The analysis of dynamic measurements provides numerous challenges that significantly limit the use of existing calibration facilities and mathematical methodologies. For instance, dynamic measurement analysis requires the application of methods from digital signal processing, system and control theory, and multivariate statistics. The design of digital filters and the corresponding evaluation of measurement uncertainty for high-dimensional measurands are particularly challenging. Several international research projects involving national metrology institutes (NMIs), academia and industry have developed mathematical, statistical and technical methodologies for the treatment of dynamic measurements at NMI level. The aim of the European research project 14SIP08 is the development of guidelines and software to extend the applicability of those methodologies to a wider range of users. This paper outlines the required activities towards a traceability chain for dynamic measurements from NMIs to industrial applications. A key aspect is the development and provision of a new open-source software package. The software is freely available, open for non-commercial distribution, and contains the most important data analysis tools for dynamic measurements.

The analysis of dynamic measurements, i.e. measurements where at least one of
the quantities of interest is time-dependent, is becoming increasingly
important in metrology and industry. Dynamic measurements are encountered in
a wide range of application areas, covering, for instance, single sensors to
complex sensor networks, and measured quantities changing on scales from
picoseconds up to several minutes. Examples of dynamic measurements of
mechanical quantities can be found in, for instance,

Despite the widespread occurrence of dynamic measurements, there is a lack of
guidelines and standards for their treatment, application and analysis. For
static measurements, i.e. measurements where no quantity of interest is
time-dependent, the Guide to the Expression of Uncertainty in
Measurement (GUM) and its supplements

In contrast, the situation for dynamic measurements is more complicated.
Currently, there is a lack of harmonized vocabulary, mathematical and
statistical modelling, and measurement analysis, as outlined in

Dynamic metrology is a very active field of development, and various
approaches to the evaluation and propagation of uncertainty can be found in
the literature. For instance, on-line evaluation of uncertainty in the
application of finite impulse response (FIR) filters is addressed by

Harmonization and standardization are underpinning most of today's metrology
and industrial areas where comparability, conformity and quality assurance
play an important role. The Guide to the Expression of Uncertainty in
Measurement (GUM)

For most applications in dynamic metrology, the analysis of dynamic
measurements follows the basic workflow illustrated in
Fig.

Basic outline of a dynamic measurement with analogue-to-digital conversion (ADC) of the measured system output before processing.

The measurand is thus the sequence

Although the top-level workflow for dynamic measurements is the same as that
for static measurements, the individual low-level steps differ significantly
for the following reasons:

The measurement system input is a function of time;

The measuring device is a dynamic system (often linear and time-invariant);

The estimation task requires the solution of an ill-posed inverse problem;

The measurand is a high-dimensional multivariate quantity.

These differences pose various challenges for metrology and require the
development of a new metrological vocabulary, the adaptation of methods from
signal processing and system theory for metrological purposes, and the
harmonization of regularization methods regarding the corresponding
evaluation of uncertainty. In particular:

Quantities whose values are continuous functions of time would require the translation of the GUM uncertainty framework
to the treatment of stochastic processes as described in

Estimation of the measurand requires knowledge of the dynamic behaviour of the measurement system. Hence, a calibration has to identify and quantify the frequency-dependent characteristics of the complete measurement chain. As a consequence, measurement principles from the static case do not transfer to the dynamic case.

The estimation task is a mathematically ill-posed inverse problem, which requires some kind of regularization to obtain valuable results. Although many regularization methods can be found in the literature, the majority of the available approaches are heuristic and their application to metrology is an ongoing topic of research. In particular, evaluation of the uncertainty contribution of the regularization is a challenging task, because it incorporates prior knowledge about the measurand in the estimation process. This approach is common, for instance, in Bayesian statistical inference, but is not yet considered within the GUM uncertainty framework.

The reporting and dissemination of a dynamic measurement result cannot be carried out in the same way as for static
measurements, due to the high dimensionality of the measurand. Typically, a dynamic measurand is a time series of dimension
greater than

Several research efforts in dynamic metrology have developed initial answers
to some of the challenges listed above. For instance, primary dynamic
calibration methods for several mechanical and electrical quantities have
been developed at NMIs during the last few years – see
Sect.

Despite the availability of many publications on dynamic metrology, the
translation into international standards and guidelines is still at an early
stage. Some national guidelines, such as the draft German DKD-R 3–10 on
dynamic calibration of uni-axial testing machines, and international
standards, such as

In a first step, a harmonized vocabulary has to be determined. For instance,

The software can be downloaded free of charge from the PTB website.

. Similar situations can be found in many other applications. For instance, the parametric dynamic calibration method in ISO 16063 requires the availability of a measured frequency response with associated uncertainties and the propagation of uncertainties to the estimated transducer model parameters. The required mathematical methods are beyond the standard toolbox of most dedicated laboratories. Another example can be found in the standardMany tasks in dynamic metrology involve the application of signal processing,
for which ready-to-use implementations are available in almost all major
software packages. These software implementations, though, lack the
corresponding evaluation of uncertainty. As a consequence, uncertainty
evaluation is frequently undertaken using either rule-of-thumb methods or
time-consuming simulation approaches, or is neglected completely. The EMPIR
project 14SIP08 develops a user-friendly software environment to carry out
data processing for dynamic metrology. The software is called
PyDynamic and it implements recently published mathematical and
statistical methods required to carry out the workflow shown in Fig.

Estimation of the dynamic measurand in the workflow depicted in Fig.

