Controlling Swarms of Bandit Inspector Spacecraft

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1 SSC06-V-6 Contollng Swams of Bant Inspcto Spaccaft Jmy S. Nubau Gauat Stunt, Washngton Unsty n St. Lous Mchal A. Swatwout Aso, Washngton Unsty n St. Lous h ast fosabl bnfts of on-obt scng to cunt an futu spac systms ha spa th lopmnt of both lag-scal scng spaccaft an small-scal tchnology monstaton platfoms. h latt po th ablty to po ctan cucal tchnologs mo ffcntly than th lag countpats, u to ncas sponsnss an uc cost. Washngton Unsty s Bant s on such hcl cuntly n pogss, sgn to sach th snsoy, autonomy, an contol poblms of th multhcl clos-poxmty flght ncssay to naly all of th on-obt scng nusty s ambtons. hs stuy asssss th ablty of bhao-bas mthos to sol th mult-bant contol poblm, complcat by th hcl s hghly constan actuaton, computaton, an stat obsaton capablts. Hn, a potntal functon contol systm s talo spcfcally fo hcls opatng un such constants. A stablty analyss s that pos ths contoll wll la ach nspcto to ts fnal s qulbum stat wthn a calculabl o boun, whl multpl smulatons of th systm a us to alat ths analyss an nstgat ts ynamc chaactstcs. Rsults ncat that th sgn contoll pos sabl pfomanc n ploymnt, nzous, an staton-png scnaos, an also shows poms n aaptng to oth scng tass, such as autonomous ocng. Intoucton Extnng th opatonal lftm an lablty of spac systms qus many nw on-obt capablts, nclung th ablty to ful, upga hawa an tct an pa poblms. Sal sach pogams a aggssly pusung ths tchnologs, nclung NASA s DAR, AFRL s XSS-10 an -11, NRL s SUMO, an DARPA s Obtal Expss, to nam a fw. hs hcls, bachng 700 g n sz an $100 mllon n pc plan to monstat all of ths J.S. Nubau 1 ablts an mo an ha alay ach som succss 1,, 3, 4, 5. h hgh cost an complxty of ach of ths mssons s ctanly ncssay to accomplsh such fa achng goals n a sngl flght. How, small scal pocts can mo sponsly an ffcntly tst an po substs of ths tchnologs. Fo nstanc n Jun of 000, Suy Satllt chnology s 6.5 g SNAP-1 monstat populson, 3-axs atttu stablzaton, an tagt magng capablts wth a latly small nstmnt of soucs. Aoun th sam tm both th Spac Systms Laboatoy at MI an th Aospac Systms Laboatoy at Washngton Unsty n St. Lous bgan lopmnt of small satllt tst bs fo futu on-obt scng tchnology monstatons 6, 7. Washngton Unsty s Bant s a f-flyng, cama-cayng, nano-scal nspcto spaccaft. Onboa hawa s pt to a mnmum by lgatng long-po functons (pow gnaton an long ang communcatons) to a host hcl mang th -ocabl on small, agl an nxpns. Its mnut statu maxmzs compatblty wth both host an launch hcls, whl mnmzng assmbly an ntgaton tms. hus, low cost Bant mssons can b ntat qucly wth naly any host an launch hcl mang th platfom an xtmly spons tst b fo lopng on-obt scng tchnologs. Clos poxmty flght of on o mo sc hcls s phaps th most cucal tchnology to th succss of th nusty, as t s a ncssay stppng ston to all nson scng opatons. Accongly, ths stuy asss th contol poblm of manung multpl Bant hcls about a host. Byon th unsal guanc an collson aoanc ssus, ths matt s futh complcat by Bant s lmt obsatonal, computatonal, an actuatonal ablts. Du to th sultant n fo smplcty an nsnstty to stubancs an ynamc

