Localization in Spatiotemporal Fields via Environmental PDEs

2026-07-31

AI Summary

This study addresses the challenge of high-precision localization for autonomous vehicles in GNSS-denied aquatic environments by proposing a novel approach that integrates physics-driven environmental field information. The method employs partial differential equations—specifically the shallow water equations and convection–diffusion equations—to model spatiotemporal fields such as temperature and salinity. Numerical solutions of these PDEs generate predictive environmental fields, which are incorporated as multimodal observations into a Rao–Blackwellized particle filter (RBPF). This framework uniquely leverages PDE-governed environmental fields as localization features while decoupling nonlinear vehicle pose estimation from linear sensor bias correction, substantially reducing the required number of particles and effectively compensating for sensor drift. Both simulations and real-world experiments demonstrate that the proposed method consistently outperforms standard particle filtering across varying particle counts, and that measured environmental fields exhibit sufficient spatial variability to enable reliable localization.

Abstract

This paper proposes a localization framework that uses spatiotemporal fields governed by partial differential equations (PDEs) as localization signatures. Two PDE classes are considered: the shallow water equations, which describe free-surface flows in coastal and riverine environments, and the advection-diffusion equation, which models the transport and mixing of scalar quantities such as temperature, salinity, and dissolved oxygen. A numerical PDE solver provides predicted fields over the domain, and multiple field channels are fused as multimodal measurements to improve localization accuracy. We formulate the problem within a Rao-Blackwellized particle filter (RBPF) that partitions the vehicle state into a nonlinear component sampled by particles and a linear sensor bias component tracked analytically via per-particle Kalman filters. This factorization reduces the required number of particles compared to a standard particle filter while accounting for realistic sensor drift. Simulation studies on both PDE scenarios show that the RBPF consistently outperforms a standard particle filter in terms of final position error and Root Mean Square Error (RMSE) across varying particle counts. Field experiments with an autonomous surface vehicle measuring salinity, temperature, and dissolved oxygen validate that PDE-governed environmental fields provide sufficient spatial variability for practical localization. Related experimental videos are available at https://localization-environmental-pdes.github.io/.

Problem

• localization
• spatiotemporal fields
• partial differential equations
• environmental sensing
• autonomous surface vehicle

Innovation

• spatiotemporal fields
• partial differential equations (PDEs)
• Rao-Blackwellized particle filter
• multimodal environmental sensing
• autonomous vehicle localization