A locally exact strain-to-displacement approach for shape reconstruction of slender objects using fiber Bragg gratings

This paper presents an algorithm for determining three-dimensional displacement of thin rod- or tether-like structures from a set of scalar strain measurements, for arbitrarily large deformations. The approach employs a material-adapted reference frame and a local linearization approach that results in an exact local basis function set for the displacement and for the material frame evolution. The basis set is shown to be robust to potential singularities from vanishing bending and twisting angle derivatives and from vanishing measured strain. Validation of the approach is performed through comparison with both finite element simulations and an experiment, with average root mean square reconstruction error of 0.01%-1% of the total length depending upon the number of sensors used. Analysis of error due to extraneous noise sources and boundary condition uncertainty shows how error propagates under those effects. (Approved for release: LA-UR-13-21066)