Evaluation of the Large-Length Indoor Standard Installation Based on Single Point Scanning

Abstract

The large length indoor standard installation (LLISI) serves as a standard traceability system for large-scale measuring devices, and hence evaluating the accuracy of its measurements during motion is of great importance. A laser tracker, as a single-point scanning method with strong anti-interference ability and high accuracy for 3D measurement, can meet the measurement requirements of LLISI. In this study, a laser tracker was used to evaluate the accuracy of LLISI during motion based on the single-point scanning method. Fifteen measurement combinations including five measurement intervals and three measurement speeds were selected in the experiment and the results were compared with the measurement data of a laser interferometer. By analyzing the local movement speed of the ISS platform, the uncertainty in local measurement speed of the platform was evaluated. The results showed that the laser tracker has high measurement accuracy and good repeatability for the measurement of LLISI, which can provide strong support for data resource protection of LLISI.

1. Introduction

With the development of industrial measurement technology, large measurement devices such as laser interferometers, laser trackers, laser radars and scanners have become indispensable measurement tools of data measurement in the various fields of national defense, aerospace, shipbuilding, and manufacturing. Large measurement devices need to be tracked back to LLISI. Internationally, seven countries including Germany, the United States, Finland, Japan, and China have established a measurement system of their own for LLISI. In China, an 80-meter LLISI was established by the Chinese Academy of Metrology and Science [1], and several other 50-meter LLISIs were established by the Institute of Metrology in Shanghai, Guangzhou, and Guangxi. In addition, the Great Wall Institute of Metrology and Testing Technology in Beijing established a 35-meter LLISI [2]. The domestic LLISIs all adopt the air-floating guide operation mode.

LLISI is widely used in the measuring field, particularly for the calibration of laser tracking three-dimensional (3D) coordinate measurement systems [3], ground-based laser scanners [4], laser interferometers [5], range finders, and ballbar-based measurement [6]. It can also be used for the calibration of linear displacement sensors, liquid level meters [7,8], invar baseline wires and other non-standard length devices. A laser tracker system (LTS) is a large measurement device with high accuracy used in industrial measurement systems. It collects various advanced technologies such as laser interference ranging technology, photoelectric detection technology, fine mechanics technology, computer and control technology, modern numerical calculation theory, etc., to track the space motion target and measure the target’s spatial 3D coordinate [9]. It is characterized by high accuracy, high efficiency, real-time tracking and measurement, quick installation, and easy operation, which is suitable for real-time measurement of geometric quantities in a large space [10,11].

Laser trackers can be used in combination with sphere targets to realize single-point measurement and single-point scanning, and can work with an intelligent probe system to realize single-point measurement, and with a linear scanning system to realize multi-point scanning. In a study by Yang et al. a laser tracker was combined with a servo control system to greatly improve the performance and stability of the dynamic measurement of the laser tracker [12]. Yang et al. proposed a laser tracker automatic guidance method based on the rotating laser scanning angle measurement technique, which proved that the laser tracker has better robustness for fast motion measurement of the target [13]. Wan et al. proposed a closed iterative combination weighting method using laser trackers to calibrate and compensate the volumetric errors of machine tools, which provided a research direction for the use of laser trackers to measure high-accuracy linear guides [14]. Adamczak, S. et al. proposed a novel method for measuring the waviness of cylindrical surfaces, which proved that the measurement accuracy of the V-block method used to measure the waviness deviation of cylindrical parts depends on the measurement parameter [15]. Xu et al. introduced a sampling strategy to avoid data gaps and obtain evenly distributed data points in the case of an over-fitting deformation analysis in the engineering measurement field [16]. Recently, several studies have investigated various indexes for evaluating LLISI, such as straightness and flatness, while no research has been carried out to evaluate its operation speed which can directly affect measurement accuracy.

In this paper, to improve the speed performance of the linear guide of LLISI and analyze its common measurement intervals, we compared a laser tracker method based on single-point scanning and the laser interferometer method under three nominal measurement speeds, and evaluated the local speed of the linear guide using the laser tracker method. The results showed that the laser tracker method based on single-point scanning has strong robustness, which provides a reasonable evaluation of the operation speed of the linear guide of LLISI.

