Effective exploration and coverage under resource limitations is crucial for many applications such as planetary exploration and search and rescue. Optimizing the use of limited resources while effectively exploring an area is vital in scenarios where sensing is expensive, exhaustive, or has adverse effects. In this thesis we present a novel, sparse sensing motion planning algorithm for autonomous mobile robots in coverage problems with limited sensing resources. We approach this problem using ergodic search processes, which produce trajectories that drive robots to spend time in areas in proportion to the expected amount of information in those regions. We recast the ergodic search problem as a mixed-integer optimization problem in order to determine when and where a sensor measurement should be taken while optimizing the agent’s trajectory for coverage. We further employ a continuous relaxation of the presented sparse ergodic optimization problem to reduce computation time. We show that our approach performs comparably to dense sampling methods in terms of coverage performance, by gathering information-rich measurements while adhering to sensing resource constraints. We extend our formulation to problems involving multiple agents, and experiments demonstrate the capability of our approach to automatically distribute sensing resources across a team. Additionally, experiments demonstrate the applicability of our approach to both synthetic and real-world data