We apply the well-known homotopy continuation method to address the
motion planning problem (MPP) for smooth driftless control-affine
systems. The homotopy continuation method is a Newton-type procedure
to effectively determine functions only defined implicitly. That
approach requires first to characterize the singularities of a
surjective map and next to prove global existence for the solution of
an ordinary differential equation, the Wazewski equation. In the
context of the MPP, the aforementioned singularities are the abnormal
extremals associated to the dynamics of the control system and the
Wazewski equation is an o.d.e. on the control space called the Path
Lifting Equation (PLE). We first show elementary facts
relative to the maximal solution of the PLE such as local existence and
uniqueness. Then we prove two general results, a finite-dimensional
reduction for the PLE on compact time intervals and a
regularity preserving theorem. In a second part, if the Strong Bracket
Generating Condition holds, we show, for
several control spaces, the global existence of the solution of the PLE,
extending a previous result of H.J. Sussmann.