Consider a control-affine nonholonomic system (Sigma). We present in this paper a new desingularization procedure which constructs explicitly from (Sigma) a regular control-affine nonholonomic system. The main idea is to add new variables, thus augmenting the dimension of the state space, such that its projection coincides with the original control system (Sigma). This construction will allow us to devise a global and fully constructive motion planning method for general driftless and control-affine nonholonomic systems under the sole assumption of the Lie Algebraic Rank Condition.