Shape sensing using a multi-core optical fiber having an arbitrary initial shape in the presence of extrinsic forces

1. A method of determining the shape of a multi-core optical fiber, the method comprising:

initializing a controller, including:

positioning the multi-core fiber in an arbitrary initial shape;

determining an initial curvature vector of the fiber that describes the arbitrary initial shape; and

initializing strain measurements using the controller; and

determining the shape of the fiber, including:

positioning the fiber in a desired shape to be measured;

measuring strain over a length of the fiber using a plurality of strain sensors while the cable is held in the shape to be measured, wherein the strain sensors are distributed along a length of each of the cores;

defining a three-coordinate p-vector for each of the cores as a function of the distance of each respective core from a center point of the fiber and a bending angle of each respective core with respect to a local axis of the fiber, wherein the coordinates of the p-vector define a plane;

calculating, via the controller, an applied strain value of the fiber using the p-vector and the measured strain;

calculating strain due to bending as a function of the measured strain and the applied strain value;

defining an apparent local curvature vector for each core of the fiber as a function of the calculated strain due to bending;

calculating curvature and bend direction of the fiber using the apparent local curvature vector and the initial curvature vector; and

determining the shape to be measured via the controller using the calculated curvature and bend direction.

2. The method of claim 1 , further comprising:

executing a control action with respect to the fiber using the determined shape.

3. The method of claim 1 , wherein positioning the fiber in an arbitrary initial shape includes positioning the fiber in a non-straight initial shape.

4. The method of claim 1 , wherein determining the shape of the fiber includes accounting for at least one extrinsic force acting on the fiber.

5. The method of claim 1 , wherein measuring strain includes measuring a corresponding Bragg wavelength of each of a plurality of Fiber Bragg Gratings at different axial locations along a length of each of the cores.

6. The method of claim 1 , wherein determining the shape of the fiber via the controller includes using the controller to solve Frenet-Serret equations.

7. The method of claim 1 , further comprising:

recording discrete measured sets for each of: curvature, bending, twisting, and strain of the fiber;

performing a curve-fitting operation on each measured set to generate a curvature function, a bending function, a twisting function, and a strain function.

8. The method of claim 7 , further comprising:

using the controller to calculate a function of torsion vs. fiber length, including differentiating the bending function and then adding the twisting function.

9. The method of claim 8 , wherein determining the shape of the fiber via the controller includes solving a set of Frenet-Serret equations using each of the curve-fitted sets to thereby derive the position of the fiber in a three-dimensional space.

10. The method of claim 9 , wherein solving the set of Frenet-Serret equations includes determining the position, tangent vector, normal vector, and bi-normal vector of the cable with respect to the three-dimensional space.

11. The method of claim 1 , further comprising:

attaching a cable containing the fiber to a device;

wherein determining the shape of the fiber includes calculating the position of the device in three-dimensional space.

12. A system for determining a three-dimensional (3D) shape of a multi-core optical fiber, the system comprising:

a fiber optic cable having an optical fiber with a plurality of cores, and further having a plurality of strain sensors, wherein the strain sensors are configured to measure strain at different axial locations of each of the cores; and

a controller in communication with the plurality of strain sensors, wherein the controller is configured to:

calculate, using the measured strain, an applied strain value as an axial strain component experienced by all of the cores due to at least temperature and an axial load acting on the cable;

calculate strain due to bending of the fiber as a function of the measured strain and the applied strain value;

define an apparent local curvature vector for each core as a function of the calculated strain due to bending;

calculate curvature and bend direction of the fiber using the apparent local curvature vector and the initial curvature vector;

generate a curvature function, a bending function, a twisting function, and a strain function by performing a curve-fitting operation;

calculate a function of torsion vs. fiber length, at least in part by differentiating the bending function and adding the twisting function to the differentiated bending function; and

determining the shape of the cable via the controller in part by solving a set of Frenet-Serret equations using each of the curvature function, the bending function, the twisting function, and the strain function to thereby derive the shape of the cable in a three-dimensional space.

13. The system of claim 12 , wherein the controller is configured to calculate the applied strain value using a p-vector and the measured strain values for each core.

14. The system of claim 12 , further comprising:

a device connected to one end of the cable;

wherein the controller is configured to determine the position in three-dimensional space of the device using the determined shape of the cable.

15. The system of claim 12 , wherein the arbitrary initial shape is a non-straight shape.

16. The system of claim 12 , wherein the cable includes a sleeve which circumscribes the fiber, and wherein the sleeve is at least partially bound to the fiber.

17. A method of determining the shape of a multi-core fiber optic cable in the presence of extrinsic twisting and stretching forces, the method comprising:

recording discrete measured sets for each of: curvature, bending, twisting, and strain of the cable;

performing a curve-fitting operation on each measured set to generate a curvature function, a bending function, a twisting function, and a strain function;

using a controller to calculate a function of torsion vs. fiber length, including differentiating the bending function and then adding the twist function; and

determining the shape of the fiber via the controller in part by solving a set of Frenet-Serret equations using each of the curvature function, a bending function, a twisting function, and a strain function to thereby derive the shape of the fiber in a three-dimensional space.

18. The method of claim 17 , wherein solving the set of Frenet-Serret equations includes determining the position, tangent vector, normal vector, and bi-normal vector of the cable with respect to the three-dimensional space.

19. The method of claim 17 , wherein the fiber is bound to a sleeve of the cable such that the extrinsic forces include a twisting force imparted to the fiber by the sleeve.

20. The method of claim 17 , further comprising:

winding the cable onto a spool such that the initial shape of the cable is helical.