Optimal linear and quadratic estimators for tracking from distance measurements

We consider the tracking problem of a point moving in a three-dimensional space using only measurements of distance from a set of reference points. The approach followed in this paper is to derive a linear map with multiplicative noise through a quadratic transformation of the distance measurements. A suitable rewriting by means of an output injection term makes the multiplicative noise of the linear map amenable to be processed by recursive estimators. These estimators are guaranteed to be internally stable and the variance of the estimation error is estimated. We compare the performance of the resulting algorithm for the linear and quadratic case with standard alternatives.