2. Measurement Platform

LLISI is composed of the guide system, the spatial triangular laser length measuring system, the real-time environment parameter correction system, the control system, the targeting and image acquisition system and the data processing system. The first linear guide LLISI in China that was developed by our institution is shown in Figure 1b. The length of the linear guide was about 57 m, and the two guides were placed on a granite bearing platform, with a distance of about 600 mm and an approximate height of 1.3 m. The laser tracker was aimed at the optical target mirror [17] on the LLISI, which was fixed on the platform via magnetic force. According to the loading conditions, the bearing platform was controlled to move forward, and the coordinates in the coordinate system of the laser tracker were recorded. Since the relative spatial position between the optical target mirror and the measuring platform remained constant, the speed, displacement and positioning accuracy of the optical target mirror directly reflect the motion parameters of the LLISI platform.

The control system of LLISI includes the software control system and the electrical control system. In the software control system, the 60-meter LLISI software based on the Labview environment can indirectly implement the function of the electrical control system, and can set the parameters such as the velocity of the measurement system, time and so on. There are some function modules for controlling the system and monitoring the status of system in the software, including experimental environment monitoring, operation status monitoring of the stress system, the electric control system trigger and data acquisition. The electrical control system is composed of a power supply control system, control handle, and bed electrical module (such as drive motor, sensor, limit switch, etc.). Pulse width modulation (PWM) method is utilized to control the DC motor drive of x-axis for modulating the movement mode of Sanyo motor. In order to accurately control the measuring platform, the PWM method can not only reduce 2-axis inertia but also enhance the response rate of motor. In addition, controlling the initial value of timer of microprocessor can transform the input analog control signal to the digitally coded signal to implement the PWM method. It can instantly generate pulse waveforms with different duty ratios, thereby achieving the adjustment of the average speed of the platform. The average speed of the continuously adjustable measuring platform ranged from 0 to 200 mm/s, which can be calculated by the following equation: In order to accurately control the measuring platform, the PWM speed control system was used to adjust the speed of the electric platform of LLISI. The initial value of the internal timer of the single-chip computer was adjusted to generate pulse waveforms with different duty ratios, thereby achieving the adjustment of the average speed of the platform. The average speed of the continuously adjustable measuring platform ranged from 0 to 200 mm/s, which can be calculated by the following equation:

The average speed of the measuring platform:

$${\mathrm{V}}_{\mathrm{d}}={\mathrm{V}}_{\max }·\frac{{\mathrm{t}}_1}{\mathrm{T}}$$

(1)

$${\mathrm{t}}_1=\left(2^{\mathrm{n}}-{\mathrm{T}}_{\mathsf{\omega }}\right)\times \frac{\mathrm{N}}{\mathrm{f}\times {10}^6}$$

(2)

where ${\mathrm{V}}_{\mathrm{d}}$ is the average speed of the motor, ${\mathrm{V}}_{\max }$ is the maximum rotational speed of the motor, ${\mathrm{t}}_1$ is the length of the timer, ${\mathrm{T}}_{\mathsf{\omega }}$ is the initial value of the timer, $\mathrm{N}$ is the clock value in one machine cycle, $\mathrm{f}$ is the clock frequency of the single-chip computer.

As shown in Figure 1a, the specific steps of the method are as follows:

(1)

An optical target mirror for aiming of the laser tracker was installed on the LLISI. There was no relative motion between the target mirror and the platform during the aiming of the laser tracker, that is, the optical target mirror remains in the local coordinate system o₁x₁y₁z₁ of the platform;

(2)

The laser tracker was installed at a proper position while ensuring that the optical target mirror was within the measurement range of the laser tracker and the field range of the lens of the laser tracker. The position of the laser tracker remained unchanged during measurement, that is, the measurement coordinate system o_gx_gy_gz_g of the laser tracker does not change relative to the earth coordinate system during measurement;

(3)

The laser tracker was adjusted to aim at the optical target mirror on the multi-functional bearing platform, and the measurement parameters of the laser tracker were set. The automatic tracking mode was enabled. The multi-functional bearing platform was controlled to move in the axial direction x1 of its coordinate system o₁x₁y₁z₁, and data were collected;

(4)

The data measured by the laser tracker were the spatial coordinates of the optical target mirror while moving with the multi-functional bearing platform. Since the relative position between the optical target mirror and the multi-functional bearing platform remained unchanged, the motion trajectory of the optical target mirror was that of the multi-functional bearing platform. Through data processing such as fitting and difference, the speed, displacement, and positioning accuracy of the multi-functional bearing platform was obtained.

