Optimal Allocation of Multi-Platform Sensor Resources for Multiple Target Tracking

Transcription

1 4th Internatona Conference on Informaton Fuson Chcago, Inos, USA, Juy 5-8, 0 Optma Aocaton of Mut-Patform Sensor Resources for Mutpe Target Tracng Gary Asns Space and Arborne Systems Raytheon Company E Segundo, CA, USA Gary_I_Asns@raytheoncom Sam acman Space and Arborne Systems Raytheon Company E Segundo, CA, USA ssbacman@raytheoncom Abstract - Ths paper presents graph agorthms for computaton of the exact ma souton of the Radar Resources Dstrbuton Probem These agorthms can be apped for snge and mutpe vehces on a target fed and for snge and mutpe target tracng Ths probem nvoves the computaton of the ma fght paths for a number of observng vehces (arcrafts or UA) on a target fed for radar observatons of mutpe targets (or trac custers for mut-target tracers) Each vehce had restrcted resources of tme or fue and fxed start and fnsh poston Each target or custer was assgned a prorty and poston on the target fed Optmzaton crtera were: maxmze number of observed targets or custers wth hghest prortes for a vehces on target fed wth the constrant that each target coud be observed by ony one vehce The exact ma souton was found as the resut of a fxed number of teraton steps on a mut-eve tree, whch represents the fu partay-orented graph of fght paths wth tme-weghted connectvty matrx and prorty-weghted vertces The agorthms are desgned to wor n rea-tme: ma fght paths are recacuated for each vehce each tme that the stuaton on the target fed changes - targets or tracs can be added, removed, moved or ther prortes and postons can be changed These agorthms can be apped for statonary and movng targets, as we as for combnatons of ground and arborne targets Separate agorthms were desgned for cases of snge and mutpe vehces on the target fed Each target can be observed by a vehce once or severa tmes durng fght path Agorthms can be apped for snge and for mutpe-target tracer radars Cosey spaced targets or tracs can be grouped n custers to speed up computatons Aso, for mutpe targets tracng of ground targets these agorthms can be apped for computaton of ma fght path to cover ntersectons wth hgh numbers of trac ambgutes The same agorthms can be apped for the Combat Management Probem: to mze fght paths of vehces (arcraft or UA) on the combat fed to dstrbute troops, weapons, and ammunton to dfferent areas of the combat fed wth dfferent prortes The agorthms were mpemented n C-code Resuts are shown to ustrate the methods Keywords: Resources Management, Optma Souton of the Radar Resources Dstrbuton Probem, Optma Fght Path, Mut-Patform Sensor Resources, Mutpe Target Tracng, Graph Agorthms Introducton Ths paper presents graph agorthms for computaton of the exact ma souton of the Radar Resources Dstrbuton Probem These agorthms can be apped for snge and mutpe vehces on the target fed and for snge and mutpe target tracng Ths probem nvoves the computaton of the ma fght paths for a number of observng vehces (arcrafts or UA) on a target fed for radar observatons of mutpe targets (or trac custers for mut-target tracers) Each vehce had restrcted resources of tme or fue and fxed start and fnsh poston Each target or custer was assgned a prorty and poston on the target fed Optmzaton crtera were: maxmze number of observed targets or custers wth hghest prortes for a vehces on target fed wth the constrant that each target coud be observed by ony one vehce Dfferent methods and approaches for the Radar Resources Dstrbuton Probem are desgned n []-[3] The exact ma souton was found as the resut of a fxed number of teraton steps on a mut-eve tree (MLTA mut-eve tree agorthms), whch represents the fu partay-orented graph of fght paths wth tmeweghted connectvty matrx and prorty-weghted vertces The MLTA agorthms are desgned to wor n rea-tme: ma fght paths are recacuated for each vehce each tme that the stuaton on the target fed changes targets or tracs can be added, removed, moved or ther prortes and postons can be changed These agorthms can be apped for statonary and movng targets, as we as for combnatons of ground and arborne targets Separate agorthms were desgned for cases of snge and mutpe vehces on the target fed Each target can be observed by a vehce on one or severa tmes durng fght path ISIF 88

