivot from the bottom tothe top. This is done as a precaution to improve stability in the physicaldesign for better performance in FPS. Analytically, the performance of theT-Beam does not change. The analysis of performance similarity is coveredin Appendix A.2.905.1. DESIGN B: THE T-BEAM ABSORBER5.1.4 Evaluating the T-Beam in FPSThe evaluation of the T-Beam in the FE DOF and the subsequent bandwidthoptimization has resulted in a T-Beam that is capable of multi-frequencytremor reduction. Now, we evaluate the optimal configuration of the T-Beamin the FPS DOF to determine whether the same parametric combination forthe T-Beam is effective in both FE and FPS DOFs. We begin by representingthe FPS and T-Beam complex in Figure 5.16.Figure 5.16: Dynamic representation of the T-Beam in the FPS DOFWe derive the EOM for this system using Newton’s 2nd law and the resultingEOMs are given by Equation 5.3 and Equation 5.4 where Ib = 2 mT a2 (notethe change in inertia definition due to the change in pivot location).J5θ¨5 + cθ˙5 + kθ5 + 2kb1a2 sin(θ5 − θb) cos(θb) + cb(θ˙5 − θ˙b) = T (t) (5.3)Ibθ¨b + cb(θ˙b − θ˙5) + 2kaa2 sin(θb − θ5) cos(θb) = 0 (5.4)915.1. DESIGN B: THE T-BEAM ABSORBERWith the equations of motion, we numerically study the response of theT-Beam. At this juncture, it becomes important to understand the contributionof the added T-Beam mass to the reduction of tremor. This is because byadding inertia, we can reduce vibrations as explained in Appendix A.1. Thus,we now and hereinafter compare the absorbers to the absorberless system aswell as the response due to the added inertia in a concept we call ”sham”.The performance of the T-Beam and the sham for the FPS DOF is given inFigure 5.17.The RMS reduction for both the T-Beam and the Sham is 7.3% indicatingthat this reduction is as a result of the added inertia and the dynamic systemis ineffective. This is the fundamental challenge in designing a multi-degreeof freedom tremor attenuator since the optimized T-Beam configuration inthe FE DOF reduced tremor by 92% but completely fails in the FPS DOF.Figure 5.17: T-Beam Performance in FPS compared to Sham andAbsorberless system. The T-Beam does not perform as well in FPS.925.1. DESIGN B: THE T-BEAM ABSORBERThe T-Beam’s movement is approximately 1o in the steady-state of the FPStremor which is much less than the theoretical maximum of 22.5o thus it isno surprise the performance is abysmal. We highlight two potential factorscontributing to this performance:∗ The T-Beam has been optimized for FE tremors over a wide frequencyrange. Additionally, the optimization resulted in two new resonantpeaks below the FE resonant frequency which is at 2.5Hz. This is acontributing factor to the poor performance in the FPS DOF since theT-Beam performance in this frequency range is extremely poor.∗ The lower inertia and tremor of FPS compared to the FE DOF mayresult in a mismatched combination of the T-Beam parameters andarm hence the poor performance.Logically, we optimize the T-Beam for the FPS tremors to determine thenecessary parametric configuration for excellent performance in this DOF.Using the PSO algorithm with the same parameter bounds, the optimalparameters for FPS are:∗ m = 120g∗ kb1 = kb2 = 53.2913Nm/rad∗ cb = 0.1Nsm/rad∗ a = 30mmThe performance of the optimized T-Beam is given in Figure 5.18. TheT-Beam now has an RMS reduction of 48% in FPS (41% more than the FEOptimized configuration). It also reduces both transient and steady-statetremors and outperforms the sham scenario. Although this is an improvementcompared to the previous version, this is still below the performance thresholdof 50%. Additionally, this comparably lower performance of the optimizedT-Beam in FPS further highlights the glaring concern that two differentT-Beams are needed for tremor reduction to be achieved in bothFPS and FE tremors.935.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.18: Optimized T-Beam Performance in FPS compared toAbsorberless system.The sub-par performance of the optimized T-Beam in FPS also highlightsthat it is likely the optimal parameters for the T-Beam to perform wellin the FPS DOF may lie outside the bounds used for optimization in theFE DOF. We investigate this by changing the limits for mass and dampingin the PSO optimizer such that the new limits are are mT = 10 → 140gcb = 0.01→ 2Nms/rad. Conducting the optimization with these limits resultsin an optimal mass of 140g and damping of 0.01Nms/rad. With this mass,the total T-Beam weight is approximately 340g presuming the fabricationmaterial mass is still approximately 60g. This is a heavier device than whatis outlined in the guidelines. Even with these parameters, the T-Beamperformance improves only by 5% at resonance indicating that the 16%increase in mass does not drastically improve the performance. We thereforeexplore different approaches to MDOF tremor attenuation using the T-Beam.945.1. DESIGN B: THE T-BEAM ABSORBER5.1.5 Exploring the T-Beam for MDOF TremorsWe have established that the optimal parametric T-Beam configurations forFPS and FE are different. This becomes a challenge in the design of theT-Beam since no single configuration has performed acceptably in both DOFs.Therefore, we explore three potential solutions to combat this challenge andproduce a combination capable of tremor suppression in both multiple DOFs.Option 1: Dual T-Beam ConfigurationThe preceding analysis has shown that both DOFs require different configurationsto achieve acceptable tremor reduction. As we aim to design a single solutioncapable of attenuating both tremors, we evaluate the impact of having bothT-Beams. That is, studying the response in both DOFs when using a twoT-Beam configuration although it is worth noting that the only differencesare the mass and stiffness of the two T-Beams.Flexion-Extension: For the FE DOF, Figure 5.19 shows the setup withtwo T-Beams. Note here that the first T-Beam is optimized for FE and theother is optimized for FPS.Figure 5.19: Dual T-Beam configuration for the FE DOF using the FE andFPS optimized T-Beams. The FE optimized T-Beam is furthest away fromthe shoulder.955.1. DESIGN B: THE T-BEAM ABSORBERWe model this system as a 3 DOF system with the EOMs as follows:I1θ¨1 + k1l2cθ1 + c1l2c θ˙1 + kb1(x1 − a sin θb1)(l − d1 + a) cos θ1+ kb2(x2 − a sin θb1)(l − d1 − a) cos θ1 + cb(θ˙1 − θ˙b1)(l − d1) cos θ1 + . . .+ kb1(x3 − a sin θb2)(l − d2 + a) cos θ1 + kb2(x4 − a sin θb2)(l − d2 − a) cos θ1+ cb(θ˙1 − θ˙b2)(l − d2) cos θ1 =M(t) (5.5)Ibθ¨b1 + cb(θ˙b1 − θ˙1) + kb1(a sin θb1 − x1)(a cos θb1)+ kb2(a sin θb1 − x2)(a cos θb1) = 0 (5.6)Ibθ¨b2 + cb(θ˙b2 − θ˙1) + kb1(a sin θb2 − x1)(a cos θb2)+ kb2(a sin θb2 − x2)(a cos θb2) = 0 (5.7)Where :x1 = (l − d1 + a)θ1,x2 = (l − d1 − a)θ1,x3 = (l − d2 + a)θ1,x4 = (l − d2 − a)θ1,I1 =Ml2c ,Ib = 2mTa2.(5.8)We set the distance d2 = 2× d1 and with this configuration, the performanceis shown in Figure 5.20.965.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.20: Dual T-Beam response for the FE DOF using the FE and FPSoptimized T-Beams at resonance.First, we note that the dual sham does not significantly reduce tremor with ameagre 0.1% reduction, thus, we observe the dynamic system. The resultingtremor reduction for this configuration is 95.4% which is a negligible increasefrom the 92% recorded using the single T-Beam. However, closer inspection ofthis system shows the dual T-Beam configuration response to be completelyout of phase with the primary tremor response whereas the single T-Beamresponse is only partially out of phase. This is result suggests that using twoT-Beams is excellent for phase shift which improves system stabilityand performance bandwidth when well tuned.975.1. DESIGN B: THE T-BEAM ABSORBERPronation-Suppination: Implementing the 2 T-Beam configuration forthe FPS DOF, we generate the following EOMs:J5θ¨5 + cθ˙5 + kθ5 + kb1a sin(θ5 − θb1)a cos(θb1) + kb1a sin(θ5 − θb1)a cos(θb1)+ cb(θ˙5 − θ˙b1) + · · ·+ kb1a sin(θ5 − θb2)a cos(θb2) + kb1a sin(θ5 − θb2)a cos(θb2)+ cb(θ˙5 − θ˙b2) = T (t) (5.9)Ibθ¨b1 + cb(θ˙b1 − θ˙5) + kb1a sin(θb1 − θ5)a cos(θb1)+ kb2a sin(θb1 − θ5)a cos(θb1) = 0 (5.10)Ibθ¨b2 + cb(θ˙b2 − θ˙5) + kb1a sin(θb2 − θ5)a cos(θb2)+ kb2a sin(θb2 − θ5)a cos(θb2) = 0 (5.11)The sham reduction for this configuration is 13.89% which is approximatelytwice the sham reduction using a single T-Beam, an expected result. Theresponse of the two T-Beam configuration for FPS is provided in Figure 5.21.The dual T-Beam performance in FPS is an RMS reduction of 13.8%,practically equal to the sham reduction. This performance is worse comparedto the single FPS optimized T-Beam case implying a dual T-Beam, despiteits increased inertia, is detrimental to the performance in the FPS DOF.This prompts further investigations on the applicability of the T-Beam forMDOF tremors.985.1.DESIGNB:THET-BEAMABSORBERFigure 5.21: Dual T-Beam response for the FPS DOF using the FE and FPS optimized T-Beams.995.1. DESIGN B: THE T-BEAM ABSORBEROption 2: Changing the size of the T-BeamUsing 2 T-Beams increases the overall mass of the tremor suppression deviceand so far, the tremor reduction from this exercise do not warrant the massincrease. Therefore, we explore the effect of changing the T-Beam mass andoverall size to the performance of the dual T-Beam system. We can anticipatethat modifications to the current T-Beam configuration could result in asmaller, lighter T-Beam that can be combined in various configurationsto achieve better performance than a single T-Beam. Additionally, thisreduction in size means we could also implement the T-Beam in MDOFtremor situations in various locations on the arm without it becoming bulkyand looking out of place.Flexion-Extension: We begin with the FE DOF and modify the T-Beamby reducing its size by half. We do this for all its parameters of mass,stiffness, damping, and the T-Beam arm length. Since this is a dual T-Beamconfiguration, we modify both FE and FPS optimized T-Beams. The changeof the T-Beam size in the FE DOF results in the performance depicted inFigure 5.22.This size reduction results in an RMS tremor reduction of 90.5% which isslightly lower than the single optimized T-Beam or dual full size T-Beams.However, this is still a substantial reduction in tremor and the system ismore stable due to the phase difference. This shows that the T-Beam haspotential for miniaturization while still performing incredibly in the FE DOF.1005.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.22: Dual half T-Beam response for the FE DOF using the FE andFPS optimized T-Beams.Pronation-Suppination: For the FPS DOF, the reduction in size of theT-Beams to half their size results in an abysmal performance as shown inFigure 5.23. The size reduction results in 1.9% tremor reduction which isworse than the single FPS optimized T-Beam or the dual full size T-Beams.1015.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.23: Dual half T-Beam response for the FPS DOF using the FE andFPS optimized T-Beams.This is the expected result since we have see that the FPS DOF requires amuch larger inertia to mitigate the tremor. The use of dual T-Beams forMDOF tremors appears unsuccessful in all these attempts. However, wehave identified a scenario in which the FPS DOF has promising performance.This is the scenario using a single T-Beam optimized for the FPS DOFresulting in 48% tremor reduction. Thus, we investigate another approachfor combating MDOF tremors.1025.1. DESIGN B: THE T-BEAM ABSORBEROption 3: Changing the Orientation of the T-BeamWe have established that the FPS optimized T-Beam is capable of attenuatingtremors in FPS while changing the size of the T-Beam does not appear tohave an impact to the performance in the FE DOF. Combining these twoapproaches, we explore the performance when changing the orientation of thetwo T-Beam setup to target both DOFs. We understand that the maximumreduction occurs when the T-Beam is oriented perpendicular to the directionof tremor. However, optimization using this approach results in a T-Beamthat is effective in just the single DOF leading the challenges previouslyobserved. We will call this orientation 1, depicted in Figure 5.24 to beperpendicular to the FPS and FE tremors in light grey. Now we explorehow a change from this orientation to orientation 2 shown in black affectsperformance. Note that we only consider this analysis for 2 T-Beams placedat an angle with the reasoning being that 2 T-Beams allow for the reductionin both size and total mass of the T-Beams consequently resulting in lessloading of the arm. Logically, since the center of the T-Beam remains in theneutral axis of the arm (both longitudinally and radially), the only affectedcomponent of the T-Beam with this orientation change is the stiffness ofthe T-Beam acting on the arm. This is reflected in the EOMs given below.Additionally, we only use the FPS optimized T-Beam for this analysissince the FE optimized T-Beam performs poorly in FPS.Figure 5.24: Representation of original orientation in grey and proposedorientation change in black.1035.1. DESIGN B: THE T-BEAM ABSORBERFlexion-Extension: For the flexion-extension DOF with 2 T-Beamsoriented at 45o and −45o degrees from neutral, the EOM becomes (orientationhere is marked by ϕ):I1 θ¨1 + k1 l2c θ1 + c1 l2c θ˙1+ kb1 (x1 − a sin θ b1) (l − d1 + a) cos θ1 cos ϕ+ kb2 (x2 − a sin θb1) (l − d1 − a) cos θ1 cos ϕ+ cb(θ˙1 − θ˙b1)(l − d1) cos θ1+ kb1 (x3 − a sin θ b2) (l − d2 + a) cos θ1 cos ϕ+ kb2 (x4 − a sin θb2) (l − d2 − a) cos θ1 cos ϕ+ cb(θ˙1 − θ˙b2)(l − d2) cos θ1 =M (t) (5.12)Ib θ¨b1 + cb(θ˙b1 − θ˙1)+ kb1 (a sin θb1 − x1) (a cos θb1)+ kb2 (a sin θb1 − x2) (a cos θb1) = 0 (5.13)Ib θ¨b2 + cb(θ˙b2 − θ˙1)+ kb1 (a sin θb2 − x1) (a cos θb2)+ kb2 (a sin θb2 − x2) (a cos θb2) = 0 (5.14)Note that all variable definitions remain the same as earlier. The performanceof the 2 T-Beam at 45o from neutral is provided in Figure 5.25.The results show that the orientation change of the T-Beam has no significantimpact on performance, reducing it from 95% when using two perfectlyperpendicular T-Beams to 92% when at a 45o angle. This is true whenevaluated between 25 to 45 degrees. Additionally, changing the size of theT-Beam in terms of dynamic mass and arm length a has a much lower impacton performance compared to the original perpendicular orientation.1045.1. DESIGN B: THE T-BEAM ABSORBERFigure 5.25: Performance of dual deviated T-Beam at 45o from neutral forthe FE DOF.Pronation-Suppination: Changing the orientation of the T-Beams inFPS do not improve performance, with reduction teetering in the 45%→ 48%reduction range. This supports the original observation that for the T-Beamto perform well in FPS, it must be designed and optimized using a different setof parametric boundaries. This is a common challenge in vibration absorptionwhen using linear systems where changes in inertia and stiffness of the primarysystem significantly change the natural frequency thus requiring a change inparameters of the absorber.This is especially noteworthy for the case of tremors as the change of primarysystem dynamic parameters can occur with changing patient tasks. In fact,1055.1. DESIGN B: THE T-BEAM ABSORBERit can occur even when the patient is performing the same activity. Forinstance, the action of drinking requires the arm to switch through multiplepositions as shown Figure 5.26.Figure 5.26: Movement of the human arm when performing the act ofdrinking. The movement from the lower position to the higher positionengages different muscles changing the effective DOF dynamic parameters.This switch through postures means different DOFs are engaged at differentpoints of the activity. This poses significant challenges to the T-Beamsince the changing DOFs mean changing natural frequencies thus differentperformance ranges. This a key limitation of linear systems. One potentialsolution is the use of non-linear springs which can increase the effectivefrequency bandwidth for the T-Beam allowing it to be effective even inchanging DOFs. The use of non-linear springs is not a novel concept invibrations engineering and has been implemented in various applicationsincluding the original T-Beam design from the HVPL depicted in Figure 4.3.In Appendix B, the concepts of non-linear vibration absorber, perturbationmethods, and the most common depiction of nonlinear springs, the Duffingequation [84] are covered in depth. Understandably, the field of non-linearmechanics is vast so the concepts and derivations covered in this body ofwork are limited to what is relevant for our analysis.In the sections that follow, we introduce the concept of non-linear vibrationabsorption and evaluate the hypothesis that non-linear vibrations systemscan achieve wider tremor reduction bandwidths compared to linear systems.We will do this using a simpler system to observe the range of validity forthis claim before implementing it in the T-Beam for analysis.1065.2. NON-LINEAR VIBRATION ABSORPTION5.2 Non-Linear Vibration AbsorptionIn 1952, Roberson [85] published an article on the synthesis of a non-linearDVA . They argued that a linear DVA became unsuitable in cases where theexcitation frequencies are variant over a range. This claim is readily verifiedwhen one observes the frequency response of a linear DVA in Figure A.2provided in Appendix A. Their implementation of non-linearity involved theuse of the Duffing Oscillator defined by Equation 5.15 where β describes thedegree of non-linearity.k(x) = α+ β x3 (5.15)Roberson is one of many researchers who have researched a better absorbercapable of vibration absorption over a wider bandwidth. The lack ofrobustness with linear DVAs has been noted by Stephan at al. [86], Borgesand Steffen [83] and arguments for the use of non-linear DVAs not onlyincludes their robustness but also their reliability when well optimized. It isimportant to understand some fundamentals on the impact of non-linearityto any system before evaluating its applicability to vibrations absorption. Wenote the following: Non-linearity generates harmonics and causes frequencyshifts in a system [87]. This is depicted by the FRCs in Figure 5.27.The FRC in Figure 5.27 d where β = 2.5 highlights an important aspectwhen studying non-linear dynamics. For certain combination of parameters,non-linear systems experience bifurcation which implies the solution branchesand the relevance of each solution depends on its stability [87]. Additionally,the accuracy of simulation for non-linear systems depends on the approachused for the problem at hand. Generally, one begins with a weakly non-linearsystem and then observes the response compared to the true response ofthe system. This concept is called Perturbation and there are numerousPerturbation methods some of which are covered in Appendix B.2.1075.2. NON-LINEAR VIBRATION ABSORPTIONFigure 5.27: Frequency response curves for the Duffing oscillator with varyingnon-linearity. (a) β = 0 is a purely linear system, (b) β = −0.25 is a weaknon-linear system with a softening spring, (c) β = 0.5 weak non-linearsystem with a hardening spring, (d) β = 2.5 strong non-linear system with ahardening spring causing multivalued responses for certain ω.We will introduce the theory of non-linear vibration absorption using a simpleinertial non-linear absorber just as we did for the linear systems. This allowsus to compare its performance to that of the Den Hartog optimized linearinertial absorber introduced earlier and evaluate the claim that a non-linearT-Beam may be the solution to the challenges faced when attempting toattenuate MDOF tremors with the T-Beam.1085.2. NON-LINEAR VIBRATION ABSORPTION5.2.1 A simple Non-Linear AbsorberThe concept behind vibration absorption is adding a secondary systemto a primary system experiencing problematic vibrations and tuning thesecondary system to shoulder the burden efficiently. Let us consider thesystem depicted in Figure 5.28 where both the primary and absorber systemhave linear damping coefficients and non-linear springs modelled as Duffingtype springs.Figure 5.28: Basic representation of a single DOF non-linear dynamicvibration absorber system with k(x) = k1 + knl1 x3.We are analyzing the efficacy of the non-linear absorber, especially in theneighbourhood of the primary resonance. We wish to determine the necessaryabsorber parameters that minimize the resonance response. We begin byderiving the EOMs for this system which are given below. Note that x1denotes the primary system motion and xa denotes the movement of theabsorber and only the primary system is subjected to harmonic forcingF (t) = fo cos(ωt).m1x¨1+c1x˙1+ca (x˙1 − x˙a)+k1 x1+knl1 x31+ka (x1 − xa)+knla (x1 − xa)3 = F (t)ma x¨a + ca (x˙a − x˙1) + ka (xa − x1) + knla (xa − x1)3 = 01095.2. NON-LINEAR VIBRATION ABSORPTIONTo simplify the solution process, some algebraic manipulations becomenecessary. This should be done whether we intend to study the behaviour ata given frequency or the overall frequency response. We begin by adding thetwo equations which results in Equation 5.16.m1x¨1 + c1x˙1 + ca(x˙1 − x˙a) + k1x1 + knl1 x31 + ka(x1 − xa) + knla (x1 − xa)3 + · · ·· · ·max¨a + ca(x˙a − x˙1) + ka(xa − x1) + knla (xa − x1)3 = F (t) (5.16)This equation simplifies to:m1x¨1 +ma x¨a + c1x˙1 + k1 x1 + knl1 x31 = F (t)The second equation remains:ma x¨a + ca (x˙a − x˙1) + ka (xa − x1) + knla (xa − x1)3 = 0For simplicity, we introduce the parameter x2 = xa − x1 which representsthe relative displacement of the absorber to the primary system. Thus, ourEOMs become Equation 5.17 and Equation 5.18.m1x¨1 +max¨1 +max¨2 + c1x˙1 + k1x1 = F (t)− knl1 x31 (5.17)max¨2 +max¨1 + cax˙2 + kax2 = −knla x32 (5.18)The non-linear components are treated as part of the forcing to the system.This is done to allow for a few things:∗ We can study the underlying linear system and observe incrementalthe influence of non linearity to the system.∗ We understand that non-linearity in a dynamic system shifts frequenciesand generates harmonics. Therefore, by first treating the system aslinear, we can observe the response of the system and study the impactof non-linearity on absorber effectiveness.∗ We treat the non-linearity as well as the force as perturbations to oursystem.1105.2. NON-LINEAR VIBRATION ABSORPTIONWe non-dimensionalize our EOMs by introducing the following parameterswhere the subscript denotes the primary or absorber system:ω21 =k1m1= α1, ζ1 =c12√k1m1, δ1 = 2ζ1ω1,ω2a =kama= αa, ζa =ca2√kama, δa = 2ζaωa,xc =ak1, ν =ω2ωa, µ =m2ma, r =ωω1,ϵ1 =knl1 x2cm1ω2, ϵ2 =knla x2cmaω2, y1 =x1xc, y2 =x2xc,P =fom1ω2xc, γ =Pα1, τ = ωt.Making these substitutions, we obtain the following for the new EOMs:(1 + µ) y¨1 + δ1y˙1 + α1 y1 = P cos τ − ϵ1 y31µ y¨2 + µy¨1 + µ δa y˙2 + µ αa y1 = −ϵ2 y32With these substitutions, we have successfully represented the non-linearityas a disturbance to the system using the parameters ϵ1 and ϵ2. We canrepresent the EOMs in matrix form as follows:M y¨ +C y˙ +K y = fWhere the matrices representing each term are:M =[1 + µ µµ µ], C =[δ1 00 µδa],K =[α1 00 µαa], y =[y1y2], f =[P cos τ − ϵ1y31−ϵ2y32].1115.2. NON-LINEAR VIBRATION ABSORPTIONWith this representation of the system of equations, we have various avenuesto consider for solving to obtain the response. The primary course of actionis to use perturbation methods to study the behaviour of the system aroundthe neighbourhood of the primary system resonance. Various perturbationtechniques exist that are capable of performing the analysis leading to thestudy of the frequency response for various driving frequencies. One suchtechnique is the method of Harmonic balance applied by (cite here) to studythe response of a classic Duffing oscillator. There are numerous perturbationtechniques that are based on averaging. These are covered in the review ofperturbation techniques section. For the purposes of this problem, we willuse the Krylov-Bogoliubov [83, 88, 89] method described in Appendix B.1(hereinafter KB method) as demonstrated in Nayfeh’s Perturbation Methods[90].From the procedure in Appendix B.1, we obtain the equations for n˙ and m˙by integrating over the period(τ ∈ [0, 2π]). Doing so, we obtain the followingexpressions:M n˙ =12(K−M) m−[12δ1 n1 − 38(m31 ϵ1 − n21 m1 ϵ1)12 µ n2 δ2 − 38(µ m32 ϵ2 − µ n22 m2 ϵ2) ]M m˙ =12(K−M) n−[−12P + 12δ1 m1 − 38(n31 ϵ1 −m21 n1 ϵ1)12µ m2 δ2 − 38(µ n32 ϵ2 − µ m22 n2 ϵ2) ]These two equations represent four first order ODEs resulting from theKB method’s averaging over a period for the original system. To obtainthe steady state response from these two equations, we set n˙ = m˙ = 0as they are time invariant in steady state. The result is the non-linearsystem of algebraic equations in terms of n1, n2,m1,m2. Note here that thecorresponding amplitude of each mass is given by:ai =√n2i +m2i1125.2. NON-LINEAR VIBRATION ABSORPTIONThe four equations are:(1 + µ− ω21)n1 + µ n2 − 2 ζ1 ω1 m1 −3 ϵ1(n21 +m21)n14+ γ ω21 = 0µ n1 +(µ− µ ν2 ω21)n2 − µ(2 ζ2 ν ω21 m2 +3 ϵ2(n22 +m22)n24)= 0(ω21 − 1− µ)m1 − µ m2 − 2 ζ1 ω1 n1 +3 ϵ1(n21 +m21)m14= 0−µ n1 +(µ ν2 ω21 − µ)m2 − µ(2 ζ2 ν ω21 m2 −3 ϵ2(n22 +m22)m24)= 0With these equations, we use Matlab’s ”fsolve” function by Coleman toobserve the frequency response of this system with focus on how varying thenon-linearity of the absorber ϵ2 affects the vibration reduction amplitude andbandwidth of performance. We are interested in finding out whether the use ofnon-linear springs is advantageous in vibration analysis hence we only observethe frequency response. For completeness, we compare the performance of aDen Hartog optimized linear absorber to the proposed non-linear systems.In Figure 5.29, the FRCs for various non-linear configurations are providedand compared to the aforementioned Den Hartog optimized linear absorber.1135.2.NON-LINEARVIBRATIONABSORPTIONFigure 5.29: Response of various non-linear vibrations absorbers with varying degrees of non-linearity compared tolinear Den Hartog optimized absorber.1145.2. NON-LINEAR VIBRATION ABSORPTIONFrom the response curves above, it is evident that non-linear vibrationabsorbers are capable of broadband vibration reduction compared to theoptimized linear absorber especially in the frequencies above resonance. Thisis even more evident in Figure 5.30 where the percentage reduction for thedifferent absorbers is shown. There are some points to note here about theperformance of non-linear systems:∗ Non-linear systems may experience stability challenges when poorlyoptimized. This is evident in Figure 5.29 where the FRC has suddensignificant changes in amplitude.∗ Increasing or decreasing non-linearity does not have an immediatelydiscernible impact on performance. Thus, each non-linear system mustbe studied separately to determine if the degree of non-linearity isappropriate the expected for optimal performance.The results of the non-linear system simulation show the potential benefitsin implementing non-linear springs for the T-Beam to achieve broadbandtremor reduction for MDOFs. It is also worth noting that the method usedabove is one of many perturbation methods and Appendix B.2, a detailedreview of Perturbation methods is provided for reference.The revelation that non-linear absorbers can in fact achieve broadbandtremor reduction lends credence to the idea proposed in several publicationsfrom the HVPL. This design of the T-Beam now becomes the focus of ouranalysis and we now model and analyze the magnetic stiffness conceptualdesigns from Chapter 4.2.1155.2.NON-LINEARVIBRATIONABSORPTIONFigure 5.30: Reduction percentages using varying degrees of non-linearity compared to linear Den Hartog optimizedabsorber.1165.3. NON-LINEAR T-BEAM5.3 Non-Linear T-BeamWe have seen that non-linear absorbers are capable of much wider performancebandwidth compared to linear systems. We model and analyze the updatedversion of the T-Beam shown in conceptual Design B iteration 2 wherestiffness is controlled by adjusting the overlap area. The logic and decisionbehind modeling this system is further explained in Chapter 7.3.1.The modeling for this design is done by first modeling the effective magneticsprings then generating the T-Beam EOM before mathematically simulatingthe performance of the T-Beam in both FE and FPS DOFs.5.3.1 Modeling the T-Beam Magnetic SpringsCharacterizing the magnetic spring is a two-step process; first we generatethe axial force-distance (F − d) relationship between the magnetic facesresulting from the magnetic filed interactions. Then we determine the truestiffness due to the change in overlap area for the range of motion of theT-Beam. The result of this process is a F − θ relationship which we take thederivative to obtain our desired k − θ relationship.The setup used to characterize the force-distance relationship for a pair ofmagnets is shown in Figure 5.31. This setup measures the force due to therepulsion between magnets A and B (two similar poles). This is the forceresponsible for suspending the weights at the top. The force is then measuredby a scale at the bottom. As weights are added, the height of the top magnetchanges and a ruler measures the distance between the two magnets. Theforce on the scale is recorded as well as the distance and the process isrepeated multiple times to generate the axial force between the magnets ata given distance F − d. To ensure that the measured force is purely axial,vertical guides are added to constrain the motion of the magnets to an axialinteraction.1175.3. NON-LINEAR T-BEAMThe collected F − d data is plotted on a chart and resembles the sketch inFigure 5.32a. Notice that this is not a smooth fit since there is a considerablegap between the data points. Using Matlab’s curve fit program, a best fitcurve is created for the data while minimizing the SSE, RSME, R2 values.Figure 5.32b shows a sketch similar to the result from a curve fit.Figure 5.31: Setup to determine the axial force-distance F − d relationshipbetween repelling magnetic faces. The force is measured on a scale at thebottom and the distance is measured using the ruler.1185.3. NON-LINEAR T-BEAMFigure 5.32: (a) Plotting data collected from magnetic setup; (b) Fittingdata using Matlab based on minimum SSE, RSME, R2 values.The generated curve fit equation is now used to interpolate for all valuesof the magnetic F − d relationship. This is the first part to characterizingthe magnetic spring and for the current T-beam configuration, NeodymiumN42 are used magnets for springs. The F − d relationship is given the chartin Figure 5.33. The data for this chart is obtained from K&J magnetics aswell as verified through the experimental setup in Figure 5.31.From Figure 5.33, the exponential fit is chosen over the polynomial fit. Whileboth models have similar goodness of fit measures as detailed in Table 5.1,the exponential model has fewer terms thus simpler for implementation insimulation. Generally, polynomial fits are preferred in simulation for theircomputational simplicity but since we will not simulate this T-Beam, a betterfit is preferred.1195.3.NON-LINEART-BEAMFigure 5.33: N42 Neodymium magnetic F − d profile (blue); Exponential curve fit (in red); Polynomial curve fit(in orange). Observe the better fit from the Exponential fit compared to the Polynomial coefficients approach.1205.3. NON-LINEAR T-BEAMTable 5.1: Goodness of fit data for Exponential vs. Polynomial fitsModel Type SSE RMSE R2Exponential 0.6209 0.3858 0.9995Polynomial 0.5529 0.2629 0.9996The selected Exponential fit model is described by Equation 5.19:F (x) = a× eb x + c× ed x (5.19)Where the parameters a, b, c, & d obtained with 95% confidence bounds are:a = 16.84(14.12, 19.56)b = −1342(−1634, −1049)c = 31.09(28.24, 33.95)d = −231.3(−249.7, −212.8)With a model for the axial F − d, we can generate the corresponding F − θrelationship for the T-Beam over its range of motion. For this step, we willintroduce the parameters defined in Figure 5.34. In Table 5.2, a descriptionof the physical interpretation of each of the defined parameters is providedand the values for the current T-Beam configuration are given in inches.Table 5.2: Description of parameters used in the magnetic spring model. Thevalues for the current T-Beam configuration are given in inches.Parameter Description VariableNameCurrent Value (in)Midpoint to center of top magnet Ainit 0.65Midpoint to center of bottom magnet b 0.797Midpoint to end of top magnet a 1.15Midpoint to end of bottom magnet x 1.297Radius of magnet R 0.5Height of T-Beam q 0.61215.3. NON-LINEAR T-BEAMFrom Figure 5.34 and Table 5.2, some relationships between the dimensionsare defined in below. x = b+Rz = x−Ra = Ainit +Rx =√a2 + q2Figure 5.34: Parameters used in calculating the magnetic spring stiffnessover the range of motion of the T-Beam1225.3. NON-LINEAR T-BEAMIn order to calculate the overlap area for the range of motion, we make theassumption that the top magnetic face projects a circle that intersects withthe bottom magnetic face as shown in Figure 5.35.Figure 5.35: Overlap area of top magnet (in blue) over bottom magnet (inred). The overlap area is given by the green hatched region formed by thetwo intersecting segments.From this result, the area of overlap is calculated as follows:Know :p2 + h22= R2 ⇒ h22= R2B − p2(p− d)2 + h22= R2T1235.3. NON-LINEAR T-BEAMCombine the two equations :⇒ (p− d)2 +R2B − p2 = R2Tp2 − 2 d p+ d2 +R2B − p2 = R2T∴ p = d2+R2B−R2T2d & d− p =d2−R2B+R2T2dThe height of the segment is given by :Letk = h2k2 = R2B − p2 = R2B −(d2+R2B−R2T2d)2∴ k = 12d√[4d2R2B −(d2 +R2B −R2T)2]⇒ h = 2k = 1d√[4d2R2B −(d2 +R2B −R2T)2]Note that the area of overlap for each circle is a segment given by Figure5.36. The formula for calculating the area of a segment is given by Equation5.20.Figure 5.36: Parameters for calculating are of a segmentSegment AreaAsegment = Asector −AtriangleAsegment = 12R2 (θ − sin θ) = R2 cos−1 (h′R)− h′√R2 − h′2(5.20)1245.3. NON-LINEAR T-BEAMUsing the formulations presented earlier, we can determine the respectiveradius and triangle height for each segment and the calculate resulting totaloverlap area which is given by Equation 5.21. Note that the magnets haveequal radius hence the formula replaces this with R.A = 2R2 cos−1(d2R)− d2√4R2 − d2 (5.21)The calculated area represents the total region of overlap between themagnetic faces. However, the orientation of the T-Beam means that theheight of overlap varies over the entire range of overlap. This means thatwe cannot approximate the force by taking the fraction of the total area.Instead, we discretize the area based on the Domain Theory of Magnetismwhich states that there exists small regions within a magnet whereall of the atoms are aligned in the same direction. These regionsare called domains [91]. Note that this is an approximation and the resultsmust be experimentally validated.With this approximation, the parameters necessary to calculate the force ateach position of the T-Beam are defined in Figure 5.37. The total force at agiven position θ is then calculated as follows:Calculate theoretical maximum angles : αmax = tan−1 ( qa); θmax =π2 − αmaxDetermine end postion of overlap : y = q cos θ + a cosαDetermine the width of the overlap : d = y − zCalculate the total overlap area : A = 2R2 cos−1(d2R)− d2√4R2 − d2Determine height at farthest point of overlap : hmin = q sin θ − a sinαDetermine height at nearest point of overlap : hmax = q sin θ −[(z − q cos θ) tanα]With these parameters, the force is calculated over the entire area using theExponential curve fit of the force-distance relationship for the N42 magnetsdiscussed earlier. Note that the number of discrete areas is controlled toreduce computing time. Finally, an F − θ curve is produced as shown inFigure 5.38.1255.3.NON-LINEART-BEAMFigure 5.37: Discretizing the magnetic overlap area at different T-Beam positions defined by θ. The total force ateach position is the sum of all the forces due to each discrete area.1265.3. NON-LINEAR T-BEAMFigure 5.38: F − θ profile for N42 magnets for the motion of the T-Beamfrom rest (θ = 90o) to maximum displacement shown. The maximum force(in red) occurs at the extreme position where θ = 66.5o.The force profile shows the T-Beam stiffness (inferred from force) at rest willbe low making it responsive to low amplitudes. As the T-Beam moves to theextreme position, the force increases thus restoring the T-Beam to its neutralposition and attenuating the tremors. While it is possible to numericallygenerate a stiffness profile for the T-Beam, it is inconsequential since we areultimately interested in the restoring force at an angle. Instead, we modelthe force profile using a two term exponential fit described by Equation 5.22.The coefficients are based on a 95% confidence boundary of fit. The stiffnessis given by the derivative of the force with respect to the position.F (θT ) = a× eb θT + c× ed θT (5.22)1275.3. NON-LINEAR T-BEAMWhere: a = 9.797× 1013b = −27.62c = 2672d = −4.953Finally, the stiffness is given by:k(θT ) = a b× eb θT + c d× ed θTThe work conducted by Masoumi [65] experimentally showed that the originalT-beam was capable of attenuating high frequency low amplitude tremorsin the pronation-suppination DOF. However, to the author’s knowledge,the original T-Beam was not modeled and evaluated for FE tremors orlow frequency high amplitude tremors. The proposed design iterations andretrofitting of the non-linear springs result in the updated version of theT-Beam shown in Figure 5.39. This T-Beam has a dynamic mass of 200gwith two N42 magnetic spring pairs. The bearing is a low friction bearingwith c = 0.5Nms/rad. The mathematical model and simulation for thisT-Beam are provided in the subsequent section.Figure 5.39: Final model of the modified T-Beam to combat both HFLAand LFHA tremors. The magnets are off-center and the bearing providesdamping. Dimensions shown in inches.1285.3. NON-LINEAR T-BEAM5.3.2 Modeling the Non-Linear T-BeamThe magnetic force profile