Positioning in Wireless Networks- Non-cooperative and Cooperative Algorithms -

Giuseppe Destino

Centre for Wireless CommunicationsUniversity of Oulu

October 12, 2012

1

Location-Based Service Market- Toward context-awareness -

Reveneu

PyramidResearch Forecasts 2008-2015

2

Location-Based Service Market- Toward a pervasive platform -

Navigation devices

PyramidResearch Forecasts 2008-2015

3

Geo-Social Network- Location and information sharing -

• Find your friend

• Get direction

• Information sharing

4

Shopping- Personalized Advertising -

• Real-time sale

• Indoor navigation

• Find objects

• Information sharing

5

Smart and Safe Navigation- In-car augmented reality -

• Navigation

• Localization of accidents

• Alert

• Information sharing

6

Industrial Monitoring- Safety and Security -

• Monitoring of the environment

• Tracking of personnel

• Tracking of assets

7

... and Many Others

• Health-care: remote monitoring, find personnel, track assets, etc.

• Indoor-sport: person tracking

• Indoor navigation: get directions, estimate travel time, etc.

• Sensor networks: measurement maps

• Surveillance: detect intruder, anomaly localization, etc.

• Warehouse: asset tracking and monitoring

8

Location-aware Network Optimization

9

Network Planning and Expansion

• Location-based RSS map

• Use mobile nodes for monitoring

• Adaptive network expansion

10

Cognitive Radio in the TV-White Space

• Location-based primary user database

• Allocate free TV-white space based on location information (FCC’ 10)

• 50 m, minimum distance between primary and secondary users

11

Positioning System

12

System Architecture

System Centric

• Measurements convey to theradio access network

• Centralized calculations

Node Centric

• Measurements convey to themobile nodes

• Local calculations

13

Network

Nodes• NA anchors, known fixed location

• NT targets, unknown location

Topology

• Star-like, non-cooperative scheme

• Mesh, cooperative scheme

Syncronization

• Global, synchronous

• Local, asynchronous

A1 A2

A3A4

X

14

Internode Interaction- Measurement system -

Power Profile• Channel Impulse Response (CIR), wideband signal

• Power Delay Profile (PDP), wideband signal

Angle

• Angle-of-Arrival (AoA), multiple-antenna

Distance (Ranging)

• Received Signal Strength Index (RSSI), always available

• Time-of-Arrival (ToA), technology dependent, asynchronous network

• Time-Difference-of-Arrival (TDoA), technology dependent, synchronousnetwork

15

Source of Errors- Example of an indoor propagation channel -

• Noise

• Multipaths

• Blockage

• Mobility

S-V Indoor Propagation Modeling A. In Section VIII-A we will see, from another point

of view, that 1 / A 5: 300 ns.

B. The Ray Arrival Rate, X

By resolving the individual rays in about 200 power profile measurements similar to those in Figs. 3 and 4, we

estimate 1 / X to be in the range of 5-10 ns. The range uncertainty comes from the fact that our ray-resolving al- gorithm, coupled with our measurements sensitivity, is unable to detect many weak rays, in particular, those fall- ing near strong rays. The higher the sensitivity, the more (weak) rays we would find, and hence, the larger the value of X. At the same time, the probability distribution of the path gains Pk would be increased for small values of 6’s: Thus, the appropriate choice of X is strongly coupled to the probability distribution of the P ’ s . We find that. a consistent choice is 1 / X = 5 ns‘coupled with Rayleigh- distributed P ’ s having the appropriate mean-square value (see Section VII-D). Actually, as will be discussed later (see Section VIII-B), smaller values of 1 / X could have been employed, even down to the limiting value of zero

(i.e., continuous ray arrival process). This, however, is beyond our measurement time resolution.

C. The Ray and the Cluster Power-Decay Time

Constants, y and I’

As mentioned in Section VII-A, the number of arrival

times of clusters, say, To, T I , - * , TL, are the same for all locations within a given room. It follows from (26) that, within that room, the expected value of the ray power as a function of time, measured from the arrival point of the first ray of the first cluster, is given by

L

p20 = ~ ~ ( 0 , 0 ) C e - f i / r e - ( r - f i ) / y u ( t - T ~ ) , 1=0

(27 1 where U ( t ) is the unit step function, which equals one for t L 0, and zero for t C 0. A sketch of (27) is shown

in Fig. 7(a). - An estimate of P 2 ( t ) , and hence of y and r, was ob-

tained for a given room by aligning the time origin and

taking the average of many measured power profiles, s ( t ) ,

within that room. Actually, what one obtains from this process is an estimate of the time convolution of P 2 ( t ) and p 2 ( t ) , the square of the transmitted pulse. However, since this pulse is narrow, the-effect of the convolution is negligible. Four different examples of such space-aver- aged s ( t ) are shown in Fig. 8.. More than 20 power pro- files were averaged in each case. Fig. 8(a) indicates that only one cluster reached the receiver in that case. Each of the remaining cases shows two arriving clusters. The time spread evident in the leading edges of a few clusters in Fig. 8 is mainly due to fundamental uncertainty in align-

ing the time origins of the various measured power pro- files.

