# A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calibration System

## 3. Calibration Method

#### 3.1. Calibration of Composite Module Array

#### 3.1.1. Angle Calibration

#### 3.1.2. Position Calibration

#### 3.2. Influence of Array Placement Error on Calibration

#### 3.2.1. Influence of Array Placement Error on Angle Calibration

- 1.
- The placement error angle of the composite module array around the Z-axis is $\gamma $.

- 2.
- The placement error angle of the composite module array around the X-axis is $\beta $.

- 3.
- The placement error angle of the composite module array around the Y-axis is $\alpha $.

#### 3.2.2. Influence of Array Placement Error on Position Calibration

## 4. Experiment

#### 4.1. Angle Calibration Experiments

#### 4.2. Position Calibration Experiments

#### 4.3. Verification Experiments

- Since the composite reference piece was made of metal, there was an oxidation problem with an extended time of use, which affected the surface shape of the composite reference piece.
- The measurement with higher accuracy put forward higher requirements on the quality of the light source and other optical components of the angle sensor.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Repeatability of the calibration experiment: (

**a**) angle around the X-axis; (

**b**) angle around the Y-axis.

**Figure 11.**Verification experiments with the uncalibrated composite module array: (

**a**) comparison experiments for the angle; (

**b**) comparison experiments for the position.

**Figure 12.**Verification experiments with calibrated composite module array: (

**a**) comparison experiments for the angle; (

**b**) comparison experiments for the position.

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}={\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

180 | 50 | 0.044 |

180 | 100 | 0.087 |

180 | 150 | 0.131 |

180 | 200 | 0.174 |

180 | 300 | 0.262 |

180 | 500 | 0.436 |

180 | 800 | 0.698 |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}={\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

360 | 50 | 0.087 |

360 | 100 | 0.174 |

360 | 150 | 0.262 |

360 | 200 | 0.349 |

360 | 300 | 0.523 |

360 | 500 | 0.872 |

360 | 800 | 1.395 |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}={\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

720 | 50 | 0.174 |

720 | 100 | 0.348 |

720 | 150 | 0.523 |

720 | 200 | 0.697 |

720 | 300 | 1.045 |

720 | 500 | 1.742 |

720 | 800 | 2.788 |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

180 | 50 | 8.00 $\times {10}^{-6}$ |

180 | 100 | 4.00 $\times {10}^{-6}$ |

180 | 150 | 3.70 $\times {10}^{-5}$ |

180 | 200 | 9.30 $\times {10}^{-5}$ |

180 | 300 | 2.67 $\times {10}^{-4}$ |

180 | 500 | 8.67$\times {10}^{-4}$ |

180 | 800 | 2.40$\times {10}^{-3}$ |

$\mathit{\gamma}$/arcsec | ${\theta}_{xi}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

360 | 50 | 5.50 $\times {10}^{-5}$ |

360 | 100 | 6.70 $\times {10}^{-5}$ |

360 | 150 | 3.80 $\times {10}^{-5}$ |

360 | 200 | 3.40 $\times {10}^{-5}$ |

360 | 300 | 3.05 $\times {10}^{-4}$ |

360 | 500 | 1.35 $\times {10}^{-3}$ |

360 | 800 | 4.20 $\times {10}^{-3}$ |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{y}}$/arcsec |
---|---|---|

720 | 50 | 2.66 $\times {10}^{-4}$ |

720 | 100 | 4.40 $\times {10}^{-4}$ |

720 | 150 | 5.33 $\times {10}^{-4}$ |

720 | 200 | 5.42 $\times {10}^{-4}$ |

720 | 300 | 3.05 $\times {10}^{-4}$ |

720 | 500 | 1.18 $\times {10}^{-3}$ |

720 | 800 | 5.96 $\times {10}^{-3}$ |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}$/arcsec |
---|---|---|

