Ad Hoc Positioig System (APS) Usig AOA Dragoş Niculescu ad Badri Nath DATAMAN Lab Rutgers Uiversity email: {dicules,badri}@cs.rutgers.edu Abstract Positio iformatio of idividual odes is useful i implemetig fuctios such as routig ad queryig i ad-hoc etworks. Derivig positio iformatio by usig the capability of the odes to measure time of arrival (TOA), time differece of arrival (TDOA), agle of arrival (AOA) ad sigal stregth have bee used to localize odes relative to a frame of referece. The odes i a ad-hoc etwork ca have multiple capabilities ad exploitig oe or more of the capabilities ca improve the quality of positioig. I this paper, we show how AOA capability of the odes ca be used to derive positio iformatio. We propose a method for all odes to determie their orietatio ad positio i a ad-hoc etwork where oly a fractio of the odes have positioig capabilities, uder the assumptio that each ode has the AOA capability. Idex Terms ad hoc etworks, positioig, orietatio, digital compass, AOA. I. INTRODUCTION The mai features of ew adhoc etworks are large umber of uatteded odes with varyig capability, lack or impracticality of deployig supportig ifrastructure, ad high cost of huma supervised maiteace. What is ecessary for these types of etworks is a class of algorithms which are scalable, tuable, distributed, ad easy to deploy. With recet advaces i small device architectures [1], it ca be foresee that cheap, or eve disposable odes, will be available i the future, eablig a array of ew agricultural, meteorological ad military applicatios. These large etworks of low power odes face a umber of challeges: cost of deploymet, capability ad complexity of odes, routig without the use of large covetioal routig tables, adaptability i frot of itermittet fuctioig regime, etwork partitioig ad survivability. It is a give that i may of these etworks, due to cosideratios of cost, size, ad power requiremets, idividual odes will have varyig capabilities. A geeral questio is how to export capabilities to various odes i the etwork so that the overall capability ca be icreased i the etwork. For example, may ad hoc etwork applicatios ad protocols assume the kowledge of geographic positio of odes. The absolute positio of each etworked ode is a assumed fact by most sesor etworks which ca the preset This research work was supported i part by DARPA uder cotract umber N-6661--1-8953 the sesed iformatio o a geographical map. Also, the availability of positio would eable routig i sufficietly isotropic large etworks, without the use of large routig tables. However, ot all odes have the capability of locally determiig their positio by meas of GPS. Fidig positio without the aid of GPS i each ode of a ad hoc etwork is importat i cases where the GPS service is either ot accessible, or ot practical to use due to power, form factor or lie of sight coditios such as idoor sesors, sesors hidde uder foliage, etc. A similar argumet holds for orietatio as compasses face erratical behavior i the viciity of large metal objects or electrical fields. Orietatio, or headig, is used i remote avigatio, or remote cotrol of specialized sesors, such as directioal microphoes or cameras. I this paper, we address the problem of self positioig ad orietatio of the odes i the field, which may provide a geeral framework for exportig capabilities i a etwork where more capable odes cooperate i dispersig iformatio to less capable odes. What is ecessary for ad hoc deploymet of temporary etworks is a method similar i capability to GPS ad magetic compasses, without requirig extra ifrastructure, or extesive processig capabilities. What we propose is a method by which odes i a ad hoc etworks collaborate i fidig their positio ad orietatio uder the assumptios that a small fractio of the etwork has oly the positio capability. A compass is ot ecessary i ay ode, but if it is available, either at the ladmarks, or everywhere, it will ehace the accuracy of the positioig algorithm. Previous positioig methods used so far used either TDOA, like i Cricket[2] ad AhLOS[3], or sigal stregth (RADAR[4], APS[5]). What makes our approach differet from previous oes is that it is based o the capability of the odes to sese the directio from which a sigal is received, which is kow as agle of arrival (AOA). AOA sesig requires either a atea array, or several ultrasoud receivers, but besides positioig, it also provides the orietatio capability. This is curretly available i small formats i wireless etworked odes such as the oe developed by the Cricket Compass project[6] from MIT. I fact, APS usig AOA is part of a larger effort to provide positioig based o multimodal sesig. The
aim is to show that ad hoc positioig is possible usig various localizatio capabilities (ragig, AOA, compasses), idepedetly, or together. Oe sceario ivolvig sesor etworks frequetly metioed i literature is that of aircraft deploymet of sesors followed by i flight collectio of data by simply cruisig the sesor field. This ad other meteorological applicatios are implicitly assumig that the data provided by the sesor is accompaied by the sesor s positio. It is thus possible to attach the sesed iformatio to a geographical map of the moitored regio. If this is a absolute ecessity for makig sese of the observed data, accurate positio might also be useful for routig ad coordiatio purposes. For some ad hoc etworks, algorithms such as Cartesia routig[7], or geocast[8], eable routig with reduced or o routig tables at all, ad are appropriate for devices like the Ree mote[1], with oly half a kilobyte of RAM. A improvemet