## Shape sensing using multi-core fiber optic cable and parametric curve solutions |

Optics Express, Vol. 20, Issue 3, pp. 2967-2973 (2012)

http://dx.doi.org/10.1364/OE.20.002967

Acrobat PDF (1752 KB)

### Abstract

The shape of a multi-core optical fiber is calculated by numerically solving a set of Frenet-Serret equations describing the path of the fiber in three dimensions. Included in the Frenet-Serret equations are curvature and bending direction functions derived from distributed fiber Bragg grating strain measurements in each core. The method offers advantages over prior art in that it determines complex three-dimensional fiber shape as a continuous parametric solution rather than an integrated series of discrete planar bends. Results and error analysis of the method using a tri-core optical fiber is presented. Maximum error expressed as a percentage of fiber length was found to be 7.2%.

© 2012 OSA

## 1. Introduction

2. M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. **36**(2), 120–121 (2000). [CrossRef]

3. G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. **28**(6), 387–389 (2003). [CrossRef] [PubMed]

4. R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE **6530**, 65301S, 65301S-11 (2007). [CrossRef]

4. R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE **6530**, 65301S, 65301S-11 (2007). [CrossRef]

5. S. M. Klute, R. G. Duncan, R. S. Fielder, G. W. Butler, J. H. Mabe, A. K. Sang, R. J. Seeley, and M. T. Raum, “Fiber-optic shape sensing and distributed strain measurements on a morphing chevron,” in *Proceedings of the 44 ^{th} AIAA Aerospace Sciences Meeting and Exhibit* (Reno, Nevada, Jan. 9–12, 2006).

## 2. Frenet-Serret formulas for curves in three dimensions

*s*represent the length along a curve in 3-D space andrepresent the location in R

^{3}of a point at length

*s*along the curve. Then the Frenet-Serret

*frame*can be defined by the tangent, normal, and binormal vectors. The tangent unit vector,

**T**(

*s*), defining the “pointing” direction of the curve is defined bywhere differentiation with respect to

*s*is denoted by a prime ('). The normal unit vector,

**N**(

*s*), pointing in the direction of curvature, is defined by

**B**(

*s*), which points in the direction of zero curvature, and is orthogonal to both

**N**(

*s*) and

**T**(

*s*), is defined by

## 3. Elastic tube theory and Frenet-Serret formulas

7. J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. **38**(4), 605–618 (1996). [CrossRef]

7. J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. **38**(4), 605–618 (1996). [CrossRef]

## 4. Bending measurements in three dimensions along the fiber

*r*, and an angular separation of 2π/3 with

*θ*

_{1}representing the offset of core 1 from the

*θ*= 0 axis. Not shown in Fig. 1 are the cladding boundary and fiber coating. The orientation of the cores relative to the material frame of the fiber is set by

*θ*

_{1}and, as explained prior, is assumed to be constant throughout the fiber. The value of

*θ*

_{1}is typically set using the simple calibration method of bending the fiber in a known single plane, measuring the strain values in the cores, and determining the value based on the measured strain values and knowledge of the bending plane.

*ε*

_{1}and

*ε*

_{2}are the strain values of core 1 and 2 FBG’s, respectively,

*R*is the bend radius of the fiber, and

*d*is the distance between the two cores respective to the bending plane of the fiber [2

2. M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. **36**(2), 120–121 (2000). [CrossRef]

*κ*= 1/R) gives a general expression for strain in a given core due to bending:where

*ε*is the axial strain at the

_{i}*i*

^{th}core,

*r*is distance from the

_{i}*i*

^{th}core to center of fiber,

*θ*is the angular offset from the local y-axis to the fiber bending direction, and

_{b}*θ*is the angular offset from the local y-axis to the core. Though this paper deals with the results of a symmetric tri-core fiber, Eq. (10) applies to any number of cores in any arrangement, symmetrical or asymmetrical. Previous works calculate curvature by comparing strain measurements in pairs of cores. While this method is mathematically sufficient, strain measurement error can lead to large curvature errors. To calculate curvature using information from all cores and reduce the sensitivity to error in a single strain measurement, we first define an apparent curvature vector which points in the direction of the

_{i}*i*

^{th}core from the center of the fiber:where unit vectors

**and**

*j***align with the local**

*k**y*- and

*z*-axes, respectively. Note that the magnitude of each core’s apparent local curvature vector is dependent on its measured strain and radial distance from the center of the fiber while the vector direction depends on the angular offset of the core. For