180 | 50 | 4.00 $\times {10}^{-6}$ |

180 | 100 | 2.10 $\times {10}^{-5}$ |

180 | 150 | 4.20 $\times {10}^{-5}$ |

180 | 200 | 6.70 $\times {10}^{-5}$ |

180 | 300 | 9.30 $\times {10}^{-5}$ |

180 | 500 | 2.94 $\times {10}^{-4}$ |

180 | 800 | 5.71 $\times {10}^{-4}$ |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}$/arcsec |
---|---|---|

360 | 50 | 1.50$\times {10}^{-5}$ |

360 | 100 | 7.80 $\times {10}^{-5}$ |

360 | 150 | 1.60 $\times {10}^{-5}$ |

360 | 200 | 2.47 $\times {10}^{-4}$ |

360 | 300 | 3.38 $\times {10}^{-4}$ |

360 | 500 | 9.67 $\times {10}^{-4}$ |

360 | 800 | 1.75 $\times {10}^{-3}$ |

$\mathit{\gamma}$/arcsec | ${\mathit{\theta}}_{\mathit{x}\mathit{i}}={\mathit{\theta}}_{\mathit{y}\mathit{i}}$/arcsec | ${\Delta}_{\mathit{x}}$/arcsec |
---|---|---|

720 | 50 | 6.10 $\times {10}^{-5}$ |

720 | 100 | 3.08 $\times {10}^{-4}$ |

720 | 150 | 6.03 $\times {10}^{-5}$ |

720 | 200 | 9.51 $\times {10}^{-4}$ |

720 | 300 | 1.28 $\times {10}^{-3}$ |

720 | 500 | 3.46 $\times {10}^{-3}$ |

720 | 800 | 5.94 $\times {10}^{-3}$ |

$\mathit{\gamma}$/arcsec | (x, y)/mm | $(\text{}\Delta \mathit{x},\Delta \mathit{y}\text{})$/mm |
---|---|---|

100 | $\left(50,0\right)$ | $\left(5.88\times {10}^{-6},-2.42\times {10}^{-2}\right)$ |

100 | $\left(100,0\right)$ | $\left(1.18\times {10}^{-5},-4.85\times {10}^{-2}\right)$ |

100 | $\left(150,0\right)$ | $\left(1.76\times {10}^{-5},-7.27\times {10}^{-2}\right)$ |

100 | $\left(200,0\right)$ | $\left(2.35\times {10}^{-5},-9.70\times {10}^{-2}\right)$ |

100 | $\left(250,0\right)$ | $\left(2.94\times {10}^{-5},-1.21\times {10}^{-1}\right)$ |

100 | $\left(300,0\right)$ | $\left(3.53\times {10}^{-5},-1.45\times {10}^{-1}\right)$ |

100 | $\left(350,0\right)$ | $\left(4.11\times {10}^{-5},-1.70\times {10}^{-1}\right)$ |

$\mathit{\gamma}$/arcsec | (x, y)/mm | $(\text{}\Delta \mathit{x},\Delta \mathit{y}\text{})$/mm |
---|---|---|

200 | $\left(50,0\right)$ | $\left(2.35\times {10}^{-5},-4.84\times {10}^{-2}\right)$ |

200 | $\left(100,0\right)$ | $\left(4.70\times {10}^{-5},-9.70\times {10}^{-2}\right)$ |

200 | $\left(150,0\right)$ | $\left(7.05\times {10}^{-5},-1.45\times {10}^{-1}\right)$ |

233 | $\left(200,0\right)$ | $\left(9.40\times {10}^{-5},-1.94\times {10}^{-1}\right)$ |

200 | $\left(250,0\right)$ | $\left(1.18\times {10}^{-4},-2.42\times {10}^{-1}\right)$ |

200 | $\left(300,0\right)$ | $(1.41\times {10}^{-4},-2091\times {10}^{-1}$) |

200 | $\left(350,0\right)$ | $(1.65\times {10}^{-4},-3.39\times {10}^{-1}$) |

$\mathit{\gamma}$ | (x, y)/mm | $(\text{}\Delta \mathit{x},\Delta \mathit{y}\text{})$/mm |
---|---|---|

