# Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems

^{*}

## Abstract

**:**

## 1. Introduction

_{x}, PE

_{y}, PE

_{z}) can be expressed as follows:

_{x}= PEF × nx

_{y}= PEF × ny

_{z}= PEF × nz

- nx, ny, nz—approach direction cosines
- PEF—value of probe head error function

## 2. Developed Simulative Model and Steps to Its Implementation

#### 2.1. Description of Simulative Model

- A—rotation angle along the horizontal axis of the probe head,
- B—rotation angle along the vertical axis of the probe head,
- α—angle in which the touch-trigger module is working,
- PE—probe error given as a result of PEF usage for considered angles.

_{s}, B

_{s}and α

_{s}denote the values of angles for which the simulation of errors has to be performed. A

_{s−1}, B

_{s−1}and α

_{s−1}are the values of angles for the nearest node with angles lower than A

_{s}, B

_{s}and α

_{s}, respectively, and A

_{s+1}, B

_{s+1}and α

_{s+1}are the values of angles for the nearest node with corresponding angles higher than A

_{s}, B

_{s}and α

_{s}. The values of PEs in nodes surrounding the point defined using A

_{s}, B

_{s}and α

_{s}have to be used. In order to simulate the PE value for the point defined using A

_{s}, B

_{s}and α

_{s}, the values of PEs in nodes surrounding this point have to be simulated using the Monte Carlo method. For all of the simulations presented here, the Monte Carlo method uses the scaled and shifted t-distributions with parameters ($\overline{x}$, s, ν), where $\overline{x}$ denotes the mean radial PE, s is the standard deviation associated with $\overline{x}$ and ν is the number of degrees of freedom. The parameters of these distributions are determined using the experiment presented in Section 2.2. Hence, the simulation should be performed for nodes (A

_{s−1}, B

_{s−1}, α

_{s−1}), (A

_{s−1}, B

_{s−1}, α

_{s+1}), (A

_{s−1}, B

_{s+1}, α

_{s−1}), (A

_{s−1}, B

_{s+1}, α

_{s+1}), (A

_{s+1}, B

_{s−1}, α

_{s−1}), (A

_{s+1}, B

_{s−1}, α

_{s+1}), (A

_{s+1}, B

_{s+1}, α

_{s−1}) and (A

_{s+1}, B

_{s+1}, α

_{s+1}). Then, a trilinear interpolation according to Formula (6) should be performed in order to obtain the PE value for the simulated point.

_{s}, B

_{s}, α

_{s}) = ((A

_{s+1}− A

_{s})/(A

_{s+1}− A

_{s−1}) × ((B

_{s+1}− B

_{s})/(B

_{s+1}− B

_{s−1}) × P2 + (B

_{s}− B

_{s−1})/(B

_{s+1}− B

_{s−1}) × P4)) + ((A

_{s}− A

_{s−1})/(A

_{s+1}− A

_{s−1}) × ((B

_{s+1}− B

_{s})/(B

_{s+1}− B

_{s−1}) × P1 + (B

_{s}− B

_{s−1})/(B

_{s+1}− B

_{s−1}) × P3))

- P1 = (((α
_{s+1}− α_{s})/(α_{s+1}− α_{s−1})) × PEF(A_{s+1}, B_{s−1}, α_{s−1})) + (((α_{s}− α_{s−1})/(α_{s+1}− α_{s−1})) × PEF(A_{s+1}, B_{s−1}, α_{s+1})) - P2 = (((α
_{s+1}− α_{s})/(α_{s+1}− α_{s−1})) × PEF(A_{s−1}, B_{s−1}, α_{s−1})) + (((α_{s}− α_{s−1})/(α_{s+1}− α_{s−1})) × PEF(A_{s−1}, B_{s−1}, α_{s+1})) - P3 = (((α
_{s+1}− α_{s})/(α_{s+1}− α_{s−1})) × PEF(A_{s+1}, B_{s+1}, α_{s−1})) + (((α_{s}− α_{s−1})/(α_{s+1}− α_{s−1})) × PEF(A_{s+1}, B_{s+1}, α_{s+1})) - P4 = (((α
_{s+1}− α_{s})/(α_{s+1}− α_{s−1})) × PEF(A_{s−1}, B_{s+1}, α_{s−1})) + (((α_{s}− α_{s−1})/(α_{s+1}− α_{s−1})) × PEF(A_{s−1}, B_{s+1}, α_{s+1}))

