Stochastic Modeling of the Expected Time to Search for an Intermittent Signal Source Under a Limited Sensing Range

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1 1 Stochstic Modeling of the Epected Time to Serch for n Intermittent Signl Source Under Limited Sensing Rnge Dezhen Song, Chng-Young Kim, nd Jingng Yi Abstrct A mobile robot is deploed to serch for sttionr trget tht intermittentl emits short durtion signls. The serching mission is ccomplishe soon s the robot receives signl from the trget. However, the robot cnnot perceive the signl unless the trget is within its limiteensing rnge. Therefore, the time to serch the trget is inherentl rndom nd hence unknown despite its importnce in mn serching nd rescue pplictions. Here we propose the epecteerching time EST s metric to evlute different robot motion plns under different robot configurtions. We derive closed form solution for computing the EST. To illustrte the EST model, we present two cse studies. In the first cse, we nlze two common motion plns: slp method nd rndom wlk. The EST nlsis shows tht the slp method is smptoticll fster thn the rndom wlk when the serching spce size increses. In the second cse, we compre tem of n low-cost equllconfigured robots with super robot tht hs the sensing rnge equl to tht of the summtion of the n robots. The EST nlsis shows tht the low-cost robot tem tkes Θ1/n time nd the super robot tkes Θ1/ n time s n. In both cses, our EST model successfull demonstrtes its bilit in ssessing the serching performnce. The nlticl results re lso confirmed in simultion. I. INTRODUCTION Mobile robots re often emploed to perform serching tsks such s finding blck bo in remote re fter n irplne crsh, serching victims fter n erthquke or mine collpse disster, or locting rtifcts on the ocen floor. In mn cses, the trget cn intermittentl emit short durtion signls to ssist serching. For emple, n irplne blck bo trnsmits rdio signls periodicll. An erthquke victim m knock the rubble from time to time. The serching tsk is ccomplished once the robot detects the signl emitted b the trget. However, the robot usull hs limiteensing rnge nd cnnot detect the trget tht is out of the rnge. It seems strightforwrd tht we cn use the trditionl covergebsed motion plns to guide the robot to cclicll scn the serching spce to locte the trget. However, the time to serch the trget is inherentl rndom nd hence remins unknown despite its importnce in mn serching nd rescue pplictions. To ddress this new problem, we propose the epected serching time EST s metric for the serching bilit. We model the serching process s deled lternting renewl process nd derive the EST s function of the serching spce size, the signl trnsmission rte, nd the This work ws supported in prt b the Ntionl Science Foundtion under CAREER grnt IIS nd MRI D. Song nd C. Kim re with CSE Deprtment, Tes A&M Universit, College Sttion, TX 77843, USA, emils: dzsong@cse.tmu.edu nd kcoung@cse.tmu.edu. J. Yi is with MAE Deprtment, Rutgers Universit, Pisctw, NJ 8854 USA, emil: jgi@rutgers.edu. O Fig. 1. A robot ttemps to serch for trget the red dot tht intermittentl emits short durtion signls in squre. The gr circle is the region tht the robot cn sense the signl from the trget. The dshed line is robot trjector. robot sensing rnge. The resulting closed-form solution of the EST cn be used to nlze the serching efficienc for different robot configurtions nerching plns. Since the model components cn be obtined from online mesurements nd known robot prmeters, gret benefit of the resulting model is tht it is cpble of predicting the EST for n ongoing serching process. This chrcteristic is importnt for timecriticl serching nd rescue pplictions. The contributions of the pper re tring to bring nlticl results to interpret well-known serching strtegies. The EST nlsis not onl cn revel our common believes bout eisting serching methods but lso predict how the effectiveness of those methods chnges s trjector selection, sensing rnge, serching spce size, nd robot distribution chnge. Building on the ltest development in rndom wlk in constrinepce from stochstic modeling communit, our nlsis for the first time show tht trditionl slp method Θ 2 ctull is smptoticll fster thn the rndom wlk O 2 log for squreerching spce of the size length, which is never known before. In the second cse, we compre tem of n identicll-configured low-cost robots with super robot tht hs the sensing coverge equl to the summtion of the n lowcost robots. The EST nlsis shows tht the low-cost robot tem outperforms the super robot becuse its EST is Θ1/n while the EST for the super robot is Θ1/ n s n. Agin, this new nlticl result hs not been seen before nd is importnt for developing new serch strtegies. The nlticl results re confirmed in simultion for both cses. The EST model successfull demonstrtes its bilit in ssessing the serching performnce under different robot configurtions nd motion plns. The rest of the pper is orgnize follows. We begin with the relted work in Section II. We define the problem in Section III. We derive the EST in closed form in Section IV.