3. Methods

In this study, the displacement and speed of LLISI were measured using the laser tracker based on single-point scanning and compared with the measurement results of the laser interferometer. The laser tracker obtained the speed between two points by calculating the spatial distance between them, whereas the laser interferometer calculated the instantaneous speed between two points through a one-dimensional measurement of the distance between the two points. The measurement range of the laser tracker was set to (0~80) m, and the uncertainty of the length measurement indication error was U = (0.80 + 0.11 L) µm, k = 2 (L = length; m = meter). The laser tracker was set to the equal time-interval mode to measure the multi-functional motion platform along with its movement. A difference operation was performed on the data, and the overall average speed and local average speed were then obtained.

3.1. Selection of Measurement Intervals

The effective measurement range of LLISI is between 0 and 54.3 m and the measurement interval can be selected randomly within the effective range. The distance interval was determined based on the calibration standards for commonly used measurement devices, e.g., handheld laser range finder [18], standard steel tape [19], short-range sensors [20], laser interferometers [21], ground laser scanners [22] and laser trackers [23], as shown in

Table 1. Selection Standard for Measurement Intervals.

No.	Measurement Intervals	Measurement Subjects
1	0.1 m	Handheld laser range finder, standard steel tape measure, and short-range sensors
2	0.3 m	Handheld laser range finder
3	0.5 m	Standard steel tape measure
4	1.0 m	Handheld laser range finder, laser trackers, and laser interferometers
5	5.0 m	Handheld laser range finder, ground laser scanners, and laser trackers

. The calibration in this study was carried out using the measurement intervals of 0.1 m, 0.3 m, 0.5 m, 1 m, and 5 m.

3.2. Selection of Nominal Measurement Speed

The nominal speed of LLISI can be adjusted by the control system. The speed used in the calibration of the handheld laser range finder and the laser tracker is usually 150 mm/s, the speed was 50 mm/s for a standard steel tape measure, and 100 mm/s and 150 mm/s for laser interferometer. In this study, three measurement speeds, i.e., 50 mm/s, 100 mm/s, and 150 mm/s were selected for data collection and analysis. There is a 200 kg z-axis on the bearing platform of the linear guide. The guide can be worn quickly when the speed is above 150 mm/s.

4. Results and Analysis

The measurement results under different measurement intervals and nominal speeds were analyzed to evaluate the accuracy of LLISI measurements. Moreover, in order to reduce the influence of environmental factors on the accuracy and stability of the measurement data, the laser tracker and the laser interferometer were warmed up for more than half an hour, and the environmental parameters of the two devices were set at the same standard state, namely, the environment temperature at 20.0 °C, humidity at 60% RH, air pressure at 1013 kPa, and additional data of 1 to 2 s were collected after the linear guide stopped. Three repeated data collection and analyses at 0.1 m, 0.3 m, 0.5 m and 1 m were performed at three speeds, i.e., 50 mm/s, 100 mm/s, and 150 mm/s. At 5 m, another three data collections were carried out with the speeds of 50 mm/s and 100 mm/s, and ten repeated data collections at 150 mm/s.

4.1. Comparative Analysis of Single Measurement Results

The constant speed stage refers to the time from the first acceleration to the nominal speed, to the last; the local average speed is the mean value of the speed in the constant speed stage; and the total length of the movement period is the summation of three stages, i.e., acceleration, constant speed and deceleration. The global average speed is the mean value of speed collected during the whole movement period. In the comparative experiment of the single measurement, the results of the two methods are shown in Figure 2, both of which were divided into acceleration, constant speed and deceleration stages. The acceleration and deceleration stages were instantaneous changes. In the constant speed stage, there were slight fluctuations in the measurement results of the two methods, with the data of the laser interferometer having more obvious oscillation. In terms of the number of measurement points with a speed over 150 mm/s, the laser tracker and the laser interferometer had 313 and 314 points, respectively. In addition, the local average speeds of the two were 149.9985 mm/s and 150.0052 mm/s, respectively. Thus, by comparing the single measurement results of the two methods, the laser tracker was more stable in the submillimeter-level and had strong resistance to environmental interferences, which is more suitable for the measurement of LLISI.

