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1 Mcnns, C.R. (003 Vlocty fld path-plannng for sngl and multpl unmannd arl hcls. Aronautcal Journal, 07 (073. pp SSN Strathprnts s dsgnd to allow usrs to accss th rsarch output of th Unrsty of Strathclyd. Copyrght and Moral Rghts for th paprs on ths st ar rtand by th nddual authors and/or othr copyrght ownrs. You may not ngag n furthr dstrbuton of th matral for any proftmakng actts or any commrcal gan. You may frly dstrbut both th url ( and th contnt of ths papr for rsarch or study, ducatonal, or not-for-proft purposs wthout pror prmsson or charg. You may frly dstrbut th url ( of th Strathprnts wbst. Any corrspondnc concrnng ths src should b snt to Th Strathprnts Admnstrator: prnts@cs.strath.ac.uk

2 Vlocty fld path-plannng for sngl and multpl unmannd aral hcls C. R. Mcnns Dpartmnt of Arospac Engnrng Unrsty of Glasgow Glasgow, UK ABSTRACT Unmannd aral hcls (UAV ha sn a rapd growth n utlsaton for rconnassanc, mostly usng sngl UAVs. Howr, futur utlsaton of UAVs for applcatons such as bstatc synthtc aprtur radar and stroscopc magng, wll rqur th us of multpl UAVs actng coopratly to ach msson goals. n addton, to d-skll th opraton of UAVs for crtan applcatons wll rqur th mgraton of path-plannng functons from th ground to th UAV. Ths papr dtals a computatonally ffcnt algorthm to nabl path-plannng for sngl UAVs and to form and r-form UAV formatons wth act collson aodanc. Th algorthm prsntd xtnds classcal potntal fld mthods usd n othr domans for th UAV path-plannng problm. t s dmonstratd that a rang of tasks can b xcutd autonomously, allowng hgh ll taskng of sngl and multpl UAVs n formaton, wth th formaton commandd as a sngl ntty. NOMENCLATURE A ctor potntal C st of obstacls unt ctor f, F shapng functons G goal pont Q fld strngth R UAV S start pont u locty ctor V, potntal fld, locty ctor W workspac x UAV poston ctor poston rlat to lad UAV λ controllr nrs tm constant υ UAV spd ψ UAV hadng.0 NTRODUCTON Unmannd aral hcls ha sn a rapd growth n us for rconnassanc applcatons wth a wd rang of hcl typs and capablts fldd. Most currnt UAV typs rqur a fxd bas-staton to up-lnk way-ponts to th UAV, whch ha bn dtrmnd by a human oprator, or possbly by path-plannng softwar hostd by th bas-staton. As th us of UAVs bcoms mor commonplac, thr s a rqurmnt to d-skll th opraton of UAVs to allow untrand oprators to us ths systms n th fld wth a mnmum of ground qupmnt. Ths wll rqur much of th path-plannng capablty to rsd on th UAV wth th oprator up-lnkng only hgh-ll goals, whch must thn b autonomously xcutd. Such autonomous path-plannng must oprat n nar ral-tm, must b computatonally ffcnt and must b aldatd to nsur that th UAV safly achs ts goal. Futur UAV systms wll also s th us of multpl UAVs usd n formaton for tasks such as bstatc synthtc aprtur radar and stroscopc magng. Ths systms wll rqur path-plannng algorthms whch can form and r-form th UAVs whl nforcng collson aodanc btwn mmbrs of th formaton. n ordr to plan such rconfguratons as a top-down procss s unwldy for larg numbrs of UAVs. For xampl, a group of N UAVs would rqur N(N constrant chcks for collson aodanc n addton to N goal satsfacton chcks, so that th problm scals as N. Howr, for a dstrbutd algorthm th problm scals as N for ach mmbr of th formaton. n addton, dstrbutd plannng algorthms ar nhrntly mor robust and can cop wth addng or rmong mmbrs from th formaton wthout sgnfcantly modfyng th mannr n whch th algorthm oprats. Th partcular scnaro nsagd hr s for UAVs (possbly mcro-ar hcls usng an on-board dgtal map along wth GPS and nrtal nagaton to manour n a complx and cluttrd nronmnt. Th algorthm prsntd can transform a map of path constrants and goals nto a locty fld, wth any path through th locty fld guarantd to b collson-fr and to rach th goal. Th algorthm s abl to gnrat such paths for a sngl UAV or for

