{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T04:57:09Z","timestamp":1776056229276,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"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":["51775405, 51490664, 51505356, 51305323"],"award-info":[{"award-number":["51775405, 51490664, 51505356, 51305323"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Defense Basic Research Program","award":["JCKY2016210B002"],"award-info":[{"award-number":["JCKY2016210B002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper investigates the problem of an optimal sensor placement for better shape deformation sensing of a new antenna structure with embedded or attached Fiber Bragg grating (FBG) strain sensors. In this paper, the deformation shape of the antenna structure is reconstructed using a strain\u2013displacement transformation, according to the measured discrete strain data from limited FBG strain sensors. Moreover, a two-stage sensor placement method is proposed using a derived relative reconstruction error equation. In this method, the initial sensor locations are determined using the principal component analysis based on orthogonal trigonometric (i.e., QR) decomposition, and then a new location is sequentially added into the initial sensor locations one by one by minimizing the relative reconstruction error considering information redundancy. The numerical simulations are conducted, and the comparisons show that the proposed method is advantageous in terms of the sensor distribution and computational cost. Experimental validation is performed using an antenna experimental platform equipped with an optimal FBG strain sensor configuration, and the reconstruction results show good agreements with those measured directly from displacement sensors. The proposed method has a large potential for the strain sensor placement of complex structures, and the proposed antenna structure with FBG strain sensors can be applied to the future wing-skin antenna or flexible space-based antenna.<\/jats:p>","DOI":"10.3390\/s18082481","type":"journal-article","created":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T11:22:34Z","timestamp":1533122554000},"page":"2481","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Efficient Sensor Placement Optimization for Shape Deformation Sensing of Antenna Structures with Fiber Bragg Grating Strain Sensors"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9978-1251","authenticated-orcid":false,"given":"Jinzhu","family":"Zhou","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiheng","family":"Cai","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pengbing","family":"Zhao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baofu","family":"Tang","sequence":"additional","affiliation":[{"name":"Nanjing Research Institute of Electronic Technology, Nanjing 210039, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1177\/1077546312472927","article-title":"Optimal piezoelectric sensor\/actuator placement of cable net structures using H2-norm measures","volume":"20","author":"Wang","year":"2014","journal-title":"J. Vib. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1177\/1475921713481015","article-title":"Principal component analysis and perturbation theory-based robust damage detection of multifunctional aircraft structure","volume":"12","author":"Hajrya","year":"2013","journal-title":"Struct. Health Monit."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.actaastro.2017.08.025","article-title":"Optimal sensor placement for deployable antenna module health monitoring in ssps using genetic algorithm","volume":"140","author":"Yang","year":"2017","journal-title":"Acta Astronaut."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, N.-L., Jiang, S.-F., Wu, M.-H., Shen, S., and Zhang, Y. (2018). Deformation monitoring for chinese traditional timber buildings using fiber bragg grating sensor. Sensors, 18.","DOI":"10.3390\/s18061968"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2416","DOI":"10.1177\/1045389X16641204","article-title":"Fiber bragg grating plate structure shape reconstruction algorithm based on orthogonal curve net","volume":"27","author":"Zhang","year":"2016","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/j.mechatronics.2011.10.005","article-title":"Spatial shape reconstruction using orthogonal fiber bragg grating sensor array","volume":"22","author":"Yi","year":"2012","journal-title":"Mechatronics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.2514\/1.J052215","article-title":"Strain-based deformation shape-estimation algorithm for control and monitoring applications","volume":"51","author":"Derkevorkian","year":"2013","journal-title":"AIAA J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.sna.2016.09.027","article-title":"In-situ calibrated deformation reconstruction method for fiber bragggrating embedded smart geogrid","volume":"250","author":"Wang","year":"2016","journal-title":"Sens. Actuators A Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kefal, A., and Yildiz, M. (2017). Modeling of sensor placement strategy for shape sensing and structural health monitoring of a wing-shaped sandwich panel using inverse finite element method. Sensors, 17.","DOI":"10.3390\/s17122775"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.cma.2004.03.015","article-title":"A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells","volume":"194","author":"Tessler","year":"2005","journal-title":"Comp. Methods Appl. Mech. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"045027","DOI":"10.1088\/0964-1726\/23\/4\/045027","article-title":"An inverse finite element method for beam shape sensing: Theoretical framework and experimental validation","volume":"23","author":"Gherlone","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3100","DOI":"10.1016\/j.ijsolstr.2012.06.009","article-title":"Shape sensing of 3D frame structures using an inverse finite element method","volume":"49","author":"Gherlone","year":"2012","journal-title":"Int. J. Solids Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/j.optcom.2013.11.027","article-title":"Research on non-uniform strain profile reconstruction along fiber bragg grating via genetic programming algorithm and interrelated experimental verification","volume":"315","author":"Zheng","year":"2014","journal-title":"Opt. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1412","DOI":"10.1364\/AO.55.001412","article-title":"Inverse problem of determining periodic surface profile oscillation defects of steel materials with a fiber bragg grating sensor","volume":"55","author":"Piotr","year":"2016","journal-title":"Appl. Opt."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.jsv.2007.04.037","article-title":"Estimation of dynamic structural displacements using fiber bragg grating strain sensors","volume":"305","author":"Kang","year":"2007","journal-title":"J. Sound Vib."