The Joint Calibration Method of Multi-line Laser and Tracking System based on Conjugate Gradient Iteration

Huiming Huang; Guihua Liu; Jiajia Liu; Xueyin Liu; Lei Deng; Tao Song; Fuping Qin

doi:10.2478/msr-2025-0027

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The Joint Calibration Method of Multi-line Laser and Tracking System based on Conjugate Gradient Iteration

Measurement Science Review

Volume 25 (2025): Issue 5 (October 2025)

By: Huiming Huang , Guihua Liu, Jiajia Liu, Xueyin Liu, Lei Deng, Tao Song and Fuping Qin

Open Access

|Sep 2025

Figures & Tables

Traditional multi-line laser system and tracking ball joint calibration schematic diagram.

Hardware controller, (a) hardware circuit board, (b) circuit board structure diagram.

Signal timing diagram of the laser controller.

Binocular tracking system and cross pole.

The conversion relationship between the multi-line laser system and the tracking ball cage.

Joint calibration of physical structure drawing.

Schematic diagram of stitching the tracking system.

Accuracy verification of joint calibration.

Table 1.

Binocular hardware parameters.

Camera		Cross pole
Resolution [pixel²]	2248×2048	Size [mm²]	1200×1200
Focal length [mm]	8	Coded point number	16
Frame rate [s⁻¹]	90	Coded point size [mm²]	80×80
Field range [mm]	3000~4000
Distance [mm]	2000~3000

(a) Cross pole; (b) Three-dimensional coordinates and encoded values of points on the cross pole.

The results of left and right image coding point detection.

Table 2.

Calibration parameters results.

Intrinsic and distortion parameter	External parameter
$C_{left} = \begin{matrix} 3667.05796 & 0.0000594 & 1315.3227 \\ 0 & 3667.57804 & 910.788437 \\ 0 & 0 & 1 \end{matrix}]$
$C_{right} = \begin{matrix} 3672.2510 & - 0.0001125 & 1335.044848 \\ 0 & 3672.77186 & 908.559738 \\ 0 & 0 & 1 \end{matrix}]$	$R = \begin{matrix} 0.9685 & - 0.0441 & - 0.24486 \\ 0.0456 & 0.99895 & 0.000594 \\ 0.2445 & - 0.01176 & 0.96955 \end{matrix}]$
$D_{left} = \begin{matrix} - 0.0837 & 0.37011 & 0.002683 & 0.000038 & 0.0025 \end{matrix}]$	$T = \begin{matrix} 526.39 & 9.781 & 46.10 \end{matrix}]$
$D_{right} = \begin{matrix} 0.0752 & - 0.21367 & 0.001970 & 0.000041 & 0.000361 \end{matrix}]$

The images of the ball cage were captured from various perspectives.

(a) Effect of coding point recognition, (b) 3D coordinates of all reflecting points on the ball cage.

Multi-line laser and tracking ball cage joint calibration.

The 3D data and the transformation relationship of the cross pole within the multi-line laser system and the binocular tracking system.

The conversion relationship between the tracking ball cage and the binocular tracking system.

Table 3.

The intermediate and final results of joint calibration.

$T_{c}^{t} = \begin{matrix} 0.915432 & 0.068235 & 0.396645 & - 238.572 \\ - 0.134755 & 0.980606 & 0.142311 & - 313.702 \\ 0.379242 & 0.183726 & - 0.906874 & 2354.21 \\ 0 & 0 & 0 & 1 \end{matrix}]$
$T_{c}^{m_{1}} = \begin{matrix} 0.913486 & - 0.134567 & - 0.234568 & - 137.654 \\ 0.167536 & 0.936780 & 0.223134 & - 151.456 \\ 0.23786 & - 0.34278 & 0.945893 & 795.234 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$T_{b}^{t_{1}} = \begin{matrix} 0.871258 & - 0.18524 & 0.16452 & - 782.452 \\ 0.245785 & 0.752896 & 0.32587 & - 875.745 \\ - 0.35287 & - 0.357482 & 0.924752 & - 1514.457 \\ 0 & 0 & 0 & 1 \end{matrix}]$
⋯	⋯
$T_{c}^{m_{1}} = \begin{matrix} 0.92584 & - 0.14025 & - 0.22547 & - 129.145 \\ 0.17258 & 0.94253 & 0.202452 & - 171.528 \\ 0.22578 & - 0.362584 & 0.92458 & 810.452 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$T_{b}^{t_{1}} = \begin{matrix} 0.972245 & - 0.132809 & 0.192618 & - 415.108 \\ 0.0291214 & 0.885565 & 0.463602 & - 921.649 \\ - 0.232146 & - 0.445125 & 0.864854 & - 1421.11 \\ 0 & 0 & 0 & 1 \end{matrix}]$
$T_{b}^{m} = \begin{matrix} 0.986325 & - 0.11425 & 0.11475 & 120.1425 \\ 0.24758 & 0.975425 & 0.45728 & 35.1425 \\ - 0.15368 & - 0.24758 & 0.99442 & 150.140 \\ 0 & 0 & 0 & 1 \end{matrix}]$

