AUTONOMOUS FORMATION FLIGHT USING BIFURCATING POTENTIAL FIELDS

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1 7 TH ITERATIOA COGRESS OF THE AEROATICA SCIECES ATONOOS FORATION FIGHT SING BIFRCATING POTENTIA FIEDS suk Suuk, Knj chm Dmn of Aosc Engnng, hon nvs Kwos: AV, Fomon Flgh, Swm, Afcl Ponl Fl, Bfucon Absc Ths ms o sgn gunc lw usng bfucng onl fls fo swm of uonomous nmnn Al Vhcls (AVs). W cons n uonomous flgh ssm h cn c ffn swmng ns so s o gun obscl n vhcl collson vonc. Th gunc lw, whch s v fom sng n ulsv onl fl, cn ss vbl gomc ns fo fomon flgh of AVs. Th ssm cn nson bwn ffn fomon ns b w of sml m chng. umcl smulon s fom o vf h vl of h oos gunc lw. Inoucon A flgh ssm usng mull nmnn Al Vhcls (AVs) hs cnl c ns fo lcons such s scnfc collcon n connssnc fo cvl o ml uoss. Flgh conol ssms hv bn su [-5] n o o chv mo vnc lcons h cnno b chv b mns of sngl AV. Howv, gunc n conol lws bcom mo coml s h numb of AVs ncss. Afcl onl fls hv bn l o h sgn of conolls fo swmng ssms [5-]. Th bsc conc bhn hs ho s o c woksc n whch ch AV s c ow n ulbum s, h sbl of whch s gnll gun b unov s scon mho. Howv, h conol ssm m bcom coml fo h cs n whch h gol s s llow o v ung msson. Bnn n cinns hv l clsscl bfucon ho o h onl fl o ovcom hs oblm [4, 5]. Th mho, whch s sml n cn b uckl cu, cn llow fo ffn confguons o b fom hough sml m chng of h onl funcon. Svl mhos usng fcl onl fls hv bn oos n cn s wh g o fomon conol [-]. os of hs mhos us fcl onl fls sml o vo collsons n fo gunc o s oson. In h sn, w n h vous gunc mho usng bfucng onl fl so s o b bl o chv h-mnsonl flgh confguon b lng nw chnu fo usng fcl onl. Th gol of h sn su s o vlo gunc mho fo msson, such s h-mnsonl fomon flng of AVs, n o vf h vl of h nwl vlo gunc mho hough numcl smulons. Fomon Flng. Gunc w Fgu shows h nl coon ssm o- n oson vcos of h AVs. o Fg.. Dfnon of oson vcos j j h AV j h AV

2 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS W cons swm of homognous AVs, whch s cls, ch of whch ncs v vloc fl v ( =,, n) usng sng onl F S n ulsv onl F R govn b h followng uon: v S R = F ) F ( ). () ( j j Th s commn s, hng ngl, n ch ngl of ch AV hn obn s follows: v = v v v, (),,,, v, =, n, (3) v, v, =, n, v, (4) wh v,, v,, n v, h l comonns of h s commn s v,. Th s oll ngl commn, s fn s follows: ( ) =, (5), K, wh K s h gn of h oll ngl commn, n s h hng ngl. Th sng onl [4] F S s fn s follows: F S ( ρ ) C = µ C h ( ρ ρ ) ( ρ ρ ) σ, (6) wh C h sn h mlu of h hbolc onl funcon, µ s h bfucon m, C n sn h mlu n lngh scls, scvl, of h onnl onl funcon, n ρ n σ h ms of h s fomon n, s no b subsc. W fn ρ so s o b bl o fom ng o ln fomon n σ so s o b bl o fom h-mnsonl fomon b h followng uons: fo ng fomon ρ = o o foln fomon, σ = b c, (7) (8) wh, b, n c sn h vco comonns of σ.. Th-mnsonl fomon flng cn b chv b mnulng ms, b, n c whch cn chng h ngl of h fomon n. If =, b =, n c =, hn fomon n h s lll o h - ln s gn. In h cs of =, b =, n c =, fomon n h s ncln n ngl of 45 gs o h -, -, n - lns s gn. Dnng on h vlu of µ, h sng onl cn chng h osons of son ons n k on vous foms. Fgus () n (b) show mls of h vloc fl whn µ s ngv n osv, scvl. F s F s Son s ρ ρ () µ < Son ss ρ ρ (b) µ > Fg.. Bfucng onl fl In cul, h sng onl hs on s of nmcl ulbum s snc ρ, s shown n Fg. () whn h

