NDLI: Solving positioning problems with minimal data

Content Provider	Springer Nature Link
Author	Caravantes, Jorge Gonzalez Vega, Laurea Piñera, Alejandro
Copyright Year	2016
Abstract	Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels’ resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques relying on closed-form solutions for the solution set of a system of linear equations. We suppose the receiver just detects or chooses minimal data, i.e., four satellites in global positioning systems (GPS) or three stations in LPS. Both methods proceed by parameterizing the line joining two solution points to later solve a nonlinear univariate equation, either quadratic or with degree smaller than 6. The first one uses the Generalized Cramer Identities, which is a different presentation of the generalized Moore–Penrose inverse, and ends with a degree 6 univariate equation for GPS and a degree 4 univariate equation for LPS. The second one solves the system by dealing with a more geometric way, ending with a quadratic equation. Our approach covers all possible cases with a finite number of solutions, while Bancroft’s method cannot be applied when the four satellites, taking the clock bias as fourth coordinate, and the origin lay in the same hyperplane of $${\mathbb{R}}^{4}$$ , and the method by Grafarend and Shan fails when the four satellites are in the same plane in $${\mathbb{R}}^{3}$$ . The two proposed methods fail only when the pseudorange 4 point problem has infinite solutions.
Starting Page	149
Ending Page	161
Page Count	13
File Format	PDF
ISSN	10805370
Journal	GPS Solutions
Volume Number	21
Issue Number	1
e-ISSN	15211886
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-01-06
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	P4P configurations Generalized Gram polynomials Generalized Moore–Penrose inverse Geophysics/Geodesy Atmospheric Sciences Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics) Automotive Engineering Electrical Engineering
Content Type	Text
Resource Type	Article
Subject	Earth and Planetary Sciences

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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