derived in the previous section is fundamentalto the performance of the T-Beam. Although the derivation explicitly usesN42 magnets, the general procedure to characterize any magnetic springbehaviour would follow a similar process. To derive the EOM, we begin witha free body diagram (FBD) of the T-Beam under a disturbance with allacting forces shown in Figure 5.40.Figure 5.40: T-Beam Free body diagram showing the spring force Fk andthe line of action of the force. The bearing damping cb as well as the massesmT . The mass is a sum of the added brass weight and the magnet.In the FBD from Figure 5.40, the restoring force due to the magneticsprings,Fk is given in red. Note that since the magnets are always repelling,the force is always in the vertical direction. Additionally, since the magneticforce is due to the interaction between the two faces, it is assumed to acteffectively from the center of the top magnetic face. This assumption isbased on the fact that the magnets are overlapping beyond their halfwaypoints based on geometric constraints when the T-Beam is at rest. A final1295.3. NON-LINEAR T-BEAMnote on the damping coefficient of the bearing which is taken to be cb: thebearing has elements of both viscous and coulomb damping. In the initialanalysis, the effect of coulomb damping will be assumed as negligible. Thiswill be adjusted based on the experimental results to revise the model tobetter represent the physical parameters of the T-Beam. The masses mTaccount for the added mass and the masses of the magnets. They are treatedas point masses for this analysis.Given that the T-Beam rotates around the bearing pivot point, we write themoments using Newton’s equation:ΣMo = Ioθ¨The magnetic faces are in repulsion and as such, the forces due to the springsalways act to restore the T-Beam to its equilibrium position. Note that at agiven instant, the moments produced by each spring are opposing each other.Thus, we describe the general moments by:{Mk1 = k1(θ) · θ · (q cos θ + a cosα)Mk2 = −k2(π − θ) · (π − θ) · (a cosα− q cos θ)Since the two magnetic springs are similar, k1 = k2 thus the same equationfor the stiffness profile is used but adjusted for the direction to account forthe entire range of θ. Additionally, when the T-Beam is at rest, the twomoments due to each spring are equal and the system is at equilibrium.Finally, The inertia around the rotating pivot is is the sum of the two massesmT and is determined using the equivalent radius of rotation such that:Io = 2mT(a2 + q2)Combining these moments, we generate the following equation:2mT(a2 + q2)θ¨ + k1(θ) · θ · (q cos θ + a cosα)− k2(π − θ) · (π − θ) · (a cosα− q cos θ) = 0We can simplify this equation further to generate the equivalent parametersfor the inertia and stiffness of the T-Beam. With some manipulation of theabove equation, the EOM for the T-Beam is given by Equation 5.23.1305.3. NON-LINEAR T-BEAM2mT(a2 + q2)θ¨+(k1(θ) · l1 + k2(π − θ) · l2)θ−k2(π−θ) ·π · l2 = 0 (5.23)Where:l1 = q cos θ + a cosα; l2 = a cosα− q cos θNote that the terms k1(θ)& k2(π− θ) are to be taken as function evaluationsand should be read as k1 of θ and k2 of π − θ; that is, we are evaluatingthe value of stiffness k at the respective position based on the characterizedmagnetic springs. We can now evaluate the performance of the T-Beam onthe flexion-extension and pronation-supination models. Note that since theforce of the magnetic springs is non-linear, the solution follows an iterativeprocess which is discussed in Chapter 5.3.3.5.3.3 Numerical SimulationThe T-Beam has been now been modeled as a dynamic vibration absorberwith the updated non-linear magnetic springs. From the study of thelinear system, we understand that its performance depends on its geometricconfiguration and dynamic parameters. Similar to the linear T-Beam, it isnecessary to evaluate the impact of changing the height and added mass tothe performance of the non-linear T-Beam. This is analysis is conductedusing the parameter given in Table 5.3.Table 5.3: Possible Mass (g) and height (in) values for evaluating the T-Beamperformance.Mass (g) Height (in) mT (kg)90 0.5 0.212110 0.6 0.252130 0.7 0.292The performance of this new T-Beam model will be evaluated at lowerfrequencies such are 2Hz, 3Hz, and 4Hz. These cover the lower frequencyspectrum for both Parkinson’s Disease and Essential Tremor. The purposeit to investigate whether the modified T-Beam now performs better in thelow frequency spectrum. The T-Beam will also be evaluated in the higher1315.3. NON-LINEAR T-BEAMfrequencies ranging from 5 → 8Hz to study its performance over a widerfrequecny range. As a final note, the performance in the FPS DOF may needto be evaluated separately i.e. with a different set of T-Beam parameterssince the T-Beam has been shown to primarily be a SDOF absorber.Simulation Challenges: The effective stiffness due to the magneticsprings is highly non-linear such that traditional approaches using numericalmethods to evaluate the response fail to accurately represent the problem.We are not able to linearize the T-Beam around its equilibrium since itsmotion occurs over a large range i.e. −22o ≤ θ ≤ 22o. Thus, smallangle approximations are not applicable here. The consequence of theseapproximations is oversimplification of the problem and misrepresentationof the true physical interactions of the system. Initial attempts to employsimple numerical techniques resulted in unstable and unreliable results suchas the one shown in Figure 5.41.Addressing Simulation Instabilities: The instability observed in thesimulation arises from the challenges of solving highly non-linear systemsusing standard numerical methods like the explicit Runge-Kutta family(e.g., ode45 in MATLAB). Such systems exhibit rapid dynamic changesleading to numerical errors if solver settings are not carefully chosen. Acommon solution is to use stiff solvers like ode15s or ode23t, which arebetter equipped to handle rapid variations and ensure numerical stability.Additionally, adaptive step-sizing or reduced step sizes can improve solutionaccuracy. However, purely numerical methods may still struggle in cases ofextreme non-linearity, highlighting the need for alternative approaches.Implementing Perturbation Methods: Perturbation methods, such asthe Krylov-Bogoliubov method provide a robust alternative for analyzinghighly non-linear systems. These techniques approximate the system byseparating it into a dominant linear part and smaller non-linear perturbations,allowing for the derivation of analytical solutions that capture key dynamicswithout the instabilities of numerical solvers. Perturbation methods also offerinsights into critical parameters like natural frequencies and amplitude-dependentresponses while reducing computational complexity. For the remainder ofanalysis, the perturbation techniques are implemented.1325.3.NON-LINEART-BEAMFigure 5.41: Infinite growth observed in the simulation of a non-linear system using standard numerical methods.The figure illustrates the instability caused by the inability of the chosen numerical solver to handle the strongnon-linearities in the system, resulting in unbounded oscillations over time.1335.3. NON-LINEAR T-BEAMWith the implementation of the KB Method, the simulation to study theperformance of the non-linear T-Beam starts with determining the individualT-Beam behaviour and its resonant frequency. This approach is used todetermine two things: 1)whether the method is correctly implemented; 2)the resonance region of the non-linear T-Beam in this configuration. Bysimulating the system over a range of frequencies, we rule out computationalstability challenges and identify the resonance region of the T-Beam to bearound 6Hz. In Figure 5.42, the response of the T-Beam near resonanceis shown. Notice that the T-Beam experiences beat phenomenon but theoscillations do not infinitely grow.Figure 5.42: Response of the T-Beam system around resonance. The plotillustrates the periodic oscillations near the system’s natural frequency,highlighting the characteristic amplitude modulation due to the interactionbetween the system’s resonance and non-linearities in the stiffness anddamping.1345.3. NON-LINEAR T-BEAMSimulating the T-Beam in the FE DOF, we obtain the response in Figure5.43. Without the absorber, the system undergoes significant oscillationsdue to a tremor near the resonant frequency (around 2.5Hz), as depicted bythe red curve. However, the inclusion of the T-Beam effectively mitigatesthese oscillations, as evidenced by the 89.93% reduction in tremor amplitudeshown by the blue curve.Figure 5.43: Flexion-Extension response with and without the absorber.The red curve represents the system without the absorber, exhibiting largeoscillation amplitudes indicative of resonance behaviour. The blue curveillustrates the same system with the absorber, showing significantly dampenedoscillations. This highlights the effectiveness of the non-linear T-Beam inmitigating resonance effects.1355.3. NON-LINEAR T-BEAMSimilarly for the FPS DOF, the simulation result is shown in Figure 5.44.The resonant frequency in FPS is approximately 1.93Hz which results inthe high amplitude response shown by the red curve. The addition of theT-Beam reduces the tremor by 76.31% which is much better compared tothe linear T-Beam configuration studied earlier. This is a promising resultbecause the T-Beam is meeting the minimum expectations both for the FEand FPS DOFs at their resonant frequencies.Figure 5.44: Pronation-Suppination response with and without the absorber.The red curve represents the system without the absorber, exhibiting largeoscillation amplitudes indicative of resonance behaviour. The blue curveillustrates the same system with the absorber, showing reduced oscillations.1365.3. NON-LINEAR T-BEAMThe simulation of the non-linear T-Beam for the two problematic DOFs.,FE & FPS, has shown tha the non-linear T-Beam has a better performancecompared to the linear system studied earlier. Now, we extend this studyto more frequencies and also study other configurations of the T-Beam.We intend to see whether the performance bandwidth of the non-linearsystem is better compared to the linear system. Additionally, by changingthe parametric configuration, we also study whether the performance isconsistent across both DOFs for the same device. The ultimate goal isto have a device that performs equally as well across both DOFs for thespecified frequency range. Implementing this study approach, the evaluationcombinations are in Table 5.4. Notice that the mass and geometric parametercombinations are from Table 5.3.Table 5.4: Summary of experimental factors and their correspondingtreatment levels for evaluating the T-Beam performance.Experimental Factor Treatment LevelsDegree of Freedom (DOF) FE, FPSFrequency (Hz) 2, 3, 4, 5, 6, 7, 8Mass (mT in kg) 0.212, 0.252, 0.292Height (h in inches) 0.5, 0.6, 0.7In total, there are 126 simulation scenarios from Table 5.4. It is not necessaryto simulate all scenarios for significant trends to emerge because we cananticipate certain results based on the principles of vibrations engineering.However, it is important to recall that non-linearities shift frequencies andintroduce harmonics. Thus, some phenomena may be observed due tonon-linearity.The simulation begins with the non-linear T-Beam at a height of h = 0.5insince one of the guidelines in Chapter 2.6 is having the device be small toconform to the human hand aspect ratio. At this height, we simulate differentmasses and frequencies and the results are given in Table 5.5.1375.3. NON-LINEAR T-BEAMFor the flexion-extension DOF, increasing the absorber mass (mT ) consistentlyimproves performance across all frequencies which is expected. At lowerfrequencies, such as F = 2Hz, performance increases from 44.78% for mT =0.212 kg to 65.54% for mT = 0.292 kg. At higher frequencies (F = 6, 7, 8),the performance stabilizes i.e. increasing mass does not significantly improveperformance., with values ranging between 71% and 85% across all masses.The maximum observed performance is at F = 3Hz, where mT = 0.292 kgachieves 88.41%. This is around the resonance frequency for the FE DOFwhich explains the high reduction. These results indicate that FE DOFperformance greatly benefits from increased mass, particularly at mid-rangefrequencies.In contrast, the forearm pronation-suppination DOF exhibits more variabilityin performance trends. At lower frequencies (F = 2, 3, 4), higher absorbermasses generally result in better performance. For example, at F = 4Hz,mT = 0.292 kg achieves 73.30%. However, at higher frequencies (F = 6, 7, 8),the results are less consistent. While mT = 0.292 kg maintains relatively highperformance at F = 6Hz with 68.28%, the performance for mT = 0.212 kgdrops significantly to 38.11%. This suggests that FPS performance is lessdependent on mass increases compared to FE, with higher masses providingonly marginal improvements at high frequencies.Overall, the results highlight that the absorber performs better in the FEdegree of freedom compared to FPS, particularly at higher frequencies. Whileincreasing absorber mass improves performance for both DOFs, the impact ismore pronounced for FE, where the improvements are consistent and stabilizeat higher frequencies. These findings underscore the importance of optimizingthe T-Beam mass and configuration for each specific degree of freedom toachieve optimal performance which is a similar conclusion from the linear case.1385.3.NON-LINEART-BEAMThe performance trend holds true for the other T-Beam heights. However, the significance of non-linearity in thisanalysis is noticeable from the fact that the performance drop-off is not as drastic as was the case with the linearT-Beam. In Figure 5.45, we observe the performance trend closely.Table 5.5: Percentage reduction results for both FE and FPS degrees of freedom at T-Beam height h = 0.5 in.Frequency (Hz)Flexion-Extension (FE) Forearm Pronation-Suppination (FPS)mT = 0.212 mT = 0.252 mT = 0.292 mT = 0.212 mT = 0.252 mT = 0.2922 44.78 58.19 65.54 69.72 78.29 82.103 76.40 85.09 88.41 63.68 60.07 72.294 64.03 80.19 86.33 57.17 58.01 73.305 71.92 78.94 83.12 55.68 56.93 68.006 71.06 78.58 83.07 38.11 56.73 68.287 71.05 78.66 83.01 39.36 57.48 68.678 71.72 84.98 83.22 40.49 70.98 68.511395.3. NON-LINEAR T-BEAMFor FE DOF, the non-linear T-Beam achieves superior performance nearresonance (F = 3Hz), with reductions exceeding 88% for mT = 0.292 kg.Unlike the linear T-Beam from Chapter 5.1, which exhibited a sharp fall-offin performance after resonance, the non-linear T-Beam maintains consistentperformance above resonance frequencies (F = 6, 7, 8Hz) highlighting itsbroader performance bandwidth.For FPS DOF, the non-linear absorber demonstrates a different responseprofile, where performance is more variable across frequencies but stillsurpasses that of a linear absorber in both resonance and post-resonanceconditions. This variability may be attributed to the inherent behaviourof non-linear systems where harmonics and frequency shifts may occur.Moreover, the inertia for the FPS DOF is orders of magnitude lower thanthat of the FE DOF which results in significant variation in performance fromslight changes in frequency. However, while reductions are less significantat lower frequencies (F = 2Hz), the non-linear T-Beam maintains steadyperformance at higher frequencies, especially for mT = 0.292 kg, achievingreductions of around 68% to 70% at F = 7, 8Hz.These results emphasize the capability of the non-linear T-Beam to outperformits linear predecessor by mitigating the steep performance decline and insteadproviding consistent vibration suppression across both degrees of freedom.The results of the modeling and analysis of the non-linear T-Beam revealsome key findings:∗ Even with non-linear springs, the T-Beam is partially reliable incombating MDOF tremors. Additionally, the reduction of tremorsin both FE and FPS at the same time would still require two T-Beamswhich doubles the mass of the overall device.∗ For consistent performance across the frequency spectrum, a highermass of the T-Beam is required. With the assumption that theconnecting links are of negligible weight for the sake of analysis, thefinal device may have an additional 100g from the assembly parts. Thiswould make the device heavier than the target weight in Chapter 2.6.These findings point to the need for another device that builds on theimproved performance from the non-linear T-Beam but addresses the challengesof consistency and simultaneously combating MDOF tremors.1405.3.NON-LINEART-BEAMFigure 5.45: Performance of the non-linear T-Beam for FE (top) and FPS (bottom) DOFs at h = 0.5 in. Theplots compare percentage reduction in vibration amplitude across a range of frequencies (2 Hz to 8 Hz) for threeabsorber masses (mT = 0.212 kg, mT = 0.252 kg, mT = 0.292 kg). For FE, the T-Beam achieves over 88% reductionnear resonance (F = 3Hz) for mT = 0.292 kg and maintains consistent performance at higher frequencies, withminimal decline. For FPS, the T-Beam outperforms the linear design in both resonance and post-resonanceconditions showing the superior adaptability and broad-band performance.1415.4. NON-LINEAR MDOF DEVICE5.4 Non-Linear MDOF DeviceWe have established that the T-Beam is effective as a SDOF absorber thatrequires different parametric combinations for adequate performance in eachDOF. Additionally, we have shown that the non-linear T-Beam outperformthe liner T-Beam both in terms of cross-DOF performance and reductionbandwidth. However, the requirement of two T-Beams to target multipleDOFs of tremor remains a limiting factor in our pursuit of a reliable tremorattenuation device.In an attempt to tackle this challenge of MDOF tremor attenuation, wewill now consider Design C. From Chapter 4.3, there are three proposedconfigurations for Design C but our analysis will only focus on the thirditeration that uses ring magnets for the spring and doughnut shaped mass.The other two design ideas will be discussed in the prototyping and implementationsection in Chapter 6.Note: The development of this device was done in collaboration with industrypartner Steadiwear Inc. under a MITACS agreement. Due to the intellectualproperty agreements, certain details of the the mathematical model will beomitted. This will not impact the results or the level of discussion for thisdevice.1425.4. NON-LINEAR MDOF DEVICE5.4.1 Selecting the Design IterationFrom Chapter 4.3, Design C’s iteration three, shown in Figure 5.46, isproposed. Recall that this design includes two thin ring magnets placedat the top and bottom of the device. The ring magnets provide a uniformfield such that the resulting magnetic springs are presumed to be uniform.Additionally, the dynamic mass is a doughnut shape to minimize the totalmass while keeping the dynamic inertia high. The swivel bearing is designedto be low friction allowing for maximum travel when high amplitude tremorsoccurred.Figure 5.46: Desic C iteration 3 MDOF device that uses ring magnets toprovide an even field and doughnut mass for higher inertia with low weight.This device has a variable height to change the baseline dynamic stiffnessprofile.1435.4. NON-LINEAR MDOF DEVICE5.4.2 Mathematical Model of Design CFor the mathematical model of this design, we consider tremor in each DOFindependently. This is an important assumption since we know tremor hascross-DOF properties. However, by considering each DOF separately, we cansee whether the device is capable of mitigating MDOF tremors for the sameconfiguration. In the validation section from Chapter 7.2, we will evaluatethe impact of this assumption in our analysis.With this in mind, we will consider the mathematical model for the FPSDOF first. The EOM for the MDOF device is given by Equation 5.24. Thenon-linear springs are modeled using a 5th degree Duffing oscillator.Iθ¨ + cbθ˙ + k1θ − αθ3 − βθ5 = 0 (5.24)Here, the inertial of an the annulus mass rotating in its x or y axis is givenby I = m2[R2o +R2i]where Ro and Ri are the outer and inner radius of thering respectively. The combined EOM when the absorber is placed on thearm is given by Equation 5.25.Jθ¨p + cθ˙p + kθp + cb(θ˙p − θ˙a) + k1(θp − θa)− α(θp − θa)3 − β(θp − θa)5 = T (t)Iθ¨a + cb(θ˙a − θ˙p) + k1(θa − θp)− α(θa − θp)3 − β(θa − θp)5 = 0(5.25)Simulating this mathematical model comes with the same challenges previouslydiscussed due to the degree of non-linearity. For instance, attempting tosolve the EOMs using Runge-Kutta produces unreliable results affected byminor changes in parameters. Therefore, we continue using perturbationtechniques for the mathematical modeling.1445.4. NON-LINEAR MDOF DEVICE5.4.3 Numerical Simulation of the MDOF DesignWe begin the simulation by first studying the response of the absorber undera small disturbance. This is done to check for simulation errors and determinethe natural frequency of the absorber. The mass is taken as 200g, similarto the T-Beam. The swivel bearing damping and stiffness profiles are alsosimilar to those of the T-Beam. The response of the absorber under a smalldisturbance is given in Figure 5.47.Figure 5.47: Response of the MDOF absorber concept due to a small harmonicdisturbance. The system experiences resonance at a higher frequency (f ≈9Hz). Inherent damping allows the system to settle much faster comparedto the T-Beam.1455.4. NON-LINEAR MDOF DEVICEWith the mathematical models created for the MDOF device, we perform asimilar simulation to that of the T-Beam by studying both flexion-extensionand forearm pronation-suppination at resonance. We model tremor, similar tothe T-Beam scenario, to be harmonic torque. In Figure 5.48, the performanceof the MDOF in both FE and FPS DOF resonant frequencies is shown.Figure 5.48: Result of the simulation for the flexion-extension and forearmpronation-suppination resonance response using the MDOF device. Thedevice performs similarly in both DOFs with a 88.06% and 87.29% reductionin FE and FPS respectively.1465.4. NON-LINEAR MDOF DEVICEThe simulation results show the MDOF device reducing tremor in both FEand FPS by 88.06% and 87.29% respectively. This is a significant resultbecause it shows the system working equally in both DOFs at their resonantfrequencies.This is a significant improvement compared to the T-Beam whichshowed exemplary results in FE but not always in FPS.This performance holds for the 3→ 7Hz frequency range as shown in Table5.6. At 2Hz and beyond 8Hz, the effects of harmonics due to non-linearitybegin exhibiting. The sharp decline in performance resulting in amplificationfor the FE DOF and eventually the FPS DOF is characteristic of . Recallthe analysis in Chapter 5.2, the performance of the non-linear system hasa broadband and then a sudden, almost instantaneous decline in performance.Table 5.6: Frequency sweep results for the MDOF device across FE andFPS degrees of freedom (DOF). The performance is given as percentagereduction, with ”Amplification” indicating instances where the deviceamplifies oscillations instead of reducing them.Frequency (Hz) FE Reduction (%) FPS Reduction (%)2 28.14 95.373 77.53 70.244 69.08 64.575 70.26 63.606 75.39 64.057 67.66 62.168 7.38 55.949 Amplification 45.3110 Amplification 29.2411 Amplification 12.9812 Amplification Amplification1475.4. NON-LINEAR MDOF DEVICEFrom the results of this analysis, we make the following observations:∗ For frequencies between 3 Hz and 7 Hz, the MDOF device demonstratesrobust cross-DOF performance as shown in Figure 5.49.∗ Beyond 8Hz, the performance of the MDOF device diminishes significantly,leading to amplification in oscillations for the FE DOF starting at9Hz and for both DOFs at 12Hz. This suggests the influence of higherorder harmonics resulting from the quintic term in the stiffness profile.∗ Unlike the T-Beam, which primarily targets a single DOF with higheffectiveness, the MDOF absorber exhibits better cross-DOF consistencywithin its effective range as shown by the comparison in Figure 5.50.Both FE and FPS see significant reductions with the MDOF device,whereas the T-Beam’s performance falls off more sharply in FPS.∗ While the T-Beam generally performs better at its resonance frequency,the MDOF device offers a broader effective bandwidth, particularly inthe FPS DOF.∗ The sharp decline in performance at higher frequencies (from 8 Hzonwards) and the amplification observed in both DOFs highlighta limitation of the current configuration of the MDOF absorber.Adjustments to the absorber’s non-linear coefficients (e.g., α, β) couldpotentially mitigate this issue.The performance trend of the MDOF device is similar to the characteristicperformance of a non-linear absorber shown in Figure 5.30 with a peak justbefore resonance, steady reduction bandwidth with minimal fluctuation, andfinally a decline in performance. The difference in the initial response andrate of performance decline can be attributed to inherent characteristics suchas inertia and stiffness of FE and FPS DOFs. This differences influence thenatural frequency which may affect the type of harmonics experienced i.e.subharmonics (ω/n) or superharmonics (nω wheren ≥ 1).1485.4.NON-LINEARMDOFDEVICEFigure 5.49: Percentage reductions for the MDOF device for the FE and FPS DOFs. The device is effective inboth DOFs between 3→ 7Hz with sharp declines due to harmonics on either end of the bandwidth.1495.4. NON-LINEAR MDOF DEVICEFigure 5.50: Percentage reductions for the MDOF device and T-Beam forthe device for the FE and FPS DOFs. The MDOF device is effective in bothDOFs between 3→ 7Hz at a lower weight compared to the T-Beam. Bothsystems have performance declines due to higher order harmonics.The performance fall of due to frequency shifts above 8Hz is attributed tothe non-linear stiffness characteristics of the MDOF absorber, where higherfrequencies excite cubic and quintic stiffness terms, leading to less efficientenergy dissipation and potential instability in the response. Consequently, theMDOF absorber’s performance diminishes at these frequencies, underscoringthe need for careful tuning during prototyping to address harmonic excitationand ensure consistent attenuation across the frequency spectrum.150CHAPTER 6Prototyping and ImplementationWith the successful mathematical model of the MDOF conceptual design, webegin the fabrication and iterative refinement of the final tremor suppressiondevice. This chapter outlines the process and iterations in constructingthe prototype. It emphasizes the practical challenges encountered and themodifications made to optimize performance. The author acknowledges herethat material selection is an important aspect of prototyping but due to theintellectual property rights, this aspect of prototyping is omitted.1516.1. ITERATIVE PROTOTYPING PHASE6.1 Iterative Prototyping PhaseFrom the conceptual design to the final product, several device prototypes aredeveloped and evaluated with each device building on its predecessors successand overcoming prototyping challenges. The final prototype of the MDOFdevice hereinafter Steadi-Two, is shown in Figure 6.1. In this rendering ofthe CAD model, the outer cover of the Steadi-Two is made transparent toshow the internal components. The final device weighs 400g with a dynamicmass of 250g. In the subsections that follow, we will discuss key componentsof the design and the decisions made in implementing them.Figure 6.1: Steadi-Two device with a transparent outer case showing theinner components. Multiple iterations were completed due to prototypingchallenges.1526.1. ITERATIVE PROTOTYPING PHASE6.1.1 Outer CoverThe outer cover is made from high strength plastic which is shatter resistant(not shatter proof) to protect the internal components. There is a channelin the device shown in Figure 6.2 that interfaces with a knob allowing theadjusting of device height. In doing so, the baseline stiffness of the deviceis adjusted. This allows the device to target either low or high frequencytremors.Figure 6.2: Steadi-Two outer cover showing notch for height adjustmentknob. This notch ensures the device stays locked in place when the height isadjusted.Additionally, there are marks on the outer cover with I or II to indicatewhich mode the device is on. The inclusion of multiple modes for the devicearose from iterative evaluation during prototyping. This was to combat theperformance decline seen in the modeling phase. By including two devicemodes, the Steadi-Two can combat harmonic influences during use. Chapter7 further explains the performance of each mode.1536.1. ITERATIVE PROTOTYPING PHASE6.1.2 Height Adjustment MechanismIn Figure 6.3, the outer cover has been removed to reveal the internalcomponents and the knob is shown to be connected to the swivel bearing andmass assembly to allow for height adjustment. The decision to incorporatethis height adjustment is made based on the understanding that changingDOFs require different stiffness profiles. Since the Steadi-Two incorporatesmagnetic springs, changing the height alters the magnetic field resulting ina weaker or stronger field. The lower height of the device is labelled L8symbolizing a lower height of 8mm and the higher height is labelled H12which represents a height of 12mm.Figure 6.3: Height adjustment knob for the Steadi-Two. The knob is groovedfor better grip and labelled with markers for L8 and H12.The height adjustment is made possible through the use of a high pitch screwin the middle of the device. In Figure 6.4, the knob is removed to reveal theheight adjustment screw. The device is designed such that only two turnsare required to bring the device to the desired height. At the top of thescrew, a bearing is added to minimize friction and wear from repeated turns.1546.1. ITERATIVE PROTOTYPING PHASEFigure 6.4: Height adjustment screw showing high pitch threads whichminimize required turns to achieve height changes.There are several other components of the Steadi-Two such as the customfabricated swivel bearing, bottom magnet that allows easy and fast attachmentto a glove, and the entire frame connecting the mass to the bearing andheight adjustment screw. These components have been iterated several timesto provide reliable performance and minimize potential failure in practice.1556.2. POLYMAGNETIC SPRINGS6.2 Polymagnetic SpringsA key change in the design of Steadi-Two is the implementation of Polymagnets.Polymagnets, unlike traditional magnets, are not unipolar. This meansthat a single magnetic face has multiple poles of North and South ina pre-determined pattern. This results in a unique behaviour in theirforce-distance relationship depending on the height and angle of interaction.Consider the sketch in Figure 6.5 showing a sample Polymagnet face.Figure 6.5: Sample Polymagnet face showing the arrangement of the magneticpoles as concentric rings of varying Radii. The North poles are in red andSouth poles in blue.From the Polymagnet sketch, we can infer that the axial interaction oftwo Polymagnet rings produces a non-standard force-distance relationshipcompared to the interaction of two regular permanent magnets like the onesused in the T-Beam. This is best understood through Figure 6.6.1566.2. POLYMAGNETIC SPRINGSWhen the Polymagnets interact non-axially i.e. at an angle, the response is acombination of attractive and repulsive forces. This means that the behaviourof the magnetic spring depends on the make-up of the two magnetic faces.There are significant challenges involved in attempting to quantify the forcedistance relationship of these Polymagnets which involves extensive magneticmodeling. However, the advantage of using Polymagnets is two-fold:1. Given the ability to alter the combination of magnetic ring size andcombination, it is possible to create a desired spring behaviour to coverboth HFLA and LFHA tremors as well as multi-DOF tremos.2. When a region of the Steadi-Two device is at extreme position where theinteraction may produce a primarily attractive or repulsive force, otherregions are in different interaction phases and produce a counteractingforce which results in restoring the device to equilibrium. This meansthe device does not experience magnetic locking and is more stableeven under high amplitude tremors.The nature of interactions between the Polymagnets that result in themagnetic spring for the Steadi-Two device is fairly complex such that amathematical model is computationally intensive. The models generatedfor this purpose have been classified as proprietary material together withthe Steadi-Two device under the Patent US11466745B2 titled Apparatus andmethod for suppressing oscillations.1576.2.POLYMAGNETICSPRINGSFigure 6.6: Comparison of the force-displacement relation of regular magnets (blue) vs. Polymagnet (red).Polymagnets have a non-traditional force-displacement relationship allowing for unique responses in low and highdisplacement scenarios.1586.2. POLYMAGNETIC SPRINGSFigure 6.7 is a section view of the final Steadi-Two iteration highlightingsome key components.Figure 6.7: Steadi-Two MDOF actuator highlighting key features. The swivelbearing is designed to be low friction for response to low amplitude tremors.The Base Magnets connects to a glove fitted on the arm when in use andthe dynamic mass is made of brass; a non-magnetic material.159CHAPTER 7Model Validation and Experimental EvaluationIn the previous chapters, we have introduced the theoretical frameworkfor vibration absorption for tremor attenuation, introduced and modeledlinear and non-linear conceptual designs and prototyped the final devicefor MDOF tremor reduction. This chapter evaluates the performance in acontrolled experimental setting. It describes the experimental design, testingand validation protocols, and methods for data collection and analysis. Theresults of these experiments provide quantitative measures of the effectivenessin suppressing tremors across key degrees of freedom. The findings set thestage for patient testing and real-world application, as discussed in thesubsequent chapter.The chapter begins with a detailed description of the experimental setupand describes the model validation process. Validating the model is a keystep in confirming the accuracy and applicability of the model in predictingreal-world performance. This is especially important for future developmentwhere the model can be updated using data from real patient tests providedin Chapter 8.1607.1. EXPERIMENTAL SETUP7.1 Experimental SetupA constant challenge when developing tremor attenuation devices is theinability to test devices in real time on tremor patients. This is due to amultitude of reasons including satisfying the ethical requirements necessaryfor in-vivo evaluations. Additional challenges such as rapid design changesduring prototyping make scheduling of tremor tests difficult. Moreover, theinability to predict side effects of the device in early development makeit more challenging to reliably quantify performance at the required rate.These challenges, coupled with the impact of the COVID-19 pandemic whichoccurred during the early development of this device, make the developmentof alternative tremor evaluation techniques a necessity.7.1.1 Multi-DOF Variable Parameter Tremor InducerThe successful design of a tremor attenuator begins with understanding andcharacterizing tremors. As discussed in Chapter 2, this research aims toattenuate neurological tremors resulting from both Parkinsons Disease andEssential Tremor Disease. These diseases manifest with varying frequenciesand amplitude. Therefore, to accurately represent their manifestation, anytremor inducer should be able to simulate a range and combination offrequencies and amplitudes.Tremor inducers are common in attenuator design. The size, shape, andeffectiveness of each inducer are specific to the study but it is importantthat the inducer is well fixed to the arm to ensure proper transmission ofinduced tremors. Due to the limitation of obtaining tremor test subjects andconcerns on long-term impact, tremor inducers are often used on a healthytest subject. The limitations of this approach are covered in Chapter 9.4.In the experimental phase of this study, the tremor inducer used is shownin Figure 7.1. This device consists of a rotating eccentric mass driven bya motor. The motor has an encoder that is addressable from an Arduinoto achieve variable frequencies. Variable amplitude is achieved using anadjustable radius arm to change the eccentricity of the rotating 50g mass.1617.1. EXPERIMENTAL SETUPTo determine the tremor (modeled as a harmonic torque) from the inducer,we take the force Fm due to the rotating eccentric mass, the height h ofthe motor from the glove which is fitted on the arm, and the position ofthe eccentric mass r. When the motor is rotating at a frequency of ωr, theresulting force is given by Fm = mr ω2r cos(ωrt).The resulting induced torque on the arm is modeled by:T (t) = Fm × h = mr hω2r cos(ωrt)Figure 7.1: Tremor inducer showing the adjustable radius arm and slotfor changing induced tremor direction to achieve different tremor DOF.Current configuration shown is for pronation-supination tremor inductionand r = 45mm, h = 71.82mm thus at 6Hz the induced moment amplitudeis Mo = 0.21792Nm.1627.1. EXPERIMENTAL SETUPWhen the glove is properly fitted on the arm, we assume all torque is inducedto the arm and consider human skin elasticity as negligible. This is animportant assumption as it allows modeling of the induced tremor as atorque without having to account for the changes in transmitted torquedue to the skin compliance. Additionally, it is important to recall that thehuman arm is intricate with 7 DOFs. Therefore, while our intention is tostudy MDOF tremors, this device primarily induces tremor in a single degreeof freedom and the cross-DOF tremor is due to transmission effects. Thismeans