Modeling, Design, and Performance Analysis of Plate Harmonic Reducer = 판형 조화 감속기의 모델링, 설계 및 성능 분석

다국어 초록 (Multilingual Abstract)

Modern robotic actuators face structural limitations in the moderate reduction-ratio range of 10:1 to 30:1. This range is where designers must practically balance high torque density, backdrivability for safe human interaction, and torque transparency for precise control in collaborative manipulators, rehabilitation devices, and legged robots. High-ratio transmissions provide large torque but degrade backdrivability due to increased inertia and friction, while quasi-direct-drive architectures offer excellent transparency and control performance but struggle to simultaneously achieve moderate ratios and sufficient torque density within a single stage gear train. This dissertation aims to quantitatively analyze these trade-offs and to expand the design freedom in the mid-ratio regime through a new transmission architecture.

This work introduces the Plate Harmonic Reducer (PHR), a strain-wave-based transmission that exploits axial elastic deformation of a thin, plate-type flexspline. Unlike conventional cup-type strain wave gears, which generate tooth engagement by elastically deforming a cup-shaped flexspline in the radial direction, the PHR forms a traveling wave by axially bending a plate-shaped flexspline and creating axial tooth engagement. In this mechanism, the wave generator employs a CAM-profiled groove and steel balls; by modulating the axial phase of the balls along the cam profile, localized out-of-plane bending is induced and converted into relative motion between the inner and outer teeth. This operating principle preserves the high-resolution, high-density power transmission characteristics of strain wave gears while turning plate thickness and wave shape into design variables that can independently tune stiffness, backdrivability, and efficiency within a compact mechanical structure. To analytically support this concept, an elastic model based on Kirchhoff–Love thin-plate theory is formulated, and the dependence of required deformation force on plate thickness, effective radius, and wave geometry is derived in a systematic manner. In particular, a cubic scaling law between deformation force and plate thickness is identified, showing that the large forces inherent to conventional cup-type designs can be substantially mitigated in plate-based architectures. The model is validated via finite-element analysis and used to establish a design procedure that maps target reduction ratio and allowable deformation to appropriate plate thickness and geometry.

The Tri-contacted Plate Harmonic Reducer (Tri-c PHR) is a concrete realization of this concept built on the above analytical foundation. The mechanism employs three engagement zones arranged at 120° intervals to realize a tooth count difference of ΔZ = 3, enabling reduction ratios below the lower bound of conventional two-contact strain wave gears. In addition, the profiled cam-groove wave generator with rolling elements minimizes sliding during wave formation and introduces additional kinematic design freedom through the shaping of the groove and the choice of preload conditions. Within this unified framework, reduction ratio, torque capacity, backdrivability, and efficiency can be co-designed by jointly selecting the tooth count difference, wave geometry, and preload strategy.

The proposed concept is experimentally validated through two generations of prototypes with complementary roles. In the first-generation, mid-ratio prototypes, plate thickness is systematically varied, and the relationships among torque transmission efficiency, torsional stiffness, and backdrivability are experimentally characterized. This reveals how the thin-plate scaling manifests as a practical efficiency–stiffness trade-off in real hardware and leads to design guidelines for choosing appropriate stiffness and backdrivability levels for given applications. In the second-generation, low-ratio prototypes, the Tri-c PHR architecture is implemented to evaluate positioning accuracy, backlash, and backward starting torque at reduction ratios lower than those typically achieved by strain wave gears. In this process, elastic ratcheting phenomena in aluminum plate flexsplines are identified as a key limitation, and directions for improvement through material selection, heat treatment, and tooth-geometry optimization are discussed.

In summary, this dissertation establishes the Tri-c PHR as a viable solution for the 10:1–30:1 reduction-ratio range, which is not easily addressed by existing power transmission technologies. By integrating thin-plate analytical modeling, Tri-contact mechanism design, and staged prototype validation into a coherent design methodology, the work provides a structured approach to exploring the design space of plate-based strain-wave transmissions. The axial-deformation paradigm, together with the accompanying design, manufacturing, and evaluation procedures, offers a foundation for systematically achieving desired trade-offs among torque density, backdrivability, and precision in next-generation robotic actuators for collaborative robots, exoskeletons, and legged systems.

