Metrology, Volume 6, Issue 3 (September 2026) – 11 articles

Precise measurement of large mirrors is essential for high-performance space-based optics. Traditionally, laser interferometry provides high accuracy but faces challenges related to the requirements of computer-generated holograms (CGHs) and the transformation of standard wavefronts to match complex test surfaces. This paper presents a proof-of-concept contact-based surface metrology method for large mirrors. To reduce drift errors caused by stylus instability or contact loss, a wavefront-based calibration method was developed to suppress measurement artifacts while preserving intrinsic surface errors. The method was tested on a concave aspherical mirror using a tactile coordinate measuring machine (CMM), and its accuracy was evaluated by comparing deviations between calibrated CMM error maps and laser-based measurements. Experimental results demonstrated that the XENO CMM achieved a global metric accuracy comparable to interferometric systems for relevant geometric parameters. These findings suggest that ultra-precision CMMs combined with focus point calibration can provide practical preliminary surface verification and serve as a flexible alternative to laser interferometry when global geometric accuracy is the primary concern. Future work will enhance robustness by addressing higher-order aberrations. Full article

Biofilm is known to negatively affect the hydraulic performance of open channels by increasing friction and energy losses and, consequently, economic costs. This study proposes a method to accurately quantify the effect of flow reduction caused by biofouling in open channels. The approach is based on Manning’s equation, using the Monte Carlo Method for the propagation of input uncertainties. A case study concerning a trapezoidal concrete open channel of an agricultural irrigation network is presented, showing a relative expanded uncertainty (95% confidence level) of 10% for flow rates ranging from 1 m³·s⁻¹ to 50 m³·s⁻¹. Manning’s roughness coefficient was identified as the dominant contribution to the overall dispersion of flow rate values, being probabilistically modeled by a beta distribution using minimum, expected and maximum reference values. This information was used for conformity assessment of the studied open channel, considering flow reductions due to moderate and severe biofilm effects of 12.5% and 31.3%, respectively. The corresponding standard uncertainties (4.8% and 7.8%), the adopted confidence level (95%), tolerance upper limit (33%) and decision rule led to different conformity assessment outcomes. These results highlight the importance of accurate assessment of flow reduction due to biofouling for management entities. Full article

The need to measure the geometric dimensions of a railway wheel during train motion at a speed of 120–150 km/h with an error below ±0.25 mm and to identify defects on the tread surface and flange is justified. A new optical measurement system that provides this measurement accuracy was developed. The wheel geometric parameters were measured, and defects were identified using square-wave pulsed laser radiation with a plane-parallel wavefront in the form of a 4 mm wide and 280 mm long line, with a pulse duration of 14 μs, formed using a newly designed collimator. A method was developed for selecting the laser pulse duration so that image blur would have an insignificant influence on the measurement results. Under these conditions, the wheel surface could be considered locally flat with respect to the incident laser radiation. The receiving modules were positioned relative to the rail surface so that the CMOS field of view covered a wheel-surface sector of 10°12′ from both sides and recorded the reflected radiation as a point cloud. To reduce stray illumination and reflections, a filter with a 12.6 nm bandwidth centered at 638.2 nm with a transmittance of 0.71 was installed in front of the CMOS matrix. A point-cloud processing algorithm was developed. Of 236 recorded profiles, 232 were processed successfully, corresponding to a successful-processing rate of 98.3%. Defects were detected in seven processed profiles and confirmed by control measurements. The geometric parameters were determined with a calculated error not exceeding 0.25 mm at 150 km/h. Complete wheel monitoring required 36 sequentially installed optical measurement modules developed in this work, approximately 1.4 times fewer than in comparable systems, for which the measurement error at 150 km/h was approximately ±0.6 mm and the reported defect identification reliability was 95%. Full article

Insight into spectrum measurement is provided regarding the possibility of achieving a unified definition of the measurand for important classes of phenomena, including periodic, transitory impulsive, stochastic stationary, and stochastic non-stationary ones. Furthermore, the notion of characterizing the observation window as a key feature of the measurement method is introduced, and the issue of reducing the variability of the results is discussed, with a focus on the key role that can be played by the multi-taper method. Full article

