Beacon Placement for Range-Based Indoor...

Beacon Placement for Range-BasedIndoor Localization

Niranjini Rajagopal, Sindhura Chayapathy, Bruno Sinopoli, Anthony RoweElectrical and Computer Engineering Department

Carnegie Mellon UniversityEmail: niranjir,schayapa,brunos,[email protected]

Abstract—In this paper, we address the problem of range-based beacon placement given a floor plan to support indoorlocalization systems. Existing approaches for trilateration requirethree or more beacons to determine a unique position solution.We show that with prior knowledge of the map and a modelof beacon coverage, it is possible to uniquely localize with onlytwo beacons. This not only reduces installation cost by requiringfewer nodes, but can also improve robustness. One of the mainchallenges with respect to beacon placement algorithms is defininga metric for estimating performance. We propose augmentingthe commonly used Geometric Dilution of Precision (GDOP)metric to account for indoor spaces. We then use this enhancedGDOP metric as part of a toolchain to compare various beaconplacement algorithms in terms of coverage and expected accuracy.When applied to a set of real floor plans, our approach is ableto reduce the number of beacons between 22% and 60% (33%on an average) as compared to standard trilateration.

I. INTRODUCTION

Beacon technologies for indoor localization are rapidlydecreasing in price while simultaneously improving in termsof ranging accuracy and energy requirements. Commercialsolutions range in cost and performance from Bluetooth Low-Energy (BLE) modules that use Received Signal Strength(RSS) to estimate distance to more precise Time of Flight(TOF) ranging chipsets like [1], Ultra-Wide Band (UWB) [2]and WiFi [3] radios. Other distance sensing technologies likelaser ranging and ultrasonic TOF systems are both decreasingin cost and becoming more common in consumer productslike digital range finders and consumer robotics products. Inorder to localize solely based on range to beacons, a systemwill typically perform trilateration between the receiver/mobiledevice/tag and three or more fixed beacons that are at knownpositions. The beacon deployment itself is performed manuallywhere experts working with the system determine the place-ment of beacons with empirical guidelines for spacing andcoverage.

When deploying beacons, an installer faces the conflictingobjectives of reducing the number of beacons to be placedand increasing system coverage, accuracy and resilience. Theproblem of placement indoors differs from Global Position-ing System (GPS) in two ways. First, GPS satellites coverenormous areas, whereas indoor systems require comparativelydense placement of beacons, making it challenging to reducethe number of beacons. Second, the satellites are positionedsymmetrically around the earth and a subset of them areassumed to be within Line-of-Sight (LOS) at all locations.This simplifies estimating coverage and computing metrics like

Geometric Dilution of Precision (GDOP). Indoor systems oftenhave irregular propagation models and barriers that complicatedeployment, making it challenging to define a metric to easilycompare beacon configurations.

There are also several practical issues that need to beconsidered when deploying nodes indoors. First, the coverageof a beacon depends on the ranging-technologies which vary interms of the maximum range and signal permeability throughwalls. For instance, acoustic signals are confined to walls,while RF signals exhibit high penetration. Second, indoorspaces have rich semantics that lead to different localizationaccuracy requirement across different areas. For instance,room-level accuracy might be sufficient in certain areas, whilesub-meter accuracy within a large room might be requiredfor audio guides in museums. Third, physical factors oftenconstrain the deployment. For instance, it may be preferableto place beacons with convenient access to power outlets orwhere they do not disrupt the aesthetics of the space.

In this paper, we address the question - For a range-basedlocalization system, given the floor plan, where should thebeacons be placed? In order to answer this question, we firstdiscuss how we can compare two beacon configurations. Aconfiguration refers to a unique placement of beacons for agiven floor plan.

One of the key insights in our work is that beacon coverageinformation can be used to prune incorrect location solutionsin under-defined beacon configurations. When range measure-ments are received from two beacons, it normally results intwo possible location solutions. It is possible to resolve whichis the more likely solution if each location is covered bya different set of beacons. For example, if one of the twolocation solutions does not receive a signal that was likely tobe received from a nearby beacon, it indicates that either thebeacon was blocked or the alternate solution is more likely. Formost of our results, we assume an ideal ray-tracing coveragemodel since our primary target system is based on acousticbeacons. However, this concept is also applicable to othercoverage types given a suitable propagation model. We referto regions that can be localized in this manner without anyambiguity as Uniquely Localizable and introduce a functionUL that indicates if a location is unambiguously localizedby the beacons in coverage. We use the percentage of areauniquely localizable in a floor plan to quantify the quality ofa beacon configuration, purely based on coverage.

For any realistic system with noise in the range measure-ments, in addition to beacon coverage, the beacon geometryalso affects the location accuracy. One of our main contribu-tions is that we adapt the concept of GDOP to quantify the978-1-5090-2425-4/16/$31.00 c© 2016 IEEE

2016 International Conference on Indoor Positioning and Indoor Navigation (IPIN), 4-7 October 2016, Alcala de Henares, Spain

location accuracy provided by the beacons. GDOP is a unitlessquantity that is a function of the geometry between the beaconsand the target, often used to estimate the expected accuracyof GPS due to the location of satellites [4]. When used asa metric for evaluating indoor accuracy, GDOP has two maindrawbacks. First, it cannot accurately capture cases where thereare multiple ambiguous solutions. In these cases, a localizationsolver could be completely incorrect and yet an evaluationfunction would still indicate a confident reading based ongeometry. Second, there are circumstances that cause thestandard GDOP metric to grow towards infinity. This makes itdifficult to normalize over multiple competing configurations.To overcome the first problem, we define a function GDOPULthat is a modified GDOP that also incorporates the UniqueLocalizability information. For the second problem, we usea function of the Cumulative Distribution Function (CDF)of the GDOPUL curve to quantify the quality of a beaconconfiguration across the floor plan.

