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Parallel manipulators. Part 2 : Theory. Singular configurations and Grassmann geometry

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(1)Parallel manipulators. Part 2 : Theory. Singular configurations and Grassmann geometry Jean-Pierre Merlet. To cite this version: Jean-Pierre Merlet. Parallel manipulators. Part 2 : Theory. Singular configurations and Grassmann geometry. [Research Report] RR-0791, INRIA. 1988, pp.69. �inria-00075760�. HAL Id: inria-00075760 https://hal.inria.fr/inria-00075760 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(89). Š~ L oum1K7IWijlm2† qtot—7k†Ivm/mvs7e mO jlkeFWNM-I oum/jSOVÉE‡†QWio‡IG si† qts U eSMHˆLofqtjNgPE eFW7ehk G o)s7er o m%G&j.E †RWio‡IGpsi† qts$s7eh o-e q jCUŽU j\W ehk2mv† qŒKPE e mv† jCW$gJjl†QW=m/jCWbm2sio©U j\ˆi†RE o • I2e‡M%3*1 šW‚:Se W7n mvsio#mvsi† kWnÀE †RWio#s7eCIfe qtj\UJU jCW e kvm2† qŒKPE ehm2†‡j\W g7jN†RWim&jCW m2sioˆ7e\I o • I2eFM4/Œš1G † m2s \j Wio jSO9m2sio gik2orqtorn=†RWP[ E †RWio‡I]5ƒ7jNkxoT™:eFU gPE o qtj\W7I † n=oukm2sioÀI oˆm N‚¾Æ:‚¬ÉN¿ŽE‡†QWio‡IbN¾‚ Æ!s7e o e qtjCUŽU j\W g7jN†RWimfjCW m2sio-U j\ˆi†RE o • 3 š9eSW7nºE‡†QWio‡IÐÆ=‚¬Ébe qtj\UJU jCW gJjl†QW=m9jCW m2si-o ˆ7\e I o • / š†] {¶EQE)mvsio¹E †RWio‡I-ˆJou†RWP[ n=†zI mv†QW7qtm†I † m † I m2sPK7IY†RU gJj¦I#Iv†RˆPE o m2s7ehm m2sik2ou‹o E‡Q† Wi‡o I s7er o e qtjCUŽU j\W †RWim2ouk†Ivorqtmv† jCW g7jN†QWim9oT™:qtougim-†wOmvsTo-ˆ7eCI o eSW7n!mvsi-o U Cj ˆiQ† E‡o ehk2o qtjl3g E Fe W7ehk~jlk †ÅO9m2sio ehk2mv† qŒKPE e mv† jCW?gLjl†QWim#j\W?mvsioψ7e\I o eFW7n mvsio U \j ˆiR† E o s7er No mvsi‹o IWSe U Ïo E‡jTquehm2†‡\j W • ˆPKim m2si† Ifqtj\WN—7[\KikWe mv† jCW queFWPWijlmµˆLoxkvoreNqtsior¯n ˆNM e k2orFe EU$Fe Wi†‡Pg K3E e mvjNŒk šª]&w W O eNqtm9m2sio qtjlgPE eSW7e k2†‡m%MjSO<mvsio ˆ7eCIvo eSW7n$m2sio©U \j ˆiR† E o- † I).e [NTo WioukWFe E quCe I o-Fj OI R† WP[\KPE ehk2†‡%m MŽGpsi† q s z† I oT™:qŒERK7n=orn †RW gikWeNqtmv† quSe E e giPg E‡† quehm2†‡\j W7I] ¨&sNK7ItÉ-E †RWio‡I)Sj O/'e ­Á§P§ ­ quFe WPWijlm9qt\j W7I mv† Lm Kim2oYe ² ehm gLTo W7qtR† E<Sj OE R† Wi\o ] . z.