2 The two cse studies re presented in Section V. The nlticl results re vlidted in simultion in Section VI before we conclude the pper. II. RELATED WORK Serching n object in phsicl spce is one of the most importnt tsks for robots or humns. When prior informtion such s the sptil distribution of the trget is known, this is comprble to the forging behvior of niml [1]. However prior trget informtion is often not vilble. If the trget is continuousl emitting signls, just simpl scnning the entire serching spce once enbles the robot to find the trget. Since the worst cse for the serching time is the time to cover the entire serching spce, the serching problem becomes coverge problem [2] [4]. For known environment, coverge problem for single robot often emplos different pproches to decompose the serching spce nd output continuous pth tht llows the robot to cover the entire serching spce. If the serching spce cn be modele set of w-disjoint discrete choices, serching trget with limiteensing rnge nd w-choice is known s w-lne Cow-Pth problem [5]. While the running time is well understood for the coverge problems [6], [7], this is not the cse when the serching process depends on the signl emitted b the trget becuse the colloction of the robot nd the trget does not necessril men tht the trget is found. When the trget is not emitting signl, the robot cnnot find the trget. The robot hs to keep scnning the serching spce. The deterministic coverge lgorithm becomes Ls Vegs lgorithm [8] where the trget will eventull be found but the serching time is rndom. However, the serching time cn be crucil for mn serching tsks. For emple, victims of n erthquke often hve limited survivl time. Although serching itself is ver old problem, few models eist for nlzing the effectiveness of serching strteg when the source is intermittentl emitting signls. Another set of relted work is robot eplortion nd mpping problems where the environment is not previousl known [9]. The tsk is not onl to cover the entire spce but lso to output the true representtions of the environment. Recent dvnces in using multi-robot tem to perform eplortion nd mpping tsks minl focuses on the coordintion of the robot/sensor tem [1] [15] under vrious dnmics, communiction, sensing, nd energ constrints. Although not directl pplicble to our problem, reserchers hve ccumulted interesting empiricl results tht re using tem of low-cost robots usull performs fster nd more fult-tolernt [14] thn single epensive robot. This rell inspires our problem becuse we wnt to see if our nlticl model cn show similr results under similr constrints/conditions. Our group hs built eperience in serching for trgets tht intermittentl trnsmit signls b developing lgorithms nd sstems to detect n unknown wireless sensor network [16] [18]. In these problems, the robot cn ccumulte the informtion bout the trget loction over time through the signl strength redings nd ntenn models. The serching problem is less difficult becuse the robot cn utilize the informtion in the plnning process. However, such informtion is often not vilble in mn serching tsks, which is the focus of this pper. III. PROBLEM DEFINITION As illustrted in Fig. 1, single robot serches for single trget in squred 2D Eucliden spce with side length of. Define s the mimum sensing distnce of the sensor on the robot. The robot trvels t the constnt speed of v. To formulte the problem nd focus on the most relevnt issues, we mke the following ssumptions, 1 There is no prior informtion bout possible loctions of the trget. Therefore, the trget is ssumed to be uniforml distributed in the serching spce. This is ctull the most difficult serching cse. 2 The trget trnsmits short durtion signls periodicll ccording to Poisson process with known rte λ. The signl durtion is short due to energ concerns. A Poisson process is good pproimtion to generl rndom rrivl process in stochstic modeling [19]. In some cses, the trget m be continuous becon; but it is ver difficult to be detected due to environment conditions or unrelible sensing, which cn lso be modele trget with intermittent signls. 