Minimal curvature-constrained networks

TL;DR

This paper presents an exact algorithm for constructing shortest curvature-constrained networks, called minimum Dubins networks, with applications in underground mining tunnel design.

Contribution

It introduces an exact algorithm for 2D and iterative methods for 3D minimum Dubins networks, addressing a curvature-constrained Steiner problem.

Findings

Steiner point traces a limaçon when varying one terminal.

Algorithm effectively finds shortest Dubins paths with curvature constraints.

Application to underground tunnel network optimization.

Abstract

This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and an iterative method for doing so in 3D space. Such a network will be referred to as a minimum Dubins network, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins network appears in the context of underground mining optimisation, where the aim is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins network problem is similar to the Steiner tree problem, except that the terminals are directed and there is a curvature constraint. We propose the minimum curvature-constrained Steiner point algorithm for determining the optimal location of the Steiner point in a 3-terminal network. We show that when two terminals are fixed and the third varied, the Steiner point traces out a lima\c{c}on.