*N*number of cores in the fiber, the vector sum of the apparent curvature vectors is formulated as

*r*from the center of the fiber, substituting Eq. (10) for the strain and applying trigonometric identities gives

## 5. Solving for shape

*κ(s)*and bend direction

*θ(s)*vs. fiber length. The bend direction function is then differentiated with respect to length to give an explicit torsion function:

*s*

_{0}= 0,

*κ*

_{0}=

*θ*

_{0}= 0 is specified prior to curve fitting the curvature and bend direction measurements, and initial conditions of the Frenet-Serret frame and fiber position are specified at

*s*

_{0}= 0. The initial condition for position at fiber length zero is dictated by the actual physical position of the fiber in the global reference frame. The trivial choice is

**T**

_{0}, is also dictated by the actual physical position of the fiber at (

*x*

_{0},

*y*

_{0},

*z*

_{0}), in that

**T**

_{0}must be “pointing” in the direction of increasing fiber length at

*s*

_{0}= 0 in the global reference frame. A trivial choice is along a single axis; such is the case for the data presented in this paper, where

**T**

_{0}is set equal to unit vector

**. The normal vector initial value,**

*i***N**

_{0}, is determined from the curve-fitted value of

*θ*(0) The specification of

**N**

_{0}is made in the global reference frame and acts to relate the fiber local reference frame to global reference frame at

*s*

_{0}= 0. For the data presented in this paper,

**T**

_{0}points along the x-axis so

**N**

_{0}must be located somewhere in the y-z plane. Arbitrarily chosen,

*θ*= 0 points along the y-axis and

*θ*= π/2 points along the z-axis. Therefore

**N**

_{0}is found by

**B**

_{0}, is found using the Frenet-Serret frame relationship of Eq. (4) and substituting the initial tangent and normal vectors. With initial conditions specified, Eqs. (5) and (8) can be solved using numerical methods to give location of the fiber vs. fiber length as well as the tangent, normal and binormal vectors along the fiber.

## 6. Experimental measurements

9. B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE **4332**, 133–142 (2001). [CrossRef]

^{−1}. The grooves are located on the surface such that the fiber experiences a region of zero curvature followed by a single bend and then another region of zero curvature. The second template is made from a 0.12 m radius, 0.6 m long cylinder and has three grooves, denoted C1, C2, and C3, machined into its surface. The arc-lengths of curves C1, C2, and C3 are 1.03, 1.11, and 1.68 m, respectively. The grooves are defined by parametric curves in cylindrical coordinates with constant radius. The Y-X and Z-

*θ*relationships for the two- and three-dimensional reference curves are shown in Figs. 2(a) and 2(b) along with a three-dimensional representation of shape C1 in Fig. 2(c).

## 7. Results

## 8. Conclusion

## References and links

1. | R. Kashyap, |

2. | M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. |

3. | G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. |

4. | R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE |

5. | S. M. Klute, R. G. Duncan, R. S. Fielder, G. W. Butler, J. H. Mabe, A. K. Sang, R. J. Seeley, and M. T. Raum, “Fiber-optic shape sensing and distributed strain measurements on a morphing chevron,” in |

6. | A. Gray, |

7. | J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. |

8. | C. de Boor, |

9. | B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(120.3890) Instrumentation, measurement, and metrology : Medical optics instrumentation

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 2, 2011

Manuscript Accepted: January 18, 2012

Published: January 24, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Jason P. Moore and Matthew D. Rogge, "Shape sensing using multi-core fiber optic cable and parametric curve solutions," Opt. Express **20**, 2967-2973 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-3-2967

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### References

- R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Elsevier, 2010).
- M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett.36(2), 120–121 (2000). [CrossRef]
- G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett.28(6), 387–389 (2003). [CrossRef] [PubMed]
- R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE6530, 65301S, 65301S-11 (2007). [CrossRef]
- S. M. Klute, R. G. Duncan, R. S. Fielder, G. W. Butler, J. H. Mabe, A. K. Sang, R. J. Seeley, and M. T. Raum, “Fiber-optic shape sensing and distributed strain measurements on a morphing chevron,” in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit (Reno, Nevada, Jan. 9–12, 2006).
- A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 1993), Chap. 7.
- J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev.38(4), 605–618 (1996). [CrossRef]
- C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001), Chap. 14.
- B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE4332, 133–142 (2001). [CrossRef]

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