Fundamental Theory About Fully Symmetric Parallel Mechanism With Lower-Mobility Based On The Theory Of Differential Manifold

Based on the theory of differential manifold, the theory and methodology for modeling of the fully symmetric parallel mechanisms with lower-mobility are studied by considering of the characteristics of both kinematics and constraint, including kinematics, accuracy, static and stiffness. The research intends to provide the theoretical guide for distilling a universal performance index about kinematics, accuracy, static and stiffness. The dissertation is organized as following.1) Preliminary theory of differential manifold describing the motion of a rigid body is analyzed, which shows that the tangent and the cotangent space consist of two orthogonal subspaces respectively when some degrees-of-freedom (DOFs) of the body are limited. In the unified frame, the relationship between twist and tangent space, wrench and cotangent space, equivalent relationship between D-H method and product-of-exponentials (POE) are presented respectively. A 2R mechanism is taken as an example to illustrate the above analysis.2) According to the fully symmetric parallel mechanisms (FSPM) with lower-mobility, a general method for kinematic modeling of the mechanisms is presented by integrating D-H method, translation operation on the Group with the theory of reciprocal screw, in which the complete 6×6 Jacobian matrix is obtained including kinematic and constraint sub-matrices. It lays the unified foundation for both error and static stiffness modeling. Tsai's 3-UPU parallel mechanism is taken as an example to illustrate the method.3) By using the theory of differential manifold, an explanation is presented for the controllable and uncontrollable error of the FSPM with lower-mobility. A new method is proposed to modeling error of the mechanism based on the translation operation and reciprocal screw theory, in which the linear mapping between the controllable and uncontrollable error and their source can be set up. The former can be used to instruct tolerance design of parts, and the latter mainly act as a fundamental formula. The detailed illustration is made through an example of Tsai's 3-UPU. 4) The explanation about transmissions of actuation and constraint force of the FSPM with lower-mobility is first presented based on the theory of differential manifold. By using the theory of virtual work, an approach is proposed to obtain the complete static and stiffness, in which constraint force and constraint stiffness mapping are also obtained besides traditional actuation force and actuation stiffness mapping. It is demonstrated in detail through an example of general 3-UPU. The analysis of constraint force and constraint stiffness is indicated to be necessary for geometric design of these parallel mechanisms.