국문 초록 (Abstract)

현대 로봇 액추에이터는 10:1에서 30:1 사이의 중간 감속비 영역에서 구조적 한계에 직면해 있다. 이 구간은 협동 로봇, 재활 장치, 다리형 로봇에서 요구되는 높은 토크 밀도, 사람과의 상호작용을 위한 역구동성, 그리고 제어 관점에서의 토크 투명성 사이의 균형을 설계자가 실제로 조정해야 하는 핵심 영역이다. 고감속비 기어는 높은 토크를 제공하지만 관성 증가와 마찰로 인해 역구동성이 저하되고, 준직접구동 방식은 투명성과 제어 성능은 우수하나 단일 기어열 구성에서는 중간 감속비와 충분한 토크 밀도를 동시에 확보하기 어렵다. 본 논문은 이러한 상충 관계를 정량적으로 분석하고, 새로운 감속기 구조를 통해 중간 감속비 영역에서 설계 자유도를 확장하는 것을 목표로 한다.

본 논문은 얇은 판형 플렉스 스플라인의 축방향 탄성 변형을 활용하는 판형 조화 감속기(Plate Harmonic Reducer, PHR)를 제안한다. PHR은 기존 컵형 스트레인 웨이브 기어가 경방향으로 플렉스 스플라인을 탄성 변형시켜 치 맞물림을 만들어내는 방식과 달리, 판 형상의 플렉스 스플라인을 축방향으로 탄성 변형시켜 축방향 치 맞물림을 생성함으로써 파동을 형성한다. 이 과정에서 파동발생기는 CAM 원리로 설계된 프로파일드 홈을 따라 배치된 강구의 축방향 위상을 제어함으로써, 원하는 위치에 국부적인 굽힘 변형을 유도하고, 이 변형이 내·외치의 상대 운동으로 변환된다. 이러한 작동 원리는 스트레인 웨이브 기어 특유의 고해상도·고밀도 동력 전달 특성을 유지하면서도, 판 두께와 파동 형상을 통해 강성, 역구동성, 효율을 상대적으로 독립적인 설계 변수로 다룰 수 있게 해 준다. 이를 해석적으로 뒷받침하기 위해 Kirchhoff–Love 박판 이론에 기반한 모델을 정식화하고, 판 두께, 유효 반경, 파동 형상 등 설계 변수에 따라 요구 변형력이 어떻게 변하는지를 체계적으로 도출하였다. 특히 변형력과 두께 사이에 존재하는 세제곱 스케일링 법칙을 활용하여, 기존 컵형 구조에서 필연적으로 요구되던 큰 변형력을 판형 구조에서 크게 완화할 수 있음을 보인다. 제안된 모델은 유한요소해석을 통해 검증되었으며, 이를 바탕으로 목표 감속비와 허용 변형 한계가 주어졌을 때 적절한 판 두께와 형상을 결정하는 설계 절차를 수립하였다.

삼점 접촉 판형 조화 감속기(Tri-c PHR)는 이러한 해석적 기반 위에서 설계된 구체적 구조이다. 본 구조는 120도 간격으로 배치된 세 개의 맞물림 영역을 통해 치수차 ΔZ = 3을 구현함으로써, 기존 이점 접촉 스트레인 웨이브 기어가 가지는 감속비 하한을 넘어서는 저감속비 구성을 가능하게 한다. 또한 프로파일이 정의된 CAM 홈 파동발생기와 구름 요소를 결합하여, 파형 생성 과정에서 발생하는 슬라이딩을 최소화하고, 접촉 패턴을 설계 변수로 포함시킬 수 있는 기구학적 자유도를 확보하였다. 이를 통해 감속비, 토크 용량, 역구동성, 효율 간의 목표 사양에 따라 치수 차이, 파동 형상, 프리로드 조건을 공통의 설계 프레임워크 안에서 조정할 수 있다.

