Safe region prediction and constraint design for INS–polarization–acoustic hybrid navigation

Accurate and robust navigation in complex underwater environments requires the effective integration of heterogeneous sensors. Inertial navigation systems (INS) suffer from drift accumulation, polarization-based heading is sensitive to environmental disturbances, and acoustic positioning is limited by low update rates and measurement uncertainties. Despite recent progress in multi-sensor fusion, systematically modeling and handling uncertainties induced by dynamic sensor availability, environment-dependent degradation, and stochastic measurement errors remains challenging. This paper proposes a probabilistic hybrid navigation framework that tightly fuses INS, polarization, and acoustic information by modeling the system as a stochastic hybrid process with Markovian mode transitions. We further employ forward stochastic reachability to predict the future state distribution under switching sensor configurations and disturbances, from which probabilistic occupancy functions and dynamic confidence sets are derived. These safe regions are integrated as real-time probabilistic constraints in the fusion process to enable adaptive sensor weighting and enhance robustness under degraded or intermittent sensing. Extensive simulations under realistic uncertainty settings demonstrate improved positioning accuracy, reduced constraint violation rates, and better uncertainty bounding, indicating the potential of the proposed framework for long-duration autonomous underwater navigation.