A Local Material Basis Solution Approach to Reconstructing the Three-Dimensional Displacement of Rod-Like Structures From Strain Measurements

Abstract

This paper presents a new approach for determining three-dimensional global displacement (for arbitrarily sized deformation) of thin rod or tetherlike structures from a limited set of scalar strain measurements. The approach is rooted in Cosserat rod theory with a material-adapted reference frame and a localized linearization approach that facilitates an exact local basis function set for the displacement along with the material frame. The solution 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 a comparison with both finite element simulations and an experiment, with average root mean square reconstruction error of 0.01%-]% of the total length, for reasonable sensor counts. An analysis of error due to extraneous noise sources and boundary condition uncertainty shows how the error scales with those effects. The algorithm involves relatively simple operations, the most complex of which is square matrix inversion, lending itself to potential low-power embeddable solutions for applications requiring shape reconstruction.

A Generalized Approach for Reconstructing the Three-Dimensional Shape of Slender Structures Including the Effects of Curvature, Shear, Torsion, and Elongation

“…This paper extends the theory of shape sensing from strain measurements presented by Todd et al [1]. Unlike Ref.…”

“…This paper extends the ideas presented in Ref. [1] and related works by including far more comprehensive mechanics in the model, allowing for applicability to a greater range of slender structures (such as nonnegligible cross-sectional shear deformation). The remainder of this paper is presented as follows: Section 2 explains the kinematics of the approach, obtaining the uniaxial strain as a function of the deformation and geometric parameters (shear angles, curvatures, elongation, and the location of strain gauge), states the appropriate assumptions, and presents a locally exact solution basis on which to build a global deformed shape including the orientation of the cross section.…”

“…Unlike Ref. [1], which uses a relationship between a Frenet coordinate system and a body-centered coordinate system on an unsheared cross section, we directly exploit the body-centered material frame in a way that captures the complete effects of multiple curvatures, shear, elongation, and torsion, consistent with the Cosserat theory. We assume the cross-sectional shape to be geometrically unchanged after deformation, and thus, we ignore Poisson's and warping effects in the present study.…”

“…Like Ref. [1], this paper focuses on the latter group of sensing methods, since a large number of applications do not lend themselves to the strict limitations of noncontact methods, such as line-of-sight or ranging restrictions. Conventional contact sensors typically involve measurement of kinematic/kinetic properties such as strain or acceleration (distributed or discrete) and then obtain the global displaced shape by means of some inverse model.…”

“…As mentioned in Ref. [1], distributed sensing may be grouped into two types: noncontact (the sensing mechanism is remote, such as using laser Doppler vibrometry (LDV)) and contact (the sensing mechanism is affixed to the object for in situ measurement). Like Ref.…”