Tree-based Representation of Locally Shortest Paths for 2D k-Shortest Non-homotopic Path Planning

A novel algorithm to solve the 2D k-shortest non-homotopic path planning (k-SNPP) task is proposed in this paper. The task is of practical significance as a sub-module for higherlevel planning and scheduling tasks, and is gaining increasing attention and focus in recent years. There have existed algorithms that explicitly characterised non-homotopic paths using topological invariants such as ℎ-signature and winding number. However, these algorithms are inefficient due to their separate treatment of topology and geometry: Topological invariants are singularly utilised for distinguishing non-homotopic property among paths, which significantly increases the volume of the robot configuration space. Meanwhile, distance-optimal path planners search for locally shortest paths in the augmented space, which becomes extremely time-consuming. In this paper, a topological tree is proposed to simultaneously leverage topology and geometry. The tree grows from the starting location and explores all topological routes, until the best k of its leaves reach the goal. It is proven that different branches of the tree explore different homotopy classes of paths, and all the branches are locally shortest. Comparative experiments for k-SNPP are conducted in challenging grid-based simulated environments to validate the performance of the proposed algorithm. The C++ implementation of the proposed algorithm is released for the benefit of the robotics community.