Theory of Parallel Mechanisms

Part I Screw Theory.- Chapter 1 Basics of Screw Theory.- Introduction.- 1.1 Equation of a Line.- 1.2 Mutual Moment of Two Lines.- 1.3 Line Vectors and Screws.- 1.3.1 The line vector.- 1.3.2 The screw.- 1.4 Screw Algebra.- 1.4.1 Screw Sum.- 1.4.2 Product of a scalar and a screw.- 1.4.3 Reciprocal Product.- 1.5 Instantaneous Kinematics of a Rigid Body.- 1.5.1 Instantaneous Rotation.- 1.5.2 Instantaneous Translation.- 1.5.3 Instantaneous screw motion.- 1.6 Statics of a Rigid Body.- 1.6.1 A Force Acting on a Body.- 1.6.2 A Couple Acting on a Body.- 1.6.3 A Twist Acting on a Body.- References.- Chapter 2 Dependency and Reciprocity of Screws.- 2.1 Concept of Screw Systems.- 2.2 Second-order screw system.- 2.2.1 Linear combination of two screws.- 2.2.2 Special two-screw system.- 2.3 Third-order screw system.- 2.3.1 Principal screws.- 2.3.2 Special three-screw systems.- 2.4 Grassmann line geometry.- 2.5 Screw dependency in different geometrical spaces.- 2.5.1 Basic concepts.- 2.5.2 Different geometrical spaces.- 2.6 Reciprocal screws.- 2.6.1 Concept of a reciprocal screw.- 2.6.2 Dualism in the physical meaning of reciprocal screws.- 2.7 Reciprocal screw system.- 2.8 Reciprocal screw and constrained motion.- 2.8.1 Three skew lines in space.- 2.8.2 Three lines parallel to a plane without a common normal.- 2.8.3 Three non-concurrent coplanar lines.- 2.8.4 Three coplanar and concurrent line vectors.- 2.8.5 Three line vectors concurrent in space.- 2.8.6 Three line vectors parallel in space.- References.- Chapter 3 Mobility Analysis (Part 1).- 3.1 The Concept and Definition of Mobility.- 3.2 Mobility Open Issue.- 3.2.1 Grubler-Kutzbach Criterion.- 3.2.2 Mobility Open Issue.- 3.3 Mobility Principle based on Reciprocal Screw.- 3.3.1 Mechanism Can Be Expressed as a Screw System.- 3.3.2. Development of Our Unified Mobility Principle.- 3.3.3 The Modified G-K Formulas.- 3.4 Constraint Analysis based on Reciprocal Screw.- 3.4.1 The Common Constraint.- 3.4.2 Parallel Constraint.- 3.4.3. Over-constraint.- 3.4.4. The Generalized Kinematic Pair.- 3.5 Mobility Property Analyses.- 3.5.1 Translation and Rotation.- 3.5.2 Rotational Axis.- 3.5.3 Instantaneous Mobility and Full-cycle Mobility.- 3.5.4 Full-field Mobility.- 3.5.5 Parasitic Motion.- 3.5.6 Self-motion.- References.- Chapter 4 Mobility Analysis Part 2.- 4.1 Mobility analysis of simple mechanisms.- 4.1.1 Open Chain linkage.- 4.1.2 Roberval mechanism.- 4.1.3 RUPUR mechanism.- 4.1.4 Herve 6.bar mechanism.- 4.1.5 Spatial 4P mechanism.- 4.1.6 Delassus H-H-H-H Mechanism.- 4.1.7 Herve's CCC Mechanism.- 4.2 Mobility Analysis of classical mechanisms.- 4.2.1 Bennett mechanism.- 4.2.2 Five-bar Goldberg Linkage.- 4.2.3 Six-bar Goldberg linkage.- 4.2.4 Myard linkage with symmetrical plane.- 4.2.5 Bricard with symmetrical plane.- 4.2.6 Altmann Abb.34 Mechanism.- 4.2.7 Altmann six-bar linkage.- 4.2.8 Waldron six-bar linkage.- 4.3 Mobility Analysis of Modern Parallel Mechanisms.- 4.3.1 4-DOF 4-URU Mechanism.- 4.3.2 3-CRR mechanism.