2 Agorthms can be apped for snge and for mutpetarget tracer radars Cosey spaced targets or tracs can be grouped n custers to speed up computatons Aso, for mutpe target tracng of ground targets these agorthms can be apped for computaton of the ma fght path to cover ntersectons wth hgh numbers of trac ambgutes The same agorthms can be apped for the Combat Management Probem: to mze fght paths of vehces (arcraft or UA) on the combat fed to dstrbute troops, weapons, and ammunton to dfferent areas of the combat fed wth dfferent prortes Agorthms were mpemented n C-code Tme to appy: approxmate computaton tme s 40 sec on a typca PC for target fed wth 00 targets (or targets /tracs custers) and three vehces on target fed Statement of the Graph Optmzaton Probem We consder a partay orented graph (networ) G wth n vertexes: vertexes are nputs, m vertexes are outputs and ntermedate vertexes, where n + m + (some vertexes can be nputs and outputs at the same tme) The connectvty matrx of graph G s non-symmetrc - C nxn Each vertex of the graph has a weght (prorty of the vertex) - p Each edge (,) aso has a weght (trave tme from vertex to vertex pus observaton tme of vertex ) - t (n genera t t ) We defne the path n the graph as a sequence of connected vertexes: Q = {,,, u} Where - s nput, u - s output and each vertex n the path can be repeated up to s tmes (but ony for ntermedate vertexes and not-mmedate vertex repettons) The prorty of path Q s the sum of prortes of vertexes n the path: P = u p = The tme of path T s the sum of tmes of edges n the path: T = t (, ) Q Suppose we have r traveers on the graph: r mn(, m) (f r =, t s one traveer on the graph) Each traveer moves from one nput to one output usng one path Each traveer can vst the same vertex n the path up to s tmes (e we have s-oops n the path, but f s = - no oops n the path), but dfferent traveers can not vst the same vertex n the graph (no ntersectons between paths of traveers) Each traveer has restrcted resources of trave tme (or fue) T durng hs path The probem s to fnd the ma path for each traveer on the graph We defne the set of ma paths for a vehces as the ma pan The ma pan maxmzes the sum of a vertex prortes, beongng to the paths of a traveers: r = P max () wth restrcted tme (fue) resources for each traveer: T for =,,, r () T We consder two appcatons of probem () - () and agorthms for computng the ma souton of ths probem Note The graph probem () - () s dfferent from the Traveng Saesman Probem [4]: gven a st of ctes and ther parwse dstances, the tas s to fnd a shortest possbe tour that vsts each cty exacty once 3 Radar Resources Dstrbuton Probem and Combat Management Probem We descrbe the Radar Resources Dstrbuton Probem as foowng: Suppose we have m vehces wth radars (arcrafts or UAs) on the target fed - {,,, m} These vehces shoud mae observatons of n targets - C, C,, C } We group the cosey spaced targets or { n tracs n custers The approxmate ocatons of these targets (or custers) are nown Each vehce has a start pont r and a fnsh pont wth fxed coordnates on the target fed (start and fnsh pont can be the same pont) Aso each vehce has restrcted resources of fght tme (or tme schedue) or restrcted fue resources T Each target (custer) C ocaton on the target fed s assgned coordnates ( x, y ) and a prorty p Aso every par of targets ( C, C ) has an assocated tme t Ths tme s a fght tme from target C to target C pus observaton tme for target C (matrx [ t ] n n s notsymmetrc - ( t t ) ) If no fght path between targets C and C exst, then t = The stuaton on the target fed changes dynamcay: targets can be added, moved, or removed from the target fed Target prortes can change any tme durng observatons Thus the number of targets n, arrays {( x, y )}, { p }, and matrx [ t ] n n are not constant f 89

