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Accuracy of Range-Based CooperativePositioning: A Lower Bound Analysis

Liang Heng, and Grace Xingxin Gao, Senior Member, IEEE

Abstract—Accurate location information is essential formobile systems such as wireless sensor networks. A location-aware sensor network generally includes two types of nodes:sensors whose locations to be determined and anchors whoselocations are known a priori. For range-based cooperativepositioning, sensors’ locations are deduced from anchor-to-sensor and sensor-to-sensor range measurements. Positioningaccuracy depends on the network parameters such as networkconnectivity and size. This paper provides a generalizedtheory that quantitatively characterizes such relation betweennetwork parameters and positioning accuracy. We use theaverage degree as a connectivity metric and use geometricdilution of precision (DOP) to quantify positioning accuracy.Under the assumption that nodes are randomly deployed,we prove a novel lower bound on expectation of averagegeometric DOP (LB-E-AGDOP) and derives a closed-formformula that relates LB-E-AGDOP to only three parameters:average anchor degree, average sensor degree, and number ofsensor nodes. The formula shows that positioning accuracy isapproximately inversely proportional to the average degree,and a higher ratio of average anchor degree to average sensordegree yields better positioning accuracy. Furthermore, thepaper shows a strong connection between LB-E-AGDOP andthe best achievable accuracy. Finally, we demonstrate thetheory via numerical simulations with three different randomgraph models.

Index Terms—Range-based positioning, cooperative posi-tioning, accuracy, network connectivity, dilution of precision(DOP), sensor networks

I. INTRODUCTION

LOCATION awareness is a fundamental requirementfor mobile systems such as wireless sensor networks

(WSN) [1] and mobile robot networks (MRT) [2] becauselocation information facilitates navigation, path planning,resource allocation, and networking operations such as geo-graphic routing and topology control [3]. Global navigationsatellite systems (GNSS) such as GPS and GLONASS havebeen widely used to make location awareness possible.However, GNSS cannot guarantee accurate and reliablepositioning anytime and anywhere, particularly indoors andin urban canyons. For many WSN and MRT applications,it is cost and energy prohibitive to include a GNSS receiveron every device.

To enable location awareness in mobile systems, a widevariety of positioning schemes have been explored over thepast decade. According to the measurements used to esti-mate locations, these schemes can be generally classified

Authors’ current addresses: L. Heng, Baidu USA LLC, Sunnyvale,CA 94089; G. X. Gao, University of Illinois at Urbana-Champaign, 315Talbot Lab, 104 S. Wright St., Urbana, IL 61801. Corresponding authoris G. X. Gao, E-mail: gracegao@illinois.edu.

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Fig. 1. A scenario of range-based cooperative positioning. The outdoorphones (slanted, greenish) denote anchor nodes whose locations are knownthrough GNSS positioning. The indoor phones (upright, reddish) denotesensor nodes whose locations are to be estimated. The blue dashed linesrepresent ranging links which provide inter-node distance information.Nodes are randomly distributed and ranging links are randomly establishedaccording to a certain random graph model. This paper aims for a gener-alized theory that characterizes the connection between system parameters(namely network connectivity and network size) and positioning accuracy.

as range-based [4], [5], angle-based [3], [6], proximity-based [7], [8], and event-driven [9], [10]. Besides, thepositioning schemes can be categorized as either nonco-operative or cooperative [11], [12]. In a noncooperativescheme, the unknown-location nodes (hereinafter referredto as sensor nodes or simply sensors, following the nomen-clature in [13]) make measurements with known-locationreferences (hereinafter referred to as anchor nodes or an-chors), without any communication between sensor nodes.In a cooperative scheme, in addition to anchor-to-sensormeasurements, each sensor also makes measurements withneighboring sensors; the additional information gained fromsensor-to-sensor measurements enhances positioning accu-racy, availability, and robustness.

This paper focuses on range-based cooperative position-ing. As illustrated in Fig. 1, anchor nodes are aware of theirlocations, and sensor nodes determine their locations usinginter-node distance information. Range-based cooperativepositioning is essentially a graph embedding problem [14]–[16]. Connectivity of the graph exerts significant influenceon many performance metrics, such as accuracy, energy ef-ficiency, localizability, robustness, and scalability. Althoughlocalizability has been extensively studied with respectto connectivity [14], [15], [17], the relationship betweenaccuracy and connectivity has not yet been theoreticallytreated. The objective of this paper is a generalized theorythat quantitatively characterizes the connection between

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network parameters (namely, network connectivity and net-work size) and positioning accuracy. The theory providescompendious guidelines on the design and deployment oflocation-aware mobile systems.

A. Related work

As previously mentioned, range-based cooperative po-sitioning is a graph embedding problem. Saxe [18] hasshown that testing the embeddability of weighted graphs(equivalently, testing localizability) is strongly NP-hard.Aspnes et al. [19] have further proven that positioning insparse networks is NP-hard. However, when the network isdensely connected such that O(N2) pairs of nodes knowtheir relative distances, where N is the number of nodes,there are efficient algorithms such as multidimensionalscaling (MDS) [20] and semidefinite programming (SDP)[13] for solving the positioning problem.

Cooperative positioning can also be seen as a high-dimension optimization problem that finds a vector of nodelocations such that inter-node distances are as close to rangemeasurements as possible. In general, this optimizationproblem may have many local optima. The MDS andSDP algorithms [13], [20], as well as some stochasticoptimization algorithms [21], are able to find a solutionclose to the global optimum under certain conditions (e.g.,dense connectivity). The solution is not necessarily veryaccurate but can be treated as an initial guess. Then, thelocation solution can be improved using an iterative algo-rithm such as lateration (also referred to as trilateration,multilateration, and even mistakenly triangulation) [4], [5],[22]–[25].

