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This article is an research for Wireless Sensor Networks about location/position estimation. The main method is TOA (Time of Arrival) using the parameter RSSI (Received Signal Strength Indication). Download this paper and enjoy! ### 相对位置估计在无线传感器网络中的应用 #### 概述本文研究了无线传感器网络（Wireless Sensor Networks, WSNs）中的位置或位置估计问题，特别关注了一种称为到达时间（Time of Arrival, TOA）的方法，并利用接收信号强度指示（Received Signal Strength Indication, RSSI）作为参数进行位置估计。该研究通过理论分析和实际测试来验证其有效性。 #### 自配置与Cramér-Rao界自配置是无线传感器网络中一个重要的概念，它涉及一系列估计问题，这些问题是通过Cramér-Rao界（CRB）来研究的。具体而言，在传感器位置估计中，传感器可以测量它们之间的接收信号强度（RSS）或到达时间（TOA）。网络中只有一小部分传感器具有已知的位置，而其他传感器的位置则需要被估计。文章中推导了基于高斯模型和对数正态模型下的TOA和RSS测量的最大似然估计器（Maximum Likelihood Estimator, MLE）以及相应的Cramér-Rao界。 #### 实验验证为了验证理论分析的有效性，作者们在一个室内办公环境中进行了广泛的TOA和RSS测量活动。这些实验数据用来评估最大似然估计器的表现。此外，相对位置估计算法也在无线传感器网络的实际测试平台中得到实现，并在室内和室外环境下部署。测量结果和测试床实验显示，使用TOA时可以达到1米的均方根误差（Root Mean Square Error, RMSE），使用RSS时则可以达到1到2米的RMSE。 #### 相对位置估计的实现在相对位置估计过程中，只有少量的设备（称为参考设备）拥有事先知道的位置信息，而其余设备（称为盲设备）需要根据相互间的距离估计来确定自己的位置。这种类型的估计被称为“相对位置”估计，因为收集的距离估计主要是在没有绝对坐标知识的设备对之间进行的。 #### 相关技术和理论基础 - **Cramér-Rao界（CRB）**：在统计学中，Cramér-Rao界提供了一个关于任何无偏估计量方差的下限，即对于任何无偏估计量来说，其方差不能低于这个界限。 - **最大似然估计（MLE）**：是一种统计方法，用于估计未知参数的值，使得观察到的数据发生的概率最大化。 - **到达时间（TOA）**：指信号从发送者传播到接收者所需的时间，通常用于精确测距。 - **接收信号强度指示（RSSI）**：反映了接收端接收到的信号强度，常用于粗略的距离估计。 #### 实验环境与结果 - **室内实验**：通过对办公室环境中的TOA和RSS测量，验证了最大似然估计器的性能。 - **测试床实验**：在室内和室外环境下部署了相对位置估计算法，结果显示使用TOA时定位精度可以达到1米的RMSE，而使用RSS时可以达到1至2米的RMSE。 #### 结论与展望本文提出了一个全面的相对位置估计框架，该框架不仅包括理论分析，还结合了实际测试验证了其有效性。通过使用TOA和RSS进行测量，结合最大似然估计方法，可以在无线传感器网络中实现较高精度的位置估计。这一研究成果对于未来无线传感器网络的设计、部署和优化具有重要意义。

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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

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

2138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 8, AUGUST 2003

determination if the accuracies necessary for particular applica-

tions are possible.

We begin in Section II by considering CRBs for network

self-calibration estimators. Next, we state the relative location

estimation problem and derive CRBs and MLEs in Section III.

In Section IV, measurements of TOA and RSS are used to il-

lustrate estimator performance. Finally, real-time operation of

relative location is demonstrated in Section V. Photos of the ex-

periments are included in an extended electronic version of this

paper in [19].

II. N

ETWORK ESTIMATION BOUNDS

In network self-calibration problems, parameters of all de-

vices in a network must be determined. Information comes both

from measurements made between pairs of devices and a subset

of devices that know a priori their parameters. A network self-

calibration estimator estimates the unknown device parameters.

For example, distributed clock synchronization in a network

could be achieved by devices observing pair-wise timing off-

sets when just a small number of devices are synchronous.

Specifically, consider a vector of device parameters

. Each device has one parameter. Devices

are blindfolded devices, and devices

are reference devices. The unknown parameter vector is

, where for . Note that

are known. Devices and make

pair-wise observations

with density .

We allow for the case when devices make incomplete ob-

servations since two devices may be out of range or have

limited link capacity. Let

device makes pair-wise

observations with device

. By convention, a device cannot

make a pair-wise observation with itself, so that

.By

symmetry, if

, then .

We assume by reciprocity that

; thus, it is suf-

ficient to consider only the lower triangle of the observation

matrix

when formulating the joint likelihood

function. In practice, if it is possible to make independent ob-

servations on the links from

to and from to , then we as-

sume that a scalar sufficient statistic can be found. Finally, we

assume that

are statistically independent for . This

assumption can be somewhat oversimplified (see [20] for the

RSS case) but necessary for analysis. Using measurements like

those in Sections IV and V remains important to verify true per-

formance. The log of the joint conditional pdf is

(1)

The CRB on the covariance matrix of any unbiased estimator

is cov , where the Fisher information matrix (FIM)

is defined as

(2)

As derived in Appendix A, the diagonal elements

reduce to a single sum over since there are

terms in (1) that depend on . The off-diagonal elements

can be further reduced: When

, there is at most one sum-

mand in (1) that is a function of both

and ; thus

(3)

where

is an indicator function: 1 if or 0

otherwise.

A. Conditions for a Decreasing CRB

Intuitively, as more devices are used for location estimation,

the accuracy increases for all of the devices in the network. For

-devicenetwork, there are parameters but vari-

ables

used for their estimation. The analysis of this sec-

tion gives sufficient conditions to ensure the CRB decreases as

devicesare added to the network.Consider a network of

blind-

folded devices and

reference devices. Now, consider adding

one additional blindfolded device. For the

and blind-

folded device cases, let

and be the FIMs defined in (2),

respectively.

Theorem 1: Let

be the upper left block

. If for the blindfolded device case 1)

, and 2)

device

makes pair-wise observations between itself and

at least one blindfolded device and at least two devices, in total;

then two properties hold: 1)

in the positive

semi-definite sense and 2) tr

tr .

Theorem 1 is proven in Appendix B. The Gaussian and log-

normal distributions in Section III meet condition 1). Property

1) implies that the additional unknown parameter introduced by

the

st blindfolded device does not impair the estimation

of the original

unknown parameters. Furthermore, property 2)

implies that the sum of the CRB variance bounds for the

un-

known parameters strictly decreases. Thus, when a blindfolded

device enters a network and makes pair-wise observations with

at least one blindfolded device and at least two devices in total,

the bound on the average variance of the original

coordinate

estimates is reduced. Note that properties 1) and 2) of Theorem

1 would be trivially satisfied by the data processing theorem if

adding a device into the network did not increase the number of

parameters.

III. R

ELATIVE LOCATION ESTIMATION

In this section, we specialize for device location estimation

using pair-wise RSS or TOA measurements in a wireless

network. Specifically, consider a network of

reference

and

blindfolded devices. The device parameters

, where, for a two-dimensional (2-D) system,

(although extension of these results to 3-D is

also possible). The relative location problem corresponds to the

estimation of blindfolded device coordinates

(4)

given the known reference coordinates

. In the TOA case, is the mea-

sured TOA between devices

and in (seconds), and in the

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