Determining robot s maximum dynamic load carrying capacity in point-to-point motion by applying limitation of joints torque

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1 Globl Journl o Scenc Reserches Avlble onlne gjsr.blue-.org GJSR Journl. Vol.. - June E-ISSN: -7 Deernng robo s u ync lo crryng ccy n on-o-on oon by lyng lon o jons orue H. R. She M. Bhr A. Kl E n A. M. She - Mechnc Engneerng Deren Arkbr nversy o echnology ehrn Irn - Mech. Eng. Arkbr nversy o echnology ehrn Irn - Mech. Eng. Arkbr nversy o echnology ehrn Irn - Deren o Mechncl Engneerng Shh Bhonr nversy o Kern Kern Irn Corresonng Auhor: A. M. She Receve: My Accee: June Publshe: June A B S R A C hs rcle seeks o eerne wo-lnk robo s u ync lo crryng ccy DCC n on-o-on oon by lyng orue ls on s jons. he eho resene here s bse on oen-loo ol conrol n uses nrec roch o erve oly conons. he Ponrygn's nu rncle PMP hs been use o obn he oly conons h les o wo-on bounry vlue roble PBVP. wo ses o erenl euons n one lgebrc euon re obne whch re solve by usng BVPC con n MAAB sowre. In hs er robo s DCC n on-o-on oon hs been eerne n wo wys. In he rs cse no orue l consrn hs been consere n he Hlonn uncon or he jons; whle n he secon cse hs consrn hs been ncorore no he Hlonn uncon n ers n he euons obne by usng PMP whch cuses hs consrn o show u n he se n cose euons. In boh cses sulons hve been erore. he sulon resuls nce h when orue l consrn s consere n he Hlonn uncon he ngulr osons n velocy o robo s jons re he se bu he orue o jons re eren. Keywors: Roboc r; Mu lo crryng ccy; Ou rjecory; orue ls. GJSR Journl All rghs reserve. INRODCION he ync lo crryng ccy o robo s ene s he u lo h roboc syse cn crry rove h he oors' orues o no ecee he suron ls. DCC s one o he crer or selecon o roboc syses. Generlly wo ehos rec n nrec es or solvng he roble o DCC -. Drec eho: hs eho s bse on he screzon o syse s ync vrbles se n conrol vrbles h les o reer ozon roble. hen lner ozon ehos nonlner ozon ehos evoluonry echnues or Sochsc echnues 6 re eloye o obn he ol vlues o he reers. he vrbles y be clsse s se vrbles conrol vrbles or boh 7. he lnerzng roceure n rec eho n s convergence s chllengng ssue eseclly when nonlner ers re lrge n lucung 7 8. In hs wy he obne nswer s n roe soluon whch s recly rele o he orer o olynol uncon. Wng e l. hve solve n ol conrol roble by usng Bslne uncons o clcule he u lo o e nulor. he n e o her reserch s o screzon he jons rjecores by usng Bslne uncons n hen eernng he reers hrough nonlner ozon so s o obn locl nu whch yels he consrns. A shorcong o hs eho s h ls he soluon o -orer olynol 9.

