Persistent manifolds of the special Euclidean group SE(3): A review

Mechanisms and robots often share the following fundamental property: the instantaneous twist space generated by the end-effector at a generic pose is a rigidly-displaced copy of the one generated at the home configuration, i.e., the tangent spaces at all points of its motion manifold (a manifold of the Lie group of rigid displacements SE(3)) are mutually congruent. A manifold of this kind, hereafter denoted as persistent, can be seen as the envelope of a persistent twist subspace rigidly moving in SE(3). In this paper, we shall summarize three important classes of persistent manifolds that have so far been discovered and systematically investigated in the literature, namely the Lie subgroups, the persistent product-of-exponential (POE) manifolds, and the symmetric subspaces. In each case, the persistence property arises from a distinct manifold structure, which dictates the ensuing classification and underlies the framework for the synthesis of mechanical devices that are capable of generating such manifolds. In this regard, we attempt to offer a guideline to classification and mechanism synthesis of persistent manifolds for a general audience.