TOA/FOA Geolocation Solutions Using Multivariate Resultants

Jeff Mason and Louis Romero

Peer Reviewed

Abstract:	This paper gives an algorithmic procedure for solving any system of n polynomials in n variables as long as one of the variables is second degree or less in each polynomial. We show that the time-of-arrival (TOA) and frequency-of-arrival (FOA) navigation equations are easily put into this form. The solution procedure is applied to five specific positioning problems, only one of which has been solved previously. First we give an improved solution of the two-satellite problem in which both TOA and FOA measurements are used to localize an emitter on an ellipsoidal earth. Algebraic solutions are then given for the 3-satellite localization problems using either TOA or FOA measurements, where the emitter is assumed to be on the surface of an ellipsoidal earth. Next an algebraic solution is given for the location of a radio frequency (RF) emitter using 4 FOA measurements. Finally, the 4 -TOA problem, although previously solved, is covered for the sake of completeness.
Published in:	NAVIGATION: Journal of the Institute of Navigation, Volume 52, Number 3
Pages:	163 - 178
Cite this article:	Mason, Jeff, Romero, Louis, "TOA/FOA Geolocation Solutions Using Multivariate Resultants", NAVIGATION: Journal of The Institute of Navigation, Vol. 52, No. 3, Fall 2005, pp. 163-178.
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