Optimized Robotic Assembly Sequence using ACO

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1 S.Shr Lhlr I Prvt Lt., Th (Wst), Mu, , I surjtt@gl.o Opt Root Assl Squ usg ACO B.B.Bswl Dpt. of Mh. Egg. NIT Rourl Orss I hut.swl@gl.o P.Dsh Dpt. of Mh. Egg. IACR, Rg Orss I prswr_sh@hoo.o B.B.Chouhur Dpt. of Mh. Egg. IGIT Srg Orss I t@gl.o Astrt A root ssl squ s osr to optl wh t s ssl ost stsfs th pross ostrts. Th ssl ost rlts to ssl oprtos, ssl otos ssl rto hgs. Th prst wor utls t olo optto (ACO) tho for grto of opt root ssl squs. Th tho rflts th ssl ost to rg futo ssot wth th ssl squ. Th rg futo s trtvl ACO to grt th sr opt ssl squ. A s stu s prst to show th fftvss of th propos tho. I. INTRODUCTION Assl squ rtl flus th proutvt of th pross, prout qult th ost of prouto. Th ost of ssl o vrg s 0-30% of th ufturg ost of orl prout. Th rto tw ost prfor of ssl hs grull rs wth rspt to th othr phss of th ufturg pross rt rs, ths ft hs us growg trst rsrhrs. A portt spt of ths vlopg pross s rprst th to utotll grt th ssl pl tfg th optu squ of oprtos wth rspt to ts ost orrtss. Prouts wth lrg ur of prts hv svrl ltrtv fsl squs. A vrt of optto tools r vll for pplto to th prol, ut thr sutlt /or fftvss r lso ur sr. Trtol oputtol thos oft prou otorl plosos of ltrtvs, wth uptl oputtol ts. Stu of vrous optto thos rvls tht t olo optto (ACO) thqu vtgousl us to solv suh prols. So of th jor wor th r rss th ssus o grtg orrt squs whl so othr fous t grtg th squs ltrtv ut splr thos. D Fo Wht [] vlop splf tho for grto of ssl squs usg srs of qur--swr. D Mllo Srso [2] vlop stt trsto grph whh splfs th vrg ur of oprtos fro th prvous o. Cho Cho [3] vlop tho usg rtol prt ott lvl grphs whh ots th forto o rtol otos for h pr of tg prts. Cho, Sh Cho [4] sr th prt oto stlt whh s ssr for stlt of prts urg th pross of ssl. Hog Cho [5] us th url twor tho for grto opt root ssl squs. But ost of th t ths tho fls to tr th glol opt. Hog Cho [6] tr rs ur of ssl ostrts for th s tho ot supror rsults. But t stll hs th prol of gttg trpp lol optu pot. Th prst of tho tht osrs ssl ostrts ssl osts s or pproprt root ssl squ grto ts th trrltoshp tw th tg prts vr possl rto of ssl. II. WORK DESCRIPTION A orrt squ for ssl grt fro th sssl squ whh s rprst s orr lst of sssl oprtos (DOs).Th os r DO (, ), whr, s th ur of prts {,,, } rprsts th ssl rtos. B osrvg th vlu of sssl oprtos, th frst o to sssl s slt. Strtg fro th frst o, ts srh th fsl sssl squs trvllg ll th os. Squs whh stsf pr ostrts, otvt ostrts rg qutos r th solutos to th prol. I th gg, th tours ostrut th ts r th tl fsl squs for th prol. Th prol s ru to grt tt pplg th otos of stlt. Thrftr th stl opt o s ot orrspog to th ssl ostrts. Th of t lgorth, prst hr, grts th optl ssl squ orrspog to th rg tr of th orr of 5 X 5. Th srh sp rus to th orr of 5 X 5 trs vr t vst of th ts. Th pr ostrts tw lsos th orrspog rg tr to ot two lsos r opt to pross th t lgorth. A phroo τj s us s th shr or of ll ts tht orrspos to th rg tr s upt urg pross. Ths gvs th u rg pth whh hs to follow urg sssl. Th ojtv of th prst wor s to grt stl optl root ssl squ wth u ssl ost whl stsfg th ssl ostrts. Th prst rsrh uss ACO thqu osrg of th stlt of ssl otos /or rtos. III. METHODOLOGY A. Prout Molg A prout s osr to sutl for root ssl ol wh () ll th vul opots r

