Controlling Swarms of Mobile Robots for Switching between Formations Using Synchronization Concept

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1 7 IEEE Iol Cof o Roos Auomo Rom, Ily, -4 Apl 7 hb9. Coollg Swms of Mol Roos fo Swhg w Fomos Usg Syhozo Cop Dog Su C Wg Dpm of Mufug Egg Egg Mgm Cy Uvsy of Hog og, Hog og l: (85) Fx: (85) Eml: {msu, wg}@yu.u.hk As hs pp pss syhoous ool ppoh o swms of mol oos swhg w fomos. Aog o h s fomo, syhozo ool gol s v, s o whh h poso syhozo o s f s ffl poso os w vy p of wo ghog oos. A lz joy kg ooll s h vlop wh fk of oh poso syhozo os, fom wh omo of ffow fk ools. I s pov h hs kg ooll sympolly ovg oh poso syhozo os o zo. Smulos pfom o goup of wy fully-u mol oos swhg sk w ff llps uvs. h smulo suls mos h ffvss of h popos syhoous ool sg fo h fomo ool. ywos fomo, mol oos, syhozo. I. INRODUCION Suy o ooo of mulpl mol oos hs v sg os ys. I hs pp, w suss h fuml ssus ulyg joy ools of swm of oos whl mg s fomos h my m-vyg. h h ovol ppohs o ooo of mulpl mol oos po h lu: hvo-s mho [-5]; vul suu hqus [6-9]; l-followg sgy [-4]. I hvo-s ool, svl s hvos ps fo h g, h fl ool s v fom wghg of h lv mpo of h hvo. h vg of hs sgy s h h goup yms o fomo fk y ouplg h wghgs of h os. h svg s h s fful o s h yms of h goup o gu h sly of h whol sysm []. I h vul suu ppoh, h fomo s s sgl y. Ds moo s ssg o h vul suu whh s ou jos fo h mm of h fomo o follow. h vg s h h mho s sy o ps fomo hs wok ws suppo y h Rsh Gs Coul of h Hog og Spl Amsv Rgo, Ch, u Poj CyU 976, y g fom h Cy Uvsy of Hog og u Poj 793. sgy hs sly gus fo h oos. h svg s h h ooll s o lz hu my ou ffuly som pplos. I h l-followg sgy, som oos sg s ls, whl ohs sg s follows. h vg of hs sgy s h s sy o ool mulpl oos s fomo usg oly wo oolls s sul fo sg h fomo of oos. h svg ls h ffuly o os h ly gp of oo []. I hs pp, w popos o us syhozo ool sgy o ss muloo ooo, ulzg h op of oss-oupl ppoh [5]. h s s o l m of mol oos k s jos whl usg oss-oupl ools o syhoz moos mogs h oos so h kms loshp m fo s fomo. h oss-ouplg hology [6] povs vgs oppous o sg suh syhoz ooll. Ov h ps, h oss-ouplg op hs wly us mul-xs moos oh pplos suh s ug oou o of CNC mhs [7-]. Rly, h oss-ouplg op ws opo o pv ool hu o solv poso syhozo of mulpl xs []. h oss-ouplg hology hs us oos, suh s ools of mol oos [] oo mpulos [3]. h soluos fo oou kg polm fou [4-5]. o vo usg of molg, mol f vl-g oss-ouplg ooll ws ou fo gl lss of oous [6]. h ffo o xm h sly ousss of h oss-oupl ool sysm ws po [7]. Applos of h syhozo ppoh o ssg fomo ool polm hv o po y h lu. h vgs of usg h syhozo ool fo h fomo ool hfol. Fs, h syhozo ool gol s m og o h s fomo, h v o um of su-gols fo h vul oo whou smo. By hs wy, h fomo sgy wll ps h ly of h oo s o lm. So, syhozo ooll h gus sympo sly of oh poso kg syhozo os sg lz hu []. h, /7/$. 7 IEEE. 3