3 Fgur. Schmatc gomtry of th UAV workspac. multpl UAVs, whr ach UAV has nformaton on th stat of othr mmbrs of th formaton through cross-lnks. For multpl UAVs, ach mmbr of th formaton ws th othr mmbrs as mobl obstacls, whl th formaton confguraton s dfnd through a st of mobl goals towards whch ach UAV manours. ntally a sngl UAV R wll b consdrd mong n a structurd workspac W R, whr th UAV poston n W s rprsntd by a ctor x (x, x, as shown n Fg.. Th workspac has a st of statc obstacls C: x C ( x, x wth boundary C, a start pont S: x s (x s, x s and a goal pont G: x G (x G, x g towards whch R manours. Th UAV wll b consdrd to b pont-lk wth bank-to-turn control whch can track a hadng command, up to som turnng rat lmt, and a throttl control whch can track a spd command. Snc W s n R th UAV wll manour at a fxd alttud, although xtnsons of th mthod to R 3 ar possbl. A path-plannng algorthm s now sought whch can translat R from S to G wthout crossng th obstacl boundars C. An xtnson of th classcal potntal fld mthod wll b consdrd hr. To broadn th analyss to multpl UAVs n formaton, th st of obstacls C may also b mobl, wth ach UAV wng othr UAVs n th formaton as an obstacl. n addton, th goal G may b also b mobl and may b rfrncd to th lad UAV n th formaton, to nabl ladr-followr bhaour, or th goal may b rfrncd to som functon of th stat of th UAV formaton to prod d-cntralsd control. Th classcal applcaton of th potntal fld mthod rqurs an artfcal potntal functon V to b suprmposd on W (-4. Th potntal functon s chosn such that t has a sngl global mnmum at G and ts gradnt fld V drcts R safly away from C towards G for any start pont S W. Howr, for hurstcally gnratd potntal functons thr may b a st of local mnma whch can trap R n an qulbrum poston ( V 0 othr than G. To aod such local mnma V can b gnratd as a soluton to th Laplac quaton ( V 0 (3,4. Th rsultng artfcal potntal functon thn has th appalng proprts of lnarty and unqunss, whl ts maxma and mnma can only occur on th boundars of th doman. Thrfor, th potntal s fr of local mnma and so trappng at postons othr than G cannot occur. A mor aggrss form of path-plannng has also bn proposd by utlsng ortx functons (5-9. Th ortx potntal s also a soluton to th Laplac quaton and s usd to gnrat a solnodal gradnt fld whch s agan fr from local mnma. Vortx functons also prod a consstnt, prfrrd drcton for crcumnagatng th obstacl st C. Howr, prous applcatons of ortx mthods ha rld on hurstcs to blnd th gradnt and ortx flds. Such hurstc approachs do not lnd thmsls to rgorous aldaton. n ths papr ortx functons ar xtndd and usd as th bass for a dstrbutd path-plannng algorthm for sngl and multpl UAVs. Th frst nw fndng s that th ortx fld can b shapd by a scalar functon to rapdly truncat ts ffcts byond th cnty of th obstacl boundary. Th rsultng locty fld has zro drgnc and so can stll n prncpl satsfy th Laplac quaton wth ts appalng proprts. t s also dmonstratd