Low-cost air temperature and relative humidity sensors are increasingly deployed in dense urban monitoring networks for the characterization of urban heat islands and heat exposure. However, measurement accuracy strongly depends on the performance of the radiation shield protecting the sensor from solar heating. This study evaluates five low-cost radiation shield designs, including naturally ventilated, forced-ventilated, spherical, and chimney-type configurations, under tropical outdoor conditions on Reunion Island. Five calibrated SHT31 sensors were deployed simultaneously alongside a reference meteorological station over a five-week measurement campaign. Shield performance was assessed using standard metrological indicators, daytime–nighttime analyses, error distributions, and two-dimensional irradiance–wind diagnostics. Temperature RMSE values ranged from 0.68 to 1.18 °C, while relative humidity RMSE ranged from 2.65 to 7.39%. The forced-ventilated shield provided the best overall temperature performance, whereas the chimney-type design exhibited the largest errors. Combined irradiance–wind analyses showed that measurement errors were primarily governed by the balance between radiative forcing and convective cooling, with maximum temperature biases exceeding 2.5 °C under high-irradiance and low-wind-speed conditions. Based on these findings, several correction approaches were evaluated. A physically interpretable semi-empirical model reduced RMSE by 50%, while a Random Forest model achieved reductions of up to 66%. These results suggest that low-cost meteorological measurements can be substantially improved through appropriate shield design and meteorologically informed calibration procedures, particularly under tropical conditions characterized by strong solar radiation and limited precipitation. Full article

This closing editorial summarizes and introduces the Special Issue, Applied Industrial Metrology: Methods, Uncertainties, and Challenges, highlighting recent advances in traceability, uncertainty evaluation, intelligent metrology, sensor technologies, flow measurement, and standardization. The collected contributions demonstrate how digitalization, artificial intelligence, and advanced sensing are transforming industrial measurement while reinforcing the enduring importance of measurement traceability, comparability, and uncertainty for reliable, sustainable industrial applications. Full article

Anisotropic polymers fabricated via additive manufacturing exhibit complex thermal transport profiles that are challenging to characterize using steady-state techniques. We present a transient thermal method utilizing a short rectangular current pulse excitation to determine the directional thermal diffusivity and conductivity of anisotropic materials. The measurement is conducted on finite specimens, where the low diffusivity of the polymer media results in a highly attenuated and dispersed rear-side temperature profile over an extended transient window. Conduction losses to the adjacent coolers are accounted for by solving the one-dimensional heat conduction equation on an asymmetric multi-layer sandwich structure using the implicit Crank–Nicolson method. Because thermal diffusivity and conductivity are not independent quantities ($\lambda =a\rho c$), the inverse problem is deliberately formulated to estimate the diffusivity alone: the volumetric heat capacity is predetermined and held fixed, and the conductivity follows directly as $\lambda =a\rho c$. This removes the ill-conditioning that would otherwise arise from treating $\lambda$ and a as free, independent parameters in the fit. A two-parameter non-linear least-squares fit is applied to the rear-side temperature rise following Savitzky–Golay noise filtering to estimate the directional diffusivity and effective heat flux. The method is validated using an isotropic reference standard to rule out false system anisotropy, and is subsequently applied to additively manufactured polymer specimens to resolve print-induced directionality through sequential, axis-aligned (unidirectional) measurements along the axial and transverse printing directions. The validity of the one-dimensional reduction is confirmed quantitatively by two- and three-dimensional anisotropic simulations of the exact geometry, which bound the lateral-spreading bias below $0.01\%$ even for the highest-anisotropy specimen, and the robustness of the method to sensor thermal response, signal filtering, and effective-flux estimation is quantified. A rigorous evaluation of the expanded metrological uncertainty demonstrates the high accuracy and reliability of this low-energy excitation technique for highly dispersing media, making it a viable and highly accessible alternative for evaluating material anisotropy. Full article

An interlaboratory comparison of RF power measurements using automatic power measurement software was carried out among partners of the European Partnership on Metrology project titled “Development of RF and microwave metrology capability II” (RFMicrowave2). The absolute RF power output of a travelling signal generator was measured at six frequencies between 2 GHz and 18 GHz and at three nominal power levels of −10 dBm, 0 dBm, and 10 dBm. Comparison results were evaluated in terms of comparison reference values, degrees of equivalence, and normalised errors. Uncertainty evaluations were performed using both the Guide to the Expression of Uncertainty in Measurement and Monte Carlo simulation methods. The results demonstrate an acceptable level of agreement among participating laboratories and confirm the applicability of automatic power measurement software for interlaboratory RF power comparisons. Full article