We integrate our evaluation metric into a software toolchainthat is able to evaluate a variety of beacon placement algo-rithms given a floor plan. The system first pre-process the floorplan and segments it into regions based on the coverage of can-didate beacon locations. It then uses a greedy approach to placebeacons in the largest regions first. We then apply algorithmsfor placing beacons to optimize for (1) Unique LocalizabilityUL and (2) both GDOP and Unique Localizability GDOPUL.The objective in both cases is to place the minimal number ofbeacons. In the case of GDOPUL, the algorithm also accountsfor predicted accuracy. We then evaluate this across ten floorplans and the number of beacons placed by our scheme is22% to 60% (33% on an average) lesser than when coverageby three or more beacons is required. In practice, this resultsin a large saving in infrastructure and installation cost whenprovisioning large indoor spaces with range-based beacons.

In summary, this paper makes four main contributions:

1) We introduce and define the concept of Unique Lo-calizability and show how we can localize with twobeacons instead of three by exploiting the beaconcoverage information. We introduce GDOPUL, amodified GDOP metric that incorporates the ULinformation.

2) We show how we can quantify the quality of abeacon configuration based on these metrics, wherethe quality is a function of the localization accuracyacross the floor plan.

3) We present an algorithm that provides beacon loca-tions and compare the number of beacons againstminimal placement.

4) We design a toolchain that implements the algorithmwhere the user can provide or draw a floor plan,and specify the design space, such as the accuracyrequirement, number of beacons available and beaconcoverage information.

II. RELATED WORK

In order to find beacon configurations that minimize thelocalization error at a given target location, several authorshave studied optimal beacon geometries that minimize theCramer-Rao bound (CRB) [5] or the GDOP [6], [7]. For N ≥ 3beacons, optimal placement can be obtained when the adjacentbeacons subtend an angle of 2π

N or πN about the target, and for

N = 2 beacons, optimal placement is when the two beaconssubtend an angle of π

2 about the target [8], [9].Though the optimal placement for a single target is well

understood, the optimal placement for multiple target loca-tions, a target trace and a target area for range-only systemsare still open problems. [10] shows that the generalized sensorplacement problem is at least as hard as the k-center problem,which is NP-complete. They present a solution based oninteger linear programming for triangulation-based system,but the complexity for trilateration is similar. Due to thecomplexity of the problem, most proposed solutions involvedesigning heuristics and utilizing optimization techniques. [11]explores computational-geometry based heuristics for deter-mining location of beacons given a predefined trace of a robot.They define a utility metric that is a function of the numberof beacons in range and the convex hull of beacons. Theyevaluate the quality of a configuration using the integral ofthe utility function. Our work is similar in the respect that wealso account for the coverage of the beacons but differs sincethe metric they propose is purely based on coverage and notaccuracy. [12] proposes a heuristic for Quality of Trilaterationthat is based on the probability that a location estimate iswithin a given radius of the true location. However, thisapproach is for a network of nodes where a node localizes itselfwith respect to three other nodes given a prior distribution onthe expected inter-node distances. [13] and [14] propose theirown metrics to quantify the quality of a beacon placement,which is a function of the GDOP over the desired localizationarea and ratio of area which cannot be localized as well aspropose a placement algorithms to optimize the metric. [13]classifies areas that are localizable based on whether the GDOPis above or below a threshold and use a genetic algorithmapproach for placement. [14] considers the average GDOPover the areas that are localizable and implements a meta-heuristic optimization strategy. These approaches are similarto our work, but do not consider the minimal beacon countplacement for areas and require three or more beacons toprovide coverage.

[15] looks at the problem of beacon placement whilelocalizing with range-based beacons with limited field of view.They propose a heuristic based on the GDOP being belowa certain threshold and propose a placement algorithm forthe same, but they consider only a single room case and thebeacons are placed on the periphery of the room, avoiding anyambiguity in solution.

The Art Gallery Problem from computational geometry isclosely related to our beacon placement problem. Its goal is tofind the minimal set of guards such that every region in a floorplan is covered by at least one guard. The range-based beaconplacement problem discussed in this paper differs since werequire coverage by multiple beacons to localize the target, andthe localization accuracy is a function of the beacon locations,even when sufficient coverage is achieved.