2 rg, () ssl oprto prfor utull prpulr rtos sp ptg Z rto, ()h prt ssl spl srto or srwg. A prout wth prts s rprst s: A (P, L) () whr, A s prout hvg prts P {pα α, 2 }, trot lsos L {lαβ α, β, 2 r. α β}. Hr rprsts th ur of prts of th prout r s th rltoshp tw th ot prts (-) r (-)/2. Th lso lαβ rprsts th otv rltoshp tw pr of prts pα pβ, l αβ lso ( p α, C αβ, f αβ, p β ) (2) whr, Cαβ s th ott-tp oto tr fαβ s ft-tp oto tr r rprst C C C C C C C f f f αβ f αβ f f f Th ssl rtos for root ssl r t to, whr, {,, }. Th rprstto of th lts of ott-tp ft-tp r: 0: oott th rtotw pα C r: rl ott th rtotw pα v: vrtul ott th rtotw pα 0: o ft th rtotw pα sw: srwg th rtotw pα rf : rou pg hol ft th rtotw f pα p: ultpl rou pg hol ft th rto tw pα pβ Eh lt of f lso rprst s polgo ft (pf), tght ft (tf), ulg (), rvtg (r), vrtul ft (vf) or o ft (0). Th oto tr quto (2) s rprst th followg fort. 0 r r 0 0 rf l αβ lso pα, p r r β L L 6 L 2 L 5 L 3 L 4 () () () Fgur (). A spl pl of prout (Grr ssl); (). Drtos for ssl or sssl; (). Lso grph ol of grr.[ -shft; -l; -ut; -l, -ut] ) Epl Stu: A grr ssl s show Fgur () s osr for grtg th ssl squ vltg th propos tho. Fgur () shows th rtos for ssl or sssl Fgur () rprsts th lso gr of th opots. Th lsos of th opots r rprst s: o r r o o o l lso r r r rf o o o o o o o o l lso v o o sw o o r r r rf o o l lso o r r o o o v o o sw o o l lso o o o o o o o o o o o o l lso, r o o o o o r o o o o o l lso, o o o o o o B. Assl Costrts A pr ostrt of lso lαβ s rprst st of prts, p, tht ust ot for two prts pα pβ r ot. Th pr ostrt ( ) ostrt PC ( p f ) of th prt pf r prss PC l αβ of lso lαβ th pr PC( l ) { p,,..., } αβ γγ γ γ, PC( p ) P( l ) γ 2 P f l U q αβ l l whr, P ( l αβ ) s pr ostrt of lso l αβ. Th pr ostrts of th lsos r: PC(l ){ Φ }, PC(l ){ }, PC(l ){ Φ }, PC(l ){ }, PC(l ){ Φ }, PC(l ){ Φ }. H, th pr ostrts of th prts r: PC(P ){, }, PC(P ){ Φ }, PC(P ){}, PC(P ){ Φ }, PC(P ){ }. IV. OPTIMIZATION OF ASSEMBLY SEQUENCE USING ACO A. Th Erg Futo Erg futo, Esq, s ssot wth ssl squ, t s rprst s: E sq E J + E P + E C (3) whr, Esq, EJ, EP EC r th rg futos ssot wth ssl squ grto (ASG), ssl ost, pr ostrts otvt ostrts rsptvl. Th rg ssot wth pr ostrts s: E P C P μ (4) whr, CP s postv ostt µ s th pr (µ0, f t stsfs th pr ostrts, othrws ). Th rg ssot wth otvt s; E C C C λ (5)

3 I slr r otvt λ s frr o th ss of lso rltoshps. Cosrg CJ s rg ostt rlt to ssl squ ost J, th vlu of J s; : f t voltsth ostrt s or susutl J ρs Cs + ρtct : othrws Th ojtv ftor, whh s l wth hurst vlus of t olo lgorth, t prss s; Esq CJ J + ( CPμ + CCλ ) (6) {,,,, } DL... {,,,, } DL Strt B. Dgr of Moto Istlt Th possl ssl rtos frr fro th lsos tw th trrlt prts. It wll t grl for s DS s(,,, ). Th orr lsts of possl ssl rtos orrspog to th ssl squ prss : DL (,2,... ). Bs upo ths prssos, th orl gr of oto stlt C s th orl ur of ssl rto hgs, C t r vlut. C ( S{ }) ) 2 j s BA j whr BA j ( j,2,..,5) s th -sussl for t th jth ssl stp, S{BAj} s th gr of oto stlt of th jth sussl. A ro gr of oto stlt s th prts logg to th sussl r opltl f to h othr, whrs twlv grs s th prts r fr to ov rto. Slrl C ( ) j t NT j whr, (NT j ) s t s f rto hg of BAj ours for orr lst DL, othrws t s 0. If th squ s ustl for ll ( NT j ) ( j,2,3,4,5 ) t s t s. So th ur for ll Css Cts l tw 0. Th ro s, th squ opltl stsfs th ostrts, whrs o s, th rltoshp s ustl. A possl squ sq { } s osr. Th ssl rtos of tg prts frr fro lsos tw th prts. Eh st of possl ssl rtos r rprst s: DS { φ}, DS { }, DS {,, }, DS {,, }, DS {,, } Th orr lst of possl ssl rtos orrspog to th ssl squ ot fro quto () quto (2). Th orr lsts for th s prout r gv ; (7) (8) Th hrrhl strutur of th ssl rtos th orr lst of possl ssl rtos r show Fgur 2() Fgur 2() rsptvl. A ssl squ ossts of ur of prtl squs wth hvg rg vlus. Th su of ths vlus s th ojtv rg of vul ssl squ. Th rg tr for ths stu s 25 X 25,.. 