Abstract
This paper investigates the problem of time-optimum movement planning in two and three dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macro-scale motion planning for ships, submarines, and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the meso-scale and micro-scale robots that have been constructed recently, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that an optimum path for the 2D version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the 3D version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the 2D flow path problem, but this decision algorithm has very high computational complexity. We also give the first known efficient approximation algorithms with bounded error.
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Reif, J., Sun, Z. Movement Planning in the Presence of Flows. Algorithmica 39, 127–153 (2004). https://doi.org/10.1007/s00453-003-1079-5
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DOI: https://doi.org/10.1007/s00453-003-1079-5