On Sensor Fuson n the Presence of Packet-droppng Communcaton Channels Vjay Gupta, Babak Hassb, Rchard M Murray Abstract In ths paper we look at the problem of multsensor data fuson when data s beng communcated over channels that drop packets randomly. We are motvated by the use of wreless lnks for communcaton among nodes n upcomng sensor networks. We wsh to dentfy the nformaton that should be communcated by each node to others gven that some of the nformaton t had transmtted earler mght have been lost. We solve the problem exactly for the case of two sensors and study the performance of the algorthm when more sensors are present. For the two-sensor case, the performance of our algorthm s optmal n the sense that f a packet s receved from the other sensor, t s equvalent to recevng all prevous measurements, rrespectve of the packet drop pattern. I. INTRODUCTION In recent years, there has been a lot of nterest n sensor networks and sensor webs. Orgnally motvated by mltary survellance applcatons, they are now beng employed n a wde varety of applcatons e.g., see [], [23] and the references theren). Compared to usng one bg sensor, typcal advantages of usng sensor networks nclude relatvely lower costs, nherent robustness and greater moblty. Moreover greater accuracy s possble not only due to more observatons beng taken for the same source but also snce multple types of sensors can potentally be used that generate observatons related to dfferent facets of the source. However, the prce to be pad for these advantages s the need for communcaton and greater complexty n the estmaton algorthms. A basc problem n sensor networks s how to fuse the observaton data from many nodes. The classcal Kalman flter s a centralzed flter that assumes all observatons comng to a central computng faclty. Over the years technques have been proposed to decentralze the flter computatons and to mnmze the amount of nformaton that needs to be transmtted among the nodes. An early contrbuton was [24] where nformaton obtaned from the local sensors s combned to generate the global estmate. However t requred that data about the global estmate be sent from the fuson node to the local sensors. A smlar requrement was mposed n the successve orthogonalzaton of measurement subspaces algorthm proposed n [3]. Ths dffculty was frst overcome n [22], [7] n whch each local node sends ts own local estmate based on ts own data and communcates ths estmate and the error covarance Dvson of Engneerng and Appled Scence, Calforna Insttute of Technology, Pasadena, CA 925, USA {gupta,hassb,murray}@caltech.edu Work supported n part by the AFOSR grant F49620-0--0460. data to the fuson center. Smlar results for contnuous tme systems were presented n [25]. These results were further extended n [2] where both the measurement and tme update steps of the Kalman flter were decentralzed. An alternatve approach for data fuson from many nodes usng the Federated flter was proposed n [5]. A Bayesan method was used and some algorthms presented n [4], [4] whch are optmal when there s no process nose. A scatterng framework [7] and algorthms based on decomposton of the nformaton form of the Kalman flter [9], [3] have also been proposed for data fuson. A scheme based on the concept of dynamc consensus s proposed n [2], but t assumes multple communcaton rounds per tme step of the system evoluton. For some other approaches proposed n the lterature e.g. those based on tracklets [9]), see [8], [8]. However these approaches assume a fxed communcaton topology among the nodes wth a lnk, f present, beng perfect. Wth wreless channels beng used for communcaton between sensor nodes that are moble, ths assumpton needs to be questoned. In such a case, packets of nformaton from one node to another wll be dropped randomly by the communcaton channel present between them. Ths random loss of nformaton rentroduces the problem of correlaton between the estmaton errors of varous nodes [] and renders the approaches proposed n the lterature as suboptmal. We wsh to address ths problem of fndng the optmal global estmate for each node n the case when there are communcaton channels present between the nodes and packets of nformaton are beng randomly dropped. We dsallow approaches such as sendng all the measurements taken by each node across the entre network each tme communcaton s possble because they can potentally ental transmttng arbtrarly large amounts of data. An approach to solve ths problem was proposed n [2] n the context of track-to-track fuson through exchange of state estmates based on each sensor s own local measurements but the specfc scheme that was used was not proven to be optmal. Moreover, as was found n [20], the algorthm for fusng the local state estmates that was proposed s not optmal n the mean square sense. It was subsequently proved n [6], [8] that the technque was based on an assumpton that was not met n general. The man contrbuton of ths paper s to pose the problem of optmal codng for estmaton n the presence of packet droppng channels and to propose an alternatve algorthm whch s optmal n the mean square sense for the case of two sensors beng present. For more than two sensors, the algorthm no