3 As prevousy mentoned, cosey spaced targets are grouped n custers For each custer we cacuate average poston and average prorty of custer targets If the stuaton on the target fed has been changed, then szes of custers, ther postons and prortes shoud be recacuated We cacuate a set of fght paths { F } for vehces ( =,,, m) on the target fed Each fght path F ncudes start pont r, sequence of targets for observaton { C, C,, C } and fnsh pont f One target C can be observed one or severa tmes durng fght path, so each sequence C, C,, C } can { ncudes the same target up to s tmes (s-oops) We defne the ma fght path OFP for vehce fght path targets: F wth maxmum prortes of observed C F n restrcted fght tme p max (3), ( C, C ) F T : t T (4) as In (3) we add the prortes of a targets { C, C,, C } n fght path F r In (4) we cacuate fght and observaton tme for a targets n F as the sum ( t ) + t + + t 3 r ncudng the fght tme from r the start pont to frst target t r, as we as the fght tme from the ast target to the fnsh pont t r, f To sove the Radar Resources Dstrbuton Probem we have to fnd the ma fght paths OFP for a vehces {,,, m} on the target fed These ma fght paths do not have any overaps or ntersectons (common targets or custers) Thus we do not observe the same target (custer) more than one tme on the target fed by dfferent vehces In Fg we ustrate the probem defned by equatons (3)-(4) on partay orented graph G wth nne targets and two vehces (number of vertexes on graph N G n + m ) ertexes -9 are targets (or custers) on the target fed ertexes 0 and are start ponts for vehces and, vertexes and 3 are fnsh ponts Each target vertex has weght prorty p (start and fnsh ponts have prortes 0) A vertexes on graph G are connected by edges (a edges from start ponts to targets and edges from targets to fnsh ponts are orented) Each edge (, ) between vertex and vertex has weght t The connectvty matrx for graph G s A = [ a ] ( n+ m) ( n+ m) (where a = 0 f vertex s not connected wth vertex, and a = f vertex s connected wth vertex ) In Fg vehce has ma fght path OFP : and vehce has ma fght path OFP : OFP and OFP do not have ntersectons (common targets) Another appcaton of graph mzaton probem () - () s the Combat Management Probem: to mze fght paths of vehces (arcrafts or UAs) wth restrcted resources on the combat fed to observe and attac targets, whe dstrbutng troops, weapons, and ammunton to dfferent areas of the combat fed wth dfferent prortes Optma fght path ehce - Start 0 ehce - Start t 0, p t,7 3 6 t,9 p t 6,9 t 7, t 9,6 Optma fght path p p 9 8 t 9, ehce - Fnsh 3 ehce - Fnsh Fgure Radar Resources Dstrbuton Probem on a Graph 4 MLTA: Mut-eve tree agorthm Case : One vehce on target fed Frst we consder the probem defned by equatons (3)-(4) wth one vehce on target fed We suppose aso for smpcty now that one target C can be observed ony one tme durng the fght path, so there are no dupcated targets n sequence { C, C,, C } (no s-oops) In Fg we deete vertexes and 3 - start and fnsh ponts for vehce from graph G Ths forms the subgraph G for vehce wth connectvty sub-matrx A = [ a ] ( n+ ) ( n+ ), where n s the number of targets (custers) on the target fed ( we deete rows and coumns and 3from connectvty matrx A) 90

4 In genera, we descrbe a mut-eve tree agorthm on graph G for cacuaton of the ma fght path for one vehce on target fed Frst we transform graph G to a tree (Fg ) The root vertex of the tree s the start pont r for vehce Ths s eve 0 n tree Leve n tree ncudes a vertexes of graph G OFP connected wth root r by edges accordng to connectvty matrx A So, eve ncudes a vertexes, whch are one step away n graph G from root of tree Leve ncudes a vertexes connected by edges wth vertexes from eve, and they are two steps away from root of tree, etc Last eve (n + ) ncudes a vertexes connected by edges wth vertexes from eve n, and they are (n + ) steps away from the root vertex (t ncudes fnsh pont f for vehce ) We defne branch of tree = { r, v, v,, v } as