III. QUANTIFYING THE QUALITY OF A BEACONCONFIGURATION

In this section, we discuss our approach for evaluatingbeacon placement in order to compare candidate placementsin terms of expected localization accuracy. We formulate theproblem with the following inputs:

• X = {Xi|i ∈ [1, Nx]}: Set of points in the discretized

2


A

A’

B1 B2

(a)

A

A’

B1 B2

B1

B3

CB:{B1,B2}

B2

A

A’

(ii)(i)

CB:{B1,B2,B3}

CB:{B1,B2}

CB:{}

(b)

A’

B1 B2

B3

A

CB={B1,B2}

CB={B1,B2,B3}

CB={B1}

CB={B2}

CB={B3}CB={B2,B3}CB={B1,B3}

(c)

CB:{}

A

B1

Z

A’ Z’

CB:{B3}

CB:{B1}

B2

B3

CB:{B2}

CB:{B1,B2}

CB:{}

CB:{B2,B3}

(d)

Figure 1. Localizing with two beacons based on beacon coverage set. (a)Localization ambiguity with two beacons (b) Ideal ray-tracing: Infinite range,permeability =0 (c) Unit disk coverage: Fixed finite range, permeability=1(d) Arbitrary coverage: 0<Permeability<1

floor plan, where Xi denotes the (x, y) coordinates ofthe ith point, and Nx is the total number of points. Inour implementation, we discretize the real-world floorplans using a 20cm× 20cm grid.

• P = {Pj |j ∈ [1, Np]}: Set of candidate beaconlocations, where Np is the total number of locationsconsidered as beacon candidates. In our implementa-tion, we consider all corners or vertices in the floorplan as candidate beacon locations.

• CP (Xi)∀i: Set of candidate beacon locations thatprovide coverage to each location Xi, CP (Xi) ⊆ P .

Our objective is to find a beacon configuration, or a subsetof the candidate locations B = {Bj |j ∈ [1, Nb], B ⊆ P,Nb ≤Np}, such that the localization accuracy at Xi,∀i satisfiescertain criteria based on how we define the quality of a beaconconfiguration. The set of beacons that provide coverage toXi, is denoted as CB(Xi), CB(Xi) ⊆ B. In Section III-A,we show how we localize with two beacons by using thebeacon coverage information and introduce the concept ofUnique Localization. In Section III-B, we provide the back-ground and definition of GDOP and describe its significancein quantifying the location accuracy. In Section III-C, wecombine the concepts of Unique Localizability and GDOPand propose a new function GDOPUL. The UL, GDOP andGDOPUL metrics are defined for a single location (Xi). InSection III-D, we propose heuristic metrics for quantifying thequality of a beacon configuration which is a function of theoverall localization accuracy provided by the beacons acrossthe entire floor plan (Xi,∀i). In Section IV, we propose abeacon placement algorithm built on these concepts.

A. Unique LocalizabilityIn this section, we formalize our two beacon localization

using the concept of Unique Localizability.Coverage models and localizing with two beacons:

Consider the two-beacon scenario in Figure 1(a), with beaconsB1 and B2, with the receiver’s true location as A. The

location A′ also receives the same range measurements andthus we cannot disambiguate between A and A′. However, bymaking use of beacon coverage information, it is possible todisambiguate between the two locations under the conditionthat the set of beacons providing coverage is different for Aand A′. We explain this with three types of coverage models (1)ray-tracing (2) unit disk and (3) arbitrary coverage, as seen inFigure 1(b)-1(d). The coverage of a beacon is technology andsignal-dependent and can vary in terms of range and the pen-etration/permeability through walls. Figure 1(b) shows idealray-tracing coverage, where the beacon has no permeabilitythrough walls and the range can be considered to be infinite,or larger than the maximum distance between the beacon andthe walls. This type of coverage is common in acoustic orultrasonic ranging systems. In Figure 1(b)(i), we see that theregions inside the floor plan have CB = {B1, B2} and theregions outside the floor plan have CB = {}. If the truelocation is A and the ranges from B1 and B2 are received,we can disambiguate the location A from A′ since A′ cannotreceive measurements from B1 and B2. In Figure 1(b)(ii),all regions are covered by B1 and B2 and in addition, someregions are covered by B3. Here, though both A and A′ receivethe same ranges from B1 and B2, we can disambiguate themsince A′ would receive a range measurement from B3 aswell. Figure 1(c) shows a unit disk coverage model whereall points within a finite distance of the beacon are within itscoverage. This is common for proximity-based technologiessuch as BLE, or a system that provides a range measurementwhen the SNR is above a threshold and the SNR is a functionof distance from the beacon. Figure 1(d) shows an arbitrarycoverage model where the range reduces when the signalpenetrates through walls, as is common for several RF-basedtechnologies. In both Figure 1(c) and Figure 1(d), we canresolve the location A from A′, and Z from Z ′ since the twoambiguous locations have a different beacon coverage. In thismanner, if the localization solver incorporates the coverageinformation of the beacons, we can localize using two beacons,instead of three. Though a simple concept, this is the keyinsight that results in around 33% reduction in the numberof beacons required across a building floor plan by usingour approach for beacon placement. Note that we currentlyonly consider the coverage to be deterministic and binary (incoverage, out of coverage) and not statistical.

Experimental validation of ray-tracing: We adopt a2D ray-tracing coverage model that most closely matchesan acoustic ranging system. Figure 3 shows experimentallyobtained results using ultrasonic beacons [16] ranging to aphone, in two different real-world deployments in buildingson our university campus. Figure 3(a) and Figure 3(b) showthe coverage of a single ultrasonic beacon with measurementstaken uniformly across the entire floor plan. The beacon undertest is shown as a blue square. Locations that received a mea-surement from the beacon are shown with a circle, with radiusproportional to the received range measurement. The locationsin LOS and NLOS are shown in green and red respectively.In practice, the ultrasonic signals reflect off walls resulting inNLOS measurements, but the ideal ray tracing model assumesno NLOS measurements are received. Although the ray tracingassumption is not perfect, there are techniques to distinguishLOS from NLOS measurements, which we can apply beforesolving for the final location [16], [17].