The accuracy of lateration has been widely studied usingthe Cramer-Rao (CR) bound [26]–[32], which is the recip-rocal of the Fisher information matrix [33]. For indepen-dent and identically distributed (iid) Gaussian range errors,the CR bound can be simplified to dilution of precision(DOP), a metric widely used in the GNSS community [23].Many of previous studies on CR bound ended up with acomplicated Fisher information matrix (or an equivalentform) involving node locations, without any closed-formexpression that can characterize positioning accuracy withrespect to network connectivity.

There have been some papers (e.g., [28], [34]) usingMonte-Carlo simulations to reveal that (a) for one sensornode and m randomly-distributed anchor nodes, the CRbound is inversely proportional to m; (b) a larger proportionof anchors results in better accuracy. However, there is stilla dearth of theory to describe these relationships preciselyfor more generalized settings.

Two papers [35], [36] deserve special mentions herebecause they contain a similar flavor to this paper. Shenet al. [35] presented scaling laws of positioning accuracyfor randomly-deployed nodes. The scaling laws indicatethat sensors and anchors “contribute equally” to positioningaccuracy. However, this statement is correct only for densenetwork (a larger number of nodes); in this paper, we shallshow that anchors generally contribute more than sensors

do, and the sensors and anchors tend to contribute equallyas the number of sensors increases. In [36], Javanmard andMontanari offered neat upper and lower bounds of posi-tioning accuracy for random geometric graphs. However,the bounds are only applicable to random geometric graphs,and require range errors to be uniformly bounded.

Our previous work [37] has derived the lower boundof average positioning accuracy in a random network, anddiscussed its scaling laws. Extending that work, this papergives the lower bound more rigorous theoretical treatment,relates the lower bound to the best achievable positioningaccuracy, and demonstrates the effectiveness using morenumerical simulations.

B. Our method and main results

In range-based cooperative positioning, positioning accu-racy is affected by three factors: range measurement errors,geometry of nodes, and network parameters (connectivityand size). In order to establish a quantitative relationshipbetween network parameters and positioning accuracy, we(a) decouple range measurement errors and the rest byassuming iid Gaussian range errors, and (b) decouplegeometry and network parameters by randomizing nodelocations.

In the paper, we define average degree, and use aver-age anchor degree and average sensor degree to quantifynetwork connectivity in cooperative positioning. We usegeometric dilution of precision (DOP), equivalent to the CRbound, to quantify positioning accuracy. We have proved alower bound on the expectation of average geometric DOP(E-AGDOP) under the assumption that nodes are randomlydistributed, and nodes are randomly connected such that thegraph of network can reach a certain average degree.

We have further derived a closed-form formula thatrelates the lower bound to only three network parameters:average anchor degree, average sensor degree, and numberof sensor nodes. The formula shows that (a) positioningaccuracy is approximately inversely proportional to theaverage degree, and (b) average anchor degree contributesmore to positioning accuracy than average sensor degreedoes. Our numerical examples and simulations have verifiedthe formula and further shown that (a) the lower boundis strongly connected to the best achievable positioningaccuracy, and (b) the lower bound is applicable to manyrandom graph models.

C. Outline of the paper

The remainder of this paper is organized as follows.Section II formulates the cooperative positioning problemand introduces the assumptions, definitions, and notationsused throughout this paper. Section III analyzes positioningaccuracy and connects it to DOP. Section IV derives aclosed-form expression LB-E-AGDOP, which is a functionof network connectivity and size. Section V shows thestrong connection between LB-E-AGDOP and the bestachievable accuracy. Numerical simulation results are pre-sented in Section VI. Finally, Section VII concludes the

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0.4

0.6

0.8

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Fig. 2. Graph model of the range-based cooperative positioning problem.Black squares denote anchor nodes, red circles denote sensor nodes,and blue lines represent ranging links. The sensor nodes are randomlydistributed in [0, 1]2 and connected according to the random geometricgraph (RGG). The anchors are fixed at four corners. All the sensor-to-sensor links and anchor-to-sensor links satisfy coordinate symmetry.

paper. Proofs of key theorems and equations are providedin Appendices A and B.

II. PRELIMINARIES

A. Problem formulation

In this paper, we consider the cooperative positioningproblem in a sensor network and follow the nomenclaturein [13]. The discussion and results are not limited to sensornetworks but applicable all scenarios using range-basedcooperative positioning.

As shown in Fig. 2, a sensor network can be modeled asa simple graph1 G = (V,E), where V = {1, 2, . . . , N} is aset of N nodes (or vertices), and E = {e1, e2, . . . , eK} ⊆V ×V is a set of K links (or edges) that connect the nodes[15].

All nodes are in a d-dimensional Euclidean space (d ≥1), with the locations denoted by pn ∈ Rd, n = 1, . . . ,N . The first NS nodes, labeled 1 through NS , are sensornodes (or mobile nodes), whose locations are unknown;the rest NA = N −NS nodes, labeled NS + 1 through N ,are anchor nodes (or beacon nodes). Anchors are awareof their exact locations through built-in GPS receivers ormanual pre-programming during deployment.

There is a link ek = (ik, jk) in E if and only if thereexists a direct ranging link between nodes ik and jk. Letrk = ‖pik − pjk‖ be the Euclidean distance between nodesi and j. The link ek provides the range measurement

ρk = rk + εk, (1)

where εk is the range measurement error.The cooperative positioning problem is to determine the

locations of sensor nodes pn, n = 1, . . . , NS , given afixed network graph G, known locations of anchors pn,n = NS + 1, . . . , N , and range measurements ρk, k = 1,. . . , K.

1A simple graph, also known as a strict graph, is an unweighted,undirected graph containing no self-loops or multiple edges [38].