By fitting decaying exponentials to each cluster in Fig. 8, as well as to similar measurements taken in other

Fig. 8. Four spatially averaged power profiles within various rooms. The dashed lines correspond to exponential power decay profile of the rays and the clusters.

rooms, we find that, on the average, rays within a cluster decay with an approximate time constant of y = 20 ns

(see dashed lines in Fig. 8). Similarly, by fitting decaying exponentials through the leading peaks of successive clus- ters, we find that the clusters themselves decay, on the average, with an approximate time constant of J? = 60 ns (see dashed lines in Fig. 8).

The use of exponentials in (26) and (27) to represent

the decays of the powers of the rays and clusters as func- tions of time has an intuitively appealing interpretation. Consider, for example, our physical picture of the rays bouncing back and, forth in the vicinity of the receiver and/or the .transmitter to form a cluster. On the average, with each bounce, the wave suffers some average delay (say, equivalent to the width of a room), and some aver- age decibels of attenuation (which depends on the sur- rounding materials of the walls, furnitures, etc.). In this case, the power level in decibels of each successive ray would be proportional to the time delay of that ray, which results in our exponential power decay characteiistics‘.

Note that from this picture, y and would be increased if the building walls were more reflective and/or if the sizes of the rooms and the building itself were increased.

D. The Probability Distribution of the Path Gains, Pkl

So far, ouy_model gives the expected value of the path power gain Pil as a function of the associated cluster and ray delays Tl and T ~ ~ . We now make the assumption, which is reasonably supported by our observations, that the prob&ility distribution of the normalized power gain P L / P i l is independent of the associated delays, or for that matter, of the location within the building. Under this as- sumption, we can put the measured power gains of all our resolved paths into one, supposedly homogeneous, data pool by simply normalizing by the appropriate (i.e., mea- sured within the same room and having the same delay) spatially averaged power profiles, such as those shown in

Fig. 8. The cumulative distribution of Pil/z, obtained as de-

scribed above, is shown by the solid line in Fig. 9. The dashed line in that figure is the unity-mean exponential.

Room 1

Room 2

Room 3

Room 4

16

Ranging with Bluethooth- Measurement result with AP Class 1 and MT Class 2 -

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

-10

-5

0

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iry-b

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Distance vs. RX power level

Figure 3: Relationship between various Bluetooth signal pa-rameters & distance.

which is carried by the experimenter, is a Pentium-based TabletPC. All the desktops (connected to the Bluetooth adapters byUSB cables) together with our Tablet PC run Fedora Core 4,with the latest BlueZ protocol stack [7].

B Data Collection, Results and Discussion

During the experiments, our mobile host is connected using“SSH” (secure shell) to the desktops controlling the Bluetoothadapters. This facilitated the experimenter to have completecontrol over the whole system from the mobile host. Whilestanding at a specific grid position, the experimenter could runBluetooth signal extractor programs at both the mobile host andany AP over the network.

We now present the results from our various experiments:

1) Signal parameters’ correlation with distance

For this experiment, we carefully chose five different grid posi-tions where we took readings from each of the 3 APs, thus re-sulting in 15 data points. We adopted this methodology, ratherthan choosing 15 distinct distances from a single AP, becausewe wanted to correlate distance with signals originating fromAPs that were placed at different locations and surroundings.

In our experiments, we discovered that the Bluetooth wire-less signal strengths tend to vary quite significantly dependingon the user’s orientation. Therefore, for every chosen grid po-sition, we took 30 readings from every AP for each of the fourdifferent orientations. We then calculated the average of these120 readings to obtain the signal parameter’s value for that par-ticular AP at the specific grid position. Since we know thedistances of all grid positions from any AP, the signal strengthvalues are simply mapped against the corresponding distancesto generate Fig. 3.

In order to acquire the connection-based status parameterreadings (i.e., RSSI, LQ, and TPL), we maintained connectionsat the HCI level from the APs to our mobile host.

From Fig. 3, the following observations can be made:

• As anticipated in our earlier analysis, RSSI turns out tocorrelate poorly with distance, as shown in Fig. 3(a).

• Fig. 3(c) shows a horizontal straight line for TPL values.This is because our Class 2 adapter at the mobile hostwhich uses Broadcom’s BCM2035 chip does not supportpower control feature. As a result, the TPL at the AP re-mained at its default value, which happens to be 0 dBmfor the Bluetooth adapter used.

• From Fig. 3(b), we see that LQ correlates with distancemuch better than RSSI and TPL, although the LQ read-ings obtained at smaller distances show very little varia-tion. Note that these readings were taken at the AP side,rather than at the mobile host side, as the LQ perceivedat our mobile host was always 255 at any grid position,which is the highest possible LQ value. This is due to ourClass 1 APs’ large transmit power. The measurements atthe AP side, on the other hand, show variations becauseour mobile host uses a Class 2 adapter.

• Our BT-2100 Class 1 adapters provide absolute RX powerlevel through inquiry, instead of the relative RSSI valuesas suggested by Bluetooth specification. As the parameter“Inquiry Result with RSSI” also suffers from the GRPR-related zero-RSSI problem (just like the “connection-based RSSI”), we believe that making RX power levelavailable should augur well in terms of distance. Fig. 3(d)certainly establishes this claim since the RX power levelshows the best correlation with distance, compared to theother three signal parameters.

2) Effect of GRPR on RSSI

Fig. 5(a) illustrates the adverse effects of wider GRPR on thereported RSSI. From the figure, it is seen that BT-2100’s RSSIreadings (GRPR ! 80 dB ) showed little variation comparedto our Broadcom’s adapter, which has a narrower GRPR. Be-cause of the combined effect of large GRPR and power control,BT-2100’s RSSI readings always remained at or above 0. Onthe contrary, Broadcom’s adapter gave negative RSSI valuesat greater distances, although we did not have many such gridpositions owing to our testbed’s size.

