Polygon Exploration with Time-Discrete Vision

With the advent of autonomous robots with two- and three-dimensional scanning capabilities, classical visibility-based exploration methods from computational geometry have gained in practical importance. However, real-life laser scanning of useful accuracy does not allow the robot to scan continuously while in motion; instead, it has to stop each time it surveys its environment. This requirement was studied by Fekete, Klein and Nuechter for the subproblem of looking around a corner, but until now has not been considered in an online setting for whole polygonal regions. We give the first algorithmic results for this important algorithmic problem that combines stationary art gallery-type aspects with watchman-type issues in an online scenario: We demonstrate that even for orthoconvex polygons, a competitive strategy can be achieved only for limited aspect ratio A (the ratio of the maximum and minimum edge length of the polygon), i.e., for a given lower bound on the size of an edge; we give a matching upper bound by providing an O(log A)-competitive strategy for simple rectilinear polygons, using the assumption that each edge of the polygon has to be fully visible from some scan point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some details (title, figures and text) for final journal revision, including explicit assumption of full edge visibilit

A simple proof for visibility paths in simple polygons

The purpose of this note is to give a simple proof for a necessary and sufficient condition for visibility paths in simple polygons. A visibility path is a curve such that every point inside a simple polygon is visible from at least one point on the path. This result is essential for finding the shortest watchman route inside a simple polygon specially when the route is restricted to curved paths

Approximation Algorithms for the Two-Watchman Route in a Simple Polygon

The two-watchman route problem is that of computing a pair of closed tours in an environment so that the two tours together see the whole environment and some length measure on the two tours is minimized. Two standard measures are: the minmax measure, where we want the tours where the longest of them has smallest length, and the minsum measure, where we want the tours for which the sum of their lengths is the smallest. It is known that computing a minmax two-watchman route is NP-hard for simple rectilinear polygons and thus also for simple polygons. Also, any c-approximation algorithm for the minmax two-watchman route is automatically a 2c-approximation algorithm for the minsum two-watchman route. We exhibit two constant factor approximation algorithms for computing minmax two-watchman routes in simple polygons with approximation factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively, where n is the number of vertices of the polygon. We also use the same techniques to obtain a 6.922-approximation for the fixed two-watchman route problem running in O(n^2) time, i.e., when two starting points of the two tours are given as input.Comment: 36 pages, 14 figure

Minimizing Turns in Watchman Robot Navigation: Strategies and Solutions

The Orthogonal Watchman Route Problem (OWRP) entails the search for the shortest path, known as the watchman route, that a robot must follow within a polygonal environment. The primary objective is to ensure that every point in the environment remains visible from at least one point on the route, allowing the robot to survey the entire area in a single, continuous sweep. This research places particular emphasis on reducing the number of turns in the route, as it is crucial for optimizing navigation in watchman routes within the field of robotics. The cost associated with changing direction is of significant importance, especially for specific types of robots. This paper introduces an efficient linear-time algorithm for solving the OWRP under the assumption that the environment is monotone. The findings of this study contribute to the progress of robotic systems by enabling the design of more streamlined patrol robots. These robots are capable of efficiently navigating complex environments while minimizing the number of turns. This advancement enhances their coverage and surveillance capabilities, making them highly effective in various real-world applications.Comment: 6 pages, 3 figure