B. Assumptions

1) Range measurement errors: The range measurementsρk can be obtained by a variety of methods, such as one-way time of arrival (ToA), two-way ToA, or received signalstrength indication (RSSI) [39]. Range measurement errorsin RSSI-based methods are usually treated as having alog-normal distribution [40]. For most ToA-based methods,line-of-sight range measurement errors can be modeledas zero-mean Gaussian random variables [4]. One-wayToA usually results in biased range measurements due tounsynchronized clocks [23].

In this paper, we assume that range measurement errorsare zero-mean Gaussian distributed. Our model is applica-ble to two-way ToA and one-way ToA with perfect clocksynchronization.

2) Coordinate symmetry: In a Cartesian coordinate sys-tem, for all links ek = (ik, jk) ∈ E, we assume that thereis no direction of preference among the coordinate axes.Mathematically speaking, the direction vector, defined as

vk = r−1k (pik − pjk) = [vk,1, . . . , vk,d]T, (2)

satisfies the following coordinate symmetry assumption:

E(v2k,1) = E(v2k,2) = · · · = E(v2k,d), (3)

where E(·) is the expectation operator.The coordinate symmetry assumption requires nodes to

be deployed in a “symmetric area.” The area can be finiteor infinite. For example, this assumption holds for thefollowing scenarios:• Sensors uniformly deployed in a square, and one

anchor at each corner of the square, as shown in Fig. 2.• All nodes uniformly and independently deployed in a

unit circle or a sphere;• All nodes uniformly and independently deployed in a

unit square or a cubic;• All nodes independently following a circularly-

symmetric Gaussian distribution.This assumption does not hold for the scenarios such as allnodes uniformly distributed in non-square rectangular.

The coordinate symmetry assumption requires links to beestablished in a “proper way.” There are three well-studiedmodels of random graphs.• Erdos–Renyi random graph (ERG) G(N, p): Nodes

are connected randomly regardless of the distance.Each link is included in the graph with probabilityp independent from every other link [41].

• Random geometric graph (RGG) G(N, r): Two nodesare connected if and only if the distance between themis at most a threshold r [42], [43].

• Random proximity graph (RPG) G(N, k): Each nodeconnects to its k nearest neighbors. These graphs arealso denoted k-NNG [43].

It is not hard to verify that as long as nodes are properlydeployed in a “symmetric area,” when links are establishedaccording to these random graphs, the coordinate symmetryassumption holds.

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3) Correct convergence of lateration: Even when thecooperative positioning problem is overdetermined, noisyrange measurements can lead to flip ambiguity in a badgeometry, as discussed in [44], [45]. The flip ambiguity cancause incorrect initial guess or divergence of some iterativelateration algorithms. The positioning accuracy discussedin this paper (as well as many previous papers) is aboutthe lateration errors under the assumption that the initialguess is correct and the lateration converges.

C. Metrics of connectivity

For all nodes n = 1, . . . , N , we define the followingdegrees:• Anchor degree: degA(n), the number of anchor nodes

incident to node n;• Sensor degree: degS(n), the number of sensor nodes

incident to node n;• Degree: deg(n) = degA(n)+degS(n), the number of

nodes incident to node n;We assume that there are no anchor-to-anchor links, i.e.,degA(n) = 0 for n = NS + 1, . . . , N , because anchor-to-anchor links are helpless when locations of anchors areperfectly known.

In graph theory, connectivity is usually described byvertex connectivity or edge connectivity: a graph isκ-vertex/edge-connected if it remains connected when-ever fewer than κ vertices/edges are removed [46].Unfortunately, vertex/edge connectivity mainly reflectssome “minimum” properties of connectivity, such asminn∈{1,...,N} deg(n) [46], and does not distinguish be-tween sensor and anchor nodes. This paper uses averagedegrees to characterize the overall connectivity of thenetwork. Average degrees are defined as

δ∗ =1

NS

NS∑n=1

deg∗(n), (4)

where the subscript ∗ can be blank, A, or S , for the averagedegree, average anchor degree, or average sensor degree,respectively.

Let KS and KA denote the number of sensor-to-sensorand anchor-to-sensor links in the network, respectively. It iseasy to verify the equalities K = KS+KA, NSδS = 2KS ,NSδA = KA, and δ = δS + δA.

D. List of notations

d dimensionalitydeg(n) degree of node ndegA(n) anchor degree of node ndegS(n) sensor degree of node nδ average degreeδA average anchor degreeδS average sensor degreeE set of all links, {e1, . . . , eK}ε range errors of all links, ε ∈ RKε positioning errors of all sensor nodes,

ε ∈ RNSd

F inverse of DOP matrix, F = GTGG graph (V,E)G geometry matrixH DOP matrix, H = (GTG)−1

I identity matrixik head of link ek = (ik, jk)jk tail of link ek = (ik, jk)K number of linksKA number of anchor-to-sensor linksKS number of sensor-to-sensor linksL Laplacian matrix of graph G,

dΞ = [Lij ]i,j∈{1,2,...,NS}m index of dimensions, m ∈ {1, . . . , d}.N number of nodes, N = NA +NSNA number of anchor nodesNS number of sensor nodesN (µ, σ2) Gaussian distribution with mean µ and

variance σ2

p positions of all sensor nodes i, p ∈ RNSd

pi position of node i, pi ∈ Rdrk actual distance of link ekρk distance measurement of link ekΣ covariance of range errorsσk standard deviation of range errors of link ekV set of all nodes, {1, . . . , N}vk unit vector denoting the direction of link ek,

vk = r−1k (pik − pjk)Ξ conditional expectation of F given certain

links, Ξ = Elocations(F |links)Ξ submatrix of Ξ, representting one coordinate

‖z‖ Euclidean norm of vector zA � B A−B is positive semidefinite

III. POSITIONING ACCURACY

Positioning is essentially an optimization problem thatfinds coordinate vectors pn ∈ Rd, n = 1, . . . , NS , suchthat for each ranging link ek = (ik, jk) ∈ E, the distancerk = ‖pik − pjk‖ is as close to the range measurement ρkas possible.

