  This paper investigates performance guarantees on coverage-based ergodic
exploration methods in environments containing disturbances. Ergodic
exploration methods generate trajectories for autonomous robots such that time
spent in an area is proportional to the utility of exploring in the area.
However, providing formal performance guarantees for ergodic exploration
methods is still an open challenge due to the complexities in the problem
formulation. In this work, we propose to formulate ergodic search as a
differential game, in which a controller and external disturbance force seek to
minimize and maximize the ergodic metric, respectively. Through an
extended-state Bolza-form transform of the ergodic problem, we demonstrate it
is possible to use techniques from reachability analysis to solve for optimal
controllers that guarantee coverage and are robust against disturbances. Our
approach leverages neural-network based methods to obtain approximate value
function solutions for reachability problems that mitigate the increased
computational scaling due to the extended state. As a result, we are able to
compute continuous value functions for the ergodic exploration problem and
provide performance guarantees for coverage under disturbances. Simulated and
experimental results demonstrate the efficacy of our approach to generate
robust ergodic trajectories for search and exploration with external
disturbance force.
