# An Online Measurement and Calibration Method for a Radio Telescope Sub-Reflector Support Structure Using Fiber Bragg Grating

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of the Deformation Reconstruction Model for Beam Structure

#### 2.1. Inverse Finite Element Method for Beam Element

**u**, the result is simplified as

^{e}#### 2.2. Calculation of Section Strains from Measured Strains

#### 2.3. Temperature Compensation for Measured Strain

_{e}is the photo-optical coefficient of the fiber; and the strain measurement ${\epsilon}_{i}$ is expressed as micro strain.

## 3. The Establishment of Self-Architected Fuzzy Calibration Network

#### 3.1. Expansion of Data Sample

#### 3.2. Construction of the Calibration Network

- 1.
- Adding MF and generation rules
- (1)
- Error criterion

- (2)
- Completeness criteria

- 2.
- The consequence of self-adaptive rules

- 3.
- Fixing the rules and obtaining the fuzzy network

## 4. Experimental Examples

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Measurement devices: (

**a**) Strain sensor and temperature compensation device; (

**b**) Demodulation instrument.

Axial Position ${\mathit{x}}_{\mathit{k}}$ | 0.2L | 0.8L |
---|---|---|

$\left({\theta}_{1},{\beta}_{1}\right)$ | (0,0) | (0,45) |

$\left({\theta}_{2},{\beta}_{2}\right)$ | (120,0) | (−120,0) |

$\left({\theta}_{3},{\beta}_{3}\right)$ | (−120,0) | (120,0) |

Angles | $\mathbf{x}$ Direction | $\mathbf{y}$ Direction | $\mathbf{z}$ Direction | |
---|---|---|---|---|

${\mathbf{55}}^{\xb0}$ | ${u}^{act}$ | 1.86 mm | 19.01 mm | 4.76 mm |

${u}^{iFEM}$ | 1.53 mm | 17.48 mm | 4.56 mm | |

${u}^{cal}$ | 1.83 mm | 18.79 mm | 4.71 mm | |

$\delta (\%)$ | 1.61% | 1.16% | 1.05% | |

${\mathbf{40}}^{\xb0}$ | ${u}^{act}$ | 1.56 mm | 15.85 mm | 3.99 mm |

${u}^{iFEM}$ | 1.26 mm | 14.18 mm | 3.71 mm | |

${u}^{cal}$ | 1.54 mm | 15.85 mm | 3.98 mm | |

$\delta (\%)$ | 1.28% | 0% | 0.25% | |

${\mathbf{30}}^{\xb0}$ | ${u}^{act}$ | 1.29 mm | 12.70 mm | 3.25 mm |

${u}^{iFEM}$ | 0.97 mm | 10.75 mm | 2.85 mm | |

${u}^{cal}$ | 1.28 mm | 12.68 mm | 3.22 mm | |

$\delta (\%)$ | 0.78% | 0.16% | 0.92% | |

${\mathbf{15}}^{\xb0}$ | ${u}^{act}$ | 0.75 mm | 6.66 mm | 1.77 mm |

${u}^{iFEM}$ | 0.48 mm | 5.15 mm | 1.41 mm | |

${u}^{cal}$ | 0.74 mm | 6.49 mm | 1.73 mm | |

$\delta (\%)$ | 1.33% | 2.55% | 2.26% |

Angles | ${55}^{\xb0}$ | ${40}^{\xb0}$ | ${30}^{\xb0}$ | ${15}^{\xb0}$ | |
---|---|---|---|---|---|

FBG strain/με | ${\mathsf{\epsilon}}_{1}$ | −20 | −62 | −101 | −173 |

${\mathsf{\epsilon}}_{2}$ | −485 | −431 | −376 | −263 | |

${\mathsf{\epsilon}}_{3}$ | −111 | −115 | −117 | −112 | |

${\mathsf{\epsilon}}_{4}$ | −345 | −314 | −272 | −191 | |

${\mathsf{\epsilon}}_{5}$ | 418 | 331 | 251 | 96 | |

${\mathsf{\epsilon}}_{6}$ | −79 | −82 | −86 | −91 | |

FBG temperature compensation/με | ${\mathsf{\epsilon}}_{1}$ | 55 | 52 | 51 | 43 |