longer remans optmal, but we present smulatons to study the performance. Ths paper s organzed as follows. We begn wth the problem formulaton where we also defne our notatons and assumptons. Then we solve the problem for the case of two sensors. In the next secton we present smulaton results to study the performance of the protocol as the number of sensors s ncreased. We fnsh wth conclusons and outlne some avenues for future work. II. PROBLEM FORMULATION Process Sensor Sensor 2 Channel Fg.. Problem block dagram. Only two sensors are shown, but n general there can be N sensors. The basc set-up of the problem s llustrated n Fgure. Consder the dscrete-tme process that evolves accordng to x k+ = F k x k + G k u k, ) where x k R n s the process state and u k R m s zero-mean whte nose process. The ntal condton x 0 s assumed to be a zero mean random varable wth covarance matrx Π 0. The process s beng observed by N sensors N = 2 n the fgure) of the form y k = H kx k + v k, =,, N, 2) wth vk beng zero-mean whte nose processes as Rp well. Further u k u j Q k 0 0 vḳ v j... = 0 Rk 0.... δ kj, vk N vj N 0 0 Rk N where the nner product x, y s defned n the usual way see, e.g., [5]) x, y = E [ xy T ]. The sensor models and the nose statstcs are known to all sensors. From now on, we use the words sensor, sensor-estmator and node nterchangeably. The sensors are allowed to communcate through communcaton channels. The communcaton channels are modeled as swtches wth an assocated drop probablty. In other words, for every transmsson, the channel ether transmts the nput to the output wth a certan probablty or does not produce any output. Such a channel model s a natural way to model stuatons where error detecton codng s done for each packet of nformaton. We assume that there s no ntertransmsson codng present. Further we assume that a suffcent number of bts are allotted for each transmsson so that quantzaton effects can be gnored. At every tme nstant, each sensor-estmator thus has a pool of knowledge avalable to t about ts own and other sensors measurements. On the bass of ths knowledge, t constructs an estmate of the process ). The estmate of sensor at tme step k s denoted by ˆx k. We assume that to generate the estmate of x k the measurements tll tme step k can be used. Thus we are nterested n causal estmators only. We further restrct our attenton to lnear estmators. The goal of each sensor-estmator s to mnmze the covarance matrx of the error defned by e k = x k ˆx k. In other words, we are nterested n the lnear least mean square llms) estmator of x k. If there are no communcaton channel related constrants present, each sensor can transmt the latest measurement t has observed to other sensors and at tme step k, every sensor wll thus have access to all the measurements [ {yj }k j=0, {y2 j }k j=0,, ] {yn j }k j=0 taken so far by the N sensors. It can then calculate the best estmate through a Kalman flter see, e.g., [6]). Further, to reduce computaton at every node, the Kalman flter can be decentralzed through any of the technques already mentoned. The key thng to note n these technques s that all nodes are alke n the sense that they have access to an dentcal set of nformaton from whch to construct estmates. Thus every node can functon as a central node whch processes the nformaton sent by other nodes. Further snce the latest nformaton s transmtted at every tme nstant, the estmaton process can be recursve. The ntroducton of communcaton channels, breaks ths symmetry and ntroduces several nterestng ssues nto the problem. ) It s no longer clear what nformaton the nodes should transmt to each other. Transmsson of measurements alone mght not be optmal. Consder two nodes and j joned by a communcaton channel that has just dropped a packet at tme step k. If the node s sendng ts latest measurement alone, at tme step k + the node j wll have no access to the measurement yk. Thus ts estmate wll not be optmal. On the other hand, the node does not know that a packet has been dropped at tme step k. Thus t cannot retransmt the nformaton. If t transmts all the measurements t has taken so far, the amount of nformaton needed to be transmtted wll grow wthout bound as tme k ncreases. 