2 nonmnts, t s hypothsz that bhao-bas mthos may b wll sut to th tas. Aft ntoucng th Bant platfom, ths pap wll poc to w basc bhao-bas contol thoy an ts past applcatons to satllt contol. Nxt, a contoll sgn spcfcally fo Bant swams s popos an assss analytcally. Fnally, multpl smulatons of on-obt opatons a conuct an xamn. ay th total numb of ons n th systm. MEPSI bas launch contans a nclu to physcally accommoat th ons (Fg. 1), mof to allow fo -ocng an chagng n flght. h c nclus a motoz bay oo to compltly nclos th nspctos ung launch, an an lato platfom to bng ach on to th sufac fo ploymnt. h Bant Platfom Bant s a popos sach platfom fo on-obt scng tchnologs, spcfcally assng th snsoy, autonomy, an contol poblms of multhcl clos-poxmty opatons. h fst flght monstaton, cuntly un lopmnt a th Unsty Nanosat-4 sgn comptton, ncopoats a 5g Aoya host hcl an two sub 5-g ployabl an -ocabl Bant f-flys. h msson ss to monstat basc nagaton an contol ablts, pocng fom human plot opatons to lmt autonomous manus. Each Bant on cas an ght thust col-gas populson systm, wth an ntgat popllant tan capabl of pong at last 1 m/s total V. h thust confguaton was sgn to po coupl tanslaton an otaton about any sngl axs fom a mnmal numb of ts, motat by th sz an cost of th qust als. Nagaton s hanl by a combnaton of th othogonal MEMS at gyos an a CMOS cama. Instumntng th xto of th host an ach nspcto wth colo-co LEDs allows a Xlnx-bas FPGA to cont mags po by th cama nto lat poston an atttu ata at sps up to 30 fams p scon. An Atml-bas mcocontoll manags comman an ata hanlng, also pong a 418 MHz, 4800 bau wlss shot ang ( m) ata ln wth Aoya. NCa batts supply th ons wth up to 30 mnuts of lctcal pow, nclung act hatng fo th populson an magng systms. h 45 cm tall, hxagonal Aoya host contans a smla son systm an batty pac, n aton to a 5 W sola cll pow gnaton systm, magntcally stablz atttu contol systm, 5 W, 9600 bau tlmty ownln, an an aay of Atml-bas mcocontolls. h stbut natu of th comman & ata hanlng systm allows fo th samlss aton o moal of nt subsystms nclung Bant ons. hs fatu pos a smpl mans of passng ata an commans btwn th host an ons, as wll as ncas flxblty to Fg. 1 Launch Contant an On-Obt Soft Doc A mtallc hoo-an-loop systm ss as th on-obt soft oc (Fg. ), whl also pong a conuct path fo chagng. h cm hmsphcal, hoo co pusho tp on th lato platfom an loop co Bant xto gs a lag magn n ocng angl an offst up to 75 an 3 cm. Fg. Dmonstaton Soft Doc o tst th pfomanc of th abo hawa po to flght, a th g of fom a bang tst b has bn lop (Fg. 3). o at ths systm has bn mploy to show succssful motly-plot poxmty manung (wthn 3 m) an soft ocng, wth lat closng locts angng fom 0.5 to 100 cm/s. In aton, a Jaa-bas ntgat opato wostaton s un lopmnt to sgn an tst aous Bant contol algothms fom human plot sons to compltly autonomous ons. Bult upon a sx g of fom smulato nclung th J.S. Nubau

