A method for locating targets using range only

Suppose that we have the distances to
p + 1
targets obtained from a moving platform at discrete times with the (
p
+ 1)st target being a reference beacon. Assume that the distances are measured with an additive error which has a zero mean and that the errors are uncorrelated in time and space. Given the knowledge of the error structure, the position of the reference target, and the set of the successive simultaneous ranges, we present a statistical method for accurately determining the positions of the remaining targets in a computationally reasonable manner, assuming that the platform positions satisfy a certain constraint. Alternatively, reversing the roles of targets and observer, the method can be used to track a moving target platform using ranges simultaneously measured from p observers whose fixed positions are known, provided that there is a (
p
+ 1)st observer at a known position. As long as the target track is sufficiently variable, the discrete-time range data can be used to estimate the position of the p observers and the target track. In practice the method could be used to refine the location estimates of the observers based on successive simultaneous range measurements of the target while tracking the target. If {\sl two} observer positions {sl are known}, {sl no constraint} has to be put on the target track. The technique is analyzed using several artificial data sets for one reference target (or observer). We show that when the error is small we obtain an accurate map of ten fixed targets from 500 platform locations, and the map is also reasonably accurate even when the mean error is 70 percent of the average true distance. We also analyze another model that has 14 targets observed from 1000 locations but where the location of two of these targets are known. In any of the scenerios, as the number of successive range measurements whose errors are uncorrelated grows to infinity, the coordinate estimates of the p target (observer) locations converges in probability to the true coordinates.