Kinematic analysis of Five-DOF (3T2R) parallel mechanisms with identical limb structures

This dissertation is presented specifically to an audience composed of two separate groups working on the kinematics of parallel mechanisms: mechanical engineers and geometricians. Originally, this work was based solely on engineering concepts. However, during the course of this work, a theoretical and practical algebraic geometry approach has been proposed, and now, this thesis is a (hopefully judicious) mixture of these two approaches. The work presented in this thesis does not favour one method over another but rather uses the synergies between the methods of engineering mechanics and algebraic geometry. By keeping the kinematic analysis of symmetrical parallel mechanisms with fivedegree-of-freedom—three translations and two rotations—as a case study, this thesis can be regarded as a guideline of the application of algebraic geometry in the kinematic analysis of parallel mechanisms. This choice, i.e., the selection of symmetric 5-degreeof-freedom parallel mechanisms, has been appreciated since the kinematic properties of this type of architecture have proven to be quite remarkable. In this context, Chapters 2 to 4 are devoted to the adaptation of some algebraic geometry techniques to the kinematic analysis of parallel mechanisms while keeping in the background the study of symmetrical 5-degree-of-freedom parallel mechanisms. The major contribution of Chapter 2 is the development of a systematic approach for the kinematic modelling of symmetric parallel mechanisms which is applied in Chapter 3 to symmetric 5-degree-of-freedom parallel mechanisms. In Chapter 3, the application of the framework presented in Chapter 2 leads to some astonishing results for the number of solutions for the FKP: 1680 finite solutions and for a given design 208 real solutions. All these solutions are in terms of Study parameters, i.e., in the sevendimensional kinematic space. In Chapter 4, the mapping from the seven-dimensional to three-dimensional kinematic space is introduced which allows to obtain the Cartesian coordinates and the corresponding angles for each solution. Moreover, in this chapter the first-order kinematic properties are also investigated which results in a better understanding of the mechanism. The reader will notice in Chapters 5 and 6 a kinematic investigation which is based on the three-dimensional kinematic space. The main concern of Chapter 5 is the geometric constructive approach for the workspace analysis in which an algorithm previously proposed for the constant-orientation workspace of 6-degree-of-freedom parallel mechanisms is extended to the symmetric 5-degree-of-freedom parallel mechanisms. The CAD model of the workspace is also presented. The results of this chapter reveal that the workspace of symmetric 5-degree-of-freedom parallel mechanisms can have small isolated part. Chapter 6 completes the discussion initiated in Chapter 3 for the FKP in which the FKP is investigated for some simplified designs having either a closed-form solution or a univariate expression. For a nearly general design a univariate expression of degree 220 is found. In this chapter, we veer a little from the three-dimensional kinematic space to the seven-dimensional kinematic space in order to validate and refine the obtained results, a state of the art which can be applied to other cases. The last chapter is devoted to the singularity analysis of symmetric 5-degree-offreedom parallel mechanisms which relies on Grassmann line geometry. This chapter covers extensively the study of the singular configurations of the symmetrical 5-degreeof-freedom parallel mechanisms for the simplified design proposed in Chapter 6. The main contribution of this chapter is the application of the Grassmann line geometry to the lower-mobility parallel mechanisms in which a line at infinity is among the Plucker lines under study. Finally, Chapter 8 concludes the thesis by summarizing the results obtained throughout Chapters 2 to 7. It provides also several ongoing works and future works which can be the subjects of some further studies for a new direction of research.