Geometry of Adjoint-Invariant Submanifolds of SE(3)

This article aims to extend the theory of Lie subgroups and symmetric subspaces for studying an important class of submanifolds of the special Euclidean group
$\text{SE}(3)$
whose tangent space at each point on the submanifold relates to that at the identity by an adjoint map. These submanifolds, which we call adjoint-invariant submanifolds in this article, are known in the literature as persistent submanifolds, since they are strictly related to the concept of persistent screw systems. The difference is that in this article, just as Lie subgroups and symmetric subspaces, we put forward adjoint-invariant submanifolds as independent geometric objects from mechanisms and their associated local screw systems. Adjoint invariance relaxes the strict left and right invariance of Lie subgroups and the reflective invariance of symmetric subspaces by allowing generic moving reference frame in the aforementioned adjoint map. It turns out such adjoint invariance can be studied under the framework of distributions on manifolds, which allows us to explore global geometric properties of adjoint-invariant submanifolds. We classify adjoint-invariant submanifolds into reflective-type and product-type submanifolds and derive the conditions for their adjoint invariance. We then propose geometric methods and algorithms for synthesizing the kinematic generators for reflective-type submanifolds, as demonstrated with a number of examples.