Assume that range errors follow a zero-mean Gaussiandistribution:

εk = ρk − rk ∼ N (0, σ2k), ∀k = 1, . . . ,K. (5)

The maximum-likelihood estimation of {pn}NSn=1 is equiv-

alent to the weighted least squares (LS) problem

arg max{pn}

NSn=1

f({ρk}Kk=1

∣∣ {pn}NSn=1

)= arg max{pn}

NSn=1

K∏k=1

1√2πσk

exp(− (‖pik − pjk‖ − ρk)2

2σ2k

)

= arg min{pn}

NSn=1

K∑k=1

(‖pik − pjk‖ − ρk)2

σ2k

,

(6)

where f(·) is the probability density function.

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The LS problem cannot be directly solved because thedistance rk = ‖pik − pjk‖ is a nonlinear function of thecoordinate vectors pik and pjk . Let r = (r1, r2, . . . , rK)T ∈RK and p = column{p1, p2, . . . , pNS

} ∈ RdNS . The first-order linear approximation of the distance function r(p(0)+∆p) with respect to an initial guess p(0) can be written as

r(p(0) + ∆p) = r(p(0)) +G∆p, (7)

where the geometry matrix G ∈ RK×dNS is given by

G =∂r

∂p

=

∂r1∂p1,1

. . . ∂r1∂p1,d

. . . ∂r1∂pNS,1

. . . ∂r1∂pNS,d

......

......

......

∂rK∂p1,1

. . . ∂rK∂p1,d

. . . ∂rK∂pNS,1

. . . ∂rK∂pNS,d

,(8)

where pi,m, m = 1, . . . , d, is the mth element of thecoordinate vector pi. Each element of the geometry matrixG is given by

Gk,(n−1)d+m =∂rk∂pn,m

=∂‖pik − pjk‖

∂pn,m

=

pik,m−pjk,m

‖pik−pjk‖= vk,m if n = ik,

pjk,m−pik,m

‖pik−pjk‖= −vk,m if n = jk,

0 otherwise.

(9)

Each row of G represents a link. There are only d nonzeroelements in a row for an anchor-to-sensor link, and thereare 2d nonzero elements for an sensor-to-sensor link. Giventhat each row of G has dNS elements, G is highly sparsewhen the network contains many sensor nodes.

When the network is localizable, G must be a tall matrix(i.e., K ≥ dNS [14], [15], [47]) with full column rank.Then, the weighted LS problem (6) can be solved bythe following iterative algorithm based on the Newton–Raphson method:

p(n+1) = p(n)+(GTΣ−1G)−1GTΣ−1[ρ−r(p(n))], (10)

where ρ = (ρ1, ρ2, . . . , ρK)T, Σ = Cov(ε, ε) is thecovariance of range errors, where ε = (ε1, . . . , εK)T. Thisiterative algorithm is an extension of the single-sensorpositioning algorithm widely used in GPS positioning [23].

When the initial guess p(0) is accurate enough and theiteration converges, the positioning errors ε have the fol-lowing relationship to the range errors ε = (ε1, . . . , εK)T:

ε = p(∞) − p = (GTΣ−1G)−1GTΣ−1(ρ− r(p))

= (GTΣ−1G)−1GTΣ−1(ρ− r)

= (GTΣ−1G)−1GTΣ−1ε.

(11)

The covariance of positioning errors is thus given by

Cov(ε, ε) = (GTΣ−1G)−1GTΣ−1 Cov(ε, ε)

Σ−1GT(GTΣ−1G)−1

= (GTΣ−1G)−1GTΣ−1ΣΣ−1GT(GTΣ−1G)−1

= (GTΣ−1G)−1.(12)

This has achieved the CR bound [26]–[31].If range measurement errors are iid, i.e., Σ = σ2I , where

σ = σ1 = · · · = σK , we can decouple range errors andgeometry, as given by

Cov(ε, ε) = (GTΣ−1G)−1 = σ2(GTG)−1. (13)

The matrix

H = (GTG)−1 ∈ RdNS×dNS (14)

is referred to as dilution of precision (DOP) matrix. DOPis a term widely used in satellite navigation specifying themultiplicative effect on positioning accuracy due to satellitegeometry2 [23]. The smaller DOP is, the better positioningaccuracy one would expect.

For cooperative positioning, DOP specifies the multi-plicative effect due to not only geometry the nodes butalso connectivity of the network. A diagonal elementH(n−1)d+m,(n−1)d+m is the DOP of coordinate m forsensor node n. The sum

GDOPn =

d∑m=1

H(n−1)d+m,(n−1)d+m (15)

is the geometric DOP (GDOP) of sensor node n. Thepositioning accuracy of sensor node n is given by

‖εn‖ = σ√

GDOPn. (16)

For the whole network, we define average GDOP (AG-DOP),

AGDOP =1

NS

NS∑n=1

GDOPn =tr(H)

NS, (17)

as a performance indicator of positioning accuracy due togeometry and connectivity.

For a network where nodes are deployed and connectedrandomly, the geometry matrix G is a random variable, so isAGDOP. The expectation of AGDOP (E-AGDOP) indicatesthe expected positioning accuracy because the root-mean-square positioning error is proportional to

√E-AGDOP, as

shown by

E( 1

NS

∑NS

n=1‖εn‖2

)= σ2 · E-AGDOP. (18)

We shall use E-AGDOP and its lower bound to studythe relationship between positioning accuracy and networkconnectivity in the remainder of this paper.