3) TPL Consideration

For this experiment, we recorded the stabilized TPL values aswell as the stabilization time periods for each AP’s signal atspecific grid positions using BT-2100 at the mobile host side.Fig. 4(a) indeed shows very few discrete transmit power levels,in harmony with our analysis in Section C. Moreover, the timeperiods required to reach these stabilized TPL values are alsoquite significant, as revealed in Fig. 4(b). Both these attributesmake TPL a poor candidate for localization purpose.

4) Effect of Varying Inquiry Time Period

In this experiment, the inquirer, which is the mobile host, isplaced at a location where it can hear all 9 Bluetooth devicesto be discovered. Since BlueZ’s HCI API allows us to vary theinquiry time period in increments of 1.28 sec, we varied it ac-cordingly, and took 50 readings for each distinct inquiry timeperiod. From Fig. 5(b), it is observed that, although the gap be-tween the maximum and the minimum number of discovered

[Hossain] A. Hossain and W.-S. Soh, A comprehensive study of bluetooth signal parameters forlocalization’, in Proc. IEEE 18th International Symposium on Personal, Indoor and MobileRadio Communications, pp. 1-5, September 2007

17

RSSI Ranging with Wi-Fi- Cardbus Wi-Fi, corridor environment -826 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 3, NO. 5, OCTOBER 2009

Fig. 4. Relation between distance and RSSI in a corridor.

Thus, we can impose certain constraints to the distance esti-mates between the MS and the APs sorted accordingto their average RSSI values.

In a usual homogeneous deployment of APs in a WLAN net-work, it is reasonable to assume that the distance from the firstAP, will be less than a constant, . For example, wecan choose a fraction of the known distance between and

. Also, if is the distance between and , itfollows that

(26)

Thereby, we can enclose both the maximum distance to theAP from which we receive the most average signal strength andthe minimum distance to the AP from which we receive the leastaverage signal strength.

By using (9) and (15) we can translate distance constraintsinto path loss exponents constraints, therefore

(27)

Moreover, we can obtain other type of constraints imposingrelations between different distances. It is clear that although

, , this fact does not imply that .However, it is logical that this distance difference exists if thedifference in the average signal strength is great enough. There-fore,

(28)

moreover, this difference in the distances depends on how greatthe difference in the average signal strength is. Therefore, if

thus , where is a value which de-creases when the difference between the average RSS increases.In the case that it occurs that

(29)

and therefore we can assume

(30)

where the value of will be a number slightly higher than 1and describes the relevance that we want to impose to a differ-ence of average signal strengths in terms of distance difference.

In the same way as we did before, we are going to convert thedistance constraints into path loss exponents constraints

(31)

Therefore, a feasible set of solutions is

if (32)

is a convex set, indeed it is a polyhedral set and therefore vari-ants of the Levenberg–Marquardt algorithm [27] can be appliedin order to resolve (32), where we can choose, as an initial guessin the algorithm, a rough approximation of the path loss expo-nents like or the center of the polyhedron .

VI. RESULTS WITH MEASUREMENTS AND SIMULATIONS

A. Relationship Between RSS and Distance

In this section, we are going to show the suitability of expres-sions (2) and (3). In order to do that we carried out a large cam-paign of measurements on the second floor of the Higher Tech-nical School of Telecommunications, University of Valladolid(Spain), shown in Fig. 9.

As for APs, we used eight identical wireless broadbandrouters with two antennas each, in diversity mode, which istypically found on most IEEE 802.11 WLAN routers. APs wereconfigured to send a beacon frame every 10 ms to constantpower. Diversity circuitry determines which antenna has betterreception and switches it on in a fraction of a second whileit turns off the other antenna. Therefore, both antennas arenever active at the same time. APs have omnidirectional rubberduck antennas mounted. These are vertically polarized with asymmetrical 360 radiation pattern in the horizontal plane andwith a vertical beamwidth of approximately 75 . As for MS,we used a WLAN cardbus adapter with a vertically polarizedomnidirectional external antenna, also found on most IEEE802.11 WLAN adapters.

Authorized licensed use limited to: National Taiwan University. Downloaded on August 06,2010 at 03:32:43 UTC from IEEE Xplore. Restrictions apply.

[Mazuelas] S. Mazuelas, A. Bahillo, R. Lorenzo, P. Fernandez, F. Lago, E. Garcia, J. Blas, and E. Abril,Robust indoor positioning provided by real-time RSSI values in unmodified WLAN networks,IEEE Journal of Selected Topics in Signal Processing, vol. 3, pp. 821 - 831, October 2009.