3 During the serching process, either the trget is sttic or its movements re negligible in comprison to. The serching spce is much lrger thn the sensing distnce:. Condition 1 Sensing Condition: The robot cnnot sense the signl unless n ctivel-trnsmitting trget is within distnce due to the sensing rnge limit. As illustrted in Fig. 1, this defines circle centered t the trget with the rdius of, which is the region tht robot hs chnce to sense the trget. We refer to the region s the circle in the rest of the pper. Due to the fct tht the robot does not know the loction of the trget, the ctul position of the circle in the serching spce is lso unknown. Condition 2 Termintion Condition: The serching tsk is ccomplishe soon s the robot senses signl. Condition 2 implies tht the robot cnnot find n inctive trget even it is collocted with the trget. For emple, n irplne is not be ble to notice the survivor on n islnd if the person does not send signl e.g. fire or smoke. On the other hnd, onl one signl reception is needed in the serching process. Conditions 1 nd 2 estblish new tpe of serching problem s oppose to regulr coverge problem. Let us define T s s the serching time for the robot to find the trget. Therefore, our problem is define follows, Problem 1 The EST Computtion: Given λ,, nd, clculte the EST ET s, where E denotes the epected vlue function. IV. MODELING One immedite question bout Problem 1 is whether we cn obtin the EST without referring to or being limited to prticulr motion pln. To ddress this dependenc, we first chrcterize the motion plns bsed on their outcomes before modeling the EST. 2

3 A. Chrcterizing Plnners Periodicll, the robot plnner outputs motion pln nd the sstem is nturll repetitive scnning process. We nme its trjector in ech perio tour. The dshed line in Fig. 1 illustrtes tour. Definition 1: A tour strts t the moment when the robot enters the trget circle nd ends t net moment when the robot enters the circle gin. Tours m be quite different bsed on the plnner. For emple, tour length vries ech time if the robot follows rndom wlk. As nother emple, deterministic plnner usull hs fied tour trjector. Bsed on Condition 2, we know tht the robot does not ccumulte the knowledge regrding the trget loction from tours to tours becuse no signl hs been perceived before the moment the serching mission is ccomplished. Hence we cn tret ech tour s independentl nd identicll distributed i.i.d.. This llows us to model the serching process s renewl process. When tour begins, the robot first spends some time inside the circle, which is define τ IN. After tht, the robot leves the circle npends some time before entering the circle gin, where the net tour strts. This ields n lternting renewl process. The durtion outside the circle is define τ OUT. Hence τ IN + τ OUT is the overll durtion for the tour. B. Modeling the EST Without loss of generlit, we ssume the robot strts the serching process from the origin which is on the boundries of the serching spce. It tkes some time to rech the circle where the first tour strts. Define the time s del D. From Conditions 1 nd 2, we know tht the robot cnnot find the trget in D. The serching process is deled lternting renewl process. Define T c s s the time to find the trget fter the robot enters the repetitive tours. Hence, the EST is ET s = ED + ET c s. 1 Define N s the number of signl trnsmissions during τ IN in tour. Since the rrivl process of the signl trnsmission is Poisson, N conforms to Poisson distribution, P N = k = e λτin λτ IN k, k =, 1, 2,...,. 