Life-long Informative Paths for Sensing Unknown Environments

Transcription

1 Lfe-long Infomatve Paths fo Sensng Unknown Envonments Danel E. Solteo Mac Schwage Danela Rus Abstact In ths pape, we have a team of obots n a dynamc unknown envonment and we would lke them to have accuate nfomaton about the envonment fo all tme. That s, the eo between the obots model of the envonment and the actual envonment must be bounded fo all tme, despte dynamc changes n the envonment. We pesent an adaptve contol law fo the obots to shape the paths to locally optmal confguatons fo pesstently sensng an unknown dynamcal envonment. Pesstent sensng tasks ae concened wth contollng the tajectoes of moble obots to act n a gowng feld n the envonment n a way that guaantees that the feld emans bounded fo all tme. Wth the adaptve contolle, as the obots tavels though the paths, they dentfy the aeas whee the envonment s dynamc and shape the paths to sense these aeas. A Lyapunov-lke stablty poofs s used to show that, unde the poposed contolle, the paths convege to locally optmal confguatons accodng to a Voono-based coveage task, efeed to as nfomatve paths. Smulated and expemental esults suppot the poposed appoach. I. INTRODUCTION Robots opeatng n dynamc and unknown envonments ae often faced wth the poblem of decdng whee to go to get the most elevant nfomaton fo the cuent task. Gven a dynamc envonment that s unknown, and a goup of obots, each wth a senso to measue the envonment, we want to fnd a set of paths fo the obots that wll maxmze nfomaton gatheng n long-tem opeaton. To acheve ths, the obots need to do thee thngs: 1) lean the stuctue of the envonment by dentfyng the aeas wthn the envonment that ae dynamc and the ate of change fo these aeas; 2) compute paths whch allow them to sense the pats of the envonment that have hgh ate of change; and 3) contol the speeds along these paths fo all tme so that they always have tme-accuate nfomaton of the envonment. These paths ae called nfomatve paths snce they focus on dvng the obots though locatons n the envonment whee the sensoy nfomaton s mpotant. In ths pape 1 we ntoduce the nfomatve path plannng poblem and pesent a new contol algothm fo geneatng closed nfomatve paths n unknown envonments. The key nsght s nsped by [2]. The obots move along the paths, makng the aeas they obseve as dynamc o statc. As the obots dscove the statc/dynamc stuctue of the envonment, they eshape the paths to avod vstng statc aeas and Ths mateal s based upon wok suppoted n pat by ONR-MURI Awad N , the NSF Gaduate Reseach Fellowshp Awad & The Boeng Company. D. E. Solteo and D. Rus ae wth the Compute Scence and Atfcal Intellgence Laboatoy, Massachusetts Insttute of Technology, Cambdge, MA (solteo@mt.edu, us@csal.mt.edu). M. Schwage s wth the Mechancal Engneeng Depatment, Boston Unvesty, Boston, MA (schwage@bu.edu). 1 Ths pape s content s extacted fom the fst autho s Maste s thess [1]. (a) Iteaton: 1 (b) Iteaton: 1 (c) Iteaton: 3 (d) Iteaton: 113 Fg. 1: Path shapng phase of the mult-obot system hadwae mplementaton. The paths, shown as the blue and ed lnes connect all the wayponts, shown as black ccles. The dynamc egons (ponts of nteest) ae epesented as geen dots. As tme passes by, the obots shape the paths so they can cove the dynamc egons of the envonment. focus on sensng dynamc aeas. An example of ths eshapng pocess fo two obots can be seen n Fgue 1. The adaptve contolle has two key featues. The fst s an adaptaton law that the obots use to lean how sensoy nfomaton s dstbuted n the envonment, though paamete estmaton. The second featue s a Voono-based coveage contolle (buldng upon [2]) that pefoms the eshapng of the paths based on the obots estmates of the space. The nfomatve