IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8, AUGUST 2003 2137 Relative Location Estimation in Wireless Sensor Networks Neal Patwari, Member, IEEE, Alfred O. Hero, III, Fellow, IEEE, Matt Perkins, Member, IEEE, Neiyer S. Correal, Member, IEEE, and Robert J. O’Dea, Member, IEEE Abstract—Self-configuration in wireless sensor networks is a general class of estimation problems that we study via the Cramér–Rao bound (CRB). Specifically, we consider sensor location estimation when sensors measure received signal strength (RSS) or time-of-arrival (TOA) between themselves and neigh- boring sensors. A small fraction of sensors in the network have a known location, whereas the remaining locations must be estimated. We derive CRBs and maximum-likelihood estimators (MLEs) under Gaussian and log-normal models for the TOA and RSS measurements, respectively. An extensive TOA and RSS measurement campaign in an indoor office area illustrates MLE performance. Finally, relative location estimation algorithms are implemented in a wireless sensor network testbed and deployed in indoor and outdoor environments. The measurements and testbed experiments demonstrate 1-m RMS location errors using TOA, and 1- to 2-m RMS location errors using RSS. Index Terms—Cramér–Rao bound, localization, radio channel measurement, self-configuration, sensor position location estima- tion, signal strength, time-of-arrival. I. INTRODUCTION W E CONSIDER location estimation in networks in which a small proportion of devices, called reference devices, have a priori information about their coordinates. All devices, regardless of their absolute coordinate knowledge, estimate the range between themselves and their neighboring devices. Such location estimation is called “relative location” because the range estimates collected are predominantly between pairs of devices of which neither has absolute coordinate knowledge. These devices without a priori information we call blindfolded devices. In cellular location estimation [1]–[3] and local positioning systems (LPS) [4], [5], location estimates are made using only ranges between a blindfolded device and reference devices. Relative location estimation requires simultaneous estimation of multiple device coordinates. Greater location estimation accuracy can be achieved as devices are added into the network, even when new devices have no a priori coordinate information and range to just a few neighbors. Manuscript received October 3, 2002; revised April 7, 2003. This work was supported in part by a National Science Foundation Graduate Research Fellow- ship for N. Patwari and by ARO-DARPA MURI under Grant DAAD19-02-1- 0262. The associate editor coordinating the review of this paper and approving it for publication was Dr. Athina Petopulu. N. Patwari and A. O. Hero, III are with the Department of Electrical Engi- neering and Computer Science, University of Michigan, Ann Arbor, MI 48104 USA (e-mail: npatwari@eecs.umich.edu; hero@eecs.umich.edu). M. Perkins, N. S. Correal, and R. J. O’Dea are with Motorola Labs, Planta- tion, FL, USA (e-mail: M.Perkins@Motorola.com; N.Correal@Motorola.com; Bob.O’Dea@Motorola.com). Digital Object Identifier 10.1109/TSP.2003.814469 Emerging applications for wireless sensor networks will depend on automatic and accurate location of thousands of sensors. In environmental sensing applications such as water quality monitoring, precision agriculture, and indoor air quality monitoring, “sensing data without knowing the sensor location is meaningless”[6]. In addition, by helping reduce configuration requirements and device cost, relative location estimation may enable applications such as inventory management [7], intru- sion detection [8], traffic monitoring, and locating emergency workers in buildings. To design a relative location system that meets the needs of these applications, several capabilities are necessary. The system requires a network of devices capable of peer-to-peer range measurement, an ad-hoc networking protocol, and a distributed or centralized location estimation algorithm. For range measurement, using received signal strength (RSS) is attractive from the point of view of device complexity and cost but is traditionally seen as a coarse measure of range. Time-of- arrival (TOA) range measurement can be implemented using inquiry-response protocols [7], [9]. In this paper, we will show that both RSS and TOA measurements can lead to accurate location estimates in dense sensor networks. The recent literature has reflected interest in location esti- mation algorithms for wireless sensor networks [8], [10]–[16]. Distributed location algorithms offer the promise of solving multiparameter optimization problems even with constrained resources at each sensor [10]. Devices can begin with local coordinate systems [11] and then successively refine their location estimates [12], [13]. Based on the shortest path from a device to distant reference devices, ranges can be estimated and then used to triangulate [14]. Distributed algorithms must be carefully implemented to ensure convergence and to avoid “error accumulation,” in which errors propagate serially in the network. Centralized algorithms can be implemented when the application permits deployment of a central processor to per- form location estimation. In [15], device locations are resolved by convex optimization. Both [8] and [16] provide maximum likelihood estimators (MLEs) for sensor location estimation when observations are angle-of-arrival and TOA [8] and when observations are RSS [16]. In this paper, we mention only briefly particular location es- timation algorithms. Instead, we focus on the accuracy pos- sible using any unbiased relative location estimator. The radio channel is notorious for its impairments [17], [18], and thus, sensor location accuracy is limited. The Cramér–Rao bounds (CRBs) presented in this paper quantify these limits and allow 1053-587X/03$17.00 © 2003 IEEE