Raspi$^2$USBL: An open-source Raspberry Pi-Based Passive Inverted Ultra-Short Baseline Positioning System for Underwater Robotics

Precise underwater positioning remains a fundamental challenge for underwater robotics since global navigation satellite system (GNSS) signals cannot penetrate the sea surface. This paper presents Raspi$^2$USBL, an open-source, Raspberry Pi-based passive inverted ultra-short baseline (piUSBL) positioning system designed to provide a low-cost and accessible solution for underwater robotic research. The system comprises a passive acoustic receiver and an active beacon. The receiver adopts a modular hardware architecture that integrates a hydrophone array, a multichannel preamplifier, an oven-controlled crystal oscillator (OCXO), a Raspberry Pi 5, and an MCC-series data acquisition (DAQ) board. Apart from the Pi 5, OCXO, and MCC board, the beacon comprises an impedance-matching network, a power amplifier, and a transmitting transducer. An open-source C++ software framework provides high-precision clock synchronization and triggering for one-way travel-time (OWTT) messaging, while performing real-time signal processing, including matched filtering, array beamforming, and adaptive gain control, to estimate the time of flight (TOF) and direction of arrival (DOA) of received signals. The Raspi$^2$USBL system was experimentally validated in an anechoic tank, freshwater lake, and open-sea trials. Results demonstrate a slant-range accuracy better than 0.1%, a bearing accuracy within 0.1$^\circ$, and stable performance over operational distances up to 1.3 km. These findings confirm that low-cost, reproducible hardware can deliver research-grade underwater positioning accuracy. By releasing both the hardware and software as open-source, Raspi$^2$USBL provides a unified reference platform that lowers the entry barrier for underwater robotics laboratories, fosters reproducibility, and promotes collaborative innovation in underwater acoustic navigation and swarm robotics.