18

Frequency Diversity of RSSI- TelosB Platform, IEEE 802.15.4 Compliant, d = 2m -

0 1 2 3 4 5 6 7 8 9!85

!80

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Node distance (m)

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Fig. 1. RSS measurement in different environ-ments

0 2 4 6 8 10 12 14 16 18!80

!75

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!65

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S (

dB

m)

Fig. 2. RSS measurement in different channels:node distance=2m

Fig. 3. An illustrative example; there are twopaths a and b; The signals of two frequencies!1, !2 will have different RSS values at the re-ceiver end because of the phase-shift differencebetween paths (i.e.,P (!1 != P (!2))

An interesting observation is that the frequency diversitymay help RSS provide phase information indirectly. Fig. 2depicts some RSS measurements at the same environment fora pair of TelosB sensors [1] at different spectrum channels (theband is divided into 16 channels). We can see that, the pair ofnodes may have significantly different RSS values at differentspectrum channels and the measurements are quite stable at thesame environment. This RSS difference at different channelsis essential for our work as it carries the valuable phaseinformation. By carefully analyzing these RSS, we can identifythe amplitudes and phases of signals from each path, thenderive the accurate distance according to the amplitude of LOSsignals. A clearer illustrative example is in Sec. 2.3.

We find that the analysis of RSS values is a typical non-linear curvature fitting problem with trigonometric modelfunctions. It has no close-form solutions and is typicallyapproximated by numerical iterations. We also find that theoriginal fitting problem has a bad shape. Its Hessian matrixis ill-conditioned and the solutions will be un-stable and nottrustable, which may yield great ranging errors. To deal withthis issue, we further explore the practical considerations toreform the problem. For example, certain hardware-dependentparameters, though unknown, will be unlikely to change andcan be obtained before ranging. The LOS path is always theshortest path, and reflection will absorb a lot of signal energy.Granting these facts, we design treatment techniques to theproblem. Thus, the reformed problem has greatly improvedconditioning and the solutions become much more stable.

To demonstrate the effectiveness of the ideas, we implementa real tracking system called MuD (Multipath Distinguishing).It is based on TelosB [1] platform. Compared with traditionalRSS-based ranging ( localization), MuD offers the followingkey advantages. 1) It is robust to the environment dynamics.Moving people, new furniture and etc. will not degrade itsperformance unless the LOS path is affected (e.g., blocked);2) No labor-intensive training is needed. The training ofhardware-oriented parameters is a one-time operation and canbe done in an online manner. We can also use values inthe hardware specification manual to assign these parameters.It will sacrifice about 10% accuracy; 3) Ranging in MuDis very accurate in dynamic environments. Therefore threeanchor nodes for trilateration purposes are enough for local-izations. Experimental results show that the average ranging

(localization) error is about 1 meters in complex and dynamicenvironments (five people moving around). Compared with thetraditional RSS-based approaches in dynamic environment, theaccuracy is improved by 10 times.

The rest of this paper is organized as follows. In Sec. 2,we will give some background information. Sec. 3 will givethe system model and the problem definition. Sec. 4 will showhow to use non-linear optimization to solve the above problem.Sec. 5 presents the our treatment techniques to improve theproblem conditioning. We will describe the implementation ofMuD tracking system and evaluate its performance in Sec. 6.Sec. 7 will be the related work. We will conclude this workand point out our future work in the last section.

II. MULTIPATH EFFECT AND FREQUENCY DIVERSITY

In this section, we first introduce the background of theradio propagations in free space, then describe how the signalbehaves in multipath phenomenon. At last we use a simpleexample to illustrate how to exploit the frequency diversity tocope with the multipath effects.

A. Radio Propagation in Free Space

Radio propagation describes the behavior of radio waveswhen they are transmitted from the transmitters to the re-ceivers. During the propagation, radio waves will fade out withthe distance. By Friis model [14] it can be expressed as

|!"p | =PtGtGr!

2

(4"d)2(1)

where !"p = |!"p |, # is the signal wave vector, |!"p | isits amplitude and # is its phase at the receiver, Pt is thetransmission power, Gt, Gr are the antenna gain of thetransmitter and receiver, ! is the signal wavelength and d is theLOS path length between the transmitter and receiver. Supposethe sender has the phase zero, the signals phase at the receiveris

# = (d

!! # d

!$) · 2" (2)

In the remainder of this paper, we call |!"p | the path strength,and # the path phase. For signals along the LOS path, theFriis model is appropriate to describe its behavior as there isno other fading. For NLOS paths in a multipath scenario, weneed more physical laws. #$ is the floor operator.

2202

[Zhang] D. Zhang, Y. Liu, X. Guo, M. Gao, and L. Ni, On distinguishing the multiple radio paths inRSS-based ranging, in Proc. IEEE INFOCOM 2012, pp. 2201 - 2209, March 2012.

19

Ranging based on Time Measurements- Time-of-Arrival and Time-Difference-of-Arrival -

ToA

d = c(T 2Tx − T 1

Tx)− (T 2Rx − T 1

Rx)

2

• Asynchronous method• Two-way-communication

TDoA

δ = c(TRx1 − TRx2)

• Asynchronous Tx-Rx• Synchronous Rx-Rx• One-way-communication

20

Ranging Error- IR UWB Technology -Line-of-Sight (LOS)

6 EURASIP Journal on Wireless Communications and Networking

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Figure 5: Range error (cm) versus ground-truth range (m) while varying bandwidth B (GHz) in line-of-sight conditions.

Non-Line-of-Sight (NLOS)8 EURASIP Journal on Wireless Communications and Networking

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Figure 6: Range error (cm) versus ground-truth range (m) while varying bandwidth B (GHz) in nonline-of-sight conditions.

In order to quantify the small-scale e!ects in the mea-surements, we also compute the standard deviation of the 25range errors on the grid for each experiment. The standarddeviation varied between 0.5 to 1 cm in LOS conditions forall four buildings. The mean of the standard deviation overthe ensemble of experiments in NLOS conditions rose to 3, 5,11, and 70 cm for NIST North, Child Care, Sound, and Plant,respectively. No apparent trend existed in the standard de-viation as a function of range as opposed to the increasingaverage range error observed in the figures as a function ofrange.

