Multidimensional Online Robot Motion

TL;DR

This paper investigates online algorithms for multidimensional robot movement tasks, revealing unbounded competitiveness in higher dimensions and proposing optimally competitive solutions under fixed clearance assumptions.

Contribution

It demonstrates the unbounded competitiveness of online algorithms in higher dimensions and introduces optimally competitive algorithms with fixed clearance.

Findings

No upper bound on competitiveness in 3D and higher dimensions.

Existence of optimally competitive algorithms with fixed clearance.

Analysis of robot movement problems in arbitrary dimensions.

Abstract

We consider three related problems of robot movement in arbitrary dimensions: coverage, search, and navigation. For each problem, a spherical robot is asked to accomplish a motion-related task in an unknown environment whose geometry is learned by the robot during navigation. The robot is assumed to have tactile and global positioning sensors. We view these problems from the perspective of (non-linear) competitiveness as defined by Gabriely and Rimon. We first show that in 3 dimensions and higher, there is no upper bound on competitiveness: every online algorithm can do arbitrarily badly compared to the optimal. We then modify the problems by assuming a fixed clearance parameter. We are able to give optimally competitive algorithms under this assumption.