Robotic mapping and simultaneous localization and mapping (SLAM) typically rely on sensors that produce a large number of measurements at many locations in an environment to produce an accurate map and, in the case of SLAM, the pose of the robot in that map. However, with the advent of small, low-power robots with insect-scale features, there is a need for techniques that can produce useful maps using limited capability sensors and a small number of measurements. In this work, we focus on the use of compressive sensing to extract local environment reconstructions from ultrasonic sensor measurements. We first examine a simplistic setting where a square pulse is emitted and use the returned echoes in a compressive sensing scheme to reconstruct the locations of objects inside the sensing cone. We then extend this to the more practical setting, accounting for the wave nature of the acoustic signal and corresponding issues of interference, showing that these can be accounted for in designing the measurement matrix of the compressive sensing description of the problem. We demonstrate the performance of our approach though several simulations.

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