In this paper, we address the issue of solving the motion planning problems (MPP for short) for a class of left-invariant control systems Σ whose state spaces are semisimple compact Lie groups. The sub-Riemannian metrics induced by the dynamics of Σ admit nontrivial abnormal extremals. This fact a priori represents an obstruction for the procedure we use to tackle the MPP, which consists of a homotopy (or continuation) method. We are however able to provide complete answers for the MPP. © 2002 Elsevier Science B.V. All rights reserved.