  This paper presents a rapid (real time) solution to the minimum-time path
planning problem for Dubins vehicles under environmental currents (wind or
ocean currents). Real-time solutions are essential in time-critical situations
(such as replanning under dynamically changing environments or tracking fast
moving targets). Typically, Dubins problem requires to solve for six path
types; however, due to the presence of currents, four of these path types
require to solve the root-finding problem involving transcendental functions.
Thus, the existing methods result in high computation times and their
applicability for real-time applications is limited. In this regard, in order
to obtain a real-time solution, this paper proposes a novel approach where only
a subset of two Dubins path types (LSL and RSR) are used which have direct
analytical solutions in the presence of currents. However, these two path types
do not provide full reachability. We show that by extending the feasible range
of circular arcs in the LSL and RSR path types from $2\pi$ to $4\pi$: 1) full
reachability of any goal pose is guaranteed, and 2) paths with lower time costs
as compared to the corresponding $2\pi$-arc paths can be produced. Theoretical
properties are rigorously established, supported by several examples, and
evaluated in comparison to the Dubins solutions by extensive Monte-Carlo
simulations.
