Robust TDOA Passive Location Using Interval Analysis and Contractor Programming

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1 Robust TDOA Passve Locaton Usng Interval Analyss and Contractor Programmng REYNET O. E 3 I 2 Laboratory, EA3876 ENSIETA Brest, France JAULIN L. ENSIETA Brest, France CHABERT G. Contrantes Team LINA CNRS UMR 6241 Nantes, France Abstract Ths paper presents a new approach for solvng non-lnear passve locaton problems. It s based on a hgh level nterval modelng language named Qumper. Whereas classcal passve locaton resolutons do not provde any guarantee of convergence to a soluton, nterval analyss, constrant propagaton and contractor programmng allow us to avod any approxmatons and any lnearzaton. Besdes, Qumper naturally provdes guarantees on locaton and bounded error. TDOA passve locaton confguratons are dscussed to prove Qumper s effcency. (Abstract) Keywords : Passve Locaton, TDOA, Qumper, Interval Analyss, Constrant Propagaton, Contractor Programmng, Bounded Error I. INTRODUCTION Passve locaton has been ntensvely studed n the past years. Numerous devces may actually use passve locaton ncludng wreless or cellular mobles [1,2], sonar and radar systems [3], vehcles localzaton systems. Passve locaton can take advantage of the growngly number of emtters that are present n envronment. It offers a complementary alternatve to GPS when ndoor, guaranteed or accurate locaton s requred. Moreover, quckly locatng an emtter or fndng ts own locaton s an mportant challenge for electronc warfare and unmanned autonomous vehcles (UAV) [4]. Sometmes, decsons have to be taken from results of passve locaton. When human people are nvolved, many extra-parameters can be taken nto account to choose rght actons. For example, symmetrcal mage locaton ambguty n n-lne three recevers confguraton may be removed from geographcal consderatons or past observatons. In the case of UAVs, there s no human decson maker process. Decsons must be taken from mprecse detectors measurements. That s why exact error estmaton s as mportant as exact locaton. Ths paper nts to show that Qumper software s able to produce an exact bounded error from measurements and, n the same tme, to gve guarantees on locaton. Passve locaton classcal approaches gve certanly precse results and error estmaton. Nevertheless, ths error estmaton often comes from a probablty model whch does not care about real-tme measured uncertanty. Besdes, optmzaton processes are used to reach a precse result. Theses processes guarantee nether to converge nor to avod local mnmum. That s why an UAV may beleve n ts locaton wth hgh level of precson and confdence, whle actually beng elsewhere. Ths paper demonstrates the possblty to solve passve locaton nonlnear equatons, wthout optmzaton, approxmatons or lnearzaton. Second secton of ths paper gves necessary notons about nterval analyss, constrant propagaton and contractor to understand Qumper software. Thrd secton s dedcated to a smple TDOA passve locaton confguraton whch s solved by Qumper s natural contractors. Last secton deals wth contractor programmng concept and s llustrated by an n-lne three recevers confguraton. II. INTERVAL ANALYSIS AND CONSTRAINT PROPAGATION A. Short Introducton to Interval Analyss Assume x s a random varable x of IR. An nterval approach ntroduces [x] to represent an nterval [x -,x + ] whch encloses the support of the probablty functon of x. Standard Operators +, -,., / and elementary functons lke exp, log, sn or cos can be defned and appled on ntervals. To understand basc dea of nterval analyss, consder two varables x and y. Assume that they belong to some pror ntervals [x -,x + ] and [y -,y + ]. Addton and dfference of ntervals can be defned as follow: and [ x, x ] [ y, y ] [ x y, x y ] (1) [ x, x ] [ y, y ] [ x y, x y ] (2) Multplcaton and nverson of ntervals are:

2 and [ x, x ]*[ y, y ] [mn( x y, x y, x y, x y ), max( x y, x y, x y, x y )] IR f 0 [ x, x] [ x, x], otherwse. x x Let's gve some numercal examples of nterval computatons: Addton: [ 1,5] [ 3,8] [ 2,13] Dfference: [ 1,5] [ 3,8] [ 7,8] Multplcaton: [ 1,5]*[ 3,8] [ 15,40] Inverson: 1 1 1, 3,8 8 3 Interval arthmetc tools have been developed 1 and t s now possble and smple to buld robust programs drectly handlng uncertantes [5]. B. Interval Constrant Propagaton Interval constrant propagaton combnes nterval computaton [6] and constrant propagaton [7]. The combnaton of these two tools has been frst presented ndepently by Clearly [8] and Davs [9]. Constrants can be used to contract the pror feasble domans by removng nconsstent values n the domans of the varables. For example, consder the three varables x,y, and z and assume they belong to pror feasble domans as follow: x 1,5 y 3,8 z 5,2 Assume also that these three varables are lnked by the ADD constrant: (3) (4) (5) z x y (6) Then, feasble domans for the varables can easly be contracted by takng nto account (6). As ths constrant s a prmtve constrant, a smple proecton algorthm exsts: Algorthm ADD_PROJECTION(nout: [z],[x],[y]) z: z x y x: x z y y: y z x Our numercal example leads to: x [1,5] y 3,1 z 2,2 [y] and [z] have been reduced followng ADD constrant. No soluton has been removed from these ntervals. Proecton s a smple way to contract ntal domans. In ths case, t s possble because the constrant addton s a trval constrant. Fortunately, more complex propagaton processes have been found. If constrants lke equatons or nequaltes exst between the random varables, a constrant propagaton process can produce effcent polynomal algorthms to solve constrants and compute precse soluton ntervals [5]. Besdes, ths process guarantees that these computed ntervals enclose all solutons for gven ntal ntervals. There are many algorthms to generate constrant propagaton. Nevertheless, one of the most effcent way s forward and backward algorthm [10]. Qumper Software mplements forward and backward algorthm. But, n addton, t focuses on another concept: contractors [11]. C. Qumper Software and Natural Contractor Qumper Software s a recent hgh-level language for QUck Interval Modelng and Programmng n a bounded- ERror context (QUIMPER). It s based on Profl/BIAS 2 and IBEX lbrares. It has been created to democratze contractor programmng, constrant propagaton process and nterval analyss. Even f Qumper syntax s smple, manual and examples are avalable on Qumper s web ste 3. Qumper s paver algorthm s a generc solver. It takes a lst of contractor, an ntal box and follows a classcal recurson: the contractors are successvely called on the current box untl ether t gets empty or no more contracton could be done. In the latter case, the box s bsected and contractors are called back agan. Contractor s defnton s gven n [11]. Thnkng of contractor programmng as an extenson of constrant programmng s vald to the extent that contractors help n modellng the output of a problem. But, fundamentally, there s not such an extenson snce constrants bascally tell the what whereas contractors tell the how. If no specal contractor s gven, then Qumper uses natural contractors whch result from constrants broken nto several trval constrants. Forward-backward propagaton s then appled by contractor to reduce boxes. Nevertheless, Qumper presents specfc operators to program ad-hoc contractors. A solver can then be programmed, rather than confgured, by combnng dfferent contractors. These new contractors faster reduce ntal domans. But let frst try to solve TDOA hyperbolc equatons usng Qumper language. 1 See for example INRIA COPRIN proect. An n-lne nterval analyss solver s avalable at 2 See Profl/BIAS web ste at 3 See IBEX QUIMPER ste at