(90) 

(91) . ‹  P/G)€   Œ,‚1  D=/‡ D‚#Œ"š1f€R'   w}W mvsi†zIfqueCIvo©O jCKiktE †RWio‡Ipehk2o qtjlgPE eSW7e k] ¨&sioµ—7kªI m g7jI#I †QˆPE‡o que\I o †zI j\ˆimWe †RWiorn‹G sioTWbm2sioˆ7e\I o eFW7nbm2sio©U j\ˆi†RE oYe kvo qtjNgPE eFZ. W7D eho©k]AU „˜e‡KiM m kvjljNm2m siehoum2k@o-qumve\siI oTo‡WbI qum2sieSW³o©UˆLoj\O ˆij\†RKPE oYW7n;e kv]j\L KPoˆW7mVn$K7m2IAsiIqo©KiE‡gi†QWigJoj¦Iv„"o ÆCm2„˜s7eÂJmVKPE‡†QW=Wimvo‡†QELIAm2Æ=si‚¬É=o-‚ÈÂ~g7jNe †RkvWio)m˜qt„"jNgP´HE eFE‡† W7o eh†QkW ] mvsiogPE eFiW o n=oŒ—7iW orn‰ˆPM¹=Æ ‚¬=É ‚È • ƒ †R[\iK kvo Tš†] w}W m2si† I queCI oŽE †RiW oJ:Æ ‚¾:É ‚{T ‚¾´ e kvo qtjNgPE eS7W e kT] &¨ sio e gig oT7W n=†Q™  †RW–vI orqtmv† jCW E‚ g7eFl[ o‹GÆ  ‚N[ †‡ o‡I mvsio qtj\7W n=†‡m2†‡j\W7I.pG si† q sÁUŽK7I m Lˆ o Mi M