inducing a purely FPS or FE tremor is not possible with this device.Figure 7.2: Experimental tremor setup showing the position of the tremorinducer (black box) on the forearm and measurement axes. Rotation aboutthe axes represents; X: Pronation Suppination, Y: Pronation-Suppination,Z: Radial-Ulnar Deviation. Translation on the axes represents; X,Y:Pronation-Supination, Z: Flexion-Extension.1637.1. EXPERIMENTAL SETUPThe hand tremor simulator is wired to produce tremors over a wider rangeof frequencies using a Mosfet driving circuit controlled by an Arduino. Theinducer is placed on the arm at various positions and orientations to generateDOF specific tremors. For instance, in Figure 7.2, the placement of theinducer to generate FE tremors is shown.The motor voltage is calibrated for each driving frequency for both a freehandi.e. no device and dead-weight i.e. added inertia of device. This removesthe necessity of frequency measurement for each experimental run makingevaluation easier. Additionally, calibrating the device for a freehand anddead-weight (sham) scenario is key in ensuring no stray frequencies areintroduced during experiments. The calibration is completed for a range offrequencies shown in Table 7.1. There are two additional columns showingthe upper and lower frequency bounds for a specific tremor frequency.These boundaries are obtained from a study by Steadiwear Inc on tremorfrequency band distribution for real patients.From the frequency boundary data, there is a potential overlap in differenttremors due to deviations. This implies that in real tremors, the the tremorattenuators may behave differently due to frequency variation in patientswhich is not captured in these experiments since the frequency induced usingthis device is fixed.Table 7.1: Voltage-frequency mapping comparing free-hand induced frequencyvs. sham frequency (50g inducer mass). Upper and lower frequency boundsbased on deviation data courtesy of Steadiwear Inc.Inducer Mass = 50g Frequency BoundariesVoltage (V) Freehand (Hz) Sham Freq (Hz) Upper Lower8.00 5.00 5.00 5.34 4.438.50 5.60 5.00 5.34 4.439.00 5.80 5.20 5.54 4.639.50 6.00 5.60 5.94 5.0310.00 6.40 6.00 6.34 5.4310.50 6.70 6.20 6.54 5.6311.00 6.90 6.40 6.74 5.8311.50 7.40 6.80 7.14 6.2312.00 7.60 7.00 7.34 6.431647.1. EXPERIMENTAL SETUP7.1.2 FPS Tremor Test PlatformThe impact of using a tremor inducer on a healthy human arm is not evaluatedin this research. However, we infer from other studies that have studied theeffects of long-term use of power tools that induce vibrations and whole bodyvibrations. These have been found to have detrimental side effects to limbhealth. Thus, creating a tremor platform is beneficial for the evaluation ofthe Steadi-Two and its future iterations. Additionally, the modeling sectionuses tremor induced in a single DOF without cross-DOF effects. This is notpossible when inducing tremor in the human arm.This study tackles these challenges by designing a tremor testing platform(herinafter ”RIG”) to evaluate forearm pronation-suppination tremors tremors.The RIG, shown in Figure 7.4, consists of a model of the human arm inthe FPS DOF with. The inertia is achieved using doughnut masses andthe stiffness is obtained from a calibrated torsional spring matched to thedynamic properties of the human arm. The damping in the human arm inthe FPS DOF is approximately 0.021Ns/m which is too low to obtain frommanufacturers thus no viscous damper is included. For tremor measurement,two approaches are implemented for the RIG:1. A sensor is attached on the palm to measure the displacement of thearm on the longitudinal axis.2. A pen is placed at the end of the rig and a paper is used at the bottomof the pen to sketch the response displacement from equilibrium. Thisis similar to seismograph measurements and a sample result is shownin Figure 7.3.The rig is modeled in Matlab as a shaft under torsion. This is to evaluatethe Steadi-Two computationally and physically for validation. Using therig for validation is the most reliable approach for validation since it limitsparametric variability compared to the human arm. The EOM of the rig isgiven by Equation 7.1.Iθ¨ + carmθ˙ + karmθ = T (t) (7.1)1657.1. EXPERIMENTAL SETUPFigure 7.3: A sample showing the displacement of the pen tip attached atthe end of the rig. This approach, similar to seismograph output, allows forquick visual reference when determining the amplitude reduction for a giventest.The torque function for the rig is given by Equation 7.2 where A is a frequencyrelated amplitude parameter and f is the driving frequency in Hz. Similar tothe tremor inducer, the rig’s voltage-frequency relationship is also calibrated.Tt =3A(−0.0981f + 0.8717) cos(2π ft) (7.2)1667.1.EXPERIMENTALSETUPFigure 7.4: Tremor test platform to induce purely FPS tremors. The ”RIG” includes doughnut masses for inertia,a torsional spring for stiffness and two measurement approaches; a) a gyroscopic/accelerometer sensor., b) a penand paper to produce ”seismograph” measurements.1677.1. EXPERIMENTAL SETUP7.1.3 Instrumentation and Data ProcessingSensors: The deflection of the palm’s top from its equilibrium positiondue to induced tremors is measured using two distinct sensors described inTable 7.2. These sensors are mounted on the palm to track the inducedoscillations. It is important to note, however, that the measurements obtainedfrom the sensors are not a direct representation of the motion of the palm’scenter of mass. Since the sensors are placed at the top of the palm, thedeflection they capture may differ from the true movement of the palm’scenter, potentially introducing some error in the interpretation of the tremordynamics. Despite this limitation, the chosen sensor configuration provides areliable means of monitoring the tremor response, with sufficient accuracy toobserve the relative motion between the palm and the surrounding structures.In the context of this research, the primary goal is to design and evaluate atremor attenuation device. Therefore, the instrumentation is not only usedto capture the tremor-induced deflections but also serves as a critical tool incomparing the performance of the tremor attenuation system. Specifically, thesensors are employed to measure and quantify the response of the arm bothwith and without the presence of tremors. These comparative measurementsallow for an assessment of the device’s effectiveness in attenuating tremormotion, which is key to evaluating the device’s overall performance. Thedata collected from these sensors will thus form the basis for performanceanalysis, enabling the comparison of various tremor attenuation strategiesunder controlled experimental conditions.Table 7.2: Specifications for the Bosch Accelerometer (BMI160) and STEVALWeSU (LSM6DS3) sensor package used in the experimental phase.Sensor Bosch (BMI160) WeSU (LSM6DS3)Description Small, low power 16-bit IMU Evaluation board.Range ±8 g ±125→ ±2000dpsResolution ≈ 10−6 g ≈ 10−3.Axes X, Y, Z X, Y, ZSampling Rate 250 Hz 1600 Hz1687.1. EXPERIMENTAL SETUPUsing multiple sensors capable of measuring both acceleration (which can benumerically integrated to obtain vertical displacement) and angular velocity(which can be numerically integrated to obtain angular position) is excellentfor measurement confirmation. Once the data is collected, it is processedin Matlab using the built in Fast Fourier Transform function, FFT . Thistransforms the data into the frequency domain where the induced frequencycan be confirmed with the measured frequency. This is also important foridentifying disturbances or noisy signals which result in unreliable data.Figure 7.5 shows the results of a sample FFT analysis with a 6Hz inducedtremor. For this sample, it is clear that all three axes peak around 6Hz.These FFT curves are from accelerometer data which has been transformedfrom the time domain. In Appendix C.3, the code script used to implementthis process is provided.Closer inspection of Figure 7.5 reveals that the induced dominant tremor isclose but not exactly 6Hz. This may be as a result of inaccuracies in the sensorresulting in noisy data. It could also be as a result of inducer misalignmentduring the experiment due to skin compliance which would generate a dampedfrequency. From the FFT Matlab script, the peak frequency is identified tobe exactly 6.0334Hz which is less than 1% deviation from 6Hz and thus isan acceptable for analysis. However, in situations where the collected datashows a considerable amount of noise, a 4th order Butterworth bandpassfilter is implemented in Matlab using the code provided in Appendix C.3.1.Following the proper protocols of digital signal processing, filtering is onlyapplied if the Signal to Noise Ratio, SNR, is 60dB or higher. The SNR iscalculated using Equation 7.3.SNR = 10 · log10(PsignalPnoise)(7.3)Note: The sensors operate at sufficiently high sampling rates, ensuring thatanti-aliasing concerns, as described by the Nyquist sampling theorem, arenot applicable. Specifically, the sensors sample at 250 Hz and 1600 Hz, whilethe highest frequency signal of interest is 12 Hz.1697.1. EXPERIMENTAL SETUPFigure 7.5: Fast Fourier Transform of raw experimental accelerometer datato confirm that induced tremor is at 6Hz. The X-axis represents FPS tremorand Y-axis represent FE tremor. The Z-axis represents the WRUD tremor.1707.1. EXPERIMENTAL SETUPFiltering Noise: Figure 7.6 shows an example of accelerometer datacollected for the flexion-extension DOF. Using the Butterworth bandpassfilter, the noise is removed to reveal the true system behaviour whichcloselyt resembles the near resonance response of the flexion-extension modelintroduced in Chapter 3. Additionally, the filtered data follows the expectedresponse of a damped-driven vibrating system which contains both transientand steady state responses.Figure 7.6: Comparing raw and filtered data using a 4th order Butterworthbandpass filter for a 6Hz signal with cut-off frequencies at 5.5Hz and 6.5Hzfor the Z-axis accelerometer measurement1717.1. EXPERIMENTAL SETUPNumerical Integration: When necessary, the filtered data for all threeaxes is numerically integrated to obtain the displacement of the center ofmass of the palm. This is generally avoided when dealing with gyroscopedata since gyroscopic drift can occur leading to misrepresented results. Foraccelerometer data, Figure 7.7 shows numerical integration results with thevertical axis showing the displacement in meters for all three axes whichrepresent the FPS, FE, and WRUD DOFs. From this response, both thetransient and steady-state responses are visible with a 3s settling time. Closerobservation of the system response shows an increase in the displacement.This is quite visible for the Y axis (which represents the FE DOF) responseand is evidence of a beat phenomenon in the system. This was alsoobserved during the modeling of the human arm in the FPS and FE DOFs.Filtering is done for all the collected signals to eliminate high frequencysampling noise. To obtain the displacement of the arm under the inducedtremor, a numerical integral can be applied using Matlab’s built in functioncumtrapz which uses the trapezoidal method for integration. Note thatthe trapezoidal method is employed here for its convenience as a built infunction but other methods such as the Midpoint rule or Simpson’s rule canbe readily applied and yield similar results. To demonstrate this, Figure7.8 shows a comparison between the results of numerical integration usingthe Trapezoidal method and Simpson’s rule. It is clear that the results areidentical. As a final note, the methods named here are one-dimensionalnumerical integration methods and are deemed sufficient for the scope ofthis work.1727.1.EXPERIMENTALSETUPFigure 7.7: Displacement of the Palm center of mass for X, Y , and Z axes under induced tremor at 6Hz. Thearm-inducer complex has an approximately 3s transient state and experiences a beat phenomenon implying asystem natural frequency around the 6Hz region.1737.1.EXPERIMENTALSETUPFigure 7.8: Comparison of the Trapezoidal Method and Simpson’s Rule for One-Dimensional Numerical Integration.The two methods produce identical results.1747.2. MODEL VALIDATION7.2 Model ValidationWith the completed experimental setup, we can begin experimental evaluationof the absorbers. However, we must first validate the simulation models usedin this study. In this section, we consider the validity of the arm model forforearm pronation-suppination by comparing their simulated responses tothe ones obtained by inducing tremor. We also validate tremor test platformby comparing the RIG response to the modeled FPS response. Finally, wevalidate the Steadi-Two response by comparing its modeling response to thatmeasured using the RIG.The goal of validation is to compare the predictions obtained from themathematical models with the experimental results obtained from theprototype tests. This comparison ensures that the theoretical frameworkaligns with real-world performance, identifying any discrepancies and providinginsights into potential design improvements. By evaluating the modelaccuracy, we gain confidence in its predictive capabilities which is crucialbefore further experimental evaluation.7.2.1 Validation ProcessThe validation of the model is carried out by comparing its predictions withdata obtained from the prototype. Measurements taken from the prototypeunder controlled tremor-inducing conditions are compared to the modelsimulated under similar parameters. The comparison focuses on two keyaspects:1. The tremor response including the the RMS values of the all systemsduring the transient and steady-state phases of the response. Thisis both a statistical and visual evaluation to compare the predictivemodels to the real experimental systems.2. A frequency domain analysis of the predictive systems and the measuredsystems and determine whether the tremor magnitude and dominantfrequencies are similar.1757.2. MODEL VALIDATION7.2.2 Tremor Response AnalysisIn the tremor response analysis, the first comparison involves the timeresponse of three systems listed below:∗ The FPS model of the human arm introduced in Chapter 3.1.2. This isthe model through which all device analysis has been done so validationis fundamental.∗ The response of the human arm with an induced FPS tremor. Thisis the real FPS response which is obtained from the averaging ofthree experimental runs using a gyroscope to measure the angulardisplacement of the arm.∗ The response of the RIG test under tremor. The RIG is modeled afterthe FPS DOF and is the platform through which device evaluationis done throughout the experimental phase. Therefore, validating theRIG response before implementing it in further use is necessary.For the three systems listed above, the validation is conducted at 5Hz andthe time response is given in Figure 7.9. We observe from the response thatall three systems have unique transient responses which is attributed to thevarying damping characteristics for each system. However, the responsesconverge in steady-state which is is the region of significance in our analysis.This result is significant since it lends credence to the model validity andshows that the RIG is a reliable model of the FPS DOF.In addition to the time response, we implement a unique technique to furthercompare the responses and improve our understanding of the similaritybetween the systems. We do this by comparing the RMS responses of all thesystems. However, unlike traditional RMS calculations which obtain a singlevalues, we split the signals to specific response windows and evaluate theRMS values in those windows. This approach allows us to better view thevariation in transient and steady-state responses and is a better visualizationof the convergence of the different systems. Figure 7.10 displays the result ofthis approach.1767.2.MODELVALIDATIONFigure 7.9: Comparison of the FPS arm model and RIG model to induced FPS tremor. All systems have differenttransient response but the steady-state amplitude converges.1777.2.MODELVALIDATIONFigure 7.10: Comparison of the FPS arm model and RIG model to induced FPS tremor using a windowed RMSapproach. The unique transient response (attributed to damping effects of each system) is much more clearlyobserved as well as convergence in steady-state. This approach further validates the models.1787.2. MODEL VALIDATIONNow that we have observed the similarity of the FPS arm model and RIGmodel to the human arm under tremor, we can be confident in validating theresponse of the Steadi-Two by comparing the model prediction to measuredresponses using the RIG. Note here that using the RIG allows for inducingpurely FPS tremors with repeatable measurements of limited variation. Wemust acknowledge the limitation of this approach at this stage of validationbut in Chapter 8, we will review the predicted reductions and compared itto real patient data.The validation of the Steadi-Two focuses on two things:1. The windowed RMS approach instead of the raw time response. Nowthat we have seen the significance of this approach (i.e. easier visualdiscernment of the result), we can use it to quickly compare theresponses.2. Overall predicted tremor reduction. This is the goal of this studyand is the ultimate comparison tool when looking at predicted versusreal results. It can be confirmed using both visual and kinematicmeasurements and can transcend all disciplines involved in this study.In Figure 7.11, the windowed RMS response of the predicted Steadi-Tworesponse and the measured L8 and H12 responses are compared. Here we seethat the responses are different for the L8 and H12 which is expected sincethey represent two unique stiffness profiles of the Steadi-Two system. Thepredicted response of the Steadi-Two is significantly different when comparedto either of the two systems. However, this is expected for the followingreasons:∗ Modeled and real non-linear systems experience complex dynamics andexternal experimental factors such as the overall device mass, coulombdamping, and imperfect contact can influence the outcome.∗ The Steadi-Two model does not capture the Polymagnet profile ateither L8 or H12 in its implementation of the quintic Duffing oscillator.This shows the challenge in accurately modeling non-linearities evenwith the cautious approach taken in this study. This likely explainsthe simulated result being in between the two device modes.1797.2. MODEL VALIDATIONIn addition to the windowed RMS approach, we also consider the predictedresults from our simulation in Chapter 5.4.3. The simulated tremor reductionat 5Hz was calculated as 63.60% from Table 5.6. On the other hand, measuredexperimental data from the prototype evaluation reported tremor reductionsof 33.92% for the L8 mode and 66.10% for the H12 mode. The RMS reductiondata shows that the variance between the Steadi-Two prediction and themeasured H12 data is approximately 3.78% which is within acceptable boundof uncertainty.However, when looking at the results of the windowed RMS analysis inFigure 7.11, the simulated Steadi-Two response appears to converge towardthe H12 solution over time. This highlights the importance of separatingthe transient and steady-state responses from a computational perspective.When this is done, the Steadi-Two settles to approximately 65.30% which ismuch closer to the measured H12 reduction. With this result, the model isrefined to take the steady-state response for the Steadi-Two predictions.These results offer an important caution for the next step of analysis.First they implore us to refine the model which is described in Chapter7.2.5. Second, they signal that further analysis is required, particularlyinto the frequency domain to understand what the variation means fromthe perspective of tremor power and frequency shifts. From the theory ofvibration absorption in Appendix A.1, the addition of an absorber shiftsfrequencies and the in the case of non-linear absorbers, this also includes theintroduction of harmonics. Therefore, by looking at the frequency domain,we can gain further perspective on the variation in performance between thepredicted and measured results.1807.2.MODELVALIDATIONFigure 7.11: Comparison of the FPS arm model and RIG model to induced FPS tremor using a windowed RMSapproach. The unique transient response (attributed to damping effects of each system) is much more clearlyobserved as well as convergence in steady-state. This approach further validates the models.1817.2. MODEL VALIDATION7.2.3 Frequency Domain ComparisonThe frequency domain analysis is an important consideration when validationsimulation models. For our analysis, the frequency domain will show thedominant frequency peaks using FFT analysis and with the PSD analysis, wecan understand the tremor power distribution. The FFT and PSD analysisof the predicted and measured Steadi-Two responses are given in Figure7.12.The FFT analysis highlights the amplitude of frequency components for boththe simulated (MDOF) and measured data (L8 and H12). In the measureddata, a dominant frequency peak is observed around 5Hz, representing theprimary tremor frequency. This peak is sharp and well-defined, showingconsistent tremor behaviour in the experimental setup. The simulation alsopredicts a peak at approximately the same frequency, demonstrating thatthe model captures the induced tremor frequency. However, the peak inthe simulation is narrower and has a slightly lower amplitude compared tothe measured data, which suggests a slight underestimation of the tremorresponse. Beyond the dominant frequency, the simulation shows minimalamplitude, while the measured data have slight variability. This is possiblydue to noise in the mechanical system that the model does not account for.The PSD analysis provides insights into the energy distribution acrossthe frequency spectrum. Both L8 and H12 datasets exhibit a sharp andconcentrated power peak at approximately 5 Hz, indicating a periodic tremor.Similarly, the simulation identifies a dominant power peak at 5 Hz. However,the power magnitude predicted by the simulation is slightly lower than thatof the measured data, suggesting that the model may not fully replicate theenergy dynamics of the tremor. Outside the dominant frequency range, thesimulation shows minimal power distribution, while the measured data havea slightly broader power spread, likely reflecting real-world factors such asnoise or higher-order dynamics not captured in the simulation.Therefore, while the simulation effectively identifies the dominant tremorfrequency and accurately predicts tremor reduction, discrepancies in theamplitude and power distribution highlight areas for refinement.1827.2.MODELVALIDATIONFigure 7.12: Comparison of FFT (upper plot) and PSD (lower plot) between the predicted (MDOF simulation)and measured results (L8 and H12).1837.2. MODEL VALIDATION7.2.4 Error AnalysisThe discrepancies between the model’s predictions and experimental dataare briefly analyzed using error metrics. We use the root mean square error(RMSE), which quantifies the difference between the predicted and actualtremor reduction. The RMSE is calculated as follows:RMSE =√√√√ 1nn∑i=1(y(i)pred − y(i)exp)2where y(i)pred and y(i)exp are the predicted and experimental tremor reductions,respectively, and n is the number of data points.Table 7.3: RMSE analysis of predicted and measured results. Lower valuesindicate similarity while higher values highlight variation.Comparison RMSEFPS Simulation vs. Induced Tremor 0.26943Rig vs. Induced Tremor 0.23917Predicted vs. L8 0.64052Predicted vs. H12 0.36537From the error analysis, we make the following observations:∗ FPS Simulation vs. Induced Tremor: Relatively low RMSE indicatingaccurate prediction from the FPS simulation when compared to theinduced tremor on a human arm.∗ RIG vs. Induced Tremor: Similar to the FPS simulation, the RIG alsohas a low RMSE indicating an accurate representation of the humanarm in the FPS DOF.∗ Predicted vs. L8: Highest RMSE value indicating a greater discrepancyto the measured value. This indicates that the simulation does notcapture the lower height mode of the Steadi-Two prototype. Modelrefinement is required to better represent this mode.∗ Predicted vs. H12: Lower RMSE value compare to the L8 mode. Thisimplies the model is much closer to the H12 mode although not asaccurate. However, in steady-state analysis, this error might be furtherreduced since the predicted and H12 result converge over time.1847.2. MODEL VALIDATION7.2.5 Model RefinementsFrom the results of this brief validation of the Steadi-Two and the results ofthe error analysis, several refinements to improve the the model accuracyare proposed.1. Adjusting the damping and stiffness parameters within the modelto capture the different device modes (L8 and H12) will align thesimulation with the experimental data, particularly in capturing theamplitude and energy dynamics of the tremor reduction.2. Incorporating noise or higher-order dynamic behaviour could furtherenhance the model’s realism. This would better capture tremor powerdistribution across frequencies.For this study, mathematical modeling has been used to identify fundamentaldesign parameters that influence performance. To achieve better modeland reality agreement, iterative and continuous refinement is required. Inpractice, the model was continually refined after each device iteration in theexperimental phase of analysis. The experimental evaluation process and theresults are covered in the sections that follow.A note on the use of Mathematical Models: A mathematical model isa representation in mathematical terms of the behaviour of real objects andsystems. Models can help describe what behaviour is observed, explain whythat specific behaviour is observed, or predict what behaviour will be observedwhen conditions are changed. One should keep in mind the model limitationsand more importantly, the conditions in which the model predictions are true.Before engaging in a mathematical modeling or refinement exercise, we mustconsider why we are creating the model, what do we already know and/or areseeking to find with this model, assumptions we are making for the modelto work, how is the model defined i.e. governing principles, predictions themodel can make about the outcome and validity of these predictions. Modelrefinement is not always necessary, sometimes, prototyping can ease outconcerns just as well. It is also a good practice to verify if the mathematicalmodel is able to fulfil more than its original purpose to avoid over extendingits use.1857.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPES7.3 In-Vitro Evaluation of Absorber PrototypesIn this section, we present the experimental evaluation of the attenuatorprototypes. The experiments are conducted with the attenuators integratedinto the system, and the previously outlined data processing steps are appliedto the resulting signals. The performance of the attenuator prototypes isquantified through a direct comparison of the absorberless system, thedead-weight (sham) system, and the absorber-equipped system, providing acomprehensive assessment of their effectiveness.7.3.1 T-Beam Experimental EvaluationIn Chapter 4.2, there are four design iterations shown for the T-Beam. Theseare: i) the linear system, ii) the original non-linear T-Beam from HVPL, andthe proposed modifications for the non-linear T-Beam in; iii) iteration 1 andiv) iteration 2. The performance of the linear T-Beam in attenuating FE DOFmodeled by a simplified arm model has been experimentally studied in theHVPL associated publication by Shah at al. [92]. Additionally, the originalnon-linear T-Beam, reported to weigh approximately 120 g [66], has also beenstudied and found to be limited in attenuating LFHA tremors as documentedin the publication by Masoumi [65]. For the two proposed iterations ofthe non-linear T-Beam, the prototyping and experimental evaluation arediscussed here.Design B Iteration 1: This iterations aims to maximize stiffness byachieving perpendicular magnetic interaction at maximum deflection usingan angled base for the bottom magnets. However, the following flaws werediscovered during prototyping:1. The low magnetic field interaction at rest highly impacts the equilibriumstability of the desing. This makes it responsive to non-tremor motionwhich is an important performance criteria.2. For the magnetic interaction at maximum deflection to function asintended, the T-beam height must be significantly higher than previousiterations. However, this requires a higher magnetic force to restoreequilibrium which makes the system stiffer. This in turn means thesystem is not responsive to low frequency tremors which is the purposeof modification.1867.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESDesign B Iteration 2: The flaws of iteration 1 do not impact iteration2. In fact, this design functions as intended with stable performance duringtremor. Therefore, the experimental evaluation for this T-Beam involved testsat multiple tremor frequencies for three DOF; Forearm pronation-suppination(FPS),wrist flexion-extension (WFE), and wrist radial-ulnar deviation (WRUD).The instrumentation, sampling and data processing have already been coveredin Chapter 7.1. The aforementioned proposed T-Beam modifications were toaddress the challenge of low frequency tremors which the original T-Beamfailed to address. Thus, iteration 2 is evaluated at 3 induced frequencieswhich are 2Hz, 3Hz, and 4Hz to determine whether the modified T-Beamis now capable of tackling lower band tremor frequencies in multiple DOF.Note here that success is defined as a minimum 50% reduction in all DOFs.The response for the 2Hz tremor is in Figure 7.13. The modified T-Beammodel meets the performance expectations in the WFE DOF but fails inFPS and WRUD. In fact, the T-Beam causes an amplification in the FPSDOF. This is similar to the results seen in the modeling phase where theT-Beam performed when in one DOF but not the other. This performancetrend holds for both 3Hz and 4Hz as shown in Table 7.4.Table 7.4: Percentage tremor reduction for the T-Beam in FPS, WFE,WRUD. The T-Beam struggles to achieve cross-DOF performance.F (Hz) FPS WFE WRUD2 -8.11 52.5 16.123 8.6 53.14 27.24 2.76 43.04 26.671877.3.IN-VITROEVALUATIONOFABSORBERPROTOTYPESFigure 7.13: T-Beam response for FPS, FE, and WRUD for a 2Hz induced tremor comparing the tremor andT-Beam performance. The T-Beam significantly reduces tremor in WFE but amplifies FPS tremor.1887.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESThe results of the experimental evaluation for the modified T-Beam highlighta fact that has become clear throughout this process: the T-Beam is a singledegree of freedom absorber with limited performance for lower frequencytremors. We make the following final observations on the experimentalevaluation of the T-Beam:1. The T-Beam performance in the WFE DOF is satisfactory for the 2→ 4Hz range. This contrasts the original T-Beam whose performance wasin the higher frequency ranges for neurological tremors. We can inferthat the changes to the design and alteration of the magnetic springshave improved the lower frequency band performance in WFE. However,the drop in performance at 4Hz indicates a bandwidth limit whichrequires further investigation.2. The T-Beam is unable to adequately reduce tremor in the FPS andWRUD DOFs and in some cases, amplifies the tremor. This wasobserved when analysing the linear T-Beam model in Chapter 5.1.Although the non-linear T-Beam model showed promising performance,the measured results indicate otherwise which imply the model requiresfurther refinement.In the section that follows, we will turn our attention to the Steadiu-Twofor experimental evaluation. However, there are some lessons learned fromthe process of designing the non-linear T-Beam. These will be discussed inChapter 9.1897.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPES7.3.2 Steadi-Two Experimental EvaluationThe evaluation iteration of the Steadi-Two device becomes a necessitynow that the T-Beam has been empirically shown to be inadequate forMDOF tremors. The target tremor reduction for the Steadi-Two is aminimum of 50% tremor reduction for WFE and FPS DOFs for the 4− 12Hzfrequency bandwidth. The final Steadi-Two device was developed in atripartite collaboration between Mitacs, SteadiWear and the Heart ValvePerformance Laboratory at UBC Okanagan. The design of the Steadi-Twobegan in 2019 and concluded in 2024. The fundamental step in the design ofthis device was identifying the shortfalls of the T-Beam and compensatingfor them in both design and functionality.From the several conceptual designs in Chapter 4.3, there are three iterationsfor a potential MDOF device. The first iteration faced the following challengesduring prototyping:∗ The imbalance caused by unequal masses on each system posed stabilitychallenges during operation.∗ The device experienced magnetic locking for certain frequencies. Thisis because the magnetic field interaction resulted in an uneven field.These flaws invalidated iteration one of the design. The second iterationprototype showed initial promising results but experienced magnetic lockingfor low frequency tremors. This manifested as opposing magnetic polesattracting and locking the device in place. However, high frequency tremors(which generally have low amplitudes) were often well attenuated for bothDOFs. This challenge of magnetic locking required a uniform field to befully resolved. Iteration three was the first successfully tested prototype andsubsequent iterations modified various parameters to improve performance.Hereinafter, this iteration will be considered the first prototype since it isthe first Steadi-Two prototype to be successfully designed and tested bothin-vitro and in-vivo.1907.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESProof of Concept EvaluationThe first Steadi-Two prototype was a proof of concept device. It did not havethe full mechanism shown in the final prototype. As a result, its evaluationwas simplistic and focused on easily verifiable results. The qualitative scaledescribed in Table 7.5 was used to rank the performance of the Steadi-Twoprototype in attenuating induced tremors on a hand holding a cup of water.The poor, acceptable, and excellent criteria were based on visual confirmationof water movement or spillage in the cup. This experimental procedure isavailable for viewing as a video recording through SteadiWear Inc.. In thevideo, the eccentric motor tremor inducer is used to induce the tremors in anon-tremogenic human hand.Table 7.5: Performance ranking criteria for the Steadi-Two device. Theranking is done through visual inspection of the cup of water and rankedbased on the amount of spillage.Rank Performance Description1 No function: water is spilling2 Acceptable performance: water slightly jumpy3 Excellent performance: StillwaterThe initial prototype of the Steadi-Two had of four device heights rangingfrom 9mm to 12mm. The evaluation was used to establish a performanceenvelope for each height of the device and determine the ideal heightcombinations that produce a device capable of broadband tremor attenuation.The proof of concept test was only conducted for FPS induced tremors sinceno device had combated low frequency FPS tremors successfully. The imagein Figure 7.14 shows the experimental setup for this evaluation process.Note: The COVID-19 global pandemic impeded real patient evaluation inearly development. This approach was an improvisation to allow developmentto continue but is not meant to replace real patient evaluations duringiterative development.1917.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESFigure 7.14: Steadi-Two initial qualitative test setup. The tremor inducergenerated tremogenic movements and the performance of the Steadi-Twowas evaluated by visually inspecting the water line (marked in red). Thedevice was shammed using foam to compare dead-weight (DW) which is thesham performance versus the dynamic performance.In Table 7.6, the qualitative performance for the 12mm height is shown. Thismode is the highest height of the Steadi-Two device tested and it was designedto target the lower tremor frequencies by having a low baseline stiffness dueto the separation between the magnets. With a goal to minimize the size ofthe device, testing the highest acceptable device height was crucial to ensureperformance at the lowest frequencies was achievable. From the results inTable 7.6, the 12mm height had better performance at lower frequencies butpresented unreliable performance at higher frequencies during repeated runs.This prompted tests at 11mm whose results are provided in Table 7.7.1927.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.6: Qualitative performance for the 12mm mode of the Steadi-Twodevice.Mode (mm) Frequency (Hz) Run 1 Run 2125 3 25.6 3 26 2 36.2 1 36.4 1 36.8 1 17 1 1This mode has a single test as it was included in the second run of tests forconfirmation. While the performance is incredible at the higher frequencies,this mode experienced stability issues where the system would completelydisengage during tests. The 10mm evaluation was inconclusive similar to the11mm performance where two separate runs we conflicting in their results.Thus, the final mode of testing for the proof of concept device phase was the9mm mode and the results are provided in Table 7.8.Table 7.7: Qualitative performance for the 11mm mode of the Steadi-Twodevice.Mode (mm) Frequency (Hz) Run 1 Run 2115N/A15.6 16 26.2 26.4 26.8 37 3The performance of the Steadi-Two proof of concept device showed promisingtremor reduction for a wide range of frequencies. Although the test wasqualitative, there was sufficient evidence of tremor reduction to necessitatefurther development real patient tests. This was the next critical step indevelopment since tremor patients would have varying tremor characteristicswhich would help in identifying gaps in performance.1937.