2 Glob. J. Sc. Res. : - Ierve ner Progrng IP s noher rec eho by whch rjecory ozon roble becoes lner rogrng roble. he rs orulon o hs eho or clculng he u lo crre by sle robo hs been rove by Wng n Rvn. he lnerzon oeron n he IP eho n s convergence owrs he ol h re cul sks eseclly when syse hs lrge egree o reeo or conns lrge n luonry nonlner ers. Korye n Ghrblu use he IP eho o eerne he DCC o roboc r wh elsc lnks n lso wh elsc jons or on-o-on oon n lso or oon long sece rjecory. hey orule he DCC roble s n ozon roble n hen eloye he IP eho rec eho o solve he roble. In her work he bounry conons re hrly sse n here s n los % error n he nl soluon. Inrec eho: hs s noher eho or obnng he ol rjecory o he u ylo. he nrec eho whch s bse on he PMP ws nlly use o solve ol conrol robles. hs eho ws eloye o solve he robles o obnng he nu e o oon long sece rjecores. In hs eho he oly conons re erce s se o erenl euons whch long wh he gven bounry conons or PBVP. hese ses o erenl euons re solve by ens o nuercl echnues such ulle shoong eho or Gren eho GM. By solvng hs roble n ec soluon cn be oun. hrough hs roch he ol rjecores or e n reunn robos cn be clcule by conserng eren objecve uncons such s he zon o he lo crryng nzon o he oveen e n nzon o orue ec. By lyng n nrec eho whch yele PBVP Korye n Nkoobn obne wo-lnk robo s DCC n on-o-on sk 6. In hs rcle rs wo-lnk robo s DCC s eerne by lyng orue consrn on he jons s ws revously obne by Korye n Nkoobn 6. hen by revsng he eho use n Re. 6 n conserng he lcon o ync orue he roble s resolve n hen he obne resuls re core o ech oher. So he res o he er s orgnze s ollows. In Secon he hecl oelng o he roble wll be escrbe. Secon s evoe o erc oly conons n he PBVP. In Secon rs wo-lnk robo s DCC s obne usng he eho le n Re. 6 n hen hs roble s resolve by ssung ync orue o ech jon. An nlly n Secon he conclusons ro he resen work re surze. Proble orulon: he yncl oel o robo s escrbe n he grngn orulon s: D C G where s he orue vecor o he jons D s he ner r C reresens he cenrel n Corols orces n G eresses he eecs o grvy. By usng he se vecor s: In he se sce or E. s eresse s: F where F s ene s: F F F N N D Z Z D [ C G ] hus he ync euons o oon n se-sce re obne. P g e

3 Glob. J. Sc. Res. : - Eressng he roble o ol conrol Suosng h roer jon orue vlue ess n he sce generl cos uncon s nze s ollows: he gol s o eerne so h he J o e ev h 6 where e ev n re ene s e e v where n 7 8 W W R re he nl n nl es n s he u lo h cn be crre by he roboc syse. W reresens unor n ervble uncon. n Wv re syerc osve se-ene rces. W W n R re syerc osve ene r. n re he esre vlues or he oson n ngulr velocy o jons resecvely. he cos uncon eerne by E. 6 hrough E. 8 s nze whn he enre rjecory o he robo. In E. 6 he rs n secon ers re rele o nzon o errors o robo s oson n velocy he nl on resecvely. E. 8 s rele o nzon o jon s ngulr oson velocy n orue hroughou he enre rjecory o he robo. 9 E. 9 enes he ngulr oson n ngulr velocy o ech jon he nl n nl on. he erssble boun o ech oor cn eress s ollows: In nrec eho by nroucng cose vecor he Hlonn uncon s eresse s: F where bse on he PMP here ess non-zero cose vecor or sece oly conons us be sse. ylo or whch he ollowng Η ψ ψ Η ψ ψ P g e

4 Glob. J. Sc. Res. : - P g e ψ Η h H h In hese relons he sybol nces he erels o he ses coses n conrols. he obne oly conon s rele o se n whch he se vrbles n he jons orue vlues re no boune. In orer o ly consrn on he conrol vrbles we shoul conser he ollowng: Η ψ Η ψ 6 n ll or 7 In E. 6 s erssble conrol vlue. By ssung o be cose vecor he oly conons re obne s E. 8 o E.. Z N 8 ψ N ψ ψ Z N ψ 9 ψ Z δ δ δ E. 8 n E. 9 eress he reure oly conons. he cheve soluon s cne ol soluon. Snce he uer n lower orue ls hve been ene by E. he orue vlue o ech oor cn be obne ro E.. Z R Z R Z R Z R Whch n hs relon hgh n low ls o orues vlue s eerne s ollows. K K K K