3


B1 B2CB:{B1,B2,B3}

B3

A

A’B1

B2

CB:{B1,B2}

B1 B3CB:{B1,B2,B3}

B2

A

A’

B1 B2

B1

B2

Configura4on

2

3

4

5

GDOPCoverage GDOPUL

1

CB:{B1,B2}

CB:{B1,B2}

CB:{}

(a) Rectangle floor plan

Configura)on

2

3

GDOPCoverage GDOPUL

1

B1 B2

B3

B1 B2

CB:{B1}

CB:{B1,B2}

CB:{B1,B2}

CB:{B1,B3}

CB:{B1,B2,B3}

A

A’B1

B2

(b) L-shaped floor plan

Figure 2. Comparison of multiple beacon configurations. Legend forCoverage column is in Table I

Unique Localizability: We define a location to beUniquely Localizable if in the absence of noise, it can belocalized without any ambiguity, when range measurementsare received from the beacons that provide coverage to thelocation. We define the function UL(Xi, CB(Xi)) ∈ {0, 1}which has a binary output, as:

UL(Xi, CB(Xi)) =

1,#CB(Xi) ≥ 3

1,#CB(Xi) = 2, CB(Xi) 6= CB(X′i)

0,#CB(Xi) = 2, CB(Xi) = CB(X′i)

0,#CB(Xi) ≤ 1

where #CB denotes the cardinality of the set CB andX ′i is the reflection of Xi about the line joining the twobeacons in the set CB(Xi). Note that X ′i is defined only when#CB(Xi) = 2. We subsequently, use the notation UL(Xi)instead of UL(Xi, CB(Xi)).

Table I shows the color coding for the beacon coveragethat we have used in Figure 2 and Figure 5. The fourthcolumn shows the value of the binary UL function. Notethat the algorithm does not distinguish between Class 2 and3 points, but they are shown here for visual purposes. Class2 points have the ambiguous location outside of the floorplan and Class 3 points have the ambiguous location insidebut covered by different beacons. We see from the second

Class Colorcode Descrip.on UL1 #CB(Xi)≥3 12 #CB(Xi)=2,CB(X’i)={} 13 #CB(Xi)=2,CB(X’i)≠CB(Xi),CB(X’i)≠{} 14 #CB(Xi)=2,CB(X’i)=CB(Xi) 05 #CB(Xi)=1 06 #CB(Xi)=0 0

Table I. COVERAGE CLASS LEGEND

column of Figure 2(a) that configurations 1, 2, 4 and 5 haveall locations to be Uniquely Localizable either due to coverageby three beacons or in case of two beacons, the ambiguouslocation is outside the floor plan. In ideal scenarios, any ofthese configurations will provide exact location estimates andthe configurations with fewer beacons then would naturally bedesirable. However, in realistic scenarios, these configurationswould not all provide the same location accuracy across thefloor plan. When range measurements are noisy, the locationaccuracy is dependent on the error in ranging as well asgeometry between beacons and the receiver, as elaborated inthe next section.

B. Geometric Dilution of PrecisionA useful guideline to quantify the location accuracy is the

Cramer-Rao bound (CRB), the lower bound on the variancein the location that can be achieved by an unbiased locationestimator [5]. The results presented in this section are derivedfrom [6], [7], [18]. For 2D trilateration systems, it has beenshown [6] that the variance of the positional error σ2(Xi)at location Xi, as defined by σ2(Xi) = σ2

x(Xi) + σ2y(Xi),

corresponding to the CRLB when an unbiased estimator isused is given by:

σ2(Xi, CB(Xi)) =

∑Nb

k=1 σ−2r,i∑Nb−1

k=1

∑Nbj=k+1 σ

−2r,kσ

−2r,jA

2kj

Akj = |sin(θk − θj)|

where σ2r,k is the variance in range measurement of beacon

k, Nb is the number of beacons in CB(Xi), θk is the anglebetween Bk and the Xi.

Under the assumption that the range measurements areindependent and have zero-mean additive Gaussian noise withconstant variance σ2

r , this reduces to:

σ(Xi, CB(Xi)) = σr ×√

Nb∑Nb−1k=1

∑Nbj=k+1Akj

σ(Xi, CB(Xi)) = σr ×GDOP (Xi, CB(Xi))

The GDOP [4] is a function of the angles between thetarget Xi and beacons CB(Xi), and is given by :

GDOP (Xi, CB(Xi)) =

√Nb∑Nb−1

k=1

∑Nbj=k+1Akj

The CRB is directly proportional to the GDOP and asseen in Section II, several authors have used GDOP to quan-tify the location accuracy. We subsequently use the notationGDOP (Xi) instead of GDOP (Xi, CB(Xi)). The third col-umn GDOP in Figure 2 shows the GDOP map of the sixfloor plans. The GDOP is the worst (highest) along the linejoining two beacons, and is in general better (lower) when

4


x (m)-10 -5 0 5

y (m

)

-6

-4

-2

0

2

4

6

BeaconLOS locationNLOS location

(a) Coverage of beacon (Multi-Corridor map)

x (m)0 2 4 6 8 10 12

y (m

)

0

2

4

6

8

10

BeaconLOS locationNLOS location

(b) Coverage of beacon (Map 1)

True Range (m)0 5 10 15

Exp

erim

enta

l Ran

ge (

m)

0

5

10

15

20

25

30

35

40LOSNLOS

(c) Range measurement error(Multi-Corridor map)

True Range (m)0 2 4 6 8 10 12 14

Exp

erim

enta

l Ran

ge (

m)

0

5

10

15

20

25

30

35LOSNLOS

(d) Range measurement error(Map 1)

Figure 3. Experimental characterization of coverage and range error foracoustic beacons in real-world deployments

the regions are covered by more beacons. As a numericexample, for ranging system with standard deviation of therange measurements of 10cm, if two beacons subtend an angleof 90◦ at a target location, the resulting GDOP is 1.414 andthe 2-D location estimate would have a standard deviation of14.14cm.

