An alternative closed-form solution to the GPS pseudo-range equations

@article{Leva1995AnAC,
  title={An alternative closed-form solution to the GPS pseudo-range equations},
  author={Joseph L. Leva},
  journal={IEEE Transactions on Aerospace and Electronic Systems},
  year={1995},
  volume={32},
  pages={1430-1439}
}
  • J. L. Leva
  • Published 20 January 1995
  • Mathematics
  • IEEE Transactions on Aerospace and Electronic Systems
In the four satellite Global Positioning System (GPS) problem, the system of pseudo-range equations is shown to be equivalent to a system of two linear equations together with a range difference and pseudo-range equation. The formulation represents the user's position as the intersection of two planes and a hyperbola branch of revolution. The formulation is three-dimensional and includes almost all degenerate and special case geometries. It provides geometric insight into the characteristics of… 

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References

SHOWING 1-9 OF 9 REFERENCES

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

Simple solutions for hyperbolic and related position fixes

Navigation fixed from range differences to three stations and an additional piece of information are investigated. It is shown that if the additional information is the navigator altitude, or the

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

On the exact solutions of pseudorange equations

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

Solution and performance analysis of geolocation by TDOA

One method of geolocation is based on measuring the time difference of arrivals (TDOAs) of a signal received by three or four geostationary satellites. The received signals are cross-correlated to

A Direct Solution to GPS-Type Navigation Equations

  • Lloyd O. Krause
  • Mathematics
    IEEE Transactions on Aerospace and Electronic Systems
  • 1987
One solution to the navigation equations involves iteration on the 4 by 4 augmented range-direction-cosine matrix beginning with an assumed position and so assumed direction cosines, of which there