5 Glob. J. Sc. Res. : - K Whch n E. he reer o s s sn K g s sn n n re ene. Regrng h se vecors re e nl n nl eros hus he E. kes he or o E. n bounry conons re ene s ollows. Here he E. 8 o eress hree cegores o relons. So he E. 8-9 ene syse s ync oel. Ou conons re obne by E. n bounry conons by E.s n. In orer or solvng he wo-on bounry vlue roble erve lgorh s use. By nserng E. n E. 8 9 n o erenl euon s obne. n o bounry conon s obne ro E.. So wo-on bounry vlue roble s consruce. In hs lgorh he obne error vlue ro E. 7 shoul be lower hn vore vlue o. hereore: Where n hs relon s he vore vlue n he nl e n s he clcule se vecor vlue he nl on. he relve sgncnce o oson n velocy error vlue or ech o he jons s eerne usng W v n W rces. he eree vlue o oors ccy - n + s use or eernng orble u lo. So h he lo vlue ecees he u orble lo by he robo he oor o ech jon reures lyng orue greer hn he ere l whch cuses he jons orue o ecee her ls. Sulon In hs r we el wh he sulon o lnr wo-lnk ync r wh he seccons rove n ble. Fg. shows hs robo n he horzonl lne. ble. sulon reers 6 reer vlue engh o lnks M Mss Kg Moen o ner I I.66 Kg. M. no lo see s s.6 R s Acuor sll orue s s N. 6 P g e

6 Glob. J. Sc. Res. : - Fgure. Schec o robo n ol h 6 s s Regrng he reerence o 6 he nl oson o he en-eecor n he Z ln n he nl oson s.7 s. Moreover he velocy o nl en eecor n he begnnng n en o he rjecory s zero. Jons oson n velocy vlues re obne ro nverse knec soluon s ollows Se vrbles o n ro E. re ene s ollows. u u Where n re he rs n secon lnk ngles n show he lnks ngle velocy. u n u show he orue o he rs n secon lnk oors. sng E. 8 our euons rele o ync euons se sce or s erce s ollows C - G - - C - G C - G - - C - G 8 Where j C G : I j = s rele o wo-lnk robo whch s rove n he en secon. Now by enng enly rces s w w w w r R r W ; W ; 9 An by nserng E. 8 n E. 8 rge uncon: =. ru +r u w +w w w 7 P g e

7 Glob. J. Sc. Res. : - Fro E. Hlonn uncon s obne s ollows. H Where n ro E. 8 n re nsere. sng E. ervng ro Hlonn uncon he euons rele o us-ses re obne s ollows. H H H H ; ; ; Now usng E. n ervng or Hlonn uncon n rooron o conrol vlues he ollowng wo relons resul. H = ru u H = ru u - r - - r - By solvng hese euons he conrol vlues re obne n he cceble rnge. - - = r - - ; u = r - u hereore ou conrol rule or E. s wren s ollows. u u u u Whch he lons o conrol vlues o ech oor s clcule s ollows. k k k k ; ; k k k k 6 Where he k j j = vlues s clcule ro oor seccons rove n ble. Deernng ou rjecory o u lo or wo-lnk nulor usng hyohess o he rcle 6 In hs r or he gven wo-lnk nulor n bounry conons n reerence 6 u o lo crryng ccy n corresonng ou rjecory re clcule. Vlues o W W n R re selece n hs wy. So h W = W = n R= g - -. For hs nulor he gven bounry conons n nene rge uncon he obne u lo s / kg. In orer o show he lgorh erornce or clculng u lo he resuls o sulon or ve eren lo vlues hve been resene n grs. In Fg. he oson o he en eecor n Z lne or eren lo vlues s shown. Fgs. n b show he jons oson wh resec o e. Also Fgs. n b vrons o ngulr velocy wh resec o e wh lo crryng vrons re shown. 8 P g e

