{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T13:54:34Z","timestamp":1762005274755,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T00:00:00Z","timestamp":1609372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Autonomous vehicles need to localize themselves within the environment in order to effectively perform most tasks. In situations where a Global Navigation Satellite System such as the Global Positioning System cannot be used for localization, other methods are required. One self-localization method is to use signals transmitted by beacons at known locations to determine the relative distance and bearing of the vehicle from the beacons. Estimation performance is influenced by the beacon\u2013vehicle geometry and the investigation into the optimal placement of beacons is of interest to maximize the estimation performance. In this article, a new solution to the optimal beacon placement problem for self-localization of a vehicle on a two-dimensional plane using angle-of-arrival measurements is proposed. The inclusion of heading angle in the estimation problem differentiates this work from angle-of-arrival target localization, making the optimization problem more difficult to solve. First, an expression of the determinant of the Fisher information matrix for an arbitrary number of beacons is provided. Then, a procedure for analytically determining the optimal angular separations for the case of three beacons is presented. The use of three beacons is motivated by practical considerations. Numerical simulations are used to demonstrate the optimality of the proposed method.<\/jats:p>","DOI":"10.3390\/sym13010056","type":"journal-article","created":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T14:31:49Z","timestamp":1609425109000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9752-9346","authenticated-orcid":false,"given":"John","family":"McGuire","sequence":"first","affiliation":[{"name":"University of South Australia, Mawson Lakes, SA 5095, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5665-0980","authenticated-orcid":false,"given":"Yee Wei","family":"Law","sequence":"additional","affiliation":[{"name":"University of South Australia, Mawson Lakes, SA 5095, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6496-0543","authenticated-orcid":false,"given":"Javaan","family":"Chahl","sequence":"additional","affiliation":[{"name":"University of South Australia, Mawson Lakes, SA 5095, Australia"},{"name":"Joint and Operations Analysis Division, Defence Science and Technology Group, Melbourne, VIC 3207, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3373-6295","authenticated-orcid":false,"given":"Kutluy\u0131l","family":"Do\u011fan\u00e7ay","sequence":"additional","affiliation":[{"name":"University of South Australia, Mawson Lakes, SA 5095, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1109\/98.878533","article-title":"GPS-less low-cost outdoor localization for very small devices","volume":"7","author":"Bulusu","year":"2000","journal-title":"IEEE Pers. Commun."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.robot.2010.01.006","article-title":"Reliable and efficient landmark-based localization for mobile robots","volume":"58","author":"Loevsky","year":"2010","journal-title":"Robot. Auton. Syst."},{"key":"ref_3","unstructured":"Niculescu, D., and Nath, B. (April, January 30). Ad hoc positioning system (APS) using AOA. Proceedings of the INFOCOM 2003, Twenty-Second Annual Joint Conference of the IEEE Computer and Communications, IEEE Societies, San Francisco, CA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.automatica.2009.12.003","article-title":"Optimality analysis of sensor-target localization geometries","volume":"46","author":"Bishop","year":"2010","journal-title":"Automatica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/j.sigpro.2007.11.013","article-title":"Optimal angular sensor separation for AOA localization","volume":"88","author":"Hmam","year":"2008","journal-title":"Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1109\/TRA.2002.805663","article-title":"On mobile robot localization from landmark bearings","volume":"18","author":"Shimshoni","year":"2002","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2361","DOI":"10.1109\/TAES.2014.130461","article-title":"Self-localization from landmark bearings using pseudolinear estimation techniques","volume":"50","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_8","unstructured":"Esteves, J.S., Carvalho, A., and Couto, C. (2003, January 9\u201311). Generalized geometric triangulation algorithm for mobile robot absolute self-localization. Proceedings of the ISIE\u201903, 2003 IEEE International Symposium on Industrial Electronics, Rio de Janeiro, Brazil."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.oceaneng.2017.04.047","article-title":"Survey on advances on terrain based navigation for autonomous underwater vehicles","volume":"139","author":"Melo","year":"2017","journal-title":"Ocean Eng."},{"key":"ref_10","unstructured":"LaPointe, C.E. (2006). Virtual Long Baseline (VLBL) Autonomous Underwater Vehicle Navigation Using a Single Transponder. [Master\u2019s Thesis, Massachusetts Institute of Technology]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/TAES.2008.4560210","article-title":"Position error bound for UWB localization in dense cluttered environments","volume":"44","author":"Jourdan","year":"2008","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.adhoc.2012.05.009","article-title":"Optimal trajectory determination of a single moving beacon for efficient localization in a mobile ad-hoc network","volume":"11","author":"Miles","year":"2013","journal-title":"Ad Hoc Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1080\/002071700417876","article-title":"Optimal sensor location for parameter estimation of distributed processes","volume":"73","author":"Ucinski","year":"2000","journal-title":"Int. J. Control"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bishop, A.N., Fidan, B., Anderson, B.D., Pathirana, P.N., and