Sage Journals: Discover world-class research

Abstract

The multilateration measurement with laser tracker has become a popular method for measuring the volumetric error of machine tools, owing to its advantages of high efficiency and large measurement range. However, the main problem for the multilateration measurement is that its identification accuracy is unsteady. As the identification accuracy is greatly influenced by the laser tracker positioning, in this article, we aim to optimize the positions of the laser trackers and conduct the optimization based on the genetic algorithm to solve this problem. In order to ensure the feasibility of the optimized measurement system, the position of the laser tracker is constrained by the range of viewing angle of the reflector and the practical measurement environment. Compared to the measurement systems with different laser tracker positioning, the optimized measurement system achieves the best identification accuracy both in the simulation and in the experiment.

Keywords

Multilateration measurement laser tracker genetic algorithm uncertainty analysis volumetric error

Introduction

The development of advanced manufacturing industry has brought forward higher requirements to the machine tools. The accuracy is one of the most important criteria for machine tools and the error is usually considered as the evaluating indicator of the accuracy. The error sources of machine tools could be further classified into two categories, namely, quasi static error and dynamic error. Quasi static error accounts for about 60%–70% of the total error.¹ Volumetric error is the most common type of quasi static errors. Therefore, proposing an accurate and efficient method to detect the volumetric error is of great importance to improve the accuracy of a machine tool.

Many devices have been developed to measure the volumetric error of the machine tools. Laser interferometer (LI) has been widely used among the machine tool manufacturers and research institutions, but the adjustment of the laser path is time-consuming.^2–4 Double ball bar (DBB)^5,6 and cross grid encoder (CGE)⁷ are low-cost devices. However, the multi-axis movement of CNC is necessary when using these two devices to measure the volumetric errors, and the measurement range of them is quite limited. Laser tracker (LT) is a kind of portable measurement device. It could automatically follow the movement of the target and provide the three-dimensional coordinates. Compared with the other measurement devices, the LT can keep the merits of convenience and have a large measurement range as well. In recent decades, a growing number of researchers and companies have conducted in-depth researches on volumetric error measurement with LT.

The measurement with LT includes single-station measurement and multi-station measurement.⁸ The single-station measurement⁹ adopts the spherical coordinate system and measures the radial distance, azimuth angle, and elevation angle to calculate the three-dimensional coordinates of the measuring points. However, the measurement accuracy of angle is limited and decreases with increasing distance. As a result, the single-station measurement is not an appropriate method for the measurement aiming at high-precision requirement.

The multi-station measurement, also known as the multilateration measurement, was proposed by Brown et al.¹⁰ in 1986. At least four LTs are used to measure the distance between the measuring point and the LTs. The multilateration measurement is available to calculate the coordinates of the measuring points without the influence of angle-measurement uncertainty. The uncertainty assessment, such as the Monte Carlo method,¹¹ is one of the most important ways to evaluate the identification accuracy. Based on the uncertainty assessment, many researches have been conducted to look for the factors affecting the identification accuracy, including the identification algorithm,^12,13 the number of the measuring points,⁸ and the arrangement of the LTs.

The influence of LT positioning on multilateration measurement is significant and complicated. Takatsuji et al.^14,15 put forward that four LTs should not be located on the same plane. Otherwise, the equation set of the self-calibration in multilateration will have no solutions. Ibaraki et al.¹⁶ compared different sets of LT positions with respect to the multilateration uncertainty. Considering the optimization of the LT positions, Lin et al.¹⁷ calculated the optimal positions with Lagrange Multiplier method. But there was only one measuring point in his research. As the number of the measuring points grows, the difficulty in solving the optimal solutions by Lagrange Multiplier method increases significantly. In practical measurement of volumetric error, the coordinates of multi-points need to be identified in one measurement. Although a lot of researches have concerned on the LT positioning of multilateration measurement, few can provide an effective way to optimize LT positions, especially in multi-point measurement. Therefore, the LT positions in multilateration measurement are usually selected intuitively, but not scientifically.

In this article, the genetic algorithm (GA) is used to optimize the LT positions of multilateration measurement. The optimization is realized based on uncertainty assessment. Several constraint conditions of the LT positions have been considered to ensure the feasibility of the optimized measurement system. Finally, the comparisons of the performance of different measurement systems have been presented from both the simulation and the experiment.

