## Abstract

A laser beam used as a visualizing measuring axis is an important technique in 3D shape measurement. A highly accurate calibration method of a laser beam based on discrete point interpolation is proposed in this paper. A flexible control field constructed by a laser tracker, a theodolite and a target plane with 5 high-precision machining holes is presented. The discrete point interpolation model is established by the coordinates of holes measured by a laser tracker and the angles of holes measured by a theodolite. The coordinates of laser spots on the target plane are obtained based on the angles and discrete point interpolation model, and the direction vector of the laser beam is obtained by linear fitting. The optimal measurement pose of a theodolite is analyzed by the simulation results. The experimental results show that the RMSE of linear fitting of laser beams is no more than 14 µm within a 5 m distance, the RMSE of the spatial points is 0.09 mm and the RMSE of the reconstructed distance is 0.09 mm.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

There is an increasing demand for the efficiency and accuracy of 3D precision measurement [1–2]. The traditional contact probes or telescopes of coordinate measurement systems have been replaced by the collimated lasers or laser displacement sensors [3–5]. Moreover, these laser measurement systems can realize automatic object recognition and measurement by combining with machine vision [6–9]. Therefore, calibrating the spatial pose of the laser beam is a key aspect for 3D precision measurements, and the measurement accuracy depends heavily on the calibration method [10–11].

Sun et al. presented a vision measurement model of laser displacement sensors and proposed a corresponding calibration method of the parameters [12]. A planar target with featured lines is moved by a 2D moving platform to some preset known positions and the linear array camera is used for collecting target images. The parameters of the laser line are gained by linear fitting. Yang et al. proposed a calibration method of the direction of the laser beams for the inner diameter measuring device [13]. The laser beams are rotated and translated in the plane to constitute the rotary rays. The direction calibration of the laser beams is achieved by the sensors’ distance information and corresponding data processing method. Bi et al. presented a calibration method to determine the laser beam direction based on a standard sphere for the combination of optical sensor and CMM [14]. The sensor moves at an equal step along X, Y and Z axes respectively and the equations are established to calculate the unit direction vector of laser beam. Wu et al. proposed a calibration method of laser beam for articulated laser sensor [15]. A 1D linear displacement optical calibration device and CMM are employed to calibrate the direction vector and fixed point of laser beam. Shen et al. proposed a robust and efficient calibration method for spot laser probe [16]. During the calibration procedure, hundreds of points are scanned and mapped into a sphere for refining the initial laser beam direction. The above calibration methods has high accuracy. However, the calibration distance is limited, which is only applied for the small-scale 3D measurement.

Ma et al. presented a large-scale laser plane calibration system [17]. This plane is composed of a 35 m long horizontal granite guide rail and a 1.5 m long vertical metal guide rail. The calibration system is too complex to operate. Shao et al. proposed a calibration method for a vision guiding-based laser-tracking measurement system [18]. The points on the laser beam are obtained with the help of a normal planar target and the root mean square error of the calibration result is 1.46 mm within a range of 10 m. Hu et al. used a trirectangular trihedron as the calibration target to calibrate the 2D laser rangefinder and camera [19]. This method requires only single shot of the target and the simplified perspective-three-point problem is solved to determine the pose of laser rangefinder. Liu et al. proposed a method for calibrating the relative position and orientation of laser beam and single camera [20]. The proposed method requires a checkerboard to establish constraint equation, and the internal parameters of cameras must be calibrated. Wu et al. proposed a calibration method for non-orthogonal shaft laser theodolite measurement system [21]. The points on the laser beam are measured by manual aiming. Miao et al. presented a dynamic calibration method of the laser beam for non-orthogonal shaft laser theodolite measurement system [22]. In this calibration procedure, the rotation errors of rotary tables are introduced. The above calibration methods are suitable for the regular-scale and large-scale 3D measurement. However, the calibration and application conditions have limited the calibration accuracy.

Aiming at the highly-accurate calibration method of laser beam for 3D precision measurement, a novel calibration method of laser beam base on discrete point interpolation is proposed in this paper. The global coordinate system is provided by laser tracker, which is highly-accurate 3D coordinate measurement instrument. The angles of local interpolation model is provided by theodolite, which is highly-accurate angle measurement instrument. Moreover, the relative pose of theodolite and calibration target plane is analyzed to improve the accuracy of discrete point interpolation model. The novel calibration method is simple to operate and does not need complex optical and imaging systems.

