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Hardness of 3D Motion Planning Under Obstacle Uncertainty

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Algorithmic Foundations of Robotics XIII (WAFR 2018)

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Abstract

We consider the problem of motion planning in the presence of uncertain obstacles, modeled as polytopes with Gaussian-distributed faces (PGDF). A number of practical algorithms exist for motion planning in the presence of known obstacles by constructing a graph in configuration space, then efficiently searching the graph to find a collision-free path. We show that such a class of algorithms is unlikely to be efficient in the domain with uncertain obstacles. In particular, we show that safe 3D motion planning among PGDF obstacles is \(NP-\)hard with respect to the number of obstacles, and remains \(NP-\)hard after being restricted to a graph. Our reduction is based on a path encoding of \(3-\)SAT and uses the risk of collision with an obstacle to encode the variable assignment. This implies that, unlike in the known case, planning under uncertainty is hard, even when given a graph containing the solution.

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Funding

We gratefully acknowledge support from the Thomas and Stacey Siebel Foundation; from NSF Fellowship grant DGE-1656518; from NSF grants CCF-1763299, 1420316, 1523767 and 1723381; from AFOSR grant FA9550-17-1-0165; from ONR grant N00014-18-1-2847; from Honda Research; and from Draper Laboratory. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of our sponsors.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Luke Shimanuki

  2. Stanford University, Stanford, CA, 94305, USA

    Brian Axelrod

Authors
  1. Luke Shimanuki
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  2. Brian Axelrod
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Correspondence to Brian Axelrod .

Editor information

Editors and Affiliations

  1. Departamento de Sistemas Digitales, Instituto Tecnológico Autónomo de México, México, Mexico

    Marco Morales

  2. Department of Computer Science, University of New Mexico, Albuquerque, NM, USA

    Lydia Tapia

  3. Universidad Politécnica de Yucatán, Yucatán, Mexico

    Gildardo Sánchez-Ante

  4. Georgia Tech, Atlanta, GA, USA

    Seth Hutchinson

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Shimanuki, L., Axelrod, B. (2020). Hardness of 3D Motion Planning Under Obstacle Uncertainty. In: Morales, M., Tapia, L., Sánchez-Ante, G., Hutchinson, S. (eds) Algorithmic Foundations of Robotics XIII. WAFR 2018. Springer Proceedings in Advanced Robotics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-44051-0_49

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