  Most of the existing path-following navigation algorithms cannot guarantee
global convergence to desired paths or enable following self-intersected
desired paths due to the existence of singular points where navigation
algorithms return unreliable or even no solutions. One typical example arises
in vector-field guided path-following (VF-PF) navigation algorithms. These
algorithms are based on a vector field, and the singular points are exactly
where the vector field diminishes. In this paper, we show that it is
mathematically impossible for conventional VF-PF algorithms to achieve global
convergence to desired paths that are self-intersected or even just simple
closed (precisely, homeomorphic to the unit circle). Motivated by this new
impossibility result, we propose a novel method to transform self-intersected
or simple closed desired paths to non-self-intersected and unbounded
(precisely, homeomorphic to the real line) counterparts in a higher-dimensional
space. Corresponding to this new desired path, we construct a singularity-free
guiding vector field on a higher-dimensional space. The integral curves of this
new guiding vector field is thus exploited to enable global convergence to the
higher-dimensional desired path, and therefore the projection of the integral
curves on a lower-dimensional subspace converge to the physical
(lower-dimensional) desired path. Rigorous theoretical analysis is carried out
for the theoretical results using dynamical systems theory. In addition, we
show both by theoretical analysis and numerical simulations that our proposed
method is an extension combining conventional VF-PF algorithms and trajectory
tracking algorithms. Finally, to show the practical value of our proposed
approach for complex engineering systems, we conduct outdoor experiments with a
fixed-wing airplane in windy environment to follow both 2D and 3D desired
paths.
