The Weighted Markov-Dubins Problem

In this letter, a variation of the classical Markov-Dubins problem is considered, which deals with curvature-constrained least-cost paths in a plane with prescribed initial and final configurations, different bounds for the sinistral and dextral curvatures, and penalties
$\mu _{L}$
and
$\mu _{R}$
for the sinistral and dextral turns, respectively. The addressed problem generalizes the classical Markov-Dubins problem and the asymmetric sinistral/dextral Markov-Dubins problem. The proposed formulation can be used to model an Unmanned Aerial Vehicle (UAV) with a penalty associated with a turn due to the required additional thrust to maintain altitude and airspeed while turning, or a UAV with different curvature bounds and costs for the sinistral and dextral turns due to hardware failures. Using optimal control theory, the main result of this letter shows that the optimal path belongs to a set of at most 21 candidate paths, each comprising of at most five segments. Unlike in the classical Markov-Dubins problem, the
$CCC$
path, which is a candidate path for the classical Markov-Dubins problem, is not optimal for the weighted Markov-Dubins problem. Moreover, the obtained list of candidate paths for the weighted Markov-Dubins problem reduces to the standard
$CSC$
and
$CCC$
paths and the corresponding degenerate paths when
$\mu _{L}$
and
$\mu _{R}$
approach zero.