New iterative algorithm for hyperbolic positioning used in an Ultrasonic Local Positioning System

Ultrasonic local positioning systems (US-LPS) utilize hyperbolic trilateration to avoid the necessity of synchronizing the beacons and the tag. In actual systems the beacons emit coded signals which are processed in the tag to obtain the time-differences of arrival, with one of the beacons as reference. The absolute position is computed using an iterative method to solve the non-linear equation system as the Newton-Gauss algorithm: it is time-consuming and need a good initialization of the estimated position. In this work, a new and simpler iterative algorithm is presented to be used in hyperbolic trilateration derived using the Cayley-Menger bideterminant. Instead of use the iterative algorithm to compute the coordinates of the tag position (three unknowns in 3D), here is used to calculate the distance from the tag to the reference beacon (only one unknown).