that ca be applied to some ad hoc routig schemes whe positio is available, Locatio Aided Routig [9] limits the search for a ew route to a smaller destiatio zoe. Our positioig ad orietatio algorithm is appropriate for idoor locatio aware applicatios, whe the etwork s mai feature is ot the upredictable, highly mobile topology, but rather temporary ad ad hoc deploymet. These etworks would ot justify the cost of settig up a ifrastructure to support positioig, like proposed i [1], [4], or [2]. The orietatio ad positioig problems have bee extesively studied i the cotext of mobile robot avigatio [11]. However, may methods proposed by the robotics commuity make extesive use of image processig ad preset ifrastructure, such as recogizable ladmarks. Our aim is a positioig method that is robust, but relies o less computatioal resources ad ifrastructures. The rest of the paper is orgaized as follows: the ext sectio describes the assumptios of the problem ad the basic properties of AOA capable odes. Sectio III presets our proposed approach, with the orietatio forwardig scheme, ad sectio IV discusses some error cotrol issues. Sectios V ad VI preset simulatio results ad discuss some mobility related issues, ad VII summarizes with some cocludig remarks. II. AOA THEORY The etwork is a collectio of ad hoc deployed odes such that ay ode ca oly commuicate directly with its immediate eighborig odes withi radio rage. I the ideal case, whe radio coverage of a ode is circular, these etworks are modeled as fixed radius radom graphs. Each ode i our etwork is assumed to have oe mai axis agaist which all agles are reported ad the capacity to estimate with a give precisio the directio from which a eighbor is sedig data. We assume that after the deploymet, the axis of the ode has a arbitrary, Fig. 1. Nodes with AOA capability ac A 11 C ab 11 ca cb bc ba b 11 1 1 11 North ukow headig, represeted i figure 1 by a thick black arrow. Some of the odes, from here o called ladmarks, have additioal kowledge about their positio from some exteral source, such as a GPS receiver or huma iput. The term bearig refers to a agle measuremet with respect to aother object. I our case, the AOA capability provides for each ode bearigs to eighborig odes with respect to a ode s ow axis. A radial is a reverse bearig, or the agle uder which a object is see from aother poit. We will use the term headig with the meaig of bearig to orth, that is, the absolute orietatio of the mai axis of each ode. I figure 1, for ode B, bearig to A is ba, radial from A is âb, ad headig is b. The problem to be solved is: give imprecise bearig measuremets to eighbors i a coected ad hoc etwork where a small fractio of the odes have self positioig capability, fid headigs ad positios for all odes i the etwork. The difficulty of the problem stems from the fact that the capable odes (ladmarks) comprise oly a small fractio of the etwork, ad most regular odes are odes are ot i direct cotact with eough ladmarks. What we are lookig for is a hop by hop method to export capabilities from the capable odes to the regular oes. Whe iteractig with two eighbors, as show i figure 1, a ode ca fid out the agle betwee its ow axis ad the directio the sigal comes from. Node A sees its eighbors at agles âc ad âb, ad has the possibility of iferrig oe agle of the triagle, CAB = âc- âb. For cosistecy all agles are assumed to be measured i trigoometric directio. Node A ca also ifer its headig, if headig of oe of the eighbors, say B, is kow. If ode B kows its headig (agle to the orth) to be b, the A may ifer its headig to be 2π ( ba + π âb)+ b. Thisi fact a way to export the compass capability from B to A. If however, o compass is available i ay ode, but each ode kows its positio, headig ca still be foud because the orietatios for the sides of the triagle ca be foud from positios of its vertices. What is therefore eeded is B
Fig. 2. Basic priciple of Cricket compass (adapted from[6]). Fig. 3. Positioig by measurig agles to ladmarks A(x a,y a ) D(x, y) x1 x x2 B(x b,y b ) C(x c,y c ) θ L/2 θ a positioig algorithm based o AOA, orietatio beig available by oe of the meas metioed above. A. AOA capable odes AOA capability is usually achieved by usig a atea array, which might be prohibitive i size ad power cosumptio. A small form factor ode that satisfies coditios outlied i the previous sectio has bee developed at MIT by the Cricket Compass project[6]. Its theory of operatio is based o both time differece of arrival (TDoA) ad phase differece of arrival. Time differece is used i a similar maer i other projects, such as AhLOS[3] ad Cricket Locatio[2], ad is based o the six orders of magitude differece betwee the speeds of soud ad light. If a ode seds a RF sigal ad a ultrasoud sigal at the same time, the destiatio ode might ifer the rage to the origiatig ode based o the time differece i arrivals. I order to get the agle of arrival, each ode may use two ultrasoud receivers placed at a kow distace from each other, L (figure 2). By kowig rages x 1, x 2, ad distace L, the ode is able to ifer the orietatio θ, with a accuracy of 5 whe the agle lies betwee ±4. Medusa, used i AhLOS project[3] from UCLA, is aother wireless etworked ode with small size which makes use of several ultrasoud receivers, but without actually employig them to detect agle of arrival. These icipiet realizatios prove that it is feasible to get AOA capability i a small package that would be appropriate for future pervasive ad hoc etworks. B. Triagulatio Usig AOA The cetral observatio suggestig that positioig usig AOA is