8 Glob. J. Sc. Res. : - Y. = Kg = Kg = Kg =. Kg =. Kg Fgure. En eecor rjecory n Y ln =kg - =kg. =kg =kg -. =kg =kg r =.kg =.kg r - =.kg =.kg sec sec Fgure. ngulr oson o jon Fgure b. ngulr oson o jon 6 o r/sec - =kg =kg =kg =.kg =.kg sec sec Fgure. Angulr veloces o jons Fgure b. Angulr veloces o jons o r/sec - =kg =kg =kg =.kg =.kg In Fgs. n b he gr o he rs n secon jon orues n rooron o e or eren lo vlues hs been shown. As you cn see wh he ncrese o he lo orue vlues ncrese oo n rogress owr orue ls unl hey rech o her vlues. As cn be seen he orues hve reche suron or / kg lo n lwys re on hgh n low ls. In hs suon he lo ecees / kg necesses he orues o ecee her ls whch s no ossble n when he conon s no ullle n he roble soluon o bounry vlue hs c s subsne. he obne =. kg s he u lo crryng ccy or he nene rge uncon. he obne resul s ully conssen wh reerence 6. 9 P g e

9 Glob. J. Sc. Res. : - =.kg =kg N. - =kg =kg =kg =.kg N. - =.kg =kg =kg =.kg sec sec Fgure. orues o jon Fgure b. orues o jon Deernng he ou rjecory o wo-lnk nulor by lyng ync orue ssuon In Sec. he u lo n he corresonng ou rjecory were obne. In Re. 6 orue s hgh n low ls bse on uncon o robo s jons velocy E. 6 s consere. Regrng h orue s velocy vres by jons velocy n Hlonn uncon s no erve n rooron o jons velocy. Here he oreenone roble s consere by lyng jons orue ls ynclly. sng E. 6 we rrve he ollowng relon. k k k k k k k k 7 Fro he bove wo neules he ollowng our neules re obne. k k k k k k k k 8 9 hus wo new se vrbles re ene s ollows 7. 6 Where H sgnes Hevse uncon. So Hlonn uncon s obne s ollows. H 66 P g e

10 Glob. J. Sc. Res. : - Now usng ervon ro Hlonn uncon n rooron o se n us-se vrbles wo ses o erenl euons re obne s ollows C - - G C - G - - C - - G - C - G - An us-se euons by ervng ro Hlonn uncon n rooron o se vrbles s obne s ollows. H H H ; H 6 6 H ; ; H In E. he ls wo senences eue zero. hus n 6 vlues wll be e. By ervng ro Hlonn uncon n rooron o conrol orue o E. s obne s ollows. H u H u In orer o eror he corson beween he resene eho n Re. 6 n he eho oere n hs er sulon s erore or he wo-lnk robo whch lo eul o kg s lce on s en eecor. In wh ollows he resuls o sulon re resene n Fgs. 6 o 7b.. - r. Re [6] -. Re [6] hs work hs work sec sec Fgure 6. Corson ngulr oson o jon Fgure 6b. Corson ngulr oson o jon r -. - P g e