5. CONCLUSIONS

Our nominal ranging system at 6 GHz bandwidth and 5 GHzcenter frequency delivers a mean range error of 6 cm in line-of-sight conditions up to a range of 45 m throughout allfour buildings tested. This error increases to 24, 38, and84 cm for sheet rock, plaster, and cinder block wall materi-als, respectively, in non-line-of-sight conditions; the systemranges within 390 cm up to 15 m in the steel building, but theperformance degrades rapidly thereafter. The ranging preci-sion improves significantly when raising the bandwidth from

[Gentile] C. Gentile, and A. Kik, “A Comprehensive Evaluation of Indoor Ranging UsingUltra-Wideband Technology”, EURASIP Journal on Wireless Communications andNetworking, vol. 2007, pages 10.

21

Fundamentals of Positioning Algorithms

22

Positioning via Connectivity- Proximity Positioning -

System model

• Single target

• Connectivity/proximityinformation

Position estimation• Logic intersection

• Centroid:

z =

NA∑i=1

wiai

NA∑i=1

wi

wi ∝ 1/di.

A1 A2

A3A4

X(1,2,3,4)

(1,2,4)

(1,4)

23

Finger-Printing- Signal Space based Positioning -

System model

• Single target

• Measurement phase

• Real-time localization

Position estimation• Fingerprint: f ipq = (zp, φq, f(rix))

• Database: Ω , f ipq, |Ω| = PQNA

• Real-time measurement:s , f(rix)NA

i=1

• Search method, e.g. Nearest neighbor

z = arg minfipq∈Ω

NA∑i=1

(f ipq − f ipq)2

s.t. f ipq = (zp, φq, f(rix))

0.2 0 0.2 0.4 0.6 0.8 1 1.20.2

0

0.2

0.4

0.6

0.8

1

A1 A2

A3A4

X

24

Triangulation- Angle-based and Positioning -

System model

• Single target

• AoA measurements

Position estimation

pML(θ) = pR+(GTθΣ−1

θ Gθ

)−1GθΣ

−1θ

(θ − θR)

)

Gθ ,

− sin(θR1)/dR1 cos(θR1)/dR1

......

− sin(θR1)/dRNA cos(θRNA )/dRNA

A1 A2

X

H

25

Multilateration- TDoA-based Positioning -

System model

• NA ≥ η + 1 anchors

• Single target

• Differential distance estimation

• Syncrhonous system

Position estimation

‖z− ai‖F − ‖z− aR‖F = ∆diR,

∀i ∈ IA rR A1 A2

A3A4

Z1

Anchor nodeTarget node

26

Trilateration- ToA-based Positioning -

System model

• NA ≥ η + 1 anchors

• Single/Multi-targets NT ≥ 1

• Distance estimation

Position estimation• Single target

‖z− aj‖F = dij , ∀j ∈ IA

• Multi-target‖zi − aj‖F = dij , ∀i ∈ IT , j ∈ IA...‖zi − zj‖F = dij , ∀i ∈ IT , j ∈ IT

A1 A2

A3A4

Z1

Anchor nodeTarget node

27

Trilateration Using AoA- Hybrid Angle-Distance Positioning -

System model

• NA ≥ η + 1 anchors

• Multi-targets NT ≥ 1

• Differential-AoA βiki=1

• k > N(N+1)/2+2N2

Position estimation• Differential angle β

• Angle-to-distance

dXA1 =√

2r2o − (1− cos(2β))

• Position estimation‖zi − aj‖F = dij , ∀i ∈ IT , j ∈ IA...‖zi − zj‖F = dij , ∀i ∈ IT , j ∈ IT

A1 A2

X

O!c

!

28

Positioning Accuracy

29

Range-based Position Estimation Problem- System model -

Network• NA anchor nodes

• NT target nodes

• Connectivity range RMAX

Measurement

• connectivity, cij

1, dij ≤ RMAX

0, dij > RMAX

• ranging, dij ∼ fij(dij |dij), if cij = 1

• channel, Edij =

dij , LOSdij + bij , NLOS, bij > 0

30

Rigid-Bar Model

Graph

• Connectivity

• Distance values

a b

c d

e

f

Is the graph unique?

31

Mirroring Ambiguity- Symmetric ambiguity -

The white node is subject to a flip-ambiguity.

32

Swing Ambiguity- Folding in the η-dimensions -

a b

c

a b

c

a b

c

d

e

f

d

e

f

a b

c d

e

f

d

e

f

Equivalent graphs with incongruent nodesNotice the distances between (c,e) and (c,f)

33

Higher-dimensional Ambiguity- Folding in higher dimension -

2D Network Representation

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

3D Network Representation

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

34

Position Information

Theorem: Generalized Information Matrix Decomposition (Destino, ’12)

In a network with NA = η + 1 anchors, NT targets and connectivity C, theposition error bound to the location of the k-th node is given by the inverse of

Sk ,NA∑n=1

ζnkΥnk︸ ︷︷ ︸anchor-to-target information

+

k−1∑n=NA+1

ζnkΥnk

︸ ︷︷ ︸target-to-target information

−QTkG−1

k−1Qk︸ ︷︷ ︸equivocation

,

where Gk−1 and Qk are obtained by partitioning Fd as

Fd =

[Gk−1 Qk

QTk Fkd,

].