possible is that the followig: if we kow the positios for the vertices of a triagle ad the agles at which a iterior poit sees the vertices, we ca determie the positio of the iterior poit. This problem, called triagulatio, is somewhat similar to the trilateratio problem, used i GPS[12]. The differece is that the iterior poit kows agles towards triagle sides istead L/2 of distaces to vertices. I figure 3, if beside coordiates of A, B ad C, ode D kows distaces DA, DB ad DC, it ca use trilateratio to ifer its positio.o the other had, if it kows the agles BDA, ADC, ad CDB it ca fid its positio usig triagulatio. This is doe by fidig the itersectio of the three circles determied by the ladmarks ad the kow agles. Iformatio from several ladmarks ca be used to get a least square error solutio, because i the geeral case, AOA measuremets do ot have perfect accuracy. There are several possibilities to compute this estimated poit of itersectio. For this explaatio it is useful to review how positioig is doe usig trilateratio, whe distaces to kow ladmarks are kow. Give (x i,y i ) ad d i, the coordiates of ad respectively the distace to ladmark i, we build the oliear system (x x i ) 2 +(y y i ) 2 = d 2 i i =1,..., I Global Positioig System[12], the system is solved usig oliear methods based o successive approximatios, but it also ca be solved by reductio to a liear system by subtractig oe equatio from the rest. I this latter case, we obtai equatios 2x(x i x 1 )+2y(y i y 1 )= = d 2 1 d 2 i + x 2 i x 2 1 + yi 2 y1 2,i=2,..., This liear system ca be solved usig stadard methods for over-determied systems, such as the pseudo-iverse. Gettig back to our triagulatio problem, it ca be reduced to trilateratio by some simple trasformatios. If for example a ode D kows the agle to a pair of ladmarks A ad B, it may ifer that its positio is somewhere o the circle determied by the agle ad the positio of the two ladmarks (figure 4). What is fixed i this picture is the ceter of the circle, O, whose positio may be determied whe x a,y a,x b,y b ad agle ADB are kow. This may help i trasformig a triagulatio problem of size ito a trilateratio problem of size ( ) 2 if for each pair of ladmarks observed by a ode we create a trilateratio equatio usig x, y, x,y ad the radius of the circle as the distace. Aother possibility is to form all triplets of obtaied ladmarks ad fid the ceter of the
Fig. 4. Geometric locus of a poit seeig two kow ladmarks at a give agle, is a circle. ÂOB =2ÂDB D(x, y) O(x,y ) D(x, y) B(x b,y b ) A(x a,y a ) circumscribed circle for each such triplet ad the ukow poit D. This leads to the solvig of ( ) 3 trilateratio problems of size 3, oe for each circle, with ( ) 3 solutio poits. For small umbers of ladmarks, the ( ) 3 has a similar CPU complexity as the ( ) 2 oe. However, the problems solved are all of size 3, thus requirig much less memory, whereas the ( ) 2 approach eeds to hadle 2 2 x2 sized matrices. A solutio liear i the umber of ladmarks, proposed i [11], makes efficiet use of the represetatio of ladmarks as complex umbers. I our simulatio we used the simple ( ) 3 implemetatio, as it gives the same quality estimates as the liear solutio preseted i [11], but it is much more simple to implemet ad has a low pealty for small. Aother method of positioig usig agles is VOR (VHS Omi-directioal Rage), which is curretly still the mai aid for aircraft avigatio. Its priciple is very simple: a ladmark seds two sigals, oe that is periodic ad omi-directioal, while the aother oe is directioal ad is rotated about the ladmark. The airbore equipmet receives both sigals, ad iterprets the differece as a radial from the statio. The coordiates of the statio are kow, therefore placig the mobile aywhere o a give lie. A secod VOR readig provides a secod lie to be itersected with the first. Give (x i,y i,r i ) the coordiates ad the radial to the ladmark i, a ode ca build the equatio of the lie a i x + b i y = c i o which it places itself. if cos(r i )= a i =1;b i =;c i = x i else a i = ta(r i ); b i = 1; c i = y i + x i ta(r i ) Combiig all such lies to ladmarks, the liear system to be solved for a locatio is: [ a T b ] [ ] T x = [ c ] y T This approach is less expesive computatioally, for ladmarks requirig just a weighted least square liear system solvig. What makes it slightly differet form the previous oe is the fact that the ladmark should be equipped with a compass, so that it reports all radials agaist a well kow directio, such as orth. The bearig method, o the other had does ot require ay compass at all, but still provides positioig ad orietatio for the Fig. 5. Node A ifers its bearig to L by corroboratig B s ad C s bearigs to L A c b all odes. C B III. AD HOC POSITIONING SYSTEM (APS) ALGORITHM The problem i a ad hoc etwork is that a ode ca oly commuicate with its immediate eighbors, which may ot always be ladmarks (ladmarks are odes which kow their positio ad possibly their headig). APS [5] is a hybrid betwee two major cocepts: distace vector (DV) routig ad beaco based positioig (GPS). What makes it similar to DV routig is the fact that iformatio is forwarded i a hop by hop fashio, idepedetly with respect to each ladmark. What makes it similar to GPS is that evetually each ode estimates its ow positio, based o the ladmark readigs it gets. The origial APS cocept has bee show to work usig rage measuremets, but is i fact extesible to agle measuremets. What we propose is a method to forward orietatio so that odes which are ot i direct cotact with the ladmarks ca still ifer their orietatio with respect to the