4.2. Comparative Analysis of Multiple Measurement Results

Figure 3 shows the results of 10 multiple measurements of the laser tracker with a measurement interval of 5000 mm. The ten groups of data show great similarity, reflecting good data repeatability. Due to an error in the initial position in the experimental, there were differences in the total number of points of each measurement.

Table 2. Results of ten measurements of the laser tracker with a measurement interval of 5000 mm.

Time (s)	Time of the Constant Speed Stage (s)	Mean Value of Local Speed (mm/s)	Time of the Movement Period (s)	Global Average Speed (mm/s)	Time (s)	Time of the Constant Speed Stage (s)	Mean Value of Local Speed (mm/s)	Time of the Movement Period (s)	Global Average Speed (mm/s)
1	31.3	149.9985	35.9	139.2755	6	31.4	149.9985	35.8	139.6646
2	31.1	149.9979	35.8	139.9945	7	31.3	149.9980	35.6	140.449
3	31.7	149.9982	35.6	140.4493	8	31.2	149.9988	35.5	140.8448
4	31.5	149.9978	35.6	140.4492	9	31.3	149.9988	35.4	141.2427
5	31.5	149.9984	35.4	141.2428	10	31.2	149.9984	35.9	139.2754

shows the results of 10 measurements of the laser tracker with a measurement interval of 5000 mm and a nominal speed of 150 mm/s. According to the data, the local average speed was stable within a distance range of 5000 mm, which was between 149.9978 mm/s and 149.9988 mm/s for the ten measurements. The number of measurement points that met the requirements ranged 312–317, and the maximum time of the movement was 35.9 s, and the average time was 35.7 s.

Table 3. Measurement results of the laser tracker at different measurement intervals and speeds.

Measurement Interval (mm)	Nominal Speed (mm/s)	Average Time of the Constant Speed Stage of Multiple Measurements (s)	The Local Average Speed of Multiple Measurements (mm/s)	Average Time of the Movement Period (s)	Maximum Time of the Movement Period (s)	Global Average Speed (mm/s)
100	50	1.2	50.0124	3.4	3.4	29.4103
100	100	/	/	2.7	2.7	32.9874
100	150	/	/	3.2	3.5	31.5788
300	50	5.4	50.0003	7.5	7.9	40.0002
300	100	1.7	100.0028	5.8	6.5	52.0232
300	150	0.2	150.0122	4.6	4.9	65.6937
500	50	9.2	50.0000	11.2	11.3	44.7756
500	100	3.7	100.0060	7.2	7.4	69.4442
500	150	1.2	150.0150	5.6	6.0	88.7562
1000	50	19.2	49.9992	21.9	22.2	45.7312
1000	100	8.6	99.9989	11.8	11.9	84.7452
1000	150	4.8	149.9967	9.2	9.6	108.3020
5000	50	99.2	49.9995	101.4	101.7	49.3258
5000	100	47.4	99.9987	52.5	53.4	95.2984
5000	150	31.4	149.9983	35.7	35.9	140.2853

shows the results of the laser tracker at different measurement intervals and speeds. With an interval of 100 mm, and the nominal speeds were 100 m/s and 150 mm/s, and the measurement speed did not reach the set value, thus there was not a constant speed stage. Under the same measurement intervals, the greater the nominal speed, the greater the absolute value of the difference between the actual speed and the nominal speed. The last row in Table 3 is the summary of data in Table 2. When the measurement interval and the speed was 5000 mm and 150 mm/s, the average time of the 10 measurements was 31.4 s and 35.7 s in the constant speed stage and the whole movement period, and the maximum time of the movement period was 35.9 s. The local average speed of multiple measurements is $\overline{\mathrm{v}}=\frac{{\sum }_1^{\mathrm{n}}{\mathrm{v}}_{\mathrm{i}}{\mathrm{t}}_{\mathrm{i}}}{{\sum }_1^{\mathrm{n}}\mathrm{ti}}$, where n = 10, v_i is the local average speed in the ith measurement, t_i is the time of the constant speed stage in the ith measurement, and the value of i is from 1 to 10, and the calculation of the global average speed was the same. When the nominal speed was 100 mm/s and the measurement interval was 100 mm, the measurement requirement was not met, leading to the loss of measurement results and the difficulty in determining the measurement boundary. In the last two columns of Table 3, when the measurement interval was the same, the greater the nominal speed, the shorter the total time in the whole movement period. In addition, when the nominal speed was 150 mm/s with an interval of 5000 mm, the average time in the whole movement period was 35.7 s, and the maximum time was 35.9 s. The average time and the maximum time in the movement period can be an important reference basis for the measurement of LLISI.

