Classification of the three-dimensional persistent POE manifolds of SE(3)

The set of rigid-body displacements of the end-effector of a mechanism or robot is ordinarily a manifold of the special Euclidean group SE(3). This manifold is usually endowed with properties that have physical relevance, related to the application at hand. The most usual properties concern the instantaneous motions of the end-effector, which form vector spaces of twists, namely the tangent spaces of the manifold. The twist space generated at any generic pose is often required to be a rigidly-displaced copy of the twist space generated in the home configuration, since this guarantees uniform instantaneous motion capabilities throughout the workspace. When this happens, the twist space is said to be persistent, and the corresponding manifold, also called persistent, can be thought of as the envelope of a rigidly-moving twist space. There are three known families of persistent manifolds of SE(3). The first and most common one comprises the Lie groups of SE(3), for which the twist space at each pose is invariant and coincides with a subalgebra of the Lie algebra $se(3)$ of SE(3). The second family is composed of the symmetric spaces of SE(3), for which the twist space is a persistent Lie triple system. The third family includes a subset of the product-of-exponential (POE) manifolds of SE(3). The latter emerge by taking the product of two or more Lie groups, and they naturally describe the motion capabilities of serial chains. While the classification of Lie groups and symmetric spaces of SE(3) is state-of-the-art, the exhaustive classification of persistent POE manifolds is yet to be completed. This paper provides the detailed derivation and the comprehensive classification of persistent POE manifolds of dimension three.