path poblem has many applcatons. Fo example, a suvellance task of a cty whee the obots could lean the egons whee cme s fequently commtted and geneate paths so that they can sense these cme egons moe fequently and sense the safe egons less fequently. In ths pape we ae nteested n usng t to acheve and mpove pesstent sensng tasks [3] n unknown envonments, whee the obots ae assumed to have sensos wth fnte footpnts. Moe specfcally, n a pesstent sensng task we wsh fo the obots wth fnte senso footpnts to conduct the nfomaton gatheng whle guaanteeng a bound on the dffeence between the obots cuent model of the envonment and the actual state of the envonment fo all tme and all locatons. Snce the sensos have fnte footpnts, the obots cannot collect the data about all of the envonment at once. As data about a dynamc egon becomes outdated, the obots must etun to that egon epeatedly to collect new data. In pevous wok [3], a pesstent sensng contolle calculates the speeds

2 of the obots at each pont along gven paths (efeed to as the speed pofles) n ode to pefom a pesstent sensng task,.e. to pevent the obots model of the envonment fom becomng too outdated. The ntuton behnd ths speed contolle s to vst faste changng aeas moe fequently than slowe changng aeas. The pesstent sensng poblem s defned n [3] as an optmzaton poblem whose goal s to keep a tme changng feld as low as possble eveywhee fo all tme. We efe to ths feld as the accumulaton functon. The accumulaton functon gows whee t s not coveed by the obots sensos, ndcatng a gowng need to collect data at that locaton, and shnks whee t s coveed by the obots sensos, ndcatng a deceasng need fo data collecton. A stablzng speed pofle s one whch mantans the heght of the accumulaton functon bounded fo all tme. In ths pape we show that ou computed nfomatve paths can be used n conjuncton wth the stablzng speed pofles fom the pesstent sensng contolle to locally optmze a pesstent sensng task. The contbutons of ths pape ae: a povably stable adaptve contolle fo leanng the locaton of dynamc events n an envonment and smultaneously computng nfomatve paths fo these events n ode to locally optmze a pesstent sensng task. We call ths the pesstent nfomatve contolle; smulaton and hadwae mplementaton. A. Relaton to Pevous Wok Most of the pevous wok n path plannng/geneaton focuses on eachng a goal whle avodng collson wth obstacles, e.g. [4], o on computng an optmal path accodng to some metc, e.g. [5]. Othe woks have focused on pobablstc appoaches to path plannng, e.g. [6]. Po wok n adaptve path plannng, e.g. [7], consdes adaptng the obot s path to changes n the obot s knowledge of the envonment. In ths pape, we focus on geneatng paths that allow the obots to pepetually tavel though egons of nteest n an unknown envonment. We use adaptve contol tools to ceate a novel algothm fo computng nfomatve paths. We use an appoach based on Voono pattons, smla to [8]. Howeve, contay to [8], the envonment, although unknown, s not andom, and we ae not concened wth optmzng the tajectoy of the agents to mnmze the pedctve vaance, but athe optmzng the locaton of agents fo a coveage task n an unknown envonment. The adaptve contolle elevant to ths thess was ntoduced n [2], whee a goup of agents wee coodnated to place themselves n locally optmal locatons to sense an unknown envonment. Ths pape bulds upon ths pevous wok, but wth some sgnfcant changes and addtons. The Voono pattons ae used by the obots to change the locaton of the agents, whch ae now wayponts defnng the obots paths. The paths must be closed paths and must povde good sensng locatons fo the obots. Also, we wsh the obots to pefom pesstent sensng along these paths. The pesstent sensng concept motvatng ths pape was