A simple intuitive method for seeking intersections of hyperbolas for acoustic positioning biotelemetry.

Area of Science:

• Biotelemetry
• Navigation Systems
• Computational Geometry

Background:

• Hyperbolic positioning is crucial for biotelemetry but faces challenges.
• Intersection calculation of hyperbolas is a complex, often undescribed, procedure.
• Clock synchronization is another significant obstacle for biotelemetry users.

Purpose of the Study:

• To propose a simplified hyperbolic positioning method.
• To introduce the mathematical concept of a 'pencil' for analytical calculations.
• To address the major obstacle of hyperbola intersection calculation in biotelemetry.

Main Methods:

• A novel hyperbolic positioning method representing hyperbolas using a discrete parameter θ.
• Analytical calculations incorporating the mathematical concept of a 'pencil'.
• Comparison of the novel method, analytical method, and Newton-Raphson approximation based on correctness, accuracy, and speed.

Main Results:

• The novel method and analytical method provide correct and accurate positioning.
• Positioning accuracy in the novel method improves with finer parameter θ tick-width.
• The Newton-Raphson approximation method shows limitations in correctness, especially with dual hyperbola intersections.

Conclusions:

• The proposed hyperbolic positioning method simplifies calculations and enhances understanding.
• This study offers a solution to a major difficulty in biotelemetry positioning.
• The method provides biotelemetry users with a more intuitive approach to hyperbolic positioning.