300 | $\left(50,0\right)$ | $\left(5.29\times {10}^{-5},-7.27\times {10}^{-2}\right)$ |

300 | $\left(100,0\right)$ | $\left(1.06\times {10}^{-4},-1.45\times {10}^{-1}\right)$ |

300 | $\left(150,0\right)$ | $\left(1.59\times {10}^{-4},-2.18\times {10}^{-1}\right)$ |

300 | $\left(200,0\right)$ | $\left(2.12\times {10}^{-4},-2.91\times {10}^{-1}\right)$ |

300 | $\left(250,0\right)$ | $\left(2.64\times {10}^{-4},-3.64\times {10}^{-1}\right)$ |

300 | $\left(300,0\right)$ | $\left(3.17\times {10}^{-4},-4.36\times {10}^{-1}\right)$ |

300 | $\left(350,0\right)$ | $\left(3.70\times {10}^{-4},-5.09\times {10}^{-1}\right)$ |

Composite Module | Standard Deviation of the Angle around the X-axis /arcsec | Standard Deviation of the Angle around the Y-axis /arcsec |
---|---|---|

1 | 0 | 0 |

2 | 0.28 | 0.25 |

3 | 0.32 | 0.13 |

4 | 0.27 | 0.19 |

5 | 0.28 | 0.14 |

6 | 0.24 | 0.28 |

7 | 0.21 | 0.15 |

8 | 0.28 | 0.16 |

9 | 0.15 | 0.25 |

Composite Module | Angle around the X-axis /arcsec | Angle around the Y-axis /arcsec |
---|---|---|

1 | 0 | 0 |

2 | 35.53 | 5.28 |

3 | 56.27 | −50.69 |

4 | 45.66 | −22.78 |

5 | 29.48 | −37.01 |

6 | 58.17 | 6.40 |

7 | 44.61 | −16.44 |

8 | 43.01 | −27.25 |

9 | 41.08 | −3.26 |

Composite Module | Standard Deviation of the Position in the X Direction /µm | Standard Deviation of the Position in the Y Direction /µm |
---|---|---|

1 | 0 | 0 |

2 | 0.28 | 0.49 |

3 | 0.29 | 0.61 |

4 | 0.40 | 0.59 |

5 | 0.31 | 0.48 |

6 | 0.35 | 0.58 |

7 | 0.30 | 0.52 |

8 | 0.36 | 0.52 |

9 | 0.35 | 0.54 |

Composite Module | Position in the X Direction /mm | Position in the Y Direction /mm |
---|---|---|

1 | 0 | 0 |

2 | 50.0972 | −0.0421 |

3 | 100.0294 | 0.0484 |

4 | 149.9785 | 0.0374 |

5 | 199.9603 | 0.0261 |

6 | 250.0125 | 0.0424 |

7 | 299.9843 | −0.0393 |

8 | 349.9831 | 0.008 |

9 | 399.9664 | −0.0281 |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Li, J.; Wei, X.; Yang, X.; Su, Z.; Liang, J.; Yang, Z.; Fang, F.
A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array. *Appl. Sci.* **2020**, *10*, 4358.
https://doi.org/10.3390/app10124358

**AMA Style**

Li X, Li J, Wei X, Yang X, Su Z, Liang J, Yang Z, Fang F.
A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array. *Applied Sciences*. 2020; 10(12):4358.
https://doi.org/10.3390/app10124358

**Chicago/Turabian Style**

Li, Xinghua, Jue Li, Xuan Wei, Xiaohuan Yang, Zhikun Su, Jiaqi Liang, Zhiming Yang, and Fengzhou Fang.
2020. "A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array" *Applied Sciences* 10, no. 12: 4358.
https://doi.org/10.3390/app10124358