ntoduced n [3], whee a lnea pogam was desgned to calculate the obots speed pofles n ode fo them to pefom a pesstent sensng task, that s, mantan the heght of the gowng accumulaton functon bounded. The obots wee assumed to have full knowledge of the envonment and wee gven a pe-desgned path. In ths pape we allevate these two assumptons by havng the obots lean the envonment though paamete estmaton, and use ths nfomaton to shape the paths nto useful paths. These two allevatons povde a sgnfcant step towads pesstent sensng n dynamc envonments. In Secton II we set up the poblem, pesent locatonal optmzaton tools usng a Voono-based appoach, and pesent a bass functon appoxmaton of the envonment. In Secton III, we ntoduce the pesstent nfomatve contolle and pove stablty of the system unde ths contolle. Fnally, n Secton IV we povde smulated and hadwae esults. II. PROBLEM FORMULATION We assume we ae gven multple obots whose tasks ae to sense an unknown dynamcal envonment whle tavelng along the ndvdual closed paths, each consstng of a fnte numbe of wayponts. The goal s fo the obots to dentfy the aeas wthn the envonment that ae dynamc and compute paths that allow them to jontly sense the dynamc aeas. A fomal mathematcal descpton of the poblem follows. Let thee be N obots, dentfed by {1,..., N}, n a convex, bounded aea Q R 2. An abtay pont n Q s denoted q. Robot s located at poston p Q and tavels along ts closed path consstng of a fnte numbe n() of wayponts. The poston of the th waypont n obot s path s denoted p, {1,..., n()}. Let {p,..., p n() } be the confguaton of obot s path and let V be a Voono pattons of Q, wth the th waypont poston n obot s path as the geneato pont, defned as = {q Q : q p q p, (, ) (, )},, {1,..., N}, {1,..., n()}, {1,..., n( )}, (1) whee, denotes the l 2 -nom. We assume that the obot s able to compute the Voono pattons based on the waypont locatons, as s common n the lteatue [2]. Snce each path s closed, each waypont along obot s path has a pevous waypont 1 and next waypont + 1 elated to t, whch ae efeed to as the neghbo wayponts of. Note that fo each, +1 = 1 fo = n(), and 1 = n() fo = 1. Once a obot eaches a waypont, t contnues to move to the next waypont along ts path, n a staght lne ntepolaton. 2 We assume that the netwok of obot s n the system s a connected netwok,.e. the gaph whee each obot s a node and an edge epesents communcaton between two nodes s a connected gaph. Ths connected netwok coesponds to the obots abltes to communcate wth each othe. 2 We assume that the wayponts do not outun the obots unde the acton of the nfomatve path contolle.

3 A. Envonment Stuctue A sensoy functon, defned as a map φ : Q R (whee R efes to non-negatve scalas) detemnes the constant ate of change of the envonment at pont q Q. The functon φ(q) s not known by the obots, but each obot s equpped wth a senso to make a pont measuement of φ(p ) at the obot s poston p. The ntepetaton of the sensoy functon φ(q) may be adjusted fo a boad ange of applcatons. It can be any knd of weghtng of mpotance fo ponts q Q. In ths pape we teat t as a ate of change n a dynamc envonment. B. Locatonal Optmzaton Buldng upon [2], we can fomulate the cost ncued by the mult-obot system ove the egon Q as n() N W s H = 2 q p 2 φ(q)dq =1 =1 n() + N =1 =1 W n 2 p p +1 2, (2) whee q p can be ntepeted as the unelablty of the sensoy functon value φ(q) when obot s at p, and p p +1 can be ntepeted as the cost of a waypont beng too fa away fom a neghbong waypont fo obot s path. W s and W n ae constant postve scala weghts assgned to the sensng task and neghbo dstance, espectvely. Note that unelable sensng and dstance between neghbong wayponts ae expensve. The fst tem couples all obots and the espectve