The Weighted Markov-Dubins Problem

TL;DR

This paper extends the classical Markov-Dubins problem to include asymmetric turn costs and penalties, providing a comprehensive set of optimal path candidates for UAV path planning with curvature and turn penalties.

Contribution

It introduces a generalized formulation of the weighted Markov-Dubins problem, characterizes the optimal paths, and shows the classical $CCC$ path is not optimal under the new weighting.

Findings

Optimal paths are among at most 21 candidates with up to five segments.

The classical $CCC$ path is not optimal in the weighted problem.

Candidate paths reduce to classical paths as penalties approach zero.

Abstract

In this article, 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 a loss in altitude while turning or a UAV with different costs for the sinistral and dextral turns due to hardware failures or environmental conditions. Using optimal control theory, the main result of this paper 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.