Basic theory

The multilateration method

In general, there are two types of distance measurement in LT, namely, absolute distance measurement (ADM) and interferometer measurement (IFM). ADM measures the distance $l_{Bi}$ between the LT and the first measuring point. IFM measures the change of distance $Δ l_{ij}$ between the LT and the measuring points and holds higher accuracy compared with ADM. Therefore, the identification model of multilateration method is established with $Δ l_{ij}$ measured by IFM. The multilateration method derives from the method of GPS, as shown in Figure 1. The basic equation of the model could be expressed as follows

\sqrt{{(x_{Bi} - x_{j})}^{2} + {(y_{Bi} - y_{j})}^{2} + {(z_{Bi} - z_{j})}^{2}} = l_{Bi} + Δ l_{ij}

(1)

where $P_{Bi} (x_{Bi}, y_{Bi}, z_{Bi})$ and $P_{j} (x_{j}, y_{j}, z_{j})$ stand for the coordinates of LT and measuring point, respectively. $l_{Bi}$ is the distance between the ith LT and the first measuring point, and it is measured by ADM. $Δ l_{ij}$ is the change of distance measured by IFM. For the sake of economy and time-saving, the number of LTs used in multilateration is usually 4.

Figure 1.

Measurement system: (a) principle of GPS and (b) multilateration measurement.

Equation (1) could constitute an overdetermined set which is usually solved by least square method (LSM).⁸ In this article, Levenberg–Marquardt (LM) method, a numerical method of LSM, is used to solve the equation set. The objective function F is defined as the residual

F = \sum {f_{i, j}}^{2}

(2)

f_{i, j} = \sqrt{{(x_{Bi} - x_{j})}^{2} + {(y_{Bi} - y_{j})}^{2} + {(z_{Bi} - z_{j})}^{2}} - l_{Bi} - Δ l_{ij}

(3)

The calculation of the measuring system involves two procedures: the self-calibration for system parameters and the calculation for the coordinates of the measuring points. The former one uses the ideal positions of the measuring points to calculate the system parameters $P_{Bi} (x_{Bi}, y_{Bi}, z_{Bi})$ and $l_{Bi}$ . The latter one uses the changes of distance $Δ l_{ij}$ measured by IFM to calculate the accurate coordinates. Both procedures could be conducted by the following steps of LM method:

Step 1. Give the initial value and parameters of iteration;

Step 2. Calculate the Jacobian matrix J of function F and $ρ = α / β$ ;

Step 3. Solve the functions

x_{k + 1} = x_{k} + d_{k}

(4)

d_{k} = - {(J {(x_{k})}^{T} J (x_{k}) + ρ I)}^{- 1} J (x_{k})^{T} F (x_{k})

(5)

Step 4. If $F (x_{k + 1}) < F (x_{k})$ , $k = k + 1$ ; otherwise, $ρ = α β$ and back to Step 3.

Step 5. As long as $J (x_{k})^{T} F (x_{k}) \leq δ$ , the iteration should be stopped; otherwise, back to Step 2.

Optimization of the LT positions

The accuracy of LSM is sensitive to the calculation parameters, especially to the LT positions. The GA is used to optimize the LT positions. GA is an intelligent optimization algorithm built on the natural genetics. The optimization procedure is based on the uncertainty analysis of the multilateration measurement. The optimization object F, defined as the fitness function in GA, is the root mean square error (RMSE) of the predicted coordinates $P_{jk} (x_{jk}, y_{jk}, z_{jk})$

F = \sqrt{\frac{\sum_{k = 1}^{m} \sum_{j = 1}^{n} (Δ {x_{jk}}^{2} + Δ {y_{jk}}^{2} + Δ {z_{jk}}^{2})}{3 nm - 1}}

(6)

where n is the number of the measuring points, and m is the number of simulations.

The basic evolution steps of GA include selection, crossover, and mutation. In the steps of crossover and mutation, the offspring individuals, namely, the new LT positions, will be generated. The constraint conditions of LT position follow after the operators of crossover and mutation (Figure 2). The constraint conditions, presented in section “Constraint condition of the LT position,” are used to evaluate the feasibility of measurement at the new LT position.

Figure 2.

The flow chart of the proposed method.