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1088\/0964-1726\/5\/6\/005","article-title":"Shape and vibration mode sensing using a fiber optic bragg grating array","volume":"5","author":"Davis","year":"1996","journal-title":"Smart Mater. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"025006","DOI":"10.1088\/0964-1726\/18\/2\/025006","article-title":"Displacement field estimation for a two-dimensional structure using fiber bragg grating sensors","volume":"18","author":"Rapp","year":"2009","journal-title":"Smart Mater. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"035011","DOI":"10.1088\/0964-1726\/20\/3\/035011","article-title":"Shape estimation with distributed fiber bragg grating sensors for rotating structures","volume":"20","author":"Kim","year":"2011","journal-title":"Smart Mater Struct"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"251","DOI":"10.2514\/3.20635","article-title":"Sensor placement for on-orbit modal identification and correlation of large space structures","volume":"14","author":"Kammer","year":"1991","journal-title":"J. Guid. Control Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1016\/j.jsv.2007.05.004","article-title":"The connection between effective independence and modal kinetic energy methods for sensor placement","volume":"305","author":"Li","year":"2007","journal-title":"J. Sound Vib."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1177\/107754630000600508","article-title":"Entropy-based optimal sensor location for structural model updating","volume":"6","author":"Papadimitriou","year":"2000","journal-title":"J. Vib. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1002\/eqe.2486","article-title":"Efficient bayesian sensor placement algorithm for structural identification: A general approach for multi-type sensory systems","volume":"44","author":"Yuen","year":"2015","journal-title":"Earthq. Eng. Struct. Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bertola, N., Papadopoulou, M., Vernay, D., and Smith, I. (2017). Optimal multi-type sensor placement for structural identification by static-load testing. Sensors, 17.","DOI":"10.3390\/s17122904"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1922","DOI":"10.2514\/3.11868","article-title":"Sensor placement for on-orbit modal identification via a genetic algorithm","volume":"31","author":"Sethares","year":"1993","journal-title":"AIAA J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"15525","DOI":"10.3390\/s140815525","article-title":"Efficient sensor placement optimization using gradient descent and probabilistic coverage","volume":"14","author":"Akbarzadeh","year":"2014","journal-title":"Sensors"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"075003","DOI":"10.1088\/0964-1726\/25\/7\/075003","article-title":"3D sensor placement strategy using the full-range pheromone ant colony system","volume":"25","author":"Feng","year":"2016","journal-title":"Smart Mater. Struct."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhang, X., Li, J.L., Xing, J.C., Wang, P., Yang, Q.L., Wang, R.H., and He, C. (2014). Optimal sensor placement for latticed shell structure based on an improved particle swarm optimization algorithm. Math. Probl. Eng.","DOI":"10.1155\/2014\/743904"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3993","DOI":"10.1007\/s11771-014-2387-4","article-title":"Optimal sensor placement for structural response estimation","volume":"21","author":"Chen","year":"2014","journal-title":"J. Cent. South Univ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1142\/S0219455411004221","article-title":"Integrated optimal placement of displacement transducers and strain gauges for better estimation of structural response","volume":"11","author":"Zhang","year":"2011","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.mechatronics.2013.05.007","article-title":"Dual-type sensor placement for multi-scale response reconstruction","volume":"24","author":"Zhang","year":"2014","journal-title":"Mechatronics"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.jsv.2015.09.018","article-title":"Optimal multi-type sensor placement for response and excitation reconstruction","volume":"360","author":"Zhang","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2469","DOI":"10.1016\/j.jsv.2013.12.014","article-title":"Sensor placement method for dynamic response reconstruction","volume":"333","author":"Wang","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.ymssp.2011.07.022","article-title":"Sensor placement for modal identification","volume":"27","author":"Stephan","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"095015","DOI":"10.1088\/0964-1726\/22\/9\/095015","article-title":"Optimal sensor placement for large structures using the nearest neighbour index and a hybrid swarm intelligence algorithm","volume":"22","author":"Lian","year":"2013","journal-title":"Smart Mater. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"025011","DOI":"10.1088\/1361-665X\/26\/2\/025011","article-title":"Development and coupling analysis of active skin antenna","volume":"26","author":"Zhou","year":"2016","journal-title":"Smart Mater. Struct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"409","DOI":"10.3233\/JAE-140184","article-title":"Performance of structurally integrated antennas subjected to dynamical loads","volume":"48","author":"Zhou","year":"2015","journal-title":"Int. J. Appl. Electromagn. Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1109\/7.144579","article-title":"Performance of phased-array antennas with mechanical errors","volume":"28","author":"Wang","year":"1992","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Zhu, L.H., Dai, J., and Bai, G.L. (2015). Sensor placement optimization of vibration test on medium-speed mill. Shock Vib.","DOI":"10.1155\/2015\/690196"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1007\/s13042-012-0131-7","article-title":"Principal component analysis using QR decomposition","volume":"4","author":"Sharma","year":"2013","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.2514\/1.J053503","article-title":"Clustering of sensor locations using the effective independence method","volume":"53","author":"Friswell","year":"2015","journal-title":"AIAA J."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/8\/2481\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:15:41Z","timestamp":1760195741000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/8\/2481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,1]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["s18082481"],"URL":"https:\/\/doi.org\/10.3390\/s18082481","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,8,1]]}}}