that complx, nonsymmtrc obstacls can also b rprsntd as a solnodal locty fld, whch gratly nhancs th us of ortx functons as a pathplannng tool. n partcular, obstacls can b addd to or dltd from th workspac wthout r-plannng th ntr UAV path, as would b rqurd by othr path-plannng algorthms. Ths s an mportant capablty, whch allows th addton of nw path plannng nformaton as t bcoms aalabl. n addton, snc th global locty fld can b gnratd analytcally, th on-board computatonal orhad to mplmnt th algorthm s mnmal, allowng autonomous opratons wth on-board path-plannng. Whl not ncssarly ful or tm optmum, (0 th bnfts of th potntal fld mthod ha bn rcognsd for othr problm domans ( and ndd th mthod has bn appld to rlatd ar traffc managmnt problms (9,. Agan, th ky adantag for UAV applcatons s th ablty of th mthod to gnrat guarantd collson-fr paths n complx workspacs wth a mnmal computatonal orhad. Lastly, t s shown that th ortx locty fld can b drd from a ctor potntal whch can also b xtndd to rprsnt complx, non-symmtrc obstacls by a solnodal fld. Hlmholtz s thorm s thn nokd to dmonstrat that th solnodal fld componnt rprsntd by th ctor potntal, usd to nforc collson aodanc, can b systmatcally combnd wth gradnt flds gnratd from scalar potntal functons, usd to manour R towards G, n a rgorous way. Th utlty of th algorthm s xplord by consdrng a numbr of path-plannng problms for sngl and multpl UAVs n formaton..0 VORTEX FUNCTONS Th Laplac quaton has bn wdly usd to gnrat artfcal potntal functons for mobl robot path-plannng thr by a numrcal grd soluton or by utlsng a lmtd class of analytcal solutons (3,4. Solutons to th Laplac quaton ha th usful proprty that only a sngl global mnmum wll xst at th goal pont of th workspac G. Th Laplac quaton s dfnd by ` V( x 0, +.. ( wth two typs of analytc soluton n R. Typ solutons rprsnt an rrotatonal sourc or snk whras typ solutons rprsnt a solnodal sourc. Th typ solutons wth a snk trm can b usd to gnrat a potntal whch wll dr R towards G, whl th typ sourc and typ functons can b usd to drct R away from C and so nforc collson aodanc usng V V Q Tan Q ln å Q å + d ( x ~ x + ( x ~ x, x ~ x x ~ x d ( ~ x x + ( ~ x x G G.. (a.. (b.. (c

4 Fgur (a. Rlatonshp btwn a symmtrc solnodal fld and shapng functon ƒ. Fgur (b. Rlatonshp btwn an asymmtrc solnodal fld and shapng functon ƒ. t can b shown that th strngth of th typ potntal functon Q may b chosn to satsfy an xcluson zon of radus ε about an obstacl usng Equaton (c (4. Snc th typ potntal functon s not sngl-alud thr s howr no corrspondng rlatonshp for th strngth Q. Ths scalar functons ar thn usd to gnrat locty flds through th gradnt oprator V as ( x ~ x Q ( ~ x x Q.. (3a ~ ~ ~ ~ x x + x x x x + x x ( ( ( x x Q ( x ~ x + ( x ~ x ~ ( ( whr t can b shown that both locty flds ha null drgnc. 0 V. 0 V 0 0 snc thy ar both gnratd from solutons to th Laplac quaton. Although thr s no rlatonshp for th strngth Q, t wll now b shown that t s possbl to truncat th typ locty fld by a scalar shapng functon ƒ whl nsurng that th rsultng nw locty fld stll has null drgnc. Thrfor, f w postulat that th nw locty fld s drd from th gradnt of som potntal, ths potntal wll satsfy th Laplac quaton. Snc only th gradnt of th fld s to b usd, th potntal can howr rman unknown. A nw locty fld ƒ wll thrfor b consdrd wth ƒ chosn such that th fld has null drgnc so that.( f f. + f (. 