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.8: Qualitative performance for the 9mm mode of the Steadi-Twodevice.Mode (mm) Frequency (Hz) Run 1 Run 295 1 15.6 1 16 3 16.2 3 16.4 3 26.8 3 37 3 3Qualitative Patient EvaluationWith the positive results from the proof concept evaluation, SteadiWearInc. organized qualitative in-vivo tests with John Kirkconnell and BettyFaulkner. The videos of the tests are hyper-linked to their name for viewing.These tests were fundamental since the device was still in alpha developmentstage. These videos show a breakthrough in performance and substantialimprovement in quality of life for the patients. The significance of thisbreakthrough is reflected in the linked article where SteadiWear secured a1.1 Million CAD investor funding for the continued development.From the development perspective, the key takeaway from the proof ofconcept device testing is a need for a further in-vitro evaluation. This beginswith formulating a Design of Experiments (DoE) by identifying parametersinfluencing performance and their treatment levels. Below is a detailed listof all factors, their associated impacts and identified levels.∗ Tremor Frequency (Hz): This is the most fundamental factor influencingperformance since a wide performance band results in a wider group ofpatients targeted. Hereinafter, this bandwidth is set to 4→ 8Hz.∗ Tremor Amplitude: The experimental tremor is induced by an eccentricmass on a motor whose properties a covered in Chapter 7.1. The masson the motor is placed on an arm with a variable slider for its position.By adjusting the position, the induced amplitude can be manipulated.Thus, the treatment levels are a half length and full length of the leverarm (eccentricity).1947.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPES∗ Hand Posture: In Chapter 3, the posture of the hand is discussed as afactor influencing performance. The proof of concept evaluation andinitial patient tests did not evaluate hand posture directly but throughvarious tasks. However, it is important to determine the impact ofrepeated tasks that result in the hand being positioned at differentpostures. The identified postures are:1. Fully stretched arm perpendicular at the shoulder joint. Thismimics the FE model introduced earlier in the study.2. Elbow at right angle to upper arm and forearm parallel to theground.3. Posture of the arm when taking a drink.4. Posture of the arm when picking an item.∗ Mass/size of the human hand: The inertia of the human arm is criticalin determining the transmitted tremor when using an inducer. It isideal to have various hand size of small, medium, large to evaluate theperformance of the device.∗ Steadi-Two Mode: As introduced in the proof of concept tests, themode of the Steadi-Two device determines the effective frequencyrange. However, having more modes requires complex implementationfor reliable operation. Thus, the ideal solution is a single mode or twothat cover the entire frequency band for all postures and DOFs. Thisis a significant undertaking since no passive attenuator,to the authorsknowledge, has yet to achieve this. Additionally, fewer modes result inless complicated operation for patients.∗ Magnetic Profile: The Steadi-Two uses Polymagnets for its non-linearsprings. The design and strength of these Polymagnets impacts thestiffness which changes the effective frequency bandwidth. The variationof the magnetic profile is limited by what is manufacturable (a conceptoutside the scope of this study). However, evaluating various stiffnessprofiles is key in identifying the best profile for improved performance.∗ Dynamic Mass: The dynamic mass of the Steadi-Two directly impactsthe total weight of the device as well as inertia. This affects bothpatient comfort for long term use as well as device performance. Theremust be a balance between these conflicting outcomes. For the mass,two levels of 250g and 300g are selected.1957.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESA full factorial experimental run would require excess of 3000 experimentswhich is not possible within the time constraint for this study. Therefore,a partial run is conducted with key parameters evaluated at all levels.These are the dynamic mass, tremor frequency, Steadi-Two mode, andthe magnetic stiffness profile. From this step of evaluation, the developmentof the Steadi-Two combines experimental and design work. This means thateach iteration builds on the success of the previous device.Therefore, the rest of the experimental evaluation results are presented instages according to the device iteration. Given the quantity of collected data,it is impossible to include all frequencies from the evaluations. Therefore,only data from approximately 6Hz tremor is provided in text. Additionalexperimental results are provided Appendix D.Iteration One: Magnetic Springs and Additional InertiaFollowing the results of the proof of concept device and initial patient tests,a new iteration of the device is created. This device is implements newPolymagnets (N42) with a stronger magnetic field resulting in stiffer springsboth at baseline and throughout the dynamic motion range. Additionally, thedynamic mass is increased from 250g to 300g. The goal of these modificationis to target higher frequency tremors with increased stiffness while improvingtremor reduction with the added inertia. In Figure 7.15, the response forFPS, WFE, and WRUD tremors at 10V corresponding to 6.4Hz is shown forall four scenarios of analysis. From the response, we observe that the devicemildly attenuates tremors in the WFE DOF for the H12 mode. The otherconfigurations experience amplification and the DW response is similar tothe H12 response for all DOFs. This indicates that the dynamic system doesnot respond to tremor at this frequency.1967.3.IN-VITROEVALUATIONOFABSORBERPROTOTYPESFigure 7.15: Time response response showing an induced tremor in FPS, WFE, and WRUD for freehand (FH),deadweight (DW), lower height mode at 8mm (L8), and higher height mode 12mm (H12) at 10V corresponding to6.4Hz.1977.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESThe performance evaluation continues for other frequencies and the amplitudereduction is calculated for each test. In Figure 7.16, the results of this stepare reported. First, we note that the added inertia does not improve tremorreduction. Second, we observe that the dynamic device appears to amplifylow frequency tremors for both the L8 and H12 when compared to the DWmode. Finally, only the H12 mode attenuates with 40% or higher tremorreduction for tremor frequencies ranging from 6.2Hz to 6.8Hz.These results indicate that the changes made for this iteration do not producethe expected outcome. This may be due to the combination of stiffness andinertia changes which pushed the performance envelope for the device tomuch higher frequencies than the the ones evaluated here.This performance is not limited to the FPS DOF as the results in Table7.9 show the 6Hz response for WFE and WRUD. In addition to low tremorreduction, the device shows inconsistent results across multiple runs withreduction and amplifications between similar runs. This could be theresult of external factors such as fatigue on the arm from the inducer,inconsistent placement of the absorber or sensor during experimental runs.These inconsistencies in performance are discussed further in Chapter 9.Table 7.9: Results of Steadi-Two iteration two experimental evaluation.Gyroscopic (Gyro) and Accelerometer (Acc.) data from multiple runs showthat the performance is inconsistent.FPS WFE WRUDDW L8 H12 DW L8 H12 DW L8 H12Gyro. #1 -113.2 27.4 -35.8 -5.3 -7.2 1.7 -110.7 -25.5 -119.4Gyro. #2 -82.6 37.6 71.5 4.2 0.9 40.6 23.1 69.9 66.3Acc. FE 11.7 38.1 24.6 14.6 30.8 40.1 -26.9 -15.0 6.3Acc. FPS -16.9 -20.5 56.5 -14.8 -32.3 -17.3 1.4 28.0 -2.61987.3.IN-VITROEVALUATIONOFABSORBERPROTOTYPESFigure 7.16: Frequency RMS response for the Steadi-Two iteration one using WeSU gyro-sensor for the FPS DOF.1997.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESIteration Two: Swivel Bearing Damping and Postural AnalysisThe inconsistent performance of the previous iteration identified a challengeof energy dissipation. The time response from Figure 7.15 does not show aclear transient phase which indicates low inherent damping in the system.This prompted a new iteration which implemented a new swivel bearingmade out of Nylon instead of the original plastic bearing. The nylon bearinghas improved coulomb damping which can improve energy dissipation.In this stage of development, we also make the decision to study the influenceof posture to the performance of the Steadi-Two. Posture is an importantaspect of analysis as it can provide a more realistic evaluation of performancecomparable to real patient use. We identify four postures which are listedbelow:∗ Straight Arm: The arm is outstretched at the shoulder and parallelto the ground. This posture is selected as it is simple to replicate whenevaluating the device with patients.∗ Elbow Bend: 90o bend of the elbow with forearm parallel to theground. this posture is selected as it mimics the orientation of the armwhen writing, a common challenge for tremor patients.∗ Picking posture : The orientation of the arm when attempting topick up an object from a table. This posture may vary for patients butwe establish a baseline in analysis.∗ Drinking posture: The orientation of the arm when drinking from acup. In this posture, the arm is close to the mouth and any tremorssignificantly hinder activity.2007.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESStraight Arm: For the straight arm posture, the performance of the thirdSteadi-Two iteration at 6.4Hz is given by Figure 7.17. We immediately noticethe impact of the nylon swivel bearing with the faster setting time. This isespecially true for the WFE tremor. This improvement is also captured in thereductions for this evaluation which are shown in Table 7.10. The performanceof the Steadi-Two in the straight arm posture is statistically better comparedto the previous iteration. Specifically, the multi-DOF performance for thelower height and higher height is better than the dead-weight which is anotable improvement from the previous iteration.Figure 7.17: Performance of the Steadi-Two iteration two device at 6.4Hz fora straight arm posture with an FPS induced tremor. Improved performancecompared to previous iteration with the addition of the nylon swivel bearing.2017.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.10: Straight arm reductions for a 6.4Hz tremor.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Straight -38.43 -39.8 21.57 6.5 43.47 32.68 17.35 41.88 34.13Elbow Bend: The performance in the elbow bend posture is shown inFigure 7.18. The lower height (L8 mode) amplifies the FPS DOF tremorwhich also happened in the straight arm posture. The percentage reductionfor this posture is given in Table 7.11.Figure 7.18: Performance of the Steadi-Two iteration two device at 6.4Hzfor 90o elbow bend posture with an FPS induced tremor.2027.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.11: Percentage reduction for the elbow bend posture for a tremorfrequency of 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Elbow Bend -16.88 -20.05 56.53 -14.78 -32.27 -17.27 1.37 27.99 -2.56We observe that for the FPS DOF, only the H12 performance meets theminimum threshold. However, the WFE and WRUD performance are stillbelow the acceptable levels. Further evaluation of the FPS DOF for allfrequencies yields the chart in Figure 7.19.Figure 7.19: Frequency RMS response for the Steadi-Two iteration threeusing WeSU gyro-sensor for the FPS DOF in the elbow bend posture.2037.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESFrom the frequency response chart, we notice that the reduction for the FPSDOF is primarily due to the higher height (H12). Moreover, this performs ispredominantly in the higher frequency range with amplification occurringin the lower frequencies. This suggests that the changes to stiffness of theSteadi-Two introduced in iteration one may have shifted the baseline stiffnesssuch that is unable to respond to low frequency tremors for the elbow bendposture. The implications of these observations are discussed in Chapter 9.Picking posture: The performance of the Steadi-Two at 6Hz for thepicking posture is shown by Figure 7.20 and the percentage reductions atthis frequency are given in Table 7.12.Figure 7.20: Performance of the Steadi-Two iteration two device at 6.4Hzfor the arm in the picking posture with an FPS induced tremor.2047.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.12: Percentage reduction for the picking posture for a tremorfrequency of 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Picking 11.34 42.86 24.44 61.27 -18.87 66.77 13.71 -3.46 10.18The results for the picking posture highlight a significant performanceimprovement, especially in the FPS DOF where the lower height mode (L8)outperforms the higher height mode. Recall that by design, the purpose ofthe higher height mode is to attenuate lower frequency tremors and the lowerheight mode targets high frequency tremors. This appears to the be case here.Additionally, while the performance does not meet the minimum performancerequirement in all DOFs, there is a significant improvement where at least onemode always attenuates tremors. Additionally, the amplification of tremorin the picking posture is less egregious compared to the other postures. Thisis a milestone result since it is the first quantitative indication that theSteadi-Two can mitigate tremor for patients as they perform their dailyactivities.Observing the frequency response curve for the FPS DOF in Figure 7.21, wesee that the two modes, L8 and H12, now perform effectively within theirtarget frequency regions. The H12 mode is effective in the 4.4→ 6Hz rangeand the L8 mode is effective from 5.8 → 7.5Hz. The measured reductionsare not always above the reduction threshold but their is indication that theSteadi-Two has the potential to achieve wideband tremor reduction with themodes targeting sections of the tremor spectrum.2057.3.IN-VITROEVALUATIONOFABSORBERPROTOTYPESFigure 7.21: Frequency RMS response for the Steadi-Two iteration two using WeSU gyro-sensor for the FPS DOFfor the picking posture.2067.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESDrinking posture: For the drinking posture, the performance at 6.4Hzis given by Figure 7.22. The percentage reduction for this configuration isgiven in Table 7.13. From the response and the associated statistics, we notethat the performance is only acceptable in the FPS DOF with a reductionof 64.42% from the L8 mode. The performance in other DOFs is primarilytremor amplification. This can be attributed to the significant inertia changeassociated with the drinking posture. Drinking is a significant challenge fortremor patients and the performance of the Steadi-Two in this posture iscritical.Figure 7.22: Performance of the Steadi-Two iteration two device at 6.4Hzfor the arm in the drinking posture with an FPS induced tremor.2077.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.13: Percentage reduction for the drinking posture at 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Drinking 19.45 64.42 37.23 4.95 -27.49 -46.56 -55.61 -74.42 -96.84The multi-frequency evaluation for the FPS DOF shown in Figure 7.23reveals that the performance in the FPS DOF is not always above theminimum threshold for the entire frequency spectrum. There is amplificationnear the 6Hz region just before the point of analysis shown in Figure 7.22.This response implies the existence of a resonance region, likely due to thedynamics of the arm in which the tremor is induced. This highlights theprimary challenge in the mechanical attenuation of tremors since they varyfrom patient to patient and even for the same patient.Figure 7.23: Frequency RMS response for the Steadi-Two iteration to usingWeSU gyro-sensor for the FPS DOF for the drinking posture.2087.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESFinal Iteration: Patient DeviceFollowing the results of the second iteration, the final patient device wasdesigned to address the challenge of broadband tremor reduction for the FPSand WFE DOFs. This is based on the understanding of tremor propagation(proximal to distal as described in Chapter 9.4) which recognizes the WRUDas a distal DOF. Additionally, the final iteration addressed amplificationchallenges observed in iteration one and two. The objective is not tocompletely avoid amplification but ensuring that if any amplification isto occur, it is not catastrophic.The acknowledgement of minimizing amplification is a realization that thecharacteristic of tremor may change during activities. This may cause thedevice to briefly amplify tremor before settling and minimizing tremor. Forinstance, in circumstances where the posture is transnational and not afixed posture like the drinking activity, the inertia and stiffness of the armcan change resulting in a temporary performance decline. While the goalis to reduce tremor throughout the motion, vibration absorber analysis isbased on systems that oscillate about an equilibrium. The movement ofthe arm during a drinking activity would be classified as rigid body motionand different principles would be required for evaluation. Therefore, theprimary change for the final iteration is a modification of the Steadi-Twoto achieve steady-state performance as quickly as possible so parametricchanges experienced in various postures are effectively mitigated.With the implementation of the final changes, a similar evaluation is conductedwith the patient device where the posture is the lens of performance analysis.However, the elbow bend posture, which primarily manifests in the forearmpronation-suppination tremor, is eliminated from analysis. In its place,the biomechanical model of the human arm in the FPS DOF (the RIG)presented in Chapter 7.1.2 is used to evaluate the patient test device. Thisis an important change since it allows the Steadi-Two to be evaluated in apurely FPS tremor without cross-DOF influences.2097.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESStraight Arm: For the patient device performance in the straight armposture at 6.4Hz, the results are displayed in Figure 7.24 and the associatedpercentage reduction statistics are given in Table 7.14. This performancenot significantly different from the the previous device iteration. However,the iterations for the patient device focused on its performance in commonpostures such as drinking and picking, we must evaluate other postures forcompleteness.Figure 7.24: Performance of the Steadi-Two patient device at 6.4Hz for thestraight arm posture with an FPS induced tremor.2107.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.14: Percentage reduction for the straight arm posture at 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Straight -38.34 -28.62 -16.34 6.5 -3.98 -5.42 17.35 33.51 34.51Drinking Posture: For the drinking posture, the performance at 6.4Hz isshown in Figure 7.25 and the percentage reductions are given in Table 7.15.Notice the improvement in performance for the FPS DOF compared to theprevious iteration which was amplifying tremor. Admittedly, the WRUDDOF now amplifies tremor but this is a better trade off since the inertiaof this DOF is much smaller so the manifested tremor is not problematic.Moreover, with it being a distal DOF, the impact of amplification would benegligible in practice.Figure 7.25: Performance of the Steadi-Two patient device at 6.4Hz for thedrinking posture with an FPS induced tremor.2117.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.15: Percentage reduction for the drinking posture for a tremorfrequency of 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Drinking 19.45 34.08 24.48 4.95 43.11 38.97 -55.61 -35.77 -21.14The frequency response in the FPS DOF for the drinking posture is givenby Figure 7.26. Here, we can observe a challenge that has been previouslynoted where an amplification occurs around the 6Hz region which is likein the neighbourhood of resonance for the arm used in analysis. Since thistest uses an induced tremor on a non-tremogenic human arm, real patienttests can help determine if this an experimental artefact or a performancechallenge for the Steadi-Two.Figure 7.26: Frequency RMS response for the Steadi-Two patient deviceusing WeSU gyro-sensor for the FPS DOF for the drinking posture.2127.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESPicking Posture: Evaluating the picking posture produces the result inFigure 7.27 with the reductions in Table 7.16. This posture shows the mostimprovement compared to the previous iterations at 6.4Hz. There is noamplification from the patient device. This is a promising result since pickingis a one of the challenges faced by patients in their daily life. As a fine motortask, its impact to the quality of life for tremor patients is significant. Noticeas well that the effective device mode changes based on the DOF. This meanspatients are able to adjust the device and tackle the problematic DOF. Thisis a significant improvement to the device performance.Figure 7.27: Performance of the Steadi-Two patient device at 6.4Hz for thepicking posture with an FPS induced tremor.2137.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESTable 7.16: Percentage reduction for the picking posture at 6.4Hz.FPS WFE WRUDPosture DW L8 H12 DW L8 H12 DW L8 H12Picking 11.34 70.76 46.35 61.27 10.03 50.87 13.71 27.57 16.15The frequency response for this posture in FPS is provided in Figure 7.28.Observe that the L8 mode consistently performs better than the H12 modefor entire tremor range. This is likely due to the dynamics of the arm andtype of induced tremor used in experimental evaluation. We know thatdifferent tremor properties, even at the same frequency, can have differentamplitudes which can impact the performance of the device. Notably, theSteadi-Two is now capable of high reduction rates for the entire frequencyrange. This result is a milestone in the development of the device since wehave now seen it is possible to conduct a successful test without amplification.Figure 7.28: Frequency RMS response for the Steadi-Two patient deviceusing WeSU gyro-sensor for the FPS DOF for the picking posture.2147.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESRIG Test Results: The RIG presented in Chapter 7.1.2 is invaluable forexperimental evaluation because it eliminates variability in induced tremor.Tremor variability is a noted challenge when using the tremor inducer since thehuman arm can be affected by many uncontrollable factors. Therefore, usingthe RIG for evaluation allows for repeatability in experimental analysis whichis key for eliminating variance in the results. For experimental completeness,three separate runs are conducted using the RIG and since the primarytremor is in the FPS DOF, only the FPS results are relevant. To confirmthat the tremor is primarily in the FPS DOF, Figure 7.29 shows a plot of thetime response at an induced frequency of 5.5Hz. Here, we can see that theFPS tremor (now placed in the middle graph for comparison), is an order ofmagnitude higher than the other tremors.Figure 7.29: Performance of the Steadi-Two patient device at 5.5Hz for thepicking posture with an FPS induced tremor.2157.3. IN-VITRO EVALUATION OF ABSORBER PROTOTYPESThe performance of the patient device on the RIG is summarized in Table7.17. We see that the patient device significantly reduces in all runs withlow variance between the runs. This is the benefit of using the RIG as itgenerates a similar tremor for each run allowing for comparative evaluationsbetween runs. Additionally, since the RIG is a pure FPS tremor, it is likelyeasier to attenuate compared to the inducer on a human arm which generatesMDOF tremors. Reviewing the performance at 5.5Hz which is on the lowerend of the frequency spectrum, we see the higher mode (H12) outperformingthe lower mode (L8). This is the intention of the dual mode design. Thefrequency response is shown in Figure 7.30.Table 7.17: Percentage reduction on the RIG for the Steadi-TwoFPSRun # DW L8 H12Run 1 NIL 31.44 63.59Run 2 NIL 52.6 81.94Run 3 NIL 46.45 70.34From the frequency response, we can see the trend for both the H12 and L8 asthe tremor frequency increases. We can presume, based on the results shownthus far, that at higher frequencies, the L8 mode will begin to outperform theH12 mode. The result is a device capable of broadband tremor attenuationnot just in the FPS DOF but also the WFE DOF.The results of the RIG evaluations for the Steadi-Two are promising andat the time of compiling this manuscript, this iteration of the Steadi-Twodevice is in its production phase. With the successful iterative developmentof the Steadi-Twothe and subsequent conclusion of in-vitro evaluations, thenext step is to evaluate the performance of the Steadi-Two for real tremorpatients. The findings and implications of the in-vitro evaluations are coveredin Chapter 9. The chapter that follows will cover the performance of theSteadi-Two for real tremor patients and discuss in detail the process ofpatient evaluations.Note: The amalgamated percentage reduction results for the 6.4Hz evaluationpresented here are given in Table D.2 in Appendix D.2167.3.IN-VITROEVALUATIONOFABSORBERPROTOTYPESFigure 7.30: Frequency response for the Steadi-Two patient device on the RIG for the lower frequency spectrum.217CHAPTER 8Patient Tests and Real-World ApplicationsThe promising results of the in-vitro tests and the successful iterationsof the Steadi-Two provide a basis for the next step in development. Wenow consider the performance of the Steadi-Two on real tremor patients.Given that development coincided with the COVID-19 pandemic quarantinelockdowns, the quantitative patient evaluations were only conducted withthe final iteration of Steadi-Two.Therefore, this chapter examines the Steadi-Two performance in real-worldscenarios through patient testing. It covers the testing protocols, participantselection criteria, and feedback collection methods. The outcomes of thisphase provide valuable insights into the usability, adoption, and impact onimproving patient quality of life for the Steadi-Two. These findings furthersolidify the practical contributions of the research while identifying areas forfuture refinement.2188.1. PATIENT ON-BOARDING AND DATA COLLECTION8.1 Patient On-boarding and Data CollectionThe data collection procedure was streamlined to simplify the process andminimize confusion to patients. It is as follows:∗ A patient walks into the office for the test and is on-boarded through thepersonal form provided in Appendix D.2. They are then instrumentedwith a sensor on top of the palm for the duration of the test. Thissensor is the WeSU sensor introduced in Chapter 7. This is a bluetoothsensor which eliminates any need for wires on the patient.∗ Now, the postures for the test are described to the patient to ensure theyare able to perform them and the sensor is calibrated here. The posturesare: straight outstretched arm, drinking posture, writing posture, restposition. These postures are shown in Figure 8.1.Figure 8.1: Posture definitions for patient evaluations. The rest and writingpostures are similar although the actions are different.2198.1. PATIENT ON-BOARDING AND DATA COLLECTIONThe patients hold the posture for 10s and their tremor data is recorded.Once, the time is over, they rest their arm momentarily (approximatelyanother 10s) before holding a different posture. This is repeated until all thedata is collected for the particular test run. In total, there are four test runs:∗ First, we conduct a baseline tremor analysis to characterize the patientstremor for all postures. This characterization identifies the tremorfrequency and amplitude characteristics.With this information, wedetermine the appropriate mode of the Steadi-Two (L8 or H12) to beused for each posture based on the results from the in-vitro tests.∗ We then conduct the second test run (hereinafter called alpha test)using the predetermined mode of the Steadi-Two. From the results ofthe alpha test, we infer whether the initial guess for the device modeis correct. If the results are opposite of the expected values, we switchthe mode and redo the test.∗ We then conduct the beta and gamma test and store all test data in adatabase for future analysis. This process is summarized in the processflow diagram in Figure 8.2.The collected data is stored in a SQL database which is then queried fordifferent types of analysis. The analysis involves a comparison of the tremorresponse with and without the absorber for all three test runs and determiningthe DOFs where the device is effective and where it may be amplifying tremor.As observed during the in-vitro tests, there may be some scenarios wherethe device reduces tremor in one DOF while amplifying tremor in anotherDOF. For such instances, we review the results to get a better understanding.The patient evaluation reported here was conducted in two phases. PhaseI testing was brief with only 5 patients. The level of data collection andevaluation for this phase was solely focused on the reduction percentages forthe identified activities. Phase II testing was a lot more detailed analysisinvolving a clinical investigation protocol based on the lessons learned fromphase one. We will now cover the data analysis from both phases of evaluationand compare them to both the modeling and in-vitro results.2208.1.PATIENTON-BOARDINGANDDATACOLLECTIONFigure 8.2: Process flow diagram describing the patients test conducted for the Steadi-Two including the decisionmaking steps during sampling.2218.2. PHASE I EVALUATIONS8.2 Phase I Evaluations8.2.1 Patient Characterization and EvaluationWe will first demonstrate the data analysis process flow using the resultsfrom a single patient and then we will review the amalgamated results forall patients in Phase I. The raw data collected using the aforementionedWeSU sensor, which is a is triple axis sensor capable of collecting data forthe FPS, WFE, and WRUD DOFs. In Figure 8.3, the baseline tremor forthe WFE DOF is shown for a patient including the characterization usingthe frequency domain analysis.Figure 8.3: Characterizing patient tremor using frequency analysis for thecollected raw data. This data shows the patient’s WFE DOF tremor for theright arm in the rest posture with peaks at 4.8Hz, 5.2Hz, and 8.8→ 9.8Hz.2228.2. PHASE I EVALUATIONSFor the response shown in Figure 8.3, the patient tremor shows multiplepeaks in the frequency spectrum with dominant peaks at 4.8Hz, 5.2Hz,and in 8.8→ 9.8Hz range. This implies that there are multiple tremors ofvarying frequencies present. From our understanding of tremor propagationin Chapter 2, we can infer there is tremor propagation from the more proximalFPS DOF to the distal WFE DOF. Since this tremor response is measured forthe WFE DOF which also has a lower inertia compared to the more proximalFPS DOF, it is likely that the lower frequency dominant peaks result frompropagation in the proximal WFE DOF. To evaluate this hypothesis, let uslook at the FPS response for this patient in Figure 8.4.Figure 8.4: Characterizing patient tremor comparing FPS DOF response tothe WFE response. The FPS response has an inverted amplitude responsecompared to the WFE response which is indicative of proximal to distaltremor propagation.2238.2. PHASE I EVALUATIONSThe FPS response supports the hypothesis that the tremor is propagatingfrom proximal DOFs to distal DOFs. We can observe this from Figure 8.4where the 4.8 → 5.2Hz range of tremor is less dominant compared to the8.8→ 9.8Hz frequency range. We infer from these results that the patientlikely has a small-to-medium sized palm which results in the high frequencypeak propagated from the FPS DOF. If the patient had a larger sized palm,its high inertia would likely result in a diminished propagated peak in the8.8→ 9.8Hz frequency range.Now that we have characterized the patient tremor, we decide which mode ofthe device to use for this patient. The two device modes, L8 and H12, targethigh and low frequency tremors respectively. We must decide which DOF toattempt to target first since the dominant frequencies for each DOF are inopposite tremor ranges. In this instance, we decide to target the FPS DOFwith the hypothesis that tremors propagate from proximal to distal DOFs.We presume that attenuating the proximal FPS DOF will also attenuatedistal WFE DOF. Therefore, we run the rest posture test with the deviceset to the L8 mode to target higher frequencies in an attempt to attenuatethe dominant peaks in the FPS DOF. The response of this test is given inFigure 8.5.The results show that tremor is reduced in all 3 DOFs with significantreduction in the FPS and WFE DOFs. Evaluating the RMS tremor reduction,the calculated values show 72.92% reduction in the FPS DOF, 74.69%reduction in the WFE DOF, and 14.13% reduction in the WRUD DOF.These results further support the hypothesis that reducing proximal DOFtremor can reduce the distal DOF tremors without specifically targetingthe distal DOFs. This is further evident in the frequency response analysiscomparing the results with and without the absorber in Figure 8.6.2248.2.PHASEIEVALUATIONSFigure 8.5: Evaluating patient tremor reduction using the Steadi-Two. The device is set to target the high frequencytremor in the FPS DOF. Tremor reduction is observed in all 3 DOFs.2258.2.PHASEIEVALUATIONSFigure 8.6: Frequency domain analysis of tremor reduction with the absorber compared to the baseline tremor.These results are from targeting the high frequency proximal FPS DOF. Tremor is reduced in both all the DOFswhile primarily targeting the FPS DOF.2268.2. PHASE I EVALUATIONSThe frequency domain analysis shows that targeting the higher frequenciesusing the H12 mode for the FPS DOF mitigates the tremor significantlyevident by the reduced tremor peak for the absorber case. This also mitigatesthe higher frequency tremors in the other DOFs which implies that highfrequency tremors are not propagated to the distal DOFs. Additionally,although the lower frequency tremors are not specifically targeted in theWFE and WRUD DOFs, we can observe that they are also mitigated albeitminimally. This further supports the hypothesis that reducing proximalDOF tremors can attenuate distal DOF tremors and also shows that theSteadi-Two is capable of targeting specific tremor frequency. This is afundamental result because it corroborates the theoretical framework andprototyping approaches for the Steadi-Two. In Chapter 9, we will discussthe agreement between the mathematical model, the in-vitro, and the in-vivoevaluations.We have now seen that the evaluation results support the initial hypothesisfor selecting mode H12 on the Steadi-Two. Consequently, we continue withdata collection for the other postures and other patients. Hereinafter, weonly present the final tremor reductions instead of covering the entire processflow for each postural analysis since the approaches are similar.For this first patient, the reduction results of the patient evaluation in otherpostures are given in Figure 8.7. We observe that the excellent performancein the rest posture does not transcend to other postures. This is especiallytrue for the straight arm posture where the FPS and WRUD DOFs show near100% amplification. This shows that the Steadi-Two device is not infalliblein its performance and there will be scenarios in which its effectiveness islimited. Similar results were observed both in the modeling and in-vitroevaluation results so this is to be expected, especially for the straight armposture.2278.2.PHASEIEVALUATIONSFigure 8.7: Percentage reduction for all DOFs for all postures for a single patient. Observe the inconsistency inreductions for other activities across DOFs.2288.2. PHASE I EVALUATIONSA closer inspection of this particular amplification result is shown in Figure8.8. This figure shows that for this patient, in the straight arm posture,tremor amplification is experienced in the FPS DOF. The specificity of thisstatement is important since it is potentially unique to this patient. Thisfailure likely stems from the increased inertia in the straight arm posture incomparison to the rest posture as well as the changing dynamic parametersin that posture. During this patient test, changes to different modes of theSteadi-Two did not result in better performance implying that the device isincapable of targeting these specific tremors.Notably, the performance in the drinking and straight arm postures arealmost opposites. We can note that when drinking, the patient has an addedinertia due to holding a cup. Therefore, depending on their tremor and theirarm inertia, the device performance can improve or become worse. On theother hand, the writing activity is the most similar to the rest posture and wecan see from the performance, it is the most similar in reduction percentages.From these results, we can anticipate that the Steadi-Two will be effectivefor this patient only if they use it for tasks that do not require posturalholds for extended periods. Activities like drinking and straight arm holdingrequire the arm to be in an elevated position for extended periods whichmay alter the dynamic parameters of the arm as the patient becomes tired.2298.2.PHASEIEVALUATIONSFigure 8.8: Graphs showing the failure of the Steadi-Two in the FPS DOF for a patient in the straight arm posture.2308.2. PHASE I EVALUATIONS8.2.2 Overall Analysis for Phase IWe now consider the other patients evaluated during Phase I. Consider theFPS response for patients 2 to 4 shown in Figure 8.9. These results showthat the Steadi-Two consistently performs well for the rest and writingpostures, which are the two most similar postures. We can also observe goodperformance in the drinking posture from patients 2 and 3. This suddenimproved performance is not an inconsistency but instead highlights thevariation in tremor between patients. This is the fundamental challengein designing a tremor attenuator, such as the Steadi-Two. It requiresacknowledgement that the device may be ineffective for some patients.Consider the measured performance from another patient in Figure 8.10.We see that in addition to the poor performance in the FPS DOF, we canalso observe that initially, the Steadi-Two mitigates tremor but eventuallycauses amplifications. This may be attributed to the existence of harmonicsand frequency shifts (characteristic phenomenons of non-linearity). Thisresult highlights another potential shortcoming of the Steadi-Two whereharmonics can result in unstable performance regions. It is possible to mapthe harmonics and avoid these specific scenarios but that does not guaranteethat patients using the device will not experience these situations in theirday to day use of the device. This failure is easily observable in the frequencydomain result shown in Figure 8.11.Phase I consisted of 5 patients, which is a small sample size to make conclusivestatements about the Steadi-Two. However, some valuable insights havebeen gained from this exercise.