• ζnk, Ranging Information Intensity (RII)

• Υnk, Ranging Direction Matrix (RDM)

35

Equivocation Matrix

Theorem: Decomposition of the Equivocation Matrix (Destino, ’12)

Consider a network with NA anchors and NT targets, the equivocation matrixof the k-th target node, denoted by Ek, with k = N can be decomposed as

Ek =

k−1∑i=ma

ζeikΥik

︸ ︷︷ ︸link uncertainty

+

k−1∑i=ma

k−1∑j=maj 6=i

κkjikΥkjik

︸ ︷︷ ︸coupling uncertainty

,

where ma = NA + 1 and sa = max (i, j) + 1.

36

Impact of the Information Coupling- Benefits of node cooperation -

!10 !8 !6 !4 !2 0 2 4 6 8 10!10

!8

!6

!4

!2

0

2

4

6

8

10

A1

A2

A3

Z1

Z2

Z3Z4

A1

A2

A3

Z1

Z2

Z3Z4

Investigation of the Information Coupling

- cooperative network -

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeError with couplingError w/o coupling

Decoupling by disconnection

(c46 = 0, c56 = 0) → (ζ46 = 0, ζ56 = 0) → κ7567 = 0.

37

Impact of the Information Coupling- Anchor placement -

!15 !10 !5 0 5 10 15!10

!8

!6

!4

!2

0

2

4

6

8

10

12

De-coupling via Anchor Nodes

- cooperative network -

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeError Ellipse for m1

Error Ellipse for m2 < m1

Z22 Z21

Z20Z19

!15 !10 !5 0 5 10 15!10

!8

!6

!4

!2

0

2

4

6

8

10

12

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeError Ellipse for m1

Error Ellipse for m2 < m1

Z22 A6

Z20Z19

Decoupling by anchor replacement

• κkjik = ζikζjkχij , i = 20, j = 21, k = 22.

• χij =j−1∑

t=NA+1

ζtj(vik[G−1

j ]ηitvTtj

) (vtjS

−1j vT

kj

)• Z21 → A6 → S−1

j = 0→ χij = 0→ κkjik = 0

38

Impact of RII in the Position Information

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

100

101

102

Ranging Information Intensity

Maximum bias, bMAX [m]

Ran

ging

Info

rmat

ion

Inte

nsi

ty,! i

j

"ij = 0.1 [m]

"ij = 0.25 [m]

"ij = 0.5 [m]

"ij = 1 [m]

Sk ,NA∑n=1

ζnkΥnk︸ ︷︷ ︸anchor-to-target information

+

k−1∑n=NA+1

ζnkΥnk

︸ ︷︷ ︸target-to-target

− Ek︸︷︷︸equivocation

39

Ranging in NLOS Channels

LOS NLOS NLOS2

TF407 TF406

TF404

TF405

eLab

A1 8

A3

A2

4

7 12

11

9

5

6 10

y

x −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

5

10

15

20

25

Ranging error (in meters)

Ranging Error Distribution

pdf

d7,12

: LOSd2,12

: NLOSd4,12

: NLOS2

Fitting

40

Information of NLOS links- Discard or do not discard? -

0 5 10 15 20 25 30 35 40 45 500.025

0.05

0.075

0.1

0.125

0.15

Probability of NLOS links, pNLOS [%]

Mea

n-s

quar

e-er

ror,

MSE

[m2]

PEBLOS

PEBNLOS

PEBLOS with incompletion

Simulation: NA = 4 anchors, NT = 10 targets, σd = 0.3 meters, bMAX = 3 meters. Full connectivity.

Metric: MSE = E(Z− Z)2

Answer: Do not discard NLOS!

41

Non-Cooperative Positioning

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

Scenario of Non-cooperative Positioning

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget node

42

Maximum-Likelihood - Weighted Least Squares

• Independent measurements

• Anchor-to-target ranging

• Gaussian model

Non-Cooperative Maximum Likelihood Formulation:

~z = max~z∈RηNT

K

NA∏i=1

N∏j=NA+1

exp

− (dij − ‖ai − zj‖2)2

2σ2ij

︸ ︷︷ ︸

anchor-to-target

Non-Cooperative Weighted Least-squares Formulation:

~z = min~z∈RηNT

NA∑i=1

N∑j=NA+1

wij

(dij − ‖ai − zj‖F

)2︸ ︷︷ ︸

anchor-to-target

43

Illustration of the Log-Likelihood Function- 2-D, 4-Anchors and 1-Target -

Perfect Measurements Noisy Measurements

44

The Least-Square Formulation Revised

NA∑i=1

(di − ‖ai − z‖F

)2=

NA∑i=1

(‖ai − z‖F + ρi − ‖ai − z‖F)2

When the variables ρis are not harmful?

45

Linear Algebra Intuition- The null space -

Property: Noise in the Null-spaceLet n denote a perturbation vector and assume that n lies in thenull-space of A, i.e. n ∈ N (A). Then,

A(x+ n) = Ax

46

Illustration of the Null-Space Analysis- Distance contraction principle -

Exact ranging

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeGlobal optimum

Positive ranging errors

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeGlobal optimum

Negative ranging errors

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeGlobal optimum

Error in the null-space of the angle-kernel

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget nodeGlobal optimum

47

Robust Non-Cooperative Positioning- Distance contraction based algorithms -

Algorithm 1 WC-DC

1: Measurements, di,2: Anchor positions, PA

3: Estimate a feasible region, BD4: z0 ← BD;5: Ω← O([PA; ~z0]);6: ρ← arg min

ρ∈RNAρ Ω ρT;

s.t. di + ρi ≤ 0 ∀i7: [ωdc]i ← ρi/di;8: z← ωdcPA.

