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There are a number of applications for which the in-situ determination of the three-dimensional shape of rod-like structures is desired. Such structures include tethers, towed arrays, antennae, tubes, pipes, hoses, flexible needles, and catheters; Figure 2 shows one important application, where it is desirable to know the location of a seafloor-crawling mine countermeasures vehicle by knowing the shape of its control tether.
In particular, many of these applications restrict the possibilities to embedded finite arrays of sensors whose indirect measurements of some kinematic property (e.g., strain) must be converted to global displacement information. The proposed solution here is two-fold: first, arrays of fiber Bragg grating (FBG) sensors are used to provide local strain information at discrete locations along optical fibers embedded inside or on the surface of the target structure. Next, the main thrust of this project was in developing an appropriate inverse algorithm to convert local strain information into global displacement. For essentially one-dimensional “rod-like” structures, a material basis vector approach was used in modeling the geometry and displacement of such structures, as in Figure 1. The use of a basis that moves with the material has the advantage of being singularity-free (unlike, for example, a Frenet basis, which is tied to the curvature evolution). Appropriate constraint assumptions and local linearization in the strain field in the neighborhood of each n-th sensor results in a linear system of equations valid for that neighborhood that are easily solved for a local basis set of material basis vector and displacement vector solutions. These local solutions are assembled by imposing continuity and boundary conditions between neighborhood segments to arrive at a consistent global solution estimate for displacement. Figure 3 shows some results comparing actual (black) displacement from one finite element simulation of a 212 m tether and the algorithm’s estimated (gray) displacement, as a function of sensor density. The complex deformed shape, which has several curvature and torsional changes, is well estimated by 10 sensors and almost perfect estimated to within FBG noise levels by 20 sensors.