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
Your Session has timed out. Please sign back in to continue.


Shield, R. T., and Im, S., 1986, “Small Strain Deformations of Elastic Beams and Rods Including Large Deformations,” ZAMP, 37, pp. 491–513. [CrossRef]
Love, A. E. H., 1944, A Treatise on Mathematical Theory of Elasticity, 4th ed., reprinted by Dover, New York.
Cosserat, E., and Cosserat, F., 1909, Theorie des Corps Deformables, Herman, Paris.
Reisner, E., 1981, “On Finite Deformations of Space-Curved Beams,” ZAMP, 32, pp. 734–744. [CrossRef]
Spillmann, J., and Teschner, M., 2009, “Cosserat Nets,” IEEE Trans. Vis. Comput. Graph., 15(2), pp. 325–338. [CrossRef] [PubMed]
Chang, J., Shepherd, D. X., and Zhang, J. J., 2007, “Cosserat-Beam-Based Dynamic Response Modeling,” Comput. Animat. Virt. Worlds, 18, pp. 429–436. [CrossRef]
Pai, D., 2002, “STRANDS: Interactive of Thin Solids Using Cosserat Models,” Comput. Graph. Forum, 21(3), pp. 347–352. [CrossRef]
Langer, J., and Singer, D., 1996, “Lagrangian Aspects of the Kirchhoff Elastic Rod,” SIAM Rev., 38(4), pp. 605–618. [CrossRef]
Lawton, W., Raghavan, R., Ranjan, S. R., and Viswanathan, R., 1999, “Ribbons and Groups: A Thin Rod Theory for Catheters and Filaments,” J. Physics A: Math. Theor., 32, pp. 1709–1735. [CrossRef]
Cao, D. Q., Liu, D., and Wang, C. H.-T., 2005, “Nonlinear Dynamic Modeling for MEMS Components via the Cosserat Rod Element Approach,” J. Micromech. Microeng., 15, pp. 1334–1343. [CrossRef]
Yang, T. Y., 1973, “Matrix Displacement Solution to Elastical Problems of Beams and Frames,” Int. J. Solids Struct., 9, pp. 829–842. [CrossRef]
Kawakubo, S., “Kirchoff Elastic Rods in Three-Dimensional Space Forms,” J. Math. Soc. Jpn., 60(2), pp. 551–582. [CrossRef]
Bishop, R. L., 1975, “There is More Than One Way to Frame a Curve,” Am. Math. Monthly, 82(3), pp. 246–251. [CrossRef]
Todd, M. D., Mascarenas, D., Overbey, L. A., Salter, T., Baldwin, C., and Kiddy, J., 2005, “Towards Deployment of a Fiber Optic Smart Tether for Relative Localization of Towed Bodies,” Proceedings of the SEM Annual Conference on Experimental Mechanics, Portland, OR, June 6–9.
Friedman, A., Todd, M. D., Kirkendall, K., Tveten, A., and Dandridge, A., 2003, “Rayleigh Backscatter-Based Fiber Optic Distributed Strain Sensor With Tunable Gage Length,” SPIE Smart Structures/NDE 5050 Proceedings, San Diego, CA, March 2–6.
Duncan, R., Froggatt, M., Kreger, S., Seeley, R., Gifford, D., Sang, A., and Wolfe, M., 2007, “High-Accuracy Fiber Optic Shape Sensing,” SPIE Smart Structures/NDE 6530 Proceedings, San Diego, CA, March 19–22.


Grahic Jump Location
Fig. 1

Model geometry and material-adapted frame

Grahic Jump Location
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

Grahic Jump Location
Fig. 3

Geometry for the determination of strain at arbitrary surface point Q

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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)

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
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)

Grahic Jump Location
Fig. 10

Schematic showing the location of sensor failures

Grahic Jump Location
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.

Grahic Jump Location
Fig. 12

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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In