Algorithm 2 NLS-DC

1: Measurements, di,2: Anchor positions, PA

3: Estimate a feasible region, BD4: z0 ← BD;5: Ω← O([PA; ~z0]);6: ρ← arg min

ρ∈RNAρ Ω ρT;

s.t. di + ρi ≤ 0∀i

7: z← arg minz∈Rη

NA∑i=1

(di− ρi− di)2.

48

Non-Cooperative Positioning- Algorithm comparisons -

0 1 2 3 4 50.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

3.5

3.75

Comparison of Di!erent Localization Algorithms

- non-cooperative network -

Maximum Bias, bMAX [m]

Loca

tion

accu

racy

,! !

[m]

TS-WLSWC-DCNLS-DCPEBNLOS

Network: Area (14.14× 14.14) [m2], 4 anchors (square location), 10 targets (inside the anchors).

Noise: σd = 0.3 [m], pNLOS = 1. Location accuracy: ε` =

√E(Z− Z)2 [m].

49

Non-Cooperative Positioning- Algorithm comparisons -

0 0.25 0.5 0.75 10.25

0.5

0.75

1

1.25

1.5

1.75

2

Comparison of Di!erent Localization Algorithms

- non-cooperative network -

Probability of NLOS, pNLOS

Loca

tion

accu

racy

,! !

[m]

TS-WLSWC-DCNLS-DCPEBNLOS

Network: Area (14.14× 14.14) [m2], 4 anchors (square location), 10 targets (inside the anchors).

Noise: σd = 0.3 [m], bmax = 3 [m]. Location accuracy: ε` =

√E(Z− Z)2 [m].

50

Cooperative Algorithm

!8 !6 !4 !2 0 2 4 6 8!8

!6

!4

!2

0

2

4

6

8

Scenario of Cooperative Positioning

x-coordinate, [m]

y-c

oor

din

ate,

[m]

Anchor nodeTarget node

51

Maximum-Likelihood - Weighted Least Squares

• Independent measurements

• Target-to-target cooperation

• Gaussian model

Cooperative Maximum Likelihood Formulation:

~z = max~z∈RηNT

K

NA∏i=1

N∏j=NA+1

exp

− (dij − ‖ai − zj‖2)2

2σ2ij

︸ ︷︷ ︸

anchor-to-target

N∏j=NA+1

N∏j=NA+1j 6=i

exp

− (dij − ‖zi − zj‖2)2

2σ2ij

︸ ︷︷ ︸

target-to-target

Cooperative Weighted Least-squares Formulation:

~z = min~z∈RηNT

NA∑i=1

N∑j=NA+1

wij

(dij − ‖ai − zj‖F

)2︸ ︷︷ ︸

anchor-to-target

+N∑

i=NA+1

N∑j=NA+1j 6=i

wij

(dij − ‖zi − zj‖F

)2︸ ︷︷ ︸

target-to-target

52

Facts of the WLS Optimization Problem

Number of local minima grows with:

• number of nodes N ,

• the lack of connections,

• the noise.

Robustness to measurement errors can be achieved by:

• adding constraints (hard mitigation method),

• using weights (soft mitigation method).

Optimization complexity grows with:

• number of nodes N ,

• the lack of connections,

• the lack of a priori information,

• number of costraints.

53

Global Optimization- Smoothing continuation method -

!5 0 5 10

!8

!7

!6

!5

!4

!3

!2

!1

0

Optimization via GDC Technique

- sum of Gaussian functions -

Optimization variable, x

Obje

citv

efu

nct

ion,!g

" !

Estimated minimumSmoothed objectiveOriginal objective

smooth

ing

param

eter,!

#0

• Smooth, g(x)→ 〈g〉λ(x)

• Minimize, xm = minx〈g〉λ(x)

• Continue (minimum tracking), λ′ < λ and x0 = xm

54

Efficient Implementation of the Optimization Method- Range-Global Distance Continuation -

• Smoothed function and gradient in closed-forms

• The first smoothed function is convex

• Random initialization

• Minimization without matrix inversions (BFGS)

• Decreasing smoothing paramters

55

Cooperative Positioning- R-GDC optimization performance -

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

R-GDC Performance in Large Scale Networks

- cooperative network -

Meshness ratio, m

Loca

tion

accu

racy

,! !

[m]

R-GDCPEBLOS

NT = 50

NT = 100

NT = 200

Network: Area (14.14× 14.14) [m2], 4 anchors (square location), NT targets (inside the anchors).

Noise: σd = 0.3 [m]. s Location accuracy: ε` =

√E(Z− Z)2 [m].

56

Weighing Function- Heuristic strategies -

“Weights are chosen to reflect differing levels of concernabout the size of the squared error terms.

Higher weights to more reliable measurements, less to othersand zero the unmeasured.”