3 ffcts of obtal ynamcs an local stubancs, th motly accssbl pogam accuatly poucs th aalabl tlmty ata an constucts tual mags of th obtal scn ( Fg. 4). otaton at an lat poston, locty, an ontaton, whl th mag pocssng ncssay to attan ths ata consums th maoty of th aalabl computatonal soucs, lang lttl pocssng pow to ffcntly manag th ght thusts that tghtly stct th aalabl actuaton spac an coupl tanslaton to otaton. hfo a smpl yt obust contol algothm s n whch may b foun wthn th thoy of bhao-bas mthos. Bacgoun Fg. 3 Bant an A Bang stb Bhao-bas contol mthos wo on th pncpl that complx global tam bhaos can as fom smpl nual hcl bhaos, as sn n many bologcal systms. In fact, a pomnnt bhaobas contol statgy n th ltatu follows th fom of foagng tactcs of th E. col bacta. It has bn popos that ths bacta, sng to maxmz th ngy nta o tm ffctly calculat a potntal functon bas on th local stbuton of foo, toxns, an nghbong bacta, thn scn ths potntal to ach th most hosptabl nonmnt 8, 9. h applcaton of such potntal functons (also call gant functons an atfcal o stuctual potntals) to th contol of autonomous hcl tams has foun much succss. Sal sachs ha us th mtho to ctly contol hcula locty, whl oths ctly contol acclaton. In th cas th contol acton s calculat to maxmz th at of scnt of th potntal functon. h sultant tactos s th potntal functon s mnmum alu whch colats to th pf qulbum stat of th systm 10, 11, 1, 13. Fg. 4 GUI fo Intgat Opato Wostaton Bhao-Bas Contol Futu Bant mssons wll qu th autonomous opaton of multpl ons about a pass host fo xampl th nspcton of a fogn hcl o obct. h absnc of an act supsoy host an th n fo obustnss to nual systm falus ctats that ach on must opat n a compltly cntalz fashon. hs obcts, coupl wth Bant s hawa constants, ma fo a challngng contol poblm. Dcntalzaton an mnmal hawa lmts th aalabl stat nfomaton to absolut J.S. Nubau 3 h potntals a most oftn fn by nt-hcl stancs an a homognous acoss all hcls, lang to unfomly tangulat gomtc fomatons. In th gn of atfcal physcs, potntal functons a bas on th lat poston an spn of th nghbong hcls, thus catng mo complx mgnt spatal stuctus. h mtho s smlaty to th mchancs of atoms has sult n bhaos an to thos sn n cystalln stuctus, nclung slocatons an phas-tanstons 14, 15, 16. A fw sachs ha appl potntal-bas mthos to th contol of satllts. McInns has xamn th matt on sal occasons, stuyng mpuls nzous n th psnc of obstacls, as wll as lag angl slw manus, tmnal scnt guanc an fomaton flght wth contnuous actuatos 17, 18. Rn an Ba ha also nstgat a

4 smla tual stuctu tchnqu appl to th fomaton flyng poblm, lopng a cntalz contoll that mantans accuat lat postonng of spaccaft usng low-banwth communcatons 19. All of ths bhao-bas appoachs ha yl smla sults n gant scnt an fomaton ynamcs, showng that complx systm bhaos can n fact ol fom smpl nual contol laws. Yt t s mpotant to not that tmntal goup bhaos can b ust as lly to occu as bnfcal ons, hghlghtng th lagst shotcomng of bhaoal bas contol mthos fw tools xst to analytcally assss th stablty an tansnt pfomanc of th oall systm. As a sult, tunng paamts of complx potntal functon contolls typcally manats a tal an o appoach, an can b qut tm consumng. But thy can off sal aantags n obustnss to unctan nonmnts an hcl falus (thy a not mol-bas), tolanc of banwth lmtatons (thy can pfom wthout nfomaton of th total systm stat), an smplcty of mplmntaton (thy ntnscally ha smpl contol laws at th nual ll, qung lttl pocsso pow). hus, thy appa wll sut to Bant hcl systms. Bant Applcatons h smplcty an obustnss of potntal functon mthos ma thm goo canats fo a Bant contol systm. Yt th applcaton of cunt potntal functon