IV. LOWER BOUND ON E-AGDOP

In this section, we shall prove that [E(GTG)]−1 is alower bound on E[(GTG)−1]. Furthermore, we shall showthat it is possible to evaluate this lower bound analyti-cally for a random network (randomly-deployed nodes andrandomly-established links) that achieves a certain level ofconnectivity.

2The DOP is usually defined in the form of√

tr[(GTG)−1] [23], [34].In this paper, we define DOP in the form of tr[(GTG)−1] for simplicityin calculation and analysis.

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Theorem 1 (Lower bound on DOP matrix): For a ran-dom, localizable sensor network with a geometry matrixG defined in (8),

E[(GTG)−1] � [E(GTG)]−1, (19)

where the operator X � Y denotes that X − Y is positivesemidefinite.

Proof: Detailed in Appendix A.The matrix F = GTG is a function of node locations

and links, both of which have been assumed to be random.Let us calculate EF by the following two steps:

1) Ξ = Enodes(F |links), conditional expectation of F forrandomly-deployed nodes given certain links;

2) EF = Elinks(Ξ), expectation of Ξ for randomly-established links.

A. Step 1: Randomly-deployed nodes

Recall (9) which describes the elements in G. Note thatwhen link ek connects to node n, i.e., n ∈ {ik, jk},

d∑m=1

( ∂rk∂pn,m

)2=

∑dm=1(pik,m − pjk,m)2

‖pik − pjk‖2= 1. (20)

By the coordinate symmetry assumption (Section II-B2),we have

E( ∂rk∂pn,1

)2= E

( ∂rk∂pn,2

)2= · · · = E

( ∂rk∂pn,m

)2. (21)

To satisfy (20), we must have

E( ∂rk∂pn,m

)2=

1

d, ∀m = 1, . . . , d. (22)

Therefore, the elements of matrix F = {Fij} ∈RdNS×dNS have the conditional expectation

Ξij = Enodes(Fij |links) = EK∑k=1

∂rk∂pi,m1

∂rk∂pj,m2

=

1d deg(i) if i = j and m1 = m2,

− 1d if (i, j) ∈ E and m1 = m2,

0 otherwise,

(23)

where i = (i−1)d+m1, j = (j−1)d+m2, 1 ≤ m1,m2 ≤d. For instance, let us consider a very simple sensor networkshown in Fig. 3. The matrix Ξ for this network is given by

ΞFig. 3 =

32 0 − 1

2 0 − 12 0

0 32 0 − 1

2 0 − 12

− 12 0 3

2 0 0 00 − 1

2 0 32 0 0

− 12 0 0 0 3

2 00 − 1

2 0 0 0 32

. (24)

As shown by the red- and blue-colored elements in (24),we have Ξ = Ξ ⊗ I , where ⊗ denotes the Kronecker

1

23

5

6

4

Fig. 3. A localizable sensor network comprised of 6 nodes and 7 linksin 2 dimensions. Nodes 1 to 3 are sensors; nodes 4 to 6 are anchors(NS = 3, NA = 3, KS = 2, KA = 5). Eq. (24) shows the matrix Ξfor this network.

product, and I is the identity matrix of size d. The elementsof the matrix Ξ ∈ RNS×NS are given by

Ξij =

1d deg(i) if i = j,

− 1d if (i, j) ∈ E,

0 otherwise.(25)

For the sensor network shown by Fig. 3, the matrix Ξ isgiven by

ΞFig. 3 =

32 − 1

2 − 12

− 12

32 0

− 12 0 3

2

. (26)

The matrix Ξ (as well as Ξ) indicates a relationshipbetween positioning accuracy and graph Laplacians [48].Let L denote the Laplacian matrix of the graph G. It can beseen that Ξ = d−1[Lij ]i,j∈{1,2,...,NS}, i.e., dΞ is the matrixof L obtained by deleting its last NA rows and columnsthat are related to the anchor nodes.

The lower bound on E-AGDOP (LB-E-AGDOP) can becalculated by inverting F = EΞ or, equivalently, invertingF = E Ξ, because tr[(EΞ)−1] = d tr[(E Ξ)−1].

B. Step 2: Uniformly-established links

Given an average degree δ, the trace of Ξ is given by

tr(Ξ) =

NS∑i=1

Ξii =

NS∑i=1

deg(i)/d = NSδ/d. (27)

Given an average sensor degree δS , there are KS =NSδS/2 sensor-to-sensor links in the network, and thusΞ includes NSδS off-diagonal elements with a non-zerovalue of −1/d. Assume that the sensor-to-sensor links arechosen uniformly at random from the set {(i, j)|1 ≤ i <j ≤ NS , i, j ∈ Z}. Then, each off-diagonal element Ξij ,i 6= j satisfies the Bernoulli distribution

Ξij =

{−1/d with probability δS

NS−1 ,

0 with probability 1− δSNS−1 .

(28)

Then, the expectation of F is given by

E Fij = Elinks(Ξij) =

{δd if i = j,

− δSd(NS−1) otherwise.

(29)

Appendix B shows that

tr[(E F )−1] =NSη

(1 +

ζ

1−NSζ

), (30)

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where η = d−1[δ + δS/(NS − 1)] and ζ = δS/[δ(NS −1) + δS ]. Therefore, LB-E-AGDOP is given by

LB-E-AGDOP =tr[(EF )−1]

NS=d tr[(E F )−1]

NS

=d

η

(1 +

1

ζ−1 −NS

)=

d2

δ + δS/(NS − 1)

(1 +

δSδA(NS − 1)

)=d2

δ

NS − 1 + δS/δANS − 1 + δS/δ

.

