Equivalence relation and partial ordering of parallel mechanisms with applications to mechanism synthesis and mobility analysis

Abstract

Motion type (or motion pattern) of a mechanism is defined as the set of all rigid motions achievable by the mechanisms’s end-effector; the motion type of a parallel mechanism equals the intersection set of all subchain motion types. The motion type of a non-instantaneous parallel mechanism locally agrees with a regular submanifold (or a Lie subgroup in particular) of the special Euclidean group SE(3). Based on submanifold germs of SE(3), we can define an equivalence relation and a partial order relation for both motion types and parallel mechanisms: two motion types are equivalent if and only if they agree on an open neighborhood around the identity element of SE(3); two motion types are comparable if and only if one is a submanifold of the other on an open neighborhood around the identity of SE(3). It is also possible to define equivalence relation and partial ordering on the collection of parallel mechanisms. In this paper, we first study properties of the equivalence and partial order relation of both motion types and parallel mechanisms, then we discuss their application in type synthesis, mobility analysis and non-overconstrained ness realization of parallel mechanisms.