#### 2.2. Implementation Measurements

#### 2.3. Verification Measurements

#### 2.4. Correction of Probe Head Errors Using the Developed Model

_{x}, PE

_{y}and PE

_{z}components have to be calculated using Equations (2)–(4). Then, the values of these components should be subtracted from the actual values of point coordinates x, y, z, giving the corrected point coordinates x

_{corr}, y

_{corr}and z

_{corr}from Equations (7)–(9).

_{corr}= x − PE

_{x}

_{corr}= y − PE

_{y}

_{corr}= z − PE

_{z}

_{corr}, y

_{corr}and z

_{corr}. In the presented research, the process of correction of the probe head errors was assisted by a script written in the Python programming language cooperating with macro prepared in Modus software. The raw measurement data including point coordinates, approach vectors and stylus orientation angles was sent to the Python script, which performed the correction of probe head errors and sent back corrected point coordinates to the metrological software, in which the calculation of measured features was done once again.

## 3. Results

#### 3.1. Results of Identification of Errors

#### 3.2. Results of Model Verification

#### 3.3. Example of Probe Head Error Correction

## 4. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CMM | Coordinate Measuring Machine |

LCM | Laboratory of Coordinate Metrology |

PEF | Probe Error Function |

PE | Probe Error |

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**Figure 1.**Construction of the neural network used for probe error modeling. α: probe deflection angle; PEF: Probe Error Function.

**Figure 2.**Articulated probe head used in five-axis coordinate systems with revolution axes marked. A: rotation angle along the horizontal axis of the probe head; B: rotation angle along the vertical axis of the probe head.

**Figure 4.**Results of identification of errors for a B angle equal to 120° and A changing in the range 0–90°. All angles given in degrees and errors in mm.

**Figure 5.**Results of mean Probing Error (PE) values obtained for the chosen position during verification measurements and measurements repeated after a relatively long time of probe head functioning.

**Figure 6.**Comparison between results of experimental probe head errors identification and simulation using developed model for A = 75 and B = 80.

**Figure 7.**Comparison between results of experimental probe head errors identification and simulation using developed model for A = −100 and B = 30.

**Table 1.**Example of input data for simulative model for A = 30° and B = −120°. A, B, α given in degrees; $\overline{x}$, s in mm.

A | B | α | $\overline{x}$ | s | A | B | α | $\overline{x}$ | s |
---|---|---|---|---|---|---|---|---|---|

30 | −120 | 0.000 | −0.00072 | 0.00048 | 30 | −120 | 180.000 | −0.00017 | 0.00016 |

30 | −120 | 5.625 | 0.00007 | 0.00019 | 30 | −120 | 185.625 | −0.00034 | 0.00019 |

30 | −120 | 11.250 | 0.00007 | 0.00020 | 30 | −120 | 191.250 | −0.00024 | 0.00020 |

30 | −120 | 16.875 | 0.00015 | 0.00019 | 30 | −120 | 196.875 | −0.00034 | 0.00017 |

30 | −120 | 22.500 | 0.00022 | 0.00013 | 30 | −120 | 202.500 | −0.00010 | 0.00019 |

30 | −120 | 28.125 | 0.00029 | 0.00016 | 30 | −120 | 208.125 | 0.00005 | 0.00029 |

30 | −120 | 33.750 | 0.00015 | 0.00016 | 30 | −120 | 213.750 | 0.00016 | 0.00027 |

30 | −120 | 39.375 | 0.00010 | 0.00017 | 30 | −120 | 219.375 | 0.00032 | 0.00019 |

30 | −120 | 45.000 | 0.00020 | 0.00023 | 30 | −120 | 225.000 | 0.00018 | 0.00019 |

30 | −120 | 50.625 | 0.00019 | 0.00016 | 30 | −120 | 230.625 | 0.00010 | 0.00012 |