(31)

Thus far, we have obtained a closed-form expression forLB-E-AGDOP. It depends on two parameters of networkconnectivity, δS and δA (note δ = δS + δA), and oneparameter of network size, NS . In (31), the first termd2/δ shows that LB-E-AGDOP is approximately inverselyproportional to the average degree, and grows quadrati-cally with dimensionality. The second term (NS − 1 +δS/δA)

/(NS − 1 + δS/δ) shows that a higher ratio of

average anchor degree to average sensor degree leads tobetter positioning accuracy. However, this effect diminisheswhen number of sensor nodes increase. As NS → ∞, theLB-E-AGDOP approaches d2/δ regardless of the ratio ofaverage anchor degree to average sensor degree.

V. LB-E-AGDOP AND THE BEST ACHIEVABLEACCURACY

The previous section has proven that the LB-E-AGDOPgiven by (31) is a lower bound on E-AGDOP. In thissection, we shall show that LB-E-AGDOP describes thebest achievable accuracy with certain network connectivity.We first prove that LB-E-AGDOP is equal to the minimumAGDOP for one sensor node. For multiple sensor nodes, weuse several numerical examples to show that LB-E-AGDOPis less than and very close to the minimum AGDOP.

A. Minimum AGDOP for one sensor node

Let us first consider the simplest case that there is onlyone sensor node, i.e., NS = 1. By (31), the lower boundbecomes

LB-E-AGDOP =d2

δ

δS/δAδS/δ

=d2

δA=

d2

NA. (32)

In this subsection, we shall show that the lower boundd2/NA represents the best achievable performance.

When NS = 1, the geometry matrix can be written as

G =

cos θ1 sin θ1cos θ2 sin θ2

......

cos θNAsin θNA

, (33)

where θi, i = 1, . . . , NA is the angular coordinate of anchornode i in the polar coordinate system poled at the sensornode. Fig. 4 shows a scenario of NA = 3.

Noting that

GTG =

[ ∑NA

i=1 cos2 θi∑NA

i=1 cos θi sin θi∑NA

i=1 cos θi sin θi∑NA

i=1 sin2 θi

], (34)

we obtain a closed-form expression of (GTG)−1 as

(GTG)−1 =1

det(GTG)·[ ∑NA

i=1 sin2 θi −∑NA

i=1 cos θi sin θi−∑NA

i=1 cos θi sin θi∑NA

i=1 cos2 θi

],

(35)

where the determinant of GTG is given by

det(GTG) =

(NA∑i=1

cos2 θi

)(NA∑i=1

sin2 θi

)

−(NA∑i=1

cos θi sin θi

)2

=

NA∑i=1

NA∑j=1

(cos2 θi sin2 θj

− cos θi sin θi cos θj sin θj)

=

NA∑i=1

NA∑j=1

cos θi sin θj sin(θj − θi)

=∑

1≤i<j≤NA

sin2(θj − θi).

(36)

Therefore, the AGDOP is given by

tr[(GTG)−1] =NA∑

1≤i<j≤NAsin2(θj − θi)

, (37)

which agrees with the result obtained in [49], [50]. Theminimum AGDOP is attained when θi = 2πi/NA [50].The corresponding maximum of S is given by∑1≤i<j≤NA

sin2(θj − θi) =∑

1≤i<j≤NA

sin2 (j − i)2πNA

=1

2

NA∑i=1

NA∑j=1

sin2 (j − i)2πNA

=1

2

NA∑i=1

NA/2 =N2A

4.

(38)

Therefore, the minimum AGDOP is 4/NA, equal to thelower bound given by (32).

θ1θ2

θ3

Fig. 4. Geometry of one sensor node and three anchor nodes.

8

Fig. 5 compares the LB-E-AGDOP calculated from (31)and the AGDOP obtained from simulations with the pa-rameters NS = 1, NA = 3, 4, . . . , 9. The green solidcurve shows the LB-E-AGDOP. The box-and-whisker plotshows the sample minimum, lower quartile, median, upperquartile, and sample maximum of the AGDOP. The red plusmarks denote statistical outliers. It can be seen that our LB-E-AGDOP is equal to the sample minimum, and is closerto the median when NA is greater.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3 4 5 6 7 8 9N

A

AG

DO

P

LB−E−AGDOP

Fig. 5. Comparison between LB-E-AGDOP from (31) and AGDOP fromsimulations with the parameters NS = 1, NA = 3, 4, . . . , 9. In thesimulation, the anchors are randomly distributed so that the bearing anglesare independently uniformly distributed in [0, 2π). The green solid curveshows the LB-E-AGDOP. The box-and-whisker plot shows the sampleminimum (lower whisker, black), lower quartile (lower edge of the box,blue), median (central mark, red), upper quartile (upper edge of the box,blue), and sample maximum (upper whisker, black; outliers excluded).Our LB-E-AGDOP is equal to the sample minimum, and is closer to themedian when NA is greater.

B. Minimum AGDOP for multiple sensor nodes

For multiple sensor nodes, the minimum AGDOP canbe hardly derived from a theoretical analysis. Instead, wepresent several results obtained from numerical optimiza-tion. From the following results it can be seen that formultiple sensor nodes, our LB-E-AGDOP is less than theminimum achievable AGDOP. The gap between LB-E-AGDOP and the minimum AGDOP is smaller when theLB-E-AGDOP is smaller.

Case 1: NS = 2, δS = 1, δA = 2: One of the bestgeometries is given by

98.68◦

The geometry has up-down and left-right reflection sym-metry. The minimum AGDOP is 1.633, greater than theLB-E-AGDOP 1.500.