The use of industrial and collaborative robots in tasks requiring high precision places increasing demands on the evaluation of their performance. In practice, parameters such as positioning accuracy, repeatability and stability are typically assessed according to ISO 9283, based on repeated measurements and comparison of commanded and measured positions. This paper presents a measurement and analysis system developed for this purpose. The system combines selected measurement hardware with a software solution that covers the full workflow from data acquisition to result evaluation. A Python-based backend is used to handle communication with measuring devices, data processing and storage, while a web-based interface provides access to system control, real-time monitoring, and visualization of results. The separation of these components allows the system to remain stable even if the user interface is interrupted. Measured data are evaluated using statistical methods based on repeated measurements, with results presented in both numerical and graphical form. This approach simplifies interpretation and reduces the need for additional external tools. The proposed solution provides a practical and extendable framework for evaluating robot performance in laboratory as well as industrial conditions. Based on the obtained data, the robot’s performance can be evaluated in terms of pose accuracy and pose repeatability. In addition, robot parameters can be monitored and evaluated over an extended period, which allows the proposed solution to be used in predictive maintenance. This article primarily focuses on verifying pose accuracy, since these data were required by the robot user, who specified a minimum of 30 measurement repetitions. The maximum allowable deviation was ±0.01 mm. In the case of pose repeatability and drift of pose characteristics, the calculated value obtained from the measured data must not exceed ±0.02 mm, which is the value declared by the robot manufacturer. Full article

Accurate calibration of anisotropic magnetoresistive (AMR) sensors under varying environmental conditions is crucial for reliable magnetic field measurements in geophysics, navigation, and space applications. Traditional physics-based calibration models offer interpretability but are limited in modeling complex nonlinear effects and typically lack trustworthy uncertainty estimation, whereas purely data-driven approaches often suffer from poor physical consistency and uncalibrated uncertainty estimates. This study proposes a hybrid calibration approach that integrates a physics-based analytical model for primary calibration with a neural network used exclusively for residual error correction, together with explicit uncertainty quantification. A temperature-compensated analytical calibration model is first estimated using nonlinear multivariate regression, with physics-based aleatoric uncertainty quantified via Quasi-Monte Carlo sampling and epistemic uncertainty propagated through a combination of parameter sampling and a Jacobian-based covariance approach. Residual errors not captured by the analytical model are learned using heteroscedastic neural networks, while Monte Carlo DropConnect is employed to quantify neural network epistemic uncertainty. The final calibrated output is computed by integrating physics-based predictions with data-driven error corrections and their associated uncertainties. Experimental results and simulation studies exhibit improved calibration accuracy and statistically consistent confidence interval coverage. Full article

A robust evaluation of predictive uncertainty is essential for deploying machine learning models in high-risk sectors. While various techniques such as Gaussian Processes and Bayesian Neural Networks have been considered to address model uncertainty, the measurement uncertainty associated with input data, particularly regarding heteroscedasticity and autocorrelation, is often overlooked. This work introduces the Generalized Least Squares Support Vector Machine (GLS-SVM), a kernel-based regression model designed to integrate the full variance–covariance matrix of the response variable into the training process. A GUM-consistent methodology was developed for evaluating prediction uncertainty, including a correction for model bias. The model’s performance was validated against standard Least Squares Support Vector Machines (LS-SVMs) and Gaussian Processes (GPs) through two case studies: a simulated regression problem with correlated data and the calibration of a mass flow controller. Performance was quantified using a comparability index ($C_{\mathrm{index}}$), defined as the absolute error of the prediction weighted by its expanded uncertainty. Results demonstrated that, in the simulated case study, the GLS-SVM achieved a $C_{\mathrm{index}}$ consistently below 0.65, indicating that its predictions are statistically consistent with the ground truth. In contrast, competing models significantly exceeded unity, with peak values near 8, indicating a failure to provide physically consistent estimations. For the calibration of the mass flow controller, the GP models produced uncertainties one to three orders of magnitude smaller than the measurement uncertainties, whereas GLS-SVM yielded uncertainties that were more physically consistent with the underlying measurement process. Eventually, the proposed approach offers a versatile, metrologically informed framework for data-driven regression tasks where measurement covariance information is available and rigorous uncertainty quantification is required. Full article