2 hb9. h ooll gly smplf y syhozg h moo of h oo wh s wo ghos oly [3,]. I oh wos, h oo ool oly qus h fomo of wo y oos u o ll oh os. hs pp hs h followg wo ouos. Fs, w sussfully sfom h wo smulous os of joy kg fomo o moo syhozo sk. A syhozo quo s m og o h s fomo, s o whh syhozo ool gol v. I sml m o [], w v h oo o ppoh s g poso whl syhozg s moo wh s wo ghos. o vlu h syhozo ool ff fo h fomo, h op of h syhozo o s ou, whh s f fo h oo s h omo of h poso os of wo y oos wh ouplg pms. So, w vlop lz oss-oupl ooll o slz mul-oo moos whl syhozg posos of h oos fo h s fomo. Af fo of oupl poso o y omg h poso o h go of h syhozo os, smpl kg ooll s osu wh ffow fk ools, wh fomo qus of wo ghog oos oly. I s pov h h popos ooll gu sympo ovg o zo of oh poso syhozo os. Smulos pfom o swm of oos swhg w ff llps uvs, o mos h ffvss of h popos ppoh. h popos syhozo ool ppoh s mo sul o h sks of m of fomo shp wh goup oos mov s whol, swhg w fomos. II. FORMAION IA SYNCHRONIZAION Cos h ool polm of gug posog goup of pl fully u oos log h ouy (uv) of wo-msol omp s. h yms of h oo s gv y: q ( ~ ) () wh q [ x, y ] u s vo og oos x-y pl, u os h ool pu of h oo. Iou m-vyg s shp, S (q), wh [ q q ] q os h ofguo of h swm of oos. h ouy of S (q) s pmz y wo msol pl uv, S (q). Cos sk of swhg w ff fomos S (q). Assum h h oo s ssg y g posos q, ll g posos lo h uv S(q ). h gol s o m h ppop ool pus fo ym () suh h h oos ovg o hs g posos whl mg h fomo shp S (q). Df h poso o of h oo s q q. h gol of h oo poso ool s o v. Mwhl, h oos o hv h gol of h fomo ool, mly q, S( q ). hs fomo ool polm solv y usg h syhozo ool op. h s s o gul moos of h oos wh hy ppoh h g posos q, so h ll oos m h qu ouy uv S (q). Df f ( q q ) s fuo of syhozo h ll oos qu o ssfy smulously. hs fuo s m s o h fomo ool gol suj o h os S (q). I ully pss w sk-p qum kms. A xmpl s gv low o show how suh syhozo fuo s v fom S (q). Exmpl: Cos h oos qu o m llps uv ug h moo. h oo q of oo s suj o h followg quo: x osϕ S ( q ) : q A () y sϕ wh o h logs h shos s of sα h llps, spvly, ϕ, osα y α os h gl of h oo loo o x h llps. Assum h h oos o pl h logs o h shos xs of h llps so h h vs of A xss. Fo h s fomo, ll oos o ssfy h followg syhozo fuo smulously: f ( q q ) : A q A q A q. Fom h ov xmpl, s s h h syhozo fuo glly ps h fom: f ( q q ) : q q q (3) wh os h ouplg off of h oo s ssum o ozo. (h quls zo ms h oo hs o fomo qum hus s o h fomo ool.) No h s h os o m-vyg. S q. (3) hols ll s oos q, w hv q ) : q q q f ( q. (4) 3