wayponts, makng them wok togethe to cove the envonment. The second tem couples the wayponts, makng a path fo each obot. A fomal defnton of nfomatve paths fo multple obots follows. Defnton II.1 (Infomatve Paths fo Multple Robots). A collecton of nfomatve paths fo a mult-obot system coesponds to a set of waypont locatons fo each obot that locally mnmze (2). Next we defne thee popetes analogous to massmoments of gd bodes. The mass, fst mass-moment, and centod of V ae defned as M = W V s φ(q)dq, L = W V s qφ(q)dq, C = L M, espectvely. Also, let e = C p. Fom [1], we have = M e W n(p +1 + p 1 2p ), and an equlbum s eached when H =. Assgnng to each waypont dynamcs of the fom H ṗ = u, (3) whee u s the contol nput, we popose the followng contol law fo the wayponts to convege to an equlbum confguaton: u = ( K (M e + α )) /β, whee α = W n(p +1 + p 1 2p ), β = M + 2W n and K s a unfomly postve defnte matx and potentally tme-vayng to mpove pefomance. Note that β > and has the nce effect of nomalzng the weght dstbuton between sensng and stayng close to neghbong wayponts. C. Sensoy Functon Appoxmaton We assume that the sensoy functon φ(q) can be paametezed as an unknown lnea combnaton of a set of known bass functons. That s, Assumpton II.2 (Matchng condton). a R m and K : Q R m, whee Rm s a vecto of non-negatve entes, such that φ(q) = K(q) T a, (4) whee the vecto of bass functons K(q) s known by the obot, but the paamete vecto a s unknown. Let â (t) be obot s appoxmaton of the paamete vecto a. Then, ˆφ (q) = K(q) T â s obot s appoxmaton of φ(q). Then, we defne the mass moment appoxmatons as ˆM = W s ˆφ (q)dq, ˆL = W s q ˆφ (q)dq, Ĉ = ˆL. ˆM Addtonally, we can defne ã = â a, and the sensoy functon eo, and mass moment eos as φ (q) = ˆφ (q) φ(q) = K(q) T ã, (5) M = ˆM M = W s K(q) T dq ã, (6) V L = ˆL L = W s qk(q) T dq ã, (7) C = L. (8) M Fnally, n ode to compess the notaton, let K p (t) and φ p (t) be the value of the bass functon vecto and the value of φ at obot s poston p (t), espectvely. III. INFORMATIVE PATHS FOR PERSISTENT SENSING A. Relaton to Pesstent Sensng Tasks Pevous wok [3] developed a way to calculate the speed of obots tavelng along known statc paths n ode to sense the envonment and mantan the heght of the accumulaton functon bounded fo all tme. We assume each obot s equpped wth a senso wth a fnte footpnt F (p ) = {q Q : q p ρ} 3 when the obot s at locaton p, whee ρ s a constant postve scala. The accumulaton functon gows whee t s not coveed by any obot senso, and shnks othewse. Mathematcally, the accumulaton functon, efeed to as Z(q, t) at tme t fo pont q, evolves accodng to φ(q) c (q), f Z(q, t) >, N q(t) Ż(q, t) = (φ(q) N q(t) ) +, (9) c (q) f Z(q, t) =, 3 Any footpnt shape can be used, and the footpnt sze does not have to be the same fo all obots. Fo smplcty, we use a ccula footpnt wth same sze fo all obots.

4 whee N q (t) s the set of obots whose senso footpnts ae ove the pont q at tme t,.e. N q (t) = { q F ( p (t) ) }. We would lke to use the stablty cteon fo a pesstent sensng task, defned n [3], and plug t nto the adaptve contolle, such that the contol acton nceases the stablty magn of the pesstent sensng task though tme. Howeve, snce the obots do not know the envonment, but have estmates of t, each obot uses the estmated veson of ths stablty cteon, gven by ˆφ (q, t) N =1 τ c (q, t) T (t) c (q) = ŝ (q, t) <, q ˆφ (q, t) >. (1) whee ˆφ (q, t) (the estmated sensoy functon) s the estmated ate at tme t at whch the accumulaton functon gows at pont q, the constant scala c (q) s the ate at whch the accumulaton functon shnks when obot s senso s coveng pont q (and s known by the obots), T (t) s the tme t takes obot to complete the path at tme t, and τ c (q, t) s