Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements

Part I Prerequisites.- 1 Generalities on parallel robots.- 1.1 Introduction.- 1.2 General definitions.- 1.3 Types of PKM architectures.- 1.4 Why a book dedicated to the dynamics of parallel robots?.- 2 Homogeneous transformation matrix.- 2.1 Homogeneous coordinates and homogeneous transformation matrix.- 2.2 Elementary transformation matrices.- 2.3 Properties of homogeneous transformation matrices.- 2.4 Parameterization of the general matrices of rotation.- 3 Representation of velocities and forces / acceleration of a body.- 3.1 Definition of a screw.- 3.2 Kinematic screw (or twist).- 3.3 Representation of forces and moments (wrench).- 3.4 Condition of reciprocity.- 3.5 Transformation matrix between twists.- 3.6 Transformation matrix between wrenches.- 3.7 Acceleration of a body.- 4 Kinematic parameterizing of multibody systems.- 4.1 Kinematic pairs and joint variables.- 4.2 Modified Denavit-Hartenberg parameters.- 5 Geometric, velocity and acceleration analysis of open kinematic chains.- 5.1 Geometric analysis of open kinematic chains.- 5.2 Velocity analysis of open kinematic chains.- 5.3 Acceleration analysis of open kinematic chains.- 6 Dynamics principles.- 6.1 The Lagrange formulation.- 6.2 The Newton-Euler equations.- 6.3 The principle of virtual powers.- 6.4 Computation of actuator input efforts under a wrench exerted on the end-effector.- Part II Dynamics of rigid parallel robots.- 7 Kinematics of parallel robots.- 7.1 Inverse geometric model.- 7.2 Forward geometric model.- 7.3 Velocity analysis.- 7.4 Acceleration analysis.- 7.5 Singularity analysis.- 8 Dynamic modeling of parallel robots.- 8.1 Introduction.- 8.2 Dynamics of tree-structure robots.- 8.3 Dynamic model of the free moving platform.- 8.4 Inverse and direct dynamic models of non-redundant parallel robots.- 8.5 Inverse and direct dynamic models of parallel robots with actuation redundancy.- 8.6 Other models.- 8.7 Computation of the base dynamic parameters.- 9 Analysis of the degeneracy conditions for the dynamic model of parallel robots.- 9.1 Introduction.- 9.2 Analysis of the degeneracy conditions of the IDM of PKM.- 9.3 Avoiding infinite input efforts while crossing Type 2 or LPJTS singularities thanks to an optimal trajectory planning.- 9.4 Example 1: the five-bar mechanism crossing a Type 2 singularity.- 9.5 Example 2: the Tripterion crossing a LPJTS singularity.- 9.6 Discussion.- Part III Dynamics of flexible parallel robots.- 10 Elastodynamic modeling of parallel robots.- 10.1 Introduction.- 10.2 Generalized Newton-Euler equations of a flexible link.- 10.2.3 Matrix form of the generalized Newton-Euler model for a flexible clamped-free body.- 10.3 Dynamic model of virtual flexible systems.- 10.4 Dynamic model of a flexible parallel robot.- 10.5 Including the actuator elasticity.- 10.6 Practical implementation of the algorithm.- 10.7 Case Study: the DualEMPS.- 11 Computation of natural frequencies.- 11.1 Introduction.- 11.2 Stiffness and inertia matrices of the virtual system.- 11.3 Stiffness and inertia matrices of the PKM.- 11.4 Including the actuator elasticity.- 11.5 Practical implementation of the algorithm.- 11.6 Case Studies.- 11.7 Conclusion.- Appendices.- A Calculation of the number of degrees of freedom of robots with closed chains.- A.1 Introduction.- A.2 Moroskine's Method.- A.3 Gogu's Method.- A.4 Examples.- B Lagrange equations with multipliers.- C Computation of wrenches reciprocal to a system of twists.- C.1 Definitions.- C.2 Condition of reciprocity.- C.3 Computation of wrenches reciprocal to a system of twists constrained in a plane.- C.4 Computation of wrenches reciprocal to other types of twist systems.- D Point-to-point trajectory generation.- E Calculation of the terms facc1, facc2 and facc3 in Chapter 10.- E.1 Calculation of the term facc1.- E.2 Calculation of the term facc2.- E.3 Calculation of the term facc3.- F Dynamics equations for a clamped-free flexible beam.- F.1 Shape functions for a free flexible beam.- F.2 Stiffness m