Profile Synthesis of Planar Variable Joints Brian James Slaboch Marquette University

Marquette University e-Publications@Marquette Dissertations (2009 -) Dissertations, Theses, and Professional Projects Profile Synthesis Of Planar Variable Joints Brian James Slaboch Marquette University Recommended Citation Slaboch, Brian James, "Profile Synthesis Of Planar Variable Joints" (2013). Dissertations (2009 -). Paper 283. http://epublications.marquette.edu/dissertations_mu/283 PROFILE SYNTHESIS OF PLANAR VARIABLE JOINTS by Brian J. Slaboch, B.S., M.S. A Dissertation Submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Milwaukee, Wisconsin August 2013 ABSTRACT PROFILE SYNTHESIS OF PLANAR VARIABLE JOINTS Brian J. Slaboch, B.S., M.S. Marquette University, 2013 Reconfigurable mechanisms provide quick changeover and reduced costs for low volume manufacturing applications. In addition, these mechanisms provide added flexibility in the context of a constrained environment. A subset of planar reconfigurable mechanisms use variable joints to provide this added adaptability. In this dissertation, the profile synthesis of planar variable joints that change from a rotational motion to a translational motion was investigated. A method was developed to perform automated profile synthesis. It was shown that combinations of higher variable joints can be used to create kinematically equivalent variable joints that are geometrically different. The results were used to create two new reconfigurable mechanisms that utilize the synthesized variable joints. The first reconfigurable mechanism is a four-bar mechanism that performs a rigid body guidance task not possible using conventional four-bar theory. The second mechanism uses variable joints in a 3-RPR parallel mechanism for singularity avoidance without adding redundant actuation. i ACKNOWLEDGEMENTS Brian J. Slaboch, B.S., M.S. I could not have done this without the help and support from many people. First, I would like to thank Dr. Philip Voglewede for his guidance throughout this process. I would also like to thank my family and friends for their love and encouragement. Mom and Dad, you have been with me from the beginning, and I couldn’t have done this without you. In addition, I would like to thank the guys in the lab with special thanks to Jinming for being a great friend for so many years. My committee also played an important role, and I thank them for pushing me to do my best work. Finally, I would like to thank Kate for being there for me when I needed her the most. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ......................... i TABLE OF CONTENTS ........................... ii LIST OF FIGURES .............................. vii CHAPTER 1 Introduction ......................... 1 1.1 Introduction................................ 1 1.2 ResearchPurpose............................. 2 1.3 JointProfileSynthesis .......................... 3 1.4 Profile Synthesis of an Example Variable Joint . 4 1.5 OrganizationoftheDissertation. 5 CHAPTER 2 Motivation and Literature Review ........... 6 2.1 ABriefHistoryofKinematics . .. .. 6 2.2 ReconfigurableMechanisms . 8 2.3 TypesofReconfigurableMechanisms . 10 2.3.1 Kinematotropic Mechanisms . 11 2.3.2 MetamorphicMechanisms . 12 2.3.3 Mechanisms with Variable Topology . 13 2.4 Summary of Reconfigurable Mechanisms . 15 2.5 Joints ................................... 15 2.5.1 LowerPairs ............................ 16 2.5.2 HigherPairs............................ 16 2.5.3 Variable Joints . 17 iii TABLE OF CONTENTS — Continued 2.5.4 Practical Considerations for Variable Joints . 18 2.6 Summary ................................. 19 CHAPTER 3 Higher Variable Joints ................... 21 3.1 Overview of Higher Variable Joints . 21 3.1.1 Motivation for Higher Variable Joints . 22 3.1.2 Mobility of Bodies in Contact . 24 3.1.3 ConfigurationSpaceTerminology . 25 3.2 Type Ru Variable Kinematic Joints . 26 3.2.1 Case 1: Type R1 Higher Variable Joints . 27 3.2.2 Case 2: Type R2 Higher Variable Joints . 31 3.2.3 Case 3: Type R3 Higher Variable Joints . 32 3.2.4 Review of Type Ru Higher Variable Joints . 33 3.3 Type Pv HigherVariableJoints . 33 3.3.1 Case 1: Type P1 Higher Variable Joints . 35 3.3.2 Case 2: Type P2 Higher Variable Joints . 37 3.3.3 Review of Type Pv Higher Variable Joints . 39 3.4 Summary ................................. 40 CHAPTER 4 Profile Synthesis of RuPv Variable Joints ....... 41 4.1 Profile Synthesis of an Example R1P1 Variable Joint . 41 4.2 Enumeration of Rotational to Translational Variable Joints . .. 42 4.3 Design Configuration Space of Variable Joints . 45 4.4 Profile Synthesis Equations for RuPv Variable Joints . 47 4.4.1 Assumptions............................ 47 4.4.2 ProfileofLink1.......................... 48 iv TABLE OF CONTENTS — Continued 4.4.3 ProfileofLink2.......................... 