Case 2: NS = 2, δS = 1, δA = 3: One of the bestgeometries is given by

TABLE ILB-E-AGDOP ESTABLISHES A LOWER BOUND ON AGDOP

NS NA δS δA minimum AGDOP LB-E-AGDOP

1 n 0 n 4/n 4/n

2 4 1 2 1.633 1.500

2 6 1 3 1.124 1.067

3 3 2 1 2.667 2.000

3 6 2 2 1.313 1.200

112.59◦

The geometry has up-down and left-right reflection symme-try. The minimum AGDOP is 1.124, slightly greater thanthe LB-E-AGDOP 1.067.

Case 3: NS = 3, δS = 2, δA = 1: One of the bestgeometries is given by

60.00◦

The geometry has ±120◦ rotational symmetry. The min-imum AGDOP is 2.667, greater than the LB-E-AGDOP2.000. It should be noted that the connectivity of this casedoes not ensure unique localizability [5], [15].

Case 4: NS = 3, δS = 2, δA = 2: One of the bestgeometries is given by

104.15◦

The geometry has left-right reflection symmetry and ±120◦

rotational symmetry. The minimum AGDOP is 1.313,slightly greater than the LB-E-AGDOP 1.200.

Table I summaries the minimum AGDOP and LB-E-AGDOP calculated for the cases mentioned in this section.Although the LB-E-AGDOP is derived from the expecta-tion of AGDOP, we observe that LB-E-AGDOP may alsoestablish a lower bound on AGDOP.

VI. SIMULATION RESULTS

In this section, we conduct numerical simulations toverify the LB-E-AGDOP formula obtained in Sections IVand V. All our simulations are based on two dimensions(d = 2). The first set of simulations (as presented in Fig. 6)assumes• Sensor nodes are independently and uniformly dis-

tributed in the unit square [0, 1]× [0, 1];

9

10−1

100

10−1

100

101

ERGE

−A

GD

OP

, sim

ulat

ed

10−1

100

10−1

100

101

RGG

LB−E−AGDOP, theoretical10

−110

010

−1

100

101

RPG

NS = 8

NS = 16

NS = 24

NS = 32

(a) Comparison between the LB-E-AGDOP from (31) and the sample mean of AGDOP from simulations.

10−1

100

10−1

100

101

ERG

Min

imum

AG

DO

P, s

imul

ated

10−1

100

10−1

100

101

RGG

LB−E−AGDOP, theoretical10

−110

010

−1

100

101

RPG

NS = 8

NS = 16

NS = 24

NS = 32

(b) Comparison between the LB-E-AGDOP from (31) and the sample minimum of AGDOP from simulations.

Fig. 6. Simulation results based on 8 randomly deployed anchors and NS ∈ {8, 16, 24, 32} randomly deployed sensors. Links are established basedon ERG, RGG, and RPG, respectively. Each marker represents a network configuration with certain NS , δS and δA. The analytical lower bound wasverified, as all the markers are above and close to the y = x black line.

• Four anchors (NA = 4) located at the corners of theunit square, i.e., (0, 0), (0, 1), (1, 0), and (1, 1);

• Links are established according to ERG, RGG, andRPG.

Fig. 2 has shown a snapshot excerpted from the simulationof the RGG model with the parameters r = 0.495 andNS = 16.

The second set of simulations (as presented in Fig. 7)is based on a different setting in which both sensors andanchors are randomly distributed in an unlimited area.

• Sensor nodes independently follow the Gaussian dis-

tribution N([

00

],

[1 00 1

]);

• Eight anchors (NA = 8) independently follow the

Gaussian distribution N([

00

],

[4 00 4

]).

• Links are established according to RGG.

A. Comparison between the LB-E-AGDOP and E-AGDOP

Figure 6 (a) shows the results from the first set of sim-ulations, in which there are four fixed anchors. The figurecompares the LB-E-AGDOP calculated using (31) and thesample mean of AGDOP obtained from simulations. Thesimulations are based on the parameters NS = 8, 16, 24, 32

10

and the three random graph models. Each marker in Fig. 6(a) represents a network configuration with certain NS , δSand δA.

Our analytical lower bound is verified by the simulationresults as no markers are below the black line y = x.Although the relationship between LB-E-AGDOP and E-AGDOP is nonlinear, if two different network configura-tions result in the same LB-E-AGDOP value, they alsolead to very close E-AGDOP values. Therefore, our derivedlower bound, LB-E-AGDOP, can be used as a performanceindicator of the expected accuracy of range-based cooper-ative positioning in random sensor networks.

Furthermore, it can be seen that our lower bound is ver-ified for all the three random graph models, with resultingsimilar gaps between LB-E-AGDOP and E-AGDOP. Thisdemonstrates that our lower bound is applicable to variousrandom graph models, as long as the coordinate symmetryassumption holds.

Fig. 6 (a) also shows that the lower bound is tighterwhen LB-E-AGDOP is smaller. When LB-E-AGDOP isgreater than 1, the network connectivity is so low thatthe distribution of AGDOP has a very long tail. Thus, E-AGDOP does not exist. To see this, consider the case ofone sensor and NA anchors. By (37), we can verify that

E-AGDOP = E(tr[(GTG)−1]

)<∞ (39)

if and only if NA ≥ 4, which means LB-E-AGDOP ≤4/NA = 1. Therefore, if LB-E-AGDOP is greater than 1,then E-AGDOP does not exist. In our simulation, the non-exist E-AGDOP appears to diverge exponentially from thelower bound.

Figure 7 (a) shows the results from the second set ofsimulations, in which there are eight randomly deployedanchors. Similar to the results from the first set of simu-lations, the markers are above and close to the black liney = x, especially when LB-E-AGDOP is small.