The results of ten measurements of the laser interferometer with a measurement interval of 5000 mm are shown in Figure 4. Compared with the results of the laser tracker, there was obvious oscillation in the constant speed stage. The reflector was rigidly fixed on the cylinder, and thus, has low oscillation resistance during motion. In addition, there were uncertainties in terms of the distance difference and time difference of the laser tracker and the laser interferometer. Since both methods were non-coaxial measurements, the Abbe error was introduced. Since the laser tracker method has small interference, it is more advantageous in the measurement of LLISI.

The measurement interval and the nominal speed were 5000 mm and 150 mm/s, respectively. The measurement results of the laser interferometer are shown in

Table 4. Results of ten measurements of the laser interferometer with a measurement interval of 5000 mm.

Time (s)	Time of the Constant Speed Stage (s)	Mean Value of Local Speed (mm/s)	Time of the Movement Period(s)	Global Average Speed (mm/s)	Time (s)	Time of the Constant Speed Stage (s)	Mean Value of Local Speed (mm/s)	Time of the Movement Period(s)	Global Average Speed (mm/s)
1	31.4	150.0052	36.0	138.8888	6	31.4	150.0041	35.6	140.4494
2	31.5	150.0045	35.7	140.0559	7	31.7	150.0043	35.4	141.2429
3	31.7	150.0040	36.1	138.5041	8	31.7	150.0036	35.5	140.8450
4	31.7	150.0040	35.7	140.0560	9	31.8	150.0033	36.0	138.8888
5	31.5	150.0047	35.9	139.2759	10	31.4	150.0045	35.6	140.4494

. The mean value of the local average speed was stable within the distance range of 5000 mm, with all values between 150.0033 mm/s and 150.0052 mm/s. The number of measurement points that met the requirement was between 314 and 318. When the sampling frequency is the same, the three-dimensional measurement value is theoretically greater than that of one-dimensional measurement, and the increase in the instantaneous speed leads to an increase in the local average speed. Comparing the measurement results of the two methods, the mean value of the local speed of the laser interferometer (0.0056 mm/s) was higher than that of the laser tracker. During dynamic measurement, there were micron-level differences between the two methods, which verified the effectiveness of the laser tracker. Compared with the laser tracker method, the average time in the whole movement period of the laser interferometer increased by 0.1 s to 35.8 s, and the maximum time increased by 0.2 s to 36.1 s.

5. Uncertainty of Local Speed Measurement

Local measurement speed is a key parameter to determine whether the LLISI is operating normally, it reflects the stability of the LLISI and is closely related to the measurement environment, measurement repeatability, and error of the target mirror. It is of scientific value to evaluate the local measurement speed of the single-point laser tracker method using the measurement uncertainty. The local speed of the LLISI was evaluated based on methods used in relevant studies [24,25,26].

Mathematical model: Speed calculation equation:

$$\mathrm{v}=\frac{\mathrm{s}}{\mathrm{t}}$$

(3)

where v is the measurement speed, s is the measurement interval or range, and t is the total operation time.

The synthetic standard uncertainty ${\mathrm{u}}_{\mathrm{c}}\left(\mathrm{v}\right)$ can be obtained by ${\mathrm{u}}^2\mathrm{y}={\sum }^{}{\left[\frac{\partial \mathrm{f}}{\partial \mathrm{x}}\right]}^2{\mathrm{u}}^2\left({\mathrm{x}}_{\mathrm{i}}\right)$, which satisfies the following equation:

$${\mathrm{u}}_{\mathrm{c}}{\left(\mathrm{v}\right)}^2={\mathrm{c}}_1^2{\mathrm{u}}_1^2\left({\mathrm{s}}_{\mathrm{i}}\right)+{\mathrm{c}}_2^2{\mathrm{u}}_2^2\left({\mathrm{t}}_{\mathrm{i}}\right)$$

