The framework of the single point uncertainty developed at the Institute of Manufacturing Metrology (FMT) presents a methodology to determine and evaluate the local measurement uncertainty for a measurement setup by local comparison of a measurement series with an associated reference geometry. This approach, which was originally developed and optimized for the processing of complete areal measurements of work pieces using industrial X-ray computed tomography, was now also extended to line scans found in dimensional testing using tactile coordinate measuring machines (CMMs). The targets of the investigation are spur (involute) steel gear wheels, which can be dimensionally characterized by both helix and profile scans using a CMM in scanning mode in combination with a rotatory table. A second measurement procedure is characterized by a single scan of the complete gear profile without the usage of a rotatory table, using the “free-form scan” CMM functionality. The modification of the single point uncertainty framework in order to determine the single point precision of repeated gear wheel measurements was implemented successfully for gear measurements using the Zeiss Gear Pro evaluation software in combination with a rotatory table as well as unassisted free-form scans of the same gear. The examinations yielded abnormally high random measurement errors, which could not fully be explained within our examinations and was for the most part caused by the accuracy of the used rotatory table of the CMM. The alternative measurement method showed that the CMM system is capable of measuring very precisely in scanning mode if the changes in the curvature of the scan trajectory are favourable.

The framework of the single point uncertainty describes a methodology to statistically evaluate the local measurement uncertainty of a measurement series consisting of

Visualization of the different sampling strategies

This contribution presents various suitable adjustments to the single point uncertainty framework in order to also be able to process line scans from coordinate measuring machines (CMMs) using the example of gear wheel measurements. Because of the fact that no reference measurements with a lower measurement uncertainty were available for the used CMM, only the

In the context of the subsequent descriptions of this article,

For the following demonstration purposes, a wire eroded, spur (involute) steel gear wheel characterized by 17 teeth, a face width of 8 mm and tip circle diameter of 19.4 mm (module 1 mm) is used. The measurement setup is characterized by the tactile CMM Zeiss UPMC 1200 CARAT S-ACC with a built-in rotatory table. In contrast to areal measurements, which are typically represented by or easily converted into a triangle mesh representation, CMM line scans consist of point clouds with additional meta-information (e.g. probing vectors). Two different measurement sequences which were both utilizing the

Visualization of a complete tactile gear wheel measurement consisting of profile and helix scans.

With respect to gear wheel inspection, the VDI/VDE 2612 guideline states that “

The second measurement series was created based on the

Visualization of repeated gear profile scans (

As mentioned above, the single point uncertainty framework was developed to evaluate the uncertainty parameters from repeated areal measurements of measurement objects. In contrast to that, the tactile gear evaluation is characterized by line scans, and thus the sampling strategy using the normal vector of the sampling start surface of the reference geometry was not feasible. The reason for that is that in general a ray-tracing algorithm can only test the intersections of a ray with areal targets, which is not the case for line scans. Put differently, a ray-tracing test in three-dimensional space (shooting a ray in a defined direction starting from a defined location) cannot hit targets with zero area, such as lines. Even though some cases can mathematically be constructed where an intersection is still possible, in general the calculations yield no solution, ultimately because of the limited numerical accuracy. Consequently, the “shortest distance” sampling strategy had to be used. The following subsections describe the slightly different data processing pipelines for the evaluation of the profile and helix scans according to VDI/VDE 2612 (using Zeiss Gear Pro) and the measurement of the gear profile using the

In the following,

Visualization of the calculated nominal geometry from repeated helix scans and the result of the linear regression. Caution: axes are not equally scaled.

That means that deviations from the nominal geometry perpendicular to both the probing vector and the scan direction can in principle not be recorded for straight-line scans using only one vector

Projection of the calculated nominal geometry into the

A very similar data processing pipeline was implemented to determine the nominal profile scan geometry. Here, the SVD was used to identify the two main axes of the point cloud

Projection of the calculated nominal geometry into the

The evaluation of the scan on curve outline varies slightly from the previously described profile and helix scans. Because of the fact that each measurement consists of a complete profile of the gear wheel, a global registration approach was used. Thus, the transformation instructions to align the point cloud consisting of all measurement repetitions with the nominal geometry (CAD) of the gear wheel were determined using PolyWorks Metrology Suite 2018 IR5. A high-density point cloud representing the nominal gear profile was created by converting the CAD model of the gear wheel into a high-resolution triangle mesh using Autodesk Inventor Professional 2018, followed by our own implementation of a post-processing routine. Each point was associated with the normal vector defined by the vertex normal vectors defined within the underlying triangle mesh. After that, the following sampling problem can be reduced to a 2-D problem by projecting the coordinates onto the

Projection of the calculated nominal geometry into the

Single point precision for helix and profile scans of repeated gear wheel measurements.

Figure

Circle scan containing the flick (height 15

The location of the flick is clearly visible at around 10

Robust line regression at the flick location; same coordinate system as Fig.

Figure

Single point precision (standard deviation) values for all sampling points of the nominal contour. See also the histogram visualization of the shown values in Fig.

Histogram visualization of the single point precision values (standard deviation) and the mean values of

Visualization of the mean values of

As already mentioned during the discussion of Fig.

The determination of the single point precision of a complete gear wheel profile scan using the

In this paper, an adjustment of the single point uncertainty framework, which had primarily been developed to evaluated areal measurements, was presented. Now, locally resolved uncertainty values can also be calculated for different CMM examinations using the scanning mode. Repeated measurements of a steel gear wheel were used to generate the measurement data. In the case of the measurement of profile and helix scans, the adjustment was characterized by the additional reconstruction of the underlying nominal geometry from the measurement data. The measurement series exhibited unexpectedly low precision, which is in general untypical of tactile CMM measurements. A large part of the observed precision could be assigned to the uncertainty of the used rotatory table, although the observation could not be fully explained. The second method,

In this paper, we present a methodology to determine the single point uncertainty of steel gear wheels using tactile coordinate measuring machines in scanning mode. The algorithms used for this purpose, the parameters for the generation of the measurement data and all data processing steps are described in detail in the paper. Additionally, publications cited in the paper describe the algorithms used to determine the single point uncertainty. Thus, all information needed for a reproduction of the presented results is available to the interested reader.

AMM contributed to the conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, writing of the original draft and writing, review and editing. TH contributed to the conceptualization, funding acquisition, project administration, formal analysis, supervision and writing, review and editing.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Sensors and Measurement Systems 2019”. It is a result of the “Sensoren und Messsysteme 2019, 20. ITG-/GMA-Fachtagung”, Nuremberg , Germany, 25–26 June 2019.

The authors would like to thank the German Research Foundation (DFG) for supporting the

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. HA 5915/9-2).

This paper was edited by Sascha Eichstädt and reviewed by two anonymous referees.