Position Estimation Using Only Multiple Simultaneous Range Measurements

Three-dimensional generalized position-measurement systems are analyzed in this paper. In these systems, target position is obtained by trilateration using only range data collected by a group of v stations located in an arbitrary geometry. The method of maximum likelihood is used to obtain a joint estimator for the target coordinates which makes optimal use of the redundant data when the noise is Gaussian. A simple recursion formula for the estimator is obtained for this purpose and is shown to be convergent. This formula makes it possible to add data from a redundant number of stations at will and in proportion to their relative reliability. Further, it is shown that the recursion formula can be written entirely in terms of the changes in the successive iterative target position estimates. This technique offers a new means of obtaining tracking data on a moving target since it permits changes in target position to be computed directly as new data are obtained. The covariance matrix of the joint three-dimensional estimator is obtained in the case in which the measurement noise is small compared to the distances measured. The mean-square position error, namely, the trace of the covariance matrix, is shown to have a simple form for the general two-dimensional system in which the target and stations are coplanar. The geometry enters the variance expression only through the angles of cut θ
_ii
, which are the angles between the lines joining the target and the stations.