appoachs to Bant psnts sal poblms, pmaly u to Bant s hghly constan actuato. Publsh mthos fo calculatng a contol an pong stablty consstntly assum a contnuous tm actuato, unconstan n both cton an magntu whas Bant s thust systm s bst cons sct n tm an s hghly lmt n both th cton an magntu of ts contol actons. hfo a spcalz contoll must b sgn accountng fo th spcfc ns an constants of th Bant spaccaft. h contoll lop n ths stuy shas two y ponts wth th publsh potntal functon mthos of bhao-bas contol thoy: (1) ach hcl computs potntal functons bas on th cunt stat of th systm, an () ach hcl ss th mnmum alu of ths potntal functons. h tals of th contol logc thn ff substantally. Most notably, potntal functons a fn by a locty o ath than a poston o, an th t slcton pocss s ntgat wth th guanc algothm. Contol Logc Much l th mthos n th ltatu ach hcl calculats ts own Lyapuno functon, fn smply as th sum of all ptnnt potntal functons. Fo ths stuy, potntals fo atttu ( P ), poston lat to th host ( P ), an poston lat to oth nspctos H P ) a nclu as n Eq. ( 1 ). ( I Φ = P + P H + P I ( 1 ) Each of ths potntals s n by a locty o. A typcal fomulaton s that of Eq. ( ), wh s th cunt hcula locty, an s a s locty cto bas on th stat of th systm, sgn to la th hcl to ts pf qulbum stat. ( ) ( ) P = ( ) h pncpl obct of th contoll s to uc Φ, thby ucng locty o. hs fomat s chosn to complmnt th mpuls natu of Bant s actuatos, wh contol actons a fn by th nstantanous changs n tanslatonal an otatonal locty ( an ) fom th fng of a sngl t fo ts mnmum puls wth. hus, contol csons can b ma upon th nstantanous chang of th Lyapuno functon u to an mpuls (Eq. ( 3 )) an th cost of that mpuls (σ ) at ach tm stp. h t to f s fn as that whch satsfs an mnmzs th tggng functon of Eq. ( 4 ). If no thust can satsfy ths laton, no contol acton s tan. ( + + ) Φ( ) Φ = Φ ( 3 ) Φ 0 ( 4 ) In th cas of an unconstan actuato, calculatng a contol as abo s no asy tas gn multpl nonlna potntals, fnng th optmal contol mpuls qus an tat soluton. Fotunatly, th Bant hcl s constan to as fw as ght mpuls optons, ntalng that as fw as ght aluatons of th tggng functon a ncssay to slct th bst aalabl t. J.S. Nubau 4

5 Stablty h stablty poof fo ths contoll has two man pats. Fst, congnc to wthn a fn o boun s shown. hn, congnc to th qulbum poston s shown, tang nto account sa o boun on locty. Fo ths poof, a gnal Lyapuno functon of th fom of Eq. ( 5 ) s cons wh tanslatonal locty os a ncat by, otatonal locty os a ncat by, an, an, a all stctly post. h chang n Φ fom an mpuls s thn gn by Eq. ( 6 ). = + Φ ( 5 ) + Φ =,, ( + ) ( + ) ( 6 ) Combnng Eqs. ( 4 ) an ( 6 ) pos a tal loo at th ncssay cta fo fng a thust (Eq. ( 7 )). + +,, 0 J.S. Nubau 5 ( 7 ) Whn ths statmnt s tu a thust s f that cass Φ, ntalng th cas of, an congnc to th s locts. Rcognzng that th σ,, an tms a ach gat than zo fo all systm stats, only two cass xst fo whch a thust wll not b f. Fst, t coul b that a contol acton s not aalabl to ma < 0,,, mplyng that, som combnatons of locty os (no matt th magntu) a not coctabl. hs scnao must b ao by th pop sgn of th thust confguaton, a conton scb mathmatcally n Eq. ( 8 ), wh s th ffct of fng thust. x : < x x ( 8 ) Mtng ths qumnt ntals that th unon of