Localization of mobile robots seems to be, at first sight, a simple subproblem of distance geometry: the goal is to localize one single moving device, in either two or three dimensions. Comparing this problem with the joint localization of multiple entities in high dimensions, we would expect that simple solutions and guarantees exist. However, limitations in terms of available hardware, measurement accuracy and bandwidth constitute real challenges that push existing distance geometry techniques to their limits.

In this presentation, we will present two different applications from mobile robotics that showcase that these problems are 1) not trivial and 2) very interesting to study. First, we discuss the localization of an autonomous lawn mower measuring only one distance at a time, from fixed anchors. Despite the apparent simplicity of this problem, no theoretical guarantees exist for the recovery of the trajectory in this setting. Using a relaxation, we propose a novel algorithm to recover the trajectory, and we prove sufficient recovery conditions. Second, we discuss echo-location of walls on a mobile drone. We present a working solution for distance and angle recovery "on the fly", respecting the limited hardware capabilities. Finally, we present a number of open problems in distance geometry for mobile robotics and interesting research directions.