∗ We have identified that there are performance limitations as a result oflimiting the height change to only two modes.∗ Tremor variability impedes excellent performance for all postures andpatients. Thus, further parametric refinement may be required toimprove the performance bandwidth.∗ It is important to identify scenarios where amplification consistentlyoccurs for all patients, such as the straight arm posture during thisevaluation. This helps better definition of the user conditions wherethe device will succeed.2318.2.PHASEIEVALUATIONSFigure 8.9: Percentage reduction for the FPS DOF for all postures for multiple patients. Different patients havevarying performance responses for all posture although consistency is observed in the rest and writing postures.2328.2.PHASEIEVALUATIONSFigure 8.10: Performance for the Steadi-Two for another patient in the straight arm posture. Observe the initialreduction in the FPS DOF which turns into amplification over time.2338.2.PHASEIEVALUATIONSFigure 8.11: Frequency domain analysis for second patient results in the straight arm posture. Observe that thedevice under performs in the FPS DOF indicated by the increased amplitude. The device reduces tremor in theWFE and WRUD DOFs.2348.3. PHASE II EVALUATIONS8.3 Phase II EvaluationsThe lessons learned from Phase I patient evaluations necessitated a secondin depth evaluation with careful selection of patients for the test. Thisphase of evaluation included a clinical investigation plan which stipulatedthe inclusion/exclusion criteria and included the collection of video data forreview by neurologists. This step is important as it allows experienced medicalprofessionals to offer insight on the tangibility of the device performance i.e.does the device improve the patient quality of life.8.3.1 Clinical Investigation ProtocolThere are four primary activities that make up the clinical investigationprotocol.Patient Eligibility ScreeningIn the first phase of evaluation, the patients were not screened and the devicewas tested on all five patients without considering some external factors thatwould influence the performance. Through discussions with neurologists andreview other studies in the field, the following exclusion criteria was devised:1. Implanted electrical medical device, such as a pacemaker, defibrillator,or deep brain stimulator.2. Swollen, infected, inflamed areas, or skin eruptions, open wounds, orcancerous lesions of skin at stimulation site.3. Presence of any other neurodegenerative disease like Parkinson-plussyndromes suspected on neurological examination. These include:∗ Multisystem atrophy∗ Progressive supranuclear palsy∗ Dementia with Lewy bodies∗ Alzheimer’s disease4. Anyone suspected to have the diagnosis of idiopathic Parkinson’sdisease (PD). This includes excluding anyone with the presence of2358.3. PHASE II EVALUATIONSparkinsonian features including bradykinesia, rigidity, or posturalinstability. Subjects who exhibit only mild resting tremor but noother symptoms or signs of PD may be included.5. Botulinum toxin injection for hand tremor within 6 months prior tostudy enrolment.6. Significant alcohol or caffeine consumption within 12 hours of studyenrolment, which may confound the results of the study, where significantcaffeine is considered more than 95 mg (equivalent to a cup of coffee),and significant alcohol is considered more than 14 g (equivalent to 5 ozof wine, 12 oz of beer, or 1.5 oz of distilled spirits).7. Subjects unable to communicate with the investigator and staff.8. Any health condition that, in the investigator’s opinion, should precludeparticipation in this study.Video and Kinematic Data CollectionThe collaboration with neurologists required the collection of videos duringthe test for future review. These videos will be used to rate the effectivenessof the device based on the Fahn-Tolosa-Marin (FTM) . There are sevenactivities in total but only three will have both video and kinematic datacollection.1. Video and Kinematic Data:∗ Posture 1 – outstretched arm with palm in line with the forearm,i.e., the palm faces the ground.∗ Posture 2 – outstretched arm with palm perpendicular to theforearm, i.e., palm is sideways in relation to the elbow joint.∗ Posture 3 – drinking hold position with a cup full of water. Inthis posture, the participant holds the cup as one would whenpicking up a cup from the table.2. Video Only:∗ Drinking activity – instruct the participant to take a sip of waterfrom the cup. Ensure the video captures the entire motion fromraising the cup to taking a sip and returning the cup back down.2368.3. PHASE II EVALUATIONS∗ Pouring activity – instruct the participant to pour the water fromthe full cup into an empty cup. Video recording should focus onthe two cups and the patient’s forearm.∗ Kinetic tremor – instruct the participant to put their arm inposture 1 with all fingers except the index finger folded into thepalm to make a fist. Have the participant move their arm so thattheir index finger touches their nose and then move it back to theoriginal position. Have the participant repeat the motion at leasttwo more times where possible.∗ Spiral test – Administer the Archimedes spiral test.Evaluation FeedbackThe participants will be invited to complete an evaluation of the device forfeedback. The feedback form template is attached in Appendix D.This phase evaluation was conducted over a month and a half with morethan 10 patients initially available. Based on the exclusion criteria, the list ofpatients was narrowed down to 10 patients. However, initial analysis of theirtremor characteristics revealed certain tremor patterns to be inconsistentwith tremors defined within the scope of this study. An example of sucha tremor is shown in Figure 8.12. We can notice from the graph that thetremor is not harmonic as per the models introduced earlier in the analysis.Additionally, the frequency response shows two peaks with one below the3Hz threshold and the other above the 12Hz limit. Therefore, this patientwas excluded from analysis.With these restrictions, the final analysis includes a total of 7 patients. Theage, gender, and tremor characteristics from the initial characterization ofthe freehand tremor evaluations are provided in Table 8.1.2378.3.PHASEIIEVALUATIONSFigure 8.12: An example of an excluded tremor during analysis. The irregularity of the patient tremor does notalign with the scope of analysis.2388.3. PHASE II EVALUATIONSTable 8.1: Patient Demographics and Tremor ClassificationPatient ID Age Gender Frequency (Hz) Tremor Severity1 70 F 5.12 Medium2 60 F 5.02 Medium4 61 F 5.03 Mild5 63 M 5.69 Mild6 66 M 6.69 Mild7 63 F 5.09 Medium8 66 F 5.72 Severe8.3.2 Data Collection ProcedureThe characterization of patient tremors is followed by data collection. Theprocess is as follows/:1. Instrument the participant’s dominant hand with a mobile device withthe Sensor Data App open. In the Sensor Data App, enter the UserID as “Patient # - Posture # - F/S/R - DDMM” where the optionsF/S/R represent whether the test is a freehand, sham, or real test.The trial number should be entered as 1, 2, or 3 representing eitherthe freehand, sham, or real respectively. For example, a real test inPosture 1 for the third patient in May 24 will have User ID: Patient3 – Posture 1 – 2405, Trial: 3. Note here that the Sensor Data Appshould be set to record for 10 seconds at a time.2. Ensure that the video recording is centered on the participant’s armat a position where the tremor is discernible, i.e., when the posturechanges, it might be necessary to adjust the camera angle to capturethe tremor.3. Conduct the KINEMATIC TESTS FIRST for Posture 1, Posture2, and Posture 3.4. The data collection for kinematic tests should be done as follows:Freehand (Trial 1), Sham (Trial 2), Real (Trial 3). This meansconducting freehand tests for all postures, then sham tests for allpostures, then real tests for all postures. The experimental procedureshould look like the table below.2398.3. PHASE II EVALUATIONSTable 8.2: Experimental naming convention for Phase II.Test User ID TrialPosture 1 - Freehand Patient # - Posture 1 - F - DDMM 1Posture 2 - Freehand Patient # - Posture 2 - F - DDMMDrinking Hold - Freehand Patient # - Posture 3 - F - DDMMPosture 1 - Sham Patient # - Posture 1 - S - DDMM 2Posture 2 - Sham Patient # - Posture 2 - S - DDMMDrinking Hold - Sham Patient # - Posture 3 - S - DDMMPosture 1 - Real Patient # - Posture 1 - R - DDMM 3Posture 2 - Real Patient # - Posture 2 - R - DDMMDrinking Hold - Real Patient # - Posture 3 - R - DDMM5. When the Kinematic tests have been completed in the order presentedabove, conduct the visual tests in the following order ensuring thatthe camera is adjusted appropriately for each test. Maintain the sameorder of test as for the kinematic tests by conducting all freehand testsfirst, then sham tests, and then the real device tests. Since it is notpractical to record the video for each test individually, maintaining thefreehand, sham, real test procedure makes post processing easier.8.3.3 Phase II Data AnalysisThe collected kinematic data and videos of the process and the images ofthe administered Archimedes spiral tests are then consolidated into a shareddrive for analysis and post processing. Currently, the analysis has beencompleted for individual patients kinematic tremors but the Neurologistscoring is pending completion. The scale used by the Neurologists is theFahn-Tolosa-Marin (FTM) scale by Fahn S, Tolosa E, and Marin C. TheFTM scale provides a standardized method to evaluate tremor characteristicsand their impact on daily activities. It is divided into three main sections:∗ Rest Tremor: Evaluates the presence and severity of tremor whenthe patient is at rest.∗ Action/Postural Tremor: Assesses tremor during voluntary movementsand while maintaining specific postures.∗ Functional Disabilities: Rates the impact of tremor on activities ofdaily living, such as writing, eating, and speaking.2408.3. PHASE II EVALUATIONSThis scale will be used by Neurologists to rate the patient tremor withand without the Steadi-Two to quantify its effectiveness. While neurologistreview is in progress, the analysis of the collected kinematic data will offersa tangible way to evaluate performance and compare it to the modeling andin-vitro results. Similar to the evaluations presented so far in this study, theanalysis of Phase II data will be presented based on postural performance.Posture 1In this phase of experiments, posture 1 describes an outstretched arm heldwith the palm parallel to the ground. In Figure 8.13, a test participant isshown with their dominant hand in posture 1.Figure 8.13: A test participant with an outstretched arm held in posture 1where the palm is parallel to the ground.The patient is then instrumented with a sensor as described in the clinicalinvestigation protocol and data collection begins. In Figure 8.14, raw resultfrom a moderate/medium tremor severity patient is shown.2418.3.PHASEIIEVALUATIONSFigure 8.14: Graphs showing the freehand, sham, and real results for the FPS (x-axis), WFE (y-axis), and WRUD(z-axis) DOFs. The initial 3 seconds of data are ignored since the sensor is set to active during this time. Anadditional 2 seconds are also ignored when calculating tremor reduction since only the steady-state performance isevaluated.2428.3. PHASE II EVALUATIONSFor this participant, tremor reduction is observed in the FPS and WRUDDOFs while amplification is observed in the WFE DOFs. We can also observethat the sham and real results are not distinct enough to categorically classifythe performance of the device as a success. For this reason, we conduct RMSreduction calculations for all patients for posture 1 and present the resultsin a heatmap shown in Figure 8.15.The RMS results show varied performance which is expected. Varying tremorpatterns result in varying performance even for the same DOFs. Real tremorreduction (shown by the comparisons F vs. S and F vs. R) indicates tremorreductions for all patients except patient 7. Primarily, the reduction is in theFPS DOF which is expected based on both the simulations, in-vitro tests,and phase I patient tests. Comparing the freehand to real results (F vs. R),which is what most patients will experience, the Steadi-Two shows tremorreductions for all patients except patient 7. The comparison of sham andreal (S vs. R) reveals that some reductions are primarily due to the addedinertia of the device and not the dynamic absorption of the device. This isevident in patient 2 where all instances of S vs. R show negative reductionsimplying that the real device causes amplification of tremor compared to anadded mass. We note the following for tremor severity:Medium Severity: Patient 1 and Patient 2 show positive reductions inmost comparisons, suggesting that medium severity tremor patients mayhave better reductions. Patient 7 shows amplification indicating variabilityin response within the same severity category.Mild Severity: Patient 4, Patient 5, and Patient 6 show mixed results,with both positive and negative reductions. This implies a less consistentresponse for patients with mild severity tremors.Severe Severity : Patient 8 shows mixed reductions, with high positivevalues in F vs R x and F vs R z, but negative in F vs S y. This suggeststhat severe tremor patients might have more variable responses.The heatmap indicates that performance varies significantly among patients,with some showing positive RMS reductions (improvements) and othersshowing negative reductions (amplification). There is no clear patterncorrelating the severity of tremor with RMS reduction values, suggestingdependency on individual patient tremor.2438.3.PHASEIIEVALUATIONSFigure 8.15: RMS reduction for posture 1 in a heatmap showing all patients and all configurations. The legend onthe right describes the freehand, sham, real tests and maps x,y,z to the FPS, WFE, and WRUD DOFs respectively.Tremor reduction (F vs S, and F vs R) is observed in most patients in varying levels except for patient 7. Thelast three columns compare sham to real (S vs R) to determine the performance of the actual device compared tothe added weight. Significant reductions are observed in FPS followed by WFE then finally WRUD (based onaverages). Some data for patient 1 was corrupted during transfer and is inaccessible.2448.3. PHASE II EVALUATIONSPosture 2In a similar approach to posture 1, we begin by defining posture 2 in Figure8.16. Posture 2 is similar to posture 1 with the outstretched arm with thekey difference being the orientation of the palm. In posture 2, the palm isperpendicular to the ground which engages the forearm muscles.Figure 8.16: A test participant with an outstretched arm held in posture 2where the palm is perpendicular to the ground.The response of the same patient as for posture 1 is given by Figure 8.17.The freehand (red) response shows more prominent oscillations which areconsistent over time, indicating a severe tremor type. From the results inthe FPS, WFE, and WRUD axes, the real device does not perform as wellas posture 1 amplification in FPS and WFE tremors which have the higheramplitudes.2458.3.PHASEIIEVALUATIONSFigure 8.17: Graphs showing the freehand, sham, and real results for the FPS (x-axis), WFE (y-axis), and WRUD(z-axis) DOFs for posture 2.2468.3. PHASE II EVALUATIONSThe resulting patient heatmap of RMS reductions is given in Figure 8.18. Weobserve that there is significant variability in RMS reduction values acrossdifferent patients and comparisons similar to posture 1 with some missingdata. Patients 1, 5, and 6 show high positive RMS reduction values both inthe sham and real comparisons for various axes while patients 7 and 8 exhibitsignificant amplifications especially in the real WFE and WRUD DOFs.Detailed analysis shows that Patient 1 consistently has positive reductionsacross all axes for the real device. Patient 7 consistently shows negativereductions, and Patient 8 has mixed results depending on the comparisonand axis. Additionally, the real device appears to outperform the shamdevice more consistently in posture 2 compared to posture 1 which is in linewith the hypothesis of engaged forearm muscles in posture 2. Consideringindividual patient tremor severity, we note the following:Medium Severity: Patient 1 and Patient 2 show positive reductions inmost DOFs, suggesting that medium severity tremor patients might respondwell to the Steadi-Two. Patient 7 shows negative reductions, indicatingvariability in response within the same severity category. This was alsoobserved in posture 1.Mild Severity: Patient 4 and Patient 5 show mixed results, with bothreductions and amplifications. This indicates a less consistent response tothe Steadi-Two in patients with mild severity. This was also observed inposture 1.Severe Severity: Patient 8 shows mixed reductions, with high positivevalues in F vs R x and S vs R x, but negative in F vs S y and F vs R y.This further suggests that severe tremor patients might have more variableresponses.2478.3.PHASEIIEVALUATIONSFigure 8.18: RMS reduction for posture 2 in a heatmap showing all patients and all configurations. The legend onthe right describes the freehand, sham, real tests and maps x,y,z to the FPS, WFE, and WRUD DOFs respectively.2488.3. PHASE II EVALUATIONSDrinking PostureThe results of posture 2 somewhat corroborate the analysis from posture 1where the device appears to effectively combat medium severity tremor andamplifies severe tremor and ineffective for mild tremor. We now analyze thedrinking posture to see if similar conclusions can be drawn. The drinkingposture is depicted in Figure 8.19.Figure 8.19: A test participant showing the drinking posture. The patientholds the cup in this position for 10 seconds as data is collected.The freehand, sham, and real responses for a patient in this position are givenby Figure 8.20. From the figure, it is evident that the real device reducestremor significantly when compared to both the sham and the freehandresponses. In fact, the sham response significantly amplifies tremor in allaxes.2498.3.PHASEIIEVALUATIONSFigure 8.20: Graphs showing the freehand, sham, and real results for the FPS (x-axis), WFE (y-axis), and WRUD(z-axis) DOFs for the drinking posture.2508.3. PHASE II EVALUATIONSThe RMS reductions are given in the heatmap in Figure 8.21. Similar to theprevious postures, the performance is varied among the patients for differentDOFs and configurations. A notable change however is the overall higherperformance in all patients with more consistent performance for certainDOFs.For instance, in the FPS DOF for F vs. R (column 4 from the left), theSteadi-Two shows tremor reduction throughout which has not been observedin the previous postural evaluations. This is a significant finding becausedrinking is an activity that affects daily life (when compared to posture 1or posture 2). Therefore, it is necessary to evaluate other activities definedin the FTM scale to better quantify the efficacy of the device from the realworld application perspective. We can anticipate that due to the differentmuscle groups activated when performing daily activities such as drinkingor writing, the severity of tremor (both amplitude and frequency) can shift.If this shift is favourable to the available device modes for the Steadi-Two,we will observe effective tremor mitigation. The results from the neurologistevaluations will shed more light on this.The second part of Phase II patient tests involved postures where only videorecordings were conducted for the aforementioned Neurologist evaluation.These postures are based on the activities of daily living such as drinking,pouring, writing. One of the tests conducted in this phase is the the drinktransfer test which is shown in Figure 8.22. The patient uses their dominanthand to transfer liquid from one cup to another without spilling. Theeffectiveness of the device is compared to the scenario where the patientpours without assistance. Additonally, the patient is also given a sham deviceto ensure test blindness and mitigate bias.2518.3.PHASEIIEVALUATIONSFigure 8.21: RMS reduction for drinking posture in a heatmap showing all patients and all configurations. Thelegend on the right describes the freehand, sham, real tests and maps x,y,z to the FPS, WFE, and WRUD DOFsrespectively.2528.3. PHASE II EVALUATIONSFigure 8.22: Patient demonstrating the pouring posture for video recordingsto be rated according to the FTM scale. In this test, a patient transferswater from cup A to cup B using their dominant hand and the ratings arebased on the amount of water spilled during the test.Another non-kinematic test is the Archimedes spiral depicted in Figure 8.23.The Archimedes spiral is a common test in evaluating the severity of tremorand quantifying the effectiveness of mitigating devices. Patients are told totrace the spiral and their response is recorded with and without the assitivedevice.The significance of these non-kinematic tests is their ability to evaluate theeffectiveness of the Steadi-Two in everyday quality of life activities. TheArchimedes spiral for instance evaluates a fine motor activity which impactsthe lives of many tremor patients. In Figure 8.24, the freehand response of atremor patient is shown. We can readily observe how the tremor impacts thepatients ability to trace the spiral and the result is almost indecipherable.On the FTM scale, this result would be rated either a 3 or a 4, with 4 beingthe highest severity of tremor.2538.3. PHASE II EVALUATIONSFigure 8.23: Patient taking the Archimedes spiral test. Here, the patientdraws in the spiral as closely as possible to the outline and the ratings arebased on both completeness and continuity of the drawings.Figure 8.24: Result of a freehand Archimedes spiral test for a patient withsevere tremor. Notice that the patient does not complete the test and crossesthe lines frequently. This is a rating 3→ 4 on the FTM drawing scale basedon the description.2548.3. PHASE II EVALUATIONSIn Figure 8.25, the Archimedes spiral for the same patient with the Steadi-Twoon their palm is shown. Notice the significant change in tremor amplitudeand frequency just by observing the drawing. This result would be rated 1or 2 on the FTM scale with 0 being the normal spiral. There is no benefit inquantifying this result since it is evident that the Steadi-Two is effective.Figure 8.25: Result of a real device Archimedes spiral test for the samepatient with severe tremor. Notice that the patient now completes the testwith slightly tremulous motion but no lines are crossed. This is a rating 1on the FTM scale.The results of this Archimedes spiral test (and consequently other non-kinematictests) highlight an important aspect of evaluating tremor attenuation devices.By observing low performance during kinematic tests, we cannot necessarilyinfer poor performance in quality of life tests. This is the basis of the nextsection where we evaluate the quality of life impact of the Steadi-Two devicefor patients who have used the device since its initial commercialization.2558.4. QUALITY OF LIFE EVALUATIONS8.4 Quality of Life EvaluationsThe Steadi-Two has been distributed to over 1000 patients. In the initialdistribution, a survey was conducted to provide insights into the improvementsof Activities of Daily Living (ADL) and Quality of Life (QoL) . This surveywas available for patients who have used the device for a minimum periodof three months. The survey utilized the Quality-of-Life Essential TremorQuestionnaire (QUEST) available at QUEST Score Sheet webpage. It isa comprehensive questionnaire that covers all dimensions of daily life. Forthe Steadi-Two survey, emphasis was on the physical dimension of QUEST,which assesses the interference of patient tremors to daily activities suchas writing, drinking, or eating. The QUEST scale includes five responses,ranging from ”never” (0 points) to ”always” (4 points). Patients were askedto respond to each dimensional activity twice, i.e. before and after using thedevice.For the Steadi-Two survey, patients were also asked whether they kept orreturned the device. The survey responses indicate that 73.2% of patientskept the device. Overall statistics from Steadiwear Inc. indicate that thepercentage retention is approximately 66%. This disparity may be attributedto the fact that individuals who kept the device were more likely to respondto the survey. The results of the survey are summarized in in Figure 8.26comparing the retention and rejection rate based on different activities inthe QUEST questionnaire.From the survey results, the most improvement is seen when using a computerwith a 50% QOL improvement. The posture of the hand in this positionis similar to the elbow bend posture and the primary DOF is FPS. Fromthe modeling, in-vitro, and in-vivo results, the Steadi-Two performancein FPS has been consistently above the threshold. This is especially truein the final evaluation of the Steadi-Two patient device which achievedbroadband attenuation. Another notable activity is drinking, which is aposture evaluated for multiple iterations of the device. Recall from Chapter7.3.2, the drinking posture evaluations resulted in tremor reduction with someamplification around the 6Hz tremor region. With this being the mediantremor frequency for tremor patients, the rejection rate due to drinking is tobe expected.2568.4.QUALITYOFLIFEEVALUATIONSFigure 8.26: Survey responses on the retention vs. retention rate of the Steadi-Two based on QOL activities outlinein the QUEST questionnaire.2578.4. QUALITY OF LIFE EVALUATIONSFollowing the results from both patient evaluation phases and the quality oflife survey results, we note the following:1. Due to the variation of individual patient tremors, designing a singledevice capable of broadband tremor attenuation is challenging. However,given the performance of the Steadi-Two for a wide range of frequencies,postures, and activities, the approach implemented throughout thecourse of this study tackles this challenge with promising results. Thisis evidenced by the continued adoption of the device by a wide rangeof tremor patients.2. Iterative development of the tremor suppression devices must includepatient evaluations early in the prototyping phase to better understandtremor profiles and patient needs. While this was unachievable for thedevelopment of the Steadi-Two due to COVID-19 restrictions, it mustbe a priority for future development.3. The Steady-Two appears to excel for a variety of activities includingboth fine and non-fine motor activities. With more evaluations and awider sample size, further iterations can be made to improve performancefor specific scenarios.This section concludes the kinematic evaluation of the Steadi-Two deviceand the implications of the results. In the next chapter, we will discuss theoverall outcomes from this study and review the approaches taken in arrivingat the Steadi-Two as a passive non-linear MDOF tremor attenuator.258CHAPTER 9DiscussionThroughout this work, we have explored the process of modeling, designing,and evaluating a passive multi-degree-of-freedom tremor attenuator capableof broadband attenuation. The mathematical modeling presented in Chapter3 demonstrated that a non-linear absorber is more effective for broadbandtremor attenuation. This finding was substantiated by the results fromdetailed experimental and patient evaluations discussed in Chapter 8. In themodeling and evaluation of the Steadi-Two, assumptions such as representingthe human arm using a lumped-parameter model and modeling tremors asharmonic torque functions were made. It is crucial to evaluate the validity ofthese assumptions independently of the experimental outcomes to ensure therobustness and generalizability of the findings. Moreover, the methodologyemployed during the experimental phase, including the use of a tremorinducer on a healthy hand and the RIG for inducing tremors in the FPSDOF, warrants critical analysis.This chapter provides a comprehensive discussion of key decisions madeduring the modeling, analysis, and experimental phases of this study. Italso offers a critique of these choices, addressing their potential limitationsand implications for the results. By doing so, we aim to contextualize thefindings and suggest avenues for further refinement and validation.2599.1. BIOFIDELIC MODELS OF THE HUMAN ARMThere are five key topics of discussion covered in this chapter. These are:∗ Modeling of the human arm using the lumped parameter model.∗ Using harmonic function for the induced tremors in the mathematicalmodel.∗ The use of tremor mimics such as the tremor inducer and the RIG.∗ The experimental findings for the T-Beam and the various iterationsof the Steadi-Two device.∗ Overall limitations of the study.9.1 Biofidelic Models of the human armThe lumped parameter model introduced in Chapter 3.1.1 simplifies thehuman in the FE DOF as a uniform rod rotating about the shoulder joint.The inertia and length of the arm is defined based on Contini’s [74] work onbody segment parameters. However, the stiffness and damping are calculatedsuch that a near resonant tremor occurs for the arm. This allows for simplicityin studying the effectiveness of tremor suppression devices. The naturalfrequency of this arm model is approximately 2.5Hz which, as we have nowestablished, is well outside the tremor spectrum. This does not nullify thework done using this model since we have seen the similarities betweenthe model and real tremor. However, we cannot ignore the necessity formore biofidelic models to be implemented in future studies based on theinconsistencies between the current models and the in-vivo results. Biofidelityof mechanical models is an important aspect of tremor research becauseit minimizes error between models and experiments. Looking ahead, wewill suggest the following biofidelic models for the flexion-extension andpronation-supination DOFs.9.1.1 Flexion-ExtensionFor the flexion-extension (FE) DOF, the dynamic parameters of the humanarm are defined in Table 9.1. These parameters are an aggregation from thework done by Rahnavard at al. [73] and Gebai at al. [69]. They implementeda lumped mass model of the human arm segments in their work on body2609.1. BIOFIDELIC MODELS OF THE HUMAN ARMsegment parameters by following the procedure described by Contini at al.[74].Table 9.1: Parameters for the Flexion-Extension DOFMuscle Group StiffnessCoefficient(N.m/rad)Dampingcoefficient(N.m.s/rad)Shoulder k1 170 c1 0.002k1Elbow k2 150 c2 0.002k2Biceps k4 40 c4 0.002k4Wrist k3 10 c3 0.001k3At the musculoskeletal level, there are four muscle groups responsible forthe motion of the arm in flexion-extension. The shoulder, elbow and wristmuscles are single jointed which means their response only depends on themovement of the shoulder, elbow, and wrist joints respectively. However,the biceps muscles are a double jointed and their response depends on therelative movement between the upper arm and the forearm. In Figure 9.1,the representation of the hand in flexion-extension at the musculoskeletallevel is summarized. Additionally, we define the positions of the upper arm,forearm, and wrist as θ1, θ2, θ3 respectively.For analysis, consider the equilibrium position as the instance where θ1 = 0,θ2 = π/2 and θ3 = π/2 with zero angular velocities such that the forearm andwrist are in-line. We still model tremor as a harmonic torque mi(t) acting inthe upper arm and the forearm. We wish to observe tremor manifestations onthe palm. In this setup, the problem resembles a compound triple pendulumand deriving the equation of motion is a multi-step process that beginswith identifying the position and consequently the velocities of the center ofmasses. Note that this model does not contain any form of absorber sincewe are studying the mechanics of the arm.2619.1. BIOFIDELIC MODELS OF THE HUMAN ARMFigure 9.1: Musculoskeletal hand model showing the shoulder, elbow, andwrist joints with their inherent stiffness ki and damping ci as well as thebiceps muscles connected to the shoulder and wrist muscles.We make use of Lagrange’s equations to derive the equations of motionbeginning with stating the kinetic and potential energies as well as Rayleigh’sdissipation function. The general form of the kinetic energy, which containsboth the translational and rotational components is given by Equation 9.1.T ={12m1(x˙21 + y˙21)+ 12m2(x˙22 + y˙22)+ 12m3(x˙23 + y˙23)+ · · · → translational· · ·+ 12I1θ˙21 + 12I2θ˙22 + 12I3θ˙23 → rotational(9.1)2629.1. BIOFIDELIC MODELS OF THE HUMAN ARMSimilarly, the potential energy, which results from the stiffness elementsmodelled as torsional springs is given by Equation 9.2. Note that the energystored in the biceps is an addition of the energy due to the deflection of theupper arm and the forearm. It is also important to note that the potentialenergy in each spring is only dependent on the deflection of a single part ofthe arm as opposed to the relative motion. This is an important distinctionsince each joint is capable of acting independent of other joints.V =12k1θ21 +12k2θ22 +12k3θ23 +12k4 (θ1 + θ2)2 (9.2)Finally, Rayleigh’s dissipation function is given by Equation 9.3.