The relationship between spherical and hyperbolic positioning

Spherical positioning systems determine position by measuring acoustic travel times from beacons at known locations. To make this travel time measurement, the receiver must know exactly when each beacon transmitted. Hyperbolic positioning systems determine position by measuring differences in travel time between signals from the beacons. The hyperbolic receiver does not need to know when the beacons transmitted, only that they all transmitted at the same time or with known delays relative to each other. The spherical system, which must know the transmit time exactly, and the hyperbolic system, which does not know transmit time at all can be seen as the two endpoints of a continuum of systems parameterised by the accuracy with which the transmit time is known. This paper demonstrates that the Cramer-Rao bounds for position accuracy approach those of the spherical navigation system as the variance in the transmit time estimate approaches zero, and approach those of the hyperbolic positioning system as the variance in the transmit time estimate becomes infinite. This observation has practical application in that the position accuracy of a hyperbolic navigation system which transmits at regular time intervals may be improved if it estimates not only receiver position, but also the transmit time of the beacons. As the estimate of transmit time improves, the position resolution becomes closer to that of a spherical positioning system. This improvement in position accuracy is shown for a typical underwater acoustic positioning scenario.