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(93) ¦a À

(94) ¯A ¯

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(96). Š &si†zIqtj\WN—7[CKik ehm2† jCW †zI-YWij\GtW e\I 8¶KPWim I †QWP[CKPE ehk qtj\WN—7[CKik ehm2†‡j\W] D?oÏGp†REQE¶I ouo ¨ E e mvou{¶k W?m2s7jNehgJm oTm2W?si† giIpkvj\qtj\ˆPWNE‡oT—7U [CKi†zk I ehm2m2†‡j j\W‰n=ouMEm2† ouoTkLE nPU Ip†RWieFoE I j †ÅOxm2‚j$O e'jNk I gLeºorqt[N† †‡eF E oTWqtjCUn=o‡gPI E‡†RoT[\™ŽW?O jSjNO#k9e‰m2si­Áo.Š-§P§ E‡†Q­ Wio‡I• ]† ] \o ] mvsiog jI † mv† jCW jSOm2sio ehk2mv† qŒKPE e mv† jCW g7jN†RWimªIЈLou†RWP[À[N†‡ GoTW;š ‚ eSW7nºO jlkxe‹[N†‡ To W k Se WP[No Fj O mvsioŽE‡†QWPYIE‡oTWP[Nmvs7I.IqK7q s eÀI †QWP[CK3E e kxqtjCWN—7[\KikWe mv† jCW›E †‡o †RW mvsioÏG)jlk#YE†QWP[ ehk2ore Sj O mvsio ­Á§PL § ou­·m ] K7IU$eFYGo e‹K7I oŒOKPE k2oTU$e kLY‰OZjNk#m2sio-O jCEQE jCGp†RWP[ g7ehk2mŽ¿ †QW mvsiz† I#qu\e I Žo G -o U Fe M Wijlm2† qto#mvs7e m mvsioHŠJE †RWio‡IpqtkvjI#I)jCWio E‡†QWio • E †R[\Wio 3  3  †QW mvsToYSe ˆJjh o-To ™:Se U Pg E‡Fo šª]  D=/‡ D‚#Œ"š1]€ m'   š w}Wm2si† I que\I o¶Gpo˜G †QERETs7e ˜o O Cj Kik E‡Q† Wi‡o I g7\e I#I Q† WP[ mvsik2Cj KP[Ns m2sio I2eFU L o oum gLjlK7†QIYWimk2]oTU$e kLY—7kªI m#mvs7e m Fe U Cj WP[ Àe I oumxFj OtO Cj Ki³k E‡Q† Wi‡o IG o s7e No #m G&‹j E‡Q† Wi‡o I • I2Se M ¨~‚ šGp†‡m2s e qt\j UJU Cj WbQ† Wimvouªk I orqtm2†‡\j Wbg7jNR† Wim/ 1Q\j W m2sio ˆ7\e I o Fe W7nb#m G/j • IWSe M Ȃ D š/Cj W mvsioŽU Cj ˆiQ† E‡o • g jNQ† WGm 3 Œšª]‰¨)siÏo E‡Q† Wi‡o I.ˆLouR† WP[ n= † I m2R† W7qtm mvsTTo M quFe WPWijlm s7e No Se WNM!jNmvsiouk †RW=mvouªk I orqtm2†‡\j W g7jNR† Wim ]©D?oYsLer No Ce I†IqKPU orn!mvs7e m-ou Gou#k M(/ 1 e kvo n=† Jouk2To Witm O kv\j UHmvsio%3 Sg7e W7jNnR† WimmvPs] K7I)Se WPM‹I oum Fj OOZ\j Kitk E‡Q† Wi‡o IpSj OJe ­Á§P§ ­ quFe WPWijlmfs7er No e qtCj UŽU \j W R† W=mvouªk I orqtm2†‡\j W  D=/‡ D‚#Œ"š1c€ AB P   w W mvsiz† IfquCe Iv-e E‡Q† Wio Q† W qt\j UJU Cj W‰ˆPKitm EQMEQ† WP[ R† W n= † IvmvQ† W7o qtÂ'm9EPg R† WiE Se ‡o WiIf‡o qtI \j • W7ƒ I R†m2[\†‡#mKiKikvmvo o  %m ´ G)š†]-j ² ehm~g7To W7qtQ† E I-s7er GR† WP[ MM M _ c _ _ MM MM M } c c c M M M  M __ MM MM M _M ƒ ;+!  Š  9-/:I(*+= si!$  C+! Q‘.*JP ‘D  0 „oT˜™GKi † mI}w Wm G mvm2o@sisik2s7 † ouI e "o quE‡oCe Q† IWiI "o ou‡o ToIGpWYIL˜o K7R† WU-qtsxm2K7sim2I om<s7esLg7m@ehe Nkvm2msioTo n=m2M siou NkvqtoujNCj o9mvW7orqtIvn mvjl† PgmvLm jpKiE Se m2mvW7opsTe 4o µe k IL² E‡KPQ† ehWiˆ7m ‡o I IAg7oumQToGpW7Sj siO7qt† Q† mvqtE sis]k2¨)qtouCjo Ns W7K7E R†IvIWimv†‡omvLm siI Kiz† m2m2I s7opqtehµe\j mQWN² † —7m@eh[Cn=m KijEg7k ‡o To ehIWLm2Wi†‡qt\j jlR† E}W m ] † I"Wijlm m2Jj ˆJoYqt\j W7I † n=ouk2or;n ] ¨)sioYgik2\j ˆPE‡To UCz† IpmvŽj —7W7Àn Â'E R† Wi‡o I¶Gpsi† qts ehk2oY\j W mvsi-o I2Fe U o  kvTo D[\=KP/ER‡ K7DIT‚#] Œ"š1r€ A    Œ ˆ M ¨&• S siš&o mvsi s XoMGg7O Feou#kU ˆ7Q† \j ERMbE‡j JSj n OE Fj R† WiO-‡o \j I¶WiGpÀo siI † siq ous oume kvs7-o \e I I g7Feqm G&WPWij ork2¯n To [CˆPKPM EQK7I Fe W7n Se W7• Æn š/m2si o Fe O W7Fe  n U Q†GpERMbo Sj n=OTo E WiR† Wijl‡om2oI . . . . . . . . z.