ℜ = 12c1θ˙12+12c2θ˙22+12c3θ˙32+12c4(θ˙1 + θ˙2)2(9.3)Now, we define the position and velocity of each center of mass so that thekinetic energy equation can be completed. Note that this analysis uses thesmall angle approximation under the assumption that the induced motiondoes not cause large variations in the position of the arm and as such sin θ ≈ θand cos θ ≈ 1. The position analysis yields the following:x1 = d1θ1; y1 = −d1x2 = l1θ1 + d2θ2; y2 = −l1 − d2x3 = l1θ1 + l2θ2 + d3θ3; y3 = −l1 − l2 − d3(9.4)The small angle approximation is quite significant since it reduces the y−axisdeflection to a constant such that the resulting velocity is only due to thex − axis deflection. The small angle approximation must be reviewed toensure that the model does not deviate from this assumption.The resulting Cartesian velocities are thus:x˙1 = d1θ˙1; y˙1 = 0x˙2 = l1θ˙1 + d2θ˙2; y˙2 = 0x˙3 = l1θ˙1 + l2θ˙2 + d3θ˙3; y˙3 = 0(9.5)2639.1. BIOFIDELIC MODELS OF THE HUMAN ARMTherefore, the kinetic energy becomes:T =12m1d21θ˙12+12m2(l1θ˙1 + d2θ˙2)2+12m3(l1θ˙1 + l2θ˙2 + d3θ˙3)2+12I1θ˙21 +12I2θ˙22 +12I3θ˙23Using Lagrange’s Equation A.1, we derive the equation of motion for thissystem which has a general form:[M]{θ¨}+[C]{θ˙}+[K]{θ}=[m]Where:[M]=M11 M12 M13M21 M22 M23M31 M32 M33M11 = I1 +m1d21 + (m2 +m3) l21M12 = (m2d2 +m3l2) l1M13 = m3d3l1M21 = (m2d2 +m3l3) l1M22 = I2 +m2d22 +m3l22M23 = m3d3l3M31 = m3d3l1M32 = m3d3l2M33 = I3 +m3d23[C]=c1 + c4 c4 0c4 c2 + c4 00 0 c3[K]=k1 + k4 k4 0k4 k2 + k4 00 0 k3[m]=[m1(t) m2(t) 0]T2649.1. BIOFIDELIC MODELS OF THE HUMAN ARMThe input moments, modelled as harmonic functions, result from tremors inthe upper arm and forearm for the flexion-extension DOF. We can observe theresponse due to these tremors at the palm which shows the tremor propagationfrom the proximal to distal DOF. This phenomenon has been discussed andobserved in previous analysis. For the complete flexion-extension modeldescribed here, we obtain the natural frequencies of the free system whichare:ωn1 = 4.02Hz, ωn2 = 7.80Hz, ωn3 = 12.83HzThe natural frequencies for the FE model are within the boundaries ofneurological tremors ranging betweeen 4−12Hz [3, 6, 7, 34]. We can evaluatethe impact on the movement of the arm by simulating the maximum tremorresponse in steady-state for this model over the tremor frequency range. InFigure 9.2, the response of the shoulder, elbow and wrist flexion-extensionDOFs is shown with the tremor modeled as harmonic torque inputs to theshoulder and the elbow joints. We observe that the proximal DOF torqueinput at the shoulder results in a higher amplitude (manifestation of tremor)at the distal DOF in the wrist. Similar results have been observed by Davidsonand Charles [38] in their analysis on tremor propagation. There are tworesonant peaks for the wrist flexion-extension DOF (which is a distal DOF)coinciding with the resonance for the shoulder and elbow flexion-extensionDOFs (which are proximal DOFs).This depiction of the flexion-extension DOF is analogous to a compoundtriple pendulum problem. Designing a tremor suppression device to combattremors with this degree of complexity is not only challenging but alsoa misrepresentation of real tremors based on the results of the patientevaluation analysis. This is because a dynamic triple pendulum problem;often considered chaotic motion; has not been observed to be a realisticrepresentation of the behaviour of the human arm under the effect of anyneurological tremor.2659.1.BIOFIDELICMODELSOFTHEHUMANARMFigure 9.2: Frequency response of 3 DOF flexion-extension model for the shoulder (SFE), elbow (EFE) and wrist(WFE). Tremor modeled as harmonic torque input to the shoulder and elbow joints. Observe the drop in amplitudefor SFE and EFE when WFE has a second resonant peak at f ≈ 7.80Hz2669.1. BIOFIDELIC MODELS OF THE HUMAN ARMTherefore, we consider the upper arm motion to be negligible under flexion-extensiontremor and constrain its motion. We integrate its influence at different fixedpositions which impacts the contribution of the biceps muscle introducing theconcept we have referred to as posture in this model. This means the modelcan capture the induced tremor in the elbow FE DOF at different postureswhich is a significant step in the direction of biofidelity. This approachto tremor modeling can be practically correlated to patient postures whenperforming different activities. The resulting posture related dynamic tremormodel is now a 2DOF model that considers the effect of the elbow and wristjoints in flexion-extension. In the derivation, the positions of the centres ofmass remains the same however the velocities change to the ones given here:x˙1 = 0; y˙1 = 0x˙2 = d2θ˙2; y˙2 = 0x˙3 = l2θ˙2 + d3θ˙3; y˙3 = 0(9.6)Using these formulations, the generalized equations of motion for this modelare given by: [M]{θ¨}+[C]{θ˙}+[K]{θ}=[m]Where:[M]=[M11 M12M21 M22]M11 = I2 +m2d22 +m3l22M12 = m3d3l2M21 = m3d3l2M22 = I3 +m3d23[C]=[c2 + c4 00 c3][K]=[k2 + k4 00 k3][m]=[m2(t)− k4θ1 0]TN/B: The tremor in the forearm has an additional term −k4θ1. Thisaccounts for the position of the upper arm at different postures by capturingthe potential energy stored in the biceps muscles which depends on boththe position of the upper arm and the forearm. This is how the modelincorporates posture since the biceps will be under constant tension tomaintain a given posture.2679.1. BIOFIDELIC MODELS OF THE HUMAN ARMFinally, using the characteristic equations above, the natural frequencies forthis model are calculated to be:ωn2 = 5.98Hz, ωn3 = 12.56HzObservably, the natural frequency of the wrist is slightly reduced and that ofthe forearm is significantly decreased compared to the initial 3DOF model.These frequencies are much similar to the ones observed in the second phaseof patient evaluations. Recall the characterization of patient tremor in Figure8.6, the patient tremor without the absorber shows a peak at 5Hz and 10Hz.In this new model of the FE DOF, we see a similar result with frequenciesat 5.98Hz and 12.56Hz. This acts as a validation of this proposed modelsaccuracy.In addition to this similarity, the new natural frequency of the forearmis now in the low frequency range of tremors which is often plagued withhigher amplitudes. Therefore, by considering the forearm and wrist joints inflexion-extension, this model represents the tremor challenge well. Implementingthis model to design an attenuator will likely result in an better agreementbetween the predicted and real results.When we simulate the response of the new model using variable frequencytremor and the upper arm at θ1 = 0 (posture 1), the result in Figure 9.3shows that the WFE DOF experiences higher amplitudes compared to theEFE DOF. This is a similar response to the real tremor response observedin the patient characterization section which validates this approach.2689.1.BIOFIDELICMODELSOFTHEHUMANARMFigure 9.3: Frequency response of reduced flexion-extension model showing elbow(EFE) and wrist(WFE) DOFresponse. Upper arm is fixed at θ1 = 0 and tremor is a harmonic torque to the elbow.2699.1. BIOFIDELIC MODELS OF THE HUMAN ARM9.1.2 Pronation-SupinationForearm pronation-supination (FPS) is the other problematic DOF prominentlyevaluated in this study. We still model it as a shaft under a harmonic torquethat rotates about its horizontal axis. In this DOF, tremor is best describedas the movement of the forearm around its longitudinal axis [93] such thatthe forearm and palm rotate together and the center of rotation is the elbowjoint. Figure 9.4 summarizes the musculoskeletal representation of the FPSmodel.Figure 9.4: Musculoskeletal hand model of the arm in the forearmpronation-supination degree of freedom. Tremor causes rotation motionin grey arrow path along the longitudinal axis dotted in red.The model parameters of inertia, stiffness, and damping coefficients due tothe forearm muscles in the FPS DOF are summarized in Table 9.2. These areobtained by considering the coupling matrix from the work done by Davidsonand Charles [38].2709.1. BIOFIDELIC MODELS OF THE HUMAN ARMTable 9.2: Biomechanical dynamic parameters for the FPS DOFInertia (kgm2rad ) StiffnessCoefficient(N.m/rad)Dampingcoefficient(N.m.s/rad)0.002 k5 0.756 c5 0.021Figure 9.5 shows the dynamic model of the forearm represented as a shaftwith stiffness k5 and damping c5 and overall inertia J5.Figure 9.5: Dynamic model of the forearm pronation-supination degree offreedom.From observation, this is a simple torsional vibrating system and by inspectionequation of motion is given by Equation 9.7.J5θ¨5 + c5θ˙5 + k5θ5 = m5(t) (9.7)Simulating the response of this model over the frequency range 1 − 12Hzresults in Figure 9.6. From the response, the natural frequency of this DOFis around 3Hz which is almost double the natural frequency of the modelintroduced in Chapter 3. However, this is still a low natural frequencyand falls outside of the previously defined tremor spectrum. It is closer tovolitional movement frequency which is what makes the FPS DOFs one ofthe most problematic DOFs to mitigate tremors. However, this model bettercaptures the natural frequency reducing the bandwidth requirement for anyabsorber designed using it.2719.1. BIOFIDELIC MODELS OF THE HUMAN ARMFigure 9.6: Frequency response for the FPS model of the arm. Note thatωn ≈ 3Hz with a slow decay and no double resonance peaks compared to FEmodel.9.1.3 Combined DOF AnalysisThese biofidelic models of the human arm present an opportunity forthe design, optimization, and placement analysis of tremor attenuatorsto maximize tremor reduction and improve patient quality of life. Thechallenge of tremor attenuations continues to complex when consideringMDOF tremor attenuation. Analysis of factors such as positioning of theattenuator to target both DOF is key in conjunction with tremor severity inboth DOFs.2729.1. BIOFIDELIC MODELS OF THE HUMAN ARMIn addition to the orientation of the attenuator, targeting multi-DOF tremorscould incorporate analysis of both the number and position of attenuatorsplaced on. While this was not the goal of our study, future evaluations maybenefit from such an approach. Figure 9.7 shows potential positions of theSteadi-Two attenuation if it were implemented this way. Without loss ofgenerality, we can conclude that varying orientation, dynamic and geometricparameters of attenuators becomes yet another criterion for performancesince more attenuators on the arm increases the total weight may becometoo heavy to be favoured by patients.Figure 9.7: Positions of attenuators to target multi-DOF tremors. Theseattenuators may vary in size, dynamic properties and orientation on the armto combat MDOF tremors.The models of the flexion-extension and the pronation-supination DOFpresented will improve the process of design and evaluation of the tremorattenuators. By changing the weight of the subject, the model parameters caneasily be altered to suit patient specific design and the derived mathematicalmodels can be easily implemented into any simulation. The tremor torque isdesigned as a separate harmonic torque input making it ideal for analysingvarying tremors. The biofidelity of these models depends on the type oftremors being studied. The implementation is up to the researcher whenconducting analysis.2739.2. SIMULATION OF NEUROLOGICAL TREMORS9.2 Simulation of Neurological TremorsIn addition to the models of the human arm, this study models neurologicaltremors as harmonic torques with specific frequencies and amplitudes. However,we have seen from real patient tremors in Chapter 8 that tremors vary for eachpatient depending on the severity of their tremor, type of nuero-degenerativedisease, the posture of the arm, the activity they are performing, and eventhe DOF being studied. This variation in tremor makes the design of one-sizefits-all tremor attenuators a significant challenge if the tremor models do notcapture it. We will consider how real tremors present and use this as a toolto critique the tremor models in this work.We begin with rest tremors which are introduced in Chapter 1 as predominantlyoccurring during little to no activity. Figure 9.8, shows the pronation-suppinationtremor measured for patient with Parkinsons disease wich is a rest tremor.TheRMS acceleration is 4.63m/s2 and the included subplot figure shows thepeak frequency to be 4.63Hz.This tremor closely resembles the response of the lumped parameter modelusing a harmonic torque as the underlying tremor. The main difference isthe frequency at which the tremor occurs. However, the steady-state tremoris harmonic showing that the harmonic torque tremor model used in thisstudy is an accurate representation of this patients tremor. Observing theFFT also reveals a second less dominant peak around 9Hz. This peak can beattributed to the tremor propagation from more proximal DOFs; a conceptdiscussed in the development of biofidelic tremor models which is evident inreal tremors. For the same patient, the WFE DOF is provided in Figure 9.9.2749.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.8: Rest tremor for a Parkinsonian patient for the FPS DOF. The FFT analysis shows a dominant frequencyof 4.63Hz. FPS is a proximal DOF thus a single dominant frequency peak is seen.2759.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.9: Rest tremor for a Parkinsonsian patient for the WFE DOF. The FFT analysis shows a dominantfrequency of 4.63Hz. WFE is a proximal DOF with lower amplitudes compared to FPS which results in a lowerpeak.2769.2. SIMULATION OF NEUROLOGICAL TREMORSThe WFE tremor has an RMS acceleration of 0.85m/s2 which is five-timesless than the FPS tremor. This phenomenon was witnessed in the experimentalevaluation stage using both the tremor inducer and the RIG for analysis.We can anticipate that for this patient, the Steadi-Two will be effective inmitigating the FPS tremor adequately since the severity of the WFE tremoris minimal in comparison. The WRUD tremor response for the same patientis given in Figure 9.10.The WRUD tremor response is significantly different compared to the FPSand WFE tremors, especially when looking at the frequency response. Here,we notice two dominant frequencies; one at 4.63Hz and the other at 9Hz.Recall from the development of biofidelic models, the distal DOF experienceddual resonance peaks resulting from tremor propagation from the proximalDOFs. This result confirms that the tremor propagation is indeed proximalto distal with the WRUD response experiencing dual peaks.Moreover, the RMS amplitude of the WRUD DOF is 1.56m/s2 whichis higher than the WFE DOF but a fraction of the FPS DOF. We caninfer that this increased amplitude is due to propagation of tremors fromthe FPS DOF and WFE especially when we see the location of the peakfrequencies. Therefore, we can presume that by attenuating tremors inthe FPS and WFE DOFs, we may effectively reduce the WRUD tremorswithout specifically targeting them. Admittedly, this hypothesis does nothave extensive experimental evaluation but it provides a good foundation forfuture analysis.2779.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.10: Rest tremor for a Parkinsonsian patient for the WRUD DOF. The FFT analysis shows a dominantfrequency of 4.63Hz. WRUD is a distal DOF hence two dominant peaks seen due to tremor propagation.2789.2. SIMULATION OF NEUROLOGICAL TREMORSThis analysis shows that modeling tremors as harmonic inputs to the modelsof the human arm can accurately capture real patient tremors. However, aswe have noted throughout this analysis about the variability in tremors, itwould be inaccurate to base future tremor models on single patient data. Inthe search for better tremor models, we will explore the variability in tremorusing the data provided by Adams at al. [94] through the IEEE data port.This data is the result of an extensive study on tremor patients for variousaspects of their lives. There are 5 different sensors used in their study. Theyare placed on the chest, right thigh, left thigh, left anterior arm, and rightanterior arm. These sensors collect tremor data based on several UPDRSscales and the results are provided in a .csv or .txt format through theIEEE data port for use per the outlined guidelines.Adams at al. have conducted several analyses with this data which revealedvarious trends. These findings are outside the scope of this work but areimportant foundational information for future studies. However, the rawdata provides an excellent opportunity to understand and model tremor.Therefore, through the use of a SQL database and Matlab, this study hasreorganized the data to highlight tremor variation and offer critique for thetremor models used in Chapter 3.Let us consider the results from the Postural and Kinetic tremor tests (definedin Chapter 1) for Patient 12 in the database. In Figure 9.11, the posturaltremor for this patient’s left hand is shown. We can infer from the responsesthat the FPS DOF is represented by the accely data in the middle subplotdue to the regularity (harmonic) pattern of the tremor which suggests asingle dominant frequency. Notice that the accelz appears to be experiencinga beat phenomenon which implies existence of two frequencies which, as wehave come to conclude, occurs in the WRUD DOF due to tremor propagationfrom the more proximal FPS DOF.2799.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.11: Postural tremor for Parkinsonsian patient 12 for their left hand. We can infer that accely (middleplot) represents the FPS DOF due to harmonic nature of tremor and the high amplitude.2809.2. SIMULATION OF NEUROLOGICAL TREMORSContrasting the postural left hand tremor to the right hand shown in Figure9.12, we see the variability of tremors even for the same patient. Theright hand tremor is challenging to model accurately or design an attenuatorcapable of combating. We observe that there are two frequencies present withthe primary tremor resulting form the higher frequency with a much loweramplitude. A second lower frequency with a significantly higher amplitude isalso evident. Thus, a tremor attenuator for this patient’s right hand wouldhave to target much higher frequencies compared to the left hand. However,the intermittent juts due to the lower frequency tremor would have to betreated by the device as regular disturbances and requires a fast responsetime to settle.Without delving into the process of mitigating this specific tremor, thisanalysis highlights the significant variation in tremors even for the samepatient. This is further highlighted when we consider their kinetic left andright hand tremors given in Figure 9.13 and Figure 9.14 respectively.The kinetic tremor also varies significantly when comparing their left andright hand. This highlights the challenges associated with tremor attenuation.We can see the same trend for patient 25 for their postural tremor withthe results given in Figure 9.15 and Figure 9.16 for the left and right handrespectively.2819.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.12: Postural tremor for Parkinsonsian patient 12 for their right hand. Their tremor significantly differsboth in amplitude and frequency compared to the left hand. This highlights the variation in tremor even for thesame patient.2829.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.13: Kinetic tremor for Parkinsonsian patient 12 for their left hand. This tremor differs from the posturaltremor for the same hand showing that tremor is posture dependent.2839.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.14: Kinetic tremor for Parkinsonsian patient 12 for their right hand. This highlights the variation intremor for the same patient, same posture but different hands.2849.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.15: Postural tremor for Parkinsonsian patient 25 for their left hand. This patient’s tremor for the lefthand is similar to patient 12’s tremor in all three DOFs.2859.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.16: Postural tremor for Parkinsonsian patient 25 for their right hand. The FPS DOF (middle figure)shows a similar RMS response to the left hand postural tremor but the other two DOFs a significantly different.This variation is the main challenge for broadband tremor suppression.2869.2. SIMULATION OF NEUROLOGICAL TREMORSThese results further highlight significant tremor variability even for the samepatient. This becomes more evident when comparing two patients for similarposture. Variability is influenced by factors such as inertia, age of tremoronset, and medical history which results in varying degrees of tremor severity.Therefore, it is important to classify tremors in terms of severity, in addition tothe frequency classification we have used thus far. This allows the evaluationof tremor suppression devices at various severities to better classify theirefficacy. The metric of tremor severity is not agreed upon between thedifferent fields involved in tremor mitigation. For instance, mechanicalmitigation may focus on measurable quantities to classify severity whilemedical approaches may define severity based on the medical dosage (orintensity of treatment) required to combat tremors. From the mechanicalpoint of view, severity can be easily classified using the measured tremoramplitude. However, the current consensus appears to be low frequencytremors have high amplitudes while the contrary is true for higher frequencytremors. Therefore, classifying severity based on the amplitude can haveconflicting results due to existing definitions. A suggested approach here isclassifying severity as a measure of variation from the norm i.e. if a tremorhas high frequencies and high amplitudes, it is severe since combating itrequires non-standard approaches mechanically speaking. This classificationis significant in designing tremor simulation platform such as the tremorinducer or the RIG used in evaluating the Steadi-Two for in-vivo analysis.With the understanding of varying tremor characteristics and challenges facedin classifying them, the proposed solution is to classify tremor severity basedon the characteristics. Without loss of generality, low amplitude tremorssuch as the ones in the WRUD DOF have multiple frequencies while thehigher amplitude tremors in the FPS and FE DOFs have single frequencypeaks. Implementing this logic, we can classify tremor severity based onthe variability of amplitude and frequency in the tremor. The result is asimplified model of tremor severity classification for mild, moderate, or severetremors given in Figure 9.17.With this classification (and acknowledgement of tremor severity classification),future studies can evaluate the performance of a tremor attenuation devicefor a given severity level. This may lead to better predictions for deviceperformance and minimize the iterations needed for effective performance.2879.2.SIMULATIONOFNEUROLOGICALTREMORSFigure 9.17: The proposed model of mild, moderate, and severe tremors. This model is capable of capturingamplitude and frequency variation for each tremor severity which simplifies classification and attenuationimplementation.2889.3. DISCUSSION OF THE FINAL ATTENUATOR PERFORMANCE9.3 Discussion of the Final AttenuatorPerformanceFollowing the review and critique of the human arm and tremor models inthis study, we ought to review the two final stage attenuators modeled andevaluated using these approaches. Recall that in Chapter 3, the theoreticalframework showed a simple inertial absorber is capable of tremor attenuationwhen tuned to perform at a single tremor frequency. This was deemedinsufficient and the T-Beam was designed to achieve SDOF broadbandattenuation before introducing the Steadi-Two to tackle MDOF broadbandsuppression. We will evaluate the level to which these devices achieved theirobjectives.9.3.1 Performance Discussion of the T-BeamIn Chapter 5.1, the linear T-Beam was been shown to be an effective absorberin attenuating tremors both at resonance and over a frequency range. Theresults are remarkable since the T-Beam was still a linear absorber and theseresults are often achieved using non-linear absorbers [83, 85, 86]. However,the original T-Beam design and subsequent iterations failed to mitigatelow frequency tremors. In this study, the T-Beam was modified to includenon-linear springs with the aim of tackling low frequency tremors. The modelfor the non-linear system predicted promising results for broadband tremorfrequencies.The T-Beam experimental evaluation, although limited to the flexion-extensionDOF, provided insight on the performance of the T-Beam model. In Table9.3, the results for the flexion-extension DOF for both the model and in-vitroevaluations are presented. While the model overestimates the performanceof the T-Beam for this DOF, it correctly predicts tremor reduction.The discrepancies between the predicted and experimental results can beattributed to several factors, particularly the simplifications inherent in thelumped-parameter modeling approach. While the model assumes simplifieddynamics of the human arm and tremor characteristics, a real human armintroduces distributed mass, complex damping properties, and active muscleresponses that are not fully captured. For instance, at 3Hz, which is near thenatural frequency of the flexion-extension model, the predicted reduction is2899.3. DISCUSSION OF THE FINAL ATTENUATOR PERFORMANCETable 9.3: Comparison of the predicted and measure RMS reductions for thenon-linear T-Beam in the flexion-extension DOF. The model overestimatesthe T-Beam performance but correctly predicts tremor reduction.F (Hz) Predicted In-Vitro2 44.78 52.53 76.4 27.24 64.03 43.04significantly overestimated due to these modeling assumptions. Furthermore,variations in experimental conditions, such as differences in tremor inducersettings, arm variability, and external noise, also influence the observedvalues. These discrepancies suggest the need for a more refined model thatincorporates biomechanical properties closer to those of a human arm.The experimental performance of the T-Beam in the flexion-extension DOFprovides insights into its potential application to other DOFs. While thecurrent study focuses on the T-Beam as a single DOF system, the structuralsimplicity of the T-Beam makes it an attractive candidate for MDOFextension with multiple devices on the arm. By tuning the non-linear springsto target other DOFs, a dual minimized T-Beam system can be implementedfor MDOF tremors while maintaining broadband attenuation. This mightinclude exploring advanced materials or novel spring configurations to achievea more adaptive response.These findings underscore the importance of refining both the modelingand experimental approaches to better align predicted and observed results.The T-Beam’s performance demonstrates significant promise for single-DOFapplications, particularly in addressing low-frequency tremors. By buildingon these findings, the T-Beam can serve as a foundation for future innovationsin passive tremor suppression technologies.2909.3. DISCUSSION OF THE FINAL ATTENUATOR PERFORMANCE9.3.2 Steadi-Two: Design, Functionality, and ImpactThe Steadi-Two Patient Device represents the culmination of this study’sefforts to develop a passive tremor attenuation device. By implementingnon-linear springs using Polymagnets, it achieved an expanded frequencybandwidth and tackled multi-DOF tremors. The iterative design process hasprovided key insights, and the following sections discuss its design approach,performance evaluations, and overall impact.Design ApproachThe design evolution of the Steadi-Two has been characterized by iterativeimprovements. The first iteration introduced stronger Polymagnets, whichincreased the equilibrium stiffness and improved high-frequency performance.However, the increased degree of non-linearity resulted in performanceinstability, as evidenced by inconsistent results across experimental runs.This instability could be attributed to the shifting of natural frequenciesand the introduction of harmonics. This creates regions of unpredictableperformance. Further analysis of the non-linear springs is needed to stabilizeand optimize their behaviour.Another design consideration arises from posture-dependent performancevariations. Postures with high inertia are better suited for low-frequencytremor attenuation, as the added inertia effectively reduces high-amplitudetremors. In contrast, low-inertia DOFs, particularly distal ones like WRUD,are challenging due to their high tremor amplitudes and propagation dynamics.A potential design improvement could involve targeting proximal DOFs toreduce tremor propagation downstream. This hypothesis has been brieflyevaluated and found to hold merit although further analysis is required tovalidate it.2919.3. DISCUSSION OF THE FINAL ATTENUATOR PERFORMANCEPerformance EvaluationsThe performance of the Steadi-Two was evaluated through modeling, experimentaliterations, and patient testing. Table 9.4 summarizes the results for thepronation-suppination and flexion-extension DOFs over different stages ofevaluation.Table 9.4: Percentage reductions of the Steadi-Two in flexion-extension andpronation-suppination DOFs. The performance for FPS is consistent frommodeling to the RIG while the FE performance shows significant variationover the iterations.Iteration FPS FEModel 64.04 75.39Iteration 2 56.53 -17.27Patient Device 70.76 50.87RIG 52.6 N/AThe model demonstrated strong reductions in tremor amplitudes, particularlyin the FE DOF, but overestimated the real-world performance. In Iteration2, the amplification in the FE DOF highlights the challenges introduced byincreased non-linearity. The patient device showed consistent performancein FPS compared to the model and iteration 2 but produced variable resultsin FE. The RIG evaluation, only done in FPS, further supported the FPSresults showing agreement from model to prototype.Comparing the The results of patient evaluations (Table 9.5) further highlightposture-dependent performance variations. Reductions were generally highestin FPS and postures requiring gross motor control, while fine motor taskssuch as drinking showed mixed results. The variability in results emphasizesthe influence of individual patient dynamics, tremor characteristics, andactivity type on device performance.2929.3. DISCUSSION OF THE FINAL ATTENUATOR PERFORMANCETable 9.5: Percentage reductions comparing freehand and Steadi-Two resultsfor Posture 1, Posture 2, and Drinking posture in the FPS DOF. Performanceis posture dependent.Patient ID Frequency (Hz) Posture 1 Posture 2 Drinking1 5.12 74.89 70.83 39.252 5.02 64.9 61.6 45.264 5.03 21.7 53.68 32.775 5.69 27.51 42.79 39.666 6.69 34.23 N/A 41.167 5.09 -20.92 -73.3 40.458 5.72 63.17 67.74 31.93Key Takeaways and Future WorkThe Steadi-Two demonstrates considerable potential as a passive tremorsuppression device. Its ability to attenuate multi-DOF tremors, particularlyin high-inertia postures, positions it as a viable solution for gross motortasks. However, its effectiveness in fine motor tasks is limited, as seen in thelower quality-of-life improvements for activities such as writing, dressing, andusing a phone. Furthermore, the mixed retention rates for reading-relatedactivities suggest that device adoption depends not only on performance butalso on patient priorities and goals.Future iterations of the Steadi-Two must prioritize patient-specific designs,allowing some level of customization to address individual needs. Incorporatingfeedback from real-world use and refining non-linear spring behaviour willbe critical in improving its performance and stability. Ultimately, embracingpatient-centric design principles can maximize the Steadi-Two’s impact andpave the way for its broader adoption in tremor management.2939.4. LIMITATIONS OF THE STUDY9.4 Limitations of the StudyAcknowledging the limitations of this study is essential for interpreting thefindings within their proper context and for guiding future research. Thissection outlines the constraints associated with the modeling, experimentalmethods, and practical implementation of the Steadi-Two device. Theselimitations, while not detracting from the contributions of this work, providea roadmap for refining the model, improving experimental protocols, andenhancing device performance.9.4.1 Model Assumptions and SimplificationsThe models used in this study included several assumptions outlined inChapter 5. These assumptions facilitated analytical and computationalfeasibility but traded accuracy of the final results. For instance, the assumptionof linear damping and stiffness properties for the human arm, while usefulfor analysis, does not adequately capture the non-linear behaviours exhibitedby the human arm during high-amplitude oscillations.Moreover, excluding secondary factors such as friction, material fatigue, andhuman arm dynamic non-linearity reduces the model’s similarity to real-worldenvironments. Phenomenons such as the tremor propagation betweenproximal and distal DOFs, as highlighted in Chapter 2, introduce complexinteractions that were not fully captured into the model. Future modelrefinement should focus on incorporating these non-linear and environmentaleffects, as well as improving the representation of multi-DOF interactions,to create a more accurate and robust framework for predicting tremorattenuation.2949.4. LIMITATIONS OF THE STUDY9.4.2 Experimental ConstraintsThe experimental setup provided valuable insights into the performance of theSteadi-Two device, but it was not without limitations. Measurement noise,sensor inaccuracies, and the difficulty of replicating real-world conditions ina controlled laboratory environment introduced variability into the results.Additionally, the use of simulated tremors via a tremor inducer on healthyarms and the rig may not fully represent the tremor characteristics of patientswith neurological disorders. The experimental setup also constrained therange of input forces and frequencies, limiting the ability to evaluate deviceperformance under extreme or atypical conditions. Future experiments couldbenefit from using patient-specific tremor profiles and a broader range oftesting conditions to improve the validity of the findings.9.4.3 Posture and DOF InteractionsTremor attenuation is influenced by the complex interactions between proximaland distal DOFs. As noted in Chapter 2, tremors propagate from the proximalto the distal DOFs, with distal DOFs such as WRUD experiencing thehighest displacements. However, this study primarily evaluated tremorsat the wrist and forearm, simplifying the dynamics of the upper armand its impact on tremor propagation. For example, the posture of theupper arm and the activation of biceps and triceps muscles could alterthe response at the wrist and palm. While simplifying these interactionsallowed for a more focused analysis, future models could incorporate a morecomprehensive representation of the arm, including joint coupling effects andposture-dependent dynamics.9.4.4 Scaling Effects and Prototyping ChallengesTranslating the theoretical model into a physical prototype introduceddiscrepancies due to manufacturing tolerances, material properties, andassembly errors. Even minor variations in mass, stiffness, or dampingcharacteristics during prototyping affected the device’s performance comparedto model predictions. Additionally, the physical prototype faced practicallimitations, such as operational instability for some iterations. This wasnot captured in the theoretical model. Future iterations of the Steadi-Twocould implement more precise material characterizations to reduce thesediscrepancies and improve the reliability of the physical prototype.2959.4. LIMITATIONS OF THE STUDY9.4.5 Rigid Body Motion vs. Vibration AnalysisWhen evaluating the Steadi-Two during patient activities, distinguishingbetween rigid body motion and vibrations was a challenge. Activities suchas drinking or writing involve a combination of voluntary motions andinvoluntary tremors, making it difficult to isolate the device’s effectivenessin attenuating tremors alone, especially from the perspective of vibrationabsorption. Activities that involved whole arm movement such as drinking orpicking are better classified as rigid body motion compared to postural holdswhich oscillate about an equilibrium (akin to a vibrating system). Given theimplementation of vibration techniques to design the Steadi-Two, evaluatingits impact for non-vibration scenarios may unfairly represent its effectiveness.However, it is activities such as drinking and writing that significantly impactpatient quality of life. For future evaluations, incorporating evaluationssuch as Neurologist evaluations for activities of daily living would providemore refined analysis framework that separates these two components andaccurately assess device performance in real-world scenarios.9.4.6 Fatigue Life and Long-Term PerformanceThis study did not evaluate the fatigue life of the Steadi-Two under cyclicloading conditions, which could significantly impact its long-term performance.As the device is subjected to repetitive use, material degradation, wear onPolymagnets, and non-linear spring fatigue could reduce its effectivenessover time. A systematic study of the Steadi-Two’s durability and fatigue lifewould provide critical insights into its reliability and inform design changesto enhance its longevity.In addition to the Steadi-Two’s cyclic life, this study did not evaluatethe impact of adding a vibration absorber on the human arm. This isespecially important for the Steadi-Two since its a non-linear absorber andmay introduce harmonics and sub-harmonics. Experimental and patientevaluations did not show evidence of such scenarios which is a great initial sign.However, for long-term performance analysis, this aspect of the device-arminteraction should be evaluated.296CHAPTER 10ConclusionThe objective of this research was to address the mitigation of neurologicalhand tremors through the design of a novel passive hand tremor attenuator.We began with an exploration of the underlying pathophisiology of neurologicaltremors, specifically Parkinson’s disease and Essential tremor disease. Wethen examined the limitations of available therapeutic strategies beforereviewing alternative strategies to tremor mitigation. This was the basis forthe necessity for this research. Recognizing the impact of tremors on thequality of life for patients, this work has focused on the development, analysis,and evaluation of Steadi-Two, a passive non-linear multi-DOF attenuator,to mitigate the physical manifestations of tremors.The results of this research highlight both the challenges in tackling neurologicaltremors but also the opportunity to combat them through vibrations engineeringsolutions. These results do not suggest that medical approaches are ineffective,in fact, it is likely that a combination of medical and tremor attenuator deviceswill offer the best remedy to alleviating the debilitating impact of neurologicaltremor in patient’s daily activities. This work contributes to the field ofvibrations engineering through the implementation of non-linear analyticalapproaches and the validation of computational models. It also contributes tothe field of tremor attenuations in identifying a novel alternative to existingsolutions. We will discuss these contributions in detail below.29710.1. CORE CONTRIBUTIONS10.1 Core ContributionsThis study adopts an interdisciplinary approach, combining vibrations engineering,biomechanical analysis, and design engineering to address the complexchallenge of tremor attenuation. By integrating theoretical, experimental, anddesign methodologies, this research not only advances academic knowledgein vibrations but also provides tangible contributions to patient care. Thecore contributions of this work are as follows:∗ Development of Conceptual Tremor Attenuator Designs: Thisresearch outlines the evolution of tremor attenuator designs, progressingfrom simple vibration absorbers to advanced multi-DOF devices. Theuse of magnetic springs for broadband tremor reduction marks a novelcontribution to the field.∗ Mathematical Modeling of Linear and Non-Linear Systems:Comprehensive mathematical modeling of both linear and non-linearsystems was performed using numerical and perturbation methods.This work demonstrates sequential analysis steps for solving complexvibration problems, offering a reproducible framework for future researchers.∗ Implementation of Magnetic and Polymagnetic Springs: Thestudy provides detailed methodologies for measuring magnetic behaviour,determining effective stiffness experimentally and mathematically, andperforming numerical simulations. The implementation of Polymagneticsprings in the Steadi-Two significantly enhanced its attenuation bandwidth,addressing a key challenge in tremor suppression.