2 k! We know tht event N > mens tht t lest one signl trnsmission hppens during τ IN. This mens the trget is found. Therefore, we cn compute ETs c b conditioning on N, ETs c = ETs c N > P N > +ETs c N = P N =, 3 where P N > = 1 e λτin nd P N = = e λτin ccording to 2. Now let us compute ETs c N >. Since event N > is equivlent to event Ts c τ IN, we hve ET c s N > = ET c s T c s τ IN = 1 λ τ INe λτin 1 e λτ IN 4 becuse the conditionl distribution Ts c Ts c τ IN is truncted eponentil distribution. It is worth noting tht 4 is vlid onl if τ IN >. This is gurnteed ccording to Definition 1. On the other hnd, we know ET c s N = = τ IN + τ OUT + ET c s 5 becuse the robot cnnot find the inctive trget in the current tour nd hs to strt ll over gin in net tour. Plugging 4 nd 5 into 3 nd 1, we hve the following Theorem. Theorem 1: Given the epected time ED for the robot to rech the circle, the Poisson rrivl rte of signl λ of the trget, the trveling time τ IN inside the circle, nd the trveling time τ OUT outside the circle, the EST of the trget is e λτin ET s = ED + 1 λ + τ OUT 1 e λτ. 6 IN Theorem 1 hs surprisingl succinct formt reveling the reltionship between the EST nd the corresponding vribles. To further eplin 6, let us consider the following etreme cses: Cse 1: When λ, it mens tht the trget continuousl trnsmits signls. An emple is tht lost hiker keeps fire burning. Hence the light nd the smoke of the fire become the continuous signl. Now the serching time becomes the time tht it tkes for the robot to enter the circle. The problem degenertes to the trditionl coverge problem where ET s = ED. Cse 2: When τ OUT =, it mens tht the signl emitted b the trget is so powerful tht the circle defined b cn cover the entire serching spce. In this cse, ED =. Hence ET s = 1/λ. This is sensible becuse the result mens the robot cn find the trget s soon s it emits signl. Cse 3: When τ IN, which hppens when is infinitesimll smll, we hve ET s. This result conforms to our epecttion. Remrk 1: It is worth noting tht 6 does not depend on prticulr motion pln or the shpe/dimension of the serching spce, which mkes it widel pplicble in prctice. Actull, the EST cn be lso pplied to nlze serching tsks crried b humns. In mn cses, the signl trnsmission rte λ is known; ED cn be estimted bsed on observtions; τ IN cn be estimted bsed on nd v; nd τ OUT cn be mesured bsed on observtions tht how often robot would revisit region with the sme size of the circle. Bsed on the known informtion nd online mesurements, we cn even predict the EST for n ongoing serching process regrdless its motion pln, which is of gret importnce in pplictions where the serching time literll mens life or deth. V. ANALYSIS OF COMMON SEARCHING STRATEGIES Theorem 1 cn be used to nlze the serching performnce under different robot motion plns nd configurtions. We begin with demonstrting how Theorem 1 cn revel the difference between two motion plns from common coverge methods, nmel, the slp method nd the rndom wlk. 3

4 O 2 D IN Fig. 2. A smple motion pln for the slp method. b An illustrtion of how tour line l intersects the circle of the trget. A. The Slp Method The slp method [2], lso known s the trpezoidl decomposition [21] in robotics, sequentill scns the entire serching spce bck nd forth. Fig. 2 gives the robot motion pln for the squre cse. The pln is set of - is prllel lines ppers to be verticl lines in Fig. 2 tht cover the entire serching spce. The verticl lines re inter-connected using the boundries of the serching spce to formulte complete tour. To gurntee n intersection between the circle nd the tour, the distnce between djcent verticl lines is set to be 2. The red in Fig. 2 is the strting point of the tour. Since tours re ectl the sme in the slp method, the subsequent tours strt ectl t the sme loction. The overll tour length is pproimtel 2 /2. Given the robot speed v, we know it tkes τ IN + τ OUT l D l b 7 time for the robot to finish the tour. Since the trget could be nwhere in the serching spce with equl probbilities, we know tht ED τ IN + τ OUT /2 = 4v. 8 The remining undetermined vrible is τ IN. Let us define D IN s the distnce trveled inside the circle. D IN is the length of intersection when the line intersects the circle s illustrted in Fig. 2b. Here we ignore the boundr effect where the circle is not full circle becuse. Line l in Fig. 2b is prt of the tour. When l intersects the circle, we define D l s the distnce between the center of the circle nd the line. Since the trget is uniforml distributed in the 2D spce, D l U, is uniforml distributed. From Fig. 2b, we know τ IN = D IN v = 2 d 2 s Dl 2. 