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Soft Subdivision Search in Motion Planning, II: Axiomatics

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Frontiers in Algorithmics (FAW 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9130))

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Abstract

We propose to design motion planning algorithms with a strong form of resolution completeness, called resolution-exactness. Such planners can be implemented using soft predicates within the subdivision paradigm. The advantage of softness is that we avoid the Zero problem and other issues of exact computation. Soft Subdivision Search (SSS) is an algorithmic framework for such planners. There are many parallels between our framework and the well-known Probabilistic Road Map (PRM) framework. Both frameworks lead to algorithms that are practical, flexible, extensible, with adaptive and local complexity. Our several recent papers have demonstrated these favorable properties on various non-trivial motion planning problems. In this paper, we provide a general axiomatic theory underlying these results. We also address the issue of subdivision in non-Euclidean configuration spaces, and how exact algorithms can be recovered using soft methods.

C.K. Yap—Plenary Talk at the 9th Int’l. Frontiers of Algorithmics Workshop (FAW 2015) in Guilin, China, July 3–5. This work is supported by NSF Grants CCF-0917093 and CCF-1423228.

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Notes

  1. 1.

    Except speed of light which is exactly known, by definition.

  2. 2.

    To do this, it would have to detect (probabilistically) that the sampling is dense enough, a non-trivial extension of the current PRM formulations.

  3. 3.

    A geometric predicate is a 3-valued function, with a distinguished value \(0\) called the indefinite value. The others are called definite values. This is in contrast to a logical predicate which is 2-valued.

  4. 4.

    We use the notation in, e.g., [3]. This means there is a set, denoted \(set\mathtt{-}\mathtt{Expand}(B)\), of subdivisions of \(B\), and \(\mathtt{Expand}(B)\) denotes (non-deterministically) any element of this set. We assume \(set\mathtt{-}\mathtt{Expand}(B)\) is non-empty so that \(\mathtt{Expand}(B)\) is a total function.

  5. 5.

    This term is from the interval literature. Though not strictly necessary, but it simplifies some arguments.

  6. 6.

    We are indebted to Steve LaValle for asking this question at the IROS 2011 Workshop in San Francisco.

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Acknowledgments

I am indebted to Yi-Jen Chiang, Danny Halperin, Steve LaValle, and Vikram Sharma for many helpful discussions.

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Authors and Affiliations

  1. Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Chee K. Yap

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  1. Chee K. Yap
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Correspondence to Chee K. Yap .

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Editors and Affiliations

  1. Central South University, Changsha, China

    Jianxin Wang

  2. Courant Institute, New York University, New York, New York, USA

    Chee Yap

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Yap, C.K. (2015). Soft Subdivision Search in Motion Planning, II: Axiomatics. In: Wang, J., Yap, C. (eds) Frontiers in Algorithmics. FAW 2015. Lecture Notes in Computer Science(), vol 9130. Springer, Cham. https://doi.org/10.1007/978-3-319-19647-3_2

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  • DOI: https://doi.org/10.1007/978-3-319-19647-3_2

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