1 Introduction
1.1 History and Application of Parallel Mechanisms
1.2 Type Synthesis of Parallel Mechanisms
1.2.1 The Motion-Based Methods
1.2.2 Constraint-Based Methods
1.2.3 Other Methods
1.3 Objective and Organization of This Book
References
2 Fundamental of Group Theory
2.1 History
2.2 Group and Subgroup
2.3 Lie Group
2.4 Geometry in Nonrelativistic Mechanics
2.4.1 The Projective Space and Group
2.4.2 Affine Space and Group
2.4.3 Euclidean Affine Space and Group
3 Rotation and Displacements of Rigid Body
3.1 Vector Products and Algebra
3.2 Rotation of Vectors
3.3 Operator of Displacement
3.4 Axis of a Finite Screw Motion
3.5 Lie Subalgebras
3.6 The Displacement Lie Subgroups
4 Lie Group Based Method for Type Synthesis of Parallel Mechanisms
4.1 Kinematic Pairs and Chains
4.2 Composition of Kinematic Bonds
4.3 Displacement Subgroup of Primitive Mechanical Generators
4.4 Intersection of Kinematic Bonds
4.5 Procedures of Type Synthesis
4.6 Summary
5 Type Synthesis of 5-DOF 3R2T Parallel Mechanism
5.1 Kinematic Bond Between the Base and the Moving Platform
5.2 Limb Kinematic Bonds
5.3 Mechanical Generators of Limb Kinematic Bonds
5.3.1 Mechanical Generators of {T(Pvw)}{S(N)}
5.3.2 Mechanical Generators of {G(u)}{S(N)}
5.3.3 Mechanical Generators of {G2(u)}{S(N)} and {G(u)){S2(N)}
5.3.4 Generation of 2-DOF Joints
5.4 Generation of Mechanisms
5.5 Input Selection Method
5.6 Summary
6 Type Synthesis of 4-DOF 2R2T Parallel Mechanisms
6.1 Kinematic Bond Between the Base and the Moving Platform
6.2 Limb Kinematic Bond and a Configurable Platform
6.3 Mechanical Generators of Limb Kinematic Bonds
6.4 Generation of Parallel Mechanisms
6.4.1 Conventional Parallel Mechanisms
6.4.2 Parallel Mechanisms with a Configurable Platform
6.5 Summary
7 Type Synthesis of 4-DOF Parallel Mechanisms with Bifurcation of Schoenflies Motion
7.1 Preliminaries and Notations of Displacement Group
7.1.1 Displacement Subgroup
7.1.2 {G(y)} and {G- l(y)}
7.2 Bifurcation of Schoenflies Motion in PMs
7.2.1 Displacement Set of PMs with Bifurcation of Schoenflies Motion
7.2.2 Bifurcation of 1-DOF Rotation Motion
7.2.3 A 2-PPPRR PM with Bifurcation of Schoenflies Motion
7.3 Type Synthesis of PMs with Bifurcation of Schoenflies Motion
7.3.1 Geometric Conditions for PMs with Bifurcation of Schoenflies Motion
7.3.2 {X(y)}{X(x)}: General Representation of Limb Bonds for PMs with Bifurcation of Schoenflies Motion
7.3.3 {X - i(y)} and {X -j(x)}
7.3.4 Category 1: For i = 0, {X(y)}{X(x)} ={X(Y)I{X - 3(x)}
7.3.5 Category If: For i = 1, {X(y)}{X(x)} ={X- l(y)I{X - 2(x)}
7.3.6 Category III: For i = 2, {X(y)}{X(x)} ={X- 2(y)}{X- l(x)}
7.3.7 Category IV: For i = 3, {X(y)}{X(x)} ={X - 3(y)}{X(x)}
7.3.8 Implementation of 2-DOF Joints: C and U Joint
7.4 Partitioned Mobility and Input Selection
7.5 Summary
References
8 Type Synthesis of 3-DOF RPR-Equivalent Parallel Mechanisms
8.1 RPR Motion
8.2 Limb Bond of RPR-Equivalent PMs
8.2.1 Displacement Set of the RPR-Equivalent PM
8.2.2 Limb Bond of an RPR-Equivalent PM
8.2.3 Parallel Arrangements of Three Limbs
8.3 Overconstrained RPR-Equivalent PMs
8.3.1 Subcategory 4-4-4
8.3.2 Subcategory 4-4-5
8.3.3 Subcategory 5-5-4
8.4 Non-overconstrained RPR-Equivalent PMs
8.4.1 Subcategory 1 of Non-overconstrained RPR-Equivalent PM
8.4.2 Subcategory 2 of Non-overconstrained RPR-Equivalent PM
8.5 Summary
References
9 Type Synthesis of 3-DOF PU-Equivalent Parallel Mechanisms
9.1 General and Special aTbR Motion
9.1.1 General aTbR Motion and Parasitic Motion
9.1.2 Special aTbR Motion and Parasitic Motion
9.1.3 Special Case: A 1T2R PM with Rotation

This book focuses on the synthesis of lower-mobility parallel manipulators, presenting a group-theory-based method that has the advantage of being geometrically intrinsic.Rotations and translations of a rigid body as well as a combination of the two can be expressed and handled elegantly using the group algebraic structure of the set of rigid-body displacements. The book gathers the authors' research results, which were previously scattered in various journals and conference proceedings, presenting them in a unified tbrm. Using the presented method, it reveals numerous novel architectures of lower-mobility parallel manipulators, which are of interest to those in the robotics community. More importantly, readers can use the method and tool to develop new types of lower-mobility parallel manipulators independently.