An Improved Shape Reconstruction Methodology for Long Rod Like Structures Using Cosserat Kinematics- Including the Poisson’s Effect

Abstract

In this paper, we present an improved methodology for the global shape reconstruction of rod-like structures that capture the effect of curvature, shear, torsion, axial deformation, and Poisson’s transformation. The inclusion of Poisson’s effect relaxes Euler-Bernoulli’s rigid cross-section assumption such that the cross-section could now undergo planar deformation (shrinking or expansion in the same plane). This scenario is particularly useful for the inflatable structures and pipelines subjected to large radial pressure.

The theory of shape sensing utilizes the concept of curve framing using Cosserat director triad also called as Cosserat kinematics. The idea is to develop an algorithm to reconstruct the global shape of the rods using the local differential geometry parameters (finite strain parameters) of the midcurve. The deformed configuration of the object lies in ℝ³ × SO(3) × ℝ space. The presented theory exploits localized linearization approach that helps to obtain local basis set for the approximation of the midcurve position vector and the director triad, whereas moving least square approximation is utilized to estimate the axial strain field. The uniaxial surface strain incorporating all the effects mentioned above is derived and used to develop the shape-sensing algorithm. A simulation describing the idea is presented at the end.