2) The conventonal sensor fuson algorthms are no longer applcable. Ths s so snce the fuson algorthms are usually recursve n ether the measurements or the local estmates. It s not clear what to do n the case when the nformaton s not delvered at a partcular tme step, but can potentally be delvered at some future tme when the communcaton channel allows nformaton to be transmtted. 3) It may be possble to use some routng algorthms to mprove performance. Consder three nodes, j and
k. If the communcaton channel between j and k drops a lot of packets, t mght be useful for j to rout nformaton to k through the node. We propose to look at these ssues. We begn wth the case of two sensors where we have tght results and then study the performance for the case of N sensors beng present. We assume a broadcast medum n whch each sensor can talk to every other sensor, except for packet drops. The packet droppng pattern s assumed to be ndependent from one tme step to the next, although as we shall see, for the case of 2 sensors, our algorthm s optmal rrespectve of the packet drop statstcs or model. III. TWO SENSORS PRESENT Let the measurements of the sensors at tme step k be gven by yk and y2 k. Let the estmates be denoted as ˆx k and ˆx 2 k respectvely. Moreover, let â k S denote the best possble estmate of random varable a gven the nformaton n the set S. Note that f there are two nformaton sets S and S 2 such that S S 2, then the error n the estmate â k S2 s less than or equal to the error n the estmate â k S n the mean square sense. Ths s so snce whle formng the estmate â k, we can smply not use the nformaton from S 2. Keepng ths fact n mnd, we frst wrte down the bggest set of nformaton whch can possbly be avalable to a sensor on whch t bases ts estmate. To understand the stuaton a lttle better, we vew t from the poston of sensor. For sensor, we are lookng to optmze the estmate ˆx k+ S, where S can be no bgger than the followng If a packet has been receved from sensor 2 at tme step k, then S = [ {y j }k j=0, {y2 j }k j=0]. If the last packet was receved from sensor 2 at tme step k t, then S = [ {y j }k j=0, {y2 j }k t j=0]. Smlar statements can be wrtten about sensor 2. The error covarances of these estmates wll be a lower bound on all schemes that transmt dfferent nformaton from one sensor to the other. As an example, for a scheme that only sends the latest measurements, the nformaton set would not contan those measurements that are dropped by the channel and would thus be a subset of S mentoned above. Thus ts error covarance wll be more than the one mpled by ths upper bound. In the followng, we propose an algorthm that acheves the upper bound. We wll adopt the vew-pont of sensor n the presentaton. Consder a hypothetcal centralzed estmator, that has access to all the measurements from both the sensors. For ths optmal centralzed flter defne the followng quanttes: ˆx k l : estmate of x k based on all the measurements of both the sensors up to tme step l. P k l : covarance matrx of the error correspondng to the estmate ˆx k l. Equvalently, we can say that the optmal centralzed flter s obtanng measurements from a flter of the form where y k = H k x k + v k, [ ] H H k = k H 2 k whle v k s whte Gaussan nose wth zero mean and covarance [ ] R R k = k 0 0 Rk 2. Thus we can wrte the followng equatons for the tme and measurement updates of the Kalman flter: = P k k + H T k R k H k ˆxk k = P k k ˆxk k + H T k R k y k P k k = F k P k k Fk T + G T k Q k G k ˆx k k = F k ˆx k k. Now we utlze the fact that R k s a block dagonal matrx. Thus Hk T R k H k = [ H ) T ) k R ] k H k Hk T R k y k = [ H ) T ) k R ] k y k. Thus we can rewrte the equatons for the optmal flter as = + [ ) ] ˆxk k = P k k ˆxk k + [ ] ) ˆx k k ˆx k k P k k = F k P k k F T k + G T k Q k G k ˆx k k = F k ˆx k k The frst two equatons form a measurement update step and the last two a tme update step. Note that the communcaton requrement at tme step k from sensor s as follows: For the covarance matrces, no communcaton s requred. They can be calculated by ether sensor, even off-lne. For the estmates, the contrbuton from -th sensor s P k k ˆx k k P k k ˆx k k. We can use ths observaton to obtan the nformaton that sensor should transmt to the other sensor that s used to calculate the optmal estmate. Denote Λ k = ) ˆx k k ) ˆx k k.