sequence of vertexes n the tree Here ndex s number of vehce, ndex s the number of eves n the tree and ndex s the number of branches n the tree ertex r n s root of tree (eve 0) for vehce branch v s vertex of graph, vertex G from eve So, each branch has one vertex from each eve Each branch n the tree represents a fght path (or parta fght path) for vehce on target fed We defne ength of branch as number of vertexes n (or ( + )) If sequence = { r, v, v,, v } ncudes the same vertex of graph G more that one tme (for exampe, f v = v ), then we defne sub-sequence { v,, v } n branch as a oop Leve 0 Leve Leve Leve 3 Loop(0,3,,3) Root Fgure Mut-Leve Tree Agorthm 4 Agorthm 8 Optma Fght Path - ranch(0,,4,6, ) Step Leve = 0 Create st of branches on eve 0 (ust one branch n the st): L = }, where = { r } 0 { 7 4 Create empty st of ma branches (fght paths) for vehce : L = {0} Number of branches n st L : N = 0 Step ased on st of branches L = {,,} on eve n tree we cacuate st of a branches L + on eve ( + ) For each branch from st L wth ength we create set of branches wth ength ( + ) Accordng to the connectvty matrx A, ast vertex v of branch from st L connected by edges of graph G wth ast vertexes v + of set branches from st L + (Fg ) Step 3 If st L + s empty go to step 4 Ths fact means, that we do not have more new branches on eve ( + ) n tree and we can stop computatons Step 4 Deete a branches from st L +, whch have frst vertex dfferent than r We do not consder branches wth start pont dfferent that start pont of vehce If after deeton, st L + s empty go to step 4 Step 5 Deete a branches wth oops from st L + Durng fght path we do not observe each target (or custer) on target fed more than one tme, so we deete a branches wth oops If after deeton, st L + s empty go to step 4 8 t = t + t + t + ) < ( 0,,4 4, T pmax = p + p4 + p6 + Resource Restrcton: t = ( t0,8 + t8,7 + t7, ) > T 9

5 Step 6 Cacuate fght tme t for each branch (or fght paths) n st L Fght tme for branch + = { r, v, v,, v, v + } s sum of fght and observatons tmes for par of vertexes from : t = tr v t vv t vv + Step 7 Cacuate number of branches N + n st L +, whch satsfed restrcton (4): t T for branch If N + = 0 go to step 4 Ths fact means that we do not fnd any new branches n eve ( + ), whch satsfed resource restrcton (tme or fue) for vehce We can stop computatons, because n eve ( + ) fght tme w ony ncrease Step 8 If t > T deete branch from st L + We deete a branches from st L +, whch do not satsfy restrcton (4) on resources for vehce Step 9 Cacuate prorty for each branch (or fght paths) n st L + Prorty of branch s sum of prortes of vertexes (targets or custers) from sequence = { r, v, v,, v, v + } (start and fnsh ponts have prorty 0): p = pv + p + + v p v + Step 0 Add to st L a branches from st L +, whch have frst vertex r and ast vertex f - start and fnsh ponts of These branches are canddates for the ma fght path of vehce Cacuate number of branches N n st L Step Sort a branches n st L by prortes p ( =,,, N ) n descendng order, thus the frst branch n st L has the hghest prorty Step Increase eve: = + Step 3 If = (n + ), go to step 4 Ese go to step Step 4 The frst branch from st L s the exact ma fght on target fed G wth n targets (custers) and wth one vehce on fed Lst L ncudes a fght paths from start pont to fnsh pont for vehce Step 5 Stop computatons Note Agorthm cacuates the exact ma souton for the Radar Resources Dstrbuton Probem and for the Combat Management Probem (3)-(4) wth one vehce on target fed Ths agorthm fnds the best possbe fght path for vehce It means that radar observes the maxmum possbe number of targets (custers) wth hghest prortes n restrcted tme Aso, ths agorthm creates a st of a fght paths sorted by prortes for vehce on the target fed Note Agorthm s adaptve and can be used n rea tme We recacuate the ma fght path for vehce n rea tme every tme when stuaton on target fed changed: targets (custers) are added, removed or moved on target fed, or ther prortes are changed durng fght path efore each recacuaton