Kinematic control of rigid-motions in space–time

Classically rigid-motion kinematics are formulated on the Special Euclidean Group or parameterizations of that group. However, this is an approximation of the true configuration space evolving in space–time. Moreover, the Special Euclidean Group provides a sufficiently accurate approximation to a rigid-motion when moving at non-relativistic speeds. Here we present a setting which generalizes a rigid-motion in Euclidean space to rigid-motion in space–time. A kinematic feedback law is presented and proved to almost globally stabilize a rigid-motion in space–time. This setting defines the kinematics on a class of Lie group whose inverse is a function of its transpose and associated metric tensor. It is shown that the Special Euclidean Group is a limiting case of this class of Lie group and this alternative viewpoint can be utilized in stability proofs for rigid-motion kinematic control. An example control design and stability proof on this general class of Lie group is demonstrated.