# Geometric Model and Calibration Method for a Solid-State LiDAR

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## Abstract

**:**

## 1. Introduction

## 2. Problem and Model Formulation

#### 2.1. The Problem

#### 2.2. The Model

## 3. Materials and Methods

#### 3.1. Calibration Method

#### 3.2. Distortion Mapping Equations

#### 3.3. Calibration Pattern and Algorithm

Algorithm 1: Image processing for obtaining the pixel locations of the lines’ intersections. |

#### 3.4. Prototypes

## 4. Results

#### 4.1. Model Simulation

#### 4.2. Calibration Results

#### 4.3. Impact on the Point Cloud

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Vectorial Snell’s Law

#### Appendix A.2. Geometrical Model of MEMS Scanning

## References

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**Figure 1.**Scheme of (

**a**) the Light Detection and Ranging (LiDAR) scanning system geometry depicting (

**blue**) the laser source, (

**red**) the MEMS mirror and (

**green**) the scanning reference systems with ${\widehat{i}}_{1}={\widehat{m}}_{1}={\widehat{s}}_{1}$ pointing inside the paper plane; and (

**b**) the global geometry of the LiDAR scanning setup in the general case, with the tilt angles of the MEMS magnified by additional optics.

**Figure 2.**Scheme of the LiDAR Field-of-View (FOV) and an observed point

**Q**relating (

**red**) the LiDAR reference system and its scanning angles (

**green**) in spherical coordinates.

**Figure 3.**(

**a**) Scheme of the LiDAR’s acquisition frame showing the relation between (

**black**) the pixel position (i,j) from the MEMS dynamics and (

**red**) the scanning angles of the FOV. (

**b**) Scheme of the distortion mapping functions for odd and even lines, ${\mathbf{f}}_{\mathbf{odd}}$ and ${\mathbf{f}}_{\mathbf{even}}$, relating the pixel position within the acquisition frame with the final scanning angles.

**Figure 6.**Acquired images of the calibration pattern with the (

**a**) 30 × 20${}^{\xb0}$ FOV and (

**b**) 50 × 20${}^{\xb0}$ FOV LiDAR prototypes.

**Figure 7.**Simulated probability histogram of the angular resolution (

**b**) and (

**d**) for both scanning directions assuming, respectively, (

**a**) linear motion of the MEMS and (

**c**) harmonic motion of the MEMS in its fastest direction, being the red curve of $\alpha $ the cropped linear region of the whole motion shown in blue. Homogeneous resolution is calculated with $FO{V}_{H}=27.{5}^{\xb0}$ and $FO{V}_{V}=16.{5}^{\xb0}$.

**Figure 8.**(

**a**) Simulated and (

**b**) experimental image of the grid pattern at 3.8 m using the 30 × 20${}^{\xb0}$ FOV prototype.

**Figure 9.**Grid control points on the odd image for both (

**a**) 30 × 20${}^{\xb0}$ and (

**b**) 50 × 20${}^{\xb0}$ FOV prototypes.

**Figure 10.**Probability Density Function (PDF) of both angular errors, (

**blue**) horizontal and (

**red**) vertical, of the odd lines for both (

**a**) 30 × 20${}^{\xb0}$ FOV prototype and (

**b**) 50 × 20${}^{\xb0}$ FOV prototype, using the mapping function Map 3 in Equation (13). The angular errors outside the 95% confidence interval are the filled areas.

**Figure 11.**Comparison of both (

**blue**) horizontal and (

**red**) vertical resolution variation for the odd lines. (

**a**) Simulation of the 30 × 20${}^{\xb0}$ FOV prototype with a sinusoidal dynamics for the fast scanning axis. (

**b**) Results of the mapping function Map 3 for the 30 × 20${}^{\xb0}$ FOV prototype. (

**c**) Results for the 50 × 20${}^{\xb0}$.

**Figure 12.**Comparison of respectively both horizontal and vertical angular resolution variations across the whole FOV of the 30 × 20${}^{\xb0}$ FOV prototype. (