30 | −120 | 56.250 | 0.00029 | 0.00017 | 30 | −120 | 236.250 | 0.00012 | 0.00018 |

30 | −120 | 61.875 | 0.00009 | 0.00021 | 30 | −120 | 241.875 | 0.00039 | 0.00022 |

30 | −120 | 67.500 | 0.00010 | 0.00018 | 30 | −120 | 247.500 | 0.00035 | 0.00020 |

30 | −120 | 73.125 | 0.00012 | 0.00020 | 30 | −120 | 253.125 | 0.00026 | 0.00016 |

30 | −120 | 78.750 | 0.00012 | 0.00016 | 30 | −120 | 258.750 | 0.00036 | 0.00015 |

30 | −120 | 84.375 | 0.00016 | 0.00014 | 30 | −120 | 264.375 | 0.00027 | 0.00019 |

30 | −120 | 90.000 | 0.00014 | 0.00011 | 30 | −120 | 270.000 | 0.00009 | 0.00014 |

30 | −120 | 95.625 | 0.00004 | 0.00010 | 30 | −120 | 275.625 | 0.00012 | 0.00020 |

30 | −120 | 101.250 | −0.00001 | 0.00023 | 30 | −120 | 281.250 | −0.00003 | 0.00018 |

30 | −120 | 106.875 | 0.00000 | 0.00012 | 30 | −120 | 286.875 | 0.00003 | 0.00018 |

30 | −120 | 112.500 | −0.00003 | 0.00015 | 30 | −120 | 292.500 | −0.00001 | 0.00024 |

30 | −120 | 118.125 | −0.00007 | 0.00013 | 30 | −120 | 298.125 | −0.00004 | 0.00016 |

30 | −120 | 123.750 | 0.00019 | 0.00011 | 30 | −120 | 303.750 | −0.00037 | 0.00024 |

30 | −120 | 129.375 | 0.00004 | 0.00017 | 30 | −120 | 309.375 | −0.00020 | 0.00018 |

30 | −120 | 135.000 | −0.00004 | 0.00017 | 30 | −120 | 315.000 | −0.00016 | 0.00014 |

30 | −120 | 140.625 | −0.00030 | 0.00019 | 30 | −120 | 320.625 | −0.00006 | 0.00018 |

30 | −120 | 146.250 | −0.00021 | 0.00016 | 30 | −120 | 326.250 | −0.00007 | 0.00013 |

30 | −120 | 151.875 | −0.00023 | 0.00016 | 30 | −120 | 331.875 | −0.00010 | 0.00014 |

30 | −120 | 157.500 | −0.00019 | 0.00017 | 30 | −120 | 337.500 | −0.00032 | 0.00013 |

30 | −120 | 163.125 | −0.00014 | 0.00021 | 30 | −120 | 343.125 | −0.00033 | 0.00015 |

30 | −120 | 168.750 | −0.00009 | 0.00021 | 30 | −120 | 348.750 | −0.00027 | 0.00015 |

30 | −120 | 174.375 | −0.00011 | 0.00011 | 30 | −120 | 354.375 | −0.00021 | 0.00016 |

**Table 2.**Results of roundness deviation measurements (given in mm) for standard ring with and without probe head errors correction.

Number of Points Used | Without Correction | With Correction | Calibration Certificate |
---|---|---|---|

64 | 0.0013 | 0.0010 | 0.0004 |

16 | 0.0012 | 0.0008 | 0.0004 |

8 | 0.0010 | 0.0005 | 0.0004 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gąska, A.; Gąska, P.; Gruza, M. Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems. *Appl. Sci.* **2016**, *6*, 144.
https://doi.org/10.3390/app6050144

**AMA Style**

Gąska A, Gąska P, Gruza M. Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems. *Applied Sciences*. 2016; 6(5):144.
https://doi.org/10.3390/app6050144

**Chicago/Turabian Style**

Gąska, Adam, Piotr Gąska, and Maciej Gruza. 2016. "Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems" *Applied Sciences* 6, no. 5: 144.
https://doi.org/10.3390/app6050144