Experimental validation of zero mean constant varianceranging noise assumption: Figure 3(c) and Figure 3(d) showthe distribution of the LOS and NLOS ranges of an acousticranging system [16] on two floor plans, for 5-6 beacons withat least 500 range measurements taken uniformly across allregions in the floor plan. We observe that when the beaconsare in LOS, the range measurements have nearly zero biasand almost constant variance. There is no appreciable changein variance of the range measurements with distance up to10m. This is likely due to the the SNR being sufficiently highthat the distance from the beacon does not affect the rangingaccuracy.

C. A modified GDOP metric for indoorsWe now formalize our modified GDOP metric to use

Unique Localizability, which is given by:

GDOPUL(Xi) =

{GDOP (Xi), UL(Xi) = 1

NaN , UL(Xi) = 0

The fourth column of Figure 2 shows the GDOPUL

metric. For most configurations, it is the same as the GDOPmetric. But for configuration 3 of Figure 2(a) and configuration2 of Figure 2(b), where there exist Class 4 locations with thetwo-beacon ambiguity problem, the GDOPUL is not definedwhere Unique Localization cannot be achieved. These casesare now numerically handled to avoid providing a confidenceon the location estimate when ambiguity exists in the solution.

D. Quality of a beacon configuration (Q)With our modified GDOP metric, we can now compare

beacon configurations like those found in Figure 2(a) and

GDOP0 2 4 6 8

Per

cent

age

of a

rea

in fl

oor

plan

(%

)

0

10

20

30

40

50

60

70

80

90

100

Configuration 1Configuration 2Configuration 3Configuration 4Configuration 5

(a) Rectangle floor planGDOP

0 2 4 6 8

Per

cent

age

of a

rea

in fl

oor

plan

(%

)

0

10

20

30

40

50

60

70

80

90

100

Configuration 1Configuration 2Configuration 3

(b) L-shaped floor plan

Figure 4. Comparing configurations using quality of beacon configurationQGDOPUL

Figure 2(b). We utilize this in our toolchain which has twomodes of operation where it can either optimize for UniqueLocalizability or optimize for both Unique Localizability andGDOP.

Case 1: UL-based metric QUL: In this case, we attemptto place beacons such that all regions in the floor plan areUniquely Localizable without considering the localizationaccuracy. This could be required in an ideal scenario withno ranging noise where the geometry of the beacons doesnot affect the localization accuracy, or when the locationestimate is averaged over a large number of measurements,resulting in low variance. We define the quality of the beaconconfiguration B across the floor plan X , as the percentage ofarea that is Uniquely Localized.

QUL(X,B) =

∑Nx

i=1 UL(Xi, CB(Xi))

#X× 100

where UL(Xi, CB(Xi)) is the binary function that indicatesif a location Xi is Uniquely Localizable when covered by thebeacons CB(Xi), as defined in Section III-A and #X is thecardinality or number of points in the floor plan.

In Figure 2(a), QUL is 100% for configurations 1, 2, 4and 5 and 28.7% for configuration 3. In Figure 2(b), thethree configurations have QUL as 74.6%, 48.3% and 100%respectively.

Case 2: UL and GDOP-based metric QGDOPUL : TheGDOPUL metric in Section III-C is defined for a singlelocation Xi. In order to quantify the quality of the beaconconfiguration across all locations in the floor plan, we usea heuristic based on the Cumulative Distribution Function(CDF) of the GDOPUL curve across all locations, as shownin Figure 4. For instance, to compare configuration 1 and con-figuration 2 of the rectangular room, where both configurationshave three beacons, we see from Figure 4(a) that 60% of thefloor plan has a GDOP less than 1.7 under configuration 1, andGDOP less than 3.0 under configuration 2. Alternately, 100%of the floor plan has a GDOP under 3.0 in configuration 1 butonly 60% of the floor plan has a GDOP less than 3.0 underconfiguration 2. Hence configuration 1 is better since it haslow overall GDOP than configuration 2. For configuration 3,where a large part of the floor plan is not Uniquely Localizable,we can see from the CDF plot that only around 27% ofthe floor plan is localized. We can see from these curvesthat configuration 1 is better than configuration 2 which isbetter than configuration 3. However, it is not obvious howconfiguration 4 and configuration 5 compare since the CDFcurves intersect. If the goal is to have 60% of the floor planwith lower GDOP, configuration 5 is better, but if the goalis to have 90% of the floor plan with lower GDOP, then

5


Beacon 1

Beacon 4

Beacon 3

Beacon 2

Beacon 6

Beacon 5

Beacon 1

Beacon 4

Beacon 3

Beacon 2

1

1

5 4 3

2

2 Initialization

Beacon placement process

Final placement

(a) Floor plan with candidate beacon locations

(b) Clusters generated by ray-tracing

(d) Beacons placed in GDOPUL algo. (GDOP plot)

(c) Beacons placed in UL algo. (Coverage class plot)

(e) UL algo. (GDOP plot) (f) GDOPUL algo.

