On the Relative and Absolute Positioning Errors in Self-Localization Systems

This paper considers the accuracy of sensor node location estimates from self-calibration in sensor networks. The total parameter space is shown to have a natural decomposition into relative and centroid transformation components. A linear representation of the transformation parameter space is shown to coincide with the nullspace of the unconstrained Fisher information matrix (FIM). The centroid transformation subspace-which includes representations of rotation, translation, and scaling-is characterized for a number of measurement models including distance, time-of-arrival (TOA), time-difference-of-arrival (TDOA), angle-of-arrival (AOA), and angle-difference-of-arrival (ADOA) measurements. The error decomposition may be applied to any localization algorithm in order to better understand its performance characteristics, and it may be applied to the Cramer-Rao bound (CRB) to determine performance limits in the relative and transformation domains. A geometric interpretation of the constrained CRB is provided based on the principal angles between the measurement subspace and the constraint subspace. Examples are presented to illustrate the utility of the proposed error decomposition into relative and transformation components.