Some five dimensional persistent submanifolds of the Study quadric

Abstract

This work studies persistent submanifolds of the group of rigid-body displacements. At each point of a submanifold the tangent space, translated to the identity in the group becomes a screw system. For persistent submanifolds these screws systems are mutually conjugate by a rigid-body transformation. We present novel results on persistent submanifolds composed of three subgroups and products of subgroups with persistent submanifolds. We also show that some submanifolds defined as solutions to geometric constraint problems are persistent.Examples of 5-dimensional persistent submanifolds are then studied. We show that many examples can be show to be persistent in several ways. We also show that there are large overlaps between different types of submanifolds. Many constraint varieties can be realised by open-loop kinematic chains and many of these persistent 5-dimensional submanifolds are the intersection of the Study quadric with algebraic hypersurfaces. Many of the examples considered are well known but we also introduce several novel examples of 5-dimensional persistent submanifolds.