0 Thrfor, snc. 0 th followng proprty s rqurd of th shapng functon ƒ ( x x Q ( x ~ x + ( x ~ x ~.. (3b.. (3c.. (3d.. (4 ( ~ ~ ( î, î ( x x + ( x f. 0 f f x t s clar thn that ƒ and must b normal, as shown Fg. (a. Snc s solnodal, ƒ must thrfor b a functon of th radal dstanc ξ from th cntr of th ortx. Howr, n addton to such azmuthally symmtrc functons, t s also possbl to gnrat solnodal flds for mor complx objcts through a partcular constructon of th locty fld. n partcular, f s dfnd as f f + whr thn t s clar that th condton ƒ. 0 s always satsfd for any smooth scalar functon f, as shown n Fg. (b. Thrfor, by a sutabl choc of functon ƒ, th magntud of nddual symmtrc ortx functons ƒ can b rapdly truncatd so that ortcs may b suprmposd wthout orlappng nflunc. Or ndd, mor complx solnodal flds can b gnratd through a sutabl choc of f and th constructon of th locty fld usng Equaton (6. Truncatng th ffct of th ortx locty fld n a smooth, contnuous mannr wll also allow complx locty flds to b gnratd wthout dscrt swtchng of componnts of th locty fld as obstacls ar approachd. Such dscrt swtchng lads to complx hybrd control problms whr stablty s dffcult to nsur. Th rsultng global locty fld gnratd hr howr wll b smooth and wll ha null drgnc, so n prncpl rtanng th proprts of th Laplac quaton, as dscussd abo. n ordr to us any of th locty flds dscussd abo, a fld strngth or shapng functon f must b dtrmnd whch rapdly truncats th ffct of th fld and nsurs that R dos not cross th boundary C of th obstacl st C. For a typ rrotatonal fld, such as that dfnd by Equaton (a, th fld strngth Q may b chosn to satsfy an xcluson zon about an obstacl, as dfnd by Equaton (c. An xampl of th rsultng locty fld s shown n f f f +.. (5.. (6

5 Fgur 3(a. Vlocty fld gnratd by an rrotatonal typ potntal. Fgur 3(c. Vlocty fld rsultng from a suprquadratc gnratng functon. whr ξ s th radus of nflunc of th ortx fld, whch s chosn to nclos th obstacl. f m >> thn f swtchs n a rapd, but contnuous mannr from to 0 as th radus of nflunc of th ortx fld s crossd, as shown n Fg. 3(b. For a mor complx obstacl, th obstacl boundary C may b mappd by f and a solnodal locty fld gnratd usng th constructon of Equaton (6. For xampl, a suprquadratc functon (3 may b usd to map a squar obstacl usng f H, H ~ ~ whr n >> to nsur a sharp dg to th obstacl boundary C. Usng Equaton (6, and Equaton (8 as a gnratng functon, th rsultng solnodal locty fld s found to b ( ~ n- ( ~ n- n x x n x x ~ n ~ n ~ n ~ n x x + x x x x + x x.. (9 Whl th shapd locty fld ƒ has null drgnc, as dscussd arlr, t also possbl to shap th locty fld dfnd by Equaton (9 n a dffrnt mannr to form a nw locty fld F(H, whr th functon F s a functon of H only. Wth ths constructon t can b shown that.(f 0 snc ( (.( F F. + F(. 0 ( x x n + ( x x n ( (.. (8.. (0 Fgur 3(b. Vlocty fld gnratd by a solnodal typ potntal. Fg. 3(a. For a symmtrc ortx fld, th locty fld tslf must b shapd to form ƒ. A sutabl functon for shapng a symmtrc ortx fld s found to b ( î ~ î ( x ~ x + ( x ~ x f, î m +.. (7 whr agan. 0 by dfnton. Thn, usng Equatons (6 and (8 t can b sn that F. df H dh H df H + H dh H 0.. ( H A sutabl functon for shapng th suprquadratc solnodal fld s found to b F m.. ( + H n L whr L s th lngth scal of th suprquadratc, whch s chosn to nclos th obstacl. Agan, f m >> thn F swtchs n a rapd, but contnuous mannr from to 0 as th dg of th suprquadratc sd s crossd, as shown n Fg. 3(c.