${\mathsf{\epsilon}}_{2}$ | 21 | 25 | 32 | 33 | |

${\mathsf{\epsilon}}_{3}$ | 28 | 34 | 39 | 37 | |

${\mathsf{\epsilon}}_{4}$ | 7 | 13 | 20 | 25 | |

${\mathsf{\epsilon}}_{5}$ | 29 | 36 | 37 | 34 | |

${\mathsf{\epsilon}}_{6}$ | 27 | 31 | 36 | 33 |

Angles | $\mathbf{x}$ Direction | $\mathbf{y}$ Direction | $\mathbf{z}$ Direction | |
---|---|---|---|---|

${\mathbf{55}}^{\xb0}$ | ${u}^{act}$ | 1.67 mm | 16.76 mm | 4.66 mm |

${u}^{iFEM}$ | 1.40 mm | 18.86 mm | 6.35 mm | |

${u}^{cal}$ | 1.63 mm | 16.53 mm | 4.68 mm | |

$\delta (\%)$ | 2.4% | 1.37% | 0.43% | |

${\mathbf{40}}^{\xb0}$ | ${u}^{act}$ | 1.40 mm | 14.00 mm | 3.90 mm |

${u}^{iFEM}$ | 1.16 mm | 15.53 mm | 5.22 mm | |

${u}^{cal}$ | 1.38 mm | 13.97 mm | 3.94 mm | |

$\delta (\%)$ | 1.43% | 0.21% | 1.03% | |

${\mathbf{30}}^{\xb0}$ | ${u}^{act}$ | 1.16 mm | 11.38 mm | 3.14 mm |

${u}^{iFEM}$ | 0.89 mm | 12.04 mm | 4.08 mm | |

${u}^{cal}$ | 1.15 mm | 11.36 mm | 3.11 mm | |

$\delta (\%)$ | 0.86% | 0.18% | 0.96% | |

${\mathbf{15}}^{\xb0}$ | ${u}^{act}$ | 0.68 mm | 6.12 mm | 1.57 mm |

${u}^{iFEM}$ | 0.44 mm | 6.02 mm | 2.08 mm | |

${u}^{cal}$ | 0.67 mm | 6.01 mm | 1.53 mm | |

$\delta (\%)$ | 1.47% | 1.8% | 2.55% |

Angles | ${55}^{\xb0}$ | ${40}^{\xb0}$ | ${30}^{\xb0}$ | ${15}^{\xb0}$ | |
---|---|---|---|---|---|

FBG strain/με | ${\mathsf{\epsilon}}_{1}$ | −237 | −237 | −238 | −227 |

${\mathsf{\epsilon}}_{2}$ | −484 | −438 | −386 | −282 | |

${\mathsf{\epsilon}}_{3}$ | −17 | −43 | −66 | −112 | |

${\mathsf{\epsilon}}_{4}$ | −128 | −122 | −114 | −105 | |

${\mathsf{\epsilon}}_{5}$ | 365 | 283 | 206 | 62 | |

${\mathsf{\epsilon}}_{6}$ | −560 | −485 | −404 | −252 | |

FBG temperaturecompensation/με | ${\mathsf{\epsilon}}_{1}$ | 22 | 26 | 27 | 30 |

${\mathsf{\epsilon}}_{2}$ | 16 | 19 | 26 | 30 | |

${\mathsf{\epsilon}}_{3}$ | 25 | 27 | 31 | 32 | |

${\mathsf{\epsilon}}_{4}$ | −8 | 1 | 12 | 18 | |

${\mathsf{\epsilon}}_{5}$ | 29 | 34 | 37 | 35 | |

${\mathsf{\epsilon}}_{6}$ | 9 | 14 | 19 | 23 |

Angles | $\mathbf{x}$ Direction | $\mathbf{y}$ Direction | $\mathbf{z}$ Direction | |
---|---|---|---|---|