3 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS m µ s ngv. On h oh hn, whn µ s osv, h sng onl hs wo son ss, s shown n Fg. (b). Ths suls ml h h fomon n cn b chng sl hough mnulon of h ms µ, ρ,, b, n c. Th ulsv onl [6] s fn n s follows: F R j = C j, j j, (9) wh C n sn h mlu n lngh scls of h ulsv onl funcon, scvl, n j = j. Th ol ulsv boun vloc on h h AV s nn on h oson of h oh (n-) AVs n h fomon. Th ulsv onl s hfo us o nsu h h AVs o no coll wh ch oh s h s ow h gol s. In lng h oos mhos, s mon h h sbl of h ssm b mn n o o nsu h h scb bhvos wll occu. In o o mn h sbl of h ssm, w cons wo mhos, nml, unov s Scon Thom n n gnvlu nlss of h ln uons of moon. [4, 5] Th suls of hs nlss nc h h ssm cn lws b cons o b sbl.. Conol w In o o chv s-s flgh, w us obus conoll fo ln m-nvn mul-vbl ssm [7]. W cn ss h s n ouu uons fo longunl n ll moons, whch ln oun h ulbum on, s follows: - I s K u B &, ( ) = A( ) Bu( ) ( ) = C( ). () () Fs, w fn h o () bwn ouu n nu (s Fg. 3), s shown n h followng uon: ( ) =. () B ffnng E. (), w obn &( ) = A& ( ) Bu& ( ). (3) If h ssm s n s s, hn w cn ssum h & =, n ( ) = C& ( ). (4) B combnng Es. (3) n (4), w obn h n ssm: & ( ) A = ( ) C & ( ) B u& ( ) ( ). () In o o sbl hs ssm, w cons h nk of h m n h followng uon: A B nk = n, C (6) wh n s h o of m A, n s h numb of ouus. Accongl, w cn conol onl wo vbls fo boh h longunl n ll uons, n w choos n fo longunl moon, n n fo ll moon o conol s n u. Th conol nu u fo longunl n ll moons of hs ssm s gvn b h followng uon: I s C A Fg. 3. Block gm of h conoll of h ln m-nvn mul-vbl ssm -K 3

4 4 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS = ) ( ) ( ) ( K K u, (7) wh K n K h fbck gns of hs conoll. In on, f h ssm s n nson, & s no consn. Howv, w cn conol h ssm ow nu, bcus h ols of h ssm nv chng, s &. 3 umcl Smulons 3. AV Dnmcs Fgu 4 shows h block gm of h n ssm usng oos gunc n conol lws. To smul swm conol of h AVs, w us mol [8] fo h AV h s ln bou sgh n lvl flgh conons wh fow s of.5 m/s n =.. Euons (8) n (9) show h longunl n ll uons of moon, scbl. Fgu 5 shows h fnons of h s vbls n conol nus., sn cos = w u w u w u X X W g g X X W (8), n cos = v v o v V g V (9) wh, V, W,,,,,, n vbls n,,,, n flcons of movng sufcs fom m conons. Tbls n show h ln ms of longunl n ll moons, scvl. Th subscs no h l vvs wh sc o h ms. Gunc lw Conol lw h AV - j j j j h AVs µ, ρ,, b, c Fg. 4. Block gm of h n ssm W V ϕ Fg. 5. Dfnons of s vbls n conol nus