Provided that the frequency response of the measurement system is available
at a set of frequencies, the design of a compensation filter can be carried
out by solving the linear least-squares problem for the filter coefficient
vector

Frequency response of the measurement system of the FIR deconvolution filter and the resulting compensation as a product of the measurement system and deconvolution filter.

Figure

If, instead of an FIR filter, an IIR filter is sought, the PyDynamic
function

The application of digital filters is one of the most basic tasks in the
processing of dynamic measurement data. A common example is the application
of a low-pass filter for noise attenuation or a compensation filter for input
estimation, as described above. The implementation of digital filtering is
straightforward in almost all scientific software packages, whereas the
propagation of uncertainty is typically neglected. This statement in
particular holds true when the filter coefficients have associated
uncertainty. However, the propagation of uncertainties is a prerequisite for
the final step in the workflow depicted in Fig.

Input signal for a simulated measurement, calculated output signal, and estimate obtained by the application of a FIR deconvolution filter to the system output.

Consider the FIR filter with coefficient vector

The uncertainty calculated for the FIR estimation result depicted in Fig.

Point-wise standard uncertainties associated with the output of the FIR deconvolution filter.

The application of a digital filter with IIR is
given mathematically by

The recursive structure of the IIR filter makes an analytic calculation of
the uncertainty associated with its output difficult. Therefore,

Rectangular signal and corresponding output of a sixth-order IIR low-pass filter of Butterworth type.

It is well known that uncertainty evaluation using the GUM uncertainty
framework can produce unreliable results due to the use of a linearization of
the model function. A Monte Carlo method, as described in GUM Supplement 1
(cf.

Point-wise standard uncertainties associated with the IIR low-pass filter output signal, when the filter cut-off frequency is uncertain.

The DFT and inverse DFT are common tools applied
in signal processing, and all major scientific software packages provide
corresponding implementations. Uncertainty evaluation, though, is usually
neglected due to the lack of suitable software implementations. To this end,

For instance, the propagation through the application of the DFT for the
discrete-time signal

The DFT domain methods in PyDynamic provide an end-to-end propagation
of uncertainties in many important application areas. For instance, dynamic
calibration of second-order systems based on measurement of the input and
output signal can be carried out by using (i)

With the availability of a harmonized vocabulary, a principal and general
mathematical modelling approach, together with established routines for the
evaluation of measurement uncertainties and the development of a traceability
chain for industrial end users of dynamic measurement, can finally be
achieved. The next steps in the development of PyDynamic will thus
focus on the implementation of further mathematical and statistical
approaches to common tasks in dynamic metrology. This includes, for instance,
the identification of general transfer function models to frequency response
data with associated uncertainties, the propagation of the uncertainty
associated with dynamic quantities of high dimensionality, sub-sampling and
interpolation of dynamic quantities. There is an increasing use of sensors in
distributed networks with automated data assimilation and evaluation. This
requires common data protocols in order to enable a reliable communication
for the sensor network. Therefore, we are developing a custom data format
“Signal” for PyDynamic that allows the user to carry out standard
data operations without the need to manually propagate the uncertainties.
That is, “Signals” can be added, subtracted from one another using standard
“

PyDynamic is distributed under the LGPLv3 software license which allows the incorporation of PyDynamic routines in closed source code. Together with the implemented versatile data analysis methods, this opens the possibility of intelligent sensors with embedded data analysis that provides data values with associated dynamic uncertainty. In addition, data analysis for sensor networks can then be based on PyDynamic's “Signal” data format and the implemented functions. Moreover, due to the employed object-oriented programming approach for the data structure, users can easily extend the existing code functionality to their needs.

Analysis of dynamic measurements is the topic of a growing number of research initiatives. The majority of publications in this area focus on measurements at the level of NMIs. However, dynamic measurements are routinely carried out at the industrial level and mathematical and statistical methods, guidelines and best-practice guides, which are suitable for typical industrial applications, are required. The prerequisite for the development and wide acceptance of such guidance documents, though, is the availability of well-established and approachable methodologies. At present, there is a significant lack of methods and advice, standard software tools and international standards. This lack has been acknowledged in several publications and support is being requested by a growing number of standardization groups. Therefore, in the EMPIR project 14SIP08, NMIs PTB (Germany) and NPL (UK), together with international companies HBM GmbH and Rolls-Royce Ltd., aim to develop practical guidelines, tutorials, training material and software. One of the outcomes of this project is the software package PyDynamic, which after only one year of development already provides implementations of the major tools required for the analysis of dynamic measurements. The software development will continue throughout and beyond the duration of 14SIP08. The intention is for PyDynamic to act as ready-to-use software that removes the barrier between the analysis of static and dynamic measurements, and makes dynamic measurement analysis standard practice within both NMIs and industry. We outlined, for three typical-use cases in dynamic metrology, how such a software tool can enable the application of sophisticated mathematical approaches. In many applications, the complete data analysis workflow can already be carried out with the help of PyDynamic functions, making the propagation of uncertainties through that workflow a simple task for the user. In the future, this will be improved even more by the provision of the custom data format “Signal” which allows the propagation of uncertainties without the need to know which PyDynamic function would be required for the operation on the data. Together with the cooperation of EMPIR 14SIP08 with JCGM WG1 and the publication of guidance documents, this lays the foundation for future standards and international guidelines in dynamic metrology.

The “data” used for this publication is simulated data, generated by the
code available for download at

The authors declare that they have no conflict of interest.

This work has been carried out as part of the European Metrology Programme for Innovation and Research (EMPIR) project 14SIP08. The EMPIR initiative is co-funded by the European Union's Horizon 2020 research and innovation programme and the EMPIR participating states. Edited by: K.-D. Sommer Reviewed by: three anonymous referees