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