Abstract
This paper proposes a novel solution to the problem of computing a set of topologically inequivalent paths between two points in a space given a set of samples drawn from that space. Specifically, these paths are homotopy inequivalent where homotopy is a topological equivalence relation. This is achieved by computing a basis for the group of homology inequivalent loops in the space. An additional distinct element is then computed where this element corresponds to a loop which passes through the points in question. The set of paths is subsequently obtained by taking the orbit of this element acted on by the group of homology inequivalent loops. Using a number of spaces, including a street network where the samples are GPS trajectories, the proposed method is demonstrated to accurately compute a set of homotopy inequivalent paths. The applications of this method include path and coverage planning.
Highlights
The task of path planning may be defined as computing a path or set of paths between two points in a given space
Topological path planning is a type of path planning which does not distinguish between paths which are topological equivalent
We propose a novel sample-based topological path planning method that formulates the task in terms of doing inference with respect to homology
Summary
The task of path planning may be defined as computing a path or set of paths between two points in a given space. Sample-baseds method overcome these challenges with the compromise of computing an approximate solution Such methods represent the space using a set of samples drawn from the space and subsequently search for paths within this representation. The problem of computing topologically inequivalent paths using a sample-based approach has only recently been considered with [3] proposing the first such method. We propose a novel sample-based topological path planning method that formulates the task in terms of doing inference with respect to homology. The layout of this paper is as follows: Section 2 reviews related works on topological path planning; Section 3 describes how the concept of topologically inequivalent paths may be defined formally in terms of homotopy and homology theory; and Section 4 describes the proposed sample-based topological path planning method.
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