6 3.0 VECTOR POTENTALS Th ctor potntal s usd xtnsly n lctromagntc thory to rprsnt currnts whch gnrat solnodal magntc flds (4. As such t also offrs a systmatc and rgorous mans of gnratng solnodal flds for path-plannng problms. For a ctor potntal A th rsultng locty fld wll b dfnd as A whr th ctor potntal s obtand by ntgraton from th Bot- Saart law (4. t can b sn from Equaton (3 that th ctor potntal s not unquly dfnd. Any quantty wth zro ctor curl can b addd to th potntal wthout affctng th rsultng locty fld. To llustrat th mthod, th symmtrc shapd ortx functon dscussd n Scton wll now b consdrd wth a ctor potntal dfnd by A ö, 3 3 whr, usng th Bot-Saart law (4, t s found that φ ( î f î dξ + K, î ( x ~ x + ( x ~ x whr K s an arbtrary constant of ntgraton. Th rsultng locty fld s thn obtand from Equaton (3 as ( x ~ x ö ( x ~ x ö A x ~ x + x ~ x x ~ x + x ~.. (6 x ( ( whr φ f(ξ/ξ so that th shapd ortx locty fld s rcord as xpctd such that A f From Equaton (5 t s clar that only th shapng functon f s rqurd to gnrat th ctor potntal and so th rsultng solnodal fld. Thrfor, for complx obstacls a solnodal fld may b gnratd by usng a mor complx scalar functon, such as a suprquadratc, to rprsnt th shap of th obstacl, as dscussd n Scton. Th ctor potntal thn gs a rgorous and systmatc mans of gnratng a solnodal locty fld from a scalar functon. n addton, th ctor potntal may now b usd wth Hlmholtz s thorm (a dtald proof of whch s gn lswhr (5 to gnrat global locty flds. Thorm: A ctor fld wth both sourc and crculaton dnsts anshng at nfnty may b wrttn as th sum of two parts, on of whch s rrotatonal th othr solnodal (5. Hlmholtz s thorm thrfor allows a compost locty fld wth an attract goal potntal cntrd on G and solnodal ctor potntals to rprsnt th st of obstacls C to b wrttn as V + A Th proprts of th ctor fld dfnd by Equaton (8 can b nstgatd by calculatng th drgnc of th fld as ( A. V +. Clarly th frst trm of Equaton (9 wll ansh snc th potntal V satsfs th Laplac quaton. Howr, t s a ctor proprty that th drgnc of any solnodal fld anshs so that.( A 0 Thrfor, t has bn shown that. 0 so that th global fld has null drgnc. f w agan postulat that th fld can, n prncpl, b ( (.. (3.. (4.. (5.. (7.. (8.. (9.. (0 drd as th gradnt of a scalar potntal, thn th potntal wll satsfy th Laplac quaton and so wll b fr of local mnma. Agan, snc only th gradnt s bng usd, th form of th actual potntal can rman unknown. Th gradnt fld wll thrfor ha a unqu goal pont G, whch wll b rachd by R from any start pont S W wthout trappng or collson wth th obstacl st C. W now ha a rgorous and systmatc mans of gnratng locty flds whch can b usd to prod path-plannng for ngotatng complx obstacls. Lastly, t s clar from Fgs 3(b and 3(c that for a rapdly truncatng functon f th rsultng bhaour of R as t manours from S to G s smlar to dg followng. Edg followng s a smpl and ffct mans of snsory basd nagaton usd by mobl robots for ngotatng obstacls n a closd workspac. For a st of obstacls n a workspac, topology nsurs that dg followng guarants ngotaton of th obstacls. Ths can b sn from th strong analogy wth scap from a maz wth on hand n contact wth a wall of th maz, th maz topology guarants that an xt wll always b rachd. Smlarly, th solnodal flds gnratd about obstacls prod a consstnt drcton for ngotaton along th dgs of thos obstacls wth rsultng motons about C and towards G from any pont S. Thrfor, th ablty of dg followng to prod an ffct mans of obstacl ngotaton wthout trappng can b sn as qualnt to th proprty of th solnodal ctor fld and th Laplac quaton that no local mnma ar gnratd. 