3 III. TDOA PASSIVE LOCATION WITH QUIMPER SOFTWARE TDOA passve locaton requres two stages: the frst stage requres TDOA estmaton for each par recever-recever. The second stage uses TDOA estmaton to buld a set of non-lnear hyperbolc equatons. In ths paper, we suppose that the frst stage, TDOA estmaton, s already done, for example by correlaton technques [12,18]. A. TDOA Hyperbolc Equatons Consder the classcal TDOA passve locaton confguraton composed wth one emtter and three recevers as sketched on fgure 1. Let (x,y) be the locaton of the emtter, and (x,y ) the locaton of the recevers. Dstance from emtter to recever s: If d D D D, we have: d x x y y 2 (7) 2 x x y y 2 x x 2 y y 2 2 (8) Let t be the measured Tme Dfference Of Arrval (TDOA) of the sgnal between recever and. As d ct, hyperbolc TDOA equatons are: ct x x y y 2 x x 2 y y 2 2 (9) where c s the speed of the sgnal and, 0,1, 1,2, 2,0. Solvng these non-lnear equatons s not a trval problem. Closed solutons of these equatons are known [13,14]. However, these closed forms do not provde any error estmaton. Others accurate approaches lnearze ths set of equatons through second order Taylor-seres expanson [15,16,17], but need a good ntal guess and ntensve computaton. Our approach based on nterval analyss, constrant propagaton and contractor programmng allow us to avod any approxmatons and gve true error estmaton. So, let us try to model TDOA passve locaton wth Qumper. Fgure 1. TDOA classcal confguraton : three recevers and one emtter. Emtter s located at the ntersecton of the three hyperbolas for whch t -t s a constant. B. Qumper Example of TDOA Passve Locaton Unknown varables of ths problem are x and y, or, more exactly, [x] and [y]. Indeed, more than an accurate result whch mght be obtaned by others approaches, we are lookng for domans whch mght contan solutons and domans whch do not contan solutons. Suppose that the bounded recever sensblty we are usng allows us to defne some pror feasble domans for these two varables. Let [x] and [y] belong to [ ,10000] m, whch corresponds to a 400 km 2 ntal searchng area, about four tmes Pars urban area. Suppose that transmsson medum s made of ar and that electromagnetc waves are used. The sgnal speed c s then 3e8 km.s -1. The three recevers are located at R 0 (-9000, 0) m, R 1 (9000,-9000) m and R 2 (9000,9000) m. Emtter s located at (1540,345) m. In ths smulaton, measurement errors are ntroduced by specfyng ntervals for t, nstead of a smple real value. We frst extract the exact values of tme of arrval from geometrcal knowledge. Then, fxed or random values may be added or subtracted to exact t value to generate errors. In ths frst example, we supposed that t s known wth an uncertanty of plus or mnus τ = 500 ns. Intal doman of t s then 1 µs wde. Ths uncertanty corresponds to an analog to dgtal converter wth bad precson and a basc sgnal correlaton. Therefore, ct belong to doman [c(t - τ), c(t + τ)]. We can now wrte the followng Qumper nput fle (see Scrpt I). A Qumper fle s dvded n several parts, ncludng constants and varables declaratons, and contractors lst. It should be notced that constants may be ntervals but that varables must be an ntervals. SCRIPT I: TDOA AND NATURAL CONTRACTOR Constants x0= ; y0=0.0; x1=9000.0; y1= ; x2=9000.0; y2=9000.0; ct01 n [ , ]; ct12 n [ , ]; ct20 n [ , ]; Varables x n [-10000,10000]; y n [-10000,10000]; contractor hyperbola1 sqrt((x-x0)^2+(y-y0)^2)-sqrt((x-x1)^2+(y-y1)^2) n ct01; contractor hyperbola2 sqrt((x-x1)^2+(y-y1)^2)-sqrt((x-x2)^2+(y-y2)^2) n ct12; contractor hyperbola3 sqrt((x-x2)^2+(y-y2)^2)-sqrt((x-x0)^2+(y-y0)^2) n ct20; contractor sthck maxdamgt(10) At the of ths Qumper s fle are lsted all contractors. Here, we smply put hyperbolc equatons: Qumper bulds natural contractors correspondng to these constrants. Each contractor removes boxes that are not feasble, one after another. Then, f no more doman contracton s observed,