(Coverage class plot)

Figure 5. Beacon Placement Process (Legend for coverage in (c),(f) is inTable I)

configuration 4 is better. In our toolchain, the designer canspecify the requirement, but by default we use the area underthe GDOPUL CDF curve. To compute the area, we need toprovide an upper limit on the GDOP. For the plots shown,the upper limit is conservatively chosen to be 8.0, whichcorresponds to an angle of 1.8◦ between two beacons anda target. This is equivalent to considering the regions withGDOP worse than 8.0 to not be localizable. For the L-shapedroom in Figure 2(b), we see from Figure 4(b) that amongthe configurations with 2 beacons, configuration 1 is betterthan configuration 2 even though configuration 2 has bettercoverage, and configuration 3 with 3 beacons outperforms boththese configurations. The same metric can be used even if thetarget area is a predefined path or a finite set of locations acrossthe floor plan. In these cases, the CDF would be computed onlyover the desired target locations.

IV. BEACON PLACEMENT ALGORITHM AND TOOLCHAIN

In this section, we present two beacon placement algo-rithms based on the concepts described in Section III. Theimplementation is illustrated by Figure 5 for a floor plan thatrepresents two rooms connected by a corridor. The inputsprovided are the floor plan, coverage model of the beaconsand the set of candidate beacon locations. The floor plan andobstacles are represented as multiple polygons. We also builta MATLAB-based tool where the user can draw floor plansto aid in prototyping. Figure 5(a) shows the floor plan withcandidate beacon locations shown by black circles.

The algorithm has two modes of operation based onwhether we want to optimize based on QUL or QGDOPUL .Our tools provide several design options to the user such asplacing beacons until a finite number of beacons are placedor placing beacons until some stop criteria is satisfied. This

stop criteria could be in the form of accuracy requirementsacross the floor plan, for instance GDOP ≤ 4.0 for 90% ofthe regions. The stop criteria we have used for both modesof the algorithm is achieving Unique Localization across theentire floor plan.

Step 1: Initialization• Discretize the floor plan to generate X .• Apply the beacon coverage model for each Pj to gen-

erate CP (Xi), the set of candidate beacon locationsproviding coverage of each point on the floor plan.

• Perform clustering on X such that all Xi that have thesame beacon coverage, CP (Xi) belong to one cluster.

• For each cluster, assign:◦ Localization Status = 0◦ Size = number of points in cluster

• Initialize Selected Beacons = {}Figure 5(b) shows the clusters generated by ideal ray-tracingcoverage with five of the largest clusters labeled.Repeat Steps 2-4 until Localization Status of all clusters=1.In every iteration, one beacon is placed.

Step 2: Select clusterAmong the clusters with Localization Status = 0, select thecluster with largest Size

Step 3: Select subset of candidate locationsAmong all candidate beacon locations, select the subsetCP (Xi), where Xi is any point in the cluster. Note that allpoints in the cluster are covered by the same candidates.

Step 4: Among the subset of candidates, select thecandidate that maximizes criteria• The selection criteria depends on whether we are

optimizing for UL or GDOPUL. Further, since thesemetrics are not defined for single-beacon case, wehave a different criteria when Nb, the number ofbeacons already covering the cluster is zero.

(1) UL and Nb=0: Select candidate with maximum coverage.(2) GDOPUL and Nb=0: Select candidate with maximumaverage distance from other candidate locations for this cluster,to provide good geometry.(3) UL and Nb 6= 0: Select candidate that maximizes thequality of beacon configuration QUL or the percentage of areaUniquely Localized.(4) GDOPUL and Nb 6= 0: Select candidate that maximizesthe quality of beacon configuration QGDOPUL or the areaunder the CDF of the GDOPUL curve.• Add selected candidate beacon location to the set

Selected Beacons.Figure 5(c) shows the placement of beacons in the ULmode with the coverage classes shown and Figure 5(d) showsthe placement of beacons in the GDOPUL mode with theGDOPUL values shown. As can be seen, the selection of thefirst beacon (to localize the cluster labeled 1 in Figure 5(b))and subsequent beacons is different for both the algorithms.Step 5: Re-evaluate clusters• It is possible that a cluster is partially localized by

the ambiguity being resolved when the new beacon is

6


placed. In that case, split it into two clusters beforethe next step. Assign Size and Localization Status forthe new clusters.

• For all clusters: If all the points in the cluster satisfythe stop criteria (achieving Unique Localizability),assign Localization Status of cluster=1.

As we can see from Figure 5(c), the entire floor plan islocalized with only 4 beacons while optimizing for UniqueLocalization. However, the beacons are clustered close togetherand GDOP of the final beacon placement, shown in Figure 5(e)is poor. On the other hand, Figure 5(d) shows the placementwhile optimizing for GDOP as well, and places two additionalbeacons.The final configuration has a good GDOP and cover-age across the floor plan, as seen in Figure 5(d) and Figure 5(f)respectively. If the design requirement was to only place 4beacons, the first 4 beacons would have been placed, with thecorridor area not localizable but good GDOP in the two rooms.

