Theory of Parallel Mechanisms

This book contains mechanism analysis and synthesis.

In mechanism analysis, a mobility methodology is first systematically presented. This methodology, based on the author's screw theory, proposed in 1997, of which the generality and validity was only proved recently, is a very complex issue, researched by various scientists over the last 150 years. The principle of kinematic influence coefficient and its latest developments are described. This principle is suitable for kinematic analysis of various 6-DOF and lower-mobility parallel manipulators. The singularities are classified by a new point of view, and progress in position-singularity and orientation-singularity is stated. In addition, the concept of over-determinate input is proposed and a new method of force analysis based on screw theory is presented.

In mechanism synthesis, the synthesis for spatial parallel mechanisms is discussed, and the synthesis method of difficult 4-DOF and 5-DOF symmetric mechanisms, which was first put forward by the author in 2002, is introduced in detail. Besides, the three-order screw system and its space distribution of the kinematic screws for infinite possible motions of lower mobility mechanisms are both analyzed.

Inhalt

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 Grübler-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 Hervé 6.bar mechanism.- 4.1.5 Spatial 4P mechanism.- 4.1.6 Delassus H-H-H-H Mechanism.- 4.1.7 Hervé'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 Accorda…