Do\u011fan\u00e7ay, K. (2007, January 3\u20136). Optimality analysis of sensor-target geometries in passive localization: Part 2-Time-of-arrival based localization. Proceedings of the 2007 3rd IEEE International Conference on Intelligent Sensors, Sensor Networks and Information, Melbourne, Australia.","DOI":"10.1109\/ISSNIP.2007.4496812"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0898-1221(89)90134-X","article-title":"Optimal observer motion for localization with bearing measurements","volume":"18","author":"Hammel","year":"1989","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1109\/7.784059","article-title":"Optimization of observer trajectories for bearings-only target localization","volume":"35","author":"Oshman","year":"1999","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1109\/7.705885","article-title":"Optimal observer maneuver for bearings-only tracking","volume":"34","author":"Passerieux","year":"1998","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhang, H., Dufour, F., Anselmi, J., Laneuville, D., and N\u00e8gre, A. (2018, January 4\u201311). Piecewise optimal trajectories of observer for bearings-only tracking of maneuvering target. Proceedings of the 2018 IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO.2018.8396474"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.oceaneng.2016.07.028","article-title":"Optimal design of the own ship maneuver in the bearing-only target motion analysis problem using a heuristically supervised extended Kalman filter","volume":"123","author":"Sabet","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Xu, S., Do\u011fan\u00e7ay, K., and Hmam, H. (2016, January 20\u201325). Distributed path optimization of multiple UAVs for AOA target localization. Proceedings of the 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China.","DOI":"10.1109\/ICASSP.2016.7472256"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"70117","DOI":"10.1109\/ACCESS.2019.2918299","article-title":"Optimal configuration analysis of AOA localization and optimal heading angles generation method for UAV swarms","volume":"7","author":"Wang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_22","unstructured":"Hernandez, M.L. (July, January 28). Optimal sensor trajectories in bearings-only tracking. Proceedings of the Seventh International Conference on Information Fusion, Stockholm, Sweden."},{"key":"ref_23","unstructured":"Do\u011fan\u00e7ay, K. (2010, January 23\u201327). Single-and multi-platform constrained sensor path optimization for angle-of-arrival target tracking. Proceedings of the 2010 18th European IEEE Signal Processing Conference, Aalborg, Denmark."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Roh, H., Cho, M.H., and Tahk, M.J. (2018, January 9\u201313). Trajectory optimization using Cram\u00e9r-Rao lower bound for bearings-only target tracking. Proceedings of the 2018 AIAA Guidance, Navigation, and Control Conference, Grapevine, TX, USA.","DOI":"10.2514\/6.2018-1591"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"10386","DOI":"10.3390\/s130810386","article-title":"Sensor networks for optimal target localization with bearings-only measurements in constrained three-dimensional scenarios","volume":"13","author":"Pascoal","year":"2013","journal-title":"Sensors"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1109\/TAES.2017.2667999","article-title":"Optimal sensor placement for 3D angle-of-arrival target localization","volume":"53","author":"Xu","year":"2017","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1080\/00207179.2013.792606","article-title":"Optimal sensor placement for target localisation and tracking in 2D and 3D","volume":"86","author":"Zhao","year":"2013","journal-title":"Int. J. Control"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ucinski, D. (2004). Optimal Measurement Methods for Distributed Parameter System Identification, CRC Press.","DOI":"10.1201\/9780203026786"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1111\/1467-9868.00056","article-title":"Designing experiments with respect to \u2018standardized\u2019optimality criteria","volume":"59","author":"Dette","year":"1997","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1109\/70.563647","article-title":"Mobile robot localization using landmarks","volume":"13","author":"Betke","year":"1997","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bernstein, D.S. (2009). Matrix Mathematics: Theory, Facts, and Formulas, Princeton University Press.","DOI":"10.1515\/9781400833344"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1080\/00029890.2004.11920108","article-title":"A simple proof of Descartes\u2019s rule of signs","volume":"111","author":"Wang","year":"2004","journal-title":"Am. Math. Mon."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1109\/TSP.2005.861088","article-title":"Bias compensation for the bearings-only pseudolinear target track estimator","volume":"54","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Yang, P., Freeman, R.A., and Lynch, K.M. (2007, January 10\u201314). Distributed cooperative active sensing using consensus filters. Proceedings of the 2007 IEEE International Conference on Robotics and Automation, Roma, Italy.","DOI":"10.1109\/ROBOT.2007.363820"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Chung, T.H., Gupta, V., Burdick, J.W., and Murray, R.M. (2004, January 14\u201317). On a decentralized active sensing strategy using mobile sensor platforms in a network. Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), Nassau, Bahamas.","DOI":"10.1109\/CDC.2004.1430327"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:48:22Z","timestamp":1760179702000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,31]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010056"],"URL":"https:\/\/doi.org\/10.3390\/sym13010056","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,12,31]]}}}