# Simulative investigation of point diffraction interferometry with regard to machine tool calibration

## Abstract

One major effect that can deteriorate the geometric accuracy of a calibrated machine tool is the elastic deformation of its structure due to heat transferred into it. Different solutions to this problem have been presented in the past—e.g. temperature-controlled structural parts and environments—but they are complex and expensive. In an on-going Collaborative Research Centre funded by the German Research Foundation scientists are working on model-based compensation and correction strategies. In this paper, a different approach is discussed using point diffraction interferometry to directly measure the three-dimensional distance between tool centre point and workpiece coordinate system. Commercially available three-dimensional measurement systems are introduced representing the state of the art before explaining the basics of point diffraction interferometry. Different parameters are investigated using simulation tools and the achievable accuracy of the approach is evaluated. Additionally, some suggestions are made concerning hardware requirements of a planned demonstrator.

### Keywords

Optical measurement Interferometry Distance measurement Calibration Thermal deviation## 1 Introduction

In many fields of industrial manufacturing tight tolerances are required in functional areas of the produced parts. If these tolerances are not met, cost and labour extensive rework will be necessary or the parts will have to be declared as scrap. Compliance with the requested tolerances leads to high demands regarding the manufacturing processes and the accuracy of the employed machine tools [1]. The main sources for a reduced accuracy can be classified into kinematic errors, thermo-mechanical errors, loads, dynamic forces, motion control and control software [2].

Existing solutions which try to establish a thermal equilibrium increase the energy consumption and impact both the profitability and the environment. The Collaborative Research Centre SFB/TRR 96, which was started in July 2011, will therefore focus on the thermo-energetic design of machine tools. One approach is to correct thermo-elastic deformations using small movements of the machine axes based on complex mathematical models. The other approach is to compensate for those deformations by different means (e.g. controlled temperature distribution, drive optimisation) [6].

## 2 Three-dimensional distance measurement techniques

A different approach to correct thermal deviation is based on the direct distance measurement between TCP and workpiece coordinate system and comparison of this distance to the value of the machine tool’s numerical control (NC). A difference between the two values is either caused by a remaining geometric error or by thermo-elastic deformation and can be corrected by small movements of the machine axes. In the following chapter, some commercially available three-dimensional measurement systems will be introduced representing the current state of the art.

### 2.1 iGPS (triangulation)

The transmitters emit two continuously rotating laser fans which are tilted to the vertical and to each other. The relative position of a receiver is defined as the intersection of the lines of sight towards at least two transmitters (azimuth and elevation angles) which can be calculated from simple time measurements if the geometric setup and the rotational speeds of the transmitters are known [10].

Realistically achievable accuracies fall into the range of 100–500 μm for a measurement field of about 3 × 3 m² [10, 13]. Possible error sources are multipath propagation (reflexions) of the laser fans, background noise, suboptimal sensor positioning and orientation as well as kinematic effects [11, 12].

The use of this system as a recalibration device for shop floor machining centres is questionable, since the thermally induced errors are typically only as big as or even smaller than the accuracy of the measurement device [4, 5]. In addition, the minimal distance between sensor and transmitters, required for proper operation [12], hinders the deployment in entirely enclosed machines. Furthermore, the geometric setup of the transmitters has to be known precisely and may not change between different measurements. The detection of angular displacements is only possible using a set of at least three sensors with known distances to each other.

### 2.2 μ-GPS (trilateration)

Simulative investigations and one-dimensional experiments have been carried out successfully to estimate the performance of the system under real working conditions [17]. Using improved hardware the standard deviation of three-dimensional measurements was determined to be 7.5 μm. Estimates for the overall system costs were in the range of 33.000–37.000 € [16].

Due to the high system costs, a widespread deployment as a recalibration device permanently built into a machine tool is not very likely, although the performance is sufficient with regard to repeatability of the measurements. Comparable to the iGPS device mentioned before, the geometric configuration of the reflectors must be known in advance and held constant during the measurements. Angular displacements can also only be detected using a set of sensors with known distances to each other.

## 3 Distance measurement using point diffraction interferometry

### 3.1 Basic approach and physical background

**e**is the electric field strength at the origin,

**r**is the position vector originating from the source,

*c*is the speed of light,

*ε*the electric constant,

*φ*the initial phase, Δ stands for the difference between two variables and the suffix

*0*denotes vacuum condition.

**r**between the two waves. The other variable expressions can be interpreted as offset and scale factors. The cosine in the equation causes the intensity distribution to oscillate spatially between a minimum and a maximum value resulting in the pattern of dark and bright fringes seen in Fig. 4.

Because the same intensity value can be measured at different positions, there is no bijective relationship between intensity and position for single points. This problem can be overcome by looking at two-dimensional cross-sections of the intensity pattern, e.g. captured via a charged coupled device (CCD) camera. In this case, the relationship between the waves’ origins and the position of the capture device is uniquely determined [18].

The spacing of the extrema in the intensity image (and the discontinuities in the phase image accordingly) depend on the distance between the wave sources and the camera. The lateral offset defines the curvature that can be observed in the images. The initial orientation between camera and light sources is arbitrary but has to be determined via calibration and should be held constant between consecutive measurements.

### 3.2 Mathematic algorithm for distance determination

Although the position information is also contained in each individual intensity image, these images are not very suitable for position determination due to physical constraints. First of all, the initial field strengths of the light waves are unknown and difficult to determine. In addition, the intensity measured at each point of the CCD array varies depending on the background noise, the transmissibility of the medium and the accuracy of the individual photo cells so that the absolute intensity measured at each pixel is not very reliable. These problems can be circumvented by transforming the intensity information into phase information. For further details on the mathematic algorithm and the associated techniques reference is made to [19, 20, 21, 22, 23].

