{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T20:27:56Z","timestamp":1773692876990,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,5,3]],"date-time":"2020-05-03T00:00:00Z","timestamp":1588464000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875165"],"award-info":[{"award-number":["51875165"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51927811"],"award-info":[{"award-number":["51927811"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012176","name":"Project 211","doi-asserted-by":"publisher","award":["B12019"],"award-info":[{"award-number":["B12019"]}],"id":[{"id":"10.13039\/501100012176","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The measurement accuracy of the precision instruments that contain rotation joints is influenced significantly by the rotary encoders that are installed in the rotation joints. Apart from the imperfect manufacturing and installation of the rotary encoder, the variations of ambient temperature could cause the angle measurement error of the rotary encoder. According to the characteristics of the     2 \u03c0     periodicity of the angle measurement at the stationary temperature and the complexity of the effects of ambient temperature changes, the method based on the Fourier expansion-back propagation (BP) neural network optimized by genetic algorithm (FE-GABPNN) is proposed to improve the angle measurement accuracy of the rotary encoder. The proposed method, which innovatively integrates the characteristics of Fourier expansion, the BP neural network and genetic algorithm, has good fitting performance. The rotary encoder that is installed in the rotation joint of the articulated coordinate measuring machine (ACMM) is calibrated by using an autocollimator and a regular optical polygon at ambient temperature ranging from 10 to 40 \u00b0C. The contrastive analysis is carried out. The experimental results show that the angle measurement errors decrease remarkably, from 110.2\u2033 to 2.7\u2033 after compensation. The mean root mean square error (RMSE) of the residual errors is 0.85\u2033.<\/jats:p>","DOI":"10.3390\/s20092603","type":"journal-article","created":{"date-parts":[[2020,5,4]],"date-time":"2020-05-04T14:00:43Z","timestamp":1588600843000},"page":"2603","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Compensation of Rotary Encoders Using Fourier Expansion-Back Propagation Neural Network Optimized by Genetic Algorithm"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0909-5507","authenticated-orcid":false,"given":"Hua-Kun","family":"Jia","sequence":"first","affiliation":[{"name":"School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Lian-Dong","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Yi-Zhou","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-4003","authenticated-orcid":false,"given":"Hui-Ning","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China"}]},{"given":"Jia-Ming","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,3]]},"reference":[{"key":"ref_1","unstructured":"Su, Z.K., Qiu, Z.R., Wang, C.L., and Li, X.H. 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