(97) ¦a À

(98) ¯A ¯

(99) Z. =

(100). Š Ioug7 NoueFk#WPMŽWiorE‡†Qn³WioYˆ M jFO • Æ] šd eFoTW7U n¯oTWiUJj\WiˆioHouk G m2s7† mvehs m mvorsieNo qtsHjNm2E si†RWiouo‡ktI@E‡jS†QWiO o‡• IpSšQjSOs7e\• SI šªeF] W†RWim2ouk†Ivorqtmv† jCW gJjl†QW=mVG † mvs L oum©K I IqKigTg7j¦Ivo m2s7ehmXE †RWio‡I JˆLoTE‡j\WP[¦I-mvj • Sšª]-DDs7e mvou ouk-†zIfm2sio$qtj\WN—7[\KikWe mv† Cj W jSOm2sio'­›§P§ ­ E †RWioÏÆ †QWGm2ouk†I orq m†Ifm2sioŽE‡†QWio $eFW7n m2s K7I©UJK7IvmµˆJoTE‡j\WP[ mvj • Æšª] L Q† Wi‡o I ÉO jlk/†QWGm2m2siouo-k†Ivjlorm2qtsim†I¶ouk˜E †RE Wi†RWio-o=Æ$I †‡eFm&W7† n I&qŒm2Esioroue kvkoŒO m2jNs7kvehoHm4ˆ7E‡†QoTWiE jCo‡WPIp[lIp‚¬É=mv‚¬j´.ˆJ• SoTšªE‡j\]pWP{9[ gimvgPj ERMG†RWP[ eFW7m2sin'o-E †RI2WieSoU Æ:ox{‚ ÂTkv¾‚ orŠHe\ˆ7I jCTo WiE Cj Q† WPWP[[ mvj Gps7e mvou Nouk † I m2sio qtjCWN—7[\Kik ehm2†‡j\WbjSO<mvsTo­Á§P§ ­·]3¨)s K7I † m†zI †RU gJj¦I#IvR† ˆPE o-mvs7e ˜m  E‡†QWio‡I&jFO@m2sio.­Á§P§ ­ ˆ7oTE‡j\WP[I&m2j mvsi-o I2Fe U o#kvTo [\KPERK7I] . z. . . z. .

(101) 