Thus Λ k s the contrbuton from sensor correspondng to measurement at tme step k. Thus, we can wrte ˆxk k = P k k ˆxk k + Λ k = P k k Aˆxk k + Λ k = Γ k [ Pk k 2 ˆxk k 2 + Λ k ] Global error covarance matrces P k k and P k k. Γ k = P k k Fk P k k. I k = Λ k + Γ ki k. It transmts Ik and wats for the correspondng nformaton from the other sensor. If t receves I j k from the other sensor, t obtans the global estmate as ˆxk k where + Λ k Γ k = P k k APk k. Contnung ths way, we obtan ˆxk k = Γ k Γ k Γ P 0 ˆx 0 + where I k = Λ k + Γ k Λ k + Γ k Γ k Λ k 2 + I k, + Γ k Γ k Γ ) Λ 0 Thus, the nformaton needed from sensor at tme step k s precsely Ik gven n the above equaton.note that the computaton requred for calculatng Ik does not grow wth tme snce Ik = Λ k + Γ k Ik. Ths nformaton vector washes away the effect of any prevous packet losses and leads to the calculaton of optmal estmate at tme step k as f all the measurements from both sensors were avalable, e, t calculates the estmate ˆx k S f the last communcaton from sensor j was possble at tme step k. No knowledge about packet drop probabltes s requred. Thus we have obtaned a way for sensor to obtan the lowest possble error achevable when communcaton from sensor j s possble at tme step k. If the last communcaton was possble at tme step k t, clearly, the best estmate wll be obtaned by fndng the estmate at tme step k t and then propagatng t wth sensor s measurements only. Thus we obtan the followng algorthm that acheves the estmate based on the set S outlned above. We summarze the algorthm n the followng proposton. Proposton : The algorthm outlned below acheves the optmal causal llms estmate based on all of the sensor s own measurements tll tme step k and the other sensor s measurements tll tme step k t where the last communcaton from the other sensor was possble at tme k t. Algorthm : At each tme step k, Each sensor takes ts own measurement and runs a local Kalman flter. At tme step k, t calculates ˆx k k and P from ts local Kalman flter. Λ k ) = ˆx k k P k k ˆx k k. = Γ k Γ k Γ P 0 ˆx 0 + I k + I j k. If t does not receve nformaton from the other sensor, t uses the last global estmate t has and propagates t wth ts own measurements. Note that we have made no assumptons about the statstcs of the packet droppng process or even the knowledge of such statstcs to the two sensors. Thus ths algorthm s optmal for any packet drop pattern. We have proposed a way to encode the nformaton to be sent for estmaton purpose that s optmal under any channel packet drop model. In ths sense, the scheme s smlar n sprt to the work n [0]. IV. MANY SENSORS PRESENT When more than 2 sensors are present, the scheme no longer remans optmal. The reason for ths s the followng. The problem s n calculaton of the global error covarance matrx P k k. The equaton to be used s ) = + H T ) k R k H k. The observaton terms Hk to be ncluded are those whose nformaton s beng fused. For the 2 sensor case, thus terms of both sensors are to be ncluded. For the n-sensor case nformaton from all n sensors mght not be fused at all tme steps, dependng on the partcular lnks whch drop packets at any tme. Thus the proposed method s only sub-optmal n ths case. However, f all nodes are communcatng wth each other farly regularly, the performance loss wll be lmted. Whle t wll be nce to characterze ths performance loss and/or come up wth other algorthms that are optmal for partcular packet drop patterns n say) an expected sense, we have not been able to do that so far. Smulaton results seem to suggest that the performance loss s not huge for the case of multple sensors. In the smulaton results shown below, n case a sensor fals to receve packets from all the sensors, t assumes that the contrbuton from that sensor s 0 and adds up the vectors I j from the sensors that t has been succesful n communcatng to. Ths s clearly a sub-optmal thng to do; a slghtly better scheme would be to try to project the vector I j k based on Ij k f communcaton wth sensor j was not possble at tme step k. However we do not look nto such schemes for the present.