4 domans are bsected and contractors appled on these new domans. Last contractor s a specal one called Thckness contractor whch uses bult-n operator maxdamgt: sthck removes all boxes that are not thck enough. It s useful f we want solutons to be wrapped nto ntervals wth a sze lower than a lmt. In ths example, ths lower sze lmt s 10m. It means that boxes whch contan solutons are contracted by sthck and have a maxmum dmenson of 10m. C. TDOA Passve Locaton Results Fgure 2 shows results extracted from Qumper. Tme calculaton s about s on a Intel Core Duo at 2GHz. Boxes removed by hyperbolas contractors are panted n black and grey. Indscernble boxes removed by sthck whch may contan solutons appear n whte, n the center of the fgure near exact emtter locaton (1540,345). It can be seen that wde domans are removed wth only one operaton: f a doman do not belong to any hyperbola, then t s no use to search n. The correspondng contractor removes t. All n all, 1745 boxes have been created to solve ths problem. Intal searchng area s 400 km². From ths area, hyperbolas contractors remove % and guarantee that there s no solutons n. sthck removes 0.001% of ths area whch may contan solutons and represents about m². Ths result tells that the emtter s located n a 195x195 m² area centred n (1540,342) m. Concretely, t means that we have located somebody n a bg stadum, whereas we were lookng for hm on an area four tmes greater than Pars. accelerated, snce parallel-computng may be used by nterval analyss. Secondly, removed boxes are guaranteed wthout any soluton: no global mnmum has been mssed because of a local one and no ntal guess s needed. Thrdly, error estmaton naturally results from computaton and may come from real-tme error measurements. If an sthck maxdamgt parameter of 1 m s chosen nstead of 10 m, sthck area s computed at m² nstead of m². It means that no tme consumng small bsectons are needed to reach precse results. Tme uncertanty can be tuned. If τ equals 50 ns nstead of 500 ns and maxdamgt parameter remans 10 m, then sthck area s 671 m², whch represents % of the ntal are. Locaton area s now reduced to a bg tenns court centred at (1538,344). sthck area can hence be seen as a true characterzaton of tme measurement error. IV. TDOA AND CONTRACTOR PROGRAMMING A. Constrant Lst and Bult-In Contractors Qumper s key dea s that a solver can be programmed rather than confgured by combnng dfferent contractors. The prevous example shows that natural contractors can work concurrently. Let s now desgn new contractors va Qumper language. Let s try to compute agan the three recevers and one emtter confguraton of secton III.B, but n a more effcent way (see Scrpt II). Constants and varables are the same as n frst scrpt. However, nstead of usng natural contractors bult by Qumper, three stages are proposed to buld better ad-hoc contractors. SCRIPT II: TDOA AND CONTRACTOR PROGRAMMING Fgure 2. Soluton of TDOA equatons wth natural contractors: removed boxes from hyperbola1 (dark grey), hyperbola2 (black), hyperbolas3 (lght grey) and sthck (whte). Recevers are sketched wth crosses: R 0 (-9000, 0) m, R 1 (9000,-9000) m and R 2 (9000,9000) m. Emtter s located at (1540,345) m. D. Dscusson These results are farly remarkable for several reasons: frst, no approxmatons are made, though non-lnear equatons are quckly solved. These computatons may be further Constants x0= ; y0=0.0; x1=9000.0; y1= ; x2=9000.0; y2=9000.0; ct01 n [ , ]; ct12 n [ , ]; ct20 n [ , ]; Varables x n [-10000,10000]; y n [-10000,10000]; constrant-lst hyperbolas sqrt((x-x0)^2+(y-y0)^2)-sqrt((x-x1)^2+(y-y1)^2) n ct01; sqrt((x-x1)^2+(y-y1)^2)-sqrt((x-x2)^2+(y-y2)^2) n ct12; sqrt((x-x2)^2+(y-y2)^2)-sqrt((x-x0)^2+(y-y0)^2) n ct20; contractor-lst clhyp nter =1:3; hyperbolas() contractor propinter propag(clhyp) contractor sthck maxdamgt(10)

