Virtual assembly (VA) is a method for datum definition and quality prediction of assemblies considering local form deviations of relevant geometries. Point clouds of measured objects are registered in order to recreate the objects' hypothetical physical assembly state. By VA, the geometrical verification becomes more accurate and, thus, increasingly function oriented. The VA algorithm is a nonlinear, constrained derivate of the Gaussian best fit algorithm, where outlier points strongly influence the registration result. In order to assess the robustness of the developed algorithm, the propagation of measurement uncertainties through the nonlinear transformation due to VA is studied. The work compares selected propagation methods distinguished from their levels of abstraction. The results reveal larger propagated uncertainties by VA compared to the unconstrained Gaussian best fit.

As quality demands on products increase, tolerance specifications for parts become more and more complex. With these challenging geometrical specifications, verification algorithms are required that represent the geometrical system more precisely. According to Nielsen (2003), in the last few decades, dimensional tolerances shrank due to improved manufacturing systems. However, the form deviations could not be reduced by the same extent. Therefore, their consideration should be intensified. A main deficit in the current International Organization for Standardization (ISO) standard for datum definition, ISO 5459 (Deutsches Institut für Normung e.V., 2011), is the lack of consideration of local form deviations for datum features. A datum feature is defined as a “real (non-ideal) integral feature used for establishing a single datum” (Deutsches Institut für Normung e.V., 2017, p. 2). Datum systems composed of three datum features mathematically define a coordinate system. This allows the definition of tolerance zones for extrinsic tolerances (Weißgerber and Keller, 2014). About 80 % of all measurement tasks require datum systems, so a further function-oriented datum system definition has a strong impact on geometrical verification. Hence, an assessment of the uncertainty for datum systems is of broad interest. Figure 1 shows a datum definition, where three perpendicular associated planes are considered in a nested approach. The primary datum constrains 3 degrees of freedom (DOF), the secondary datum 2 DOF and the tertiary datum 1 DOF (Gröger, 2015).

Datum definition by nested registration, using associated planes

In this paper, measurement data of physical objects are gathered from measurements using industrial computed tomography (CT). Registration is the action of aligning a data set relatively to another according to a datum definition in a common coordinate system. Virtual assembly (VA) comprises the consideration of local form deviations in the datum system computation. As shown in Fig. 1a, through VA, the physical workpiece contact is simulated by computing the contact points. The registration for VA is mathematically stated as an optimization problem, as introduced in Weißgerber and Keller (2014). In the following, matrices are marked as boldface capital, vectors in boldface italic, and scalar values in roman formatting. The signed distances

This optimization constraint can be either formulated as a hard constraint, as implemented in this work, allowing

A complete statement of a measurement result includes the measurement uncertainty. The measurement uncertainty is a nonnegative quantity expressing doubt about the measured value, defined as a “parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand” (ISO/IEC, 2008b). In ISO/IEC (2008a), the main stages of uncertainty assessment are described as

The aim of this work is to analyze the uncertainty propagation for the VA algorithm. Since only few contact points may influence the hard constraint of the optimization problem, a lower robustness compared to existing methods is initially assumed. At the moment, uncertainty propagation is commonly not considered for fitting algorithms such as the VA. However, information gathered from the uncertainty propagation could, on the one hand, be used to claim the robustness of a registration result and, hence, the derived measurements. On the other hand, this allows a reduction in the uncertainty contribution of registration algorithms, for example, by avoiding less certain registration results.

As a use case, a linear guide assembly, consisting of a slider mounted to a rail, is assessed, which is shown in Fig. 3. Measurement data were captured using the CT system Werth TomoScope HV 500, with an acceleration voltage of 180 kV, a tube current of 240 mA, an integration time of 500 ms, 1000 projection images, and a resulting voxel size of 0.2 mm.

The propagation model corresponds to the model function

Graphical decomposition of the propagation model.

Shown, from left to right, is the 3D representation of the VA, the front view of the assembly, and a technical drawing of the datum system (primary datum A, secondary datum B, and tertiary datum constrained to zero

The measurement point uncertainty

The uncertainty

For the mathematical formulation of the propagation, the variance–covariance matrix

Comparison of propagation methods concerning relative difference and normalized computation time in arbitrary units.

Uncertainty components

Uncertainty

Histogram of the initial uncertainty

The linear propagation (LP) is valid for small rotations due to the small angle approximation, where

Figure 3 shows the coordinate system (CS) orientation and the datum of A and B, respectively, that define the assembly. The CS position is centered in the barycenter of the slider. Each point

The main contribution to the propagated uncertainty is due to the uncertainty in the transformation parameters

The propagation model can be verified by analyzing repeatability studies on virtual measurements of registered assemblies. Moreover, computationally efficient minimum variance estimators, such as Kalman filters can be studied in order to evaluate the preliminary VA registration result during an iteration based on the magnitude of uncertainty. As a further approach, the Jacobian matrix

The data set is available upon request by contacting the corresponding author.

MK contributed to conceptualization, data curation, methodology, software, visualization, and writing of the original draft. IE contributed to supervision, validation, and writing (review and editing). MFH contributed to supervision, formal analysis, and writing (review and editing).

The authors declare they have no conflict of interest.

This article is part of the special issue “Sensors and Measurement Science International SMSI 2020”. It is a result of the Sensor and Measurement Science International, Nuremberg, Germany, 22–25 June 2020.

The authors thank the anonymous peer reviewers.

This open-access publication was funded by the Fraunhofer Open Access Fund.

This paper was edited by Rainer Tutsch and reviewed by two anonymous referees.