5 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS Tbl.. n ms fo longunl moon X [s - ] -.3 X W [s - ].4 [s - ] -3.7 W [s - ] -3.6 [m s - ].37 o [m].5 [m - s - ] -.95 W [m - s - ] -7.4 [s - ] -.86 X [m s - ] X [kg - ].3 [m s - ] [kg - ] [s - ] -5.5 [s - ] 3. Smulon Rsuls A numcl smulon s fom o vf h vl of h oos gunc n conol lws. W cn gn ffn fomons, sngl ln, oubl ln, sngl ng, oubl ng, n clus fomons n n ln b usng h onl funcons. Th ln n ng fomons oson AVs long ln n on ccumfnc of h ccl. Th oubl ln Tbl.. n ms fo ll moon V [s - ] -.68 [m s - ] -. [m s - ] -. V [m - s - ] -3.7 [s - ] [s - ] 9.3 V [m - s - ] 7.89 [s - ] -3.3 [s - ] -4. [m s - ] [m s - ].99 [s - ] [s - ] -6.8 [s - ] [s - ] fomon lns u h AVs n ow, n h oubl ng fomon osons AVs n concnc ccls. Ths fomons cn b fom wh ffn mnsons b chngng m of hbolc n onnl onl funcon. Th clus fomon osons h AVs usnl bs on h ms of h ulsv onl. Th-mnsonl fomon flng cn b chv b mnulng m σ. Tbl. 3. Pms of onl funcons (ln) µ ρ - Fg. 6. Flgh jcos 5 AV AV AV 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV Fg. 7. Doubl ln ( = 5) AV AV AV 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV Fg. 8. Sngl ln ( = ) 5

6 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS Fgus 6 hough show h nson of fomon of nn AVs flng wh no wn. Fgus 6 hough 8 show h nson of ln fomon. Fgus 9 o show h nson of ng fomon. Th ols of h conoll lc -5.5, h smulon ms s o C =, = 4, C h =, C = 6, = 6, =, b =, c =, n h vlus of h oh ms of h onl funcons F S ls n Tbls 3 n 4. Fgu 6 shows h cs n whch h ssm chngs fom oubl ln o sngl ln. In on, Fg. 9 shows h cs n whch h ssm chngs fom oubl ng o sngl ng o clus v 5 s, whb ch AV ns h s oson. Ths s chv hough sml m chng n s on of h vngs of usng bfucon uon s h bss fo h fcl onl funcons, bcus w o no n o conol ch AV nvull. Fgus 3 hough 8 show h hmnsonl nson of fomon of nn AVs. Th ols of h conoll lc -5.5, n h smulon ms s o C =, =, C h =, C = 6, = 6, µ =, n ρ =. Though hs numcl smulon, w chng ms, b, n c, s shown n Tbl 5, v 5 s. Fgus 4 hough 8 show h h fomon nsons fom ng n h - ln o ng n - ln b ong h fomon 45 gs. Fgu 9 shows h AVs s, h ngul vloc, h Eul ngls, n h conol nus houghou h h-mnsonl nson. Th suon lms fo h cf conol sufcs = ±.35 n, = ±.79, n h hus ng s -.35< <5.45 N. Th conoll s whn s lms bcus h mmum conol vloc ws mn usng hbolc onl funcon. Fgu shows h m sonss of h conoll vbls of h AV. Ths suls vl h ch vbl follow h commns ssfcol. Tbl. 4. Pms of onl funcons (ng) µ ρ Fg. 9. Flgh jcos AV AV AV 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV 9 AV AV AV 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV Fg.. Sngl ng ( = ) Fg.. Doubl ng ( = 5) Fg.. Clus ( = ) 6

7 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS Tbl. 5. Pms of onl funcons b c 3-3 Fg. 3. Flgh jcos AV AV AV 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV 9 AV AV AV 3 3 AV 4 AV 5 AV 6 AV 7 AV 8 AV Fg. 4. Fomon n ( = 5) Fg. 6. Fomon n ( = ) Fg. 7. Fomon n ( = ) Fg.. Fomon n ( = ) Fg. 8. Fomon n ( = 5) 7

8 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS Ss [m/s] Eul ngls [] 5 5 Tm [s] ()AV s Tm [s] (b)av ul ngls Sufc nus [] V W Angul s [/s] Tm [s] 5 ()AV ngul s Tm [s] 5 ()AV hus nu Tm [s] (c)av lvo,lon n u nus Fg. 9. Tm hsos of s, ngul, Eul ngls, n nus (AV ) Thus nu [N] S [m/s] 4 3 Roll ngl [] commn conoll Pch ngl [] 5 5 Tm [s] ()Tm hsos of s commn conoll Tm [s] (b)tm hsos of Hng Angl [] Tm [s] (c)tm hsos of commn conoll Tm [s] ()Tm hsos of Fg.. Tm hsos of s n Eul ngls (AV ) 8

