Topology of Robotic Mechanisms: Framework and Mathematics Methods – In Conjunction with a Review of Four Original Theories

Abstract

In this work, a new basic framework of robotic mechanism topology is proposed, which includes: ① two important concepts (i.e., topological structure and kinematic characteristics), ② three basic formulas (i.e., POC equation of serial mechanisms, POC equation of parallel mechanisms (PMs), and DOF formula) and ③ topological structure synthesis methods of PMs. Based upon this framework, literature review and comparative study of four original theories for topological structure synthesis of PMs (i.e., screw theory, velocity space, subgroup and POC set) are carried out. Moreover, it is found that these four theories and their computable methods (linear operation, nonlinear symbolic operation and linear symbolic operation) originate from different definitions and their mathematical representations of the two concepts. These basic formulas accounting for mechanism topology are derived based on the two concepts, which reveal the mapping relationship among topological structures, POCs and DOF of mechanisms. Further, these four theories are grouped into two categories: ① the algebraic methods related to the fixed coordinate system for screw theory, velocity space and subgroup; ② the intrinsic geometry method independent of the fixed coordinate system for POC set.