{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:44:52Z","timestamp":1760143492792,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T00:00:00Z","timestamp":1707436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The most effective method for determining the coordinates of the railway track axis is based on using mobile satellite measurements. However, there are situations in which the satellite signal may be disturbed (due to field obstructions) or completely disappear (e.g., in tunnels). In these situations, the ability to measure the value of the directional angle of a moving rail vehicle using an inertial system is useful. The directional angle is determined on a topographic map as the angle between the direction of the vehicle\u2019s longitudinal axis (or the direction of a tangent to the track axis) and the reference direction, which is the north. This article presents a method for determining the directional angle of a railway line based on appropriate measurement data. The latter should be Cartesian coordinates of the track axis, allowing for the visualization of a given railway route and permitting a general orientation of its course to be obtained. The presented proposal for solving the problem refers to the assumptions made in the method for determining the curvature of the railway track axis using the moving chord. The assumptions of the proposed method for determining the directional angle of the railway route are discussed, along with the appropriate computational algorithms. The accuracy of this method is assessed using the adopted model geometric layout. Reference is also made to the appropriate method for determining the curvature of the railway track axis. In conclusion, we provide an example of determining the directional angle based on measurement data.<\/jats:p>","DOI":"10.3390\/s24041131","type":"journal-article","created":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T03:53:46Z","timestamp":1707450826000},"page":"1131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Method for Determining the Directional Angle of a Railway Route Based on Field Measurements"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4619-7852","authenticated-orcid":false,"given":"Wladyslaw","family":"Koc","sequence":"first","affiliation":[{"name":"Faculty of Civil and Environmental Engineering, Gdansk University of Technology, ul. G. Narutowicza 11\/12, 80-233 Gdansk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,9]]},"reference":[{"key":"ref_1","unstructured":"Deutsche Bahn (2016). 883.2000 DB_REF-Festpunktfeld, Deutsche Bahn Netz AG."},{"key":"ref_2","unstructured":"(2010). Railway Applications\u2014Track\u2014Track Alignment Design Parameters\u2014Track Gauges 1435 mm and Wider. Part 1: Plain Line (Standard No. EN 13803-1)."},{"key":"ref_3","unstructured":"(2008). Code of Federal Regulations Title 49 Transportation, Federal Railroad Administration, US Government Printing Office."},{"key":"ref_4","unstructured":"(2008). NR_Reinstatement of Absolute Track Geometry (WCRL Routes), Iss. 1 (Standard No. NR\/L3\/TRK\/0030)."},{"key":"ref_5","unstructured":"(2018). Attachment ST-T1-A6: Geometrical Layouts of Tracks, Polish State Railways\u2014Polish Railway Lines. (In Polish)."},{"key":"ref_6","unstructured":"Szwilski, A.B., Dailey, P., Sheng, Z., and Begley, R.D. (2005, January 29\u201330). Employing HADGPS to survey track and monitor movement at curves. Proceedings of the 8th International Conference \u201cRailway Engineering 2005\u201d, London, UK."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.trpro.2016.05.310","article-title":"Mobile satellite measurements in designing and exploitation of rail roads","volume":"14","author":"Specht","year":"2016","journal-title":"Transp. Res. Procedia"},{"key":"ref_8","unstructured":"(2023, October 28). A Guide to Using IMU (Accelerometer and Gyroscope Devices) in Embedded Applications. Starlino Electronics. Available online: http:\/\/www.starlino.com\/imu_guide.html."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/bs.aecr.2019.06.003","article-title":"Terrestrial laser scanning reveals temporal changes in biodiversity mechanisms driving grassland productivity","volume":"61","author":"Weigelt","year":"2019","journal-title":"Adv. Ecol. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.measurement.2019.04.013","article-title":"Development of a trajectory constrained rotating arm rig for testing GNSS kinematic positioning","volume":"140","author":"Quan","year":"2019","journal-title":"Measurement"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10291-019-0943-y","article-title":"Improving ambiguity resolution success rate in the joint solution of GNSS-based attitude determination and relative positioning with multivariate constraints","volume":"24","author":"Wu","year":"2020","journal-title":"GPS Solut."},{"key":"ref_12","unstructured":"(2012). Regulation of the Council of Ministers of 15 October 2012 on the national spatial reference system. J. Laws, 1247. (In Polish)."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1007\/s001900050278","article-title":"Geodetic Reference System 1980","volume":"74","author":"Moritz","year":"2000","journal-title":"J. Geod."