V. NUMERICAL EXAMPLES We now consder some numercal examples to see the performance of the algorthm. The system that we consder s a smple scalar system, wth dynamcs gven by x k+ =.25x k + w k, where w k s zero mean whte Gaussan nose wth varance. Consder ntally that the system s observed by 2 sensors of the form y k = x k + v k, 3) where vk s zero mean whte and Gaussan wth varance. Further the noses vk and w k are all ndependent of each other. The sensors are tryng to obtan the best possble estmate of the state of the system x k, whle communcatng over a channel whch drops packets randomly wth a probablty p = 0.3. All the plots that we show below are from the vewpont of sensor. Fgure 2 shows the evoluton of the estmate error covarance as a functon of tme. The packets from sensor 2 to sensor are lost at tmes k = 3 and 8. The sold lne represents the covarance for a hypothetcal sensor that has access to all the measurements taken so far. The crcles correspond to the performance of our scheme whle the remanng curve shows the performance of sensor takng nto account ts own measurements alone. It can be seen that as packets are lost, the error covarance wth our scheme ncreases; however as soon as a sngle packet s receved, the performance s the same as that of the hypothetcal sensor whch has receved all the measurements so far. Estmate error covarance 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 Our scheme Hypothetcal sensor Sensor s own data huge amount, though t does not approach the performance of the hypothetcal deal sensor anymore. Note that the plot shows the estmate error and not the covarance; hence the performance of our sensor can be momentarly better than the hypothetcal sensor. However n a mean squared sense, the hypothetcal sensor, of course, performs the best. Estmate error 0.5 0 0.5.5 Hypothetcal sensor Our scheme Sensor s own data 2 2 3 4 5 6 7 8 9 0 Tme step Fg. 3. Performance of estmators under three stuatons n presence of packet-droppng lnk: 4 sensor case To compare the performance of our scheme wth other schemes proposed n the lterature, we use two sensors wth nose varances equal to 0. We compared our scheme wth a scheme n whch only measurements are beng exchanged and another n whch local estmates are beng fused accordng to the scheme proposed n [2]. The results are shown n Fgure 4. The crcles correspond to the estmate gven by our scheme, the sold lne from the measurement exchange scheme whle the astersk curve from the estmate fuson scheme. The remanng curve s the estmate from the hypothetcal deal sensor as dscussed above. Agan our scheme performs better than other schemes proposed n the lterature. 4 2 3 4 5 6 7 8 9 0 Tme step Fg. 2. Performance of estmators under three stuatons n presence of a packet-droppng lnk: 2 sensor case We now consder the same system wth measurements beng taken by 4 sensors of the same form as 3). The measurement noses are stll ndependent wth each havng varance. Fgure 3 shows the estmates at sensor for three schemes when the probablty of packet drop n each channel s 0.2. The sold lne shows the estmate f the sensor reled on ts own measurements. The crcles correspond to our scheme whle the thrd curve s that of a hyptothetcal sensor that has access to all the measurements. It can be seen that our scheme mproves the estmates by a Error Covarance 3.5 3 2.5 2.5 0.5 Our scheme Hypothetcal sensor Measurement fuson Estmate fuson 0 2 3 4 5 6 7 8 9 0 Tme step Fg. 4. Performance of estmators usng dfferent schemes n the presence of packet-droppng lnk: 2 sensor case
VI. CONCLUSIONS AND FUTURE WORKS A. Conclusons In ths paper we looked at the problem of optmal sensor fuson n the presence of communcaton channels. We modeled the communcaton channel as a random swtch and posed the followng problem: Gven that the channel s droppng packets, what should the sensors communcate to each other so that the best estmate s obtaned. For the case of two sensors, we solved the problem and the soluton was ndependent of the packet drop pattern. The performance was much better than the schemes proposed n the lterature. However, the scheme can not be generalzed to an arbtrary number of sensors. B. Future Work Ths work can be extended n many ways. We have smply posed an nterestng and relevant problem and solved t for a specal case. The general soluton for n sensors