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A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry
We proposed a simple hyperbolic positioning method that does not require solving simultaneous quadratic equations. Moreover, we introduced the mathematical concept of a “pencil” into analytical calculations in the hyperbolic positioning method for a better understanding. In many recent studies using...
Autores principales: | , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Public Library of Science
2022
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9645641/ https://www.ncbi.nlm.nih.gov/pubmed/36350893 http://dx.doi.org/10.1371/journal.pone.0276289 |
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author | Takagi, Junichi Kanazawa, Hirotaka Ichikawa, Kotaro Mitamura, Hiromichi |
author_facet | Takagi, Junichi Kanazawa, Hirotaka Ichikawa, Kotaro Mitamura, Hiromichi |
author_sort | Takagi, Junichi |
collection | PubMed |
description | We proposed a simple hyperbolic positioning method that does not require solving simultaneous quadratic equations. Moreover, we introduced the mathematical concept of a “pencil” into analytical calculations in the hyperbolic positioning method for a better understanding. In many recent studies using positioning biotelemetry, the specific procedure for intersection calculation of hyperbolas has rarely been described. This might be one of two major obstacles, with the other being clock synchronisation among receivers, for positioning biotelemetry users, including potential users. We focus only on the intersection calculation in this paper. Therefore, we propose a novel method and introduce the mathematical concept into analytical calculations. The computing performances of the novel method, an analytical method applying the concept of a pencil, and an approximating method using the Newton-Raphson method were compared regarding positioning correctness, accuracy, and calculation speed. In the novel method, hyperbolas were represented using the parameter θ, which was treated as a discrete variant. The finer the tick-width of the parameter θ, the more accurate was its positioning, but it took slightly longer to calculate. By setting the tick-width to 0.01°, a simulated trajectory was correctly and accurately localised, as in the analytical method which always correctly returned the accurate solution. The approximating method has a major limitation concerning correctness. It returns a single solution regardless of two intersections of hyperbolas; however, the positioning is accurate when the hyperbolas intersect at a single point. This study approached one major difficulty in positioning biotelemetry and will help biotelemetry users overcome this drawback with a simple and intuitive understanding of hyperbolic positioning. |
format | Online Article Text |
id | pubmed-9645641 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2022 |
publisher | Public Library of Science |
record_format | MEDLINE/PubMed |
spelling | pubmed-96456412022-11-15 A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry Takagi, Junichi Kanazawa, Hirotaka Ichikawa, Kotaro Mitamura, Hiromichi PLoS One Research Article We proposed a simple hyperbolic positioning method that does not require solving simultaneous quadratic equations. Moreover, we introduced the mathematical concept of a “pencil” into analytical calculations in the hyperbolic positioning method for a better understanding. In many recent studies using positioning biotelemetry, the specific procedure for intersection calculation of hyperbolas has rarely been described. This might be one of two major obstacles, with the other being clock synchronisation among receivers, for positioning biotelemetry users, including potential users. We focus only on the intersection calculation in this paper. Therefore, we propose a novel method and introduce the mathematical concept into analytical calculations. The computing performances of the novel method, an analytical method applying the concept of a pencil, and an approximating method using the Newton-Raphson method were compared regarding positioning correctness, accuracy, and calculation speed. In the novel method, hyperbolas were represented using the parameter θ, which was treated as a discrete variant. The finer the tick-width of the parameter θ, the more accurate was its positioning, but it took slightly longer to calculate. By setting the tick-width to 0.01°, a simulated trajectory was correctly and accurately localised, as in the analytical method which always correctly returned the accurate solution. The approximating method has a major limitation concerning correctness. It returns a single solution regardless of two intersections of hyperbolas; however, the positioning is accurate when the hyperbolas intersect at a single point. This study approached one major difficulty in positioning biotelemetry and will help biotelemetry users overcome this drawback with a simple and intuitive understanding of hyperbolic positioning. Public Library of Science 2022-11-09 /pmc/articles/PMC9645641/ /pubmed/36350893 http://dx.doi.org/10.1371/journal.pone.0276289 Text en © 2022 Takagi et al https://creativecommons.org/licenses/by/4.0/This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. |
spellingShingle | Research Article Takagi, Junichi Kanazawa, Hirotaka Ichikawa, Kotaro Mitamura, Hiromichi A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title | A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title_full | A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title_fullStr | A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title_full_unstemmed | A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title_short | A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
title_sort | simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry |
topic | Research Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9645641/ https://www.ncbi.nlm.nih.gov/pubmed/36350893 http://dx.doi.org/10.1371/journal.pone.0276289 |
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