opn half spacs fn by th ctos occups th nt sx mnsonal spac (xclung th ogn). hs popty s hn not as complt contol authoty. Gn a hcl wth ths popty, th only manng cas fo whch a thust may not b f s whn all locty os a small, an, although th fst tm of Eq. ( 7 ) can b ma ngat t can not compltly countact th lat post tms. hs mpls that th xsts a st of small magntu locty os that cannot b coct. hs spac n by σ an th sz of th mnmum thust bt, can b ntf by fst calculatng an upp boun to th fst tm of Eq. ( 7 ) as stat n Eq. ( 9 ). α s a post constant tmn by th spcfc confguaton of th hcl s thusts. α,, ( 9 ) Combnng ths upp boun wth th tggng functon yls a s cta fo thust actaton bas on th stat n Eq. ( 10 ). + α,, ( 10 ) h maxmum compost o boun s fn by th lagst o that can xst wthout tggng a coct thust to f an s thus gn by th qualty of Eq. ( 10 ). Inual o bouns can b calculat by consng ach potntal npnntly, sttng all oth potntals to zo, sultng n th allowabl o Eqs. ( 11 ) an ( 1 ). Byon ths o bouns, a thust that ucs total locty o

6 wll b foun an f; thfo th hcl must cong to a locty na th s locty wthn th bouns of Eqs. ( 11 ) an ( 1 ). < + α, ( 11 ) +, < ( 1 ) α, o nsu locty mans wthn ths bouns onc thy ha bn ach th tm at of chang of an must b lss than th acclaton ablts of th caft, as stat n Eqs. ( 13 ) an ( 14 ). ( ) < s ( 13 ) t ( ) < s ( 14 ) t h nstgaton of poston congnc bgns un th assumpton that th hcl has attan a locty na that s as scb abo. Fo th hcl to b hang towas th s poston, ts nstantanous locty cto must b ont such that poston o s ucng. Fo th tanslatonal cas wh only on potntal commans congnc to, ths cta s fn by Eq. ( 15 ). ( + ) ( ) 0, 1 1 > ( 15 ) Claly, th wost cas fo poston congnc s whn th locty o s ct oppost th poston o, =, an of th maxmum allowabl magntu fn by Eq. ( 11 ). Applyng ths cas to th abo quaton yls Eq. ( 16 ),1 > + α 1, ( 16 ) fl of Eq. ( 17 ), wh f ( ) 0 fn by Eq. ( 18 ). f ( ) ( ), that nlop s = f ( 17 ) +, > ( 18 ) α As t s xpct that ( ) 1 f wll ncas wth, th congnc nlop wll ncompass th nt stat spac xclung a small aa na. How, t s possbl to cat an mpty congnc spac a poo sgn of th thust confguaton an contol paamts. h otatonal poston congnc agumnt follows smla logc. Fst, t s assum that only on potntal cts congnc to th s ontaton, catng th congnc conton fn n Eq. ( 19 ). H th s ontaton s fn by th algnmnt of th nspcto s cama loo cto, nˆ, an th nomalz lat poston cto of ts tagt, ˆ. nˆ ˆ > ( 19 ), 1 1 ˆ ˆ n ( + ) 0 Applyng th sam wost cas conton as bfo las to th nw laton of Eq. ( 0 ):,1 +, ˆ ˆ n > ˆ ˆ n α, ( 0 ) h congnc nlop can now b calculat wth nowlg of. Fo as fn n Eq. ( 1 ), wh g ( nˆ, ˆ ) s stctly post Eq. ( ) fns that nlop. Agan assumng a g ( nˆ, ˆ ) that ncass wth ncasng ontaton o, th congnc spac only xclus a small gon aoun th s ontaton. Wth nowlg of, 1, Eq. ( 16 ) can b mploy to calculat a congnc nlop. Fo th locty = g nˆ ˆ ( nˆ, ˆ ) nˆ ˆ ( 1 ) J.S. Nubau 6