• Inverse of the noise variance (optimal in zero-mean Gaussian model)

• Inverse of the squared-ranging (simple and effective in small scenarios)

• Locally weighted Scatterplot Smoothing (LOESS)-based(emphasize shorter connections rather than long ones)

• Channel-based (feasible if propagation model is available)

57

Ranging in a Realistic Environment

LOS NLOS NLOS2

TF407 TF406

TF404

TF405

eLab

A1 8

A3

A2

4

7 12

11

9

5

6 10

y

x −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

5

10

15

20

25

Ranging error (in meters)

Ranging Error Distribution

pdf

d7,12

: LOSd2,12

: NLOSd4,12

: NLOS2

Fitting

• Ranging statistics are spatial-time variant

• Long distance can be more accurate than short ones, e.g d2,12 vs d4,14

• Distributions are not generally Gaussian (see S-V model)

• Channel-statistics are not practical with off-the-shelf devices

58

Stochastic-Geometric Weighing Function

... wij is the confidence that the true distance dij is within a confidence boundof ±γ of the mean estimate dij , weighted by a penalty Pij on the hypothesis

that the samples dij,k are obtained under LOS conditions.

wij = Prdij − γ ≤ dij + cij ≤ dij + γ

, ∀eij ∈ E

= Prdij − γ ≤ dij ≤ dij + γ

· Pr cij = 0

= Prdij − γ ≤ dij ≤ dij + γ

· Pij

= Dispersion · Penalty = wDij · Pij

wherePij → 1 Hypothesis of LOS is truePij → 0 Hypothesis of LOS is false

59

Stochastic (Dispersion) Weighing Function- The intuition behind: sort categorical data -

Pool of objects

Object characteristics

• shape, σ

• density, K

Metric

• weight: w = f(σ,K; γ)

Scale

Which unit:? [γ]

60

Maximum Entropy CriteriaOur context:• Categories → (K, σ)

• Sample → zij = (Kij , σij)

• Weight of zij → wij

• Population → Z = [Kmin,Kmax]× [σmin, σmax]

Diversity of the the objects by the weights is

H(γ) =

Kmax∑r=Kmin

σmax∫σmin

w(S, r; γ) · ln (w(S, r; γ)) dS

• Uncertainty analysis: measure wij and compute H (diversity)

• Weight optimization: compute wij that maximizes H (diversity)

γopt = arg maxγ∈R+

H(γ)

61

Geometric (Penalty) Weights: Concept- Handling NLOS with scarce information -

Rationale:

• Higher confidence = higher weights

• Pij → 1⇒ LOS; Pij → 0⇒ NLOS;

• Kij → 1⇒ insufficient LOS/NLOS information in dijConcept: Relate likelihood of NLOS with a geometric effect captured byneighboring nodes, e.g., obtuseness of triangles.

i j

q

i j

q

i j

q

62

Experimental Test

63

Distance Estimation Indoors- Experiment with UWB ToA-based ranging -

CWC/Oulu Installations

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

9

10

11

12

p1

p2

p3

p4p5

p6

p7

Localization Test

- network -

x-coordinate, [m]

y-c

oor

din

ate

[m]

eLAB

AnchorTarget

64

Ranging Statistics- Experiment with UWB ToA-based ranging -

Link Cluster 1 Cluster 2i-th node j-th node Ag1 µg1 σg1 Ag2 µg2 σg2

1 5 0.04 0.17 0.03 0.88 0.45 0.091 6 1.00 0.78 0.19 - - -1 7 1.00 0.00 0.02 - - -2 5 0.96 -0.03 0.005 0.01 -0.02 0.0012 6 1.00 0.27 0.37 - - -2 7 1.00 0.49 0.15 - - -3 5 - - - - - -3 6 1.00 0.11 0.02 - - -3 7 0.02 1.29 0.06 0.98 2.03 0.094 5 1.00 0.51 0.10 - -4 6 1.00 0.22 0.06 - -4 7 1.00 0.62 0.05 - -5 6 1.00 0.44 0.07 - -5 7 1.00 0.12 0.04 - -6 7 0.05 -0.08 0.002 0.88 -0.05 0.09

65

Non-Cooperative Positioning- Experiment with UWB ToA-based ranging -

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

9

10

11

12

p1

p2

p3

p4p5

p6

p7

p1

p2

p3

p4p5

p6

p7

Localization Test- non-cooperative -

x-coordinate, [m]

y-c

oor

din

ate

[m]

AnchorTargetC-NLS modifiedDC-NLSDC-WC

Metric NLS-DC WC-DC C-NLS modifiedAverage RMSE [m] 0.29 0.27 0.38Average CEP-50 [m] 0.24 0.23 0.31Average CEP-95 [m] 0.41 0.37 0.47

66

Cooperative Positioning- Experiment with UWB ToA-based ranging -

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

9

10

11

12

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

p12

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

p12

Localization Test

- cooperative -

x-coordinate, [m]

y-coo

rdinate[m

]

AnchorTargetSDP - CSDP - LOESSR-GDC - DP

Metric R-GDC - DP SDP - LOESS SDP - CAverage RMSE [m] 0.29 0.33 0.40Average CEP-50 [m] 0.16 0.27 0.37Average CEP-95 [m] 0.46 0.56 0.63

67

Thank YouPh.D. Thesis “Positioning in Wireless Networks”

Defence: 16/11/2012, Op-Sali L10

Author: Giuseppe Destino, University of OuluAdvisor: Prof. Giuseppe Abreu, Jacobs Universtiy, Germany

Supervisor: Prof. Jari Iinatti, University of Oulu

68