제안된 개념은 서로 다른 역할을 가지는 두 세대의 프로토타입을 통해 단계적으로 검증되었다. 1세대(중간 감속비 프로토타입)에서는 판 두께를 여러 단계로 체계적으로 변화시키며 토크 전달 효율과 비틀림 강성, 역구동성 사이의 관계를 실험적으로 정량화하였다. 이를 통해 박판 이론에서 예측한 스케일링이 실제 기구의 효율·강성 트레이드오프와 어떻게 연결되는지를 규명하고, 특정 응용에서 어느 정도의 강성과 역구동성을 선택해야 하는지에 대한 설계 지침을 도출하였다. 2세대(저감속비 프로토타입)에서는 Tri-c PHR 구조를 적용하여 기존 스트레인 웨이브 기어보다 낮은 감속비에서 위치 정밀도, 백래시, 역구동 기동 토크를 포함한 정량적 성능을 종합적으로 평가하였다. 이 과정에서 알루미늄 판형 플렉스 스플라인에서 발생하는 탄성 래칫팅과 같은 한계를 실험적으로 규명하고, 재료 변경, 열처리 및 치형 최적화를 통한 개선 방향을 제시하였다.

결론적으로, 본 논문은 기존 동력 전달 기술로는 달성하기 어려웠던 10:1–30:1 감속비 영역에 대해, 박판 이론에 기반한 해석 모델링, Tri-contact 구조를 포함한 기구 설계, 그리고 단계적 프로토타입 검증을 통합한 설계 방법론을 제시함으로써 Tri-c PHR을 하나의 유효한 해법으로 확립하였다. 축방향 변형을 활용하는 새로운 설계 패러다임과 이에 수반되는 설계·제작·평가 절차는, 협동 로봇, 외골격, 다리형 시스템에서 요구되는 토크 밀도, 역구동성, 정밀도의 최적 균형점을 체계적으로 탐색할 수 있는 기반을 제공하며, 향후 로봇 액추에이터 설계 지평을 확장하는 데 기여한다.

목차 (Table of Contents)

