{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:49Z","timestamp":1760240449923,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T00:00:00Z","timestamp":1560902400000},"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":["41871245"],"award-info":[{"award-number":["41871245"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National High Technology Research and Development Program of China,","award":["2013AA014103"],"award-info":[{"award-number":["2013AA014103"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This study proposes a novel model-based automatic search algorithm to realize the self-calibration of nonlinear signal model for angular position sensors. In some high-precision angular position sensors, nonlinearity of the signal model is the main source of errors and cannot be handled effectively. By constructing a signal flow network framework and by embedding a modeling search network, the parameters of the nonlinear signal model can be searched, and the calibration signal can be obtained. The convergence of the network search process was analyzed. The relationship between the optimization threshold and the convergence accuracy was also studied in simulations. Compared with the maximum angular error reduction to 47.42% after the calibration with simplified model that ignores signal nonlinearities, the proposed scheme was able to reduce this error to 0.0025% in simulations. By implementing the technique in a capacitive angular position sensor, the experimental results showed that the maximum angular error was reduced to 1.63% compared to a reduction of 86.02% achieved with the simplified model calibration. The effects of the search network order and layer number on the calibration accuracy were also analyzed, and the optimal parameters under experimental conditions were obtained. Correspondingly, the proposed scheme is able to handle calibration of nonlinear signal model and further improve sensor accuracy.<\/jats:p>","DOI":"10.3390\/s19122760","type":"journal-article","created":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T10:43:32Z","timestamp":1560941012000},"page":"2760","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Self-Calibration of Nonlinear Signal Model for Angular Position Sensors by Model-Based Automatic Search Algorithm"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9147-0873","authenticated-orcid":false,"given":"Zhenyi","family":"Gao","sequence":"first","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Bin","family":"Zhou","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4030-0716","authenticated-orcid":false,"given":"Bo","family":"Hou","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Chao","family":"Li","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3189-7562","authenticated-orcid":false,"given":"Qi","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"}]},{"given":"Rong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,19]]},"reference":[{"key":"ref_1","unstructured":"Hoang, H.V., and Jeon, J.W. (2007, January 17\u201320). Signal compensation and extraction of high resolution position for sinusoidal magnetic encoders. Proceedings of the 2007 International Conference on Control, Automation and Systems, Seoul, Korea."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Guo, C., Wu, C., Ni, F., and Liu, H. (2016, January 7\u201310). Software-based resolver-to-digital conversion and online fault compensation. Proceedings of the 2016 IEEE International Conference on Mechatronics and Automation, Harbin, China.","DOI":"10.1109\/ICMA.2016.7558586"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hou, B., Zhou, B., Song, M., Lin, Z., and Zhang, R. (2006). A novel single-excitation capacitive angular position sensor design. Sensors, 16.","DOI":"10.3390\/s16081196"},{"key":"ref_4","unstructured":"Li, X., Meijer, G.C.M., and Jong, G.W.D. (1996, January 4\u20136). A self-calibration technique for a smart capacitive angular-position sensor. Proceedings of the Quality Measurement: The Indispensable Bridge between Theory and Reality (No Measurements? No Science! Joint Conference - 1996: IEEE Instrumentation and Measurement Technology Conference and Meko Tec, Brussels, Belgium."},{"key":"ref_5","unstructured":"Le, H.T., Hoang, H.V., and Jeon, J.W. (2008, January 13\u201316). Efficient method for correction and interpolation signal of magnetic encoders. Proceedings of the 2008 6th IEEE International Conference on Industrial Informatics, Daejeon, Korea."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1134","DOI":"10.1109\/TIE.2014.2336622","article-title":"Speed error mitigation for a DSP-based resolver-to-digital converter using autotuning filters","volume":"62","author":"Hatziadoniu","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"015112","DOI":"10.1088\/0957-0233\/21\/1\/015112","article-title":"Artificial neural network-based error compensation procedure for low-cost encoders","volume":"21","author":"Dhar","year":"2010","journal-title":"Meas. Sci. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1109\/TCST.2004.841648","article-title":"Adaptive online correction and interpolation of quadrature encoder signals using radial basis functions","volume":"13","author":"Tan","year":"2005","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1088\/0026-1394\/52\/2\/244","article-title":"A simple method for high-precision calibration of an angle encoder using an electronic nulling autocollimator","volume":"52","author":"Hudson","year":"2015","journal-title":"Metrologia"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1016\/j.cirp.2010.03.127","article-title":"On-axis self-calibration of angle encoders","volume":"59","author":"Lu","year":"2010","journal-title":"CIRP Ann. Manuf. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.cirp.2007.05.119","article-title":"Self-calibration of on-axis rotary encoders","volume":"56","author":"Lu","year":"2007","journal-title":"CIRP Ann. Manuf. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3382","DOI":"10.1364\/AO.20.003382","article-title":"Determination and correction of quadrature fringe measurement errors in interferometers","volume":"20","author":"Heydemann","year":"1981","journal-title":"Appl. Opt."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chuckpaiwong, I. (2004, January 24\u201324). Ellipse fitting method in multidimensional space for on-site sensor calibration. Proceedings of the 2004 IEEE Region 10 Conference TENCON 2004, Chiang Mai, Thailand.","DOI":"10.1109\/TENCON.2004.1414513"},{"key":"ref_14","first-page":"68","article-title":"Automatic calibration of sinusoidal encoder signals","volume":"38","author":"Balemi","year":"2005","journal-title":"IFAC Proc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Faber, J. (2012, January 21\u201322). Self-calibration and noise reduction of resolver sensor in servo drive application. Proceedings of the 2012 ELEKTRO, Rajeck Teplice, Slovakia.","DOI":"10.1109\/ELEKTRO.2012.6225633"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/TIE.2007.893049","article-title":"Calibration of resolver sensors in electromechanical braking systems: a modified recursive weighted least-squares approach","volume":"54","author":"HoseIANezhad","year":"2007","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"523","DOI":"10.4028\/www.scientific.net\/KEM.605.523","article-title":"Initial calibration and online error compensation of a resolver system","volume":"605","author":"Aung","year":"2014","journal-title":"Key Eng. Mater."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/TIE.2003.822084","article-title":"High-performance speed measurement by suppression of systematic resolver and encoder errors","volume":"51","author":"Bunte","year":"2004","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gao, Z., Zhou, B., Hou, B., Li, C., Wei, Q., and Zhang, R. (2018). Self-calibration of angular position sensors by signal flow networks. Sensors, 18.","DOI":"10.3390\/s18082513"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1109\/TAC.2005.849244","article-title":"Amplitude and frequency estimator of a sinusoid","volume":"50","author":"Hou","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2883","DOI":"10.1109\/JSEN.2018.2806894","article-title":"High-accuracy automatic calibration of resolver signals via two-step gradient estimators","volume":"18","author":"Wu","year":"2018","journal-title":"IEEE Sens. J."},{"key":"ref_22","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/00029890.1949.11999350","article-title":"The story of the binomial theorem","volume":"23","author":"Coolidge","year":"1949","journal-title":"Am. Math. Mon."},{"key":"ref_24","first-page":"57","article-title":"De-moivre\u2019s formula for matrices of quaternions","volume":"21","author":"Jafari","year":"2011","journal-title":"JP J. Algebra, Number Theor. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning representations by back propagating errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0893-6080(98)00116-6","article-title":"On the momentum term in gradient descent learning algorithms","volume":"12","author":"Qian","year":"1999","journal-title":"Neural Netw."},{"key":"ref_27","unstructured":"Smart, D.R. (1974). Fixed Point Theorems, Cambridge University Press. Cambridge Tracts in Mathematics, 66."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.ces.2017.02.006","article-title":"Stage-to-stage calculations of distillation columns by fixed-point iteration and application of the Banach fixed-point theorem","volume":"164","author":"Hoffmann","year":"2017","journal-title":"Chem. Eng. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"227","DOI":"10.2529\/PIERS090906182755","article-title":"An application of a fixed point iteration method to object reconstruction","volume":"6","author":"Leem","year":"2010","journal-title":"PIERS Online"},{"key":"ref_30","unstructured":"(2019, March 11). An Overview of Gradient Descent Optimization Algorithms. Available online: https:\/\/arxiv.org\/pdf\/1609.04747.pdf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1016\/j.patcog.2013.01.018","article-title":"Computational and space complexity analysis of SubXPCA","volume":"46","author":"Kadappa","year":"2013","journal-title":"Pattern Recognit."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/12\/2760\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:59:46Z","timestamp":1760187586000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/12\/2760"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,19]]},"references-count":31,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["s19122760"],"URL":"https:\/\/doi.org\/10.3390\/s19122760","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,6,19]]}}}