System Modelling and Optimization pp 102-111 | Cite as

# Dynamic voronoi diagrams in motion planning

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## Abstract

Given a set of *n* points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant of time, these points define a Voronoi diagram which also changes continuously, except for certain critical instances — so-called *topological events*.

In [Ro 90], an efficient method is presented of *maintaining* the Voronoi diagram over time. Recently Guibas, Mitchell and Roos [GuMiRo 91] improved the trivial quartic upper bound on the number of topological events by almost a linear factor to the nearly cubic upper bound of *O*(*n* ^{2} λ_{s},(*n*)) topological events, where λ_{s}(*n*)) is the maximum length of an (*n, s*)-Davenport-Schinzel sequence and *s* is a constant depending on the motion of the sites. Each topological event uses only *O*(log *n*) time (which is worst-case optimal).

Now in this work, we present a new algorithm for planning the motion of a disc in a dynamic scene of moving sites which is based on the corresponding sequence of Voronoi diagrams. Thereby we make use of the well-known fact, that locally the Voronoi edges are the *safest paths* in the dynamic scene. We present a quite simple approach combining *local* and *global strategies* for planning a feasible path through the dynamic scene.

One basic advantage of our algorithm is that only the topological structure of the dynamic Voronoi diagram is required for the computation. Additionally, our *goal oriented approach* provides that we can maintain an existing feasible path over time. This guarantees that we reach the goal if there is a feasible path in the dynamic scene at all. Finally our approach can easily be extended to general convex objects.

## Keywords

Span Tree Topological Structure Voronoi Diagram Computational Geometry Euclidean Plane## Preview

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