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

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.

(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.

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.

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.

Standard ball reconstruction process based on the tracking system.

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

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

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	Tbm=0.97234−0.081870.14523121.08320.309830.9853200.4028434.8239−0.14239−0.268120.988123149.82340001 T_b^m = \left[ {\matrix{ {0.97234} & { - 0.08187} & {0.14523} & {121.0832} \cr {0.30983} & {0.985320} & {0.40284} & {34.8239} \cr { - 0.14239} & { - 0.26812} & {0.988123} & {149.8234} \cr 0 & 0 & 0 & 1 \cr } } \right]	Tbm=0.97234−0.081870.14523121.08320.309830.9853200.4028434.8239−0.14239−0.268120.988123149.82340001 T_b^m = \left[ {\matrix{ {0.97234} & { - 0.08187} & {0.14523} & {121.0832} \cr {0.30983} & {0.985320} & {0.40284} & {34.8239} \cr { - 0.14239} & { - 0.26812} & {0.988123} & {149.8234} \cr 0 & 0 & 0 & 1 \cr } } \right]

The intermediate and final results of joint calibration_

Tct=0.9154320.0682350.396645−238.572−0.1347550.9806060.142311−313.7020.3792420.183726−0.9068742354.210001 T_c^t = \left[ {\matrix{ {0.915432} & {0.068235} & {0.396645} & { - 238.572} \cr { - 0.134755} & {0.980606} & {0.142311} & { - 313.702} \cr {0.379242} & {0.183726} & { - 0.906874} & {2354.21} \cr 0 & 0 & 0 & 1 \cr } } \right]
Tcm1=0.913486−0.134567−0.234568−137.6540.1675360.9367800.223134−151.4560.23786−0.342780.945893795.2340001 T_c^{{m_1}} = \left[ {\matrix{ {0.913486} & { - 0.134567} & { - 0.234568} & { - 137.654} \cr {0.167536} & {0.936780} & {0.223134} & { - 151.456} \cr {0.23786} & { - 0.34278} & {0.945893} & {795.234} \cr 0 & 0 & 0 & 1 \cr } } \right]	Tbt1=0.871258−0.185240.16452−782.4520.2457850.7528960.32587−875.745−0.35287−0.3574820.924752−1514.4570001 T_b^{{t_1}} = \left[ {\matrix{ {0.871258} & { - 0.18524} & {0.16452} & { - 782.452} \cr {0.245785} & {0.752896} & {0.32587} & { - 875.745} \cr { - 0.35287} & { - 0.357482} & {0.924752} & { - 1514.457} \cr 0 & 0 & 0 & 1 \cr } } \right]
⋯	⋯
Tcm1=0.92584−0.14025−0.22547−129.1450.172580.942530.202452−171.5280.22578−0.3625840.92458810.4520001 T_c^{{m_1}} = \left[ {\matrix{ {0.92584} & { - 0.14025} & { - 0.22547} & { - 129.145} \cr {0.17258} & {0.94253} & {0.202452} & { - 171.528} \cr {0.22578} & { - 0.362584} & {0.92458} & {810.452} \cr 0 & 0 & 0 & 1 \cr } } \right]	Tbt1=0.972245−0.1328090.192618−415.1080.02912140.8855650.463602−921.649−0.232146−0.4451250.864854−1421.110001 T_b^{{t_1}} = \left[ {\matrix{ {0.972245} & { - 0.132809} & {0.192618} & { - 415.108} \cr {0.0291214} & {0.885565} & {0.463602} & { - 921.649} \cr { - 0.232146} & { - 0.445125} & {0.864854} & { - 1421.11} \cr 0 & 0 & 0 & 1 \cr } } \right]
Tbm=0.986325−0.114250.11475120.14250.247580.9754250.4572835.1425−0.15368−0.247580.99442150.1400001 T_b^m = \left[ {\matrix{ {0.986325} & { - 0.11425} & {0.11475} & {120.1425} \cr {0.24758} & {0.975425} & {0.45728} & {35.1425} \cr { - 0.15368} & { - 0.24758} & {0.99442} & {150.140} \cr 0 & 0 & 0 & 1 \cr } } \right]

Calibration parameters results_

Intrinsic and distortion parameter	External parameter
Cleft=3667.057960.00005941315.322703667.57804910.788437001 {C_{left}} = \left[ {\matrix{ {3667.05796} & {0.0000594} & {1315.3227} \cr 0 & {3667.57804} & {910.788437} \cr 0 & 0 & 1 \cr } } \right]
Cright=3672.2510−0.00011251335.04484803672.77186908.559738001 {C_{right}} = \left[ {\matrix{ {3672.2510} & { - 0.0001125} & {1335.044848} \cr 0 & {3672.77186} & {908.559738} \cr 0 & 0 & 1 \cr } } \right]	R=0.9685−0.0441−0.244860.04560.998950.0005940.2445−0.011760.96955 R = \left[ {\matrix{ {0.9685} & { - 0.0441} & { - 0.24486} \cr {0.0456} & {0.99895} & {0.000594} \cr {0.2445} & { - 0.01176} & {0.96955} \cr } } \right]
Dleft=−0.08370.370110.0026830.0000380.0025 {D_{left}} = \left[ {\matrix{ { - 0.0837} & {0.37011} & {0.002683} & {0.000038} & {0.0025} \cr } } \right]	T=526.399.78146.10 T = \left[ {\matrix{ {526.39} & {9.781} & {46.10} \cr } } \right]
Dright=0.0752−0.213670.0019700.0000410.000361 {D_{right}} = \left[ {\matrix{ {0.0752} & { - 0.21367} & {0.001970} & {0.000041} & {0.000361} \cr } } \right]

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

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

Figures & Tables

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Binocular hardware parameters_

Comparison test parameters for joint calibration_

The intermediate and final results of joint calibration_

Calibration parameters results_

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

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