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

The set of rigid displacements of the end-effector of a mechanism is ordinarily a manifold of the special Euclidean group SE(3). The tangent spaces of the manifold form vector spaces of twists describing the end-effector instantaneous motions. The twist space at any generic pose is often required to be a rigidly-displaced copy of the twist space in the home configuration: when this happens the twist space and the corresponding manifold are called persistent. There are three known families of persistent manifolds of SE(3). The first one comprises the Lie groups of SE(3), for which the twist space is invariant and coincides with a subalgebra of the Lie algebra of SE(3). The second family includes the symmetric spaces of SE(3), for which the twist space is a persistent Lie triple system. The third family is a subset of the product-of-exponential (POE) manifolds, which emerge by the product of two or more Lie groups and naturally describe the motion of serial chains. While the classification of Lie groups and symmetric spaces of SE(3) is state-of-the-art, the classification of persistent POE manifolds is yet to be completed. This paper provides the derivation and classification of persistent POE manifolds of dimension 3.