4.0 VELOCTY FELD TRACKNG Th locty fld to b usd to command th UAV wll b gnratd by normalsng to prod a unt ctor fld. Ths unt ctor fld prods a unqu hadng command, wth th UAV spd bng controlld ndpndntly, as wll b dscussd blow. Th dsrd UAV locty ctor u (u, u wll now b dfnd as ê ( x x u, whr th functon κ s a scalar functon usd to control th UAV spd. n partcular, κ can dfnd to b nrsly proportonal to th curatur of th path to allow trackng of th dsrd locty fld wthout actuator saturaton through turnng rat lmts. n ordr to gnrat commands for th UAV, th rqurd locty u wll b rsold nto a scalar spd command υ c and hadng command ψ c as ( õ ê x, x.. (a c Tanø u c u.. (b Snc th dtald rsponss of th UAV wll not b consdrd hr, ths commands ar assumd to b ffctd through a smpl frst ordr control wth saturaton, such that õ & ( õ õ f õ& õ& max ë õ c ( ø ø ø& ë ø c f ø& ø& max whr th constants λ υ and λ ψ ar th nrs tm constants of th UAV rspons to commandd changs n spd and hadng. Th moton of th UAV wll thn b propagatd through th locty fld usng Equatons ( and (3 and by ntgratng th knmatc rlatons x& õcosø x& õsnø.. (.. (3a.. (3b.. (4a.. (4b n ordr to prod a ralstc rspons, th maxmum UAV turn rat ψ. max s dfnd as 0 s and th maxmum acclraton υ. max dfnd as 0 g. Th tm constants ar dfnd as λ ψ s for th hadng controllr and λ υ 0s for th spd controllr.

7 Fgur 4(a. Sngl UAV manourng from a start pont S to a goal pont G wth a st C of thr obstacls (o - 0 s tm stps. Fgur 5(a. R-formaton of a group of thr UAVs (o 00s tm stps. rprsntd through a typ locty fld G ( x x G ( x x + ( x x G G ( x xg ( x x + ( x x t can b sn that th UAV succssfully ngotats th obstacls and rachs th goal G wthout collson. Snc a lag s ntroducd n hadng changs, th UAV dos not pass xactly through G. At th nd pont of th path th UAV passs th goal and xcuts a lft turn to pass th goal agan. Subsqunt passs ar mad as th locty fld forcs th AUV to lotr n th cnty of th goal. n ths xampl th UAV spd has bn fxd at 0ms so that th only act control s th UAV hadng, as shown n Fg. 4(b. t can b sn that a srs of turns ar commandd by th algorthm to aod th st of obstacls, wth a fnal lft turn towards th goal. G G.. (5 Fgur 4(b. UAV hadng tm hstory. 5.0 MPLEMENTATON n ordr to llustrat th us of th solnodal ctor flds gnratd n Sctons and 3, a sngl UAV wll b consdrd manourng from S to G wthout crossng th boundary C of a st of statc obstacls C. Th UAV uss th frst ordr controllr, tm constants and saturaton lmts dfnd n Scton 4. A st of thr obstacls wll b dfnd usng th typ locty fld, wth two symmtrc ortcs and a suprquadratc gnratng functon usd to nsur that th boundars of th obstacls ar not crossd, as shown n Fg. 4(a. Hr, th sz of