The remainder of this paper is organized as follows. In Section 2, the calibration principle is introduced in detail. Section 3 analyzes the relative pose of theodolite and target plane. In Section 4, the calibration method is tested and evaluated with real data experiments. The paper ends with some concluding remarks in Section 5.

## 2. Calibration principle

The laser beam can be viewed as a spatial line. The equation of laser beam can be expressed as

where $({i,j,k} )$ is the direction vector of laser beam and $({{x_{Lw}},{y_{Lw}},{z_{Lw}}} )$ is the coordinate of a fixed point on the laser beam. The calibration task is to obtain $({i,j,k} )$ and $({{x_{Lw}},{y_{Lw}},{z_{Lw}}} )$. The calibration process can be simply divided into two parts, including calibrating the direction vector of laser beam and the coordinate of a fixed point on the laser beam. Firstly, a set of fixed points are calibrated by discrete point interpolation method with the help of laser tracker and theodolite. Then the linear fitting is performed from these fixed points and the direction vector of the fitted line is regarded as the direction vector of the laser beam.#### 2.1 Calibration of the fixed point on the laser beam

The calibration principle of the fixed point on the laser beam is shown in Fig. 1. The coordinate system of laser tracker, which is regarded as the world coordinate system, is defined as ${O_W}{x_W}{y_W}{z_W}$. On the target plane, the 3D coordinates of point *A*, *B*, *C*, *D* and *E* are known in ${O_W}{x_W}{y_W}{z_W}$. The 3D coordinate system on the target plane is defined as ${O_P}{x_P}{y_P}{z_P}$. The point *A* is defined as the origin of ${O_P}{x_P}{y_P}{z_P}$. The normal vector $\overrightarrow V$ of the target plane is defined as the z-axis. The direction vector of the line connecting the points *E* and *B* is defined as the x-axis. The y-axis is obtained by the right-hand rule. The transformations from ${O_W}{x_W}{y_W}{z_W}$ to ${O_P}{x_P}{y_P}{z_P}$ is expressed as

*A*,

*B*and

*E*in the coordinate system.

The 2D coordinate system on the target plane is defined as $Oxy$. The origin, x-axis and y-axis of $Oxy$ are the same as ${O_P}{x_P}{y_P}{z_P}$. The transformations from ${O_P}{x_P}{y_P}{z_P}$ to $Oxy$ is expressed as

*A*,

*B*,

*C*,

*D*and

*E*in are converted into the 2D coordinates in $Oxy$.

It is difficult to directly and precisely measure the 3D or 2D coordinate of laser spot on the target plane by laser tracker. A discrete point interpolation model is adopted to calculate the accurate 2D coordinate of laser spot under the condition of non-accurate aiming by laser tracker. The theodolite is rotated in a small angle range. The movement trajectory *AB* of sight axis on the target plane is approximately a line, as shown in Fig. 2. The range of small angle and the accuracy of discrete point interpolation model are related to the spatial pose of sight axis and target plane, which will be analyzed in Section 3. *AB* is decomposed into two directions in $Oxy$, including x-axis and y-axis. The relationship between moving distance on each axis and the rotation angles of theodolite is approximately linear. The linear relationship is described as

The theodolite is rotated more than three times in a small angle range. And more than three sets of $({x,y} )$ and $({\alpha ,\beta } )$ are obtained. $({{A_1},{B_1},{C_1}})$ and $({{A_2},{B_2},{C_2}} )$ are derived by least square method as follows

The rotation angle $({{\alpha_L},{\beta_L}} )$, which is obtained by theodolite aiming at the laser spot on the target plane, is substituted into Eq. (4). And the 2D coordinate of laser spot is obtained and defined as $({{x_L},{y_L}} )$. The 3D coordinate of laser spot in ${O_W}{x_W}{y_W}{z_W}$ is described as

*pinv*is a mathematical symbol used to calculate the pseudo-inverse matrix.

In this paper, five sets of 2D coordinates on the target plane are adopted to perform the least square method.