11 Glob. J. Sc. Res. : - o r/sec Re [6] hs work Re [6] - hs work sec sec Fgure 7. Corson ngulr velocy o jon Fgure 7b. Corson ngulr velocy o jon o r/sec As s even ro he bove grs he oson n velocy o boh ses re slr n grs re on ech oher. Fgs. 8 n 8b show he orues o rs n secon jon. N Re [6] hs work N. 6 - Re [6] hs work sec sec Fgure 8. Corson orue o jon Fgure 8b. Corson orue o jon Frs jon orue vlue Fg. 8 n secon eho s greer hn he rs one. hs s becuse n he rs eho er he orue vlue s obne ro E. usng E.s -6 he orue vlue s le unl reches suron whle n he secon eho hgh n low ls o jons orue hve been e o Hlonn uncon s uneul consrns. In oher wors jons orues cn ncrese uner he nluence o ngulr velocy whle n he eho oere n Re. 6 hs ws no ossble. CONCSIONS In hs rcle u lo crryng ccy o wo-lnk robo hs been obne by oen chne ou conrol eho n eloyng Ponyerygen Mnu Prncle. Regrng oors lons oors orue lon consrn ws le o he roble. he reure conons o oly were obne once by Re. 6 n noher e by ync orue ssuon. In eho Re. 6 hs consrn s no e no Hlonn uncon whle usng new eho whch s eloye n hs rcle hs consrn hs been e no Hlonn uncon s n neuly n hereore s e no se n us-se euons. he obne erenl euons se long wh bounry conons ors wo-on bounry vlue euon se whch ws solve usng BVPC con o MAAB sowre. By erorng he sulon whch ws crre ou bou lnr wo-lnk robo n orer o eerne u lo crryng ccy n on- o- on oon s shown h jons oson n ngulr velocy n boh ses s he se. Aen he oon o nulor s erore n horzonl lne hereore grvy velocy s ken s zero. Iner r o wo- lnk nulor s eresse s ollows. P g e

12 Glob. J. Sc. Res. : - P g e cos cos D In whch I I c c c c c erl-corols orces o C cener n grvy orce o G re ene s ollows. ; G G G C C C In whch sn sn C cos cos cos g g G hus ync euons o wo- lnk robo n he horzon lne re obne. REFERENCES Bes J Survey o nuercl ehos or rjecory ozon J. o Gunce Con. Dyn. Vol. No Cheb ehhe HE H M n Hnch S.. Mnu cos rjecory lnnng or nusrl robos Euroen J. o Mechncs A/Sols Vol Dehl M Bock HG De H n Weber PB. 9. Fs Drec Mulle Shoong Algorhs or Ol Robo Conrol Fs oons n Boechncs n Robocs Vol... Jun. H M Cheb Hnch S n ehhe HE. 6. Ol oon lnner o oble nulors n generlze on-oon sk 9h IEEE In. Worksho on Avnce Moon Conrol Hull DG Converson o ol conrol robles no reer ozon robles J. o Gunce Conrol n Dyncs Vol. No Krk DE. 97. Ol conrol heory An Inroucon Prence-Hll Inc. Korye MH. 9. Nkoobn Forulon n Nuercl Soluon o Robo Mnulors n Pon-o-Pon Moon wh Mu o Crryng Ccy rnscon B: Mechncl Engneerng Shr nversy o echnology Vol. 6. P Korye MH n Nkoobn A. 9. Mu ylo h lnnng or reunn nulor usng nrec soluon o ol conrol roble In. J. Av. Mnu. ech. Vol Korye MH n Ghrblu H.. Mu llowble lo o oble nulor or wo gven en ons o en-eecor In. J. o AM Vol. No

13 Glob. J. Sc. Res. : - Korye MH n Grblu H.. Anlyss o wheele oble leble nulor ync oons wh u lo crryng cces Robocs n Auonoous Syses Vol. 8 No uus R.. Ierve ync rogrng Auoc Vol. 9 No Mele A. 97. Recen Avnces n Gren Algorhs or Ol Conrol Probles Journl o Ozon heory n Alcons Vol Dec. Ponrygn Bolnsk V Gkrelze R n Mchchenko E. 96. Mhecl heory o ol rocesses Iner-scence ublsher. Szyszkowsk W n Foouh R.. Irovng e-ol conrol neuvers o wo-lnk roboc nulors J. o Gunce Conrol n Dyncs Vol. No Wng n Rvn B Dync lo crryng ccy o echncl nulors-pr J. o Dync Sys. Mes. n Conrol Vol Wng CYE oszyk WK n Bobrow JE.. Pylo zon or oen chne nulor: Fnng oons or Pu 76 robo IEEE rnscons on Robocs n Auoon Vol. 7 No n-sheng GE n -Qun CH. 6. Ol oon lnnng or nonholonoc syses usng genec lgorh wh wvele roon Ale Mhecs n Couon Vol P g e