th obstacls ha bn xtndd by th turnng radus of th UAV (υ. max/ψ. max to nsur collson aodanc n th prsnc of controllr saturaton. n addton, th goal G s 6.0 EXTENSON TO UAV FORMATONS Th algorthm wll now b xtndd to a group of N UAVs flyng n formaton. Th UAV formaton-flyng problm has bn nstgatd prously, although mphass has bn on mantanng formatons (6,7 rathr than r-confguraton wth collson aodanc btwn mmbrs of th formaton. Thos mthods whch do consdr r-confguraton and rly on cntralsd plannrs ar found to b computatonally ntns (8. Th xtnson of th locty fld algorthm to UAV formatons can b achd usng a rang of schms. Howr, th smplst schm to mplmnt s for ach UAV to trat th rmanng N UAVs as a st of mobl obstacls C and to rfrnc th goal G for ach UAV to a dsgnatd lad UAV. f th lad UAV has som nstantanous poston x L (x L, x L and th formaton s dfnd such that th th UAV has a poston ( (, ( (N rlat to th lad UAV, th goal locty fld for th th UAV s dfnd by a typ locty fld of th form ( x ( xl + G ( x ( xl + + ( x ( xl +.. (6 ( x ( xl + x x + + x x + ( ( ( ( L L

8 Fgur 5(b. UAV formaton hadng tm hstory. Fgur 5(d. UAV formaton spd tm hstory. Fgur 5(c. UAV formaton sparaton tm hstory. Fgur 6. UAV formaton manour at a sngl obstacl (o 0s tm stps. wth th rmanng UAVs rprsntd as mobl typ locty flds. f rqurd, th st of ctors (, can b rotatd by th hadng angl of th lad UAV to nsur that th formaton rotats as a rgd body whn th lad UAV turns. Th spd of ach mmbr of th UAV formaton can also b rfrncd to th lad UAV to nsur that th rlat spd btwn th lad UAV and othr UAVs n th formaton wll conrg. For xampl, th followng functon for th spd of th th ( (N UAV allows th UAVs to conrg on th lad UAV whl nforcng spd constrants L ( + ñ( xp( r r õ õ ç L.. (7a õ max ñ.. (7b õl whr η s a paramtr whch shaps th rat of conrgnc of th UAV spd to th lad UAV and υ max s th maxmum nforcd UAV spd. Wth ths dfnton υ > υ L ( (N so that a mnmum spd s also nforcd. Snc t s assumd that th spd of th lad UAV wll nr fall blow stall spd, th stall spd constrant s thn nforcd through th ntr formaton. An xampl of a r-formaton manour wth thr UAVs s shown n Fg. 5(a. Hr th lad UAV (A mantans a constant hadng and spd of 0ms whl B and C swap placs n th formaton. Du to th symmtry of ths manour a collson btwn B and C would normally rsult. Howr, th UAVs turn to aod ach othr, as shown n Fg. 5(b. Th typ locty flds assocatd wth ach UAV ha bn shapd usng Equaton (7 to nforc a mnmum sparaton of 500m, as shown n Fg. 5(c. Th spd of ach UAV s also scald usng Equaton (7 rlat to th lad UAV spd of 0ms and a maxmum spd whch has bn dfnd as 30ms, as shown n Fg. 5(d. A fnal xampl s shown n Fg. 6 whr a trangular UAV formaton ngotats an obstacl rprsntd usng a suprquadratc gnratng functon. Hr t can b sn that as th lad UAV (A manours to aod th obstacl, B s dsplacd snc t s rfrncd to A and so dos not ncountr