**Φ**

_{s}, calculated for a position vector

**a**using theoretical equations, and the phase image

**Φ**

_{r}, extracted and unwrapped from the intensity images, as seen in Eq. (2).

**Φ**

_{0}is the initial phase offset between the two waves. This regression problem is of a nonlinear kind due to the characteristics of the underlying mathematical equations. Several techniques are available to cope with this kind of regression problem, e.g. Newton’s method and quasi-Newton methods [24].

### 3.3 Existing diffraction interferometer setups

For the one-dimensional case a measurement uncertainty of 0.2 μm over a distance of 150 mm with a repeatability of 30 nm was reported [21]. Further tests carried out with an optimised version a year later indicated that the expected volumetric accuracy should be better than 1 μm [22]. A three-dimensional verification of this assumption was successfully carried out using a coordinate measuring machine as reference for a working volume of 60 × 60 × 20 mm³ [23].

However, some limitations hinder the practical deployment of the fibre-based interferometer setup in machine tools. Our experiments have shown that the optical fibres are extremely susceptible to mechanical and thermal stress which causes uncontrollable changes in the optical path difference between the two waves. In harsh environments without air-conditioning, like shop floors, the result will be a drift of the interference pattern which can have a magnitude significantly bigger than the shift of the interference fringes induced by a controlled phase shift. In addition, the use of piezoelectric transducers for phase shifting adds a considerable degree of complexity and costs to the entire system. Finally, the opening angle of a spherical wave emitted from an optical fibre is limited due to its numerical aperture [25], which makes it difficult to cover bigger measurement volumes.

### 3.4 Alternative interferometer setup for the use in machine tools

In order to eliminate these problems, a new system setup has been designed which features optical lenses for spherical wave generation and a simpler phase shifting technique. Coherent light from a frequency-stabilised laser is coupled into a single optical fibre and transmitted to an emitter head. Here, the light is decoupled, split into two separate beams and guided towards focussing lenses generating spherical waves that emanate from their focal points. The phase shift is realised using a liquid crystal retarder which induces a phase shift in one of the beams depending on the applied voltage. The problem of pattern drift is also avoided as the light is separated behind the connecting fibre. This way the phase relation between the two separated beams remains constant. A patent application for the system design has been filed [26].

## 4 Simulative investigation of the system behaviour

After the principle functionality had been verified [21, 22, 23], the main task in testing the point diffraction interferometer system regarding its applicability on machine tools was to optimise the system parameters with respect to machine tool specific boundary conditions. While machine tools can be found with working volumes in almost any size, even an average value exceeds the volume tested in [23] by several multiples. Further boundary conditions result from environmental conditions within the machine tool working volume, such as air temperature and contamination or fluctuation in ambient light. Additional system constraints result from the technical equipment used to perform the diffraction measurements, like the CCD array size and resolution or the algorithm implementation and number precision.

In order to deal with the large number of influencing parameters and their wide spectrum of possible values, a simulation framework has been created enabling the user to evaluate the systematic influence of each parameter on the volumetric accuracy of the system. In the next paragraph, the simulation framework and test procedure will be presented briefly, followed by results obtained from corresponding simulations.

### 4.1 Simulation framework and test procedure

Standard simulation framework setup

Parameter | Value |
---|---|

Wavelength (nm) | 630 |

Initial amplitudes | 1 |

Distance between wave sources (mm) | 10 |

Number of CCD array photodiodes | (1,280 × 960) |

Distance between adjacent photodiodes (μm) | 5 |

Initial phase offset (rad) | 0 |

Phase modulation window (rad) | (0, 1.5π) |

Number of phase modulations | 4 |

Reduced number of CCD array photodiodes | (80 × 36) |

Initial offset (mm) | 20 |

Machine epsilon | 2 |

### 4.2 Results of the simulative investigation

Using a simulation framework as described in the previous paragraph the systematic influence of different parameter has been investigated. Some exemplary results will be presented in the following chapters. If not otherwise specified, a configuration as stated in Table 1 was used for all simulations.

#### 4.2.1 Position dependent accuracy of the algorithm

#### 4.2.2 Tilt dependent accuracy of the algorithm

#### 4.2.3 Influence of camera parameters

#### 4.2.4 Impact of wavelength disturbances

## 5 Conclusion and outlook

The results of the simulative investigation confirm the results of the experimental tests carried out with the interferometer setup described in [21, 22, 23] but expand the measurement volume being examined. They further show the principal dependency of the geometric accuracy on different parameters, for example indicating that a tilt of the CCD array around the x-axis will lead to large errors that cannot be compensated within the algorithm. However, this problem could be solved by introducing a third wave source or by moving the transmitter for a known increment which allows for two consecutive measurements with all other parameters held constant. The results indicate that a point diffraction interferometer in theory can be used as a recalibration device for machine tools if further corrections to the hardware are made.

Future work will primarily focus on the improvement of the alternative diffraction interferometer setup. Final tests have to show if the accuracy is in the range indicated by the simulations and if external disturbances can be handled sufficiently via compensation or correction. Wavefront distortion of the beam caused by the optical components (aberration) might be a problem and needs to be investigated. In addition, the phase shift quality of the liquid crystal retarder has to be evaluated. Furthermore, improvements to the simulation environment are planned.

## Notes

### Acknowledgments

The authors would like to thank the European Commission for funding this work as part of Project Chameleon within the 7th Framework Programme of the European Union.

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