∗ Development of Experimental Platforms: A tremor inducercapable of generating variable-frequency and variable-amplitude tremorswas designed for in-vitro studies. Additionally, the RIG platform wasdeveloped to enable consistent FPS testing, thereby reducing prototypeevaluation time and increasing experimental reliability.∗ Biofidelic Arm Models for Future Analysis: Biofidelic arm modelswere developed to facilitate the future design and analysis of tremorattenuation systems. These models provide a foundation for morerealistic simulations in future research.29810.2. FUTURE STUDIESThe implications of this research extend beyond neurological tremor suppression.The methodologies and insights gained in this study have broader applications,including the development of vibration absorbers for non-medical uses.Furthermore, this work sets a precedent for future innovations in the designof passive, non-linear, multi-DOF absorbers. These avenues will be exploredfurther in the following sections on future studies.10.2 Future StudiesThe development of the Steadi-Two represents an iterative process of modeling,design, and experimental analysis, as detailed in Chapters 3 through 7.1.Performance evaluations demonstrate its ability to significantly mitigatetremors in the forearm pronation-supination (FPS) and wrist flexion-extension(WFE) DOFs, with reductions exceeding 70% in some cases. Furthermore,the patient retention rates discussed in Chapter 8 highlight the device’spractical viability and its positive impact on quality of life. Despite thesepromising results, the limitations identified in this study open avenues forfurther exploration. Future research should aim to build on this foundationby addressing these limitations and advancing the design and application ofthe device. In this section, we explore potential directions for future studies,both within the context of neurological tremors and the broader field ofvibrations engineering.10.2.1 Vibrations EngineeringThis study has approached tremor attenuation through the lens of vibrationsengineering, revealing several intersections between these two fields that meritfurther exploration. One key area for future research is the application ofnon-linear Polymagnetic springs for broadband tremor reduction in multi-DOFsystems. While broadband vibration reduction is a well-known challenge invibrations engineering, it is rarely studied in contexts where both frequencyand amplitude change dynamically, as observed in tremors. Addressing thiscomplexity presents an exciting opportunity for researchers to advance thedesign of passive non-linear vibration absorbers.29910.2. FUTURE STUDIESFrom a design perspective, future studies should focus on reducing the weightof the device while maintaining or enhancing its performance. A promisingavenue is the optimization of the dynamic mass shape to increase inertia bypositioning the mass further from the centroid, thereby allowing for lowermass without compromising performance. Similarly, size reduction is critical,as smaller absorbers can be more versatile and adaptable across a widerrange of applications, making the device more practical for users.The concepts of whole-body vibration can also be explored using the armmodels developed in Chapter 9. Power tool use, for instance, is known toresult in long-term debilitating effects on the human arm. By applying thesemodels, researchers could study how vibrations propagate from the distal wristto the shoulder and analyze their amplitudes to assess potential harm. Thistype of investigation would require collaboration with biomedical scientistsbut could yield valuable insights into mitigating whole-body vibration effectsand addressing knowledge gaps in this area.Several design concepts explored in Chapter 4 were deemed unsuitable forneurological tremor attenuation but may be applicable to other domainswith minor modifications. Future studies should evaluate these designs foralternative uses, as vibration absorbers are widely implemented across avariety of systems. Additionally, researchers should continue exploringinnovative designs for tremor attenuation. The Steadi-Two should beregarded not as the definitive solution but as a foundation for furtheradvancements in passive vibration absorber technology. This iterativeapproach will ensure that the field continues to evolve, leading to betteroutcomes for both medical and non-medical applications.30010.2. FUTURE STUDIES10.2.2 Neurological TremorsThis study has demonstrated, through mathematical modeling, design,simulation, and real-world analysis, that developing a multi-degree-of-freedomtremor attenuation device is a highly iterative process requiring constantrefinement based on real-world performance. The Steadi-Two device, whileachieving notable success in mitigating tremors in certain activities, highlightsthe challenges associated with addressing fine motor skill tasks. Futureiterations of such devices must prioritize patient-centred design by integratingcontinuous feedback from end users. Partnering with organizations andpatient communities to establish collaborative development teams will ensurethat future devices meet the most critical patient needs, paving the way formore effective and inclusive solutions.In-vitro analysis has also revealed the variability in performance acrossdifferent DOFs. This variation presents an opportunity for researchersto adopt a DOF-specific approach, analyzing and optimizing each DOFindependently before integrating them into a unified multi-DOF solution.While this approach may initially appear counter-intuitive, tailoring thedevice for each DOF could lead to more effective solutions that balanceperformance across multiple DOFs. Additionally, long-term studies evaluatingthe device’s impact on the human body are essential, as sustained exposure tovibration could potentially induce muscle fatigue. Addressing these concernswill bring us closer to developing devices that not only mitigate tremors butalso enhance the overall well-being of patients.One promising avenue identified during in-vitro testing is the effect ofattenuating distal DOF tremors by targeting proximal DOFs. For example,mitigating FPS tremors may indirectly reduce WRUD tremors, potentiallysimplifying device designs and reducing their complexity. If further researchconfirms that targeting proximal DOFs effectively attenuates distal tremors,future devices could become more compact, cost-effective, and widely adopted.This simplification would improve device retention and usability, greatlybenefiting patients.30110.2. FUTURE STUDIESThe evaluation of the T-Beam absorber highlights its potential for single-DOFtremor reduction, particularly for patients with isolated tremors. Althoughthe T-Beam’s inability to simultaneously combat FPS and FE tremors witha single parametric combination is a limitation, its linear models providea valuable framework for DOF-specific solutions. Miniaturization of theT-Beam and its placement in multiple locations along the arm could opennew possibilities for targeted tremor mitigation. These findings reinforce theimportance of exploring novel, scalable designs tailored to the specific needsof different tremor profiles.The ultimate goal of neurological tremor reduction is to empower patientsand significantly improve their quality of life. By integrating insights frombiomechanics, vibrations engineering, and patient-centred design, futurestudies can bring this vision closer to reality. The journey toward moreeffective tremor mitigation is ongoing, but every step forward builds uponthe foundation of prior research, advancing both the science and the hopefor better solutions.With continued innovation and collaboration, the future of tremor mitigationlooks promising. As alternative therapies evolve and new knowledge emerges,not only will device performance improve, but the guidelines for successwill become clearer, informed by lessons from each iteration. 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Asami, “Closed-form solutions to the exactoptimizations of dynamic vibration absorbers (minimizations of themaximum amplitude magnification factors),” Journal of Vibration andAcoustics, Transactions of the ASME, vol. 124, no. 4, pp. 576–582, 2002.→ pages 335314Appendices315APPENDIXADerivationsA.1 Theory of Design and Optimization of aVibration AbsorberConsider the primary system shown in Figure A.1 under the harmonicdisturbance or excitation described by the force F (t) = F0sin(ωt). With themass m and stiffness k of the system, the natural frequency ωn of the systemis such that the disturbance causes the system to resonate thus negativelyaffecting system performance as well longevity.One viable solution to this problem is to shift either the system’s naturalfrequency or the excitation frequency to remove the resonance case. However,this is often not feasible and it is better to control the system behaviour atthe resonance case. This is achieved by adding a secondary system which isthe vibration absorber component in Figure A.1. Adding the secondary masscreates a two degree of freedom system. The equations of motion(EOM) forthis system can easily be derived using Lagrangian equations (Note: Whileforce balance using Newton’s Second Law may be used here, for the sakeof consistency in deriving EOM for more complicated systems later in thisresearch, Lagrangian derivation will be used throughout).316A.1. Theory of Design and Optimization of a Vibration AbsorberThe Lagrange derivation of the EOM begins with the formulation of thekinetic and potential energy of the system shown in Figure A.1. For simplicity,this analysis considers the damping values for c and ca to be equal zero.The Lagrangian is given by Equation A.1:L = T − V (A.1)where T is the kinetic energy and V is the potential energy. From FigureA.1, the kinetic and potential energy are calculated to be:T =12mx˙2 +12max˙2a (A.2)V =12kx2 +12ka(x2 − x1)2 (A.3)Figure A.1: Basic representation of a single DOF dynamic vibration absorbersystem317A.1. Theory of Design and Optimization of a Vibration AbsorberFrom the kinetic and potential energy in Equation A.2 and Equation A.3,the Lagrangian is given by Equation A.4.L =12mx˙2 +12max˙2a −12kx2 − 12ka(x2 − x1)2 (A.4)Using the extended Lagrange’s Equation A.5, where term ∂ℜ∂q˙irepresentsthe Rayleigh dissipation function; which is zero in this case due to theassumption of negligible damping; and Qi represents non-conservative forces,the equations of motion for the primary and absorber system are derivedand subsequently given by Equation A.8 and Equation A.9 respectively.Calculus of variation is used here:ddt(∂L∂q˙i)− ∂L∂qi+∂ℜ∂q˙i= Qi (A.5)For x(t):(∂L∂q˙i)= mx˙2,ddt(∂L∂q˙i)= mx¨, Qi = F (t) (A.6)For xa(t): (∂L∂q˙i)= mx˙2a,ddt(∂L∂q˙i)= mx¨a, Qi = 0 (A.7)Therefore, assembling the results of Lagrange’s equation yields the equationsof motion below.mx¨+ (k + ka)x− kaxa = F (t) (A.8)max¨a − kx+ kaxa = 0 (A.9)For two degree of freedom systems, it is useful to represent the equations ofmotion using a state space model as shown in Equation A.10. This is bothuseful in solving the Eigenvalue problem which is significant in identifyingboth the natural frequencies and the mode shape of the system as well ascreating the steady-state solution.318A.1. Theory of Design and Optimization of a Vibration Absorber[m 00 ma]{x¨x¨a}+[k + ka −ka−ka ka]{xxa}=[F (t)0](A.10)The significance of identifying the natural frequencies of the system is selfevident as it allows identifying potential resonance cases during excitationor disturbances to the system. In addition to the natural frequency, themode shape provides insight on the displacement relationship of the mainsystem to the absorber (in general terms deformation) during resonance.Next, the mass and stiffness matrices M and K are created and EquationA.10 is summarized to be:[M]{X¨}+[K]{X}=[F]This representation of the EOM favours solving the steady-state solutionto the problem using inverse matrix multiplication. The steady-state is theperiod during which the system’s transient response has faded away due toinherent damping in the system. By solving the steady-state solution, theresponse amplitude for both the primary and absorber system is found. Thisis achieved by first assuming that the solutions to x(t) and xa(t) are givenby Equation A.11:Let :{x(t) = Xsin(ωt)xa(t) = Xasin(ωt)(A.11)Applying Equation A.11 to the EOM in Equation A.10 and simplifying, theresulting matrix is given by Equation A.12.[k + ka − ω2m −ka−ka ka − ω2ma][XXa]=[Fo0](A.12)In Equation A.12, the amplitude of the displacement for both the primaryand absorber system can be calculated for the steady state case. A morecompact version of Equation A.12 is shown in Equation A.13 where Z iscalled the impedance matrix.[Z]· X⃗ = F⃗ ⇒ X⃗ =[Z]−1 · F⃗ (A.13)319A.1. Theory of Design and Optimization of a Vibration AbsorberWith this formulation, the steady-state solution for the displacement ofboth the primary and absorber system can be solved. Equation A.14 isthe steady-state solution for the main system while Equation A.15 is thesteady-state solution for the absorber system.X =(ka − ω2ma)Fo(k + ka − ω2m) (ka − ω2ma)− k2a (A.14)Xa =kaFo(k + ka − ω2m) (ka − ω2ma)− k2a (A.15)Using Equation A.14, the optimal absorber stiffness ka and mass ma can becalculated to make the numerator zero and thus the steady-state movementof the main system can be reduced to zero. However, this solution onlyworks for this specific combination of system natural frequency and harmonicdriving frequency. By non-dimensionalizing the expressions for both X andXa, a solution that is adaptable to numerous driving and system frequenciesis created.The process of non-dimensionalizing begins by recognizing that the currentdriving frequency ω is virtually equal to the main system’s natural frequencyωs (that is ω = ωs) and thus the system is resonating. The amplitude of thevibration for the main mass is described by Xδst where δst =Fok . With thisdefinition, the response of the main and absorber systems are rewritten inEquation A.16 and A.17.Xδst=1− ωωa[1 + kka −(ωωs)2] [1−(ωωa)2]− kka (A.16)Xaδst=1[1 + kka −(ωωs)2] [1−(ωωa)2]− kka (A.17)320A.1. Theory of Design and Optimization of a Vibration AbsorberFrom these expressions, it is rather intuitive that to eliminate the vibrationin the main system via displacement X, the frequency ωa must equalthe driving frequency ω as is evident in the numerator of Equation A.16.Therefore ω = ωa =√kama. In essence, the selected absorber massand stiffness must be optimized such that the absorber naturalfrequency matches that of the driving frequency of the main system.This solution is the core to the design of a vibration absorber system andrepresented in Figure A.2 is the relationship between the absorber naturalfrequency to the driving frequency ratio r = ωa/ω and the normalizedamplitude response of the main system XkFo .Observably, the initial response of the system without the absorber resonatesat the driving frequency ω. However, the addition of the tuned absorberreduces the amplitude response at the driving frequency to zero. This isnot without consequence as there are two new resonances at ω1 and ω2.Therefore, while the primary response at the intended operating frequency(often called machine frequency) is reduced to zero, the new resonances posea challenge, specifically during startup or in the event of added mass thatmay cause the system to operate near the new resonance points.The hatched region bound by the lines ra, rb, andXkFo= 1 represents theabsorber bandwidth. This is the region within which the absorber effectivelyreduces the main system’s amplitude response. Thus, ra ⩽ ωaω ⩽ rb. Fordesign, this range can be mathematically determined using Equation A.18where:µ =mam,β =ωaωs, Ra =ωωa, Rs =ωωs321A.1.TheoryofDesignandOptimizationofaVibrationAbsorberFigure A.2: Effect of a Tuned Mass Damper on the response of the primary system. At optimal tuning, the originalresonant peak is removed but to new resonant peaks are introduced ω1 and ω2. These must be avoided in operation;especially during machine startup.322A.1. Theory of Design and Optimization of a Vibration AbsorberThis expression is derived from Equation A.16 and the solution involvesconsidering the absolute value for the normalized amplitude XkFo .∣∣∣∣Xδst∣∣∣∣ = 1−Ra[1 + µβ2 −R2s] [1−R2a]− µβ2 (A.18)For the case where∣∣∣ Xδst ∣∣∣ = 1, the Equation A.18 is reduced to:1−Ra =[1 + µβ2 −R2s] [1−R2a]− µβ2This expression can be solved using quadratic formulation such that:R2a =1 + β2(1 + µ)2β2± 12β2√β4(1 + µ)2 − 2β2(1− µ) + 1This solution provides the lower bound for the frequency bandwidth of theabsorber. Conversely, when∣∣∣ Xδst ∣∣∣ = −1, the expression above is solved andthe upper bound for the bandwidth is then determined.It is evident from the reduced expression that the frequency ratio Ra dependson the mass ratio µ and the frequency ratio of the main system to the absorbersystem β. We have identified that for an optimized system performance atresonance, the ratio β = 1. In Figure A.3, the relationship between the ratioof selected absorber mass to main system mass µ and the frequency ratio Rais presented.323A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.3: Relationship between absorber mass ratio and effective frequencybandwidth. This indicates that a higher mass ratio increases so does theeffective bandwidth, although the absorber stiffness need to increase tomaintain the frequency ratio β which increases the settling time (this is anoptimization problem where trade-offs must be considered).From this relationship chart, consider a mass ratio µα; It is obvious thatas the mass ratio µ increases, so does the effective frequency bandwidth forβ = 1. It should be noted that the absorber mass cannot increase indefinitelyas the absorber stiffness would need to increase accordingly to keep thefrequency ratio at 1 and in turn cause the system to oscillate for a longerperiod of time before settling to zero. On the other hand, the mass ratiocannot be too low as a reduced absorber mass would not effectively handle theenergy from the main systems vibration. Therefore, a balance must be struckand as such a conventional range of the mass ratio is typically 0.05 ⩽ µ ⩽ 0.25.In addition to the mass ratio, another important design parameter is therattle space. This is determined by considering the maximum allowabledeflection of the absorber system and can be calculated from Equation A.17.At optimal operating point where the motion of the main system in thesteady-state described by X has decayed to zero, the motion of the absorbersystem is described by Equation A.19. Here, the amplitude of this motion−Foka = −Xa determines the maximum displacement of the absorber andmust be below the maximum space available during operation.324A.1. Theory of Design and Optimization of a Vibration Absorberxa(t) = −Fokasinωt (A.19)With all these considerations during design, it is clear that effective absorberdesign is an iterative process and often requires fine tuning for excellentperformance (hence the name tuned mass absorber). For instance, we considera main system whose mass is m = 1.3kg and stiffness k = 1300N/m. Thenatural frequency of this system and consequently the driving frequency ofthe harmonic excitation is f = 5.033Hz with a force amplitude Fo = 2.3N.The results in Figure A.4 display the response of the system comparingdifferent tuned absorbers.Figure A.4 a shows the response of the system at resonance without anabsorber and it is clear that the response of the system increases from rest astime goes. The response shown in Figure A.4 b is that of the tuned absorberwith a mass ratio µ = 0.2 and the corresponding stiffness to match ensurethe natural frequency is matched. The amplitude of the main system isgreatly reduced by two orders of magnitude indicating a potentially successfulabsorber design. In Figure A.4 c, the mass ratio is increased to µ = 5 andthe stiffness is adjusted to ensure a match in the natural frequency. Notice,that the stiffness is now kc = 6500N/m compared to that of the initiallyoptimized system at kb = 260N/m. However, the amplitude of this systemis about one-fifth that of the optimized system which appears to be betterperforming in comparison. In Figure A.4 d, a sanity check is performed byselecting an absorber that exactly matches the main system parameters andthe amplitude here is again much lower than the so called optimized system.325A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.4: Simulated response of a system under resonance with the effectof various absorbers: (a)No absorber → resonance; (b)Tuned absorber withµ = 0.2 → amplitude reduced to 5 × 10−3m; (c) Tuned absorber with µ = 5→ amplitude reduced to 1 × 10−3m; (d)Tuned absorber with µ = 1 →amplitude reduced to 2 × 10−3mNone of the absorbers in Figure A.4 reach steady-state which is to be expectedsince the initial assumption was no inherent damping in the primary system.A frequency analysis of the displacement in Figure A.4 b shows two frequencypeaks per Figure A.5.326A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.5: Frequency analysis of the displacement data from Figure A.4 b.The second peak is due to the main system natural frequency which persistswhen damping is not accounted for.However, damping is often present and whether it is inherent/internalstructural damping through material composition or some form of addeddamping such as viscous or coulomb damping. This changes the value ofthe main system damping c in Figure A.1 to non-zero. The considerationof the main system damping introduces a different challenge when it comesto finding the optimal mass of the absorber to successfully attenuate theinduced vibrations. Suppose that the damping present in the system aboveis only due to inherent damping and as such we do not alter the equationsof motion, the transient response dissipates over time and the system finallysettles to zero after some time as is evident in Figure A.6 b, c, & d.327A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.6: Effect of primary damping on the absorber effectiveness. In (a)Noabsorber, the system still resonates but at a reduced amplitude; (b)Tunedabsorber with µ = 0.2 reduces the initial response to 3.4 × 10−3 and thesystem reaches steady-state and rest within 7 s;(c)Tuned absorber with µ = 5reduces the initial response to 7.4×10−4 and the system reaches steady-stateand rest within > 15 s;(d)Tuned absorber with µ = 1 reduces the initialresponse to 1.6× 10−3 and the system reaches steady-state and rest within11 s.In Figure A.6 a, the system resonates almost instantaneously while theaddition of the well designed and tuned absorber in Figure A.6 b showsa clear reduction in the system amplitude to zero in approximately 7s.Moreover, the amplitude reached during the transient state is significantlyreduced when compared to that in Figure A.4 b.328A.1. Theory of Design and Optimization of a Vibration AbsorberNotably, while the consideration of the damping component for the absorbersin Figure A.6 c and d eventually reduces the main system oscillation tozero, the time taken to achieve this result is much higher (almost twice inFigure A.6 c) than that attained using a tuned system that adheres to theguidelines of vibration absorber design; specifically the mass ratio. However,the amplitude of the vibration in Figure A.4 c and d remains unchangedin Figure A.6 c and d even with the inclusion of the damping property inthe system. These two observations emphasize the importance of optimizedabsorber design in reducing the vibration response of the main system. InTable A.1, the results of this analysis are summarized and the parametersfor each system are provided.Table A.1: Comparison of DVA System performance for amplitude reductionPrimary System m = 1.3kg; k = 1300N/m; c = 3.5Ns/mDVA System Mass (kg) Stiffness (N/m) Amplitude (m) Settling time (ts)a 0.0001 0.0001 0.02 ∞b 0.26 260 0.0034 ∼ 7c 6.5 6500 0.00074 >15d 1.3 1300 0.0016 ∼ 11In addition to the factors highlighted here, another aspect of effective absorberdesign is the motion of the absorber from initial response to steady-state.In Figure A.7 a through to d, the blue lines show the main system responsewhile the red lines represent the motion of the absorber system.329A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.7: Comparison of main system response (blue) to absorber response(red). In (a)No absorber present; (b)Absorber responds to input vibrationimmediately; (c) Absorber and primary system initially oscillate in phasehence longer ts; (d)Absorber and primary system briefly in phase thus slightlylonger ts.The result in a is expected as there is no absorber added to the system. In b,the tuned absorber achieves a consistent motion at the induced frequencymuch sooner compared to c and d. On closer inspection, the absorber motionin b is out of phase with that of the main system within the first time periodof oscillation. This ensures that the motion of the main system is transferredto the absorber and thus reducing the displacement to zero. In c and d,the absorber and main system are in phase for more than one time periodhence the response time is much longer resulting in a delayed reduction ofthe amplitude. It is therefore clear that the optimized absorber in b is theviable solution to reducing the main system oscillations to zero.330A.1. Theory of Design and Optimization of a Vibration AbsorberOptimizing a Dynamic Vibration AbsorberA key assumption made in the analysis is that the inherent damping onlyserves to dissipate the transient response after some time and that bysimply selecting an absorber based on the optimization for the mass andspring stiffness to match the harmonic excitation frequency, the steady-stateresponse can be attenuated. However, this assumption does not hold true fora system where other forms of damping exist. Moreover, the optimizationbased purely on reducing the steady-state response to zero does not accountfor other responses such as the maximum initial amplitude response of thesystem, the time it takes for the transient response to dissipate or otherresponse parameters that are often of importance when designing a dynamicsystem.Generally, configuring the DVA to include a damping element is useful whentargeting multiple response parameters besides the steady-state responseof the primary system. However, the process of optimization is dependenton the targeted response parameter and it is essential to understand thesystem behaviour and constraints before selecting or creating an optimizationprocess. Considering the most general form of a 1 DOF vibrating systempresented in Figure A.1, where the main system parameters m, k, c andabsorber parameters ma, ka, ca are non-zero, the equations of motion arederived using Lagrange’s equations and given by:mx¨+ cx˙− ca (x˙a − x˙) + kx− ka (xa − x) = Foeiωt (A.20)max¨a + cax˙a − cax˙+ kaxa − kax = 0 (A.21)Logically, since the induced motion is harmonic, we suppose the resultingmotion of both the primary and absorber mass to be:Let :{x(t) = Xeiωtxa(t) = Xaeiωt(A.22)331A.1. Theory of Design and Optimization of a Vibration AbsorberSubstituting the motion of the primary and absorber mass into the EOM,the maximum displacement X and Xa can be obtained using the state-spacerepresentation in Equation A.23. icω + ica ω −mω2 + k + ka −ica ω − ka−ica ω − ka ica ω −ma ω2 + ka[XXa]=[Fo0](A.23)The primary goal is to reduce attenuate the amplitude of the steady-statedisplacement of the primary mass. Using the static displacement δ = Fok ,the response of the primary system is normalized such that it becomes thedynamic amplification factor Xδ given by:Xδ=(icaω −mω2 + ka)kmmaω4 + (−icma − imca − icama)ω3+(−cca − kma − kam− kama)ω2 + (icka + ikca)ω + kkaWhile feasible, attempting to optimize the the DVA using this expressionleads to a single frequency optimization. It is convenient to make additionalparametric substitutions as shown below to non-dimensionalize the normalizedsteady-state response.ωm =√km; ωa =√kama; ξm =c2mωmξa =ca2maωa; µ =mamMoreover, it becomes favourable to express the equation in the form of thefraction in Equation A.24 due to the lengthy nature of the resulting equationsand for simplicity when optimizing the DVA parameters [77].Xδ=2iξaA+B2iξaC +D(A.24)332A.1. Theory of Design and Optimization of a Vibration AbsorberWhere:A = µω ωa ωm2B = ωm2ωa2µ− ω2ωm2C = µ2ω3ωa + µω3ξm ωm − µω ξm ωa2ωm + µω3ωa − µω ωa ωm2D = 4µω2ξa ξm ωa ωm + µ2ω2ωa2 − µω4 + µω2ωa2 + µω2ωm2 − ωm2ωa2µ(A.25)In order to optimize the DVA, it is assumed that the effect of the primarysystem damping is negligible, that is ξm << 1. This follows the logic thatsince the DVA is optimized for the steady-state response of the main system,the influence of the inherent damping diminishes with the transient response.Den Hartog [76] and Hao at al. [95] employ this assumption to derive theoptimal DVA parameters of damping ξopta and mass-stiffness in terms ofthe frequency wopta .With this assumption, we introduce the frequency ratiosν = ωaωm and λ =ωωmand normalize the response of the system such that theDVA can be optimized to minimize the steady state response of the system.Thus the normalized steady-state response of the the main system is givenby Equation A.26.A2 =∣∣∣Xδ ∣∣∣2 = (2ξa)2A22 +B22(2ξa)2C22 +D22 (A.26)Where: A2 = λµ νB2 = −µ ν2 + λ2C2 = λ3µ2ν + λ3µ ν − λµ νD2 = −λ2µ2ν2 + µλ4 − λ2µ ν2 − µλ2 + µ ν2(A.27)333A.1. Theory of Design and Optimization of a Vibration AbsorberIn Figure A.2, it was shown that the dynamic amplification factor Xδ crossestwo new resonant frequencies ω1 and ω2. It has been shown that at thesetwo frequencies, the response is independent of the damping factor of theabsorber system [77] and thus convenient in optimizing the frequency ratioν which is often employed when conducting DVA optimization [67, 77, 95, 96].In other words, for the magnitude of the dynamic amplification factor tobe independent of the absorber damping ratio ξa, the relationshipA22C22=B22D22must hold [77]. This ratio is then used to determine the optimal absorberfrequency ωopta in terms of the frequency ratio ν. Consequently, once theoptimal frequency has been determined, the optimal damping factor can bemathematically determined using Equation A.26.As noted earlier, the addition of the absorber damper allows the optimizationof the DVA to target different response parameters for the main system.It follows that for different desired outcomes, or in other words differentoptimization parameters, the optimal ν and ξa of the DVA will differ. InTable A.2, different optimization techniques and their optimal parametersare detailed. Note that the overriding assumption here is ξm << 1.The optimization parameters and final optimal parameters provided in TableA.2 highlight the importance of identifying a desired outcome when designinga DVA. In the case of minimizing the maximum displacement of the primarysystem mass, Den Hartog [76] suggests optimal parameters purely dependenton the mass ratio µ. In fact, majority of the optimization techniques resultin optimal parameters that are related to the mass ratio. For the design ofa DVA, the damping ratio ξm of the main system must be calculated anddetermined to be much below critical damping before any of the optimizationtechniques identified here can be used. Then, an appropriate mass ratio thatadheres to conventional design can be selected and the resulting stiffness anddamping components of the absorber can be selected.334A.1.TheoryofDesignandOptimizationofaVibrationAbsorberTable A.2: DVA Optimization techniques for a SDOF where ξm << 1; ξa is the optimal damping and ν is theoptimal frequency ratio ωaωmObjective Optimization parameter Optimal Parameters Proposed by:Minimize themaximumdispalcement ofprimary massAmax =∣∣∣∣Xδ∣∣∣∣maxν = 11+µ ; ξa =√3µ8(1+µ) Den Hartog[76],Nishiharaand Asami[97]Minimize therelativedisplacementbetween the mainand absorbermassAmax =∣∣∣∣Xδ∣∣∣∣max; Aamax =∣∣∣∣Xa−Xδ∣∣∣∣maxν = 11+µ ; ξa =√µ2(1+µ) Krenk [77]Minimize thepower dissipatedby the absorberIp2 =c2E[|x˙1−x˙2|2]2πSfω1/k1ν ≈ 1√1+µ; ξa ≈√µ2 Zilletti at al.[67]335A.1. Theory of Design and Optimization of a Vibration AbsorberIf we consider the system presented earlier where the mass m = 1.3kg, thestiffness k = 1300N/m and the mass ratio µ = 0.2. The natural frequency ofthis system ωm = 5.033 Hz and the driving harmonic force has a frequencyω such that it causes the system to resonate. Using the parameters in TableA.2, the optimal stiffness ka and damping ca of the absorber are determinedans summarized in Table A.3. For simplicity in calculation, recall:ν =ωaωm=1√µ√kak; ξa =c2maωa; µ =mamTable A.3: Absorber stiffness ka in N/m and damping ca in Ns/m resultingfrom the presented DVA Optimization techniquesOptimization Method Optimal Parameters Design ParametersDen Hartog, Nishihara and Asamiν = 0.833 ka = 180.41ξa = 0.25 ca = 3.424Krenkν = 0.833 ka = 180.41ξa = 0.289 ca = 3.959Zillettiν = 0.913 ka = 216.73ξa = 0.224 ca = 3.068Using the parameters calculated in Table A.3, the resulting optimized DVAfrom each technique is evaluated when the system is at resonance. For thiscomparison, the main system is taken to have an effective damping ratio ofξm = 0.01 such that its damping coefficient c = 0.822 Ns/m. In Figure A.8,the results of this analysis are presented.336A.1. Theory of Design and Optimization of a Vibration AbsorberFigure A.8: Comparison of optimized DVA response using differentoptimization methods. In (a)No absorber (resonant case) (b) Den Hartog,Nishihara & Asami minimum displacement optimization; (c)Krenk’s relativedisplacement optimization; (d)Zilleti et. al minimizing dissipated power.Observing the results in Figure A.8, the optimized systems have a loweramplitude of response A = 5×10−3m compared to the non-optimized systemA = 0.1m. However, the optimized systems still exhibit oscillations. It isimportant to note here that an optimized DVA does not always eliminate allvibrations but often seeks to attenuate them to an acceptable level. Moreover,from Table A.2, the objective of each optimization method is defined andoften the impact of an optimized DVA is not readily evident from the resultingmotion of the primary mass and measuring other parameters is necessary toobserve the effectiveness. For instance, the method proposed by Zilleti atal. [67] seeks to minimize the power dissipated by the absorber and withoutcalculating the parameter Ip2, the performance of the DVA is not significantly337A.1. Theory of Design and Optimization of a Vibration Absorberdifferent to that optimized through the method proposed by Krenk [77] asevident in Figure A.8.Besides mathematically determining the optimal parameters ν and ξa, it isalso possible to achieve an optimized DVA through a parametric study of theabsorbers stiffness and damping values to the main system response. Often,this procedure is desirable where closed-form solutions of the EOM are notachievable or where the mathematically intensive procedure to generate theformulaic relationship is unnecessary. This option is especially useful whendealing with optimization problems involving multiple degrees of freedom ornonlinear systems with conflicting optimal parameters.Finally, the theory and design of a vibration absorber discussed here formthe basis for the design of an effective tremor suppression device. However,there are some fundamental shortcomings of the vibration absorber systempresented above which include:1. The value of the main system natural frequency must be know to areasonable degree of certainty for the optimized absorber to functionas intended. Moreover, the introduction of two new resonant frequencyposes an operational challenge where a shift in frequency may causeresonance. One alternative to combat the new resonance peaks is tointroduce a damping element in the absorber as shown in the generalform. However, not only involves extensive mathematical work but theresulting system still has residual vibration which may require furtherdesign adjustments to completely isolate.2. The use of linear springs in the system above means only a singlenatural frequency is targeted and the effective bandwidth is narrowespecially when a lower mass ratio is used due to limited rattle space.338A.2. T-Beam Pivot ChangeA.2 T-Beam Pivot ChangeThe older T-Beam has the pivot at the bottom. Stability at equilibriumhas been noted as a challenge especially under large tremor amplitudes.Additionally, the original variation of the T-Beam reacts to smaller amplitudeswhich may impede regular movements. To combat these challenges, aproposed design modification is the pivot point is placed at the top of the Tshown in Figure A.9.Figure A.9: New T-Beam Model with the pivot placed at the top to addressstability and response to volitional movements.The derivation of the equation of motion begins by displacing the systemfrom equilibrium show in Figure A.10:339A.2. T-Beam Pivot ChangeFigure A.10: New T-Beam Model displaced from its equilibrium. Notice thedefinition of the displacement θ.The FBD of this displacement from equilibrium is given by Figure A.11:Using the free body diagram, the EOM can be derived using Newton’s secondlaw for rotating systems to be:I θ¨ + k1 a2 sin θ cos θ + k2 a2 sin θ cos θ + c θ˙ =M (t)Using the double angle identity that sin 2θ = 2 sin θ cos θ:I θ¨ +k12a2 sin 2θ +k22a2 sin 2θ + c θ˙ =M (t)Where: I = 2 mT a2340A.2. T-Beam Pivot ChangeFigure A.11: New T-Beam Model Free Body Diagram.We can simulate the response of this configuration using Matlab’s built inRK4 implementation. Note here that the initial position of θ0 = 0 whereas inthe original T-Beam, θ0 =π2 . Additionally, we do not make the small angleapproximation sin(θ) ≈ θ immediately without first observing the responsesince certain inputs or disturbances can cause deviations which violate theconditions necessary for the small angle approximation when deriving theEOM above. For the sake of continuity, we use the same optimal parametersof the T-Beam obtained in the optimization covered in Chapter 5.1. Notethat the simulation of the individual T-Beam response is for a sanity check toensure correct modeling of the T-Beam. Figure A.12 shows the comparisonin response to an equal tremor for similarly configured old and new T-Beam.Their initial unoptimized performance is an RMS tremor reduction of 75%.341A.2. T-Beam Pivot ChangeFigure A.12: Comparison of the unoptimized performance of the Old (top)and New (bottom) T-Beam to qualify changing pivot location.342APPENDIXBNon-Linear WorkB.1 Krylov-Bogoliubov MethodWe consider that the solution to the variable y (t) is given by:y (t) = n (t) cos (τ) +m (t) sin (τ)Where the time dependence of n = [n1 n2]T and m = [m1 m2]T s assumednegligible for higher order terms. Additionally, the KB method takes thesolution form above with the condition that:y˙ = −n (t) sin (τ) +m (t) cos (τ)However, we know that:y˙ = −n˙ (t) cos (τ)− n (t) ω sin (τ) + m˙ (t) sin (τ)−m (t) ω cos (τ)343B.1. Krylov-Bogoliubov MethodThis means that the following must hold true when we impose the conditionfrom the KB method:− 1ωn˙ (t) cos (τ) +1ωm˙ (t) sin (τ) = 0Now, we find the second derivative given the condition to be:y¨ = −n˙ (t) sin (τ)− n (t) ω cos (τ) + m˙ (t) cos (τ)−m (t) ω sin (τ)Substituting all these equations into the original EOM and collecting liketerms, we obtain (we will drop the time notation to simplify the equationwriting process):(−ω M n+C m+K n+M m˙) cos (τ)+ (−ω Mm−C n+Km−M n˙) sin (τ)= f (n,m, τ) . (B.1)The other equation is our condition:− 1ωn˙ cos (τ) +1ωm˙ sin (τ) = 0With these two equations, we need to solve for n˙ and m˙ . This is donethrough simultaneous equation solution by multiplying the EOM by 1ω sin (τ)and the condition equation by M cos (τ). The two equations are thus:1ω(−ωMn+Cm+Kn+Mm˙) sin(τ) cos(τ)...