9 th obstacl. Howr, C dos ncountr th obstacl and falls n bhnd th lad UAV untl th obstacl s succssfully ngotatd and th trangular formaton can r-form. 8. RCHARDS, A., BELLNGHAM, J., TLLERSON, M. and HOW, J. Coordnaton of multpl UAVs, 00, AAA , AAA gudanc, nagaton and control confrnc, August 00, Montry. 7.0 CONCLUSONS A locty fld approach to UAV path-plannng has bn prsntd whch allows sngl or multpl UAVs to manour btwn a start pont S and goal pont G wthout collson wth statc obstacls or othr UAVs n a formaton. Unlk hurstc path-plannng algorthms, th locty fld algorthm prsntd hr s basd on mathmatcal rgour n that th proprts of th Laplac quaton can b nokd to dmonstrat that th fld wll gnrat a unqu path whch s guarantd to rach th goal G. n addton, snc th locty fld can b gnratd analytcally, th computatonal orhad to mplmnt th algorthm s mnmal, allowng autonomous opratons wth on-board path-plannng. ACKNOWLEDGMENTS Th work prsntd n ths papr was undrtakn wth support from th Lrhulm Trust, to whom th author xprsss hs thanks. REFERENCES. KHATB, O. Ral-tm obstacl aodanc for manpulators and mobl robots, nt J of Robotcs Rsarch, 5, (, 986, pp RMON, E. and KODTSCHEK, D. Exact robot nagaton usng artfcal potntal functons, 99, EEE transactons on robotcs and automaton, 8, (5, pp SATO, K. Dad-lock moton path plannng usng th Laplac potntal fld, Adancs n Robotcs, 993, 7, (5, pp GULDNER, J. and UTKN, V. Sldng mod control for gradnt trackng and robot nagaton usng artfcal potntal flds, 995, EEE Transactons on Robotcs and Automaton,, (, pp DE MEDO, C., NCOLO, F. and G. OROLO. Robot moton plannng usng ortx flds, Progrss n Systms and Control Thory, 99, 7, pp MASOUD, A. and BAYOUM, M. Robot nagaton usng th ctor potntal approach, /93, 993, Procdngs of th EE ntrnatonal confrnc on robotcs and automaton, Aprl 993, Mnnapols. 7. SNGH, L., STEPHANOU, H. and WEN, J. Ral-tm robot moton control wth crculatory flds, 996, Procdngs of th EE ntrnatonal confrnc on robotcs and automaton, Aprl 996, Mnnapols. 8. MASOUD, A. Usng hybrd ctor-harmonc potntal flds for mult-robot, mult-targt nagaton n a statonary fld, 996, Procdngs of th EE ntrnatonal confrnc on robotcs and automaton, Aprl 996, Mnnapols. 9. ZEGHAL, K. A Rw of dffrnt approachs basd on forc flds for arborn conflct rsoluton, 998, AAA Gudanc, nagaton and control confrnc, August 998, Boston. 0. SHAPRA,. and BEN-ASHER, J. Nar-optmal horzontal trajctors for autonomous ar hcls, J Gudanc, Control and Dynamcs, 997, 0, (4, pp MCNNES, C. Potntal functon mthods for autonomous spaccraft gudanc and control, 995, AAS , AAS/AAA astrodynamcs spcalst confrnc, August 995, Halfax, Noa Scota.. GHOSH, R. and TOMLN, C. Manur dsgn for multpl arcraft conflct rsoluton, 000, Procdngs of th Amrcan control confrnc, Jun 000, Chcago. 3. VOLPE, R. and KHOSLA, P. Manpulator control wth suprquadratc artfcal potntal functons: thory and xprmnts, 990, EEE transactons on systms, man and cybrntcs, 0, (6, pp LORRAN, P. and CARSON, D. Elctromagntc Flds and Was, 970, pp , Frman, Nw York. 5. ARFKEN, G. Mathmatcal Mthods for Physcsts, 985, pp 78-83, Acadmc Prss, San Dgo. 6. PACHTER, M., D AZZO, J. and DARGAN, J. Automatc formaton flght control, J Gudanc, Control and Dynamcs, 994, 7, (6, pp WOLFE, J., CHCHKA, D. and SPEYER, J. Dcntralzd controllrs for unmannd aral hcl formaton flght, 996, Procdngs of th AAA gudanc, nagaton and control confrnc, August 996, San Dgo.