9 v Plugging 7, 8 nd 9 into 6 nd conditioning on D l, we hve, ET s D l + 1 4v λ + 2 d 2 s Dl 2 φλ, D l, v 1 where Since, τ OUT compred with 1 φλ, D l =. 11 e 2λ d 2 s D2 l v 1 τ IN, nd 2 d 2 s D2 l v, we hve, ET s D l is negligible if + 1 4v λ + φλ, D l. 12 Hence we hve the EST for the slp method, where ET s = ds δ= g, λ = Eφλ, D l = ET s D l = δ 1 dδ + 1 4v λ + g, λ 13 ds δ= Let δ = cos θ, we cn trnsform 14 into g, λ = π/2 θ= 1 e 2λ sin θ v 1 1 φλ, D l dδ. 14 sin θdθ. 15 When λ nd /v re ver smll, 14 cn be further simplified, g, λ πv λ Remrk 2: Eq. 13 lso suggests tht fst robot lrge v with gret sensing distnce reduces the EST. This conclusion grees with our intuition tht mobilit nensing re the ke elements in serching. However, it lso tkes trget s coopertion to further reduce the EST. When the robot reches its speed nensing limit, the onl w to reduce EST is to increse λ. Of course, the trget usull hs energ constrints nd cnnot rbitrril increse λ. The nlsis ssumes the distnce between verticl lines is 2, which ensures there is onl one intersection between the circle nd the tour. When smller spcing is used, the overll tour length increses no does τ IN. The nlsis is slightl more complicted becuse it needs to be conditioned on the number of intersections between the tour nd the circle. The results ctull shre similr formt with 15 nd the sme smptoticl properties with respect to, v, nd λ. Since our focus is to compre the smptotic behvior of the slp method with tht of the rndom wlk, we omit the nlsis here. Another resonble concern is tht whether ignoring the boundr effect impcts the finl result. When the circle is locted t the boundr of the squre, distnce D IN cnnot be computed using 9. Since the trget hs to be locted within distnce of the boundr to crete the scenrio, the probbilit tht such event hppens is less thn 4 = 4d 2 s 1, since. Hence its impct to the finl EST is ignorble becuse D IN for such cse is not significntl different from tht of the non-boundr cse. Therefore, we will ignore boundr effect in the rest of the pper. 4

5 O 2 Fig. 3. An illustrtion of robot motion pln bsed on 2D lttice-bsed rndom wlk. B. Rndom Wlk Another populr motion pln is to emplo 2D rndom wlk to trverse the serching spce. As illustrted in Fig. 3, we prtition the entire serching spce using 2D finite lttice with spcing of 2 in ech dimension. Denoting N s s the number of lttice nodes, we hve N s = 4d nodes. Finer 2 s lttice is possible but usull ssocited with higher energ cost becuse the robot hs to mke lot more turns. The robot lws moves from one lttice node to its neighboring node with equl probbilities. The robot does not cross the boundries. According to [22], this is finite 2D lttice with reflective boundries. Recll tht tour strts t the moment the robot enters the circle. Since the robot might not enter the circle t the ectl sme loction in different tours, ech tour is not necessril completel closed curve s tht in the slp method cse. The closed curve tour in Fig. 3 onl hppens with probbilit of 1/4. To compute the EST in 6, we need to compute ED. Recll tht the robot lws strts t origin. Given the loction of trget X t, Y t, computing the men time tht it tkes the robot to follow the rndom wlk to rech prticulr loction X t, Y t is the men first pssge time MFPT [23], [24] problem in stochstic modeling. The ect solution to this problem is epressed in the formt of pseudo Green functions nd cnnot be eplicitl nlzed. Since, there re lrge