#### 2.2 Calibration of the direction vector of a laser beam

The target plane is moved in front of the laser and a set of points on the laser beam is obtained. The direction vector of laser beam is obtained by linear fitting from the calibrated fixed points. The equation of spatial line can be simply expressed as

where $({i,j,1} )$ is the direction vector of fitted line, $({{x_0},{y_0},0} )$ is the coordinate of a fixed point on the fitted line and $({{x_{LW}},{y_{LW}},{z_{LW}}} )$ is the coordinate of calibrated fixed point.The Eq. (7) is taken the form of a matrix equation

*i*,

*j*, ${x_0}$ and ${y_0}$ in the Eq. (8). The unknown parameters are obtained by least square method from the above calibrated fixed points as follows

*N*represents the number of the calibrated fixed points in Section 2.1.

By above the analysis, the direction vector of the fitted line is regarded as the direction vector of the laser beam. And the fixed point on the laser beam are expressed as

## 3. Relative pose of a theodolite and target plane

#### 3.1 Error of discrete point interpolation model

Based on Solidworks, a simulation environment for discrete point interpolation model is set up as shown in Fig. 3. The accuracy of discrete point interpolation model is related to the relative pose of theodolite and target plane. The angles between sight axis and vertical axis of theodolite are set as 5°, 10° and 15°, respectively. The distance between theodolite and target plane, which is defined as the measurement distance of theodolite, is set from 0.5 m to 10 m. Utilizing the simulation angles and 2D coordinates, the discrete point interpolation model is established and the 2D coordinates of the laser spot on the target plane are obtained. The true 2D coordinates are generated by Solidworks. The deviations of measured values and true values are shown in Fig. 4.

From Fig. 4, the simulation results of discrete point interpolation model are obtained as follows. Firstly, the error of discrete point interpolation model decreases with the increase of the measurement distance of theodolite. Secondly, the error increases with the increase of the angle between sight axis and vertical axis of theodolite. So the angle between sight axis and vertical axis of theodolite is set less than 5° in the real data experiments.

The corresponding relationship between the error of discrete point interpolation model and the measurement distance of theodolite can be expressed as

where ${y_{inp}}$ represents the error of discrete point interpolation model, ${x_T}$ represents the measurement distance of theodolite, and $({{a_\textrm{1}},{b_\textrm{1}}} )$ is the unknown parameters of nonlinear equations.The corresponding relationships are fitted by Levenberg-Marquardt method [23]. Taking the angle of 5° as an example, the fitting results are shown in Fig. 5 and Table 1.

#### 3.2 Measurement distance of theodolite

The error of discrete point interpolation model decreases with the increase of the measurement distance of theodolite. However, the aiming error of theodolite increases with the increase of the measurement distance of theodolite. The corresponding relationship between the aiming error and measurement distance of theodolite can be approximatively expressed as

where ${y_{aim}}$ represents the aiming error of theodolite, ${x_T}$ represents the measurement distance of theodolite, and $\delta$, which is the angle error of theodolite, is 0.5″.The combination of the error curve of discrete point interpolation model and the error curve of theodolite aiming provides an optimal measurement distance for theodolite, as shown in Fig. 6. The coordinate of intersection point of two curves is (1749.099, 0.004) mm. The distance of 1749.099 mm is utilized as the optimal measurement distance of theodolite.

## 4. Real data experiments

#### 4.1 Calibration of laser beam

The flexible control field is constructed by a theodolite, a laser tracker and a target plane with 5 high-precision machining holes, as shown in Fig. 7. The accuracy of theodolite is $0.5^{\prime\prime}$, and the accuracy of the laser tracker is $7.5{\mu}m + s{\ast}3{\mu} m / m$, where s is the measuring distance. The laser is mounted on the non-orthogonal shafting laser theodolite. The calibration distance is from 0.5 m to 5 m.

The calibration procedures are as follows:

- (1) The laser tracker measures 9 points on the target plane and the plane is obtained by planar fitting.
- (2) The laser tracker measures the 5 holes, and then the coordinates of holes are projected onto the fitted plane. The 2D coordinate system on the target plane is established as follows. The projection point of hole 1 is defined as the origin. The direction vector of the line connecting the projection points hole 5 and hole 2 is defined as the x-axis, and the downward direction is taken as positive for y-axis.
- (3) The theodolite measures the 5 holes and laser spot on the target plane, and the angles are recorded.
- (4) The discrete point interpolation model is established by the 2D coordinates and angles of the projection points of holes.
- (5) The angles of laser spot are substituted into the discrete point interpolation model, and the 2D coordinate of laser spot is obtained. Then the 2D coordinate is transformed into 3D coordinate in the laser tracker coordinate system.
- (6) The target plane is moved in front of the laser. Repeat steps (1) ∼ (5) and a set of points on the laser beam is obtained. The direction vector of the laser beam is obtained by linear fitting.

