Robust Total Least‑Squares Estimator for Registration of Laser Tracker During Changing Measuring Station

Zhipeng Lü; Wei Wang; Yelong Wei

doi:10.3788/CJL250623

Chinese Journal of Lasers, Volume. 52, Issue 17, 1704003(2025)

Robust Total Least‑Squares Estimator for Registration of Laser Tracker During Changing Measuring Station

Zhipeng Lü^1,4, Wei Wang², and Yelong Wei^3、*

¹Jiangxi Provincial Key Laboratory of Comprehensive Stereoscopic Traffic Information Perception and Fusion, Nanchang 330031, Jiangxi , China

²Institute of Advanced Light Source Facilities, Shenzhen 518107, Guangdong , China

³National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, Anhui , China

⁴School of Transportation Engineering, East China Jiaotong University, Nanchang 330031, Jiangxi , China

show less

Abstract Get PDF(in Chinese)

Figures & Tables(9)

Fig. 1. Instrument coordinate system of laser tracker

Download full size

Fig. 2. Absolute bias of 100 Monte Carlo simulations under different contamination modes. (a)(d) $Δ (T)$ ; (b)(e) $Δ (R)$ ; (c)(f) $Δ (s)$

Download full size

Fig. 3. Registration accuracy of 100 Monte Carlo simulations under different contamination modes

Download full size

Fig. 4. Errors of points after changing measuring station

Download full size

Table 0. [in Chinese]

View table

Table 0. [in Chinese]

（1） Input：

Establish the symmetric similarity transformation model Eqs. （12） and （14） based on all coordinate observations and instrument accuracy information.

（2） Initialization：

Obtain the initial values of registration parameters ${\hat{x}}^{0}$ by LS.

（3） Iteration：

Do for $i = 1$ （ $i$ is the iteration index）

1. Predict the coordinate residuals ${\tilde{e}}^{i + 1}$ according to Eq. （21）；

2. Estimate the registration parameters ${\hat{x}}^{i + 1}$ according to Eq. （22）；

3. If $‖Δ {\hat{x}}^{i + 1}‖ \leq δ$ ， end do；

4. Predict the total residuals ${\tilde{e}}_{c}^{i + 1}$ and the corresponding co-factor matrix $Q_{{\tilde{e}}_{c}^{i + 1}}$ according to Eqs. （24） and （25）；

5. Construct the co-factor factor matrix $F$ according to Eq. （28）， and then perform the re-weighting procedure

according to Eq. （30）；

6. $i = i + 1$ .

（4） Output：

1. Confirm the final registration parameters ${\hat{x}}^{i + 1}$ ；

2. Calculate the co-factor matrix $Q_{{\hat{x}}^{i + 1}}$ according to Eq. （23）；

3. Calculate the registration accuracy ${\hat{σ}}_{0}$ according to Eq. （29）.

Table 1. Simulated data of laser tracker registration during changing measuring station

View table

Table 1. Simulated data of laser tracker registration during changing measuring station

No.	Simulated values			Simulated transformation parameters
No.	d /mm	α /（°）	β /（°）	Simulated transformation parameters
1	4736.34	10.3293	63.4349	1） translation $T = {[\begin{matrix} 2189.36 & 8341.52 & - 238.91 \end{matrix}]}^{T}$ 2） rotation $\{\begin{array}{l} R_{x} = 0.000005 π \\ R_{y} = 0.000006 π \\ R_{z} = 0.214064 π \end{array}$ ， where $R_{x}$ ， $R_{y}$ ， and $R_{z}$ are Euler angles $R = [\begin{matrix} c o s R_{y} c o s R_{z} & s i n R_{x} s i n R_{y} c o s R_{z} - c o s R_{x} s i n R_{z} & c o s R_{x} s i n R_{y} c o s R_{z} + s i n R_{x} s i n R_{z} \\ c o s R_{y} s i n R_{z} & s i n R_{x} s i n R_{y} s i n R_{z} - c o s R_{x} c o s R_{z} & c o s R_{x} s i n R_{y} s i n R_{z} - s i n R_{x} s i n R_{z} \\ - s i n R_{y} & s i n R_{x} c o s R_{y} & c o s R_{x} c o s R_{y} \end{matrix}]$ 3） scale $s = 1.000002921$
2	4919.23	-10.8887	69.2323
3	5312.43	4.0792	75.8439
4	5532.12	-12.3426	81.3495
5	5763.38	34.0193	87.3027
6	5623.10	-15.0800	92.5050
7	5329.87	30.3328	99.7149
8	5089.17	22.4359	104.9301
9	4832.29	0.2149	112.4945
10	4621.12	-20.5560	117.4503

