Open Access

Ranging algebraically with more observations than unknowns

  • Joseph L. Awange1Email author,
  • Yoichi Fukuda1,
  • Shuzo Takemoto1,
  • Ismail L. Ateya1 and
  • Erik W. Grafarend2
Earth, Planets and Space201455:BF03351772

https://doi.org/10.1186/BF03351772

Received: 6 May 2003

Accepted: 16 July 2003

Published: 20 June 2014

Abstract

In the recently developed Spatial Reference System that is designed to check and control the accuracy of the three-dimensional coordinate measuring machines and tooling equipment (Metronom US., Inc., Ann Arbor: http://www.metronomus.com), the coordinates of the edges of the instrument are computed from distances of the bars. The use of distances in industrial application is fast gaining momentum just as in Geodesy and in Geophysical applications and thus necessitating efficient algorithms to solve the nonlinear distance equations. Whereas the ranging problem with minimum known stations was considered in our previous contribution in the same Journal, the present contribution extends to the case where one is faced with many distance observations than unknowns (overdetermined case) as is usually the case in practise. Using the Gauss-Jacobi Combinatorial approach, we demonstrate how one can proceed to position without reverting to iterative and linearizing procedures such as Newton’s or Least Squares approach.

Key words

Overdetermined planar rangingoverdetermined three-dimensional rangingGauss-Jacobi combinatorial algorithmGroebner basisMultipolynomial resultantLeast Squares