# Analysis of Laser Tracker-Based Volumetric Error Mapping Strategies for Large Machine Tools

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Virtual Calibration and Compensation

- Structure and dimensions of the virtual machine tool, in addition to the shape and magnitude of the geometric errors.
- Modeling of the laser tracker and characterization of the sources of uncertainty in the measurement process.
- Definition of the measurement strategy: Type of trajectory, number of measurement points, LT positioning, etc.
- Estimation of the compensation model: Choice of the mathematical algorithm, order of the adjustment model, type of polynomials to be used, etc.
- Validation of results: Evaluating the improvement in accuracy after applying the compensation model obtained in the parameter estimation process.

^{M}) along with the measurement uncertainties of LT (X

^{LT}). The volumetric error (whose quantification is defined in Section 2.4) will be the output quantity Y. Therefore, the analysis is not carried out on a single machine with specific geometrical errors, since the results of a single case may not be representative of the whole set. Instead, a Monte Carlo simulation will be carried out for each case study:

_{m}= f

_{m}(X

^{M}, X

^{LT}),

_{m}represents the whole calibration procedure and Ym represents the volumetric error distribution for case study m. This way, several virtual machines with different geometrical errors will be generated according to a set of parameters (see Section 2.2) and this case will be evaluated considering the distribution composed by the totality of the results of the simulated cases. To define the number of simulations required in each case, the Monte Carlo simulation will be performed iteratively according to Guide to the Expression of Uncertainty in Measurement (GUM) supplement 1 [19], in which the stabilization of the results will be evaluated. A diagram of the entire process, described above, can be seen in Figure 1.

#### 2.2. Modeling

_{M}),

C = VM(X, t, E

_{C}),

_{M}and E

_{C}contains the parameters associated with different error sources.

_{NN}(x) = a

_{0}+ a

_{1}· x + ∑

_{n}(b

_{n}· sin(nπ · x) + c

_{n}· cos(nπ · x)),

_{0}and a

_{1}are the 0 and 1 order terms of an ordinary polynomial, n is the order of the Fourier series terms, b

_{n}and c

_{n}are the sine and cosine amplitudes of n

^{th}Fourier term, and x represents the normalized position of the axis. With this equation, each of the 18 position-dependent geometric errors contained in a three-axis machine will be modeled [21].

_{C}defined in model C will represented the geometric errors that will be identified in the parameter estimation procedure. The modeling of these geometric errors will be carried out by the combination of Legendre polynomials of grade 6, where each error be modeled as

_{NN}(x) = a

_{0}· L

_{0}(x) + a

_{1}· L

_{1}(x) + … + a

_{6}· L

_{6}(x),

_{n}are the parameter to be estimated and L

_{n}the Legendre polynomials of degree n.

^{®}[24], in which it is considered that the systematic errors have been corrected:

_{XYZ0}· R

_{AZ}· R

_{EL}· T

_{L},

_{L}is the transformation matrix representing the laser interferometer measurement, and R

_{EL}and R

_{AZ}are the rotation matrices for azimuth and elevation articulations of the laser tracker. RT

_{XYZ0}represents the translation and rotation between the machine coordinate system and the LT, which is obtained a posteriori, fitting the coordinates measured by the LT with those recorded by the virtual machine.

#### 2.3. Parameter Estimation

_{C}of the Legendre polynomials that represent the geometric errors, in order to minimize the error between the measured deviations and those predicted by the model, adjusting the mentioned polynomials to the error forms of the original system. That is, an estimation procedure will be carried out so that the error between the real model M and the compensation model C in measured points X is minimized:

_{M}) − C(X, t, E

_{C}),

_{C},

_{C}= (A’ · W

^{−1}· A)

^{−1}· A’ · W

^{−1}· e,

^{2}

_{x}, σ

^{2}

_{y}, σ

^{2}

_{z}). The calculation of this variance is made by propagating uncertainties of the LT model described in Section 2.2.

#### 2.4. Validation of the Compensation Model

_{0}= ||M(X, t, E

_{M}) − M(X, t, 0)||,

_{v}, t, E

_{M}) − C(X

_{v}, t, E

_{C})||,

_{0}represents the initial volumetric error compared with an ideal machine behavior and VE the volumetric error after compensation is applied in all X

_{v}validation points. Equations (9) and (10) provide the volumetric error distribution for X

_{v}point set. In this paper, the RMS value of this distribution will be taken as a numeric value that defines the quality of the compensation procedure. Regarding the Monte Carlo simulation presented in Equation (1), the RMS value obtained in each simulation represents the y

_{r}draw of output variable Y that will compound the distribution of results.

## 3. Results and Discussion

- Comparison between different types of measurement strategies.
- Variation of the spacing between measurement points.
- Similar measurements placing the LT in different positions.

#### 3.1. Measurement Trajectory Strategies

#### 3.2. Spacing between Measurement Points

#### 3.3. Positioning of Laser Tracker

- Keep the LT in the same position, leaving the points out of range without measuring, hoping that a closer proximity to the other points will result in better characterization.
- Move the LT slightly away in the opposite direction, measuring all the required points, but losing some precision due to the greater distance to the points.

## 4. Discussion

- Different trajectories in tests of similar duration affect the quality of the compensation. The orthogonal planes offer slightly better results than the hexahedron, which is the usual strategy in these types of calibrations.
- Calibration quality shows an asymptotic behavior when tests with an increasing number of points are performed. When the number of measured points is high enough, stabilization of the improvement is reached.
- The positioning points of the LT studied in this paper show minor influence in the quality of the calibration process. Centered position shows slightly better results, but an active reflector is required to measure all points.