${\mathbf{55}}^{\xb0}$ | ${u}^{act}$ | 2.29 mm | 17.10 mm | −0.73 mm |

${u}^{iFEM}$ | −0.18 mm | 15.77 mm | −0.84 mm | |

${u}^{cal}$ | 2.22 mm | 16.89 mm | −0.72 mm | |

$\delta (\%)$ | 3.06% | 1.23% | 1.37% | |

${\mathbf{40}}^{\xb0}$ | ${u}^{act}$ | 1.90 mm | 14.25 mm | −0.61 mm |

${u}^{iFEM}$ | −0.25 mm | 12.79 mm | −0.69 mm | |

${u}^{cal}$ | 1.88 mm | 14.26 mm | −0.60 mm | |

$\delta (\%)$ | 1.05% | 0.07% | 1.64% | |

${\mathbf{30}}^{\xb0}$ | ${u}^{act}$ | 1.48 mm | 11.42 mm | −0.48 mm |

${u}^{iFEM}$ | −0.34 mm | 9.70 mm | −0.53 mm | |

${u}^{cal}$ | 1.51 mm | 11.39 mm | −0.47 mm | |

$\delta (\%)$ | 2.03% | 0.26% | 2.08% | |

${\mathbf{15}}^{\xb0}$ | ${u}^{act}$ | 0.78 mm | 5.99 mm | −0.25 mm |

${u}^{iFEM}$ | −0.29 mm | 4.65 mm | −0.26 mm | |

${u}^{cal}$ | 0.81 mm | 5.92 mm | −0.24 mm | |

$\delta (\%)$ | 3.85% | 1.17% | 4.0% |

Angles | ${55}^{\xb0}$ | ${40}^{\xb0}$ | ${30}^{\xb0}$ | ${15}^{\xb0}$ | |
---|---|---|---|---|---|

FBG strain/με | ${\mathsf{\epsilon}}_{1}$ | −16 | −32 | −50 | −74 |

${\mathsf{\epsilon}}_{2}$ | −369 | −335 | −295 | −212 | |

${\mathsf{\epsilon}}_{3}$ | 30 | 4 | −21 | −61 | |

${\mathsf{\epsilon}}_{4}$ | −187 | −180 | −165 | −138 | |

${\mathsf{\epsilon}}_{5}$ | 328 | 240 | 164 | 25 | |

${\mathsf{\epsilon}}_{6}$ | −359 | −309 | −261 | −173 | |

FBG temperature compensation/με | ${\mathsf{\epsilon}}_{1}$ | 14 | 22 | 28 | 28 |

${\mathsf{\epsilon}}_{2}$ | 17 | 20 | 26 | 29 | |

${\mathsf{\epsilon}}_{3}$ | 27 | 30 | 38 | 34 | |

${\mathsf{\epsilon}}_{4}$ | 9 | 18 | 24 | 27 | |

${\mathsf{\epsilon}}_{5}$ | 35 | 36 | 38 | 36 | |

${\mathsf{\epsilon}}_{6}$ | 12 | 21 | 25 | 29 |

Angles | $\mathbf{x}$ Direction | $\mathbf{y}$ Direction | $\mathbf{z}$ Direction | |
---|---|---|---|---|

${\mathbf{55}}^{\xb0}$ | ${u}^{act}$ | 1.93 mm | 16.90 mm | −1.52 mm |

${u}^{iFEM}$ | −0.22 mm | 18.91 mm | −1.39 mm | |

${u}^{cal}$ | 1.88 mm | 16.70 mm | −1.50 mm | |

$\delta (\%)$ | 2.59% | 1.18% | 1.32% | |

${\mathbf{40}}^{\xb0}$ | ${u}^{act}$ | 1.60 mm | 14.12 mm | −1.27 mm |

${u}^{iFEM}$ | −0.27 mm | 15.57 mm | −1.13 mm | |

${u}^{cal}$ | 1.58 mm | 14.11 mm | −1.26 mm | |

$\delta (\%)$ | 1.25% | 0.07% | 0.79% | |

${\mathbf{30}}^{\xb0}$ | ${u}^{act}$ | 1.25 mm | 11.47 mm | −1.03 mm |

${u}^{iFEM}$ | −0.33 mm | 12.07 mm | −0.86 mm | |

${u}^{cal}$ | 1.26 mm | 11.39 mm | −1.02 mm | |

$\delta (\%)$ | 0.80% | 0.70% | 0.97% | |

${\mathbf{15}}^{\xb0}$ | ${u}^{act}$ | 0.66 mm | 6.17 mm | −0.55 mm |

${u}^{iFEM}$ | −0.27 mm | 6.02 mm | −0.41 mm | |

${u}^{cal}$ | 0.67 mm | 6.07 mm | −0.54 mm | |

$\delta (\%)$ | 1.52% | 1.62% | 1.82% |

Angles | ${55}^{\xb0}$ | ${40}^{\xb0}$ | ${30}^{\xb0}$ | ${15}^{\xb0}$ | |
---|---|---|---|---|---|

