  The paper is concerned with elongating the shortest curvature-bounded path
between two oriented points to an expected length. The elongation of
curvature-bounded paths to an expected length is fundamentally important to
plan missions for nonholonomic-constrained vehicles in many practical
applications, such as coordinating multiple nonholonomic-constrained vehicles
to reach a destination simultaneously or performing a mission with a strict
time window. In the paper, the explicit conditions for the existence of
curvature-bounded paths joining two oriented points with an expected length are
established by applying the properties of the reachability set of
curvature-bounded paths. These existence conditions are numerically verifiable,
allowing readily checking the existence of curvature-bounded paths between two
prescribed oriented points with a desired length. In addition, once the
existence conditions are met, elongation strategies are provided in the paper
to get curvature-bounded paths with expected lengths. Finally, some examples of
minimum-time path planning for multiple fixed-wing aerial vehicles to
cooperatively achieve a triangle-shaped flight formation are presented,
illustrating and verifying the developments of the paper.