of ma fght paths we change graph G accordng to the new stuaton on target fed, and we change start pont to the current poston of vehce and tme restrcton for vehce from T to ( T t), where t s current fght tme Note 3 Agorthm can be modfed for the case n whch vehce can observe each target (or some of targets) on target fed up to s tmes durng one fght path We defne branch = { r, v, v,, v } as a s-oop, f at east one vertex n sequence { v, v,, v } s repeated s or more tmes In ths case n step 5 we deete ony branches wth (s + )- oops (nstead of snge oops), and n step 3 we defne maxmum eve n tree (where we stop computatons) as: = ( n ( s + ) + ) Note 4 In tas (3)-(4) we can use other mzaton crtera nstead of (3) For exampe we can cacuate the ma fght path for vehce, whch ncudes ony targets wth eve prortes more that certan prorty P : p max ; for : p P C F (5) Agorthm can be modfed for mzaton crtera (5) (or other crtera) nstead of crtera (3) In ths case we ust have to change cacuatons of branch prortes n step 9 accordng to crtera (5) 5 Case : Mutpe vehces on target fed Suppose we have mutpe vehces on target fed: {,,, m}, where m s the number of vehces (Fg ) As we mentoned before, to sove the Radar Resources Dstrbuton Probem or Combat Management Probem n ths case we have to fnd ma fght paths for each vehce ( =,,, m) In ths case nstead of mzaton crtera (3) we have the foowng mzaton crtera: 9

6 m ( = C F p ) max (6) In crtera (6) we sum prortes for a vehces and for a ma fght paths on target fed At the same tme for each fght path F and for each vehce we have to satsfy resource restrctons (4): For : t T (7), ( C, C ) F We cacuate the ma pan (OP), whch ncudes ma fght paths for a vehces As we mentoned before, ma fght paths for dfferent vehces can not ntersect each other, e they can not ncude the same targets (custers) The dea of a MLTA agorthm for mut-vehces s the foowng: we cacuate the st of fght paths for each vehce usng agorthm, then we put a these fght paths n one st and sort t by prortes n descendng order We tae the frst fght path wth hghest prorty from the st and we put t n OP, then we tae the second fght path wth hghest prorty, whch does not beongs to vehce wth frst fght path and whch does not have ntersectons wth the frst fght path, etc, unt we cover a targets or unt we use a vehces on the target fed 5 Agorthm Step We defne OP as an ma pan OP ncudes ma fght paths for a vehces ( =,,, m) on the target fed Create empty set OP = {0} Step Appy Agorthm for each vehce ( =,,, m) As resut of these appcatons we create st of branches of the tree L (or st of fght paths) for each vehce ranches n st L are sorted by prortes p ( =,,, N ), where N s number of branches n st L Frst branch n st L has the hghest prorty (see step n Agorthm ) Step 3 Combne a sts of branches L n one st L We put together a branches from sts L ( =,,, m) n one st L Number of branches n st L s: N = m = N Each branch n st L has the foowng structure: = {, p, r, v, v,, v, f } Here: - s branch number ( =,,, N ); - s vehce number (branch beongs to vehce ); p - s prorty of branch (see step 9 n Agorthm ); { r, v, v,, v, f} - s sequence of graph vertexes (or targets n fght path), whch beong to branch : r s start pont and f s fnsh pont for vehce, { v, v,, v } are targets for observatons, s number of targets (eve of tree) Step 4 Sort branches n st N L by prortes p ( =,,, ) n descendng order Step 5 Frst branch n st L has hghest prorty for a vehces on target fed If ths branch beongs to vehce (accordng to frst poston n branch structure), then ths branch s an ma fght path OFP for vehce We add ma fght path OFP to ma pan OP Step 6 Deete a branches of vehce from st L Step 7 Deete a branches from st, whch have ntersectons (common vertexes) wth OFP ecause ma fght paths for dfferent vehces can not ntersect or can not ncude the same targets (custers), we deete branches wth ntersectons from st L Step 8 If st L s empty (after deeton of branches n steps 6-7) then: We do not have any vehces avaabe for targets observatons n L or We do