B. Comparison between the LB-E-AGDOP and minimumAGDOP

Figure 6 (b) and Fig. 7 (b) compare the LB-E-AGDOPcalculated using (31) and the sample minimum of AGDOPobtained from simulations. Each marker in the figuresrepresents a network configuration with certain NS , δS andδA. Comparing these figures with Fig. 6 (a) to Fig. 7 (a),we can see that the analytical lower bound matches theminimum AGDOP better than matches the E-AGDOP.

Similar to our discussion about Fig. 6, it can be seen that(a) the analytical lower bound is verified by the simulationresults; (b) the lower bound can be used as a performanceindicator of the best accuracy of range-based cooperativepositioning in random sensor networks; (c) the lower boundworks for all the three random graph models as well asother models that satisfy the conditions in Section II-B;and (d) the lower bound is tighter when LB-E-AGDOP issmaller.

10−1

100

10−1

100

101

E−

AG

DO

P, s

imul

ated

LB−E−AGDOP, theoretical

NS = 8

NS = 16

NS = 24

(a) Comparison between the LB-E-AGDOP from (31) and thesample mean of AGDOP from simulations.

10−1

100

10−1

100

101

Min

imum

AG

DO

P, s

imul

ated

LB−E−AGDOP, theoretical

NS = 8

NS = 16

NS = 24

(b) Comparison between the LB-E-AGDOP from (31) and thesample minimum of AGDOP from simulations.

Fig. 7. Simulation results based on 8 randomly deployed anchors andNS ∈ {8, 16, 24} randomly deployed sensors. Links are established basedon RGG. The analytical lower bound was verified, as all the dots are aboveand close to the y = x black line.

VII. CONCLUSION

This paper has presented a generalized theory thatcharacterizes the connection between system parameters(network connectivity and size) and positioning accuracyin range-based cooperative positioning. We have proven anovel lower bound on expectation of AGDOP and deriveda closed-form formula (31) that relates LB-E-AGDOP andE-AGDOP to only three parameters: average sensor degreeδS , average anchor degree δA, and number of sensor nodesN . The formula shows that LB-E-AGDOP is approximatelyinversely proportional to the average degree, and a higherratio of average anchor degree to average sensor degreeleads to better positioning accuracy.

The simulation results agree with the theoretical results,and shown that (a) the lower bound are applicable to variousrandom graph models that satisfy our coordinate symmetryassumption; (b) E-AGDOP and minimum AGDOP are very

11

close, and both of them can be approximated by LB-E-AGDOP when LB-E-AGDOP is small. The theory andsimulation results presented in this paper provide guidelineson the design of range-based positioning schemes and thedeployment of sensor networks.

APPENDIX APROOF OF THEOREM 1

There are a few approaches to proving Theorem 1. Oneof the simplest proofs is based on a recent result about theCauchy–Schwarz inequality for the expectation of randommatrices [51], [52]:

Lemma 1 (Cauchy–Schwarz inequality [51], [52]): LetA ∈ Rn×p and B ∈ Rn×p be random matrices such thatE ‖A‖2 <∞, E ‖B‖2 <∞, and E(ATA) is non-singular.Then

E(BTB) � E(BTA)[E(ATA)]−1 E(ATB). (40)

By the definition of geometry matrix in (9), we alwayshave ‖G‖2 < ∞. For a localizable network, the geometrymatrix G must be a tall matrix with full column rank. Thus,GTG is never singular, and (GTG)−1 <∞.

With the substitutions A = G and B = G(GTG)−1 intothe above inequality, we have

U = E[(GTG)−1] � V = [E(GTG)]−1, (41)

which already proves Theorem 1.Since the diagonal elements of a positive semidefinite

matrix must be non-negative, we have

Uii ≥ Vii, ∀i = 1, . . . , dNS , (42)

where U = [Uij ] and V = [Vij ]. In particular, theexpectation of GDOP, tr(U), has a lower bound tr(V ).

APPENDIX BPROOF OF EQ. (30)

Lemma 2 (Sherman–Morrison formula [53]): SupposeA is an invertible square matrix, and u and v are vectors.Suppose furthermore that 1 + vTA−1u 6= 0. Then theSherman–Morrison formula states that

(A+ uvT)−1 = A−1 − A−1uvTA−1

1 + vTA−1u. (43)

With η = d−1[δ + δS/(NS − 1)], (29) can be written as

η−1 E F = I − uuT, (44)

where u =√ζ(1, 1, . . . , 1)T, and ζ = δS/[δ(NS−1)+δS ].

Letting u = −v =√ζ(1, 1, . . . , 1)T, by the Sherman–

Morrison formula we have

(I − uuT)−1 = I + uuT/(1− uTu), (45)

and thus

η tr[(E F )−1

]= tr[(I − uuT)−1]

= NS +NSζ/(1−NSζ).(46)

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Liang Heng received his B.S. and M.S. de-grees in electrical engineering from TsinghuaUniversity, Beijing, China in 2006 and 2008. Hereceived his PhD degree in electrical engineeringfrom Stanford University in 2012. He was a post-doctoral research associate in the Department ofAerospace Engineering, University of Illinois atUrbana-Champaign. He worked at Google, Inc.and Tesla Motors, Inc. from May 2014 to March2016. He is currently a software architect atBaidu USA, LLC. His research interests include

cooperative navigation, satellite navigation, and autonomous driving. Heis a member of the Institute of Navigation (ION).

Grace Xingxin Gao received the B.S. degree inmechanical engineering and the M.S. degree inelectrical engineering from Tsinghua University,Beijing, China in 2001 and 2003. She receivedthe PhD degree in electrical engineering fromStanford University in 2008. From 2008 to 2012,she was a research associate at Stanford Univer-sity. Since 2012, she has been with Universityof Illinois at Urbana-Champaign, where she ispresently an assistant professor in the AerospaceEngineering Department. Her research interests

are systems, signals, control, and robotics.