the tme obot s senso coves pont q along the path at tme t. These two last quanttes ae calculated by the obots wth the speed pofles. Each obot also knows ts estmated stablty magn at tme t, defned as Ŝ(t) = (max q ŝ (q, t)). A stable pesstent task at tme t s one n whch S(t) (the tue veson of Ŝ(t)) s postve, whch means the obot s able to mantan the heght of the accumulaton functon bounded at all ponts q. Note that only ponts q that satsfy ˆφ (q, t) > ae consdeed n a pesstent sensng task snce t s not necessay to pesstently sense a pont that has no sensoy nteest. Ponts that satsfy ths condton ae efeed to as ponts of nteest. 4 In [3], a lnea pogam was gven whch can calculate the speed pofles fo the paths at tme t that maxmze Ŝ(t) (o S (t) fo gound-tuth), fo all. Hence, fom ths pont on, we assume that the obots know these maxmzng speed pofles and use them to obtan ŝ (q, t), q, t. B. Pesstent Infomatve Contolle We desgn an adaptve contol law and pove that t causes the paths to convege to a locally optmal confguaton accodng to (2), whle mpovng the stablty magn of the pesstent sensng task and causng all of the obots estmates of the envonment to convege to the eal envonment descpton. All of the obots estmates of the envonment convege to a same estmate due to a consensus tem [2] used n the adaptve laws. Snce the obots do not know φ(q), but have estmates ˆφ (q), the contol law becomes u = K ( ˆM ê + α ), (11) ˆβ 4 We assume the envonment contans a fnte numbe of ponts of nteests. These fnte ponts could be the dscetzaton of a contnuous envonment. whee α = W n (p +1 + p 1 2p ), (12) ˆβ = ˆM + 2W n, (13) ê = Ĉ p. (14) Addtonally, to ncopoate the stablty cteon fo pesstent sensng tasks fom (1), let the wayponts have new dynamcs of the fom whee u I = ṗ = I u, (15) s defned n (11), and { T, f Ŝ u < and t t, u > τ dwell, 1, othewse, (16) τ dwell s a desgn paamete, and t, u s the most ecent tme at whch I, swtched fom zeo to one (swtched up ). Equatons (15) and (16) ensue that the estmated stablty magn fo the pesstent sensng task does not decease. Remak III.1. Fo postve ɛ, Ŝ (t) s not always defned when ag max q ŝ (q, t ɛ) ag max q ŝ (q, t + ɛ). In such cases Ŝ (t) efes to Ŝ (t + ɛ). The paamete â s adjusted accodng to Λ = λ = t t w (τ)k p (τ)k p (τ) T dτ, (17) w (τ)k p (τ)φ p (τ)dτ, (18) whee w (t) s a postve constant scala f t < τ w and zeo othewse, and τ w s some postve tme at whch pat of the the adaptaton fo obot shuts down to mantan Λ and λ bounded. Let n() b = W s K(q)(q p ) T dq ṗ, (19) â pe =1 = b γ(λ â λ ) ζ N l, (â â ), (2) =1 whee ζ > s a consensus scala gan, and l, can be ntepeted as the stength of the communcaton between obots and and s defned as { D max p p, f p p D max l, = (21), othewse. Snce a(j), j, we enfoce â (j),, j, by the pojecton law [2], â = Γ( â pe I poj âpe ), (22) whee Γ R m m s a dagonal postve defnte adaptaton gan matx, and the dagonal matx I poj s defned elementwse as, f â (j) >, I poj (j) =, f â (j) = and â pe (j), (23) 1, othewse,

5 Fg. 2: Envonment fo the mult-obot hadwae mplementaton. Fg. 3: Intal paths fo the mult-obot hadwae mplementaton whee (j) denotes the j th element fo a vecto and the j th dagonal element fo a matx. Theoem III.2 (Convegence Theoem fo Pesstent Sensng by Multple Robots). Unde Assumpton II.2, wth waypont dynamcs specfed by (15), contol law specfed by (11), and adaptve law specfed by (22), we have () lm t I (t) ˆM (t)ê (t) + α (t) =, {1,..., N}, {1,..., n()}, () lm t φ p (τ) =, {1,..., N}, τ w (τ) >, () lm t (â â ) =,, {1,..., N}. We efe the eade to [1] to vew the detals of the poof. Popetes () and () fom Theoem III.2 togethe mply that the obots wll lean the tue sensoy functon fo the envonment f the unon of the tajectoes whle the