Mapping and coverage with a particle swarm controlled by uniform inputs

Abstract

We propose an approach to mapping tissue and vascular systems without the use of contrast agents, based on moving and measuring magnetic particles. To this end, we consider a swarm of particles in a 1D or 2D grid that can be tracked and controlled by an external agent. Control inputs are applied uniformly so that each particle experiences the same applied forces. We present algorithms for three tasks: (1) Mapping, i.e., building a representation of the free and obstacle regions of the workspace; (2) Subset Coverage, i.e., ensuring that at least one particle reaches each of a set of desired locations; and (3) Coverage, i.e., ensuring that every free region on the map is visited by at least one particle. These tasks relate to a large body of previous work from robot navigation, both from theory and practice, which is based on individual control. We provide theoretical insights that have potential relevance for fast MRI scans with magnetically controlled contrast media. In particular, we develop a fundamentally new approach for searching for an object at an unknown distance D, where the search is subject to two different and independent cost parameters for moving and for measuring. We show that regardless of the relative cost of these two operations, there is a simple O(log D/log log D)-competitive strategy, which is the best possible. Also, we provide practically useful and computationally efficient strategies for higher-dimensional settings. These algorithms extend to any number of particles and show that additional particles tend to reduce the mean and the standard deviation of the time required for each task.