FBG strain | ${\mathsf{\epsilon}}_{1}$ | 13 | −9 | −30 | −73 |

${\mathsf{\epsilon}}_{2}$ | −369 | −338 | −299 | −220 | |

${\mathsf{\epsilon}}_{3}$ | 26 | −1 | −21 | −61 | |

${\mathsf{\epsilon}}_{4}$ | −191 | −176 | −165 | −140 | |

${\mathsf{\epsilon}}_{5}$ | −288 | −272 | −239 | −184 | |

${\mathsf{\epsilon}}_{6}$ | 421 | 337 | 258 | 100 | |

FBG temperaturecompensation | ${\mathsf{\epsilon}}_{1}$ | 11 | 10 | 18 | 28 |

${\mathsf{\epsilon}}_{2}$ | 21 | 21 | 27 | 30 | |

${\mathsf{\epsilon}}_{3}$ | 35 | 36 | 37 | 36 | |

${\mathsf{\epsilon}}_{4}$ | 5 | 8 | 15 | 23 | |

${\mathsf{\epsilon}}_{5}$ | 37 | 39 | 42 | 34 | |

${\mathsf{\epsilon}}_{6}$ | 18 | 24 | 27 | 27 |

Strain | Time | ${\mathsf{\epsilon}}_{1}/\mathsf{\mu}\mathsf{\epsilon}$ | ${\mathsf{\epsilon}}_{2}/\mathsf{\mu}\mathsf{\epsilon}$ | ${\mathsf{\epsilon}}_{3}/\mathsf{\mu}\mathsf{\epsilon}$ | ${\mathsf{\epsilon}}_{4}/\mathsf{\mu}\mathsf{\epsilon}$ | ${\mathsf{\epsilon}}_{5}/\mathsf{\mu}\mathsf{\epsilon}$ | ${\mathsf{\epsilon}}_{6}/\mathsf{\mu}\mathsf{\epsilon}$ |
---|---|---|---|---|---|---|---|

$\mathsf{\epsilon}$ | Midnight | −20.48 | −484.92 | −111.42 | −345.13 | 417.76 | −79.67 |

Noon | −327.20 | −601.49 | −118.92 | −354.49 | 45.15 | −125.02 | |

${\mathit{\epsilon}}^{\mathit{t}\mathit{e}\mathit{m}\mathit{p}}$ | Midnight | 353.65 | −315.15 | 48.26 | −152.23 | 616.02 | 86.15 |

Noon | 52.94 | −442.72 | 38.77 | −150.58 | 246.41 | 45.80 | |

${\mathit{\epsilon}}^{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ | Midnight | −374.14 | −169.77 | −159.69 | −192.91 | −198.26 | −165.83 |

Noon | −380.14 | −158.77 | −157.69 | −203.91 | −201.26 | −170.83 |

Reconstruction Value | $\mathbf{x}$ Direction/mm | $\mathbf{y}$ Direction/mm | $\mathbf{z}$ Direction/mm |
---|---|---|---|

${u}^{iFEM}$ | 1.53 | 17.48 | 4.56 |

${u}^{PM}$ | 1.47 | 18.15 | 4.73 |

${u}^{comp}$ | 1.51 | 17.42 | 4.54 |

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## Share and Cite

**MDPI and ACS Style**

Xu, Q.; Bao, H.
An Online Measurement and Calibration Method for a Radio Telescope Sub-Reflector Support Structure Using Fiber Bragg Grating. *Micromachines* **2023**, *14*, 1093.
https://doi.org/10.3390/mi14051093

**AMA Style**

Xu Q, Bao H.
An Online Measurement and Calibration Method for a Radio Telescope Sub-Reflector Support Structure Using Fiber Bragg Grating. *Micromachines*. 2023; 14(5):1093.
https://doi.org/10.3390/mi14051093

**Chicago/Turabian Style**

Xu, Qian, and Hong Bao.
2023. "An Online Measurement and Calibration Method for a Radio Telescope Sub-Reflector Support Structure Using Fiber Bragg Grating" *Micromachines* 14, no. 5: 1093.
https://doi.org/10.3390/mi14051093