Steering left-invariant control systems on matrix Lie group

In this paper we generalize a technique for eliminating the drift from the description of a control system on a matrix Lie group with left-invariant vector fields. A diffeomorphism of the state space together with an affine input transformation are used in order to put the system into an equivalent left-invariant drift- free form. Techniques developed for steering drift-free control systems may then be applied. We apply this method to the Lie group of the rigid rotations SO(3) as in the authors' previous work [6], and to a new example, the rigid motions SE(3).