3 hb9. h, h followg syhozo gol v y omg (3) (4): (5) Eq. (5) ully pss h fomo ool gol mplly. Fuh, h syhozo gol (4) v o su-gols suh s, wh ouy oo h wh,. W h ou h op of poso syhozo os, whh f s sus of ll possl ps of wo ghog oos h followg wy: 3 3 wh os h syhozo o of h oo. Ovously, f h syhozo o fo ll ~, h syhozo gol (5) hv uomlly. h syhozo o pss h g of ooo mogs h u oos h fomo, s o quvl o h ovol kg o. Employm of h syhozo o povs h oo wh moo fomo oh fom slf fom h oh oos. H, h moos of ll oos wll oo. Now h ool polm s o v oos ovg o h g poso so h, whl hvg h fomo ool gol (5) y gulg h syhozo os (6) o zo. h x ojv s o sg ooll h gus sympo ovg o zo of oh h poso o h syhozo o. I sml m o [], h oo sg o ppoh s g poso q whl syhozg s moo wh s wo ghog oos. By hs wy, h ool of h oo os o qu h fomo of ll oos xp fo wo ghos, h h mplmo s gly smplf. h ov vlopm shows how o mk fomulo o pos h fomo ool polm s syhozo ool polm. h syhozo os f s o h fomo ool gol S (q), hy ps h fomo os mplly. h ppoh s xsl o y fomo shp f os y (3). III. SYNCHRONIZAION CONROL DESIGN Df oupl poso o s E ( ) w (7) wh s posv os, w s vl fom m zo o. h syhozo o (7) s suj o h ouy oo h wh, -. No h h (6) oupl poso o E of h oo os h fomo of wo ghog oo -, whh s fom q. (7) h fo of (6). Dffg E wh sp o m yls E ( ) (8) o osu sympolly sl kg ooll o v E E, w ulz [8] o ou omm vo u h ls o om poso vloy os. Ulk [8], h h xss m-vyg ouplg pm o o (7). H, w my f h omm vo u s follows: ( ) E q u (9) wh s gol posv g mx. Dfo of u (9) ls o h followg poso/vloy vos: u q ( ) E () E E Now h ool ojv s o sg h oqu pu τ o s o l o h slg suf [8], so h h oupl os E E o zo. Fo sy mplmo, h oqu ool pu s lly sg lz hu, h oo ool oly oss h syhozo w hs oo s wo ghos u o ll ohs. Fuh, g h syhozo ool woul o ff h sly of h whol sysm. W flly sg h ooll s follows: τ ( u q ) ( ) () wh posv fk ool gs. h ls m () s us o omps fo h ff of oug h oss-ouplg ool o h ovll sysm yms, whh s qu y h sly lyss. I pps h pms (7) () wll om h ool of h syhozo o. plys mjo ol o u h syhozo o, whl sus h sly of h sysm wh g h oss-ouplg ool wh o h ovll sysm yms. Susug () o h oo yms () yls h los-loop yms: ( ) () hom h popos kg ooll () ls o sympo sly of h sysm, mly, s m, u h oo h h ool g s lg ough o ssfy λ m ( ) λmx ( ), wh λ m () λ mx () o h mmum mxmum gvlus of h ms. 3

4 Poof: Df Lypuov fuo s w w (3) Dffg wh sp o m yls w (4) Mulplyg oh ss of () y ( ) yls (5) Susug (5) o (4) yls [ ] w (6) Fom (7)~(), w hv w w (7) Susug (7) o (6) ulzg h oo of hom, w hv mx m λ λ (8) hfo, s m. h syhozo gol (5) s hv. Fom (), w fuh hv E. W ow pov wh E. Comg ll quos (7) fom o, w hv (9) Susug (5) o (9) yls. S s o zo,. hfo, h sysm s sympolly sl. I. SIMULAIONS Smulos w pfom Ml o vfy ffvss of h popos syhozo ool ppoh fo fomo. Fg. lluss wy fully-u mol oos sg o llps uv. h oos, o y ll squs, qu o mov s whol o ppoh h fl s llps uv o y sh l Fg.. Dug h swh of fomos, h oos mus m s of llps