V. EVALUATION

We evaluated our beacon placement algorithm in simula-tions on 10 floor plans, which are listed in Table II. The firstfive are smaller floor plans constructed to represent differentgeometries that could occur within larger floor plans. The nextfour are real-world floor plans in buildings on our universitycampus, and the last is a floor plan drawn using our toolchain.The Rectangle and L-shape floor plans are shown in Figure 2,the Two-rooms floor plan is shown in Figure 5 and theremaining seven floor plans are shown in Figure 7.

The first column of Table II, 3-beacon (Minimal) denotesthe minimal number of beacons if all regions are to be coveredby at least 3 beacons as required by trilateration. The secondcolumn UL (Minimal) denotes the minimal number of beaconsrequired under the proposed scheme where we localize with2 beacons. The results for both of these columns are obtainedby brute-force by iterating through every possible beaconconfiguration. Note that this is not practical for larger floorplans. We see that the proposed scheme results in reducingthe number of beacons by 22-60% (33% on an average) forthese floor plans. The most significant improvement is seenfor the Spokes floor plan, shown in Figure 7. This is nottypical of indoor spaces but we selected this to show when theproposed scheme is most effective. The floor plan is generatedsuch that all points are in LOS of the left bottom and rightbottom corners. We can extend this floor plan to generateinfinite spokes, and the 3-beacon scheme would require infinitebeacons, whereas the proposed UL scheme would require only2 beacons.

The third column shows the number of beacons placedwhile optimizing for UL and the fourth column shows thenumber of beacons placed while optimizing for the GDOPUL.We see that the UL algorithm places the same number ofbeacons as the minimal in most cases, with 1 or 2 additionalbeacons in some cases (on average 5% more beacons thanminimal). The GDOPUL algorithm usually places the samenumber of beacons as UL, in worst-case it places 2 morebeacons. In Figure 7, we show the resulting GDOP map of thefinal beacon placement with the algorithm in the GDOPULmode. Due to lack of space, we have not shown the minimalbeacon placement and result of UL algorithm for these sevenfloor plans but the number of beacons for both are shown inTable II.

Number of beacons2 4 6 8 10 12 14 16

Per

cent

age

of a

rea

loca

lized

(%

)

0

20

40

60

80

100

RectangleL-shapeSpokesTwo-roomsMulti-roomMulti-corridorMap 1Map 2Map 3Map 4

(a) Area localized with addition ofevery beacon (UL algo.)

GDOP0 2 4 6 8

Per

cent

age

of a

rea

in fl

oor

plan

(%

)

0

20

40

60

80

100

RectangleL-shapeSpokesTwo-roomsMulti-roomMulti-corridorMap 1Map 2Map 3Map 4

(b) GDOP CDF curve for final place-ment (GDOPUL algo)

Figure 6. Validation of algorithm’s greedy approach and stop criteria

Figure 6 shows the effect of the design choices we haveused in the algorithm. Figure 6(a) validates the greedy ap-proach we have adopted, where we iteratively select beacons tolocalize the largest cluster. For each of the floor plans, we cansee the % of area Uniquely Localized as an additional beaconis added. For instance in Map 4, 90% of the area is localizedwith 11 beacons and an additional 5 beacons are requiredfor the remaining 10% of the area. For practical purposes,it may be sufficient to have coverage in 90% of the region,since often tracking or filtering would be used in the locationsolver. Figure 6(b) shows the final CDF of the GDOPUL whenthe algorithm is in the GDOPUL mode. The final quality ofbeacon configuration varies across floor plans since the stopcriteria is all regions being Uniquely Localizable. We couldalso have specified a different criteria based on GDOP, suchas 90% of the floor plan having a GDOP less than 4.0. Wesee that across the floor plans, the Multi-room floor plan hasthe overall highest quality of beacon configuration but ouralgorithm placed 2 beacons more than the minimal numberrequired. The worst quality of beacon configuration is formulti-corridor floor plan and we can see from Figure 7 thatthe two end corridors have a high GDOP.

In summary, the proposed scheme based on Unique Lo-calizability places between 22% to 60% (33% on average)fewer beacons than a typical trilateration-based scheme. Theproposed algorithm in UL mode usually achieves the minimalplacement and on an average places 5% more beacons thanminimal, but 29% fewer beacons than typical. The proposedalgorithm in GDOPUL mode places on an average 14% morebeacons than minimal, 23% fewer beacons than typical, andprovides a much better quality (GDOP) of beacon placement.The final quality of beacon placement and the number ofbeacons required varies with the floor plan geometry. Thequantitative results apply even for larger floor plans at building-scale since they can be represented as a union of smaller floorplans.

VI. CONCLUSIONS

In conclusion, this paper proposes a beacon placementtechnique and a new metric to compare different indoorplacements. Our approach requires a map of the interior space,along with a propagation model for beacons that provide rangedata that can be used to estimate coverage regions of a beaconlocation. We show how it is possible to localize with twobeacons instead of three beacons and introduce the conceptof Unique Localizability. We introduce a new formulation ofGDOP that incorporates the Unique Localizability informationand use these metrics to compare and optimize beacon place-ments in indoor environments. We integrated our proposed

7


Spokes Multi-room

Multi-corridor

Map 1

Map 4

Map 2

Map 3

Figure 7. Beacon Placement result and corresponding GDOP for subset ofthe floor plans with algorithm optimizing for GDOPUL.

