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Coverage Path Planning For Minimizing Expected Time to Search For an Object With Continuous Sensing

Linh Nguyen

TL;DR

This paper proposes the quota lawn mowing problem, an extension of the classic lawn mowing problem in computational geometry, and gives constant-factor approximations for the quota lawn mowing problem, which is related to the watchman route problem.

Abstract

In this paper, we present several results of both theoretical as well as practical interests. First, we propose the quota lawn mowing problem, an extension of the classic lawn mowing problem in computational geometry, as follows: given a quota of coverage, compute the shortest lawn mowing route to achieve said quota. We give constant-factor approximations for the quota lawn mowing problem. Second, we investigate the expected detection time minimization problem in geometric coverage path planning with local, continuous sensory information. We provide the first approximation algorithm with provable error bounds with pseudopolynomial running time. Our ideas also extend to another search mechanism, namely visibility-based search, which is related to the watchman route problem. We complement our theoretical analysis with some simple but effective heuristics for finding an object in minimum expected time, on which we provide simulation results.

Coverage Path Planning For Minimizing Expected Time to Search For an Object With Continuous Sensing

TL;DR

This paper proposes the quota lawn mowing problem, an extension of the classic lawn mowing problem in computational geometry, and gives constant-factor approximations for the quota lawn mowing problem, which is related to the watchman route problem.

Abstract

In this paper, we present several results of both theoretical as well as practical interests. First, we propose the quota lawn mowing problem, an extension of the classic lawn mowing problem in computational geometry, as follows: given a quota of coverage, compute the shortest lawn mowing route to achieve said quota. We give constant-factor approximations for the quota lawn mowing problem. Second, we investigate the expected detection time minimization problem in geometric coverage path planning with local, continuous sensory information. We provide the first approximation algorithm with provable error bounds with pseudopolynomial running time. Our ideas also extend to another search mechanism, namely visibility-based search, which is related to the watchman route problem. We complement our theoretical analysis with some simple but effective heuristics for finding an object in minimum expected time, on which we provide simulation results.
Paper Structure (19 sections, 6 theorems, 9 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 6 theorems, 9 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and Related Work
  2. Preliminaries
  3. The quota lawn mowing problem
  4. The expected detection time minimization problem
  5. Lawn Mowing Problem with a Quota
  6. The quota TSP
  7. Circular cutter
  8. Minimizing Expected Time to Search For an Object
  9. Lawn mowing search mechanism
  10. Approximation algorithm
  11. Running time
  12. Analyzing the approximation factor
  13. Remark
  14. Remark
  15. Visibility-based search mechanism
  16. ...and 4 more sections

Key Result

lemma 1

For a unit square cutter, there exists a tour $\gamma_{\mathcal{G}}$ whose vertices are a subset of the vertices of $\mathcal{G}$ such that $|\gamma_{\mathcal{G}}| \le O(1)|\gamma^*|$ and $C(\gamma^*) \subseteq C(\gamma_{\mathcal{G}})$.

Figures (9)

  • Figure 1: The area of coverage $C(\gamma)$ (red) by a robot with circular operational area $\chi$ traveling along $\gamma$.
  • Figure 2: The pixels $\mathcal{P}$ (grey) and the dual grid graph $\mathcal{G}$ (red).
  • Figure 3: The hexagonal pixels for the case of a circular cutter.
  • Figure 4: The construction used in showing NP-hardness of minimizing $E[T\mid\gamma]$.
  • Figure 5: The graph of $f(t)$ (left) and the graph of $f'(p)$ (right), the latter is acquired from switching the two axes, effectively "flipping" $f(t)$ and removing some vertical segments. Both envelope the same amount of area of the Cartesian plane (shaded).
  • ...and 4 more figures

Theorems & Definitions (10)

  • lemma 1
  • proof
  • theorem 1
  • proof
  • theorem 2
  • theorem 3
  • proof
  • lemma 2
  • theorem 4
  • proof