ladmark. Here, orietatio meas bearig, radial, or both. We examie two algorithms, DV-Bearig, which allows each ode to get a bearig to a ladmark, ad DV-Radial, which allows a ode to get a bearig ad a radial to a ladmark. The propagatio works very much like a mathematical iductio proof. The fixed poit: odes immediately adjacet to a ladmark get their bearigs/radials directly from the ladmark. The iductio step: assumig that a ode has some eighbors with orietatio for a ladmark, it will be able to compute its ow orietatio with respect to that ladmark, ad forward it further ito the etwork. What remais to be foud is a method to compute this iductio step, both for bearigs ad radials. A. Orietatio Forwardig The method is show i figure 5: assume ode A kows its bearigs to immediate eighbors B ad C (agles b ad ĉ), which i tur kow their bearigs to a faraway ladmark L. The problem is for A to fid its bearig to L N L
(dashed arrow). If B ad C are eighbors of each other, the A has the possibility to fid all the agles i triagles ABC ad BCL. But this would allow A to fid the agle LAC, which yields the bearig of A with respect to L, asĉ + LAC. Node A might accept aother bearig to L from aother pair of eighbors, if it ivolves less hops tha the pair B C. A the cotiues the process by forwardig its estimated bearig to L to its eighbors which will help farther away odes get their estimates for L. Forwardig orietatios is doe i a fashio similar to distace vector routig algorithms. I our case, the ladmarks are the oes startig the update messages that are propagated throughout the etwork, for each ladmark idepedetly. Oce ode A fids its bearigs to at least three ladmarks that are ot o the same lie or o the same circle with A, it ca ifer its positio usig oe of the methods outlied i sectio II-B. If the radial method is to be used, a similar argumet holds, with the differece that ow A eeds to kow, besides bearigs of B ad C to L, the radials of B ad C from L. If the agle BLN (radial at B) is also kow, the the agle ALN (radial at A) ca also be foud sice all agles i both triagles are kow. The actual dowside for this method is i the icreased amout of sigalig - odes B ad C forward two values per ladmark (bearig ad radial) istead of just oe, as i the bearig based method. If a compass would be available i every ode, the two methods would i fact become idetical because whe all agles are measured agaist the same referece directio (orth, for example), bearig = π radial. The algorithm has similar sigallig overhead behavior with the origial APS [5] algorithm (rage based), which is roughly a TTL limited floodig per ladmark. The followig table summarizes for each method the required ode capabilities ad associated sigalig-accuracy tradeoffs. More sigallig refers to the fact that two values are eeded per ladmark, whereas less seds oly oe. I the case of a large existig packet overhead, oe extra value may be of dimiished importace. The accuracy of the two propagated methods will be quatified more precisely i the simulatio sectio (V). compass method sigalig accuracy owhere DV-Bearig less less oly at DV-Bearig less less ladmarks DV-Radial more more all odes DV-Radial less more B. Network desity The questio that arises i deploymet of the etwork is what kid of ode desity is eeded i order to achieve a certai coditio with high probability. It has bee cojectured[13] that may radom graph properties exhibit phase trasitios - sharp icreases i the probability whe the desity icreases beyod a certai poit. For example, Fig. 6. Probability for a ode to satisfy coditios ecessary for orietatio forwardig success probability 1.9.8.7.6.5.4.3.2.1 4 5 6 7 8 9 1 11 ode degree it has bee prove that a radom etwork i the plae eeds a degree of about 6 i order to have complete coectivity with high probability. We expect the degree requiremet to be higher i our case, sice more tha simple coectivity is eeded for the orietatio propagatio to work. For a bearig to be propagated, two eighbors that are also eighbors of each other should be preset for ay give ode. I figure 6, we ca see that whe the mea degree of a ode icreases beyod 9, with a very high probability it will meet the coditios to forward orietatio. This data is empirically obtaied by ruig our forwardig policy i a etwork of 1 odes with a sigle ladmark ad the cout the umber of odes which get a bearig to it. Variatio of the average degree is achieved by icreasig the radio rage of the odes. I the case of a sesor etwork, it is ofte evisioed that the deployed desity is higher tha eeded to allow for extesio i battery life by tuig the duty cycle. This meas that a iitial degree of 9 might be tolerable (5% more odes have to be deployed), as the ormal fuctioig regime ca be later lowered to 6, which has bee show to be the miimum for coectivity [13]. IV. ERROR CONTROL Beig that all bearig measuremets are affected by errors, the forwardig may actually amplify ad compoud smaller errors ito larger errors. A umber of simple techiques may be employed to reduce the propagatio of such errors, icludig: avoidig iferece based o small agles or o degeerate triagles, limitig the propagatio of DV packets with a simple TTL scheme ad elimiatio of the outliers i positio estimatio. The fact that agle measuremets are affected by error greatly iflueces the very core of our algorithm: bearig propagatio. As the eviromet we evisio for this positioig algorithm is a low power, low commuicatio capacity ode, error cotrol methods employed have to be lightweight. Together, the three metioed methods achieve a error reductio of about half.