not have any fght paths avaabe n st L, whch satsfed resources restrctons or 3 We competed observatons for a targets on target fed by fght paths from L In ths case ma pan OP s competed and we go to step Step 9 Le n step 5 the frst branch n the st L has hghest prorty for a remanng vehces If ths branch beongs to vehce (accordng to frst poston n branch structure), then ths branch s the ma fght path OFP for vehce We add ma fght path OFP to ma pan OP Step 0 Index = Go to step 6 (we repeat deeton of branches from st L for new ndex = ) Step Optma pan OP = OFP, OFP,, OFP } L { m s souton of the probem (6)-(7) wth m vehces on target fed Fght path OFP s ma fght path for vehce ( =,,, m) Step Stop computatons Note 5 Agorthm does not cacuate the exact ma souton of probem (6)-(7) We ca t condtonay ma pan COP COP s usuay good enough 93

7 approxmaton of OP and can be computed qucy wth ow number of cacuatons To fnd exact ma pan we have to oo for combnatons of sts of fght paths for a vehces and cacuate the best OP wth non-ntersected ma fght paths Ths computaton taes onger tme: 5 Agorthm 3 Step Appy Agorthm for each vehce ( =,,, m) As resut we create a st of a fght paths L for each vehce Paths n each st L are sorted by prortes: frst fght path has the hghest prorty (see step n Agorthm ) Step We cacuate pans for a vehces: each pan ncudes combnaton of fght paths for m vehces: P = { FP, FP,, FP } m Here P - s a pan of fght paths for m vehces; pan number; FP s a fght path of vehce n the pan;,,, } - are numbers of fght paths for m vehces { m n the pan Step 3 For each pan P we cacuate pan prorty as a sum of prortes of m fght paths, whch are ncuded n ths pan Step 4 Sort a set of a pans by pan prortes Step 5 We start wth the frst pan wth hghest prorty n the set of pans and contnue to search unt we fnd the pan wth hghest prorty P, whch does not have any ntersectons n fght paths ncuded n ths pan Ths s the ma pan Fght paths of a vehces n ths pan P do not have any common vertexes, so we do not cover the same targets (or custers) on target fed by dfferent vehces Step 6 Stop computatons P Optma pan s exact ma pan and exact souton of probem (3-(4) wth mutpe vehces on target fed Note 6 Agorthm can be used for cases wth arge number of vehces and arge number of targets on target fed to ncrease computaton speed of ma fght paths Note 7 Agorthm and for some cases agorthm 3 wor n rea tme We recacuate the ma fght path for each vehce ( =,,, m) n rea tme every tme when stuaton on target fed s changed: targets (custers) are added, removed or moved on target fed, or ther prortes are changed durng fght path Note 8 Agorthms, and 3 can be apped not ony for snge target radar but aso for mutpe target tracers (R- MHT), whch trace smutaneousy many targets [5] In ths case we consder a group of cosed spaced unversa target tracs of R-MHT as a custer, whch represents one vertex of the graph G The ma fght path of vehce wth R-MHT cover maxmum number of groups of target tracs wth hghest prortes n restrcted observaton tme 6 Impementaton n C and tests resuts We mpemented Agorthms, and 3 n C To mze speed of computatons we reazed mut-eve tree agorthm as an array of ned sts n C-code Each ned st represents one branch of the tree Each node n ned st represents vertex n the tree (or graph) Ths node s a structure, whch ncudes vertex number and ponter to prevous node n the branch of the tree MLTA agorthm wors fast n C We need ust one path of ned st to cacuate a nformaton for one branch of the tree: we cacuate prorty of the branch (or fght path), fght tme on ths branch (f fght tme does not satsfy vehce tme restrcton, ths branch shoud be deeted from the tree), exstence oops n the branch (f branch has oops (snge or s-oops), t shoud be deeted from the tree), frst and ast nodes of branch (ma fght path must ncude start and fnsh ponts of vehce), etc Dependng of these cacuatons we defne status of branch n the tree and mae one of the three actons: Deete branch from the tree or Deete branch from the tree and remember ths branch n st