Research Papers

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

[+] Author and Article Information
Michael D. Todd

University of California, San Diego,
9500 Gilman Drive 0085,
La Jolla, CA 92093
e-mail: mdtodd@ucsd.edu

Christopher J. Stull

Los Alamos National Laboratory,
MS T-001,
Los Alamos, NM 87545

Michael Dickerson

3 Phoenix Inc.,
4585 Avion Parkway,
Suite 200,
Chantilly, VA 20151

1Corresponding author.

Manuscript received June 26, 2012; final manuscript received October 28, 2012; accepted manuscript posted November 19, 2012; published online May 23, 2013. Assoc. Editor: John Lambros.

J. Appl. Mech 80(4), 041028 (May 23, 2013) (10 pages) Paper No: JAM-12-1262; doi: 10.1115/1.4023023 History: Received June 26, 2012; Revised October 28, 2012; Accepted November 19, 2012

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%–1% 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.

Copyright © 2013 by ASME
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Fig. 1

Model geometry and material-adapted frame

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Fig. 2

A two- (left) and three- (right) dimensional example of a material reference frame: T (dashed lines), KV (solid lines), and KH (dotted lines); the curve centerline is shown in gray

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Fig. 3

Geometry for the determination of strain at arbitrary surface point Q

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Fig. 4

A representative view of the longitudinal (T-direction) strain field response from the finite element model

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Fig. 5

Comparison of simulation 1 centerlines of exact (black solid lines) and reconstructed via strain (gray dashed lines) displacements of the steel tube using 10 sensors (upper left), 20 sensors (upper right), 50 sensors (lower left), and 100 sensors (lower right)

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Fig. 6

Comparison of simulation 2 centerlines of exact (black solid lines) and reconstructed via strain (gray dashed lines) displacements of the steel tube using 10 sensors (upper left), 20 sensors (upper right), 50 sensors (lower left), and 100 sensors (lower right)

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Fig. 7

Root mean square error for simulation 1 (left) and simulation 2 (right) for noise-free and two additive uniform noise levels, as a function of sensor count

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Fig. 8

Same data as in Fig. 6 (top right), except the boundary condition was specified at s = 100 m instead of s = 0

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Fig. 9

Average root mean square error in shape reconstruction for both simulations as a function of the uncertainty level in the initial displacement conditions at the proximal end (s = 0)

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Fig. 10

Schematic showing the location of sensor failures

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Fig. 11

Photographs of the four imposed planar displacements on the experimental test structure. The reader is informed that the view changed among the photographs from proximal to distal to proximal to distal end, from left to right.

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Fig. 12

A comparison of actual (black dots) and reconstructed (gray solid line) shapes for the four experimental test shapes of the hose structure




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