number of nodes 4d in the 2D lttice nd ech robot 2 s move tkes 2 time. Hence we cn ppl the recent results v of MFPT using its smptotic formt in [25], α + α 1 ln ED X t, Y t X 2 t + Y 2 t, 17 where α nd α 1 re constnts nd cn be determined b Monte Crlo methods. According to [25], α nd α 1 strikingl do not depend on lttice size but locl trnsitionl properties. Hence, ED = α 2 + α 1 ED X t =, Y t = 1 dd, 18 ln dd. 19 Since ln dd = 2 ln + π + 2 ln we hve ED π + 2 ln 2 6 α + α 1 ln + α e λτ IN The remining unknown term in 6 is Eτ OUT 1 e. λτ IN Given the robot speed v, τ IN is uniquel determined b the distnce in the circle D IN, which is independent of the overll trjector. Also Eτ OUT Eτ OUT + τ IN given tht. Hence, e λτ IN e λτ IN Eτ OUT 1 e λτ Eτ OUT + τ IN E IN 1 e λτ. 21 IN Since the 2D lttice-bsed rndom wlk is undirected nd smmetric in trnsitionl probbilit, we know tht the sttionr probbilit of sting inside the circle is p c = πd2 s. 2 Therefore, we know the following is true ccording to Renewl Rewrd theorem, Eτ IN Eτ OUT + τ IN = p c = πd2 s Plugging 22 into 21, we hve e λτin E τ OUT 1 e λτ e λτ IN IN πd 2 Eτ IN E s 1 e λτ IN. 23 Now, we focus on the computtion of τ IN. Since the lttice hs spcing of 2, two scenrios eist when the tour on the lttice intersects the circle: i the nerest lttice point on the tour is inside the circle nd ii the nerest lttice point on the tour is outside the circle s illustrted in Fig. 4. Let us define events tht scenrios i nd ii hppen s events E i nd E o, respectivel. Since the circle center is uniforml locted in the serching spce, P E i = πd2 s 4d 2 = π s 4 = 1 P E o. 24 When event E o hppens, we know tht the robot trjector intersects the circle s stright line s shown in Fig. 4. Hence we hve τ IN E o = D IN v, 25 where D IN is defined in 9 nd the right side of is the condition for the equlit to be true. This is nottion convention widel used in stochstic modeling [19]. Hence Eτ IN E o = π, nd 26 2v e λτ IN E 1 e E λτin o = g, λ. 27 When event E i hppens, one lttice point is inside the circle. As illustrted in Fig. 4b, the lttice point inside the circle prtitions the lttice edges inside the circle into four prts: l 1, l 2, l 3, nd l 4. When robot trjector intersects the circle, the prt of the trjector inside the circle cn be divided into two segments, which re define L nd L, respectivel. L refers to the segment tht the robot tkes to rrive t the 5

6 O X t, Y t O l 1 b l 4 l 2 l 3 X t, Y t Fig. 4. An illustrtion of how the robot trjector in solid line intersects the circle. Scenrio i: when the nerest lttice point on tour is locted outside the circle. b Scenrio ii: when the nerest lttice point on tour is locted inside the circle. The dshed line in the figure is prt of the lttice. lttice node nd L refers to the segment tht the robot tkes to leve the circle. Hence τ IN E i = L + L. v Since L nd L hve equl probbilities to tke l 1, l 2, l 3, nd l 4, there is totl of 16 combintions. Conditioning on the 16 L, L combintions nd the circle center loction X t, Y t, we get the sme results s shown in 26 nd 27. Combining those results for the E i nd E o events b conditioning on them, we hve the unconditionl epected vlues, Eτ IN = π, nd 28 2v e λτ IN E = g, λ, 29 1 e λτin where g is defined in 15. Plugging 2, 23, 28, nd 29 into 6, we cn obtin the EST for the rndom wlk cse, ET s π + 2 ln 2 6 α + α 1 ln + α λ + 2 v g, λ. 3 Compring 3 to 13, we hve the follow conclusion, Corollr 1: With the sme fielide length, the sensing rnge, nd the signl trnsmission rte λ, the ET s vlue of the slp method is smptoticll smller thn tht of the rndom wlk when. Proof: It is strightforwrd becuse ET s = Θ 2 for the slp method from 13 while ET s = Θ 2 ln for the rndom wlk ccording to 3. C. Anlsis of Different Robot Configurtions Theorem 1 cn lso be used to nlze cses under different robot configurtions. Here we compre two configurtions. A low-cost robot tem LCRT cse: We hve n