RELATIVE AND ABSOLUTE ERRORS IN SENSOR NETWORK LOCALIZATION Joshua N. Ash and Randolph L. Moses Department of Electrical and Computer Engineering Ohio State University, Columbus, OH 43210 ABSTRACT This paper considers the accuracy of sensor node location es- timates from self-calibration in sensor networks. The loca- tion parameters are shown to have a natural decomposition into relative configuration and centroid transformation com- ponents based on the influence of measurements and prior information in the problem. A linear representation of the transformation parameter space, which includes rotations and translations, is shown to coincide with the nullspace of the unconstrained Fisher information matrix (FIM). To regularize the absolute localization problem, we consider constraints on the coordinate locations and the impact of these constraints on relative and absolute location error. A geometric inter- pretation of the constrained Cram´ er-Rao bound (CRB) is pro- vided based on the principal angles between the measurement subspace and the constraint subspace. Examples illustrate the utility of this error decomposition. Index Terms— Sensor networks, Localization, Constrained estimation, Cram´ er-Rao bound, Fisher information 1. INTRODUCTION In a distributed wireless sensor network, knowledge of the sensor locations is a prerequisite to obtaining meaningful in- formation from measurements made by the sensors. As such, a diverse variety of self-localization algorithms based on some form of inter-node measurements have been proposed in the literature. In order to better understand how noise, deploy- ment geometry, and measurement type effect fundamental lo- cation estimation performance, a number of authors have con- sidered the Cram´ er-Rao bound (CRB) on self-localization per- formance (see eg. [1, 2] and references therein). In this paper we extend the general CRB analysis by providing a meaning- ful decomposition of localization error. In particular, we decompose the total localization error into a relative portion representing error in the estimated net- work shape and a transformation portion representing error in the absolute position of the relative scene. This decomposi- tion is motivated by the fact that relative information is de- rived from both measurements and prior information, while This work was supported in part by an MIT Lincoln Laboratory Graduate Fellowship. transformation information comes solely from prior informa- tion. Because the inter-node calibration measurements pro- vide no information about the transformation parameters, we regularize the problem by considering general parametric con- straints on the sensor locations. One of the main contributions of this work is an anal- ysis illustrating how the constraint subspace interacts with the measurement subspace to effect total localization perfor- mance. Along with the CRB itself, the relative / transfor- mation decomposition presented here gives insight into how external inputs effect absolute localization. This partitioning of error is also useful to higher level applications in a sensor network that utilize results of the localization service. 2. RELATIVE AND TRANSFORMATION ERROR 2.1. Formulation The absolute self-localization problem is to combine inter- node measurements collected in a measurement vector z with prior information in order to obtain estimates of the coordi- nates {p i =[x i y i ] T } N i=1 of the N constituent nodes of the network. A general measurement model takes the form z = μ(θ)+ η ∈ R M , (1) where z is the vector of M measurements, μ is the mean of the observation which is structured by the true coordinate pa- rameter vector θ =[x 1 y 1 ... x N y N ] T , and η is a zero-mean noise vector. In this paper we consider inter-node distance measurements, hence elements of μ are of the form ||p i -p j ||. Since inter-node distances are invariant to the scene’s absolute position and orientation, the measurements only inform upon the relative shape of the network. As observed in [3], this manifests itself as a singular Fisher information matrix, J J = [U J  U J ]  Λ J 0 0 0  [U J  U J ] T , (2) whose nullspace R(  U J ) is spanned by the vectors v x = α x        1 0 1 0 . . .        , v y = α y        0 1 0 1 . . .        , v φ = α φ        -(y 1 - ¯ y) (x 1 - ¯ x) -(y 2 - ¯ y) (x 2 - ¯ x) . . .        , (3)