s stll to be worked out. We can also look at more complcated channel models or codng schemes whch allow some part of the nformaton to pass through sometmes. Also we have assumed perfect knowledge about all the sensor models. It wll be nterestng to relax ths assumpton as well. Another perplexng avenue s to look at nformaton theoretc bounds on the qualty of estmate as a functon of rate of communcaton. VII. ACKNOWLEDGMENTS The authors would lke to thank Amr Farajdana, Hars Vkalo and Tmothy Chung for useful dscussons on the topc. REFERENCES [] Y. Bar-Shalom. On the track-to-track correlaton problem. IEEE Trans. Automat. Contr., AC-26:57 572, 98. [2] Y. Bar-Shalom and L. Campo. The effect of the common process nose on the two-sensor fused-track covarance. IEEE Transactons on Aerospace and Electronc Systems, AES-22:803 805, 986. [3] T. M. Berg and H. F. Durrant-Whyte. Dstrbuted and decentralzed estmaton. In Proc. of the Sngapore Internatonal Conference on Intellgent Control and Instrumentaton, volume 2, pages 8 23, 992. [4] S. Mor C. Y. Chong and K. C. Chang. Informaton fuson n dstrbuted sensor networks. In Proc. of the Amercan Control Conference, pages 830 835, 985. [5] N. A. Carlson. Federated square root flter for decentralzed parallel processes. IEEE Transactons on Aerospace and Electronc Systems, AES-26:57 525, 990. [6] K. C. Chang, R. K. Saha, and Y. Bar-Shalom. On optmal trackto-track fuson. IEEE Transactons on Aerospace and Electronc Systems, AES-33:27 276, 997. [7] C. Y. Chong. Herarchcal estmaton. In Proc. 2nd MIT/ONR C 3 W orkshop, 979. [8] C. Y. Chong, S. Mor, W. H. Barker, and K. C. Chang. Archtectures and algorthms for track assocaton and fuson. IEEE Aerospace and Electronc Systems Magazne, 5:5 3, 2000. [9] O. E. Drummond. Tracklets and a hybrd fuson wth process nose. Proceedngs of the SPIE, Sgnal and Data Processng of Small Targets, 363, 997. [0] V. Gupta, D. Spanos, B. Hassb, and R. M. Murray. Optmal LQG control across a packet-droppng lnk. IEEE Transactons on Automatc Control, 2004. Submtted. [] D. L. Hall and J. Llnas. An ntroducton to multsensor data fuson. Proceedngs of the IEEE, 85):6 23, 997. [2] H. R. Hashempour, S. Roy, and A. J. Laub. Decentralzed structures for parallel Kalman flterng. IEEE Trans. Automat. Contr., AC- 33:88 94, 988. [3] M. F. Hassan, G. Salut, M. G. Sngh, and A. Ttl. A decentralzed computatonal algorthm for the global Kalman flter. IEEE Trans. Automat. Contr., AC-23:262 268, 978. [4] M. E. Lggns II, C. Y. Chong, I. Kadar, M. G. Alford, V. Vanncola, and S. Thomopoulos. Dstrbuted fuson archtechtures and algorthms for target trackng. Proceedngs of the IEEE, 85:95 07, 997. [5] T. Kalath, A. H. Sayed, and B. Hassb. Lnear Estmaton. Prentce Hall, 2000. [6] H. Kwakernaak and R. Svan. Lnear Optmal Control Systems. John Wley and Sons, 972. [7] B. C. Levy, D. A. Castanon, G. C. Verghese, and A. S. Wllsky. A scatterng framework for decentralzed estmaton problem. Automatca, 9:373 384, 983. [8] S. Mor, W. H. Barker, C. Y. Chong, and K. C. Chang. Track assocaton and track fuson wth nondetermnstc target dynamcs. IEEE Transactons on Aerospace and Electronc Systems, AES- 38:659 668, 2002. [9] B. S. Rao and H. F. Durrant-Whyte. Fully decentralzed algorthm for mult-sensor Kalman flterng. IEE proceedngs, 38, 99. [20] J. A. Roecker and C. D. McGllem. Comparson of two-sensor trackng methods based on state vector fuson and measurement fuson. IEEE Transactons on Aerospace and Electronc Systems, AES-24:447 449, 988. [2] D.P. Spanos, R. Olfat-Saber, and R.M. Murray. Dstrbuted sensor fuson usng dynamc consensus. In Proceedngs of the 2005 IFAC World Congress to be presented), 2005. [22] J. L. Speyer. Computaton and transmsson requrements for a decentralzed lnear-quadratc Gaussan control problem. IEEE Trans. Automat. Contr., AC-24:266 269, 979. [23] R. Vswanathan and P. K. Varshney. Dstrbuted detecton wth multple sensors: Part I - fundamentals. Proceedngs of the IEEE, 85):54 63, 997. [24] D. Wllner, C. B. Chang, and K. P. Dunn. Kalman flter algorthms for a multsensor system. In Proc. of the 5th Conference on Decson and Control, pages 570 574, 976. [25] A. S. Wllsky, M. G. Bello, D. A. Castanon, B. C. Levy, and G. C. Verghese. Combnng and updatng of local estmates and regonal maps along sets of one-dmensonal tracks. IEEE Trans. Automat. Contr., AC-27:799 83, 982.