(4)

where the sensitivity coefficients are ${\mathrm{c}}_1=\frac{1}{\mathrm{t}}$, and ${\mathrm{c}}_2=-\frac{\mathrm{s}}{{\mathrm{t}}^2}$. Therefore, the synthetic uncertainty ${\mathrm{u}}_{\mathrm{c}}\left(\mathrm{v}\right)$ of the local speed is as follows:

$${\mathrm{u}}_{\mathrm{c}}\left(\mathrm{v}\right)=\sqrt{\frac{1}{{\mathrm{t}}^2}{\mathrm{u}}_1^2\left({\mathrm{s}}_{\mathrm{i}}\right)+\frac{{\mathrm{s}}^2}{{\mathrm{t}}^4}{\mathrm{u}}_2^2\left({\mathrm{t}}_{\mathrm{i}}\right)}$$

(5)

According to the mathematical model and the sensitivity coefficients, there are several factors affecting the standard uncertainty of LLISI: ➀ the standard uncertainty component ${\mathrm{u}}_{\mathrm{A}1}\left({\mathrm{s}}_1\right)$ introduced by the range measurement repeatability of the laser tracker, ➁ the uncertainty component ${\mathrm{u}}_{\mathrm{B}1}\left({\mathrm{s}}_1\right)$ introduced by the measurement uncertainty of the laser tracker, ➂ the uncertainty component ${\mathrm{u}}_{\mathrm{B}2}\left({\mathrm{s}}_2\right)$ resulted from the drift of the laser tracker, ➃ the uncertainty component ${\mathrm{u}}_{\mathrm{B}3}\left({\mathrm{s}}_3\right)$ introduced by the resolution of the laser tracker, ➄ the uncertainty component ${\mathrm{u}}_{\mathrm{B}5}\left({\mathrm{s}}_5\right)$ from the measurement environment, ➅ the uncertainty component ${\mathrm{u}}_{\mathrm{B}5}\left({\mathrm{s}}_5\right)$ introduced by the error of the center of the optical target mirror, and ➆ the uncertainty component ${\mathrm{u}}_{\mathrm{B}4}\left({\mathrm{t}}_4\right)$ introduced by the collection time of the laser tracker. The Sources of measurement uncertainty are shown in

Table 5. Sources of measurement uncertainty.

No.	Measurement Uncertainty Components	Values	No.	Measurement Uncertainty Components	Values
1	Range measurement repeatability of the laser tracker	11.45 μm (k = 1)	4	Measurement environment	0.3 μm (k = $\surd 3$)
2	Measurement uncertainty of the laser tracker	0.4 μm (k = 2)	5	Error of the center of the optical target mirror	0.5 μm (k = 2)
3	Reading drift of the laser tracker	0.12 μm (k = $\surd 3$)	6	Collection time of the laser tracker	5 μs (k = 2)

.

The calculation of the measurement uncertainty components are as follows: multiple measurements were carried out with a nominal speed of 150 mm/s, a collection time of 0.1 s, and a nominal range of 0.015 m. The standard deviation was calculated based on the Bessel equation ${\mathrm{u}}_{\mathrm{A}}\left({\mathrm{s}}_1\right)=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{(\mathrm{D}-\overline{\mathrm{D}})}^2}{\mathrm{n}-1}}=11.45\mathsf{\mu }\mathrm{m}$ The measurement uncertainty can be obtained from the calibration certificate, U = (0.80 + 0.11 L) mm, (L:m), (k = 2). Thus, the standard uncertainty component at 0.015 m was ${\mathrm{u}}_{\mathrm{B}1}\left({\mathrm{s}}_1\right)\approx 0.4\mathsf{\mu }\mathrm{m}$. Since there was drifting of the laser tracker, the maximum drift introduced by the reading time difference was not more than $0.2\mathsf{\mu }\mathrm{m}$. Assuming a uniform distribution, then, ${\mathrm{u}}_{\mathrm{B}2}\left({\mathrm{s}}_2\right)=\raisebox{1ex}{$0.2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.\approx 0.12\mathsf{\mu }\mathrm{m}$. The digital resolution of the laser tracker was 1 μm, the half width of the interval was 0.5 μm, and k =$\sqrt{3}$ by assuming a uniform distribution. Thus, the uncertainty component was ${\mathrm{u}}_{\mathrm{B}3}\left({\mathrm{s}}_3\right)=0.29\mathsf{\mu }\mathrm{m}$. Moreover, the environmental factors included temperature, humidity, and air pressure. The measurement error of the average optical temperature of the laser interferometer was 0.1 °C, the measurement error of the optical air pressure was 11 Pa, and the measurement error of the partial air steam was 40 Pa. Thus, ${\mathrm{u}}_{\mathrm{B}5}\left({\mathrm{s}}_5\right)$≈ 0.3 μm based on the assumption of uniform distributions. The optical center error refers to the deviation of the optical center and the mechanical center of the reflector within the effective acceptance angle. The uncertainty component ${\mathrm{u}}_{\mathrm{B}5}\left({\mathrm{s}}_5\right)$ = 0.5 μm was introduced by the optical center error. An equal interval (0.1 s) was used for data collection of the laser tracker. The notebook time was maintained by the crystal oscillator when there was no internet, and the uncertainty component introduced by the collection time was ${\mathrm{u}}_2\left(\mathrm{t}\right)$ = 5 μs. For the uncertainty components of the measurement repeatability and the resolution of the laser tracker, the maximum values were taken.

