Distributed Fiber-Optic Shape Sensing with Endpoint Error Compensation: Theory and Experimental Validation.

Fiber-optic shape sensing enables real-time monitoring of structural deformation across a wide range of applications. For large-scale structures, Brillouin-based distributed sensing, typically implemented through Brillouin Optical Time Domain Analysis (BOTDA), offers an extended range for quasi-static measurements, albeit its limited spatial resolution degrades reconstruction accuracy. This study addresses this fundamental limitation through the introduction of a novel error compensation algorithm, particularly suited for a Brillouin-based shape sensing system, yet agnostic with respect to the sensing technology. The method leverages both the initial and final points of the sensing path, performing both forward and backward reconstructions and fusing the two trajectories by testing several polynomial and exponential weighting strategies. The algorithm is experimentally validated on a 28.91 m four-core shape sensing fiber cable (length = L), interrogated through BOTDA operating at 50 cm spatial resolution, and reconstructed through the Frenet-Serret frame formulation. Calibration procedures include radial-offset tuning and segment alignment via a hotspot reference. A non-trivial S-shaped geometry is adopted as a case study, specifically addressing curvature discontinuities arising from mixed straight and curved segments. Reconstruction accuracy is quantified through a Euclidean-distance-based Figure of Merit (FOMs). The cubic weighting strategy demonstrates improvements exceeding 86% in all FOMs compared to classical methods without compensation. Specifically, it achieves an RMSE of 0.145 m (0.50% of L), a MAE of 0.109 m (0.38% of L), and a maximum error of 0.341 m (1.18% of L). Remarkably, these percentage errors are of the same order of magnitude as those reported in the literature for Fiber Bragg Grating (FBG) and Optical Frequency Domain Reflectometry (OFDR) systems, indicating that the proposed compensation strategy enables BOTDA-based shape sensing to achieve comparable reconstruction accuracy despite its lower spatial resolution.