uvs wh h gv m-vyg logs shos s. I pul, h oo ks s vul s joy h s m og o h s fomo h x of h oo ps y h gl x y α. Mwhl, ll oos mus m h s llps fomo (h s m-vyg) ug h kg. Fg.. h s llps fomo Df h h logs h shos s of h llps hg h followg wy: ) )( ( ) )( ( f f () wh f o h l h fl s logs s of h llps, f o h l h fl s shos s of h llps. h quo of llps uv hs gv (). h ouplg pms hy f s s os - A ϕ ϕ. Fo ompso pupos, wo ool lgohms hv ppl h smulos. O s h popos syhoous ool. h oh o s h o-syhoous ool y sg,,. S hs sgf ff o h poso o s s (8), vy lg vlu of my ff h ompso of h wo lgohms. hb9. 33

5 hb9. o ovom hs polm, w mulpl sl vlu of. o h smulos. Roo.8. Roo.9.3 Roo Roo Roo Roo 6.4. Roo Roo Roo Roo..35 Fg.. Roo jos swhg w fomos. h ool gs w hos s follows:,,,. 5. Wh usg h o-syhoous ool,. h smplg po ws s o.5s. Fg. lluss jos of ll oos, wh swhg w h l o fl fomos s lly show. Fgs. 3 4 llus h poso os of ll wy oos u h popos syhoous ool h o-syhoous ool, spvly. I s s h ll poso os ovg o zo sussfully, h os of h syhoous ool pp o smll h hos of h o-syhoous ool. hs suls h h popos syhozo ool ompls h oo joy kg sk sussfully, wh v moo pfom h h o-syhoous ool. Fgs. 5 6 llus h syhozo os wh h wo ool lgohms. No h hs syhozo os ully ps h fomo os. Apply, h syhozo os u h syhoous ool muh smll h hos u h o-syhoous ool. l gvs l ompso of h syhozo os of h vul oo u h wo ool lgohms. Smll syhozo o mpls fomo. l : h mxmum syhozo os (m) Roos o-syhoous ool syhoous ool Roo.64. Roo.3.35 Roo Roo Roo Roo 6.3. Roo Roo Roo Roo..37 Fg. 3. Poso os wh h syhoous ool Fg. 4. Poso os wh h o-syhoous ool I. CONCLUSIONS I hs pp, syhoous ool ppoh s vlop fo swhg w fomos of swms of oos. A poso syhozo o s f s ffl poso o w vy p of wo ghog oos. A lz joy kg ooll s h vlop wh fk of oh poso syhozo os, fom wh omo of ffow fk ool. I s pov h h vlop ooll gus sympo ovg o zo of oh poso syhozo os. h smulo suls mos h ffvss of h popos syhoous ool sgy fo h fomo ool. Fuu wok wll xso of hs syhozo ool ppoh o l goup of mol 34

6 hb9. oos wh osos of oo/vom uy vso/ssg sysms. Fg. 5. Syhozo os wh h syhoous ool Fg. 6. Syhozo os whou h o-syhoous ool REFERENCES []. Blh R. Ak, Bhvo-s fomo ool fo mul-oo sysms, IEEE s. Roo. Auom., vol. 4, pp , D [] J. R.. Lwo, R. W. B, B. J. Youg, A lz ppoh o fomo muvs, IEEE s. Roo. Auom., vol. 9, pp , D. 3. [3] L. E. Pk, ALLIANCE: A hu fo ful ol muloo oopo, IEEE s. Roo. Auom., vol. 4, pp. -4, Ap [4] S. Bm, Y. E, M. Hmsh, Nvgo of lz uoomous uom gu vhls ml hlg, IEEE s. Roo. Auom., vol. 9, pp , Aug. 3. [5] M. Log, A. Gg., R. Muphy,. lvs, Applo of h su fl oo hu o smul mg sk, Po. IEEE I. Cof. Roos Auomo, pp. 34-3, Blo, Sp, Ap. 5. [6] M. A. Lws,. H., Hgh pso fomo ool of mol oos usg vul suus, Auoomous Roo., vol. 4, pp , 