#### 4.2 Validation experiments

The accuracy of the proposed calibration method in this paper is evaluated by spatial 3D points and distances. The true values are the measured values of the laser tracker and the tested values are the measured values of the non-orthogonal shafting laser theodolite. 16 points are placed at different spatial positions, which are shown in Fig. 9. Figure 10 shows that the 3D spatial points are measured by non-orthogonal shafting laser theodolite and laser tracker, respectively. A reference target which can be exchanged with the standard laser tracker target precisely is measured by the non-orthogonal shafting laser theodolite. The distance between any two points is calculated in turn, and a total of 120 distances are obtained. At a working distance of 1 m ∼ 5 m, the experimental results are shown in Fig. 11, Fig. 12, Table 4 and Table 5.

From Fig. 9, Fig. 10, Fig. 11, Fig. 12, Table 4 and Table 5, the experimental results are obtained as follows. First, the RMSE of spatial points is 0.09 mm and the RMSE of reconstructed distance is 0.09 mm. The encouraging results prove that proposed method is suitable for 3D precision measurement. Second, the 3D coordinate errors increase with the increase of measurement distance.

## 5. Conclusions

In the application of 3D precision measurement, highly-accurate spatial pose of laser beam has heightened the demand for calibration. This study focuses on the highly-accurate calibration for spatial pose of laser beam. A flexible control field constructed by a laser tracker, a theodolite and a target plane with 5 high-precision machining holes is adopted to achieve calibration. With the advantages of laser tracker and theodolite, it is expected that the combined approach gives an efficient solution for the highly-accurate calibration of laser beam. Firstly, the target plane and 5 holes are measured by the laser tracker and the 2D coordinate system on the target plane is built. Secondly, the 5 holes and laser spot on the target plane are measured by the theodolite. Then the discrete point interpolation model is established by the coordinates of holes from laser tracker and the angles of holes from theodolite, and the coordinate of laser spot is obtained based on discrete point interpolation. Finally, a set of points on the laser beam is obtained and the direction vector of the laser beam is obtained by linear fitting. The optimal measurement distance of theodolite is determined by the simulation results. We demonstrate that the proposed high-accuracy calibration method is feasible and effective. In the future, we would make more efforts in the optimization of 3D coordinate calculation algorithm to increase the measurement distance and further enhance the practicality of our method.

## Funding

National Natural Science Foundation of China (61771336); Natural Science Foundation of Tianjin City (18JCZDJC38600).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## References

**1. **T. Y. Tao, Q. Chen, S. J. Feng, Y. Hu, J. Da, and C. Zuo, “High-precision real-time 3D shape measurement using a bi-frequency scheme and multi-view system,” Appl. Opt. **56**(13), 3646–3653 (2017). [CrossRef]

**2. **W. Yin, S. J. Feng, T. Y. Tao, L. Huang, M. Trusiak, Q. Chen, and C. Zuo, “High-speed 3D shape measurement using the optimized composite fringe patterns and stereo-assisted structured light system,” Opt. Express **27**(3), 2411–2431 (2019). [CrossRef]

**3. **J. L. Shi, Z. X. Sun, and S. Q. Bai, “Large-scale three-dimensional measurement via combining 3D scanner and laser rangefinder,” Appl. Opt. **54**(10), 2814–2823 (2015). [CrossRef]

**4. **B. Sun and B. Li, “Laser displacement sensor in the application of aero-engine blade measurement,” IEEE Sens. J. **16**(5), 1377–1384 (2016). [CrossRef]

**5. **E. Lilienblum and A. H. Ayoub, “A structured light approach for 3-D surface reconstruction with a stereo line-scan system,” IEEE Trans. Instrum. Meas. **64**(5), 1258–1266 (2015). [CrossRef]

**6. **F. Q. Zhou, B. Peng, Y. Cui, Y. X. Wang, and H. S. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. **45**, 1–12 (2013). [CrossRef]