The multilateration measurement uses the changes of distance measured by IFM as the inputs of the identification model. The IFM uncertainty $u_{IFM}$ is mainly derived from the mechanical error and environmental variation such as humidity, air pressure, and temperature. According to the experiments of K Umetsu et al.,¹⁸ the mechanical error is estimated to be 0.3 µm and the uncertainty of laser wavelength caused by environmental variation is $0.4 \times 10^{- 3} L$ µm. In consequence, the IFM uncertainty $u_{IFM}$ is estimated as equation (7)

u_{IFM} = \sqrt{{0.3}^{2} + {(0.4 \times 10^{- 3} L)}^{2}}

(7)

The uncertainties of ADM and angle measurement affect the initial values of LM. Taken Leica AT901 as an example, the distance uncertainty measured by ADM is $\pm 10 μ m$ , and the angle uncertainty is 0.5.

Constraint condition of the LT position

The viewing angle of the reflector

During the process of measurement, a reflector, also called cateye, is fixed on the measuring point. The reflector could receive and reflect the laser from different angles within a certain range. The viewing angle $α_{critical}$ of the reflector is shown in Figure 3. If the incident laser beam moves out of the viewing angle $α_{critical}$ , the measurement will be failed. Even if the reflector moves back and gets the laser beam again in a short time, the LT will measure the next point by ADM. On this occasion, the measuring points need to be divided into two groups and the coordinates of them will be calculated separately by the LM method. However, the decrease in the number of the measuring points reduces the identification accuracy.⁸ For higher identification accuracy, the incident laser beam must be limited within the range of the viewing angle during the overall process of measurement.

Figure 3.

Structure of the reflector.

When measuring the volumetric errors, the reflector moves within the workspace of the machine tool. The improper LT position will probably cause the incident laser beam exceeding the range of the viewing angle and result in failure of the measurement. The workspace of the machine tool is usually in the shape of cuboid as shown in Figure 4. When the reflector moves from P_j to P_i, the maximum intersection angle between the incident laser beams is the angle α between $\vec{P_{j} B}$ and $\vec{P_{i} B}$ . When the reflector moves randomly in the workspace of the machine tool, the maximum intersection angle is the intersection angle between the vectors from vertices of the cuboid to the LT. The LT position should satisfy equations (8) and (9)

α_{\max} = \max (\arccos \frac{\vec{P_{i} B} \vec{P_{j} B}}{| \vec{P_{i} B} | | \vec{P_{j} B} |})

(8)

α_{\max} < α_{critical}

(9)

where P_i and P_j stand for the vertices of the cuboid, $α_{max}$ is the maximum intersection angle. $α_{critical}$ is the viewing angle of the reflector.

Figure 4.

Measurement of the volumetric error.

The practical measurement environment

Equations (8) and (9) could be considered as the ideal criterion of the feasible LT position. Besides, the LT positions will be further constrained by the measurement environment. For example, the laser beam should not be blocked by the toolframe of the machine tool and the LT must be firmly set.

The interchangeability of the LT positions

The coordinates $P_{Bi} (x_{Bi}, y_{Bi}, z_{Bi})$ of four LTs constitute the optimization variable in GA. Although variable $V_{1} [P_{B 1}, P_{B 2}, P_{B 3}, P_{B 4}]$ and variable $V_{2} [P_{B 2}, P_{B 1}, P_{B 3}, P_{B 4}]$ present the same arrangement of LTs, these two variables will generate different offspring individuals in the steps of crossover and mutation, as shown in Figure 5. This will results in the different iteration direction of optimization. In order to prevent the chaos, an extra constraint condition should be added that each LT position is constrained in a certain region as equation (10).

x_{B 1} < x_{B 2} < x_{B 3} < x_{B 4}

(10)

Figure 5.

The interchangeability of the LT positions.