7 g ( n ˆ, ˆ ) > +, α, ( ) At ths pont, congnc to a boun aa aoun th s qulbum stat has bn shown fo a sngl hcl. Fo th cas of a sngl hcl wth only on otatonal an on tanslatonal potntal, som quantfcaton of th tansnt bhao of th systm can b ma though nowlg of th s locts, maxmum locty o bouns (Eqs. ( 11 ) an ( 1 )), an ntal contons. Fo th cas of multpl hcls, potntals, an stubancs, how, lttl can b sa of th path tan to qulbum. Unfotunatly, ths cas s of patcula ntst, as th possblty of collson xsts. In ths stuy, collson aoanc wll b ass solly though smulaton. Rotatonal Potntal h s otatonal spons fo th Bant spaccaft s to contnually pont ts cama at a pfn tagt. Attmptng to mt ths goal a a quatnon o cto o smla appoach o constans th poblm by nfocng a patcula atttu about th cama s loo cto. Accongly, a smplf appoach s tan to algn th cama loo cto wth th host s nomalz lat poston cto. Fo th stana fom potntal n Eq. ( 3 ), th s angula locty of Eq. ( 4 ) wll yl ths s spons. ( ) ( ) P = ( 3 ) ( µ 1 nˆ ˆ ) nˆ ˆ = ( 4 ) nˆ ˆ h coss pouct tms n whl th manng tm, ( 1 nˆ ˆ ) magntu. A plot of po ts cton, µ, pos ts fo a sngl otatonal g of fom cas s llustat blow n Fg. 5, wh θ = cos 1 nˆ ˆ. Not that th s locty s always towas th algnmnt of nˆ an ˆ, ncasng n magntu wth ncasng stanc fom qulbum. Host Potntal Fg. 5 Ds Angula Vlocty o ach an mantan th s magng stanc a host potntal s mploy bas on th stana fom potntal of Eq. ( 5 ). h nclu locty o, Eq. ( 6 ), s fn as a functon of a poston o cto, Eq. ( 7 ), wh s th s magng stanc btwn th tagt an nspcto. P H H ( ) ( ) = ( 5 ) µ H = 1 + ( 6 ) 1 ( 7 ) = h fom of Eqs. ( 6 ) an ( 7 ) w slct to yl a lnaly ncasng aoun = 0 wth a lmt maxmum alu of µ as H. A plot of sus n Fg. 6 llustats ths tats blow. J.S. Nubau 7

8 Paamt Slcton Gn a hcl wth a ptmn thust systm (fnng α an ach ), ght paamts a aalabl to tun systm spons:, H,, I, H,, I, an σ. h fst fou can b st ctly fom msson qumnts s tmn by th onboa cama an ncssay magng qualty, whl H,, an I, a tmn fom th maxmum s hcl locts. Maxmum bouns on H an a po by th n to mantan locty congnc as not n Eqs. ( 13 ) an ( 14 ). Combnng ths latons wth an as scb al pos Eqs. ( 30 ) an ( 31 ): Fg. 6 Ds anslatonal Vlocty µ H < ( 30 ) s Int-Vhcl Potntal Applyng ths mtho to contol nt-hcl spacng s not as staghtfowa as th two pous cass. At clos stancs, wh th chanc of collson s hgh, t s s that th hcls gnat locts away fom on anoth; but at lag stancs, no spcfc laton btwn hcl locts s ncssay. o captu ths ns, a potntal of th fom of Eq. ( 8 ) s mploy. H an a th poston lm lm an locty ctos of hcl l lat to hcl m. I, lm P = I, l lm, lm lm, m lm ( ) ( ) lm ( 8 ), lm h ncluson of th nt-hcl spacng n th potntal fnton tlls th contoll whn matchng hcl locty to s bnfcal. Whn nthcl stancs a lag th ffct of a chang n nt-hcl locts on th total Lyapuno functon s null. How, whn nt-hcl stancs a small, th ffct of such changs on Φ a amplf. h s nt-hcl locty us n ths stuy s smply a constant magntu locty ct away fom th nghbong hcl as fn n Eq. ( 9 )., lm µ = I, l lm lm ( 9 ) µ < ( 31 ) s hs las th th potntal gans an th cost functon to b slct by th contol systm sgn. Notng that th absolut alus of ths fou paamts s nconsquntal, on gan ( H ) can b st to unty. Nxt, th s stay stat spons an congnc nlop quatons can b mploy to calculat an σ. Usng th spcfc potntals scb by Eqs. ( 3 ) to ( 9 ), an assumng suffcnt nt-hcl spacng an locty to gno P I, Eqs. ( 3 ) an ( 33 ) tmn th maxmum stay stat os θ ss an ss. hn, gn s θ ss an ss alus, ths quatons can b sol smultanously to fn an σ. + 1 θ = cos 1 ss α µ ( 3 ) + =, ( 33 ) ss αµ σ H h sol manng paamt, I, mans to b tmn by tat smulaton. J.S. Nubau 8

