Optimal curvature-constrained paths with anisotropic costs in the plane

The problem of constructing an optimal curvature-constrained path between two directed points in the plane where the cost depends on the instantaneous direction of the path has applications to underground mining and other path planning problems. In particular, the anisotropic velocity in the formulation of this optimal control problem captures the geological characteristics when developing an underground mine decline in a region of directional faulting. In this paper, we present a generalised version of the Dubins result, that the optimal curvature-constrained path is of the form CSCSC where C is an arc of maximal curvature and S is a straight line.