• Abstract i
• Contents iv
• List of Tables xiv
• List of Figures xvi
• 1 Introduction 1
• 1.1 Background and Motivation 1
• 1.1.1 Robotic Actuators in Modern Applications 1
• 1.1.2 The Essential Role of Mechanical Reducers 2
• 1.1.3 Shift Toward Compliant Actuation 2
• 1.2 State of the Art in Reducer Technologies 3
• 1.2.1 Conventional Harmonic Drives 3
• 1.2.2 Alternative Reducer Technologies 4
• 1.2.3 The Emerging Quasi-Direct Drive Paradigm 5
• 1.3 Problem Statement and Research Gap 5
• 1.3.1 The Missing Ratio Range: 10:1 to 30:1 5
• 1.3.2 Current Solutions and Fundamental Limitations 6
• 1.3.3 Fundamental Design Challenges 7
• 1.4 Research Objectives and Contributions 8
• 1.4.1 Primary Objectives 8
• 1.4.2 Novel Contributions 9
• 1.4.3 Impact and Applications 11
• 1.5 Dissertation Structure 11
• 2 Background and Literature Review 13
• 2.1 Historical Development of Strain Wave Gearing 13
• 2.1.1 Origins and Fundamental Principles 13
• 2.1.2 Development of Design and Analysis Methods 14
• 2.2 Conventional Harmonic Reducer Mechanics and Analysis 15
• 2.2.1 Component Design and Function 15
• 2.2.2 Stress Analysis and Structural Design 16
• 2.2.3 Tooth Profile Optimization 18
• 2.2.4 Composite Flex Spline Designs 19
• 2.2.5 Kinematic and Dynamic Analysis 20
• 2.2.6 Efficiency Analysis and Friction Mechanisms 21
• 2.3 Alternative Reducer Technologies 22
• 2.3.1 Planetary Gear Systems 22
• 2.3.2 Cycloidal Drives 23
• 2.3.3 Crown Gear and Other Eccentric Reducers 25
• 2.3.4 Comparative Assessment of Alternative Technologies 27
• 2.4 Quasi-Direct Drive and Compliant Actuation 28
• 2.4.1 Emergence of Low-Ratio Actuation 28
• 2.4.2 Design Principles and Implementation Strategies 29
• 2.4.3 Limitations and Alternative Approaches 30
• 2.4.4 Application Domain Matching 32
• 2.5 Theoretical Foundations for Thin Plate Analysis 32
• 2.5.1 Classical Thin Plate Theory 32
• 2.5.2 Force–Deflection Relationships 34
• 2.5.3 Moderate Deflection Effects 35
• 2.5.4 Application to Harmonic Reducer Flex Splines 35
• 2.6 Finite Element Analysis in Gear Design 36
• 2.7 Research Gap Identification 37
• 3 Analytical Foundations of Plate-Type Flex Spline Deformation 39
• 3.1 Introduction 39
• 3.2 Kirchhoff–Love Thin Plate Theory 40
• 3.2.1 Assumptions and Governing Equation 40
• 3.2.2 Polar Form and Axisymmetric Reduction 41
• 3.2.3 Prototype Material Constants 41
• 3.3 Problem Formulation for Annular Plate Under Two-Point Loading 42
• 3.3.1 Geometry and Boundary Conditions 42
• 3.3.2 Two-Point Axial Loading and Symmetry 42
• 3.4 Analytical Solution 44
• 3.4.1 Homogeneous Fields 44
• 3.4.2 Particular Fields and Jump Conditions 45
• 3.4.3 Piecewise Assembly 45
• 3.5 Force–Deflection Relationship 46
• 3.5.1 Influence Function and Design Law 46
• 3.5.2 Numerical Example and Comparison to Cup-Type 46
• 3.6 Extension to Multi-Point Contact 47
• 3.6.1 General N-Point Formulation 47
• 3.6.2 Three-Point Contact Case 48
• 3.6.3 Universal Cubic Scaling Law 48
• 3.6.4 Design Implications for Three-Point Contact 49
• 3.7 Validation and Discussion 49
• 3.7.1 Finite Element Verification 49
• 3.7.2 Comparison with Three-Point Configuration 50
• 3.7.3 Moderate-Deflection Effects 50
• 3.7.4 Design Implications 51
• 3.8 Summary 52
• 4 Design Methodology for Tri-Contacted Plate Harmonic Reducer 53
• 4.1 Introduction 53
• 4.2 Design Concept and Operating Principle 54
• 4.2.1 Three-Point Contact Mechanism 54
• 4.2.2 Kinematic Analysis and Reduction Ratio 55
• 4.2.3 Geometric Constraints 56
• 4.3 Wave Generator Design 57
• 4.3.1 Design Requirements 57
• 4.3.2 CAM Groove Profile Synthesis 57
• 4.3.3 Ball–Cage Mechanism Design 60
• 4.3.4 Contact Stress Analysis 61
• 4.4 Flex Spline Design 62
• 4.4.1 Thickness Selection Criteria 62
• 4.4.2 Split Design for Stress Distribution 64
• 4.4.3 Material Selection 66
• 4.5 Tooth Transfer Angle for Enhancing Backdrivability 66
• 4.5.1 Transfer Angle Selection for Backdrivability 67
• 4.5.2 Tooth Geometry Design 69
• 4.6 Circular Spline and Housing Design 70
• 4.6.1 Circular Spline Design 70
• 4.6.2 Housing Design and Axial Constraint 70
• 4.7 Design Trade-offs and Guidelines 71
• 4.7.1 Parametric Sensitivity Analysis 71