5 X 5 tr. Eh ll rprsts th rg tw two lts tr th followg r. Dtrto of E Sq : Stp : Dtrto of E P : Slt prtl squ of prout lug rtos.. two os fro th sssl oprtos. Lt th prtl squ (, ( )) ( ( )). Th pr ostrt for h prt s th put to ths prol. Th pr ostrts for th prout s, PC(){, }, PC(){ Φ }, PC(){}, PC(){ Φ }, PC(){ }. Pr µ lult fro quto (4). Tg th t fro prvous stp, th frst posto, ( ), whh hs ull st, stsfs th pr ostrt. H, pr for frst posto s µ 0 whrs th so posto ( ) os ot o th pr ostrt. Thrfor, µ 2. Aftr gttg ths vlus th rg quto ow lult s E 2 P C P ( μ + μ2), E P 35 X 35 Stp 2: Dtrto of E C : Tg guls fro th quto (2), th lso tw prts trt s follows: o r r o o o Lso l, r r r rf o o

4 I slr r, otvt λ vlut. I ths s, λ 0, for th C tr λ 2 0, for th f tr.., oth of th stsf th otvt ostrts. Fro quto (5), th rg ow os: E C C C ( λ + λ 2), E C 45 X 0 0 Stp 3: Dtrto of C s, C t, E P Th tl prours for trtg th possl ssl rtos, orr lsts th ur of rtos hgs r suss sto-v. O th ss of tht, th possl ssl rtos for prtl squ os: DS {Φ}, DS {} Th stlt of th sussl BA whh ots th prts lult s: S { BA } Aorg to th hrrh th ur of orr lst () s fou to for th prtl squ qusto. Puttg ths vlus quto (7), w gt C s (3) j I th ssl squ t s ssu tht urg th t of hlg thr s o hg of rtos s us th hgg of rtos ussrl rs th ssl ost. So for ths s NT 0 C t j ( 0) 0 Th vlu of C J s fou fro th sulto otos t s C J 45. As th squ s vl o, th quto of ssl ost s: J ρ s C s + ρ t C t H, J ( ) + (0.5 0) Th rg rlt to th ssl ost s: E J C J Х J 45 Х Tg th t fro ov thr stps th totl rg of prtl squ (, ( )) ( ( )) s: E Sq E P + E C + E J If oth th otvt s λ λ 2 stp 2 r fou to o, th th prtl squ ts tht thr s o oto or vr poor oto vll tw two prts. Ths s ot sutl for root ssl vrot. H, t s ot ssr to f th orr lsts s th prtl squ s ustl. V. APPLYING ACO TO ASG Th s opt of t olo lgorth s to solv otorl optto prols wth rsol out of t. Artfl ts trtvl tour through loop tht lus tour ostruto s th rtfl phroo trls th hurst forto. Th ths of lgorth s tht th goo tours r th postv f gv through th phroo upt th ts. Th shortr s th tour th or s th out of phroos post o th slt pth. Ths s tht th pth hv hghr prolt of g slt th susqut trtos of th lgorth. If th ssl ossts of ur of prts, th th sssl oprtos DOs r hvg 5 ur of os.. (, ), (, ), (, ), (, ) (, ). Th sssl oprto DO s ssg vlu f thr s trfr tht rto, othrws t s 0. Tht s f DO, t ot sssl fro th prout. A phroo τ j s us s th shr or of ll ts sultousl t osrs th rg tr (5 X 5) whh s to. Th phroo τ j s upt urg th prossg s prss s 5 X 5 tr s o of th (+)v Z rtos s rstrt for root ssl. Th quto for trfr tr sssl oprtos (+)v X, Y, Z rtos r gv ; II I 2 I 2I 2 I 2 DM I I I,( ) U j j 2 I2I2 I2 I 22I 22 I 22 I 2I 2 I 2 III I 2I 2I 2 I I I DO + I (0) U j j DO,( ) I () whr, I j s qul to f opot trfrs wth th opot j urg th ov log rto -s; othrws I j s qul to 0. Th tl sssl tr sssl oprto r lult s: DM DO, DO, DO, DO, DO, 0 DO, DO, DO, DO, DO, DO, DO, DO, DO, DO, DO DO, DO 0, DO, DO, DO DO, DO, DO, DO, whr, U s th Bool oprtor OR. Th rsult wll qul to 0 f ll th lts volv th oprto r 0. Ths s th lt sssl tht rto. If th DO s qul to, th lt ot sssl. I ths stu th tl fsl sssl oprtos r (, ) (, ). A. Approhg Solutos Root ssl s s of otorl optto prol. Slr to TSP gvg shortst pth wth u ost. I t sst, ts sultousl ul soluto of th ASG. Itll ts r put frst fsl DO. At h ostruto stp, t ppls prolst stt trsto rul, ll ro proportol rul, to whh o s to vst t. Th t vst s slt o th ss of th prolt rul gv ; (9)