Schematic diagram of the joint calibration comparison, (a) Joint calibration based on a flat plate; (b) Joint calibration based on a cross pole.

Comparison of joint calibration accuracy.

Table 4.

Comparison test parameters for joint calibration.

	The method in this paper	Traditional methods
Calibrated object	Cross pole	Planar calibration board
Field of view	3000 mm × 4000 mm
Depth of field range	800 mm × 1000 mm
Object test	Cross pole
Camera – resolution	2448 × 2048
Camera – focal length	8 mm
Calibration results	$T_{b}^{m} = \begin{matrix} 0.97234 & - 0.08187 & 0.14523 & 121.0832 \\ 0.30983 & 0.985320 & 0.40284 & 34.8239 \\ - 0.14239 & - 0.26812 & 0.988123 & 149.8234 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$T_{b}^{m} = \begin{matrix} 0.97234 & - 0.08187 & 0.14523 & 121.0832 \\ 0.30983 & 0.985320 & 0.40284 & 34.8239 \\ - 0.14239 & - 0.26812 & 0.988123 & 149.8234 \\ 0 & 0 & 0 & 1 \end{matrix}]$

Standard ball reconstruction process based on the tracking system.

Table 5.

Comparison of precision between the calibration algorithm proposed in this article and the traditional flat-plate calibration algorithm.

Number	Calibration algorithm of this paper (Method 1)			Calibration algorithm of traditional plate calibration (Method 2)
	D_i	D_A	D_B	D_i	D_A	D_B
1	300.0456	60.0247	60.0182	300.1547	60.0628	60.0978
2	300.0578	60.0341	60.0138	300.1236	60.1174	60.1147
3	300.0372	60.0297	60.0142	300.0412	60.1524	60.1340
4	300.0424	60.0225	60.0150	300.0824	60.0925	60.0941
5	300.0447	60.0279	60.0118	300.1471	60.0817	60.1119
6	300.0481	60.0248	60.0171	300.0758	60.1247	60.1477
7	300.0525	60.0121	60.0121	300.110	60.0852	60.0701
8	300.0473	60.0195	60.0177	300.0914	60.1264	60.1517
Mean error	0.0454	0.0220	0.0306	0.1593	0.1030	0.1121

[i] Note: (Distance of ball center D_i = 300.0015 mm, Diameter of D_A = 60.0024, Diameter of D_B = 60.0031), ±0.0002 mm

Other objects are reconstructed based on a multi-line laser tracking system.

References

Authors

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Articles in this issue

DOI: https://doi.org/10.2478/msr-2025-0027 | Journal eISSN: 1335-8871

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Language: English

Page range: 229 - 247

Submitted on: Mar 7, 2025

Accepted on: Jul 17, 2025

Published on: Sep 17, 2025

Published by: Slovak Academy of Sciences, Institute of Measurement Science

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

multi-line laser 3D reconstruction,

binocular tracking system calibration,

tracking system 3D reconstruction

Related subjects:

Engineering,

Electrical engineering,

Control engineering, metrology and testing

© 2025 Huiming Huang, Guihua Liu, Jiajia Liu, Xueyin Liu, Lei Deng, Tao Song, Fuping Qin, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 25 (2025): Issue 5 (October 2025)

The Joint Calibration Method of Multi-line Laser and Tracking System based on Conjugate Gradient Iteration

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Table 3.

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Table 4.

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Table 5.

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Paradigm

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