Table 2. Mean values of statistical results obtained by several parameter estimation methods under different contamination modes

View table

Table 2. Mean values of statistical results obtained by several parameter estimation methods under different contamination modes

Contamination mode	Method	$Δ (T)$	$Δ (R)$ / $10^{- 4}$	$Δ (s)$ / $10^{- 5}$	${\hat{σ}}_{0}$
Multiple outliers	LS	1.6484	6.4240	5.1051	6.2548
	RLS	0.9422	3.8561	3.6149	3.1338
	TLS	0.9189	3.5330	3.6440	3.0544
	RTLS	0.4903	2.1146	1.5600	1.4330
Gaussian mixture errors	LS	1.3245	5.7626	3.4164	5.6865
	RLS	1.0231	4.4188	2.7952	3.7002
	TLS	1.0079	4.4648	2.6377	3.2772
	RTLS	0.7214	3.1027	2.0747	1.9949

Table 3. Registration observation data of laser tracker during changing measuring station from Hefei light source

View table

Table 3. Registration observation data of laser tracker during changing measuring station from Hefei light source

Point name	Laser tracker A			Laser tracker B
Point name	d /mm	α /（°）	β /（°）	d /mm	Α /（°）	β / （°）
P2	5130.34	-149.7184	84.7385	3679.01	-89.9996	55.3951
P3	1804.06	-97.0605	74.8887	3679.01	-0.0035	55.3948
P4	2728.38	13.2676	80.0798	3678.79	89.9947	55.3928
HCG07	1410.30	121.7104	174.2693	2677.13	37.3065	85.3736
HCG08	4836.53	173.0513	106.8160	1969.89	-164.6853	83.5931
HCG09	4286.10	-171.1741	109.0869	1404.26	-122.2511	81.0910
HCG10	1819.21	123.8746	140.3887	2334.53	59.5167	84.6475
HCG11	3126.54	176.5045	116.5998	317.92	137.0496	46.4115

Table 4. Registration accuracy of different parameter estimation methods
View table
Table 4. Registration accuracy of different parameter estimation methods
Method ${\hat{σ}}_{0}$
LS 6.25381
RLS 2.27019
TLS 3.2217
RTLS 0.92563

Tools

Get Citation

Copy Citation Text

Zhipeng Lü, Wei Wang, Yelong Wei. Robust Total Least‑Squares Estimator for Registration of Laser Tracker During Changing Measuring Station[J]. Chinese Journal of Lasers, 2025, 52(17): 1704003

Download Citation

EndNote(RIS)BibTex Plain Text

Set citation alerts for article

Save article for my favorites

Paper Information

Category: Measurement and metrology

Received: Mar. 20, 2025

Accepted: May. 9, 2025

Published Online: Sep. 17, 2025

The Author Email: Yelong Wei (wylong@ustc.edu.cn)

DOI:10.3788/CJL250623

CSTR:32183.14.CJL250623

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

Table 0. [in Chinese]

Table 0. [in Chinese]

Table 1. Simulated data of laser tracker registration during changing measuring station

Table 1. Simulated data of laser tracker registration during changing measuring station

Table 2. Mean values of statistical results obtained by several parameter estimation methods under different contamination modes

Table 2. Mean values of statistical results obtained by several parameter estimation methods under different contamination modes

Table 3. Registration observation data of laser tracker during changing measuring station from Hefei light source

Table 3. Registration observation data of laser tracker during changing measuring station from Hefei light source

Table 4. Registration accuracy of different parameter estimation methods

Table 4. Registration accuracy of different parameter estimation methods