Fig. 7. Error is compouded with propagatio Fig. 9. Removig the outliers from cetroid computatio 25 o 12 1 2 o 8 bearig error [deg] 15 o 6 4 True locatio Cetroid of all poits 2 Cetroid of half 1 o 5 1 15 2 25 3 TTL [#hops] 2 2 4 6 8 1 12 14 16 Fig. 8. positioig error [# of hops] 2.5 2 1.5 1.5 Agle threshold provides a tradeoff betwee error ad coverage thr=.5 thr=.25 5 1 coverage [%] thr=.15 thr=.1 The first ituitive remark is that error cumulates with distace, because of the way bearigs are propagated. We verified this fact i a etwork of 2 odes, by plottig the average bearig error as a fuctio of distace i hops to the respective ladmark (figure 7). Limitig the propagatio of the DV packets usig a TTL scheme is a good idea ot oly for error cotrol reasos, but also for reducig commuicatio complexity. If TTL is ifiite, each ladmark is floodig the etire etwork with its coordiates, thus triggerig bearig computatio at every other ode. Therefore, the TTL is the mai feature that makes the proposed algorithm scalable. As log as eough ladmarks ca be acquired from the area allowed by the TTL, the total size of the etwork does ot ifluece the amout commuicatio or the quality of the estimates. The ext key observatio is that small agles are more error proe tha large agles. It is preferable to deal with equilateral triagles tha with triagles that have two very acute agles ad oe obtuse agle. The reaso for that is that AOA measuremet errors are theoretically idepedet of the actual agle measured. For example, the same agle error of 5 o will make much more differece i a triagle with a true agle of 1 o tha i oe with 6 o.itisa aalogue situatio with the geometric dilutio of precisio (GDOP) i the GPS, i which the error amplificatio depeds o the ladmark costellatio. To address this problem we use a threshold value to elimiate triagles i which small agles are ivolved. I figure 5, sectio III- A, if ay of the two triagles ABC ad BCL have small agles, A wo t get to propagate orietatio to L. There is a tradeoff betwee coverage ad positioig error, ad this results from the orietatio forwardig policy. A coservative policy would use a high threshold, limitig the computatios with small agles but also limitig the propagatio of orietatios, ad fially reducig coverage. A relaxed policy would propagate almost all agles, ivolvig more errors, but would improve coverage. I figure 8, it is show how varyig from a very coservative forwardig policy (threshold=.5 28 ) to a very relaxed oe (threshold=.1.5 ) achieves differet levels of coverage (success rate) with differet amouts of error. The positioig error represets the average distace i hops from the true positio, obtaied after propagatio (sectio III-A) ad triagulatio (sectio II-B). The coverage represets the fractio of odes successfully obtaiig a positio. The data is obtaied by positioig with differet thresholds, idicated i the figure, i the cofiguratio show i figure 1, with a TTL of 6, 2% ladmarks ad a high AOA measuremet error (stddev =.4 23 ). Aother error cotrol method, suggested i [11], refers to the positio estimatios obtaied from the triplets of ladmarks. I all the metioed methods, several positio estimatios may be obtaied, leadig to the problem of combiig them ito oe sigle estimate. While this ca simply be the cetroid of the estimates, i practice it has bee observed that large errors are clustered together. This is caused by commo agle errors across bearig propagatio paths. The method suggested i [11] is to first compute the cetroid ad the remove the outliers before
Fig. 1. Isotropic topology Fig. 11. Positioig error 89 9 94 92 3 95 88 25 77 26 86 4 93 47 33 27 6 64 16 63 1.8 1.6 gps = 5% 49 82 9 74 99 85 71 97 42 75 7 76 6 69 72 23 65 22 14 98 79 58 73 91 57 52 7 5 39 3 21 66 8 84 59 12 13 positio mea error[# of hops] 1.4 1.2 1.8.6.4 gps = 5% gps = 1% gps = 2% gps = 35% gps = 1% gps = 2% gps = 35% 81 83 2 15 2 53 24 61.2 78 37 35 38 67 34 19 4 54 32 41 1 55 36 51 31 5 o 1 o o 22.5 measuremet error[stddev] DV Bearig DV Radial 45 o 56 68 8 28 45 18 6243 96 5 17 1 87 29 48 44 11 46 Fig. 12. Bearig error recomputig a ew cetroid with the remaiig poits (fig 9). There are more powerful methods available, such as data clusterig ad k-smallest eclosig circle, but they ivolve higher computatioal ad memory complexities, which may ot be applicable to small etworked odes, such as sesor odes. bearig mea error 45 o 22.5 o DV Bearig DV Radial V. SIMULATION We simulated a isotropic 1 map similar with the oe i figure 1 (average degree=1.5, diameter=32), but with 1 