# Reconfiguration Error Correction Model for an FBG Shape Sensor Based on the Sparrow Search Algorithm

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. FBG Shape Sensing Error Model

#### 2.1. Curvature Error and Bending Direction Error Correction Model

**a**, FBG

**b**, and FBG

**c**represent the three FBGs of the detection point, and d represents the cross-sectional radius of the shape sensor.

_{i}, the sensing curvature of each FBG is k

_{i}, and is orthogonally decomposed, the curvature and bending direction at the detection point can be expressed as Equations (1)–(3).

_{1}= 90°, θ

_{2}= 210°, and θ

_{3}= 330°, the expressions of the curvature and bending direction can be simplified as follows:

_{i}, the actual placement positions of the FBG is a′, b′, and c′. The FBGs no longer satisfy the uniform placement condition, and solving for the curvature and bending direction in Equations (4) and (5) would introduce placement angle deviations, leading to curvature and bending direction measurement errors.

_{e}is the elastic coefficient, λ

_{B}is the FBG central wavelength, and d is the cross-sectional radius of the shape sensor.

#### 2.2. Error Correction Model Verification and Analysis

**b**and FBG

**c**under different placement angle deviations. a0 is in the direction of the greatest bending force and can be used as the ideal calibration direction; hence, a1 and a2 represent the detection data of FBG

**a**under different calibration direction errors.

_{0}, y

_{0}, and z

_{0}are the actual coordinates of the detection point. In this study, we used the tail point reconfiguration error φ

_{max}and maximum relative error R

_{max}(ratio of φ

_{max}to the model length) as the evaluation indicators to compare the shape reconfiguration results.

_{max}of the method in Equations (4) and (5) were 1.40 mm and 1.42 mm, for which the R

_{max}values were 2.8% and 2.84%, respectively. The shape reconfiguration errors of the errors φ

_{max}of the correction model were 0.39 mm and 0.27 mm, for which the R

_{max}values were 0.78% and 0.54%, respectively.

## 3. Experiment

#### 3.1. Experimental System and Sensor Calibration

^{−1}).

#### 3.2. Optimization Model

_{i}and the calibration direction deviation Δα

_{i}at each detection point of the sensor. This paper proposes a method based on an improved SSA to determine Δθ

_{i}and Δα

_{i}. SSA is a new swarm intelligence optimization algorithm proposed by Xue et al. [23]. Compared with other swarm intelligence optimization algorithms, it has the characteristics of a high search accuracy, fast convergence speed, good stability, and strong robustness [24].

_{t}

_{+ 1}is the sparrow individual after the t + 1th mapping iteration, and M is an arbitrary constant. The shape sensor bending curvature was set to k

_{true}during the calibration, and Δθ

_{i}and Δα

_{i}of each detection-point FBG are substituted into Equation (10) as the optimization parameters to obtain the calculated curvature k under different parameters. Let the measurement curvature error function be the fitness function, as shown in Equation (14). When Δk reaches the minimum value, the optimization parameter is the actual deviation angle. The detailed process is shown in Figure 10.

- Enter the measured curvatures k
_{a}, k_{b}, and k_{c}for FBG a, FBG b, and FBG c in the fixed curvature state, respectively. - Initialize the inputs and set the placement angles θ
_{1}, θ_{2}, and θ_{3}of the FBG sensor. In this paper, θ_{1}= 90°, θ_{2}= 210°, and θ_{3}= 330°. - Set the FBG calibration direction deviation Δθ
_{i}and placement angle deviation Δα_{i}, and Δθ_{i}and Δα_{i}are randomly assigned and coded within a certain range. - Δθ
_{i}and Δα_{i}are substituted into Equation (10) to obtain the theoretical curvatures k_{1}, k_{2}, and k_{3}, and are optimized using Chebyshev-SSA. - The optimal parameters of Δθ
_{i}and Δα_{i}are output when the value of the fitness function ∆k is minimized or at the end of the iteration. Otherwise, steps (3) to (5) are repeated.

^{−1}as an example, the above optimization algorithm is used to automatically correct the placement angle and calibration coefficient of the FBG, and the average value of the error iteration curves of different optimization models is obtained by repeating 10 times, as shown in Figure 11. It can be seen from Figure 11 that the convergence speed and convergence accuracy of the Chebyshev-SSA optimization algorithm are the best.