Simulation

When measuring the volumetric errors, the measurement region does not usually cover all the workspace of the machine tool. The researchers try to identify the components of the volumetric errors. When moving along one coordinate axis, there are six component errors: three displacement errors $Δ x (x)$ , $Δ y (x)$ , and $Δ z (x)$ and three angular errors $Δ α (x)$ , $Δ β (x)$ , and $Δ γ (x)$ which are around the X-axis, Y-axis, and Z-axis, respectively. The identification method with nine measuring lines is adopted in our research.¹⁹ Taken the X-axis as an example, the volumetric errors could be expressed as in equation (11)

{\begin{matrix} Δ X (x) = Δ x (x) - Δ γ (x) y_{1} + Δ β (x) z_{1} \\ Δ Y (x) = Δ y (x) - Δ α (x) z_{1} + Δ γ (x) x_{1} \\ Δ Z (x) = Δ z (x) - Δ β (x) x_{1} + Δ α (x) y_{1} \end{matrix}

(11)

where $[Δ X (x), Δ Y (x), Δ Z (x)]$ is the volumetric error of the measuring point $P (x, y, z)$ . Using three points with the same traveling distance in X-axis, equation (11) could be represented in matrix form as equation (12) and solved by LSM

[\begin{matrix} Δ x_{1} (x) \\ Δ y_{1} (x) \\ Δ z_{1} (x) \\ Δ x_{2} (x) \\ Δ y_{2} (x) \\ Δ z_{2} (x) \\ Δ x_{3} (x) \\ Δ y_{3} (x) \\ Δ z_{3} (x) \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 & z_{1} & - y_{1} \\ 0 & 1 & 0 & - z_{1} & 0 & x_{1} \\ 0 & 0 & 1 & y_{1} & - x_{1} & 0 \\ 1 & 0 & 0 & 0 & z_{2} & - y_{2} \\ 0 & 1 & 0 & - z_{2} & 0 & x_{2} \\ 0 & 0 & 1 & y_{2} & - x_{2} & 0 \\ 1 & 0 & 0 & 0 & z_{3} & - y_{3} \\ 0 & 1 & 0 & - z_{3} & 0 & x_{3} \\ 0 & 0 & 1 & y_{3} & - x_{3} & 0 \end{matrix}] [\begin{matrix} Δ x (x) \\ Δ y (x) \\ Δ z (x) \\ Δ α (x) \\ Δ β (x) \\ Δ γ (x) \end{matrix}]

(12)

The simulation and measurement experiment are conducted on a heavy boring and milling machine tool as shown in Figure 6. The workspace of the machine tool is $X \times Y \times Z$ $8000 \times 4500 \times 1500 mm$ . The measuring points are distributed on the edge lines of the workspace. There is one measuring point every 1000, 500, and 100 mm along the X, Y, and Z axis, respectively (Figure 7).

Figure 6.

The heavy boring and milling machine tool: (a) overall view and (b) configuration.

Figure 7.

The optimized measurement system.

During the measurement experiment, the incident laser beam should not be blocked and the LT must be firmly set. The effective area of the LT position is shown in Figure 6(a). The adjustable height of LT is 1.5 m (from 0 to 1500 mm in Y-axis). The viewing angle of the LT reflector is $\pm 45 \circ$ .

The simulation parameters are given in Table 1. Four optimized LT positions are $P_{1} (2914.7, 421.3, - 6574.4)$ , $P_{2} (3772.6, 1354.8, - 9155.4)$ , $P_{3} (4992.6, 157.3, - 9063.5)$ , and $P_{4} (6993.1, 1287.5, - 6491.2)$ . Figure 8(a) presents the predicted Dev (equation (13)) of each measuring point, and the root mean square value of Dev is 0.0045 mm. The square dots in Figure 8(b) represent the predicted residuals $Δ x$ . Compared with the actual volumetric errors in X-axis, the predicted residuals have been dramatically reduced from $[- 0.010, 0.010]$ to $[- 0.002, 0.004] mm$

Dev = \sqrt{Δ x^{2} + Δ y^{2} + Δ z^{2}}

(13)

Δ x = x_{act} - x_{pre}

(14)

where $x_{act}$ is the actual volumetric error in X-axis, and $x_{pre}$ is the predicted volumetric error in X-axis.

Table 1.

Parameters of optimization.

Parameter	Value	Parameter	Value
Number of iterations for GA	30	Variation probability	0.01
Population size	40	Number of simulations	4
Crossing probability	0.7	Range of the volumetric error	[−0.010, 0.010] mm

GA: genetic algorithm.

Figure 8.

The optimized arrangement: (a) predicted Dev (RMS = 0.0045 mm) and (b) predicted residual $Δ x$ in X-axis.

In order to present the improvement by applying the GA, one of the four optimized LT positions is replaced for comparison:

Case 1. Replace $P_{1} (2914.7, 421.3, - 6574.4)$ with $P'_{1} (0, 421.3, - 6574.4)$ as shown in Figure 9(a).