9 ATOOOS FORATIO FIGHT SIG BIFRCATIG POTETIA FIEDS 4 Conclusons In h sn, w hv scb nwl oos gunc lw h uss nw och nvolvng n fcl onl fl n h bfucon ho. Th oos gunc lw cn chng fomon ns h-mnsonll hough sml m chngs. Numcl suls vf h vl of h gunc n conol lw bs on h onl funcon mho fo fomon flng of AVs. Rfncs [] Gu, l. Dsgn of Flgh Tsng Evluon of Fomon Conol ws. IEEE Tnscon on Conol Ssms Tchnolog, Vol. 4, No. 6, 6,. -. [] Roo C K, Km H n Thk J. Oml AV Fomon Gunc ws wh Tmng Consn. Innonl Jounl of Ssms Scnc, Vol. 37, No. 6,. 4-47, 6. [3] Km S n Km. Th Dmnsonl Omum Conoll fo ull AV Fomon Flgh sng Bhvo-bs Dcnl Aoch. Innonl Confnc on Conol, Auomon n Ssms, , 7. [4] J, l. Fomon Gom Cn bs Fomon Conoll Dsgn usng unov Sbl Thom. 8 KSAS-JSASS Jon Innonl Smosum, , 8. [5] Pul T, Kogs T R n Gvhl J T. olng of AV Fomon Flgh sng 3D Ponl Fl. Smulon olng Pcc n Tho, Vol. 6, Issu 9, , 8. [6] Shm, chm K, l. Pom nuv n Obscl Avonc Conol sng Ponl Funcon Gunc ho. Pocngs of Innonl Smosum on Sc Tchnolog n Scnc, , 6. [7] cinns C R. Ponl Funcon hos fo Auonomous Sccf Gunc n Conol. Av. Asonu Sc., Vol. 9, No. P., 996, [8] cinns C. R. Vloc fl h-lnnng fo sngl n mull unmnn l vhcls. Th Aonucl Jounl, Vol. 7, No. 73, , 3. [9] Bnn D n cinns C R. Sccf Fomon Flng sng Bfucng Ponl Fls. Innonl Asonucl Congss, IAC-8-C.6.4, 8. [] Bnn D n cinns C R. Pn Tnson n Sccf Fomon Flng v Th Afcl Ponl Fl ho n Bfucon Tho. 3 Innonl Smosum on Fomon Flng, ssons n Tchnologs, 8. [] Hung W H, Fjn B R, Fnk J R n Wn W H. Vsul Nvgon n Obscl Avonc sng Sng Ponl Funcon. Robocs n Auonomous Ssms, 54, , 6. [] Kowlck W n Kolowsk K. Afcl Ponl Bs Conol fo g Scl Fomon of obl Robos. Pocngs of h Fouh Innonl Woksho on Robo oon n Conol,. 85-9, 4. [3] n n hng. Collson vonc Plnnng n ul-obo bs on Imov Afcl Ponl Fls n Ruls. Innonl Confnc on Robocs n Bommcs Bngkok, Thln,. 6-3, 9. [4] K, Hn X n Q G. Fomon n Obscl-Avonc conol fo obl Swm Robos Bs on Afcl Ponl Fl. Innonl Confnc on Robocs n Bommcs, Chn, , 9. [] Bns, Fls n Vlvns K. nmnn Goun Vhcl Swm Fomon Conol sng Ponl Fls. nn Confnc on Conol n Auomon, Ahns-Gc,. -8, 7. [6] D'Osogn R, Chung, Bo A n Chs S. Th Ro o Csoh: Sbl n Colls n D Dvn Pcl Ssms. Phscl Rvw s, Vol. 96, No. :43-, 6. [7] Dvson E. Th obus conol of svomchnsm oblm fo ln m-nvn mulvbl ssms. Auomc Conol, IEEE Tnscons on, (),. 5 34, 976. [8] Hslo A n Humb J S. W-fl ngon mhos fo uonomous nvgon of 3-D Envonmns. AIAA Gunc, vgon, n Conol Confnc n Ehb, 8-75: 8, 8. Cogh Smn Th uhos confm h h, n/o h comn o ognon, hol cogh on ll of h ognl ml nclu n hs. Th uhos lso confm h h hv obn msson, fom h cogh hol of n h ml nclu n hs, o ublsh s of h. Th uhos confm h h gv msson, o hv obn msson fom h cogh hol of hs, fo h ublcon n sbuon of hs s of h ICAS ocngs o s nvul off-ns fom h ocngs. 9