**7. **Y. B. Zou, X. Z. Chen, G. J. Gong, and J. C. Li, “A seam tracking system based on a laser vision sensor,” Measurement **127**, 489–500 (2018). [CrossRef]

**8. **Y. B. Zou and T. Chen, “Laser vision seam tracking system based on image processing and continuous convolution operator tracker,” Opt. Lasers Eng. **105**, 141–149 (2018). [CrossRef]

**9. **Y. Zhuang, F. Yan, and H. S. Hu, “Automatic extrinsic self-calibration for fusing data from monocular vision and 3-D laser scanner,” IEEE Trans. Instrum. Meas. **63**(7), 1874–1876 (2014). [CrossRef]

**10. **H. W. Jing, C. King, and D. Walker, “Measurement of influence function using swing arm profilometer and laser tracker,” Opt. Express **18**(5), 5271–5281 (2010). [CrossRef]

**11. **D. F. Wu, T. F. Chen, and A. G. Li, “A high precision approach to calibrate a structured light vision sensor in a robot-based three-dimensional measurement system,” Sensors **16**(9), 1388 (2016). [CrossRef]

**12. **J. H. Sun, J. Zhang, Z. Liu, and G. J. Zhang, “A vision measurement model of laser displacement sensor and its calibration method,” Opt. Lasers Eng. **51**(12), 1344–1352 (2013). [CrossRef]

**13. **T. Y. Yang, Z. Wang, Z. G. Wu, X. Q. Li, L. Wang, and C. J. Liu, “Calibration of laser beam direction for inner diameter measuring device,” Sensors **17**(2), 294 (2017). [CrossRef]

**14. **C. Bi, Y. Liu, J. G. Fang, X. Guo, L. P. Lv, and P. Dong, “Calibration of laser beam direction for optical coordinate measuring system,” Measurement **73**, 191–199 (2015). [CrossRef]

**15. **B. Wu, X. D. Duan, and J. H. Kang, “A calibration method for spatial pose of a laser beam,” Meas. Sci. Technol. **30**(11), 115010 (2019). [CrossRef]

**16. **Y. J. Shen, X. Zhang, Z. Y. Wang, J. X. Wang, and L. M. Zhu, “A robust and efficient calibration method for spot laser probe on CMM,” Measurement **154**, 107523 (2020). [CrossRef]

**17. **L. Q. Ma, L. D. Wang, T. Z. Cao, J. H. Wang, X. M. He, and C. Y. Xiong, “A large-scale laser plane calibration system,” Meas. Sci. Technol. **18**(6), 1768–1772 (2007). [CrossRef]

**18. **M. W. Shao, Z. Z. Wei, M. J. Hu, and G. J. Zhang, “Calibration method for a vision guiding-based laser-tracking measurement system,” Meas. Sci. Technol. **26**(8), 085009 (2015). [CrossRef]

**19. **Z. Z. Hu, Y. C. Li, N. Li, and B. Zhao, “Extrinsic calibration of 2-D laser rangefinder and camera from single shot based on minimal solution,” IEEE Trans. Instrum. Meas. **65**(4), 915–929 (2016). [CrossRef]

**20. **Z. W. Liu, D. M. Lu, W. X. Qian, G. H. Gu, J. Zhang, and X. F. Kong, “Extrinsic calibration of a single-point laser rangefinder and single camera,” Opt. Quantum Electron. **51**(6), 186 (2019). [CrossRef]

**21. **B. Wu, F. T. Yang, W. Ding, and T. Xue, “A novel calibration method for non-orthogonal shaft laser theodolite measurement system,” Rev. Sci. Instrum. **87**(3), 035102 (2016). [CrossRef]

**22. **F. J. Miao, B. Wu, C. C. Peng, G. J. Ma, and T. Xue, “Dynamic calibration method of the laser beam for a non-orthogonal shaft laser theodolite measurement system,” Appl. Opt. **58**(33), 9020–9026 (2019). [CrossRef]

**23. **J. Shawash and D. R. Selviah, “Real-time nonlinear parameter estimation using the Levenberg-Marquardt algorithm on field programmable gatearrays,” IEEE Trans. Ind. Electron. **60**(1), 170–176 (2013). [CrossRef]