# An Improved Calibration Method for a Rotating 2D LIDAR System

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## Abstract

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## 1. Introduction

## 2. R2D-LIDAR system

## 3. Calibration Model and Strategy

#### 3.1. Measurement Model

#### 3.2. LM Optimized Algorithm

$\mathbf{A}\mathbf{l}\mathbf{g}\mathbf{o}\mathbf{r}\mathbf{i}\mathbf{t}\mathbf{h}\mathbf{m}\mathbf{1}\mathbf{:}\mathrm{Bias}\mathrm{angle}\mathrm{calibration}\mathrm{based}\mathrm{on}\mathrm{Levenberg}-\mathrm{Marquardt}\mathrm{method}$ |

$\begin{array}{l}\mathbf{I}\mathbf{n}\mathbf{p}\mathbf{u}\mathbf{t}\mathbf{:}\hspace{1em}\mathrm{Points}\mathrm{cloud}\mathrm{S}\\ \mathbf{O}\mathbf{u}\mathbf{t}\mathbf{p}\mathbf{u}\mathbf{t}\mathbf{:}\mathrm{Bias}\mathrm{angle}\alpha \mathrm{of}\mathrm{the}2\mathrm{D}\mathrm{LIDAR}\\ 1:\hspace{1em}\mathbf{N}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{s}\\ 2:\hspace{1em}\theta :\mathrm{the}\mathrm{angle}\mathrm{between}{\mathrm{LIDAR}}^{\prime}\mathrm{s}\mathrm{ray}\mathrm{and}\mathrm{center}\mathrm{line}\\ 3:\hspace{1em}\mathbf{D},\mathbf{D},\mathbf{\Theta}:\mathrm{sets}\mathrm{of}d,d\mathrm{and}\theta \\ 4:\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{c}\mathbf{e}\mathbf{d}\mathbf{u}\mathbf{r}\mathbf{e}\mathrm{FindBiasAngle}\left(\mathrm{S}\right)\\ 5:\hspace{1em}\u22b3SeparatethecoincidentPointCloudandextracttheapproriaterays\\ 6:\hspace{1em}{\mathbf{S}}_{1},{\mathbf{S}}_{2}\leftarrow Separate(S)\\ 7:\hspace{1em}{\mathbf{D}}_{1}^{\prime},{\mathbf{D}}_{2}^{\prime},\mathbf{\Theta}\leftarrow ExtractRay({S}_{1})\\ 8:\hspace{1em}{\mathbf{D}}_{3}^{\prime},{\mathbf{D}}_{4}^{\prime},\Theta \leftarrow ExtractRay({S}_{2})\\ 9:\hspace{1em}\u22b3computetheJacobianandHessianoff(\alpha )\\ 10:\hspace{1em}\mathbf{J}\leftarrow Jacobian(f(\alpha ))\\ 11:\hspace{1em}\mathbf{H}\leftarrow Hessian(f(\alpha ))\\ 12:\hspace{1em}\u22b3calculatethebiasangle\theta throughLMalgorithm\\ 13:\hspace{1em}\theta \leftarrow AngleCalculateLM({\mathbf{D}}_{1}^{\prime},{\mathbf{D}}_{2}^{\prime},\mathbf{\Theta})\\ 14:\hspace{1em}\u22b3verifytheanglevalueandcomputethelineerrorin{\mathbf{S}}_{2}\\ 15:\hspace{1em}{\mathrm{E}}_{2}\leftarrow LineErrorCalculate(\alpha ,{\mathbf{D}}_{3}^{\prime},{\mathbf{D}}_{4}^{\prime},\mathbf{\Theta})\\ 16:\hspace{1em}\mathbf{i}\mathbf{f}{\mathrm{E}}_{2}\mathsf{\epsilon}\\ 17:\hspace{1em}\hspace{1em}compute\alpha again\\ 18:\hspace{1em}\mathbf{e}\mathbf{l}\mathbf{s}\mathbf{e}\\ 19:\hspace{1em}\hspace{1em}\mathrm{return}\alpha \\ 20:\mathbf{e}\mathbf{n}\mathbf{d}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{c}\mathbf{e}\mathbf{d}\mathbf{u}\mathbf{r}\mathbf{e}\end{array}$ |