- 4.3.3 Zlatanov and Gosselin's Mechanism.- 4.3.4 Carricato's mechanism.- 4.3.5 Delta mechanism.- 4.3.6 H4 manipulator.- 4.3.7 Yang's mechanism.- 4.4 Mobility analysis of Hoberman Switch-pitch ball.- 4.4.1 Structure analysis.- 4.4.2 Three-link chain.- 4.4.3 Eight-link loop.- 4.4.4 Double loop.- 4.4.5 Three-loop chain.- 4.4.6 The whole mechanism.- 4.5 Six-hole cubiform mechanism.- 4.5.1 Double-hole linkage.- 4.5.2 Four-hole linkage.- 4.5.3 five-hole linkage.- 4.5.4 The whole six-hole mechanism.- References.- Chapter 5 Kinematic Influence Coefficient and Kinematics Analysis.- 5.1 Concept of KIC.- 5.2 KIC and Kinematic Analysis of Serial Chains.- 5.2.1 Position Analysis.- 5.2.2 First-Order KIC.- 5.2.3 Second-Order KIC.- 5.3 Kinematic Analysis of Parallel Mechanism.- 5.3.1 First-Order KIC and Mechanism Velocity Analysis.- 5.3.2 Second-Order KIC and Mechanism Accelerations.- 5.4 Virtual Mechanism Principle of Lower-Mobility Parallel Mechanisms.- 5.4.1 Virtual Mechanism Principle.- 5.4.2 Kinematic Analysis Based on Virtual Mechanism Principle.- Fig. 5.7 A virtual limb.- References.- Chapter 6 Full-Scale Feasible Instantaneous Screw Motion.- 6.1. Introduction.- 6.2. Determination of Principal Screws.- 6.2.1 The Representation of pitch and axes.- 6.2.2 Principal screws of a third-order screw system.- 6.3. Full-Scale Feasible Instantaneous Screws of the 3-RPS mechanism.- 6.3.1. Virtual Mechanism and Jacobian Matrix.- 6.3.2 Upper platform is parallel to the base.- 6.3.3 The upper platform rotates by an angle about line a2a3.- 6.3.4 General configuration of the 3-RPS mechanism.- 6.4. Full-Scale Feasible Instantaneous Screw of a 3-UPU mechanism.- 6.4.1 Mobility analysis.- 6.4.2 First-order influence matrices and kinematic analysis.- 6.4.3. Initial configuration.- 6.4.4 The second configuration.- 6.5 Full-Scale Feasible Instantaneous Screw of a 3-RPS Pyramid mechanism.- 6.5.1 First-order influence coefficient matrix (Jacobian matrix).- 6.5.2 Principal screws and full-scale feasible motions.- 6.6 A 3-DOF Rotational Parallel Manipulator without Intersecting Axes.- 6.6.1 An Open Problem of the PMs with Intersecting Axes.- 6.6.2 A 3-D revolute mechanism without intersecting axes.- 6.3.3 The Orientation Workspace.- 6.6.4 Examples.- 6.6.5 Discussions about the differences between the SPMs and the 3-RPS Cubic PM.- References.- Chapter 7 special configuration of Mechanisms.- 7.1. Introduction.- 7.2. Classification of the Special Configuration.- 7.2.1 Singular kinematics classification.- 7.2.2 Classification of the Singularity via a Linear Complex.- 7.3. Singular Kinematic Principle.- 7.4. Singularity Loci of 3/6-Stewart For special orientations.- 7.4.1 Typical Singularity Structures of 3/6-SP.- 7.4.2 Hyperbolic Singularity Equation Derived in an Oblique Plane.- 7.4.3 Singularity Equation Derived in 3D Space.- 7.4.4 Singularity Distribution in 3D Space.- 7.5. Structure and Property of THE Singularity Loci of 3/6-Stewart for General Orientations ( ).- 7.5.1. Singularity Equation Based on Theorem 3 for General Orientations.