On the Optimality of Ergodic Trajectories for Information Gathering Tasks

Abstract: Recently, ergodic control has been suggested as a means to guide mobile sensors for information gathering tasks. In ergodic control, a mobile sensor follows a trajectory that is ergodic with respect to some information density distribution. A trajectory is ergodic if time spent in a state space region is proportional to the information density of the region. Although ergodic control has shown promising experimental results, there is little understanding of why it works or when it is optimal. In this paper, we study a problem class under which optimal information gathering trajectories are ergodic. This class relies on a submodularity assumption for repeated measurements from the same state. It is assumed that information available in a region decays linearly with time spent there. This assumption informs selection of the horizon used in ergodic trajectory generation. We support our claims with a set of experiments that demonstrate the link between ergodicity, optimal information gathering, and submodularity.