9 Smulaton Rsults A sx g of fom smulaton ncopoatng lat Clohssy-Wltsh obt ynamcs was constuct n Matlab to assss th xpct pfomanc of th abo contoll. h co was sgn to allow fo us fnabl thust confguatons, potntal functons, numb of hcls, an obt nonmnts. Fo th followng sults, a tam of sx hypothtcal Bant hcls wth th popts of abl 1 an Fg. 7 was xamn n a ccula 75 m alttu obt. Mass abl 1 Bant Physcal Popts g X, Y, & Z Mass Momnt of Inta 7.5 g*m hust Magntu Samplng Po / hust Duaton Mnmum 50 mn 0.01 sc. V 0.5 mm/s Usng a nonlna sach algothm, ths thust confguaton yl α =.04-4, cosponng to bst possbl stay stat o bouns of ss 6. mm an θ ss 67 gs. h lag scpancy btwn tanslatonal an otatonal accuacy s u to th w aaton n scal btwn (0.5 mm/s) an (4.0 ma/s). In cass such as ths t s bst to cons otaton an tanslaton spaatly fo th puposs of computng a congnc nlop an slctng contol paamts. Dong so als α =.88-4 an α =.0-3, cosponng to bst possbl stay stat o bouns of ss 4.4 mm an θ ss 5.9 gs. h s magng stanc was slct as = 5 m, wth s max stay stat os of ss = 0.5 m an θ ss = 10 g. Ds locts of H = 5 cm/s, = 0.1 a/s, an I, = 5 cm/s w also slct. Applyng a nonlna sach algothm to sol Eqs. ( 3 ) an ( 33 ) fo ths alus gs = an σ = Aft a fw smulatons, th nt-hcl potntal gan was chosn as I = Fg. 7 Bant hust Ontaton wo smulaton cass a psnt. In th fst cas th sx Bant hcls w clust closly togth 0.5 mts away fom th host, ncat of an ntal ploymnt scnao. In th scon cas th hcls w agan clust togth, but at a stanc of oughly 17 mts fom th host. hs scnao s psntat of a shot ang nzous that woul b ncssay to nspct a fogn tagt. Fo ach cas th ntal tanslatonal locts w zo, whl th spn ats a n cton wth a total magntu of appoxmatly 0.1 a/s p hcl. Atttu ata fo hcl on of th nzous smulaton s xamn n Fg. 8. Not th quc ntal spn an congnc to th s ontaton, as wll as th low magntu stay stat o oscllaton. hs ata s xtmly smla fo all hcls acoss both smulatons, an thus s only psnt onc fo bty. Fg. 8 Rpsntat Atttu Rspons J.S. Nubau 9

10 Fg. 9 Intal Dploymnt actos Fg. 11 Rnzous actos Fg. 10 Intal Dploymnt Rlat Dstancs h tactos of ach hcl n th Clohssy- Wltsh coonat fam (wth th host poston at th ogn) a psnt n Fg. 9 an Fg. 11 fo both smulatons. Rlat stanc hstos a splay n Fg. 10 an Fg. 1, nclung nspcto to host an nspcto to nspcto ata. h fst commnt to b ma about th abo sults s that th potntals ct th tactos as s spt th lat obt ynamcs, constan mpuls actuatos, an xtmly lmt stat nowlg. Wth no communcaton an only lat poston an locty ata, th Bant hcls cong to th s postons an ontatons wthout collson. hs smulatons also show th alty of th analytcal mthos fo slctng contol gans an pctng congnc an stay stat o bouns. Fg. 1 Rnzous Rlat Dstancs h an σ gans po by Eqs. ( 3 ) an ( 33 ) pouc sabl host-lat manung wthout th n fo tat smulaton, ylng congnc to th spcf qulbum stat wll wthn th pct stay stat o bouns. Examnng th popllant consumpton ata n abl als that th maoty of th ful cost occus wthn th fst fw scons of flght. hs s u to th quc moal of th ntal spn at an pomptly acclatng to th s tanslatonal locty. Onc ths po s o, popllant consumpton aags m/s p hou pong a maxmum total flght tm of o 100 hous fo th mnmum Bant ful supply. Assumng a wost cas consumpton at of.38 m/s p hou, total flght tm ops to f hous. Rcallng that Bant s batty lmt to 30 mnuts of flght bfo -ocng, ths tanslats to no lss than tn nual sots. J.S. Nubau 10