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(103) ¦a À

(104) ¯A ¯

(105) Z. =

(106). Š. queSWPWijNm9qtkvjI#I)†RW mvsio-I2eFU o#mv†QU oxeSW jlm2siouk9gLjl†QWim†QW gPE eSWio l‚¬ÆN] š1eS€=W7‡ n mv"siNou‚#k2Œ,Œo 'O ƒ jl€"k2‡ o  m2š si † Ip}w qtW Cj WNm2si—7† [\IfKiquk e\ehIm2ox†‡\j mvW siz†k2I"ouoHWijlE‡†Qm Wio‡m2jI4UŽˆJoxK7I qtm9Cj W7qtj\I † W7n=I oumvk2† ormLKi;n m2] o eH² ehmfg7oTW7qt†QE jFOE †RWio‡I š1€=‡ "N‚#Œ,'ƒ €"‡   ƒ<† G©o E‡Q† Wi‡o I4UŽK7I m&g7Ce I#Ipm2sikv\j KP[ls %m G&Jj IqYGTo G E R† Wi‡o I • ƒ R† [\Kikvo ‡ Ššª] . . i eie e e ee e e i e i . . . . . . . . . . i  } f fi f   f  i f. fi  ff   f . . . ƒ ;1,!  Š ~ ?o Gp†QERELqtj\W7I † n=oˆk1G † mvsij\KimE‡jI#I jSO?[NoTWioukWeFE‡† mLMHE †RWio‡I)l‚¬Æ=‚¬É=‚ÈÂi‚¬´N] L oum K7IAWijNmv† qto"—7kªI m D mvs7e m mvsiouk2o †zI4Wij m%G&jJIqYGoTG E †RWio‡I"G si† qts queSW †RWim2oukªI orqtmm2sikvouo qtjNgPE eFW7ehktE †RWio‡IT] L oum1K7I qtjCW7I † n=ouk1E †RWio‡IpN‚¾Æ:‚¾É:‚{Â'<¿ †Å/O l‚¬=Æ ‚¬YÉ ehk2o-qtjlgPE eS7W e k1&G opqueFWPiW jlm—77W nŽIqGY oTG ‡E †QiW o‡I †RW=mvoukªI orqtm2†RWP[ N‚¾Æ:‚¾É]$wO-N‚¾Æ:‚¾É ehk2oJiW jNmxqtjlg3E eF7W ehkxeF7W n †wpO eÏE †RiW o †RiW m2ouk†vI orqtm©E †RiW o‡I TZÆSZÉ mvsioTW m2si† IE‡†QWioŽE‡† o †QW m2sio gPE eFWio'ÆFZÉ eSW7n g7e\I#I mvsik2jCKP[Ns m2sio ehk2mv† qŒKPE e mv† jCW gJjl†QW=m-jFO Gl‚¬† Æ mvs‰j\W Æ mvjCsiW o mvUsioHj\ˆiˆ7†Re\E oxI o jl•kXƒ<‡E †Q† [Co iK †Qk2W o mv siCo /š gPjNEk9eS†ziW &I o m2siZoÆ E †ReSiW 7W oxn qt†QkviW jmvI#ouI kª†QWPI or['qtm©7ˆ jNÉ mvs ehm~3 † m†~I eFqt7W j\nDUŽ3 U  j\] W gJjN†R=W m „ Tm-eJE †RiW oHpG si† qts s7eCfI e$qtj\UŽU j\W gJjN†R=W mtG † mvs ‡E †QiW o‡I¶i ‚¬´ ou† mvsioukЇE † oY†QW mvsioxgPE eSiW o K ÂiqueC‚¬´YIvofjNk)eHqtE‡†Qk2Wij¦ofI†I qto‡k2I j¦m2I†siI †Ro WP[-ehk2E‡mv†Q†WiqŒo KPE le ‚¬mvÆ:† ‚¾jCÉYbW eFgLW7jln'†QiW Âim‚¬´.qtj\U-UJK U I m1jCˆ7bW o m2qtj³jNTÂgP‚¾E x´ eFW7jCehbW k&m2m2sij.o©Âi7ˆ ‚¬´ e\I eFo\W7] n w}bW N‚¾mvÆxsTjNoµkZO jNm2j.kLU Âiou‚¬´ k E‡G&eS†Q7WiW o-no s7qt=Æe kv N‚¬jÉNoI#-¿ I I †Qm2siWPsij\"[ † GtI 3 Wb† I m2qŒs7eS‡E 7Wehormne kL†zEQ/I&!M †zIp†QUG mvsi†QgLERo#E j¦I††Q†RiW IU †Rm2ˆPg7ouE k†jo I I#oroTI qt†Q:™ mµˆPqtE‡´ÏouoxgiIm2m~oTj-[C†wO"—7U W7‡E oT†QniW iW oJm†ÆJI :Æ Iq•‚¾YG:Ƀ<oT‚{†R G [\iK e E k2†Rkvo$WTo o‡rqtI ‰jlqtš4gPk2E ˆPj¦eSiKI#7W Ivm~e†RWPkT†RW[¯] wÆ:mv7Wsi‚¾É:n=†zfI‚ÈouÂPorqu] n eCvI eo w}W m2sioHE e mvm2ouk9queCI oHE oumtK7I¶iW jlm2† qto#mvs7e m9m2sioukvo#† I¶iW jJE †RiW o#†RW mvsTo gPE eFiW o‡YI N‚¾Æ jNk©=Æ ‚¬É qtkvjI#I †QWP~[ m2sio-ehk2mv† qŒKPE e mv† jCW g7jN†RiW m qtjCUŽU j\W m2ji ‚¬´N])¨ sPK7IiW j.‡E †QiW o)†QW m2siopeFˆ jh o)gPE eSiW o queSW qtkvjI#Ipehm m2sio³IWeSU o m2†RU o N‚¾Æ:‚¾É:‚{ÂT‚¾´] &¨ sNK7pI mvsiouk2o †zpI jCWPEQM jCiW o-E †RTW o • jCiW oornNl[ o jF/O mvsio-U jCiˆ †Q‡E oFštpG si† q s queSW qtkvjI#fI e¯vI oum jSO´-E †RiW o=9I eF7W nbm2siouk2oŒZO jNkvox† m † pI †QU gLj¦I#vI †RˆPE oYm2j-—77W n‰ÆJIqGY oTG E †RiW o‡fI qtk2j¦I†I †RWP$[ eÏI oum jSO´ E‡†QWio‡I] z. .

