{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:23:22Z","timestamp":1785435802245,"version":"3.56.0"},"reference-count":29,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T00:00:00Z","timestamp":1782172800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Polish Ministry of Science and Higher Education","doi-asserted-by":"publisher","award":["PM-II\/SP\/0005\/2024\/02"],"award-info":[{"award-number":["PM-II\/SP\/0005\/2024\/02"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]},{"award":["PM-II\/SP\/0005\/2024\/02"],"award-info":[{"award-number":["PM-II\/SP\/0005\/2024\/02"]}],"id":[{"id":"https:\/\/ror.org\/05dwvd537","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This article presents a method for calibration of dynamic vehicle weighing systems (WIM\u2014Weigh-In-Motion) involving the calibration of all WIM stations operating within a given road network segment as a single process. A key assumption of the method is the presence of at least one scale with significantly higher accuracy than the calibrated systems in this part of road network. This reference scale function may be played by a static scale, slow-pass scale (LS-WIM\u2014Low-Speed WIM) for measurement of vehicle axle load or by a selected WIM system with heightened accuracy. Both the reference scale and all systems undergoing calibration must be equipped with a system for the automatic recognition of vehicle registration number plates. The reference scale makes it possible to determine axle load values considered as benchmark values. Then, for each vehicle weighed on the reference scale and subsequently on any WIM system operating within the analysed area, the relative difference between the reference result and the WIM system measurement is calculated with respect to the reference value. This difference forms the basis for the operation of the algorithm estimating the coefficients of the static characteristic of the calibrated WIM system (so-called calibration coefficients), which are then used to determine corrected weighing results. The estimation of the coefficients is updated after each identified vehicle that has previously been weighed on the reference scale is considered. The article presents both the results of simulations and experimental studies concerning the proposed spatial method of calibration. The results obtained allow for an assessment of the effectiveness of the proposed solution. As can be seen from the analyses conducted, this method leads to a significant reduction in systematic error of vehicle weight measurement. Unfortunately, it does not eliminate random errors. The spatial calibration approach described in this paper has certain limitations. The main ones include the impact of ANPR system errors on calibration effectiveness, cases where a vehicle is unloaded or loaded between WIM stations, and the propagation of systematic errors from the reference systems to the other WIM systems. A significant advantage of the proposed spatial calibration method is that it can operate effectively using weighing data from a single reference WIM system and does not require heavy traffic volumes.<\/jats:p>","DOI":"10.3390\/s26133978","type":"journal-article","created":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T01:03:23Z","timestamp":1782263003000},"page":"3978","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Spatial Calibration of Weigh-In-Motion Systems\u2014Evaluation of Metrological Properties"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4818-7663","authenticated-orcid":false,"given":"Janusz","family":"Gajda","sequence":"first","affiliation":[{"name":"Department of Measurement and Electronics, AGH University of Krakow, Al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3677-713X","authenticated-orcid":false,"given":"Ryszard","family":"Sroka","sequence":"additional","affiliation":[{"name":"Department of Measurement and Electronics, AGH University of Krakow, Al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6591-9040","authenticated-orcid":false,"given":"Piotr","family":"Burnos","sequence":"additional","affiliation":[{"name":"Department of Measurement and Electronics, AGH University of Krakow, Al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2363-7912","authenticated-orcid":false,"given":"Mateusz","family":"Daniol","sequence":"additional","affiliation":[{"name":"Department of Measurement and Electronics, AGH University of Krakow, Al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gajda, J., Burnos, P., Sroka, R., and Daniol, M. (2023). Accuracy Maps of Weigh-In-Motion Systems for Direct Enforcement. Electronics, 12.","DOI":"10.3390\/electronics12071621"},{"key":"ref_2","unstructured":"ISWIM (2026, January 14). Implementation of WIM Systems for Direct Weight Enforcement. 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