Floor Plan Beacon Placement Method3-beaconMinimal

ULMinimal

ULAlgorithm

GDOPUL

AlgorithmRectangle 3 2 2 2L-shape 4 3 3 3Spokes 5 2 2 2Two-rooms 6 4 4 6Multi-room 9 6 8 8Multi-corridor 7 5 5 5Map 1 11 8 8 9Map 2 12 8 9 10Map 3 9 7 7 8Map 4 21 15 16 16

Table II. NUMBER OF BEACONS PLACED

Quality of Beacon Configuration metrics into a toolchainthat can automatically place or refine beacon locations inorder to improve accuracy, coverage and/or beacon number.The tool exports a GDOP and coverage map for the finalbeacon configuration that can be used to visualize expectedperformance across a floor plan. We believe that these typesof automated provisioning tools are going to become criticalif indoor localization services become pervasive. In the future,we intend to extend our system to different beacon propagationmodels like UWB transmissions and validate at scale in realdeployments.

REFERENCES[1] Nanotron: http://www.nantron.com/ (viewed 10/12/2015).[2] Decawave: http://www.decawave.com/ (viewed 10/12/2015).[3] 802.11v-2011 - IEEE Standard for Information technology– Local and

metropolitan area networks– Specific requirements– Part 11: WirelessLAN Medium Access Control (MAC) and Physical Layer (PHY) spec-ifications Amendment 8: IEEE 802.11 Wireless Network Management.IEEE Computer Society, 2011.

[4] Pratap Misra and Per Enge. Global Positioning System: Signals,Measurements and Performance Second Edition. Lincoln, MA: Ganga-Jamuna Press, 2006.

[5] Steven M Kay. Fundamentals of statistical signal processing, volume i:estimation theory. 1993.

[6] Maurizio Spirito et al. On the accuracy of cellular mobile stationlocation estimation. Vehicular Technology, IEEE Transactions on,50(3):674–685, 2001.

[7] Neal Patwari, Joshua N Ash, Spyros Kyperountas, Alfred O Hero III,Randolph L Moses, and Neiyer S Correal. Locating the nodes:cooperative localization in wireless sensor networks. Signal ProcessingMagazine, IEEE, 22(4):54–69, 2005.

[8] Adrian N Bishop, Barıs Fidan, Brian DO Anderson, Kutluyıl Dogancay,and Pubudu N Pathirana. Optimality analysis of sensor-target localiza-tion geometries. Automatica, 46(3):479–492, 2010.

[9] Naveed Salman, Hemat K Maheshwari, Andrew H Kemp, and MounirGhogho. Effects of anchor placement on mean-crb for localization.In Ad Hoc Networking Workshop (Med-Hoc-Net), 2011 The 10th IFIPAnnual Mediterranean, pages 115–118. IEEE, 2011.

[10] Onur Tekdas and Volkan Isler. Sensor placement for triangulation-basedlocalization. Automation Science and Engineering, IEEE Transactionson, 7(3):681–685, 2010.

[11] Ross Allen, Neil MacMillan, Dimitri Marinakis, Rahnuma Islam Nishat,Raziur Rahman, and Sue Whitesides. The range beacon placementproblem for robot navigation. In Computer and Robot Vision (CRV),2014 Canadian Conference on, pages 151–158. IEEE, 2014.

[12] Zheng Yang and Yunhao Liu. Quality of trilateration: Confidence-based iterative localization. Parallel and Distributed Systems, IEEETransactions on, 21(5):631–640, 2010.

[13] Tilman Leune, Thorsten Wehs, Marijn Janssen, Christian Koch, andGerd von Colln. Optimization of wireless locating in complex envi-ronments by placement of anchor nodes with evolutionary algorithms.In Emerging Technologies & Factory Automation (ETFA), 2013 IEEE18th Conference on, pages 1–6. IEEE, 2013.

[14] Javier O Roa, Antonio Ramon Jimenez, F Seco, Jose Carlos Prieto,and J Ealo. Optimal placement of sensors for trilateration: regularlattices vs meta-heuristic solutions. In Computer Aided Systems Theory–EUROCAST 2007, pages 780–787. Springer, 2007.

[15] Nicolaj Kirchhof. Optimal placement of multiple sensors for localiza-tion applications. In Indoor Positioning and Indoor Navigation (IPIN),2013 International Conference on, pages 1–10. IEEE, 2013.

[16] Patrick Lazik, Niranjini Rajagopal, Bruno Sinopoli, and Anthony Rowe.Alps: A bluetooth and ultrasound platform for mapping and localization.In The 13th ACM Conference on Embedded Networked Sensor Systems(SenSys 2015), SenSys ’15, 2015.

[17] Marilynn P Wylie and Jack Holtzman. The non-line of sight problemin mobile location estimation. In Universal Personal Communications,1996. Record., 1996 5th IEEE International Conference on, volume 2,pages 827–831. IEEE, 1996.

[18] Neal Patwari, Alfred O Hero III, Matt Perkins, Neiyer S Correal,and Robert J O’dea. Relative location estimation in wireless sensornetworks. Signal Processing, IEEE Transactions on, 51(8):2137–2148,2003.

8

Beacon Placement for Range-Based Indoor...

Documents

Transcript of Beacon Placement for Range-Based Indoor...