The group SE(3) of all rigid body motions and its subgroups play a key role in the analysis and synthesis of robot mechanisms with desired motion types. The most successful subgroup-motion mechanisms are the SCARA robots ( whose motion type is descibed by the 4-D Schoenflies group X(z) ), the famous Delta robot invented by Clavel ( whose motion type is T(3)), and many 6-axis robots (whose motion type is SE(3) itself). Every robot arm/mechanism/manipulator generates a subset of SE(3). Understanding/classifying possible lower-dimensional submanifolds of SE(3) is one of the most important problems in theoretical robotics.
Lie subgroups of SE(3) have been studied and completely classified by [1],[2],[3]. They have perfect symmetries (left and right invariance). Another set of important submanifolds of SE(3) are symmetric subspaces, which satisfy inversion symmetry (w.r.t. the inversion operation on the group). Wu et al. [4][5] first observed that the motion of CV joints and some omni-wrists can be explained as some exponential submanifolds (i.e. exponential of some subspace of se(3)). In an unpublished manuscript written in 2012,2013 by GF Liu, exponential submanifolds were related to the inversion map about a generic point on SE(3) (as motivated by a chapter of John Milnor's book "Morse Theory"). The similar idea was later on also adopted in [4] although it was published first. In particular symmetric subspaces of SE(3) has been completely classified in [4], and mechanisms with symmetric subspace motion types are synthesized in [5].
A significant generalization of Lie subgroups and symmetric subspaces are adjoint-invariant submanifolds. They are extremely important submanifolds of SE(3) just as Lie subgroups and symmetric subspaces (in fact the latter two are just two special cases of the former). Adjoint-invariant submanifolds possess neither left and right invariance, nor inversion symmetry. But their tangent spaces (or the tangent bundle) are invariant under an adjoint map of a generic rigid motion, i.e. their degrees of freedom have clear and invariant physical meaning. Adjoint-invariant submanifolds are completely characterized by their tangent space at identity and a function that determines how the adjoint map are based upon. Therefore they can be analyzed and classified using the theory of distributions on SE(3). Please refer to [6] for the results on analysis and synthesis of mechanisms whose motion sets are adjoint-invariant submanifolds.
[1] J.M. Selig, "Geometric Fundamentals of Robotics". Springer-Verlag, Monographs in Computer Science, 2005.
[2] J.M. Herve, "Lie group of rigid body displacement, a fundamental tool for mechanism design", Mechanism and Machine Theory, Vol 34, No 5, PP 719-730, 1999.
[3] J. Meng, G.F. Liu, and Z.X. Li, ".A Geometric Theory for Analysis and Synthesis of Sub-six dof Parallel Manipulators", IEEE Transactions on Robotics, Vol 23, No. 4, PP 625-649, 2007.
[4] Y. Wu et al, "Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics", IEEE Transactions on Robotics, Vol 32, No 2, PP 312-326, 2016.
[5] Y. Wu and M. Carricato, "Symmetric subspace motion generators", IEEE Transactions on Robotics, Vol 34, No 3, PP. 716-735, 2018.
[6] G.F. Liu, "Geometry of Adjoint-invariant submanifolds of SE(3)", IEEE Transactions on Robotics, Oct. Issue, 2019. IEEE explore online earlier access.