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1080\/13658810701602286","article-title":"Gauss Kr\u00fcger projection for areas of wide longitudinal extent","volume":"22","year":"2008","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40864-016-0033-1","article-title":"Concurrent optimization of rail transit alignments and station locations","volume":"2","author":"Lai","year":"2016","journal-title":"Urban Rail Transit."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106105","DOI":"10.1016\/j.asoc.2020.106105","article-title":"Multi-objective railway alignment optimization considering costs and environmental impacts","volume":"89","author":"Zhang","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"295","DOI":"10.2478\/ace-2014-0020","article-title":"Determining the geometrical parameters of exploited rail track using approximating spline functions","volume":"60","author":"Lenda","year":"2014","journal-title":"Arch. Civ. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1109\/TITS.2011.2163186","article-title":"A novel geodetic engineering method for accurate and automated road\/railway centerline geometry extraction based on the bearing diagram and fractal behawior","volume":"13","author":"Gikas","year":"2012","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"04018007","DOI":"10.1061\/(ASCE)SU.1943-5428.0000257","article-title":"Automatic horizontal curve identification and measurement using Mobile Mapping System","volume":"144","author":"Luo","year":"2018","journal-title":"J. Surv. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1061\/(ASCE)TE.1943-5436.0000408","article-title":"Using naturalistic field operational test data to identify horizontal curves","volume":"138","author":"Othman","year":"2012","journal-title":"J. Transp. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"06015005","DOI":"10.1061\/(ASCE)SU.1943-5428.0000163","article-title":"New approach to arc fitting for railway track realignment","volume":"142","author":"Cellmer","year":"2016","journal-title":"J. Surv. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Skala-Szymanska, M., Cellmer, S., and Rapi\u0144ski, J. (2014, January 22\u201323). Use of Nelder-Mead simplex method to arc fitting for railway track realignment. Proceedings of the 9th International Conference \u201cEnvironmental Engineering\u201d, Vilnius, Lithuania.","DOI":"10.3846\/enviro.2014.244"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"04020001","DOI":"10.1061\/(ASCE)SU.1943-5428.0000307","article-title":"Heuristic strategies of modified Levenberg-Marquardt algorithm for fitting transition curves","volume":"146","author":"Song","year":"2020","journal-title":"J. Surv. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1061\/(ASCE)0733-9453(2007)133:1(36)","article-title":"Approximate extraction of spiraled horizontal curves from satellite imagery","volume":"133","author":"Dong","year":"2007","journal-title":"J. Surv. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1179\/003962611X12894696204669","article-title":"Fitting composite horizontal curves using the total least-squares method","volume":"43","author":"Easa","year":"2011","journal-title":"Surv. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"04021010","DOI":"10.1061\/(ASCE)SU.1943-5428.0000362","article-title":"Effect of point configurations on parameter estimation analysis of circles","volume":"147","author":"Song","year":"2021","journal-title":"J. Surv. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1111\/mice.12094","article-title":"Use of heading direction for recreating the horizontal alignment of an existing road","volume":"30","author":"Tarko","year":"2015","journal-title":"Comput. Aided Civ. Infrastruct. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1111\/mice.12392","article-title":"A method for automatically recreating the horizontal alignment geometry of existing railways","volume":"34","author":"Li","year":"2009","journal-title":"Comput.-Aided Civ. Infrastruct. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"04022046","DOI":"10.1061\/JTEPBS.0000691","article-title":"Recreating existing railway horizontal alignments automatically using overall swing iteration","volume":"148","author":"Li","year":"2022","journal-title":"J. Transp. Eng. Part A Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Koc, W. (2023). The procedure of identifying the geometrical layout of an exploited railway route based on the determined curvature of the track axis. Sensors, 23.","DOI":"10.3390\/s23010274"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1061\/(ASCE)TE.1943-5436.0000303","article-title":"Design of rail-track geometric systems by satellite measurement","volume":"138","author":"Koc","year":"2012","journal-title":"J. Transp. Eng."},{"key":"ref_32","unstructured":"Korn, G.A., and Korn, T.M. (1968). Mathematical Handbook for Scientists and Engineers, McGraw\u2013Hill Book Company."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/4\/1131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:57:39Z","timestamp":1760104659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/4\/1131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,9]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["s24041131"],"URL":"https:\/\/doi.org\/10.3390\/s24041131","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,2,9]]}}}