Virtual experiments (VEs) and digital twins (DTs), pivotal for realizing European strategic policies on sustainability and digitalization within Industry 4.0 and the European Green Deal, simulate physical systems and characteristics in a virtual environment, with DTs incorporating dynamic inputs from and outputs to the real-world counterpart. To ensure confidence in their use and outcomes, traceability and methods to evaluate measurement uncertainty are needed, topics that are hardly covered by the literature so far. This paper provides a harmonized definition of VEs and DTs and introduces a framework for evaluating measurement uncertainty. Furthermore, it discusses how to propagate the uncertainty of the contributions coming from the different parts of the DT. For the core part of the DT, the framework derived for VEs can be used. For the physical-to-virtual (P2V) connection and the virtual-to-physical (V2P) connection, additional sources of uncertainty need to be considered. This paper provides a metrological framework for taking all these uncertainty contributions into account while describing a framework to establish traceability for DTs. Two case studies are presented to demonstrate the proposed methodology considering industrially relevant measuring instruments and devices, namely, a coordinate measuring machine (CMM) and a collaborative robot arm (cobot).

Sensor networks, which are increasingly being used in a broad range of applications, constitute a measurement paradigm involving ensembles of sensors measuring possibly different quantities at a discrete sample of spatial locations and temporal points outside the laboratory. If sensor networks are to be considered as true metrology systems and the measurement results derived from them used for decision-making, such as in a regulatory context, it is important that the results are accompanied by reliable statements of measurement uncertainty. This paper gives a preview of some of the work undertaken within the European-funded ‘Fundamental principles of sensor network metrology (FunSNM)’ project to address the challenges of measurement uncertainty evaluation in some real-world sensor network applications. The applications demonstrate that sensor networks possess features related to the nature of the measured quantities, to the nature of the measurement model, and to the nature of the measured data. These features make conventional methods of measurement uncertainty evaluation, and established guidelines for measurement uncertainty evaluation difficult to apply. An overview of some of the modelling tools used to address the challenges of measurement uncertainty evaluation in those applications is given.

Virtual experiments are a digital representation of a real measurement and play a crucial role in modern measurement sciences and metrology. Beyond their common usage as a modeling and validation tool, a virtual experiment may also be employed to perform a parameter sensitivity analysis or to carry out a measurement uncertainty evaluation. For the latter to be compliant with statistical principles and metrological guidelines, the procedure to obtain an estimate and a corresponding measurement uncertainty requires careful consideration. We employ a Monte Carlo sampling procedure using a virtual experiment that allows one to perform a measurement uncertainty evaluation according to the Monte Carlo approach of JCGM-101 and JCGM-102, two widely applied guidelines for uncertainty evaluation in metrology. We extend and formalize a previously published approach for simple additive models to account for a large class of non-linear virtual experiments and measurement models for multidimensionality of the data and output quantities, and for the case of unknown variance of repeated measurements. With the algorithm developed here, a simple procedure for the evaluation of measurement uncertainty is provided that may be applied in various applications that admit a certain structure for their virtual experiment. Moreover, the measurement model commonly employed for uncertainty evaluation according to JCGM-101 and JCGM-102 is not required for this algorithm, and only evaluations of the virtual experiment are performed to obtain an estimate and an associated uncertainty of the measurand. We demonstrate the efficacy of the developed approach and the effect of the underlying assumptions for a generic polynomial regression example and an example of a simplified coordinate measuring machine and its virtual representation. The results of this work highlight that considerable effort, diligence, and statistical considerations need to be invested to make use of a virtual experiment for uncertainty evaluation in a way that ensures equivalence with the accepted guidelines.

Virtual experiments (VE) can be used to assess the measurement uncertainty of complex measurements. The typical calculation procedure implemented in such a VE, called VE-DA in this paper, is based on a Monte Carlo method involving simulating possible measurement errors and possible measurement data based on extensive modeling of the measurement instrument, followed by applying a data analysis function (DA) to evaluate the measurement data. This procedure is similar to the propagation of distributions using a Monte Carlo method (PoD) procedure presented in the written standard JCGM-101, in which the Monte Carlo method is applied to an explicit mathematical model for the measurand involving simulating and applying possible corrections to the observed measurement data. However, in this paper, we show that the uncertainty provided by the VE-DA procedure can be both larger and smaller than the uncertainty evaluated based on applying the PoD to the correct measurement model, when available. This is important to realize by users of the VE-DA procedure when claiming conformity of an uncertainty evaluation with JCGM-101.

Virtual experiments (VEs), a modern tool in metrology, can be used to help perform an uncertainty evaluation for the measurand. Current guidelines in metrology do not cover the many possibilities to incorporate VEs into an uncertainty evaluation, and it is often difficult to assess if the intended use of a VE complies with said guidelines. In recent work, it was shown that a VE can be used in conjunction with real measurement data and a Monte Carlo procedure to produce equal results to a supplement of the Guide to the Expression of Uncertainty in Measurement. However, this was shown only for linear measurement models. In this work, we extend this Monte Carlo approach to a common class of non-linear measurement models and more complex VEs, providing a reference approach for suitable uncertainty evaluations involving VEs. Numerical examples are given to show that the theoretical derivations hold in a practical scenario.