identicllconfigured low-cost robots. To coordinte the serching, we prtition the serching spce into n sub squre fields with n re of 2 /n ech nd llocte one robot for ech sub squre field. A single epensive robot ASER cse: We hve n epensive robot equipped with ver cpble sensor tht hs sensing re equl to the combintion of those of the n lowcost robots. If ech of the low-cost robot hs sensing rnge of, then the re of the combineensing region for n robots is nπd 2 s. Therefore, the sensing distnce for the epensive robot is set to d s = n to ensure the sme-sizeensing coverge t n given time. We re now red to compre these two robot configurtions. Since the slp method is smptoticll fster thn the rndom wlk, we build on the slp method results in 3. For the LCRT, onl one robot ctull hs the trget in its sub field. Hence, the rest of n 1 robots re irrelevnt in the serching process. Compring with the originl EST in 3, we just need to replce with n. Defining the serching time for the LCRT s T s, we hve ET s + 1 4vn λ + g, λ. 31 2vn Defining the serching time for the ASER s T s, we hve ET s 2 4v n + 1 λ + From 15, it is not difficult to see tht 2v n g n, λ. 32 g n, λ s n. 33 Therefore, we hve the following conclusion, Corollr 2: When trveling t the sme velocit v, the low-cost robot tem cn find the trget smptoticll fster thn the single epensive robot does when n increses, if 1/λ is not the dominting fctor in the EST. Proof: From 32 nd 33, we know ET s = Θ 1 n + 1 λ. From 31, we know ET s = Θ 1 n + 1 λ. Hence the conclusion follows. It is ctull rther surprising to see the result in Corollr 2 t the first sight. We hve not epecteuch significnt difference in the comprison. This conclusion is rther interesting becuse it shows tht n epensive robot with superior sensing cpbilit is not s goo lrge number of low-cost robots with less cpble sensors when serching for trgets tht intermittentl trnsmits short durtion signls. Remrk 3: This nlsis lso shows tht if there re cost functions ssocited with the number of robots, different sensor options, or different velocit options vilble, we cn use the EST results s n objective function to optimize the robot configurtion for the tsk. VI. EXPERIMENTS We test our results using Monte Crlo simultion. The simultion progrm is written in Microsoft Visul C++.Net 25 on Desktop PC with n 32-bit Windows XP Professionl Edition OS. The Desktop PC hs n Intel 2.13 GHz Core2Duo CPU with 2GB RAM nd 25 GB Hrd disk. The eperimentl results re illustrted in Figs. 5 nd 6. Ech dt point in both figures is n verge of 1, independent trils. At the beginning of ech tril, we reset the robot position to be t, nd generte the trget loction ccording to 2D uniform distribution. We then run the robot ccording to the selected motion pln nd finish the tril s soon s the trget is found. 6

7 EST sec. 7.E+6 6.E+6 5.E+6 4.E+6 3.E+6 SM Mes. SM Model RW Mes. RW Model EST sec. 2.5E+5 2.E+5 1.5E+5 1.E+5 SM Mes. SM Model RW Mes. RW Model 2.E+6 1.E+6 5.E+4.E m.e λ 1/sec. b EST sec. 2.5E+5 2.E+5 1.5E+5 1.E+5 5.E+4 SM Mes. SM Model RW Mes. RW Model EST sec. 1.2E+7 1.E+7 8.E+6 6.E+6 4.E+6 2.E+6 SM Mes. SM Model RW Mes. RW Model.E c m.e v m/sec. d Fig. 5. Simultion results in, b, c, nd d for vlidting Theorem 1 with respect to, λ,, nd v, respectivel. SM stnds for the slp method. RW stnds for the rndom wlk. Model mens the model prediction of the EST. Mes. mens the mesured men serching time. A. Vlidting Theorem 1 nd Corollr 1 We test Theorem 1 using both the slp method nd the rndom wlk becuse Theorem 1 is supposed to be independent of motion plns. The simultion is set up with different, λ,, nd v settings in Tble I. In ech setting, we collect both the model predicted EST nd the mesured men serching time. The mesured men serching time is the verge of the serching time over the 1k trils the Mes. vlues in Fig. 5. The model predicted ESTs, which re the Model vlues in Fig. 5, refer to the predicted ESTs ccording