odes, each havig a radom, but ukow headig. A fractio of odes are ladmarks, meaig that they have self positioig capability by a exteral method such as GPS. Gaussia oise is added to each AOA estimatio to simulate measuremet errors. Gaussia distributio has the property that 95% of the samples lie withi 1.96 stadard deviatios from the mea. What this meas for agle measuremets is that if the stadard deviatio of the oise is for example π 8, the 95% of the measuremets will be i the iterval ( π 4 ; π 4 ) of the true bearig, thus givig a total spread of π 2 for bearig measuremets. Performace will be evaluated based o the accuracy of positioig for oladmark odes, accuracy of headig, ad percetage of the regular odes which succeed the solvig for a positio (coverage). All the results preseted i this sectio are averaged from 1 rus with differet radomly distributed ladmark cofiguratios over the same etwork. Due to the fact that the proposed algorithms provide differet tradeoffs, i order to produce comparable coverage we ra DV-Bearig with a TTL of 5 ad DV-Radial with a TTL 1 isotropic = havig the same physical properties i all directios (coectivity, desity, ode degree, ladmark distributio) 1 o 5 o 5 o 1 o 22.5 o 45 o measuremet error[stddev] of 4. I both cases the agle threshold was.35 ( 2 ). All performace graphs idicate the stadard deviatio i selected poits. Positioig error (figure 11) is represeted relative to the maximum commuicatio rage of a ode. A error of 1. meas that the positio resulted from the positioig algorithm is oe (maximum sized) radio hop away from its true positio. For DV-Bearig, this positio is obtaied from the bearigs to ladmarks, applyig the triagulatio method metioed i sectio II-B. For DV-Radial, positio is obtaied from the radials by solvig a liear system. O the horizotal axis of the graphs the stadard deviatio of the measuremet oise is varied from to π 4, ad the several curves o each graph correspod to differet ladmark ratios. A larger umber of ladmarks improves both accuracy ad precisio, by solvig a larger system for each positioig problem. For reasoable errors DV- Radial provides better positioig accuracy, ad exhibits less depedece o the percetage of ladmarks. Bearig error (figure 12) is the average error of the
Fig. 13. headig mea error Fig. 14. coverage [%] 45 o 22.5 o 1 8 6 4 2 1 o 5 o Headig error Coverage 5 o 1 o DV Bearig DV Radial gps = 5% gps = 1% gps = 2% gps = 35% o 22.5 measuremet error[stddev] DV Bearig gps = 35% gps = 2% gps = 1% gps = 5% DV Radial 5 o 1 o 22.5 o 45 o measuremet error[stddev] bearig to ladmarks obtaied by regular odes after the orietatio propagatio phase stops. This is a primary measure of how the forwardig method compouds ad propagates error. Because each ladmark is treated idepedetly, bearig errors are ot affected by the umber of ladmarks available i the etwork. As expected, DV- Radial exhibits lower error, maily because of the extra value that is forwarded. Headig is the agle betwee odes axis ad the orth, as would be give by a compass. Headig error is therefore the error i the absolute orietatio averaged over all odes. I our simulatio, it is obtaied by each ode after estimatig a positio. Headig error (figure 13) is about double the bearig error, which is cosistet with the results preseted i [11]. Coverage (figure 14) represets the percetage of o ladmark odes which are able to resolve for a positio. The reasos for which a ode does t get a positio are: fewer tha three ladmarks accumulated (due to propagatio errors), colliear or co-circular ladmarks, or umerical istability i the system solvig. We aimed 45 o for similar coverage for the two algorithms i order to compare the other performace metrics. Eve if positioig is theoretically possible with two ladmarks for DV-Radial ad with three ladmarks for DV-Bearig, i practice, due to agle errors compoudig, a much higher umber of ladmarks might be eeded. The mai observatios to draw from simulatios are the followig: accuracy ca be traded off for coverage by tuig the TTL ad the threshold value. The TTL tradeoff is also betwee eergy ad coverage, as its reductio would lead to less eergy spedig but also to less coverage. Positios obtaied are usable for applicatios such as geodesic routig, as it is showed i [5], with errors of similar scale. Bearig errors follow closely the measuremet oise, but they ca be further decreased usig more sophisticated correctio methods. I order to evaluate the accuracy of positios ad orietatios for a realistic applicatio, we devised a simple example i which a mobile