Generalized Watchman Route Problem With Discrete View Cost

Abstract

In this paper, we introduce a generalized version of the Watchman Route Problem (WRP) where the objective is to plan a continuous closed route in a polygon (possibly with holes) and a set of discrete viewpoints on the planned route such that every point on the polygon boundary is visible from at least one viewpoint. The total cost to minimize is a weighted sum of the view cost, proportional to the number of viewpoints, and the travel cost, the total length of the route. We call this problem the Watchman Route Problem with Discrete View Cost or the Generalized Watchman Route Problem (GWRP). In this paper, we consider a restricted version of GWRP that arises naturally in inspection tasks in robotic applications, where each polygon edge is entirely visible from at least one planned viewpoint. We call it Whole Edge Covering GWRP. This whole edge covering restriction is not trivial in that WEC-GWRP has the same NPhardness and inapproximability as GWRP.The algorithm we propose first constructs a graph that connects O(n 12 ) number of sample viewpoints in the polygon, where n is the number of polygon vertices; and then solves the corresponding View Planning Problem with Combined View and Traveling Cost, using an LP-relaxation based algorithm we introduced in [19]. We show that our algorithm has an approximation ratio in the order of either the view frequency, defined as the maximum number of sample viewpoints that cover a polygon edge, or a polynomial of log n, whichever is smaller.

Visibility-based persistent monitoring with robot teams

“…The cost of a path can be modeled as the weighted sum of the length of the path (travel time) and the number of measurements along the path (measurement time). Wang et al [15] first introduced this objective function for WRP for the case of a single robot and termed it the Generalized WRP (GWRP). They showed that GWRP is NP-hard and presented a O(polylog n) approximation for the restricted case when each viewpoint is required to see a complete polygon edge.…”

Three-dimensional coverage planning for an underwater inspection robot

“…Although feasible solutions are of value, an optimal solution is desirable both to permit a shorter-duration inspection and to improve the ease of implementation on a field robotic system. Unfortunately, even in simple cases the coverage path planning problem is equivalent to NP-hard variants of the watchman route problem (Chin and Ntafos, 1988;Wang et al, 2010), and geometric path planning over obstacles is PSPACE-hard in general (Reif, 1979), so we do not seek a globally optimal solution. We propose an iterative improvement procedure which, given a feasible, fullcoverage inspection tour as input, gradually shortens the tour and reduces the number of view configurations, making progress toward a locally optimal solution.…”

Travelling route of mobile surveillance camera

“…The problems related to travelling route problems of mobile robots are handled in [8] and [9]. They consider that a travelling route is generated by travelling salesman problem (TSP) based method which connects fixed camera locations to cover all regions.…”