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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...

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Autores principales: Takagi, Junichi, Kanazawa, Hirotaka, Ichikawa, Kotaro, Mitamura, Hiromichi
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2022
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.
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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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