A Geometric Theory for Analysis and Synthesis of Sub-6 DoF Parallel Manipulators

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Mechanism synthesis is mostly dependent on the designer's experience and intuition and is difficult to automate. This paper aims to develop a rigorous and precise geometric theory for analysis and synthesis of sub-6 DoF (or lower mobility) parallel manipulators. Using Lie subgroups and submanifolds of the special Euclidean group <formula formulatype="inline"><tex>${\rm SE}(3)$</tex> </formula>, we first develop a unified framework for modelling commonly used primitive joints and task spaces. We provide a mathematically rigorous definition of the notion of motion type using conjugacy classes. Then, we introduce a new structure for subchains of parallel manipulators using the product of two subgroups of <formula formulatype="inline"><tex>${\rm SE}(3)$</tex></formula> and discuss its realization in terms of the primitive joints. We propose the notion of quotient manipulators that substantially enriches the topologies of serial manipulators. Finally, we present a general procedure for specifying the subchain structures given the desired motion type of a parallel manipulator. The parallel mechanism synthesis problem is thus solved using the realization techniques developed for serial manipulators. Generality of the theory is demonstrated by systematically generating a large class of feasible topologies for (parallel or serial) mechanisms with a desired motion type of either a Lie subgroup or a submanifold. </para>