• 4.7.2 Design Selection Framework 72
• 4.8 Summary 72
• 5 Finite Element Analysis 74
• 5.1 Introduction 74
• 5.2 FEM Model Development 74
• 5.2.1 Software and Analysis Configuration 74
• 5.2.2 Geometric Models 75
• 5.2.3 Mesh and Boundary Conditions 78
• 5.3 Elastic Deformation Analysis 78
• 5.3.1 Force–Deformation Relationship 78
• 5.3.2 Cubic Scaling Validation 80
• 5.3.3 Plate vs. Cup Architecture Comparison 81
• 5.4 Stress Distribution Analysis 81
• 5.4.1 Von Mises Stress Patterns 81
• 5.4.2 Safety Factors and Design Margins 82
• 5.5 Torsional Stiffness Characterization 82
• 5.5.1 Analysis Methodology 82
• 5.5.2 Thickness-Dependent Stiffness 83
• 5.6 Geometric Nonlinearity Effects 83
• 5.6.1 Theoretical Background 83
• 5.6.2 Nonlinear FEM Results 84
• 5.7 Summary and Design Implications 85
• 5.7.1 Key Findings 85
• 5.7.2 FEM-Based Design Guidelines 86
• 5.7.3 Material Substitution Projections 86
• 6 Prototype Development 87
• 6.1 Prototype Specifications 87
• 6.1.1 First-Generation: 50:1 Reduction Ratio 87
• 6.1.2 Second-Generation: 16:1 Reduction Ratio 88
• 6.2 Manufacturing Approach 89
• 6.2.1 Material Selection 89
• 6.2.2 Manufacturing Process 89
• 6.2.3 Assembly 90
• 6.3 Summary 91
• 7 Experimental Characterization 92
• 7.1 Overview of Experimental Program 92
• 7.1.1 Experimental Objectives 92
• 7.1.2 Prototype Generations 93
• 7.2 First-Generation: 50:1 Prototypes 93
• 7.2.1 3D-Printed Proof-of-Concept 93
• 7.2.2 Metal Prototypes: Parametric Thickness Study 96
• 7.3 Second-Generation: 16:1 Prototype 99
• 7.3.1 Design Specifications 99
• 7.3.2 Experimental Setup 99
• 7.3.3 Kinematic Performance 100
• 7.3.4 Dynamic Performance 101
• 7.4 Summary 102
• 8 Discussion 104
• 8.1 Performance Trade-offs and Design Space 105
• 8.1.1 Efficiency–Stiffness Trade-off 105
• 8.1.2 Technology Positioning 106
• 8.2 Validation of Theoretical Framework 107
• 8.2.1 Analytical–FEM–Experimental Consistency 107
• 8.2.2 Comparative Performance Analysis 108
• 8.3 Current Limitations and Challenges 109
• 8.3.1 Torque Capacity Constraints 109
• 8.3.2 Torsional Compliance and Control Implications 110
• 8.3.3 Manufacturing Precision Requirements 110
• 8.3.4 Prototype-Level Implementation 111
• 8.4 Fatigue Life Assessment and Reliability Considerations 112
• 8.4.1 Fatigue Prediction Framework 112
• 8.4.2 FEM-Based Life Estimation 113
• 8.4.3 Design Life Results and Rated Torque 115
• 8.4.4 Model Limitations 117
• 8.5 Miniaturization Limits and Scaling Analysis 118
• 8.5.1 Theoretical Minimum Constraints 118
• 8.5.2 Practical Structural Limits 120
• 8.5.3 Size–Performance Trade-offs and Material Implications 121
• 8.6 Practical Design Guidelines 122
• 8.6.1 Design Selection Framework 122
• 8.6.2 Application Domain Mapping 122
• 8.7 Future Research Directions 124
• 8.7.1 Near-Term Development Priorities 124
• 8.7.2 Extended Characterization 125
• 8.7.3 Application Demonstrations 126
• 8.8 Summary 126
• 9 Conclusions and Future Work 129
• 9.1 Research Summary 129
• 9.1.1 Key Achievements 129
• 9.1.2 Novel Contributions 131
• 9.2 Theoretical Contributions 133
• 9.2.1 Plate Deformation Theory 133
• 9.2.2 Design Methodology 134
• 9.3 Experimental Validation 135
• 9.3.1 Performance Characterization 135
• 9.3.2 Model Validation 137
• 9.4 Context and Limitations 138
• 9.4.1 Current Prototype Performance 138
• 9.4.2 Root Causes and Improvement Potential 140
• 9.5 Future Research Directions 141
• 9.5.1 Material and Manufacturing Development 141
• 9.5.2 Design Optimization 142
• 9.5.3 Performance Characterization 143
• 9.5.4 Application Demonstrations 144
• 9.5.5 Control Integration 145
• 9.6 Broader Impact and Significance 146
• 9.6.1 Technology Impact 146
• 9.6.2 Scientific Contributions 147
• 9.7 Closing Remarks 149
• A Detailed Plate Theory Derivations 151
• A.1 Governing Equations 151
• A.2 Three-Point Loading and Fourier Solution 152
• A.3 Solution and Influence Function 153
• B FEM Configuration Details 154
• B.1 Software and Settings 154
• B.2 Material Properties and Mesh Quality 155
• B.3 Boundary Conditions 155
• C Experimental Protocols 156
• C.1 Kinematic Characterization 156
• C.2 Dynamic Performance 157
• Abstract (In Korean) 171