weghts ae postve s ch enough. Theefoe, we can desgn the ntal waypont locatons such that, between all obots, most of the dynamc unknown envonment s exploed (see Fgue 3). The stablty magn can theoetcally wosen whle I, fo some,, cannot swtch fom one to zeo because t s watng fo t t, u > τ dwell. Howeve, τ dwell can be selected abtaly small and, n pactce, any compute wll enfoce a τ dwell due to dscete tme steps. Theefoe, t s not a pactcal estcton. As a esult, ntutvely, () fom Theoem III.2 means that lm t ˆM (t)ê (t) + α (t) = only f ths helps the pesstent sensng task. Othewse lm t I (t) =, meanng that the pesstent sensng task wll not beneft f the th waypont n obot s path moves. IV. IMPLEMENTATION AND RESULTS We smulated the system many tmes and executed an mplementaton wth two quadotos moe than 1 tmes. Hee we pesent a case fo N = 2 obots, n() = 44 wayponts,. A fxed-tme step numecal solve s used wth a tme step of.1 seconds and τ dwell =.9. The envonment paametes ae σ =.4 and ρ tunc =.2, a(j) = 6, fo j {3, 4, 5, 1, 15, 2, 23, 24, 25}, and a(j) = othewse. The envonment ceated wth these paametes can be seen n Fgue 2. The paametes â, Λ and λ, fo all ae ntalzed to zeo. The paametes fo the contolle ae K = 9,,, Γ = dentty, γ = 3, W n = 6, W s = 15, w = 3, and ρ =.12. In addton, D max s assumed to be vey lage, so that Bass Coeffcent Eo Leanng Iteatons Fg. 4: Bass coeffcent eo Mean Integ. Paam. Eo Leanng teatons Fg. 6: Integal paamete eo Consensus Eo Lyapunov lke Functon Leanng Iteatons Fg. 5: Consensus eo x Leanng teatons Fg. 7: Lyapunov-lke functon n leanng phase l, (t)ζ = 2,,, t. The envonment s dscetzed nto a 1 1 gd and only ponts n ths gd wth ˆφ (q) > ae used as ponts of nteest n (1). By only usng ths dscetzed veson of the envonment, the unnng tme fo expements s geatly educed. Fo moe senstve systems, ths gd can be efned. The envonment and, theefoe, senso measuements ae smulated. The gowng behavo of the accumulaton functon ove the envonment follows the descpton fom (9). As an mplementaton detal, although ρ =.12 fo puposes of the pesstent nfomatve contolle, a value of ρ =.126 (5% ncease) was used on the physcal obot to consume the accumulaton functon n the envonment, whch allows to ovecome the effects of small tackng eos fom the quadotos and the effects of the dscetzaton of the path fo the pesstent sensng task. The ntal paths can be seen n Fgue 3, whee each obot has a zg-zaggng path acoss a poton of the envonment, and between both obots, most of the envonment s ntally tavesed. We fst allowed the obots to go though the ntal paths wthout eshapng them so that they can sample most of the space and lean the dstbuton of sensoy nfomaton n the envonment. Theefoe, we pesent esults n two sepaate phases: 1) leanng phase, and 2) path shapng phase. The leanng phase coesponds to the obots tavelng though the whole paths once, wthout eshapng them, n ode to lean the envonment. The path shapng phase coesponds to when (15) s used to eshape the paths nto nfomatve paths, and stats afte the leanng phase s done by both obots. In the path shapng phase, w =,. A. Leanng Phase In the leanng phase, we can see fom Fgue 4 that φ (q), q Q fo one of the obots. Snce the consensus eo conveges to zeo n accodance wth () fom Theoem III.2 and shown n Fgue 5, then we can conclude that the adaptaton laws cause φ (q), q Q,. Ths means