(107) ¦a À

(108) ¯A ¯

(109) Z. =

(110) . GG.  G Æ Ji G Æ G i G JM JJ i  i d  Æ J M  G i G M MJ J  JJ   d d d d`  Š  MG G J ff iJ J ´   ` ` ` i i Li L J J G G  ffJ J    ` ` ´ L L LJ J  ffJ J   ` ` ` Ji L J ffJ J   ` ` ` f JJ ` f f J `  `  ` i `f  . XÆ . . . . i.  . . . . . . . . . . .

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(124)  . .  . ƒ (;+!  Š  IzDŽa('H ! IG@B ;L(, Š F F  w}W?m2si†z#I que\I o j\WioJE †RWio †zI n=oug7oW7n=oTWimµO k2jCUÃOZj\Kik E †RWio‡I] {µU j\WP[ mvš1si€=‡†zI O "j\NKi‚#ktŒ,'E‡†Qƒ €"Wi‡ o‡I¶=WiŒ jCWio#jSO/mvsioTU†QWimvoukªI orqtm m2sio k2oT[CKPEQK7IÐI g7eFWPWiorn¯ˆPM mvsioxm2sikvouo#jlm2siouk†I †RW e gikvjNg7oukfg7jN†RWim] L oum1K7I qtjCW7I † n=ouk4‡E †QiW o‡I N‚¾:Æ ‚¾:É ‚Èi ‚¬´N] ƒ7k2jCU m2si† I I oum4G&ofqtj\W7I † n=ouk˜É.IqYGoTG E‡†QWio‡IG si† qts I„˜g7KieFm/WPmvWisioro©nbE e†RiW kvo=oTI1[\xÆ KPERjlK7k1IÂ]˜¨&†QiW simvououk2kªo I or†zqtI&m j\jCWPiW ERo©MbE jC†RiW Wio-oHjFI Ooum m2si• o N‚¾eSÉ:Lˆ ‚¾´ jlš1 Go©siI ououm k2o eS7W m2sin omvsiE †RouWik2o‡oŒI"O jlU k2o©eSM-&G ˆJofoHqueFIqWPYNiWoTG.jNm ] —7W7n¯ÂJE †RiW o‡I"G si† qts¯U ououm mvsioxkvo.-¦iK † kvorn qtjC7W n=† mv† jCW] .

(125) 

(126)  . ‹  P/G)€   Œ,‚1 †RW‰I g7eNqto\] €=‡ D?o.= U ‚#Œ,e‡MƒK€"0%‡ K7I mv †wOM Š-jCWiE‡†Q-o Wio‡U Ifjlqtk2kvo jI#m2I)†RU mvsio oH8µI2KPeSWiU mTo«¬IÐE †RI Wi†RWPo [\| KPE ehk#qtjCWN—7[\Kik ehm2†‡j\W n=o‡I2qtk2†RˆLornÀO jlk y. ÉE‡l†Qn$Wio m%MG3 gL o 3 • ƒ  ]†R[\KikvoÆ\½šª] w}Wbm2si† I&que\I o©G&o©I ouo-m2s7ehm eSEQELm2sioI oT[CU oTWimªI g7e\I#I m2sikvj\KP[lsbm2sio ?D ops7e oI ouoTW'O jlk m2sio©ÂCˆ queCvI o-mvs7e m jCiW o©E †RiW oxqueSbW s7e No-e qtj\UJU j\W †QWimvoukªI orqtmv† jCW g7jN†RWim4G † mvs‹—7 Go©E‡†QWPYI&†wO eSW7n j\WPERM †wO"¿ Z mvsi†zI¶‡E †QiW ox† ~I eFW ornNl[ ojS/O mvsioHU jCiˆ †Q‡E o eF7W nÀ¯I oT[\U oTTW m†pI ehk2o qtjlgPE eFW7ehk • mvsio‡Ivo³IvoT[CZ U oT=WZ˜mªÉÏ&I I ehoTk2[Co-U IqoT7K W=qum†qtIpo‡I#ehI k2† No oFqtšªjl] gPE eSW7e k-eFW7n mvsiox†RW=m2ouk†I orqtm2†‡j\W E †RWio qtk2j¦I†I o‡Ipmvsio e kvm2† qŒKPE ehm2†‡j\W g7jN†RiW mªI j\W mvsTo7ˆ e\I oHG si† qWs e kvoHiW jNm9qtjCUŽU j\W m2j mvsio-I oT[\U oT=W mªI • I ouoYƒ †R[\iK kvo ˆ‰ š†] .