The synthetic uncertainty and the expanded uncertainty are as follows:

$${\mathrm{u}}_1\left(\mathrm{s}\right)=\sqrt{{\mathrm{u}}_{\mathrm{A}}{({\mathrm{s}}_1)}^2+{\mathrm{u}}_{\mathrm{B}1}{({\mathrm{s}}_1)}^2+{\mathrm{u}}_{\mathrm{B}2}{({\mathrm{s}}_2)}^2+{\mathrm{u}}_{\mathrm{B}3}{({\mathrm{s}}_3)}^2+{\mathrm{u}}_{\mathrm{B}4}{({\mathrm{s}}_4)}^2}\approx 11.49\mathsf{\mu }\mathrm{m}$$

$$\mathrm{t}=0.1\mathrm{s},\mathrm{s}=0.015\mathrm{m},\mathrm{and}\mathrm{thus},{\mathrm{u}}_{\mathrm{c}}\left(\mathrm{v}\right)=\sqrt{\frac{1}{{\mathrm{t}}^2}{\mathrm{u}}_1^2\left({\mathrm{s}}_{\mathrm{i}}\right)+\frac{{\mathrm{s}}^2}{{\mathrm{t}}^4}{\mathrm{u}}_2^2\left({\mathrm{t}}_{\mathrm{i}}\right)}\approx 115.1\mathsf{\mu }\mathrm{m}/\mathrm{s}$$

The expanded uncertainty of the local speed of the laser tracker method is U ≈ 0.23 mm/s, (k = 2).

6. Conclusions

In this study, the laser tracker method based on single point scanning was proposed to evaluate the accuracy of the LLISI. The laser tracker, as a method of measuring the LLISI, can achieve high measuring accuracy, high repeatability and large space measurement. Otherwise, the proposed method is able to dynamically collect 3D point cloud data in real time and has been used to provide a higher measurement accuracy.

Under the standard experimental environment, the measurements were carried out at measurement intervals of 0.1 m, 0.3 m, 0.5 m, 1.0 m and 5.0 m and at 50 mm/s, 100 mm/s and 150 mm/s, respectively. The measurement results of the laser tracker were compared with that of the laser interferometer through single and multiple measurements under the case of indoor large-scale measurement. The maximum time and average time in the movement period under different measurement intervals and nominal speeds were quantified. Hence, it is distinctly shown that the proposed method has the advantage of high insensitivity to noise. In addition, the evaluation method of measurement uncertainty was employed to evaluate the results of the local velocity of the LLISI measured by the proposed method. When the value of a confidence factor is k = 2, the uncertainty of the local measurement speed of the laser tracker was approximatively 0.23 mm/s evaluated in accordance with the measurement uncertainty standards. The results showed that the laser tracker has the features of minor errors, low data dispersion, and superior repeatability.

In summary, the proposed method has a high accuracy greater potential capacity in the single measurement and can be used utilized to provide support for data resource protection and health management of the LLISI. However, the large capacity of sampling data has introduced the challenge of processing data to influence the real-time result of evaluating the LLISI. For ensuring to rapidly and accurately estimate the operating state of the LLISI, the intelligent optimization algorithms will be significantly taken into account to process and analyze the measurement data in the follow-up research work.