997. [7] M. Egs, X. Hu, Fomo os mul-g ool, IEEE s. Roo. Auom., vol. 7, pp , D.. [8] R. W. B, H. Lwo, F. Y. Hgh, A ooo hu fo spf fomo ool, IEEE s. Cool Sys. hol., vol. 9, pp , Nov.. [9] W. g, N. X, A. Spks, Fomo ool of uoomous gs 3D woksp, Po. IEEE I. Cof. Roos Auomo, pp Ap.. [] H. G., S. G. Lozou,. J. ykopoulos, Noholoom vgo ool of mulpl mol mpulos, IEEE s. Roo. Auom., vol. 9, pp , F. 3. [] H. khsh, H. Nsh,. Ohsh, Auoomous lz ool fo fomo of mulpl mol oos osg ly of oo, IEEE s. Ius. Elos, vol. 5, pp. 7-79, D. 4. [] J. P. Ds,. um, P. Osowsk, Molg ool of fomos of oholoom mol oos, IEEE s. Roo. Auom., vol. 7, pp , D.. [3] A.. Ds, R. Fo,. um, J. P. Osowsk, J. Splz, C. J. ylo, A vso-s fomo ool fmwok, IEEE s. Roo. Auom., vol. 8, pp , O.. [4] J. Hug, S. M. Fo, A. Q, S. Go, Lolzo follow-h-l ool of hogous goup of mol oos, IEEE/ASME s. Mho., vol., o., pp. 5-5, Ap. 6. [5] Y. o, Coss-oupl xl ompu ools fo mufug sysms, ASME J. Dym. Sys., M., Co., vol., pp. 65-7, 98. [6] M. omzuh, J. S. Hu,. C. Chu, Syhozo of wo moo ool xs u pv ffow ool, ASME J. Dym. Sys., M., Co., vol. 4, o., pp. 96-3, 99. [7] A. Roguz-Agls H. Njmj, Muul syhozo of oos v sm s fk: oopv ppoh, IEEE s. Cool Sys. hol., vol., o. 4, Jul. 4. [8] Q. Zhog, Y. Sh, J. Mo, S. Hug, A l oss oupl ool sysm fo hgh sp mhg, h I. J. Av Mufug hology, vol. 9, pp: ,. [9] S. S. Yh P. L. Hsu, Esmo of h ooug o vo fo h oss-oupl ool sg, IEEE/ASME s. Mho., vol. 7, No., Mh. [] M.. Y, M. H. L, P. L. Y, hoy pplo of om slf-ug pv ool oss ouplg ool of mllg mh, Mhos, vol 5, o., pp. 93-, M. 5. [] D. Su, Poso syhozo of mulpl moo xs wh pv ouplg ool. Auom, vol. 39, o. 6, pp , Ju. 3. [] L. Fg, Y. o, J. Bos, Coss-ouplg moo ooll fo mol oos, IEEE Cool Sys., pp , D [3] Su D., Mlls, Apv Syhoz Cool fo Cooo of Mul-Roo Assmly sks, IEEE s. Roo. Auom., vol. 8, o. 4, pp , Aug.. [4]. C. Chu, M. omzuk, Cooug ool of mh ool f v sysms: A sk oo fm ppoh, IEEE s. Cool Sys. hol, vol. 9, o., J.. [5] R. MN. so, Mulxs oou kg: A g m hozo l qu opml ool ppoh, Dym. Sys., Co., vol., Nov [6] Y. o, C. C. Lo, l-g oss-ouplg ooll fo ooug, A. CIRP, vol. 4, o., pp , 99. [7] S. Yh, P. Hsu, hoy pplos of h ous oss-oupl ool sg, Po. Am Cool Cof., Ju [8] J. J. E. Slo, W. L, O h pv ool of oo mpulos, Iol Joul of Roos Rsh, vol. 6, o.3, pp ,

Bayesian Estimation of the parameters of the Weibull-Weibull Length-Biased mixture distributions using time censored data

Bayesian Estimation of the parameters of the Weibull-Weibull Length-Biased mixture distributions using time censored data Bys Eso of h s of h Wull-Wull gh-bs xu suos usg so S. A. Sh N Bouss I.S.S. Co Uvsy I.N.P.S. Algs Uvsy shsh@yhoo.o ou005@yhoo.o As I hs h s of h Wull-Wull lgh s xu suos s usg h Gs slg hqu u y I sog sh.