Symmetric Subspaces of Euclidean Group: Characterization and Robotic Applications

The special Euclidean group SE(3) is a symmetric space under inversion symmetry. It admits seven conjugacy classes of symmetric subspaces, which are submanifolds closed under inversion symmetry. This paper characterizes the symmetric subspaces of SE(3) in three distinct ways, each focusing on some particular applications in robotics. First, they are the exponential of Lie triple systems of the Lie algebra se(3) of SE(3), leading to a list of Lie theoretical properties which are of crucial importance in mechanism analysis and synthesis. Next, they (except one) are symmetric bundle subspaces of TSO(3), the tangent bundle of SO(3). The non-triviality of these subbundle explains the kinematic behavior of several kinesiological systems, which implies a potential application in rehabilitation robotics. Finally, they (except one) are either projective subspaces or quadrics of the Study quadric under dual quaternion representation. This third characterization is connected to the study of overconstrained linkages and line/plane symmetric motions.