Cosserat rod-based modeling and stochastic analysis for distributed fiber optics shape-sensing

Distributed Fiber Optic Sensors (DFOS) enable continuous strain measurement along a sensor path, providing data at multiple spatial locations that can be used to back-calculate structural deformations. Shape sensing technology, usually utilizing DFOS, has been explored in various applications such as surgery, motion tracking, and soft robotics. However, without a systematic framework, shape estimation in existing studies is often compromised by deviations and biases, making error analysis problematic. This research presents a comprehensive mathematical framework for accurate shape monitoring using DFOS. A system of differential equations based on Cosserat rod theory is formulated to model the spatial deformation of rods, accounting for nonlinear initial configurations. Then, an analytical solution to this system is derived under the assumption of piecewise constant strain measurements. To address measurement uncertainties, the deterministic model is extended to a system of stochastic differential equations. Due to the complexity of the stochastic system, analytical solutions are generally impractical; therefore, practical pre- and post-measurement error bounds for displacement estimation are derived using an Itô–Taylor expansion. This stochastic framework emphasizes the analysis of error propagation, supported by numerical simulations that identify the key sources of error. Practical examples are presented to demonstrate the method’s applicability, with corresponding error evaluated using the proposed framework.