(127) ¦a À

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(131) }  . . .  .  .  . . -)(. . .  . . L.  . cc ^ ^ M J ^M J J cfc f i ^ _ M M J J i gg ff M Mc cf c M M J J _ J _ _ g g c c ci c c J J J _ } gg  cc cc J J g g c c c cJ J  g g c c J c cJ g g c Jc c c . LL. LL L. . -)/. . . . . . . . .  . . . ')/. . . . . . .   - 1. . . . . .  . ' 1. ƒ ;+!  Š k(:uQhi{S@B! *+  Š F F  F F –J F  F ! 8@*+#"! L oum©K I~k2oTU$e kLYU o U$eFM s7er NoHO j\KPW7n n=† kvorqtm#ERM mvs7e mxehmXE oreCI mHɹI oT[CU oTWGmªIÐUŽK7I m ˆ7mvsio oqtILjNYEgPoTE G eSW7gJe oTkT=W] m w}eSW7N[ n=jCouW¯ornºUŽE‡K7ouI mXm1K7Jˆ Ifo k2qtoTjlU gPoTE eSUŽWLˆiehoukk~] ¨&mvs7sie o mÐ[lOZoujNj\k-U e‹ouI mvgJk#M orqtjF† eSO@E m2siqtjCoU ­ÁgP§PE‡oT§ ™ ­Ã ME G† oTouEkvn3m2†I&qto‡m2siIpoTjFW O n=†‡k2orqtmLEQM m2s7ehm9e mt‡E ore\I m9m2sioukvo U-K7I mtˆ7oHÉ$qtjNgPE eFW7ehkµI oT[\U oTWGmªI] L oumK I/qtj\W7I † n=ouk/mvsio4—7k†I mque\I o\] wOG&o)k2jlm ehm2o)mvsioµU jCˆi†QE‡o-ehk2jCKPW7n mvsio9ornN[lo*3 3  KPE‡†QWGWPm2Y†QIE ]E †R?D iW o o"N—7‚¾7WÆ n e kveSo E[ eqtjN†RW gPE 8µeF7W KPehiW ktmT«¬pG I †‡I m2†RsWP[\=Š KP‚¬ÉE e kmvsiqto©jCWNE †R—7iW [\o iK 7ˆ kWeCe vImvor† jCn$WjC] W ypmvjFsiG †zI &G oro©nNN[ pG o †REQ†RJE=W qtm2ouj\k†W7I I or† qtn=mªouI k m2orsinNol[ Šo 3 3 e\IAˆLou†RWP[ mvsio †RWim2ouk†I orqtm2†‡j\W E †RWTo)jSO´I oT[\U oTWimªI • N‚¾Æ:‚¾É:‚{ÂT‚¾´ šª]L oumK7IAWijNmv† qto)m2s7ehm †R$W mvsLehm queCI o9mvsio©E †QiW o¶7ˆ e\I orn$jC$W ornNN[ o 3 3&†QGW m2ouk†vI orqtm†pI l‚¬:Æ ‚{T ‚¾:´ ‚¾YŠ eF7W n'$U e=M †RiW m2oukªI orqtm ÉqtjC†ÅWN<O —7m2[\siKio-k orehnNm2l[†‡j\o-W †zIm2s7iW eSjlW m&g7queCehIk o eSEQ‡E oTn;E ] mvjJÉ] wW IqKPUJU e kLM m2si†z&I queCI o©GM † oTE nPI TW j³U jlk2oÐI †QWP[\KPE e k ?D o-G †QERE qtjCW7I † n=ouk iW jSG mvsio que\I o jSO mvsik2ouo$q jlgPE eS7W e kHI oT[\U oT=W m†I • 2I eFMº=Æ ‚¬=É ‚ÈÂN~š eS7W n qtjCW7vI † n=ouk mvsiot‡E †QiW o • .Qš †RTW mvoukªI orqtmv†QWP[ 3 eF7W n/=] • . š †R=W mvoukªI orqtmªII oT[\U oT=W m†)I l‚¬=Æ ‚¬=É ‚Èi ‚¬´N] L oum1K7I k2jlm ehm2o©WijSG m2sio©U j\ˆi†RE o e kvj\KPW7n† m†I o\nN[lo 3 3 =] L oum"­ ˆ7o-m2sio-†RW=mvoukªI orqtm2†‡j\W g7jN†RiW m³G † mvs mvsio gPE eFiW oÀvI g7eSWPiW orn ˆPM›:Æ ‚¾:É ‚ÈÂP] w m $U eFM Lˆ o gJjI#I †RˆPE o m2s7e mJ­ Jˆ oT‡E j\WP[I mvj • .)š eF7W n mvsiouk2oŒO jlk2o • Hš †RiW m2oukªI orqtm†)I mvsioHŠ'I oT[\U oTiW mªI • ƒ<†Q[CiK k2o.:Æ Sšª]¶&¨ sioxehgigJoT7W n=†Q™ ƆRW‰I orqtmv† jCW ‰=‚:g7eF[loHÆl€:‚3I sijSGÐI mvs7e m mvj j\ˆimWe †RW mvsi†zI&qtj\WN—7[CKik ehm2†‡j\W Gpo©U-K7I m&s7eh o z. z. . z. . . z. .