11 abl Popllant Consumpton Data Smulaton Cas Consum (m/s p h) Ag. Mn Max Dploymnt, 0 to 1h Dploymnt, 0 to 0.1 hs Dploymnt, 0.1 to 1 h Rnzous, 0 to 1 h Rnzous, 0 to 0.15 hs Rnzous, 0.15 to 1 h Conclusons & Futu Wo h mthos lop n ths stuy xtn th lmts of potntal functon contol thoy, spcfcally n ts applcaton to hcls wth constan mpuls actuatos. A stablty poof has bn as wll as analytcal appoachs to calculatng stay stat o bouns an contoll gans. hs ablts gatly smplfy th tunng pocss whch has long bn a mao stumblng bloc of potntal functon mthos. Spcfc potntals fo th Bant hcl w also lop to yl tagt pontng an collson aoanc bas on lmt snsoy ata. h complt contoll pos mmns smplcty of onboa mplmntaton, as wll as samlss scalablty to lag numbs of ons. h two cass smulat fo ths stuy ploymnt an shot ang nzous of sx nspctos fy th analytcal mthos an show sabl ynamc chaactstcs. Stat hstos al shot s an sttlng tms, as wll as stay-stat os ( 5 o an 0.5 m ) wthn th analytcal poctons of th stablty analyss. Sustan staton png manus aag m/s of popllant consumpton p hou, cosponng to a mnmum lftm of o 100 hous fo th mnmum Bant ful supply. pously achabl wthn th caft s lmt obsatonal, computatonal, an actuatonal ablts, ths potntal functon contoll s a y nablng tchnology fo Bant-class hcls. Ego, th Bant systm s now wll poston to po a spons tst b fo sachng on-obt scng tchnologs. Futu wo wll xpan on th capablts of ths gnal contol famwo wth ga to poxmty flght an on-obt scng. Fst, altnat nthcl potntals wll b xplo; spcfcally ons that account fo th actual s of a collson. Nxt, autonomous ocng wll b appoach. h contoll shows poms wth spct to ocng u to ts ablty to accuatly tac locty commans; th only aaptaton ncssay s a spcfc ocng locty fl. Fxng th fl to th host hcl coul allow succssful ocng n n th psnc of a tumbl. Anoth anu to b pusu s th fbac of mag qualty nto locty commans. Poply ncopoatng ths ata nto th host potntal coul allow Bant to ntfy th optmal magng stanc npnntly a sll ncssay to nspctng fogn obcts an aaptng to changng nonmnts. Atonally, by lagng th shot ang communcaton ablts of th nspctos, pat nspcton of th sam sufac coul b ao catng a mo ffcnt sullanc pocss. In paalll to futh contoll lopmnt, xpmntal tstng on th ASL s ntgat opato wostaton an a bang tst b s plann. h wostaton wll b us to nstgat obustnss to mol naccuacs an thust falus, as wll as opaton wth an ncas numb of hcls. stng on th a bang tst b wll concn th ntgaton of ths contoll wth hawa an th nagaton an mag pocssng systms. hs sults show that th potntal functon mtho lop hn s qut capabl of contollng swams of Bant nspcto spaccaft. As scuss abo ynamc chaactstcs ha mt o xc th tagts, whl popllant consumpton has bn pt to accptabl lls. Atonally, th obustnss of ts potntal functon co pos th ablty to ocom xpct n-flght stubancs, an ts samlss scalablty allows th opaton of on o ozns of hcls. As non of ths ablts w J.S. Nubau 11

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