**a**,

**b**) Sinusoidal dynamics for the fast scanning axis. (

**c**,

**d**) Results of the mapping function Map 3.

**Figure 13.**Results of the mapping function Map 3 for the 30 × 20${}^{\xb0}$ FOV prototype across its FOV. (

**a**) Horizontal and (

**b**) vertical angular resolution. (

**c**) Horizontal and (

**d**) vertical committed angular error.

**Figure 14.**Improvement of the presented calibration for (

**a**,

**b**) the 30 × 20${}^{\xb0}$ and (

**c**,

**d**) the 50 × 20${}^{\xb0}$ FOV prototype, compared to their respective ideal cases of homogeneous angular resolution. (

**a**,

**c**) Probability density functions of the angular error, being (

**red**) the horizontal and (

**blue**) the vertical scanning direction of (

**dashed line**) the non-calibrated and (

**solid line**) the calibrated case. (

**b**,

**d**) Norm of the final point distance error.

**Figure 15.**30 × 20${}^{\xb0}$ FOV prototype point clouds of (

**e**) the scenario. (

**a**) Previous calibration point cloud. (

**b**) Presented calibration point cloud. (

**c**) and (

**d**) Orthogonal projection on the ground plane of (

**a**) and (

**b**) respectively. (

**f**) Measured distances between road pins taken as the ground truth.

**Table 1.**Mapping results of the tested prototypes. Notice that units are millidegrees (${}^{\xb0}$/1000) and the best performance is highlighted in

**bold**for all figures except for the homogeneous FOV, which is in degrees and represents how far from the designed rectangular FOV is the distortion map.

Figure of Merit [Mdeg] | Dir. | 30 × 20${}^{\xb0}$ | 50 × 20${}^{\xb0}$ | ||||
---|---|---|---|---|---|---|---|

Map 1 | Map 2 | Map 3 | Map 1 | Map 2 | Map 3 | ||

Homogeneous FOV [${}^{\xb0}$] | Odd | 27.55 × 16.32 | 27.47 × 16.54 | 27.47 × 16.52 | 52.89 × 13.96 | 53.19 × 14.44 | 53.34 × 14.38 |

Even | 27.25 × 16.44 | 27.46 × 16.64 | 27.45 × 16.63 | 53.08 × 13.33 | 53.07 × 14.27 | 53.07 × 14.21 | |

Mean error | Odd | 25 × 28 | 21 × 9 | 20 × 8 | 101 × 89 | 45 × 40 | 37 × 31 |

Even | 23 × 24 | 22 × 9 | 22 × 9 | 108 × 118 | 47 × 64 | 46 × 37 | |

Standard deviation | Odd | 24 × 23 | 14 × 5 | 14 × 5 | 66 × 97 | 34 × 32 | 29 × 22 |

Even | 24 × 18 | 14 × 7 | 14 × 7 | 73 × 110 | 37 × 48 | 35 × 31 | |

Max. angular error (<95%) | Odd | 79 × 70 | 48 × 17 | 47 × 19 | 239 × 274 | 117 × 103 | 95 × 72 |

Even | 77 × 60 | 48 × 25 | 47 × 26 | 265 × 324 | 117 × 165 | 113 × 98 |

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**MDPI and ACS Style**

García-Gómez, P.; Royo, S.; Rodrigo, N.; Casas, J.R.
Geometric Model and Calibration Method for a Solid-State LiDAR. *Sensors* **2020**, *20*, 2898.
https://doi.org/10.3390/s20102898

**AMA Style**

García-Gómez P, Royo S, Rodrigo N, Casas JR.
Geometric Model and Calibration Method for a Solid-State LiDAR. *Sensors*. 2020; 20(10):2898.
https://doi.org/10.3390/s20102898

**Chicago/Turabian Style**

García-Gómez, Pablo, Santiago Royo, Noel Rodrigo, and Josep R. Casas.
2020. "Geometric Model and Calibration Method for a Solid-State LiDAR" *Sensors* 20, no. 10: 2898.
https://doi.org/10.3390/s20102898