#### 3.3. Shape Reconfiguration

^{−1}arc reconfiguration and Figure 13b for the spiral curve reconfiguration. The reconstructed shape of the sensor after error correction was closer to the actual shape. The tail point reconfiguration errors for the arc and spiral were 11.66 mm and 22.6 mm before the error correction, which were corrected to 5.23 mm and 11.4 mm, respectively. The average relative accuracy of shape reconfiguration is improved by 56.25% and 50.6%, respectively.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**FSS error delivery model. The bending curvature k and bending direction β at the detection point can be obtained according to the curvature in each core k

_{i}, and the discrete local k and β are converted into the curvature and torsion functions k(s) and τ(s) through interpolation.

**Figure 2.**(

**a**) Illustration of cross-section at the detection point. (

**b**) Placement angle deviation diagram.

**Figure 4.**(

**a**) Shape sensor simulation model. (

**b**) Simulation model cross-section. (

**c**) FBG placement diagram.

**Figure 5.**Calculation of curvature and bending direction obtained by different methods. (

**a**) Bending direction calculation results. (

**b**) Bending curvature calculation results.

**Figure 8.**(

**a**) System diagram for the calibration experiment. (

**b**) FBG fixtures. (

**c**) Calibration tools.

**Figure 12.**Experimental diagram for shape reconfigurations. (

**a**) Arc shape reconfiguration. (

**b**) Spiral shape reconfiguration.

Data Group | Data2 | Data3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Bending direction/° | 80 | 90 | 200 | 210 | 330 | 80 | 90 | 200 | 210 | 330 | |

R_{max}/% | Uncorrected | 2.2 | 2.7 | 2.6 | 1.9 | 0.8 | 2.8 | 2.8 | 1.3 | 0.7 | 2.0 |

Corrected | 0.4 | 0.6 | 0.1 | 0.06 | 0.4 | 0.2 | 0.1 | 0.3 | 0.2 | 0.5 |

Detection Point | Placement Angle Deviation/° | Calibration Direction Deviation/° | ||||
---|---|---|---|---|---|---|

FBGa | FBGb | FBGc | FBGa | FBGb | FBGc | |

Point1 | 0 | −3.9 | 7.5 | 3.9 | 12.5 | 2.2 |

Point2 | 7.5 | −4.5 | 4.3 | 1.6 | −8.0 | 1.8 |

Point3 | 2 | 5.3 | −6.1 | −9.2 | 8.4 | −7.4 |

Point4 | −4.1 | 10.5 | 5.8 | 11.2 | −4.3 | 8.1 |

Point5 | 2.3 | 14.8 | 9.4 | 9.1 | 9.2 | −3.5 |

Radius of Curvature r/mm | Tail Point Reconfiguration Error | |||
---|---|---|---|---|

Uncorrected Error | Corrected Error | |||

Absolute Error/mm | Relative Error/% | Absolute Error/mm | Relative Error/% | |

600 | 11.80 | 2.54 | 4.63 | 0.99 |

700 | 11.66 | 2.51 | 5.23 | 1.13 |

800 | 11.72 | 2.52 | 4.38 | 0.94 |

900 | 10.67 | 2.29 | 4.21 | 0.91 |

1000 | 12.50 | 2.69 | 5.65 | 1.22 |

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**MDPI and ACS Style**

Shang, Q.; Liu, F.
Reconfiguration Error Correction Model for an FBG Shape Sensor Based on the Sparrow Search Algorithm. *Sensors* **2023**, *23*, 7052.
https://doi.org/10.3390/s23167052

**AMA Style**

Shang Q, Liu F.
Reconfiguration Error Correction Model for an FBG Shape Sensor Based on the Sparrow Search Algorithm. *Sensors*. 2023; 23(16):7052.
https://doi.org/10.3390/s23167052

**Chicago/Turabian Style**

Shang, Qiufeng, and Feng Liu.
2023. "Reconfiguration Error Correction Model for an FBG Shape Sensor Based on the Sparrow Search Algorithm" *Sensors* 23, no. 16: 7052.
https://doi.org/10.3390/s23167052