  Vector field guided path following (VF-PF) algorithms are fundamental in
robot navigation tasks, but may not deliver the desirable performance when
robots encounter singular points where the vector field becomes zero. The
existence of singular points prevents the global convergence of the vector
field's integral curves to the desired path. Moreover, VF-PF algorithms, as
well as most of the existing path following algorithms, fail to enable
following a self-intersected desired path. In this paper, we show that such
failures are fundamentally related to the mathematical topology of the path,
and that by "stretching" the desired path along a virtual dimension, one can
remove the topological obstruction. Consequently, this paper proposes a new
guiding vector field defined in a higher-dimensional space, in which
self-intersected desired paths become free of self-intersections; more
importantly, the new guiding vector field does not have any singular points,
enabling the integral curves to converge globally to the "stretched" path. We
further introduce the extended dynamics to retain this appealing global
convergence property for the desired path in the original lower-dimensional
space. Both simulations and experiments are conducted to verify the theory.