of fght paths for a vehce, as a canddate for ma fght path or 3 Keep ths branch n the tree for the next teraton (cacuaton on the next eve of the tree) Computaton tme for the MLTA agorthms depends from the foowng parameters: Number of targets on the target fed To decrease computaton tme for target fed wth arge number of targets, we group cosey spaced targets n custers Number of vehces on target fed For arge number of vehces on target fed, we recommend to use Agorthm (nstead of Agorthm 3) for fast computaton of condtonay ma fght pan We tested C-code for MLTA agorthms on target feds wth dfferent scenaros: varous number of targets (custers), number of vehces, fght tme restrctons, types of radars (snge-target and mut-target tracers), etc Tests resuts show hgh speed for computaton of ma fght paths for vehces for most scenaros For exampe, computaton of ma fght paths for three vehces wth snge-target tracer radars on target fed wth 00 target custers (or trac custers) taes approxmatey 40 sec on a typca PC In ths case we can reaze recacuatons of ma fght paths durng target observatons for vehces n rea tme f stuaton on target fed has been changed 94

8 7 Summary The exact ma souton for graph mzaton probem was found as the resut of a fxed number of teraton steps on a mut-eve tree, whch represents the fu partayorented graph of fght paths wth tme-weghted connectvty matrx and prorty-weghted vertces MLTA agorthms can be apped for Optma Aocaton of Mut-Patform Sensor Resources for Mutpe Target Tracng Agorthms cacuate ma souton of the Radar Resources Dstrbuton Probem and for the Combat Management Probem (6)-(7) Agorthm s desgned for computaton of ma fght path for a snge vehce on the target fed wth n targets (custers) Agorthms and 3 are desgned for computaton of ma fght paths for mutpe vehces on the target fed We cacuate ma fght pan, whch ncudes ma fght paths for each vehce Optma fght paths for dfferent vehces have no ntersectons (common targets) A ma fght paths satsfy radar resources restrctons (tme schedue or fue resource) Agorthm and 3 cacuate exact ma souton of the Radar Resources Dstrbuton and Combat Management Probems Probem wth snge and mutpe vehces on target fed These agorthms compute the ma fght pan, whch ncudes ma fght path for each vehce on target fed Agorthm s desgned for fast computaton of condtonay ma fght pan, as a good approxmaton of ma pan MLTA agorthms are adaptve and wor n rea tme for many scenaros We re-cacuate ma fght path for each vehce durng fght path f stuaton on target fed has been changed: targets (custers) are added, removed, moved on target fed, or ther prortes are changed MLTA agorthms are apped for vehces wth dfferent types of radars: snge-target tracers, mutpe-target tracers or mxed We modfed MLTA agorthms for the foowng cases: vehce can mae an observaton of each target not ones but severa tmes durng one fght path (see Note 3); dfferent mzaton crtera can be apped (for exampe crtera (5)) Agorthms and 3 easy to modfy for the case: possbty of the same targets observatons by dfferent vehces on target fed We mpemented MLTA agorthms as C-functons Tests resuts show hgh computaton speed of ma fght paths for vehces on target fed n C (approxmate computaton tme s 40 sec on a typca PC for target fed wth 00 targets (or targets /tracs custers) and three vehces on target fed) [] radford S Wer, Wam S Rodney Emergent Contro of Netted Sensor Resources, Proceedngs of the 008 Tr-Servce Radar Symposum, June 008 [] Abht Snha, Thagangam Krubaraan, Yaaov ar- Shaom Autonomus Ground Target Tracng by Mutpe Cooperatve UAs, 005 IEEE Aerospace Conference, March 005 [3] Matthew Rudary, Deepa Khosa, James Guochon, P Aex Dow, arbara yth A Sparse Sampng Panner for Sensor Resource Managemant, Sgna Processng, Sensor Fuson and Target Recognton, Proc of SPIE o 635, Apr 006 [4] E L Lawer, Jan Kare Lenstra, A H G Rnnoy Kan, D Shmoys The Traveng Saesman Probem, John Wey and Sons Ltd, 985 [5] Samue acman, Robert Popo Desgn and Anayss of Modern Tracng Systems, Artech House, 999 References 95