Optimal curvature-constrained paths for general directional-cost functions

Abstract

This paper studies the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We show that there always exists a path of the form $\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$ or a degeneracy which is optimal, where $\mathcal {C}$ represents an arc of maximum curvature, and $\mathcal {S}$ represents a straight line. This result is also extended to the case where there is not only a directional-cost, but the cost of curved sections are scaled up by a factor w ^C≥1. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint, the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground, and curved sections may incur more cost due to additional support and ventilation.