50 4.4.4 Constraints for RuPv Variable Joints . 52 4.5 Adjustable Plier Designs Based on RuPv Variable Joints . 53 4.5.1 Example R1P2 Variable Joint for Adjustable Pliers . 55 4.5.2 Alternative Solution . 59 4.5.3 Kinematic Redundancies . 60 4.5.4 Summary of Adjustable Plier Designs . 62 4.6 Constraints on the Direction of Translation . 63 4.6.1 ExternalArcInterference . 63 4.6.2 InternalArcInterference . 65 4.6.3 Determining the Separation Angles for an R1Pv Joint . 66 4.7 Summary ................................. 68 CHAPTER 5 Four-Bar Mechanism with Variable Topology .... 70 5.1 ProblemSetup .............................. 70 5.2 Solution Method using RuPv Variable Joints . 71 5.3 Four-BarMechanismAnalysis . 73 5.3.1 Overview of Solution Methodology . 75 5.3.2 Two Position Rigid Body Guidance for an RRRR ....... 76 5.4 Two Position Rigid Body Guidance for an RRRP ........... 79 5.5 Determining Transmission Angles for a Reconfigurable Four-Bar . .. 80 5.6 NumericalExample............................ 83 5.7 JointDesignfortheFour-BarMechanism. 85 5.8 Discussion of Alternative Solutions . 88 5.9 Summary ................................. 90 v TABLE OF CONTENTS — Continued CHAPTER 6 Singularity Avoidance in a 3-RPR Parallel Manipula- tor ......................................... 91 6.1 Introduction................................ 91 6.2 3RPRSingularities ............................ 92 6.3 ProblemSetup .............................. 97 6.4 Solution Method using RuPv Variable Joints . 98 6.5 NumericalExample............................101 6.6 Joint Design for the 3-RPR .......................105 6.7 Summary .................................106 CHAPTER 7 Summary and Conclusions ................ 108 7.1 HigherVariableJoints ..........................108 7.2 Profile Synthesis of RuPv VariableJoints . .109 7.3 Four Bar Mechanism with Variable Topology . 109 7.4 Singularity Avoidance in a 3-RPR Parallel Manipulator . 110 7.5 Conclusions ................................110 7.6 FinalRemarks...............................113 REFERENCES ................................. 115 APPENDIX A Analysis of Reconfigurable Mechanisms ....... 121 A.1 Introduction................................121 A.2 CurrentMatrixRepresentations . .122 A.2.1 AdjacencyMatrices. .122 A.2.2 EU-MatrixTransformations . .124 A.2.3 ImprovedAdjacencyMatrix . .125 A.2.4 Directionality Topology Matrices . 126 vi TABLE OF CONTENTS — Continued A.3 MechanismStateMatrices . .127 A.3.1 MatrixNotation .........................128 A.3.2 RepresentativeExample . .130 A.4 Augmented Mechanism State Matrices . 132 A.5 Examples .................................133 A.5.1 3RRRMechanism. .. .. .133 A.5.2 PlanarMetamorphicMechanism. .134 A.5.3 2R-2PMechanism. .135 A.6 ReviewofMechanismStateMatrices . .136 APPENDIX B Lower Pairs ......................... 138 vii LIST OF FIGURES 1.1 GeneralProfileofaJoint........................... 4 1.2 Example Rotational to Translational Variable Joint . 4 1.3 Rotational to Translational Variable Joint . 4 2.1 OneDOFWrenPlatform .......................... 12 2.2 TwoDOFWrenPlatform .......................... 12 2.3 The Mechanisms Oscillate Between Pins P1 and P2. ........... 13 2.4 Classification of Mechanisms with Variable Topology . 14 2.5 Mechanism Performing Skew-Axial Motion . 15 2.6 Mechanism Performing Straight Line Motion . 15 2.7 ChangeinKinematicPair .......................... 17 2.8 ChangeinOrientation ............................ 17 3.1 HigherVariableJoints ............................ 22 3.2 Fixed and Moving Centrodes of a Variable Joint . 23 3.3 Rotation About the Z-axis is Possible . 25 3.4 Immobile Object due to Frictionless Contacts . 25 3.5 Different Cases for Revolute Higher Variable Joints . 27 3.6 One Constraint Body is Not Sufficient to Constrain Body B to Purely RotationalMotion. .............................. 29 3.7 Two Constraint Bodies, A1 and A2 are Placed on the Outside of Body B. 30 3.8 (A) The C-obstacles Intersect at the Origin. (B) Only Pure Rotation is Possible Along the θ Axis........................... 31 3.9 (A) The Only Free Motion is Rotation about the Center of body B. (B) Only Pure Rotation is Possible Along the θ axis............... 32 viii LIST OF FIGURES — Continued 3.10 (A) The only free motion is rotation about the center of body B. (B) Only pure rotation is possible along the θ axis................ 33 3.11 Different Cases for Prismatic Higher Variable Joints . 34 3.12 (A) Two Point Contacts are Needed for Translational Motion. (B) Three Point Contacts are Needed for Translational Motion. 35 3.13 Only Translational Motion is Possible due to the C-obstacles . .. 36 3.14 Cross Sections for: (A) θ = 10◦. (B) θ =0◦ (C) θ = 10◦ ........ 36 − 3.15 Cross Sections

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