to the mesured D, λ, τ IN, nd τ OUT vlues in the eperiment. In other words, we record their vlues nd verge them over the 1k trils nd to obtin the estimtion of 1/λ, ED, nd e Eτ λτ IN OUT. We then feed them into 6 to obtin the 1 e λτ IN model prediction of the EST. Figure m λ 1/sec. m v m/s Fig Fig. 5b Fig. 5c Fig. 5d Fig TABLE I PARAMETER SETTINGS FOR RESULTS IN FIG. 5 AND FIG. 6. As illustrted in Fig. 5, the model prediction is firl consistent with the mesured men serching time under ll settings. There re more fluctutions between the model prediction nd the mesured men serching time in rndom wlk-bsed results thn tht of the slp method. This is epected becuse of more rndom fctors ssocited with the rndom wlk. Under the sme tril number, the results from the rndom wlk should contin more rndomness. The curve trends with respect to, λ, nd v in Fig. 5 re lso consistent with our nlsis in 13 nd 3. The EST increses s the fielide length increses. The EST decreses s λ,, nd v increse. All figures show tht the rndom wlk is slower thn the slp method. In prticulr, Fig. 5 is consistent with the smptoticl difference in Corollr 1. B. Vlidting Corollr 2 We hve lso implemented both LCRT nd ASER robot configurtions. Agin, the prmeter settings re in the lst row of Tble I. The mesured ESTs for both the configurtions re shown in Fig. 6. It is cler tht the EST for the LCRT is lws much smller thn tht of the ASER. This is consistent with Corollr 2. Curves in the figure lso show the trend tht the EST decreses s the n increses. This is consistent with our nlsis. Also, s n gets ver big, the curve levels t non-zero vlue. This indictes tht the signl trnsmission rte domintes the serching time. On the other hnd, it is not desirble to rbitrril increse n becuse the mrginl benefit would decrese. VII. CONCLUSIONS AND FUTURE WORK We nlticll modeled the epecteerching time for robot with limiteensing rnge to serch for trget tht intermittentl emits short durtion signls. We presented the 7

8 Fig. 6. EST sec. 2.E+4 1.8E+4 1.6E+4 1.4E+4 1.2E+4 1.E+4 8.E+3 6.E+3 4.E+3 2.E+3 LCRT ASER.E n Simultion results for compring two robot configurtions closed-form model for the EST. The EST model is motionpln independent nd cn be used to nlze different motion plns or robot configurtions in two cse studies. In the first cse, we nlzed the slp method nd the rndom wlk nd found tht the slp method is smptoticll fster thn the rndom wlk. In the second cse, the EST model reveled the interesting result tht low-cost robot tem is lws smptoticll fster thn n epensive robot when the sensor coverge is the sme. In both cses, the results demonstrted the usefulness nd the cpbilit of our EST nlsis. Our theoreticll results were etensivel tested using simultion. The simultion results were consistent with the model. This work will led to rich set of eciting future work. As n etension, we cn nlze cses where multiple trgets re needed to be serched. We cn lso develop the EST metrics for the serching of moving trget, n un-cooperting trget, or multiple sensor combintions. Different sensor models cn lso be considered. Also, the serching spce m eist obstcles. Applictions of these etended results will be widernging. ACKNOWLEDGEMENT Thnks for E. Frew, J. Xio, nd R. Volz for their insightful discussions. Thnks for Y. Xu, J. Zhng, A. Aghmohmmdi, nd W. Li for their inputs nd contributions to the Networked Robots Lbortor in Tes A&M Universit. [9] S. Thrun, W. Burgrd, nd D. Fo, Probbilistic Robotics. MIT Press, 25. [1] K. Lventll nd J. Cortes, Coverge control b multi-robot networks with limited-rnge nisotropic sensor, Interntionl Journl of Control, vol. 82, no. 6, pp , June 29. [11] P. Brss, Bounds on coverge nd trget detection cpbilities for models of networks of mobile sensors, ACM Trnsctions on Sensor Networks TOSN, vol. 3, no. 2, p. 9, June 27. [12] W. Burgrd, M. Moors, D. Fo, R. Simmons, nd S. 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