- 7.5.2 Singularity Analysis Using Singularity-Equivalent-Mechanism.- 7.5.3. General Case.- 7.5.4 Five Special Cases of the Singularity Equation.- 7.6. Structure and Property of the Singularity Loci of the 6/6-Stewart.- 7.6.1 Jacobian Matrix.- 7.6.2 Singularity Analysis in 3D Space.- 7.6.3 Singularity Analysis in Parallel Principal-Sections.- 7.7 Singularity of a 3-RPS Manipulator.- 7.7.1 3-RPS Mechanism.- 7.7.2. Singularity and Its spatial distribution.- 7.7.3. Geometry and Constraint Analysis.- References.- Appendix A.- Chapter 8 Dynamic Problems of Parallel Mechanisms.- 8.1 Over-determine inputs.- 8.1.1 Influence coefficient matrices and inertia forces.- 8.1.2 The Accordant Equation for over-determinate Inputs.- 8.1.3 Optimization of Over-Determinate Input.- 8.1.4 The Weight Distribution of the input Torques.- 8.2 Kinetostatic Analysis of 4-UPU Parallel Mechanisms.- 8.2.1 Main-pair reaction Forces.- 8.2.2 Numerical Example.- 8.3 Kinetostatic Analysis of 4-R(CRR) Parallel Manipulator.- 8.3.1 4-R(CRR) Parallel Manipulator.- 8.3.2. Main-pair reaction.- 8.3.3 Active moments and reactions of other pairs in limbs.- 8.3.4 Numerical example.- 8.3.5. Discussion.- References.- Chapter 9 Constraint screw-based method for type synthesis.- 9.1 Description of constraints acting on a rigid body.- 9.2 Limb twist and limb constraint systems.- 9.2.1 Limb twist system.- 9.2.2 Limb constraint system.- 9.3 Platform twist and platform constraint systems.- 9.3.1 Platform twist system.- 9.3.2 Platform constraint system and classification of lower-mobility PMs.- 9. 4 Constraint-screw based synthesis method.- 9.4.1 Procedure of the constraint-screw based synthesis method.- 9.4.2 Generation of different architectures of PM.- 9.4.3 Discrimination for instantaneous PMs.- 9.5 Examples.- 9.5.1 Type synthesis of a 3R2T 5-DOF PM.- 9. 5. 2 Type synthesis of 2R3T 5-DOF PMs.- 9. 5. 3 Type synthesis of 1R3T 4-DOF PMs.- 9. 5. 4 Type synthesis of 3R1T 4-DOF PMs.- 9. 5. 6 Type synthesis of a 2R1T 3-DOF PM.- 9. 5. 7 Type synthesis of a 3T 3-DOF PM.- 9. 5. 8 Type synthesis of a 3R 3-DOF PM.- 9. 5. 9 Type synthesis of a 1R2T 3-DOF PM.- References.- Chapter 10 Digital Topology Theory of Kinematic Chains and Atlas Database.- 10-1 Topology Modeling of Mechanisms.- 10.1.1 Modeling of simple joint kinematic chains.- 10.1.2 Modeling of multiple joint kinematic chains.- 10.1.3 Modeling of geared (cam) kinematic chains.- 10.2 Loop operation algebra of kinematic chains.- 10.2.1 Loop and its representation.- 10.2.2 " " Operation of Loops.- 10.2.3 " " Operation of Loops.- 10.2.5 Loop analysis.- 10.2.6 Edge-based operations of loops.- 10.3 Isomorphism identification.- 10.3.1 Perimeter topological graph.- 10.3.2 Canonical perimeter topological graph.- 10.3.3 Characteristic perimeter topological graph.- 10.3.4 Examples of isomorphism identification.- 10.3.5 Analysis of computational complexity.- 10.4 Detection of rigid sub-chains.- 10.5 Digital atlas database and synthesis.- References.- Index.