5 P u C ( 0, α β [ τ ] [ η(, ] α [ τ ] [ η(, ] β ), f j C ( ) othrws whr, orrspog to th hurst tr η( (2) τ s th qult of phroo, s E Sq th prol pt hurst forto orrspog to th srh lgorth. C ( ) s th t lst grt DM ftr th opot of DO hs sssl. Th prtrsα β tr th rltv port of phroo vrsus hurst forto. Hr η s th vrs of th rg vlu tw th two prts. Aftr ll th ts hv ostrut thr tours, th phroo trls r upt. Th phroo vporto s gv τ ( ρ) τ, whr 0 ρ s th phroo vporto rt. Aftr vporto, ll ts post phroo o th rs th hv ross thr tour. Th phroo vporto s gv ; τ ( ρ) τ + Δτ (3) whr, s th ur of ts tht f th st squs through trtos Δτ ) s th out of phroo t posts o th rs t hs vst. It s prss th quto:, f squof t Δτ E sq (4) 0, othrws whr, Esq s th tour rg of th th t logg to tht tour. Durg th ostruto of squs, lol phroo uptg ourgs plorto of ltrtv solutos, whl glol phroo uptg ourgs plotto of th ost prosg solutos. B. Sulto Coto Th ssl ostrts, C P, C C C J quto (0) r ssg ro vlus fro 5 to 75 wth th rt of +5. Ths hv hos suh w tht th ovrg of th ojtv futo s or. It s ssu tht th slt ftors hv slr r othr tours. TABLE I. THE SIMULATION CONDITION FOR ENERGY CONSTANTS Erg Costts Cost Costts C J C P C C ρ t ρ t TABLE II. THE SIMULATION CONDITION FOR ANTS PARAMETERS Iflug Prtr of Phroo Trl Phroo Evporto Rt Bs Prt α β ρ Assl Drtos,, Th sulto prtrs r lst Tl I Tl II. Th rg ostrts Tl I r tr urg th sulto osrvg th ovrg t. VI. RESULTS AND CONCLUSIONS Th rsrh wor fouss o th ostrts ott rltoshps of th ssl prts, fs tho of grtg optl sssl squ s o th ov to strutur. Eprts to vrf th fftvss of th tho ofr tht th grt sssl squ s th optl or r optl soluto whh gvs out th l rg stsfg th to ostrts. Th rsult of th wor wth th spf pl prout s ot th for of (, (, (, (, (. Ths sstll rprsts th squ of sssl of th prts th rvrs of t gvs out th optl or r optl ssl squ. ( (, (, (, (,. Th prfor of th tho s fou to strogl fft th rg futo th otv rltoshps. Fro th rsult of th s stus, t s olu tht th of pproh sussfull grts root ssl squs wth u ssl ost. A ur of prouts wth vrg ur of opots ott rltoshp r tr to tst th fftvss of th tho th rsults ot though ths stus r qut ourgg. Howvr, u to ltto of sp ol th stu o o prout hs prst ths ppr. Th vr phlosoph of ACO s t or sutl for ts pplto. Th prol s suh s ot qust to thtl olg. Coprl thqus suh s GA lso ltrtv proposto for suh tp of prol. Th rsults ot pl suggst tht ACO s o of th ost sutl thqus for solvg prol of ths tp. REFERENCES [] T. D Fo, D. Wht Splf grto of ll hl ssl squs, IEEE Jourl of Root Autoto, vol. 3(6), pp , 987. [2] L. S. H. D Mllo, A. C. Srso, AND/OR rprstto of ssl pls, IEEE Trstos o Roots Autoto, vol. 6(2), pp , 990. [3] D. Y. Cho, H.S. Cho, Ifr o root ssl pr ostrts usg prt ott lvl grph, Root vol., pp , 993. [4] D. Y. Cho, C. K. Sh, H. S. Cho, Autot fr o stl root ssl squs s upo th vluto of s ssl oto stlt Root vol., pp , 993. [5] D. S. Hog, H. S. Cho, Optto of root ssl squs usg url- twor, IEEE Itrtol Cofr o Itllgt Roots Sst, pp , 993. [6] D. S. Hog, H.S. Cho, A url-twor-s oputtol sh for grtg opt root ssl squs, Elsvr, Jourl of Egrg Appltos of Artfl Itllg, vol. 8(2), pp , 995.

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