5 Frst stage gves a constrant lst whch takes nto account all lnks between varables. Here, only three equatons are used, but others equatons may easly be added n ths lst. For example, equatons from others redundant recevers, phase nterferometers or radogonometers can be ntroduced n terms of (x,y). No nstructon change s needed to handle nconsstent equatons system: addng equaton to the lst s suffcent. Moreover, the greater the number of dfferent constrants s, the faster boxes are removed from ntal doman. Second stage specfes that boxes whch do not satsfy ntersecton of the three constrants can not be soluton of our problem. Ths stage s Qumper s translaton of geometrcal ntersecton of the three TDOA hyperbolas. It s done thanks to nter bult-n contractor. In the scrpt, hyperbolas() s mplctly nterpreted as the natural contractor assocated to the th equaton. Hence, ths stage creates a new contractor clhyp whch s the ntersecton of the three natural contractors. Frst scrpt s strategy apples natural contractor only one tme and one after another. However, repeatng several tmes the same contractor can contract domans n a more effcent way. Therefore, thrd stage uses the bult-n contractor propag to repeat the ntersecton contractor untl no more contracton s observed. Then, sthck contractor s appled. As a last resort, ndscernble domans are bsected and entre process restart on these new domans. B. Contractor Programmng Results Fgure 3 shows computaton results of Qumper for the pror example, but wth contractor programmng as descrbed n Scrpt II. propinter contractor removes boxes and leads to the same result, but has created fewer boxes (1207 nstead of 1217). Computaton tme s about the same than wth natural contractors (0.088 s). It means that formula (9) provdes a way to buld an effcent natural contractor to solve TDOA equatons. C. In-Lne TDOA Passve Locaton Confguraton Consder now the followng n-lne confguraton: emtter s at (8000,7000) m and recevers are located at R 0 (-9000,0) m, R 1 (0,0) m and R 2 (9000,0) m. Ths confguraton s a bad one, because n-lne recever x-locaton generates y-ambguty: two symmetrcal solutons are ndscernble. Qumper s results are shown fgure 4. sthck maxdamgt parameter s 10 m. Tme measurement uncertanty τ s 500 ns. Ths example demonstrates that nterval analyss and contractor programmng do not remove any soluton. In a pedagogcal way, t also shows that ths confguraton generates bgger error locaton: ntal area s 400 km² and sthck area s about m² (.e %). Reasons are tme uncertanty, recevers proxmty and n-lne topology. Fgure 4. Soluton of TDOA equatons wth programmed contractors: removed boxes from propinter (dark grey) and sthck (whte). Recevers are sketched wth crosses: R 0 (-9000, 0) m, R 1 (0,0) m and R 2 (9000,0) m. Emtter s located at (8000,7000) m V. CONCLUSION Qumper s a new powerful tool. To our mnd, t s the frst tme that passve locaton non-lnear equatons as hyperbolc TDOA equatons are solved wthout any approxmatons thanks to nterval analyss and contractor programmng. Bounded-error estmaton naturally results from Qumper s computaton. Contractor programmng s a way to create adhoc and more effcent contractors. Fnally, ths hgh level nterval modelng language provdes means to mx dfferent passve locaton approaches. Hyperbolc TDOA, gonometry or phase nterferrometry may easly be ntroduced n the Qumper s smple formalsm. REFERENCES Fgure 3. Soluton of TDOA equatons wth programmed contractors: removed boxes propinter (lght grey) and sthck (whte). Recevers are sketched wth crosses: R 0 (-9000, 0) m, R 1 (9000,-9000) m and R 2 (9000,9000) m. Emtter s located at (1540,345) m [1] An Enhanced Two-Step Least Squared Approach for TDOA/AOA Wreless Locaton. Anchorage, Alaska, USA: ICC 2003, MAY 2003.

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