## 4. Experiments

#### 4.1. Simulation

#### 4.2. Calibration in Different Scenarios

#### 4.3. Comparison with Alismail’s Work

#### 4.4. Accuracy of Calibration Result

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The rotating two-dimensional light detection and ranging (R2D-LIDAR) system composed by a 2D LIDAR and a Pan-Tilt Unit (PTU).

**Figure 2.**Detection range of the 2D LIDAR expresses a sector spanning 270° with 0.25° angular resolution. The detecting plane of the LIDAR that is assembled vertically result from such structure.

**Figure 6.**Geometrical relationship of calculating the bias angle. (

**a**) denotes the centerline coincides with the ${Z}^{\prime}$ axis of ${X}^{\prime}{Y}^{\prime}{Z}^{\prime}$ coordinate system; (

**b**) denotes the centerline deviates an angle α from ${Z}^{\prime}$ axis.

**Figure 7.**Absolute errors between the calculated adding bias angle and the setting angle. The error varies in a small range.

**Figure 8.**Three typical scenarios (room, construction, and roadway) were adopted to capture the 3D point clouds.

**Figure 9.**3D point clouds captured by the R2D-LIDAR system in three typical scenarios, i.e., room, construction and roadway. The left plots present the uncalibrated point clouds, and the right plots represent the point clouds captured with bias adjustment. The differences between the left plots and the right plots highlighted by red box: (

**a**) the great bias of the roof on the left was eliminated after calibrating and some bumps on the right were the lamp holders on the roof; (

**b**) the construction on the left was obviously an inclination, which was rectified through our adjustment; (

**c**) the roadway existed crack on the left and it became continuous after adjustment.

**Figure 10.**(

**a**) is the full scan captured by our R2D-LIDAR system with a large deviation. The full scan contains two coincident scans of the point clouds marked by blue and yellow; (

**b**) is the calibration result using Alismail’s method. Alismail’s method is effective in the two coincident scans matching but fails in bias adjustment, which is reflected in the roof; (

**c**) is the calibration result employing both bias adjustment of the 2D LIDAR and Alismail’s method. The performance demonstrates the superiority of our improved calibrating method compared with Alismail’s method.

**Figure 11.**(

**a**,

**b**) are the deviation comparisons of scan_front and scan_followed, respectively. The bias angle was estimated from scan_front. The deviation under calibration condition in (

**a**) is mainly ranging from −15 mm to 15 mm and the deviation under calibration condition in (

**b**) is mainly ranging from −20 mm to 15 mm. Both deviations after calibration were greatly decreased compared to the uncalibrated condition.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zeng, Y.; Yu, H.; Dai, H.; Song, S.; Lin, M.; Sun, B.; Jiang, W.; Meng, M.Q.-H.
An Improved Calibration Method for a Rotating 2D LIDAR System. *Sensors* **2018**, *18*, 497.
https://doi.org/10.3390/s18020497

**AMA Style**

Zeng Y, Yu H, Dai H, Song S, Lin M, Sun B, Jiang W, Meng MQ-H.
An Improved Calibration Method for a Rotating 2D LIDAR System. *Sensors*. 2018; 18(2):497.
https://doi.org/10.3390/s18020497

**Chicago/Turabian Style**

Zeng, Yadan, Heng Yu, Houde Dai, Shuang Song, Mingqiang Lin, Bo Sun, Wei Jiang, and Max Q.-H. Meng.
2018. "An Improved Calibration Method for a Rotating 2D LIDAR System" *Sensors* 18, no. 2: 497.
https://doi.org/10.3390/s18020497