GPS Solutions: Closed Forms, Critical and Special Configurations of P4P

  title={GPS Solutions: Closed Forms, Critical and Special Configurations of P4P},
  author={Erik W. Grafarend and Jeffrey J. Shan},
  journal={GPS Solutions},
P4P is the pseudo-ranging 4-point problem as it appears as the basic configuration of satellite positioning with pseudo-ranges as observables. In order to determine the ground receiver/satellite receiver (LEO networks) position from four positions of satellite transmitters given, a system of four nonlinear (algebraic) equations has to be solved. The solution point is the intersection of four spherical cones if the ground receiver/satellite receiver clock bias is implemented as an unknown. Here… 
Solving positioning problems with minimal data
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
A Conditional Equation for Minimizing the GDOP of Multi-GNSS Constellation and Its Boundary Solution with Geostationary Satellites
The Walker-delta constellation has been widely used in GNSS (Global Navigation Satellite System). As a key index to measure the positioning configuration, the GDOP minimization plays an important
Efficient Ambiguity Resolution in Wireless Localization Systems
This work presents a direct approach that applies exact direct methods, and resolves the ambiguous pairs of solutions without a priori information, and its divide and conquer structure and the high computational efficiency makes it a very good candidate for fast parallel computing in distributed sensor networks.
Uniqueness and hyperconic geometry of positioning with biased distance measurements
  • M. Hou
  • Mathematics
    GPS Solutions
  • 2022
Positioning an object with biased distance measurements is exactly solvable if exact knowledge of the reference locations and noise-free range measurements are assumed. By examining the positioning
A comprehensive analysis of the geometry of TDOA maps in localization problems
The TDOA map is defined from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space, and the identifiability of the model is studied, giving a complete analytical characterization of the image of this map and its invertibility.
Elementary Mathematical Models for GNSS Positioning
In 1984, the author got the opportunity to visit the US National Geodetic Survey near Washington, D.C., and, guided by Dr. Benjamin W. Remondi, could contribute to the development of civilian
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In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane,
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Abstract Many determined nonlinear systems in geodesy are analytical. C. F. Gauss and C. G. J. Jacobi ever developed an adjustment technique, later called as Gauss–Jacobi combinatorial method, to


An Algebraic Solution of the GPS Equations
  • S. Bancroft
  • Mathematics
    IEEE Transactions on Aerospace and Electronic Systems
  • 1985
The global positioning system (GPS) equations are usually solved with an application of Newton's method or a variant thereof: Xn+1 = xn + H-1(t - f(xn)). (1) Here x is a vector comprising the user
Existence and uniqueness of GPS solutions
The existence and uniqueness of positions computed from global positioning system (GPS) pseudorange measurements is studied. Contrary to the claims of S. Bancroft (1985) and L.O. Krause (1987), in
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Abstract. The perspective 4 point (P4P) problem - also called the three-dimensional resection problem - is solved by means of a new algorithm: At first the unknown Cartesian coordinates of the
Closed form solution to the twin P4P or the combined three dimensional resection-intersection problem in terms of Möbius barycentric coordinates
Abstract. The twin perspective 4 point (twin P4P) problem – also called the combined three dimensional resection-intersection problem – is the problem of finding the position of a scene object from 4
Critical configurations (determinantal loci) for range and range difference satellite networks
The observational modes of Geometric Satellite Geodesy are discussed. The geometrical analysis of the problem yielded a regression model for the adjustment of the observations along with a suitable
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Three formulations of exact solution algorithms to the system of determined pseudorange equations are derived. It is demonstrated that pseudorange equations are hyperbolic in nature and may have two
Classroom Note: The Global Positioning System and the Implicit Function Theorem
This paper provides an example of the implicit function theorem to the accuracy of global positioning system (GPS) navigation by approximate the timing accuracy required by the GPS navigation system to locate a user within a certain degree of precision.
Numerical methods for a class of nite dimensional bifur-cation problems
For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple bifurcation point $p^ * = p(t^ * )$ with rank $H_y = (p^ *
General Relativity: A Geometric Approach
Part I. The Concept of Spacetime: 1. Introduction 2. Events Part II. Flat Spacetime and Special Relativity: 3. Flat spacetime 4. The geometry of flat spacetime 5. Energy 6. Tensors 7. Tensor fields
Die gefährlichen Örter der Pseudostreckenortung
  • Wiss. Arbeiten Fachrichtung Vermessungswesen Universität Hannover,
  • 1993