## 5. Conclusions

- The feasibility and interest of the proposed approach have been demonstrated by performing an analysis of the calibration process on a large machine tool, which has provided meaningful results.
- The calibration results obtained in the analyses indicate that laser trackers are a valid solution for the calibration of large machine tools but, also, that the measurement strategy can have a relevant influence on the calibration uncertainty and on the measurement time.
- These resulting optimization criteria cannot be considered of general application, since the optimal solution for each machine type and customer requirements will lead to different solutions. What can be taken instead, as a general solution, is the methodology and simulation tools presented here for finding the optimal solution.
- The simulation of the machine to be calibrated is critical for obtaining representative results, and requires an estimation of the expected machine errors. The proposed error modeling strategy and Monte Carlo simulation ensure that the required engineering judgement is properly considered in the optimization process.
- The uncertainty model of the laser tracker, as considered in the product catalogue, has not been considered appropriate for the requirements of some of the analyses, such as the influence of the position of the laser tracker, and a model based on results of ASME B89.4.19 tests has been successfully implemented.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Process diagram for the whole calibration and compensation procedure comparing several case studies.

**Figure 3.**Model parameter fitting of the laser tracker (LT) to the manufacturer data for (

**a**) horizontal scale bar test for angular measurement performance and (

**b**) distance measurement performance.

**Figure 4.**Measuring trajectories to be compared: (

**a**) Orthogonal planes; (

**b**) Hexahedron; (

**c**) Main diagonals.

**Figure 5.**Comparison of measured trajectories. Error distributions and mean values are shown, for values before (blue) and after (grey) compensation.

**Figure 6.**Simulation results with increasing measured points. Total number of points is shown, as well as the initial (blue) and compensated (grey) error.

**Figure 7.**Measurement process schematic view and compensation results for single position and two position measurements.

**Table 1.**Parameters used to generate the 18 geometric errors contained in a three-axis machine tool according to ISO 230.

Error Components | Distribution Parameters | |||||||
---|---|---|---|---|---|---|---|---|

a0 | a1 | n1 | b1 | c1 | n2 | b2 | c2 | |

E_{XX} | 0 | 70 | 0.5 | 20 | 20 | 5 | 10 | 10 |

E_{YX} | 0 | 0 | 0.5 | 30 | 0 | 5 | 3 | 3 |

E_{ZX} | 0 | 0 | 0.5 | 15 | 0 | 5 | 3 | 3 |

E_{AX} | 10 | 20 | 0.5 | 10 | 10 | 5 | 5 | 5 |

E_{BX} | 10 | 20 | 0.5 | 10 | 10 | 5 | 5 | 5 |

E_{CX} | 10 | 20 | 0.5 | 10 | 10 | 5 | 5 | 5 |

E_{XY} | 0 | 0 | 0.5 | 15 | 0 | 2 | 5 | 5 |

E_{YY} | 0 | 30 | 0.5 | 5 | 5 | 2 | 3 | 3 |

E_{ZY} | 0 | 0 | 0.5 | 15 | 0 | 2 | 5 | 5 |

E_{AY} | 10 | 30 | 0.5 | 10 | 10 | 2 | 5 | 5 |

E_{BY} | 10 | 30 | 0.5 | 10 | 10 | 2 | 5 | 5 |

E_{CY} | 10 | 30 | 0.5 | 10 | 10 | 2 | 5 | 5 |

E_{XZ} | 0 | 0 | 0.5 | 15 | 0 | 1 | 5 | 5 |

E_{YZ} | 0 | 0 | 0.5 | 15 | 0 | 1 | 5 | 5 |

E_{ZZ} | 0 | 30 | 0.5 | 30 | 30 | 1 | 3 | 3 |

E_{AZ} | 10 | 30 | 0.5 | 10 | 10 | 1 | 3 | 3 |

E_{BZ} | 10 | 30 | 0.5 | 10 | 10 | 1 | 3 | 3 |

E_{CZ} | 10 | 30 | 0.5 | 10 | 10 | 1 | 3 | 3 |

Machine Tool | X (mm) | Y (mm) | Z (mm) |
---|---|---|---|

Working volume | 0–5000 | 0–3000 | 0–1500 |

Kinematic chain | t–Z–Y–[X1 X2]–b–w |

Laser Tracker | Range | Uncertainty (k = 1) | |
---|---|---|---|

Laser Interferometer | 80 m | - | ±0.25 µm/m |

Azimuth | ±320° | ±10 µm | ±2.5 µm/m |

Elevation | (−59°, +79°) | ±10 µm | ±2.5 µm/m |

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## Share and Cite

**MDPI and ACS Style**

Iñigo, B.; Ibabe, A.; Aguirre, G.; Urreta, H.; López de Lacalle, L.N.
Analysis of Laser Tracker-Based Volumetric Error Mapping Strategies for Large Machine Tools. *Metals* **2019**, *9*, 757.
https://doi.org/10.3390/met9070757

**AMA Style**

Iñigo B, Ibabe A, Aguirre G, Urreta H, López de Lacalle LN.
Analysis of Laser Tracker-Based Volumetric Error Mapping Strategies for Large Machine Tools. *Metals*. 2019; 9(7):757.
https://doi.org/10.3390/met9070757

**Chicago/Turabian Style**

Iñigo, Beñat, Ander Ibabe, Gorka Aguirre, Harkaitz Urreta, and Luis Norberto López de Lacalle.
2019. "Analysis of Laser Tracker-Based Volumetric Error Mapping Strategies for Large Machine Tools" *Metals* 9, no. 7: 757.
https://doi.org/10.3390/met9070757