Metric View Planning Problem with Traveling Cost and Visibility Range Pengpeng Wang * , Ramesh Krishnamurti † , and Kamal Gupta * * RAMP Lab, School of Engineering Science Simon Fraser University Burnaby, BC, CANADA, V5A 1S6 {pwangf, kamal}@cs.sfu.ca † School of Computing Science Simon Fraser University Burnaby, BC, CANADA, V5A 1S6 ramesh@cs.sfu.ca Abstract—In this paper, we consider the problem where a point robot in a 2D or 3D environment equipped with an omnidirectional range sensor of finite range D is asked to inspect a set of surface patches, while minimizing the sum of view cost, proportional to the number of viewpoints planned, and the travel cost, proportional to the length of path traveled. We call it the Metric View Planning Problem with Traveling Cost and Visibility Range or Metric TVPP in short. Via an L-reduction from the set covering problem to a two-dimensional Metric TVPP, we show that the Metric TVPP cannot be approximated within O(log m) ratio by any polynomial algorithm, where m is the number of surface patches to cover. We then analyze a natural two-level algorithm of solving first the view planning problem to get an approximate solution, and then solving, again using an approximation algorithm, the Metric traveling salesman problem to connect the planned viewpoints. We show this greedy algorithm has the approximation ratio of O(log m). Thus, it asymptotically achieves the best approximation ratio one can hope for. I. I NTRODUCTION Imagine that a robot is asked to autonomously scan the artifacts in a historic site and build their complete surface representations. The robotic artifact “documentation” or vir- tual reality environment construction ability automates many tedious work done primarily by human thus far. See [1] and the references therein for some of the few existing works on automating this process; and see Fig. 1 for a simple illustra- tion. For such applications, especially in remote missions, the time and energy spent are a critical factor for the tasks to be successfully completed. Thus, we model this robotic object inspection task as an optimization problem of minimizing the corresponding total cost, a weighted sum of both the view cost and the traveling cost. View cost corresponds to the image processing, image registration and geometric model construction after each view is taken and is proportional to the number of viewpoint planned [16]. Travel cost is the cumulative time and energy consumption due to the robot movements and thus is proportional to the length of the total tour the robot travels. We call it the problem of view planning with combined view and traveling costs, denoted by Traveling VPP [19]. Note that in [1] and the references therein, this optimization of both view and traveling cost is not addressed. robot traveling path viewpoint sensor FOV robot start position object of interest Fig. 1. A Traveling VPP instance. It shows 6 planned sensor viewpoints that totally cover the surface of the object of interest, and the robot traveling tour to realize them. Traveling VPP combines elements of both the NP- complete view planning problem (VPP) and the watchman route problem, and thus generalizes both. Consequently, Traveling VPP is NP-hard. VPP refers to planning the minimum number of viewpoints to completely inspect an object surface. It is considered in the robot vision area [16], where often a sensor positioning system is used within a well controlled and limited workspace. These formulations do not consider the traveling cost of the robot, a critical cost, particularly for large workspaces and remote autonomous missions, where power consumption is a critical factor. On the other hand, the watchman route problem, considered in the computational geometry area [4], [17], asks for the shortest tour to inspect the interior of a two-dimensional polygonal region. It does not consider view cost, a critical cost, particularly for the inspection tasks considered here, where each sensor view and the consequent processing are time-consuming. For VPP, we refer to [16] for a detailed survey. Here we mention a related theoretical work [15], in which the authors show that VPP is reduced to the well-known NP-complete set covering problem (SCP). SCP refers to, given a universe