+1ω(−ωMm−Cn+Km−Mn˙) sin2(τ) = 1ωsin(τ)f(n,m, τ)− 1ωn˙ M cos2 (τ) +1ωm˙ M cos (τ) sin (τ) = 0344B.1. Krylov-Bogoliubov MethodSubtract the resulting two equations:1ω(−ωMn+Cm+Kn) sin(τ) cos(τ) + 1ωMm˙ sin(τ) cos(τ)...+1ω(−ωMm−Cn+Km−Mn˙) sin2(τ) + 1ωn˙M cos2(τ)...− 1ωm˙M cos(τ) sin(τ) =1ωsin(τ)f(n,m, τ)Simplify the common terms out and collect like terms:1ω(−ωMn+Cm+Kn) sin(τ) cos(τ) + 1ω(−ωMm−Cn+Km) sin2(τ)...− 1ωMn˙ sin2(τ) +1ωn˙M cos2(τ) =1ωsin(τ)f(n,m, τ)Determine n˙:Mn˙ = − sin(τ)f(n,m, τ) + (−ωMn+Cm+Kn) sin(τ) cos(τ)+ (−ωMm−Cn+Km) sin2(τ)Following similar steps, we can determine m˙ such that:Mm˙ = − cos(τ)f(n,m, τ)− (−ωMn+Cm+Kn) cos2(τ)− (−ωMm−Cn+Km) sin(τ) cos(τ)To obtain the equations for n˙ and m˙, we integrate over the period(τ ∈ [0, 2π])per the KB method. Note here that n and m are taken as time invariantover the period and we are averaging them.345B.2. Perturbation MethodsB.2 Perturbation MethodsDefinition: In mathematics and applied mathematics, perturbation theorycomprises methods for finding an approximate solution to a problem, bystarting from the exact solution of a related, simpler problem. A criticalfeature of the technique is a middle step that breaks the problem into”solvable” and ”perturbative” parts. In perturbation theory, the solution isexpressed as a power series in a small parameter .The first term is the knownsolution to the solvable problem. Successive terms in the series at higherpowers of usually become smaller. An approximate ’perturbation solution’is obtained by truncating the series, usually by keeping only the first twoterms, the solution to the known problem and the ’first order’ perturbationcorrection.B.2.1 IntroductionPerturbation theory is concerned with the art of approximations. In a casualsense (or more generic should we say), we take a problem that we know thesolution to and use a problem that is perturtably close (nearby problem)that we know the solution to help understand the solution profile of a morecomplex problem. These complex problems may be solved using simulationsand numerical methods but we would like to have some form of analyticalunderstanding of the problem. Thus, perturbation theory is concerned withtaking a problem and perturbing it to observe the nature of impact of theperturbation on the problem. Often, the problem we are perturbing is linearand we want to observe, say if the problem has a weak non-linearity, howthe weak non-linearity changes the problem. Therefore, perturbation is theart of approximation (in common tongue). There are numerous perturbationmethods devised to quantify the effect of perturbations and we will reviewthem here begining with regular perturbation. Since perturbations areconcerned with small changes (small parameters), often the problem isexpanded into a Taylor-like series thus we will review common power seriesexpansion before continuing with the review of perturbation methods.346B.2. Perturbation MethodsB.2.2 Regular PerturbationPerturbation methods are applicable to both IVPs and BVPs. For instance,we consider the experessions below where the left hand side L u is equal tosome perturbative right hand side term such that:L u = ϵ F (u, ut, · · · , t) t ∈ [0,∞)ICs : u (0) = A;ut (0) = BHere, L represents a generic linear differential operator which is time-dependentand exists on some prescribed domain [0,∞) in the case of an IVP or [0, l] forthe case of a BVP. On the right hand side, we consider F to be a non-linearforcing function and ϵ << 1. The parameter epsilon is small and that is thetrick here (we have a small perturbation). When:− ϵ = 0; we have a simple linear problem whose solution we can easilyfind.− ϵ≪ 1 ̸= 0; we can begin to observe the impact of the non-linear forcingfucntion on our linear system and there is still potential to use thesolutions obtained when ϵ = 0.This method is also applicable for BVPs as shown below with the boundaryconditions. Again when ϵ = 0, the solution may be very simple to the linearoperator problem over the prescribed domain but as becomes non-zero, theright hand side begins to introduce some complex dynamics which could benon-linear making the problem challenging.L u = ϵ F (u, ux, uxx, · · · , x) x ∈ [0, l]BCs : α1 u (0) + β1 ux (0) = 0α2 u (l) + β2 ux (l) = 0So the main idea is to develop a suite of perturbation techniques that allowfor the investigation of either IVPs or BVPs and to understand how theperturbation influence/change the solutions that we have at some place wherewe know how the system behave i.e. we solve the solution for the case whenϵ = 0 and through perturbation, begin to understand how the solution weinitially obtained is affected as we perturb it around that point by makingϵ ̸= 0 but≪ 1347B.2. Perturbation MethodsConcept: Therefore, the main idea is expansion. This means we say thatour solution (whether IVP or BVP) is a series of cascading solutions ofdifferent orders such that:u = uo + ϵ u1 + ϵ2 u2 + ϵ3 u3 · · ·ϵ≪ 1Here, the term uo is called the leading order solution obtained from the casethat ϵ = 0 and the idea is to build the series of cascading solutions with thehope that uo is the dominant term in the observed behaviour and we add onthe ”corrections” ϵi ui. Here the term ui is some function (yet to be solved)but ϵ is scallar and can vary in size. When ϵ is small, this means that theassociated impact of the additional ui is an order of magnitude smaller thanthe leading order solution uo. Therefore, what perturbation does is itapproximates the solution using the leading order solution uo andimproves the approximation through addition of the correctionterms ei ui.This holds for ϵ << 1.There are concerns whether the cascading series u = uo+ϵ u1+ϵ2 u2+ϵ3 u3 · · ·converges and infact often in perturbation theory, there is no convergence.This means unlilke a Taylor series expansion about a point where a finitevalue exists, the expansions in perturbation theory often create divergingseries expansions and we would need to investigate the properties of thefunctions uo, u1, u2, · · · to determine convegence. Often times however, weonly compute the first or second correction to the leading order soluton. Solet us look at the an example to explain Regular Perturbation.ut t − u = ϵ u2u (0) = A;ut (0) = BWe have a linear problem ut t − u with some perturbatively added quadraticnon-linearity ϵ u2. We also have some initial position and initial velocityconditons to our problem. The solution begins by expanding our non-linearproblem into a hierachy of linear problmes as shown by the expansion andsubsequent collection steps summarized below:u = uo + ϵ u1 + ϵ2 u2 + · · ·348B.2. Perturbation MethodsApplying this expansion to our problem up to O(ϵ2), we end up with:(uo t t + ϵu1 t t + ϵ2u2 t t +O(ϵ3))−(uo + ϵu1 + ϵ2u2 +O(ϵ3))=[ϵ(uo + ϵu1 + ϵ2u2 +O(ϵ3))]2. (B.2)Expanding the problem out and dropping terms O(ϵ3), we obtain:uo t t + ϵ u1 t t + ϵ2 u2 t t − uo − ϵ u1 − ϵ2 u2 = 2 ε2 u1 uo + ε uo2Where the O(ϵ3)= ε5 u22 + 2 ε4 u1u2 + ε3 u12 + 2 ε3 u2 uo.Now, we collect the terms based on the order and we obtain:O (1) L uo = uo t t − uo = 0 uo (0) = A, uo t (0) = BO (ϵ) L u1 = u1 t t − u1 = u2o u1 (0) = 0, u1 t (0) = 0O(ϵ2)L u2 = u2 t t − u2 = 2 u1 uo u2 (0) = 0, u2 t (0) = 0Note here that the initial conditions are not expanded as the process is quitesimilar and should be self evident at this point. Notice here that we haveesentially built a heirachy of cascading linear problems where we can solvefor uo which is then used to solve for u1 and combine the two to solve foru2. So, at every order, we rely on the previous order’s solution to solve it.This is how perturbation theory works and is a generic feature ubiquitous inperturbation problems.Now, getting into the solution of this problem, the leading order solution (i.e.we do not perturb the problem) is given by:349B.2. Perturbation Methodsuo =(A−B2)exp−t +(A+B2)exptWhen we begin considering the perturbation, the first order correction ifgiven by (please do not do by hand unless entirely necessary, at this pointyou know the mathematics of solving ODEs just use a software):u1 = exp−t(16A2 +13AB − 13B2)+ expt(−13AB +16A2 − 13B2)+ · · ·+(A−B)2 exp−2t12+(A+B)((A+B) exp2t−6A+ 6B)12Notice the following:∗ There are some terms with the same order of exponent as the leadingorder solution.∗ There are terms with the exponent taking the form exp (2t) which isas a result of the u2 driving term. Since the leading order solution isthe driving term of the first order correction, the quadratic term showsup here in the solution form.Now, our solution for u (t) is given by uo (t)+ϵ u1 (t). Notice how complicatedu1 (t) is. In fact, the solution to the second order correction u2 is quitecomplicated and generally the solution process in perturbation ends uphaving each step more complicated than the last in terms of solving aninhomgeneous problem and the computation time and power is intensivewith each correction. Nonetheless, u2 is given by:u2 = exp−t(− 1288A3 − 35288A2B +85288AB2 − 25288B3)+ . . .+ expt(− 1288A3 +85288AB2 +35288A2B +25288B3)+ . . .+196(20(A+B)(t+12)(A−B)2 exp2t−20(A+B)2(A−B)(t− 12)exp4t+16(A+B)(A2 − 2AB − 2B2) exp5t3+ (A+B)3 exp6t−32A3 exp3t350B.2. Perturbation Methods+16(A−B)((A2 + 2AB − 2B2) expt+3(A−B)216)3 exp−3tNotice that u2 is quite complicated and that for very small values of theperturbation parameter ϵ, it is almost unnecessary to compute higher ordercorrections. We can demonstate this using a plot of the solution in matlabfor different values of ϵ in Figure B.1.Figure B.1: Comparing leading order, first correction, and second correctionsolutions for regular perturbation. Leading order and first correction solutionare virtually equal in the window.351B.2. Perturbation MethodsThe results indicate that the addition of the first correction has an impact onthe approximation of the solution while the addition of the second correctionis not as significant for small perturbations ϵ. Additionally, notice thatfor t ∈ [0, 3. 5), the leading order solution and the first and second ordercorrection are virtually indiscernible.B.2.3 Poincare-Lindstedt MethodA common theme with all mathematical methods is that they have limitationsto what they can compute either because they fail to capture the behaviourof the system or introduce components (mathematical artefacts) in thesolutions that are not observed in practice or they are inefficient in solving theproblem at hand. For instance, the regular perturbation expansion introducedso far is insufficient when dealing with problems where non-linear behaviourexists is inherent to the system. The reason for this will be evident throughthe examples that follow. We can navigate this through modifications inorder to accommodate the real behaviour that occurs in many non-lineardynamic systems. One such modified method is the Poincare-Lindstedtmethod which introduces the concept of renormalizing or stretchingtime in order to accommodate the impact of non-linearity in a dynamicsystem which causes frequency shifts and generates harmonics.Recall here that the idea behind perturbation is to use a nearby easierproblem to solve a much more difficult problem i.e. something we do nothave an analytical solution to but we approximate through a hierachy ofproblems we can solve. Note here that it’s not that we cannot solve theproblem through computation but perturbation allows us to understand thefundamental mechanisms driving the observed behaviour as is the case fornon-linear systems. With perturbation, we can trade the complex non-linearproblem for a cascading hierachy of linear problem. This hierachy is usefulin helping understand the underlying mechanisms behind complex systemsfor which closed-form solutions do not exist.352B.2. Perturbation MethodsBreakdown of regular perturbation expansionsLet us consider the example below to showcase the breakdown of regularperturbation expansion. Note that this is an oscillator but note the positionof the small perturbation parameter ϵ:ut t + (1 + ϵ)2 u = 0u (0) = A;ut (0) = 0The solution to this is a combination of sines and cosines since its a linearoscillator. By satisfying the initial conditions, the are eliminated since motionbegins at a non-zero position. As a mater of fact, the exact solution to thisproblem is given by:u (t) = A cos((1 + ϵ) t)Notice here that the perturbation parameter ϵ causes a shift in frequencyof oscillation. However, this is a problem for regular perturbation theorysince it generates the approximate solution through expansion of u = uo +ϵ u1 + ϵ2u2+. . . which when expanded and collected becomes:(uo t t + ϵ u1 t t + ϵ2u2 t t)+ (1 + ϵ)2(uo + ϵ u1 + ϵ2 u2)= 0uo t t+ϵ u1 t t+ϵ2u2 t t+ϵ4 u2+ϵ3 u1+2 ϵ3 u2+2 ϵ2 u1+ϵ2 u2+ϵ2 uo+ϵ u1+2 ϵ uo+uo = 0Now, we collect terms with respect to orders for all powers excluding O(ϵ3).This is thus given by:O (1) uo t t + uo = 0 uo (0) = A, uo t (0) = 0O (ϵ) u1 t t + u1 = −2 uo u1 (0) = 0, u1 t (0) = 0O(ϵ2)u2 t t + u2 = −2u1 − uo u2 (0) = 0, u2 t (0) = 0353B.2. Perturbation MethodsThis hierarchy of problems has been traded in for the ”challenging” problemwe have above and we now solve it at each level. From the leading orderproblem and the corrections, the leading order solution is given by:uo (t) = A cos (t)Notice that the leading order approximation is close to the exact solutionbut fails to outright capture the frequency shift caused by theperturbation parameter. Now the next order correction is given by:u1 t t + u1 = −2 A cos (t)This is a problem here since the right hand side is a cosine term thatis in the null space of the adjoint of the linear operator on the lefthand side and thus not satisfying the Fredholm Alternative Theoremand violating solvability. If we attempted to correct for solvability here,the initial condition has to be A = 0 which is the trivial solution. If we wentahead and solved the first correction while ignoring the solvability condition,we end up with the following:u1 (t) = A t sin (t)This is a problematic solution, considered a secular growth solution sinceas t→∞, u1 (t)→∞. This a consequence of violating solvability andthe correction ”blows up” figuratively since its tends to grow while the exactsolution is clearly well behaved and does not exhibit this behaviour. The exactsolution shows that the perturbation parameter causes a frequency shift butthe regular perturbation method, being incapable of accomodating frequencyshifts, implies secular growth where it does not exist as a consequence ofviolating solvability. The secular grown seen here is completely fictitious i.e.a mathematical artefact and that is problematic as we would like for theperturbation to ease our problem solving not introduce non-existent solutions.If we continue down this path, we can see that the second correction is givenby:u2(t) = −A sin (t) + A cos (t)2− A(t sin (t) − t 2 cos (t) + cos (t))2354B.2. Perturbation MethodsNotice again the presence of secular growth here which is not the true natureof the problem, which in this case is a frequency shift and this highlightsthe failure of regular perturbation which is that it is incapable of handlingfrequency shifts which are present in non-linear problems. Remember thatwhen you perturb a non-linear problem, it causes frequency shift and generatesharmonics. We can plot the exact solution versus the failure of regularperturbation in Figure B.2.Figure B.2: Comparing the exact solution and the solution using the secondorder regular perturbation solution showing secular growth.355B.2. Perturbation MethodsMethodology of the Poincare-Lindstedt ApproachGiven that regular perturbation expansion cannot accomodate frequencyshift, we consider the Poincare-Lindested method which is an old methodoriginally used to compute frequency shifts of celestial mechanics to makecorrections to planetary orbits periods by having perturabations from othergravitational forces from other planets. The idea being that the influence ofother planets’ graviational forces would not cause a planet period to tend toinfinity but just frequency shifts in motion.The main idea behind this method is to consider a spatial temporal system andintroduce the concept of time stretch which allows for perturbing frequency.spatial temporal system - system defined in both space and time i.e. atime-space variant system.The regular perturbation expansion considered the expansion parameter tobe time but since we have seen that non-linearity causes frequency shifts, weintroduce a stretched time parameter such that:u = uo (x, τ) + ϵ u1 (x, τ) + ϵ2 u2 (x, τ) + · · ·Here, τ is the stretched time parameter such that τ = ωt. This allowsfor perturbing the frequency such that:τ =(ωo + ϵ ω1 + ϵ2ω2 + · · ·)tNotice that the by perturbing the parameter τ , we allow for time shift basedon the definition which ultimately allows for frequency shift i.e. we canperturb the frequency. The correction frequencies in the perturbation arethen obtained by eliminating terms in our cascading hierarchy of equationsthat violate sovlability via the Fredholm Alternative Theorem. This meansthat we will solve for the frequencies required to make the terms causingsecular growth to decay to zero which in turn tells us what the frequency shiftintroduced due to the non-linearity will be (a neat mathematical procedurewhen well done). Let us observe this through the example of a perturbedDuffing oscillator.356B.2. Perturbation MethodsSimple Duffing PerturbationConsider the example of the pendulum equation modeled by the Duffingequation below. This expansion is obtained by considering the second termin the expansion of sin u ≈ u− u33! + u55! + · · ·ut t + u+ ϵu3 = 0Note here that the perturbation parameter is applied to the cubic non-linearityas we intend to understand the dynamics of this system for small non-linearinfluences. The leading order solution is clearly an oscillator when theperturbation parameter is ϵ = 0. Recall the statement that in dynamicsystems, the presence of non-linearities generates harmonics and shiftsfrequencies which we will observe through this example.Let f (u) = −u3 and the solvability condition for the first correction (O (ϵ))becomes:O (ϵ) :〈f (uo)− 2 ω1 uo τ τ , sin τ〉= 0 =⇒〈A3 sin3 τ − 2 ω1 A sin τ, sin τ〉= 0Now, we intend to make this orthogonal to the adjoint of the null space ofthe operator i.e. the inner product above needs to be zero. It is importantto understand trig identities here to make the solution process convenient.Note for instance:sin3 τ = −sin (3 τ)4+3 sin (τ)4Now, we note that and are orthogonal to each other. This an inherentproperty of sinusoids. From this property, we can see that the only problemleading to secular growth are the sin τ terms. But, we have the other term−2 ω1 A sin τ which we can use to negate the impact of the secular growthterm. We collect the sin τ terms such that the inner product becomes (notethat we reduce the powers of A since the right hand side is zero):〈(2 ω1 − 3A24)A sin τ +A34sin 3τ , sin τ〉= 0357B.2. Perturbation MethodsWe know already that the sin 3τ term is orthogonal to sin τ so we nowidentify the value for ω1 for the coefficient to remove secular growth bysatisfying solvabilty. With simple algebra, we obtain that:ω1 =38A2This eliminates the sin τ term and no secular growth occurs. This is thecorrection to the frequency. The remaining term in the right hand sideforcing can now be used to solve for the first correction as follows:O (ϵ) : u1 τ τ + u1 =A34sin 3τThe resulting first order correction becomes:u1 (τ) =3A3 sin (τ)32− A3 sin (3 τ)32ω1 =38A2Notice that the resulting correction contains harmonics (sin τ & sin 3τ)and the frequency is shifted by ω1. This is the effect of non-linearity inthe dynamics of the problem. Notice that the harmonics are generated forodd powers i.e. sin τ, sin 3τ, sin 5τ, · · · . This is because the non-linearityis cubic not quadratic. This is the power of the Poincare-Lindstedt methodin that by stretching time, we are able to produce bounded solutions withfrequency shifts which is what happens in reality.we can solve for the second order correction through similar steps as thefirst one by substituting the now known variables to satisfy solvability bydetermining the second correction to frequency ω2. This is tedious and infact most approximations usually take the first correction for systems withweak non-linearity.O(ϵ2):〈u1∂f (uo)∂u− 2 ω1 u1 τ τ −(2 ω2 + ω21)uo τ τ , sin τ〉= 0358B.2. Perturbation MethodsWith the first correction, the approximate solution to the unforced Duffingoscillator becomes(we substitute τ = ωt back into the approximation here):u (τ) = A sin τ + ϵ[3A3 sin (τ)32− A3 sin (3 τ)32]∴ u(t) =A sin(1 + ϵ3A28)t+ϵA3323 sin(1 + ϵ3A28)t− sin3(1 + ϵ3A28)t . (B.3)Notice here the presence of harmonics and frequency shifts as a noted impactof non-linearity. Additionally, the noted frequency shift is small given thatϵ << 1 but it is important to capture it because it becomes notable withtime. If we were to attempt a solution using regular perturbation, the resultwould be as follows:u (t) = A sin t+ ϵ[A3 ( sin 3t− sin t)32− 3 A8t sin t]Notice the presence of the secular term which is easily observable in FigureB.3. From the result, it is evident that the regular perturbation method’blows up’ i.e. the solution grows infitely while the Poincare-Lindstedt solutionis bounded and oscillates normally.359B.2. Perturbation MethodsFigure B.3: Comparison of Poincare-Lindstedt, Regular Perturbation, andNumerical Solutions. Regular perturbation shows secular growth, whilePoincare-Lindstedt first correction closely matches the numerical solution.Furthermore, comparing the Poincare-Lindstedt solution to the exact solution,a discrepancy is observed which can attributed to:∗ The Poincare-Lindstedt perturbation is a first correction which meanswe only capture the first correction to frequency. Thus, we notice thatthe frequency is somewhat misaligned to the true solution.∗ The amplitude of the Poincare-Lindstedt is slightly lower than thenumerical solution since once again it is a first correction. Highercorrections would improve the accuracy of our perturbation.360B.3. The Duffing OscillatorHowever, we perform perturbation analysis not necessarily to simulate systemresponse but to study overall behaviour. Similar to regular perturbations,Poincare-Lindstedt is limited to handling only frequency shift (all techniqueshave their limitations). There are other phenomenon for study that thismethod becomes insufficient. There are other perturbation techniques such asthe Multiple Scales expansion, The Harmonic Balance method and a plethoraof averaging methods one of which is the Krylov-Bogoulibov technique coveredin Chapter B.1. For the scope of this work, the methods presented so far aresufficient for analysis.B.3 The Duffing OscillatorThe general damped-driven Duffing Oscillator is modeled by:ut t + δ ut + α u+ κ u3 = γ cos ωtWhere α = ω2o and δ = 2 ζωo.Note: The parameter κ determines whether the non-linear spring is ahardening or softening spring. If κ > 0 it is a hardening spring meaning itgets stiffer with larger amplitudes and if κ < 0 it is a softening spring so itbecomes less stiff at larger amplitudes.B.3.1 Determining the FrequencyWe will the method of Harmonic Balance to study the response of the DuffingOscillator in the frequency domain and understand the impact of non-linearityto the behaviour of the response. Applying the harmonic balance method,we obtain:u = a1 cos ωt+ b1 sin ωtut t = −a1 ω2 cos ωt− b1 ω2 sin ωtut = −a1 ω sin ωt+ b1 ω cos ωtPutting it all together we obtain:361B.3. The Duffing Oscillator(− a1ω2 cosωt− b1ω2 sinωt)+δ(− a1ω sinωt+ b1ω cosωt)+α(a1 cosωt+ b1 sinωt)+κ(a1 cosωt+ b1 sinωt)3=γ cosωtThis expression is expanded and we collect the harmonics to obtain:(− a1ω2 + δωb1 + αa1 + 34κa31 +34κa1b21 − γ)cos(ωt)+(14κa31 −34κa1b21)cos(3ωt)+(− b1ω2 − δωa1 + αb1 + 34κa21b1 +34κb31)sin(ωt)+(34κa21b1 −14κb31)sin(3ωt) = 0We neglect the higher order harmonics and balance the harmonics for thesin (ωt) and cos (ωt) such that:For cos ωt:−a1ω2 + δωb1 + αa1 + 34κa31 +34κa1b21 − γ = 0∴(α− ω2)a1 + δ ω b1 +34κ(a31 + a1 b21)− γ = 0For sin ωt:−b1ω2 − δωa1 + αb1 + 34κa21b1 +34κb31 = 0362B.3. The Duffing Oscillator∴(α− ω2)b1 − δ ω a1 + 34κ(b31 + b1 a21)= 0We are now left with two algebraic equations and two unknowns a1, b1. Theequations are summarized below:(α− ω2)a1 + δ ω b1 +34κ(a31 + a1 b21)− γ = 0(α− ω2)b1 − δ ω a1 + 34κ(b31 + b1 a21)= 0One of the advantages of this approach is that we can formulate this as acontinuation problem with ω as a free parameter so we can observe how thesolution evolves with ω. We would like to solve the two equations in termsof a1, b1, ω to obtain our response. We transform to polar coordinates fora1 and b1 such that:a1 = p cos θ & b1 = p sin θWhere a21 + b21 = p2. We substitute into our equations and obtain:(α− ω2)p cos θ+δ ω p sin θ+34κ((p cos θ)3 + p cos θ (p sin θ)2)= γ(α− ω2)p sin θ−δ ω p cos θ+ 34κ((p sin θ)3 + p sin θ (p cos θ)2)= 0The resultant is:(−ω2p+ α p+ 34κp3)cos θ + δ ω p sin θ = γ · · · · · · (1)(−ω2p+ αp+ 34κp3)sin θ − δ ω p cos θ = 0 · · · · · · (2)We square the two equations and subtract them to get rid of the sin θ andcos θ terms to obtain:363B.3. The Duffing Oscillator(α− ω2)p+34κ p3 + δ2 ω2 p2 = γ2We can rewrite to isolate for ω2 as follows:ω2 = α(− 1p δ2 − 1 +γ2 − 34p3 κ αp α(p δ2 − 1))We can use perturbation expansion of O (ϵ) to determine the frequency andplot the response. This is done as follows with α = ω2o :ω = ωo + ϵ ω1ω2 = (ωo + ϵ ω1)2 = ω2o + 2 ϵ ωo ω1 + ϵ2 ω21Discard the O(ϵ2)term to obtain:ω2o + 2 ϵ ωo ω1 = ω2o(− 1p δ2 − 1 +4 γ2 − 3 p3 κ ω2o4 p ω2o(p δ2 − 1))We solve for ω1 to get:ω1 =ω2o(− 1pδ2−1 +−3p3κω2o+4γ24pω2o(pδ2−1))− ω2o2ϵωoFinally, the frequency ω = ωo + ϵ ω1 is given by:ω = ωo +ω2o(− 1pδ2−1 +−3p3κω2o+4γ24pω2o(pδ2−1))− ω2o2ωoWhich is simplified to become:ω =(4δ2p2 − 3p3κ− 8p)ω2o + 4γ28ωop(pδ2 − 1)364B.3. The Duffing OscillatorNow that we have the solution for the frequency shift, we can observe thefrequency response for a hardening and softening spring. For a hardeningspring where κ > 0 and a softening spring where κ < 0, Figure B.4 shows thefrequency responses for weakly non-linear and extreme non-linear systems.We can notice bifurcations for extremely non-linear cases.Figure B.4: Frequency Response of the Duffing Oscillator for a Hardeningand Softening Spring where κ ≈ 0 i.e. weakly non-linear system and for|κ| >> 0 i.e. extreme non-linearity.365APPENDIXCModeling CodeC.1 Simplified Arm Model1 %% ARM 1 DOF SYSTEM2 % by Stephen Kimanzi34 % This code represents the simulation of the human arm for SFE5 % DOF. All other DOF fixed i.e. the arm is modeled as a6 % rotating beam pivoting at the shoulder. This system has been designed to7 % have a maximum amplitude of 1.5 degrees (0.02625 radians) oscillating at8 % 6 Hz with a settling time of 2s. Tremor is modeled as a harmonic force9 % and the parameters of maximum force, system stiffness and damping10 % coefficient are chosen to meet the specs above.1112 clc13 clear1415 %% Initialize system parameters1617 global Mo keq ceq w meq1819 m uf = 3.23; %mass of upper arm and forearm20 mp = 0.54; %mass of palm21366C.1. Simplified Arm Model22 l = 0.866; %total arm length23 Fo = 29.5421;2425 m = (m uf/3) + mp;26 meq = m*lˆ2;2728 k = 1021.175;29 keq = k*lˆ2;3031 c = 6.4865;32 ceq = c*lˆ2;33 w = 2 * pi() *6; %driving frequency3435 Mo = Fo *l;3637 % determine natural frequency to check for accuracy38 wn = sqrt(keq/meq);39 fprintf(['The natural frequency of this system is...40 ',num2str(wn/(2*pi())),'Hz.\n']);4142 zeta = ceq/(2*sqrt(keq*meq));43 fprintf(['The damping ratio of this system is ',num2str(zeta),'...44 with a settling time of ',num2str(4/(zeta*wn)),'s.\n']);4546 %% Define time step to be at least 20 times the maximum frequency47 %intialize time and response vectors4849 max freq = max(w,wn);5051 dt = (1/20)*((1/max freq)/(2*pi())); % time step52 t = 0:dt:7; %time vector with steps5354 y0=[0 0]; %initial conditions with velocity then displacement55 y = zeros(2,length(t)); %empty response vector56 y(1,1) = 0;57 y(2,1) = 0;58 h = dt; %step size for rk method5960 %% Numerically solve response of the system6162 y prime = @(t,y)odefun(t,y); %calls the state space function6364 for i = 1:(length(t)−1)6566 k1 = y prime(t(i),y(:,i));67 k2 = y prime(t(i)+.5*h,y(:,i)+.5*k1*h);68 k3 = y prime(t(i)+.5*h,y(:,i)+.5*k2*h);69 k4 = y prime(t(i)+h,y(:,i)+k3*h);70 y(:,i+1) = y(:,i)+((k1+2*k2+2*k3+k4)/6)*h;367C.1. Simplified Arm Model7172 end7374 %% plot results7576 %read in ADAMS Data for comparison77 [D,¬,¬] = xlsread('AdamsConfirmData.xlsx','Run 1');78 t read = D(:,1); % Time79 theta read = D(:,2); % theta8081 t adams = rmmissing(t read);82 theta = rmmissing(theta read);8384 M = Mo* cos(w*t);85 figure(1)86 plot(t,y(1,:)*57.2958,'r',t adams,theta,'b')87 legend('Matlab Simulation','ADAMS Simulation')88 xlabel('t(s)')89 ylabel('\theta (deg)')90 title('Response of 1 DOF arm model')91 grid on9293 % figure(2)94 % plot(t,M)95 % xlabel('t(s)')96 % ylabel('Torque (Nm)')97 % title('Tremor torque');9899 %% ODE Function100101 function yp = odefun(t,y)102103 global Mo keq ceq w meq104105 yp = [y(2); −(keq/meq)*y(1)−(ceq/meq)*y(2)+(Mo/meq)*cos(w*t)];106107 end368C.2. T-Beam PSO AlgorithmC.2 T-Beam PSO Algorithm1 function [fitness] = PSO Objective Function(X)23 Om = 4.5*2*pi;45 % define vars6 t0 = 0; ft = 20;7 Mo = 0.218;8 k = 876.2453;9 c = 1.7525;10 I = 0.462250459944476;11 lc = 0.3502;12 l = 0.8660;13 % a = 0.01;14 % q = 0.03;1516 reduc tb = zeros(length(Om),1);1718 for ii=1:numel(Om)1920 omega i = Om(ii);2122 mab = X(1);23 kb1 = X(2);24 cb = X(3);25 a = X(4);26 q = X(5);27 d = X(6);2829 kb2 = kb1;30 Ib = 2*mab*(aˆ2+qˆ2);3132 % Initial conditions33 theta1 0 = 0;34 theta1 dot 0 = 0;35 theta b 0 = pi/2;36 theta b dot 0 = 0;3738 theta0 = [theta1 0; theta1 dot 0;theta b 0;theta b dot 0];39 y0 = [0; 0];4041 % ODE solver options42 options = odeset('RelTol', 1e−6, 'AbsTol', 1e−6);4344 % ODE system function45 fntb = @(tb, yb) system eqns(tb, yb, k, c, lc, kb1, kb2,...369C.2. T-Beam PSO Algorithm46 Ib, I,cb, l, d, a, q, Mo,omega i);4748 g = @(t2, r, omega i) [r(2); (−c*lcˆ2*r(2) − k*lcˆ2*r(1) +...49 Mo*cos(omega i*t2))/I];5051 % Solve ODE system52 [¬, yb] = ode45(fntb, [t0 ft], theta0, options);5354 [¬, r] = ode45(@(t2, r) g(t2, r, omega i), [t0 ft], y0);5556 % Extract solutions57 theta1 amp = rms(yb(:, 1));%max(abs(yb(:, 1)));5859 amplitudes = rms(r(:,1));%max(abs(r(:,1)));6061 reduc tb(ii) = ((rms(amplitudes)−rms(theta1 amp))...62 /(rms(amplitudes)))*100;63 end6465 fitness = −max(reduc tb);6667 end370C.3. Fourier Transform and Butterworth Filter CodeC.3 Fourier Transform and Butterworth FilterCode1 %% Fourier analysis to check for driving frequency2 tstats = [mean(diff(t)) std(diff(t))];34 L = length(X); % Signal Length5 Ts = mean(diff(t)); % Sampling Interval6 Fs = 1/Ts; % Sampling Frequency7 Fn = Fs/2; % Nyquist Frequency89 fft x = fft(X,L)*(2/L); %fourier transform for the signal10 fft y = fft(Y,L)*(2/L); %fourier transform for the signal11 fft z = fft(Z,L)*(2/L); %fourier transform for the signal1213 abs x = abs(fft x); %absolute values of fft vecotor for ploting14 abs y = abs(fft y); %absolute values of fft vecotor for ploting15 abs z = abs(fft z); %absolute values of fft vecotor for ploting1617 % create plotting frequncy vector and identify peak frequency18 fr = 0:(1/t(end)):Fn−(1/t(end));19 [max FPS, idfr] = max(abs x);20 peak fr = fr(idfr);2122 % plot Fourier graphs23 figure(3)24 plot(fr,abs x(1:length(fr)),fr,abs y(1:length(fr)),fr,...25 abs z(1:length(fr)),'−−')26 xlabel('Frequency(Hz)')27 ylabel('Amplitude(rad/Hz)')28 legend('X−axis','Y−axis','Z−axis')29 xlim([0 10])C.3.1 Butterworth Filter1 %% Create Butterworth Filter2 [b a] = butter(4,[5.5/Fn 6.5/Fn],'bandpass');3 x filt = filter(b,a,X);4 y filt = filter(b,a,Y);5 z filt = filter(b,a,Z);371APPENDIXDSteadi-Two EvaluationD.1 DoE and In-Vitro ResultsThe initial qualitative evaluation implemented DoE analysis defined by theTable below.Table D.1: Steadi-Two DoE Factors and Treatment LevelsFactors Treatment Level Designation (Factor) Designation(Level)Mass (g)250A-300 +BearingFree RunningB-Nylon +Mode (mm)9C-12 +372D.1. DoE and In-Vitro ResultsTable D.2: Combined in-vitro results for the Steadi-Two. Red showsamplification.FPS WFE WRUDVersion DW L8 H12 DW L8 H12 DW L8 H12Ver 1 WeSU FPS Run 1 -113.22 27.41 -35.81 -5.26 -7.24 1.65 -110.74 -25.52 -119.35Ver 1 WeSU FPS Run 2 -82.6 37.6 71.45 4.16 0.94 40.56 23.09 69.87 66.28Ver 1 Accelerometer FE 11.69 38.1 24.58 14.64 30.75 40.14 -26.88 -14.99 6.29Ver 1 AccelerometerFPS-16.88 -20.53 56.53 -14.78 -32.27 -17.27 1.37 27.99 -2.56Ver 2 Gyro DrinkingPosture19.45 64.42 37.23 4.95 -27.49 -46.56 -55.61 -74.42 -96.84Ver 2 Gyro PickingPosture11.34 42.86 24.44 61.27 -18.87 66.77 13.71 -3.46 10.18Ver 2 Gyro Elbow Bend -16.88 -20.05 56.53 -14.78 -32.27 -17.27 1.37 27.99 -2.56Ver 2 Gyro Straight Arm -38.43 -39.8 21.57 6.5 43.47 32.68 17.35 41.88 34.13Patient Device GyroDrinking19.45 34.08 24.48 4.95 43.11 38.97 -55.61 -35.77 -21.14Patient Device GyroPicking11.34 70.76 46.35 61.27 10.03 50.87 13.71 27.57 16.15Patient Device GyroStraight Arm-38.34 -28.62 -16.34 6.5 -3.98 -5.42 17.35 33.51 34.51Patient Device WeSURIG Run 1NIL 31.44 63.59 NIL -73.32 54.6 NIL 43.44 43.39Patient Device WeSURIG Run 2NIL 52.6 81.94 NIL -35.48 76.12 NIL 61.71 64.2Patient Device WeSURIG Run 3NIL 46.45 70.34 NIL -43.45 61.92 NIL 59.32 55.59373D.2. Participant Information FormD.2 Participant Information FormDemographic Information∗ Age:∗ Gender:∗ Ethnicity:Medical History∗ Diagnosis of Tremor:∗ Type:∗ Date of Diagnosis:∗ Severity of Tremor:∗ Current Severity:∗ Historical Severity:∗ Other Neurological Conditions:∗ Medications:∗ List Medications forTremor or NeurologicalFunction:∗ Previous or Current Treatmentsfor Tremor:Occupational Status∗ Current Employment Status:∗ Type of Work:∗ Impact of Tremor on WorkPerformance:Daily Activities and Independence∗ Activities of Daily LivingAffected by Tremor:∗ Eating∗ Dressing∗ Writing∗ Other:∗ Use of Assistive Devices:Expectations and Motivation∗ Reason for Participating in theStudy:∗ Expectations from theSteadi-Two Glove:Consent and Understanding∗ Understanding of Study’sPurpose:∗ Understanding of Procedures,Risks, and Benefits:∗ Consent to Participate in theStudy:∗ Agreement on Data Use andPrivacy:Goals and Outcomes (Optional)∗ What are your personal goals forparticipating in this study?∗ What would be considered a winfor you in this study?374D.3. Participant Feedback FormD.3 Participant Feedback FormParticipant ID:(Please do not write your name to ensure your responses remain anonymous.)Instructions Thank you for participating in our study to evaluate theSteadiwear glove designed for individuals with tremor. Your honest feedbackis crucial in helping us improve and understand how well our glove meetsthe needs of people like you.Questions1. Recommendation Likelihood:Considering your experience with the Steadiwear glove during thislab test, how likely are you to recommend this glove to someone withsimilar needs? Please rate on a scale from 0 (not at all likely) to 10(extremely likely).Your Rating:2. Product Effectiveness:On a scale from 0 to 10, how effective do you believe the Steadiwearglove is in managing symptoms related to essential tremors based ontoday’s use?Your Rating:3. Additional Feedback (Optional):Please provide any other comments or suggestions you might haveabout the Steadiwear glove:∗∗∗Consent for Use of Feedback I consent to my feedback being usedto improve the Steadiwear glove and for related market research purposes.Please sign below to indicate your agreement.Signature:Date:375
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Mohammadi, Hadi
Modeling, design, and evaluation of a novel passive hand tremor attenuator for broadband suppression
Text