6 Estmated Tue Path Shapng Iteatons Fg. 8: Mean waypont poston eo Lyapunov lke Functon Mean Poston Eo (m) Path Shapng Iteatons (a) Iteaton Fg. 9: Lyapunov-lke functon n path shapng phase Estmated 1 Estmated 2 Tue Mean Accumulaton Stablty Magn Path Shapng Iteatons Fg. 1: Pesstent sensng task s stablty magn (b) Iteaton 2 Tme Fg. 11: Mean functon value accumulaton that the unon of both obots tajectoes was ch enough to geneate accuate estmates fo all of the envonment. Fgue 6 Rt shows that the mean ove both obots of w (τ )(φ p (τ ))2 dτ conveges to zeo, n accodance wth () fom Theoem III.2. Fnally, fo ths leanng phase, we see n Fgue 7 that the Lyapunov-lke functon V4 s monotoncally non-nceasng. (c) Iteaton 113 Fg. 12: Thee snapshots of the hadwae mplementaton of the mult-obot system dung the path shapng phase at dffeent teaton values. The paths, shown as the blue and ed lnes, connects all the wayponts coespondng to each obot. The ponts of nteest n the envonment ae shown as geen dots, and the sze of a geen dot epesents the value of the accumulaton functon at that pont. The obots ae the blue-lt and ed-lt quadotos, and the senso footpnts ae epesented by the coloed ccles unde them. B. Path Shapng Phase Fgue 12 shows snapshots of the mult-obot mplementaton dung the path shapng phase. We can see how the path evolves unde ths contolle n the path shapng phase n Fgues 1a to 1d. Fgue 8 shows quantty I (t)km (t)e (t)+ α (t)k convegng to zeo,, n accodance to () fom Theoem III.2. Fgue 9 shows the Lyapunov-lke functon V4 monotoncally non-nceasng and eachng a lmt. Fgue 1 shows the pesstent sensng task s stablty magn nceasng though tme, as expected. The chatteng n the stablty magn s due to the dscetzaton of the system, and can be educed by shotenng the length of tme steps. Fnally, Fgue 11 shows the mean ove all ponts of nteest of the value of the accumulaton functon ove tme. As shown, ths value ntally nceases on aveage due to the ntalzaton of the system. Late, t stats to decease and eaches an appoxmate steady-state behavo that coesponds to the locally optmal fnal confguaton of the system. The hadwae mplementaton was un fo moe than 1 tmes, geneatng nfomatve paths that ae pactcally dentcal to the smulated cases. V. C ONCLUSION Ths pape uses a Voono-based coveage appoach, buldng upon pevous wok n [2], to geneate an adaptve contolle fo obots to lean the envonment sensoy functon though paamete estmaton and shape the paths nto nfomatve paths that can be used fo pesstent sensng,.e. locally optmal paths fo sensng dynamc egons n the envonment and pefomng pesstent sensng tasks. The pesstent nfomatve contolle was mplemented wth two quadoto obots, geneatng esults that suppot the theoy. R EFERENCES [1] D. E. Solteo, Geneatng nfomatve paths fo pesstent sensng n unknown envonments, Maste s thess, Massachusetts Insttute of Technology, Cambdge, MA, June 212. [Onlne]. Avalable: [2] M. Schwage, D. Rus, and J.-J. Slotne, Decentalzed, adaptve coveage contol fo netwoked obots, The Intenatonal Jounal of Robotcs Reseach, vol. 28, no. 3, pp , 29. [Onlne]. Avalable: [3] S. L. Smth, M. Schwage, and D. Rus, Pesstent obotc tasks: Montong and sweepng n changng envonments, Robotcs, IEEE Tansactons on, vol. PP, no. 99, pp. 1 17, 211. [4] A. Pazz, C. Guano Lo Banco, and M. Romano, η 3 -splnes fo the smooth path geneaton of wheeled moble obots, Robotcs, IEEE Tansactons on, vol. 23, no. 5, pp , oct. 27. [5] K. Kyakopoulos and G. Sads, Mnmum jek path geneaton, n Robotcs and Automaton, Poceedngs., 1988 IEEE Intenatonal Confeence on, ap 1988, pp vol.1. [6] L. Kavak, P. Svestka, J.-C. Latombe, and M. Ovemas, Pobablstc oadmaps fo path plannng n hgh-dmensonal confguaton spaces, Robotcs and Automaton, IEEE Tansactons on, vol. 12, no. 4, pp , aug [7] C. Cunnngham and R. Robets, An adaptve path plannng algothm fo coopeatng unmanned a vehcles, n Robotcs and Automaton, 21. Poceedngs 21 ICRA. IEEE Intenatonal Confeence on, vol. 4, 21, pp vol.4. [8] R. Gaham and J. Cotes, Adaptve nfomaton collecton by obotc senso netwoks fo spatal estmaton, Automatc Contol, IEEE Tansactons on, vol. PP, no. 99, p. 1, 211.