traverses a fixed etwork ad is sesed by odes withi a certai distace. Nodes are iitially deployed radomly, with a fractio of them (2%) havig the self positioig capability. After ruig the APS algorithm to ifer their positio ad orietatio, the odes sesig the mobile report their positio ad the directio i which the mobile was observed. At a cetral locatio, reports from various odes are aggregated to produce a estimate positio of the mobile. Sice both positios ad directios reported by odes are based o APS produced positios/orietatios, ad therefore affected by errors, ad because there may be more tha two reportig odes, the estimate positio of the mobile is obtaied by solvig a over-determied liear system, i order to miimize the square error. I figure 15, the origial trajectory is show with a dashed lie, ad the restored oe with a solid lie. Stadard deviatios are idicated for each sample poit. While more complicated data fusio/predictio techiques (such as Kalma filters) may be used here to improve the estimated trajectory, the purpose of the example is merely to quatify the APS produced error i the positio ad orietatio of the odes, with o additioal processig. The etwork used (fig 1) was a isotropic topology with 1 odes, mea degree 8.18, 2 odes of which have self positioig capability. The measuremet error cosidered was white gaussia oise with a stadard deviatio of.8 radias, which is about double the error of 5 o achieved by the AOA odes realized by the Cricket compass project[6]. The algorithm used to ifer positio ad orietatio is DV- Bearig, which trades off some accuracy i order to work with less sigalig ad fewer capabilities (o compasses aywhere i the etwork). We assumed that the sesig distace is equal to commuicatio radius, so that for each poit we get about 6 or 7 readigs. The sesig agle error is assumed to be, so that all the errors i the restored trajectory quatify the errors i our positioig algorithm
Fig. 15. 88 56 74 89 94 99 71 81 3 49 82 35 25 trackig example isotropic topology 85 95 38 9 78 37 68 83 8 97 92 75 7 28 67 34 42 45 9 2 77 26 76 6 18 69 15 19 96 22 86 6243 5 14 4 65 98 4 23 47 79 2 32 33 54 17 1 87 53 73 27 58 57 (DV-Bearig). It is iterestig to ote that estimatios i the middle of the etwork are much more accurate that the oes at the edge (ad this was verified with various other trajectories). The mai cause for this is that a observatio at the edge is obtaied from agles which are clustered together i a small zoe of the trigoometric circle - for example, a corer estimatio would have all the agles i oe sigle quadrat. I fact this is true about positios obtaied by both algorithms. This would suggest that this class of algorithms (positioig, orietatio, trackig) would ru better whe the border of the etwork is reduced i size, or is directly supported by preferetial ladmark placemet. VI. NODE MOBILITY Our curret simulatio of APS oly cosiders static topologies. While highly mobile topologies, usually associated with ad hoc etworks, would require a great deal of commuicatio to maitai up to date locatio, we evisio ad hoc topologies that do ot chage ofte, such as sesor etworks, ad idoor or outdoor temporary ifrastructures, like disposable etworks. APS aims to keep a low sigalig complexity i the evet etwork topology chages slightly. Whe a ode moves, it will be able to get distace vector updates from its ew eighbors ad triagulate to get its ew positio, therefore commuicatio remais localized to odes that are actually mobile. Not eve movig ladmarks would cause a commuicatio surge i our approach because the oly thigs that idetify a ladmark are its coordiates. I fact, a movig ladmark would provide more iformatio to the positioig algorithm, as the ew positio of the ladmark acts as a ew ladmark for both mobile ad fixed odes. With a mobile ladmark, we 72 93 91 41 29 52 7 5 6 64 1 39 48 44 3 21 84 55 66 16 8 11 59 36 51 31 12 13 24 61 46 63 ca evisio a case whe a sigle, fly-over GPS eabled ode is i fact eough to iitialize a etire static etwork. Subsequet mobility of the etwork is supported as log as a sufficiet fractio of odes remais fixed at ay oe time to serve updates for the mobile odes. While APS would perform well for limited mobility, it is very likely that its DV ature would icur high sigallig costs i highly mobile scearios. Drawig from experiece i ad hoc routig, we may ifer that a o-demad positioig scheme would be more appropriate for these cases. A aveue that is explored extesively i mobile robotics research, ivolves usage of accelerometers ad gyroscopes. Situatios