Case 2. Replace $P_{2} (3772.6, 1354.8, - 9155.4)$ with $P'_{2} (3772.6, 1354.8, - 6000)$ as shown in Figure 9(b).

Figure 9.

The identification accuracy of two comparisons: (a) Case 1 (RMS = 0.0052 mm) and (b) Case 2 (RMS = 0.0064 mm).

Among four LT positions, $P_{1} (2914.7, 421.3, - 6574.4)$ has the minimal traveling distance in X-axis. In order to cover larger X-axis stroke, P₁ is replaced with the new $P_{1}^{'} (0, 421.3, - 6574.4)$ in Case 1. The simulation results show that the root mean square value of Dev increases slightly from 0.0045 to 0.0052 mm. It is possible that the longest distance between the measuring points and P₁ becomes larger after replacement, which increases the uncertainty in distance measurement.

In Case 2, we try to improve the accuracy of distance measurement. $P_{2} (3772.6, 1354.8, - 9155.4)$ is the furthest LT position in Z-axis and replaced with $P'_{2} (3772.6, 1354.8, - 6000)$ . However, the root mean square value of Dev in Case 2 increases dramatically. The restriction on the arrangement of LTs to execute a self-calibration algorithm is that four LTs should not be located on the same plane.¹⁴ The replacement of P₂ shortens the perpendicular distance between the point P₂ and the plane $P_{1} P_{3} P_{4}$ . This may also decrease the solving accuracy of self-calibration algorithm.

The LT positions have a great impact on the identification accuracy of multilateration measurement. The coupling relations among the influencing factors make it difficult to find the optimal LT positions by the analytical method. The simulation results validate that the proposed GA method could optimize the LT positions effectively.

Experiment

The multilateration measurement experiment was performed with Leica AT901 Laser Tracker. A LT was settled at four optimized positions and two alternate positions in sequence. In order to lower down the measurement uncertainty, the same motion trajectory of the machine tool was measured for four times at each LT position. In addition, the experiment was conducted in the midnight to avoid the noise and vibration from environment. The positional accuracy of LT was within 10 mm.

To further verify the feasibility and accuracy, a contrast experiment using the LI was conducted. The volumetric errors of the points on three lines along the X, Y, and Z axis were measured by LI.

The measurement results are shown in Figure 10 and Table 2. Two indexes, average deviation (equation (15)) and average deviation ratio (equation (16)), are used to evaluate the performance of the identification accuracy. The lines with circles in Figure 10 demonstrate the results measured by LI. Due to the high accuracy and stability of LI, the results of LI are considered as the actual volumetric errors of the machine tool for comparison

AD = \frac{\sum_{i = 1}^{n} | E_{LT} (i) - E_{LI} (i) |}{n}

(15)

A D R = \frac{\sum_{i = 1}^{n} | 1 - E_{L T} (i) / E_{L I} (i) |}{n} \times 100 %

(16)

where $E_{LT} (i)$ and $E_{LI} (i)$ are the volumetric error measured by LT and LI, respectively.

Figure 10.

The experiment results: (a) measurement with laser interferometer, (b) volumetric error in X-axis, (c) volumetric error in Y-axis, and (d) volumetric error in Z-axis.

Table 2.

The identification accuracy of different arrangements of LTs.

Arrangement of LTs	Average deviation (mm)	Average deviation ratio (%)
The optimized arrangement	0.0188	35.89
Case 1 (replace P₁)	0.0236	36.93
Case 2 (replace P₂)	0.0352	44.38

LT: laser tracker.

The optimized arrangement performs better than the other two cases in both average deviation and average deviation ratio. As presented in Table 2, the average deviation ratio of Case 1 increase slightly from 35.89% to 36.93% and that of Case 2 reaches 44.38%. The variation trend of identification accuracy in experiment is the same as that in simulation.

Conclusion

In this article, the GA is developed to optimize the LT positions of multilateration measurement. The proposed method contributes to the literature in the following ways: first, the proposed method is a feasible way to optimize the LT positions, especially in multi-point measurement. Second, based on the viewing angle of the reflector, the constraint condition of the LT position is proposed. This could prevent the incident laser beam moving out of the range of the viewing angle during the measurement. Third, the proposed method could provide the optimal LT positions according to the practical measurement environment.

By comparison with the measurement systems with different arrangements of LTs, the optimized one achieves the best identification accuracy both in the simulation and in the experiment. Therefore, effectiveness of the proposed model is validated in this research. The proposed optimization method helps to select the LT positions scientifically.

Footnotes

Acknowledgements

Authors Haitong Wang and Yonglin Cai are also affiliated with Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology, Ministry of Education, China.

Academic Editor: Yangmin Li

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work was supported by the National Natural Science Foundation of China (NSFC) under NSFC Grant 51375040 and the Open Foundation of the Key Laboratory of Education Ministry in China.

References

Barakat

Elbestawi

Spence

AD.

Kinematic and geometric error compensation of a coordinate measuring machine. Int J Mach Tool Manu 2000; 40: 833–850.

Okafor

Ertekin

YM.

Vertical machining center accuracy characterization using laser interferometer—part 1: linear positional errors. J Mater Process Tech 2000; 105: 394–406.

Okafor

Ertekin

YM.

Vertical machining center accuracy characterization using laser interferometer—part 2: angular errors. J Mater Process Tech 2000; 105: 407–420.

Castro

Burdekin

Dynamic calibration of the positioning accuracy of machine tools and coordinate measuring machines using a laser interferometer. Int J Mach Tool Manu 2003; 43: 947–954.

Bryan

JB.

A simple method for testing measuring machines and machine tools—part 1: principles and applications. Precis Eng 1982; 4: 61–69.

Bryan

JB.

A simple method for testing measuring machines and machine tools—part 2: construction details. Precis Eng 1982; 4: 125–138.

Knapp

Weikert

Testing the contouring performance in 6 degrees of freedom. CIRP Ann: Manuf Techn 1999; 48: 433–436.

Wang

Guo

Zhang

. The technical method of geometric error measurement for multi-axis NC machine tool by laser tracker. Meas Sci Technol 2012; 23: 45003–45013.

Aguado

Samper

Santolaria

. Identification strategy of error parameter in volumetric error compensation of machine tool based on laser tracker measurements. Int J Mach Tool Manu 2012; 53: 160–169.

10.

Brown

Merry

Wells

DN.

Coordinate measurement with a tracking laser interferometer. Laser Appl 1986; October: 69–71.

11.

Schwenke

Franke

Hannaford

. Error mapping of CMMs and machine tools by a single tracking interferometer. CIRP Ann: Manuf Techn 2005; 54: 475–478.

12.

Wang

Guo

Algorithm for detecting volumetric geometric accuracy of NC machine tool by laser tracker. Chinese Journal of Mechanical Engineering 2013; 26: 166–175.

13.

Aguado

Santolaria

Samper

. Influence of measurement noise and laser arrangement on measurement uncertainty of laser tracker multilateration in machine tool volumetric verification. Precis Eng 2013; 37: 929–943.

14.

Takatsuji

Koseki

Goto

. Restriction on the arrangement of laser trackers in laser trilateration. Meas Sci Technol 1998; 9: 1357–1359.

15.

Takatsuji

Goto

Kirita

. The relationship between the measurement error and the arrangement of laser trackers in laser trilateration. Meas Sci Technol 2000; 11: 477–483.

16.

Ibaraki

Kudo

Yano

. Estimation of three-dimensional volumetric errors of machining centers by a tracking interferometer. Precis Eng 2015; 39: 179–186.

17.

Lin

Y-B

Zhang

G-X

. Optical arrangement of four-beam laser tracking system for 3D coordinate measurement. Chinese J Laser B 2002; 11: 1000–1005.

18.

Umetsu

Furutnani

Osawa

. Geometric calibration of a coordinate measuring machine using a laser tracking system. Meas Sci Technol 2005; 16: 2466–2472.

19.

Zhang

A general strategy for geometric error identification of multi-axis machine tools based on point measurement. Int J Adv Manuf Tech 2013; 69: 1483–1497.

Application of genetic algorithm to multilateration measurement of the volumetric error in machine tools

Abstract

Keywords

Introduction

Basic theory

The multilateration method

Optimization of the LT positions

Constraint condition of the LT position

The viewing angle of the reflector

The practical measurement environment

The interchangeability of the LT positions

Simulation

Experiment

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References