may arise whe either a ode does t have eough eighbors to get sufficiet orietatio readigs, or the ode wishes to stay i a iactive state for security or power coservatio reasos. I this cases dead reckoig could be used to ifer a estimate of curret positio based o the last triagulated positio. This capability is give by accelerometers, which ca provide relative positioig after a double itegratio of acceleratio readigs. Headig ca be iferred i a similar maer whe gyroscopes is available. VII. FUTURE WORK AND CONCLUSIONS Besides the extesios to mobility, already metioed, future developmet of the project will be i the directio of improvig the positioig quality by usig error estimatio ad multimodal sesig. A error estimatio method proposed i [14] ivolves trasmittig of the error estimatio together with DV data. A ode performig the orietatio estimatio described i sectio III-A, would also compute the estimated error of the ewly computed orietatio ad forward it alog. By icreasig the sigallig overhead, the fial triagulatio method has the possibility of usig weights for each ladmark. I case of rage based APS, this provided a cosiderable reductio i positioig error. We are still ivestigatig the adaptatio of this error method for agle propagatio. Multimodal sesig ca ehace the performace of positioig algorithms. AOA ad ragig, possibly ehaced with compasses ad accelerometers, have the possibility to provide better positioig tha ay of them take separately. Both AOA ad ragig are or ca be curretly achieved usig commo hardware - time differece of arrival (TDoA), based o ultrasoud trasmitters/receivers. Not requirig additioal hardware makes multimodal based sesig a viable approach for positioig, which we pla to explore i the future. To coclude, the method we proposed ifers positio ad orietatio i a ad hoc etwork where odes ca measure agle of arrival (AOA) from commuicatio with their immediate eighbors. The assumptio is that all odes have AOA capability ad oly a fractio have self
positioig capability. Two algorithms were proposed DV- Bearig ad DV-Radial, each providig differet sigaligaccuracy-coverage-capabilities tradeoffs. The advatages of the method are that it provides absolute coordiates ad absolute orietatio, that it works well for discoected etworks, ad does t require ay additioal ifrastructure. What makes the algorithm scalable to very large etworks is that the commuicatio protocol is localized. Simulatios showed that resulted positios have a accuracy comparable to the radio rage betwee odes, ad resulted orietatios are usable for avigatioal or trackig purposes. REFERENCES [1] J. Hill, R. Szewczyk, A. Woo, S. Hollar, D. Culler, ad K. Pister, System architecture directios for etworked sesors, i ASPLOS- IX, Cambridge, MA, November 2. [2] N. Priyatha, A. Chakraborty, ad H. Balakrisha, The cricket locatio-support system, i ACM MOBICOM, August 2, bosto, MA. [3] A. Savvides, C.-C. Ha, ad M. Srivastava, Dyamic fie-graied localizatio i ad-hoc etworks of sesors, i ACM MOBICOM, 21, rome, Italy. [4] P. Bahl ad V. N. Padmaabha, RADAR: A i-buildig RF-based user locatio ad trackig system, i INFOCOM. IEEE, March 2, tel Aviv, Israel. [5] D. Niculescu ad B. Nath, Ad hoc positioig system (APS), i GLOBECOM. IEEE, November 21, sa Atoio. [6] N. Priyatha, A. Miu, H. Balakrisha, ad S. Teller, The cricket compass for cotext-aware mobile applicatios, i 6th ACM MO- BICOM, July 21, rome, Italy. [7] G. Fi, Routig ad addressig problems i large metropolitascale iteretworks, Uiversity of Souther Califoria, Tech. Rep. ISI Research Report ISI/RR-87-18, March 1987. [8] J. C. Navas ad T. Imieliski, Geographic addressig ad routig, i MobiCom 97, September 26-3 1997, budapest, Hugary. [9] Y.-B. Ko ad N. H. Vaidya, Locatio-aided routig (lar) i mobile ad hoc etworks, i MobiCom 98, October 1998. [1] N. Bulusu, J. Heidema, ad D. Estri, GPS-less low cost outdoor localizatio for very small devices, i IEEE Persoal Commuicatios Magazie, ser. Special Issue o Smart Spaces ad Eviromets, October 2. [11] M. Betke ad L. Gurvitis, Mobile robot localizatio usig ladmarks, i IEEE Iteratioal Coferece o Robotics ad Automatio, vol. 2, May 1994, pp. 135 142. [12] B. Parkiso ad J. Spilker, Global Positioig System: Theory ad Applicatio. America Istitute of Astroautics ad Aeroautics, 1996. [13] B. Krishamachari, S. Wicker, ad R. Bejar, Phase trasitio pheomea i wireless ad-hoc etworks, i GLOBECOM, November 21, sa Atoio, TX. [14] D. Niculescu ad B. Nath, DV based positioig i ad hoc etworks, Telecommuicatio Systems, Baltzer, vol. 1, 23, to appear.