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(136) }   M MM i M MM M . .  . . ¿. ƒ ;+!  • Æ6<  ' Æ6< 9. 9. . * 9. iG G   i b G  ^ ^ G  bb  ^ i ^ ^  bb ^ } ^ i i^ ^. . - (. .  . .  . . i. .  . . . . . . . . . . - ". . . . . . . . . m—b,GW q ;+!H@B*+/g";+!$9*+.  9 š • < ;  ' < ; š * 9 ; • ´6<  /*> M * 3 > M * ʽ z. . z. 5 9. ´< '. * . 9.  9 š ½ • € š • ‡½š 9. 5. š1° T€=± ‡ s7"e\NI ‚#IvŒ,si'jCƒ €"G¶‡ W  ŒˆPMºw}K7W I †QWPm2si[ † I eSWqueCI†RWioÏm#KiŠÏ†‡m2E‡†Q† Wi No‡oŽIXU I g7oum2eFsiWPjiWin?orn m2s7eÏeh[NmoTm2Wisiou† kWI-eFE Gpqt†Rj\EQE U ˆJg3o E‡oTm2™?si] o @ƒ uq wLe\¸BI o 8Іw¨¶O©Í G d o. kvjNmWe mvo#mvsioHU jCˆi†QE‡oxehk2jCKPW7n m2siox Nouk2mv† queSE eF™E†zI¶G † mvs eSW eFWP[CE o jNk '  Gps7ehm2ou Gouk†zI mvsio#g7j I †‡m2†‡\j W jFO/†‡mªI qtoTWim2oukT] ƒ<† †k I Xm G&Ho G †QEREI sijFG mvs7e m~†ÅO1G o qtj\W7I † n=ouk~j\WPERM kvjNmWe mv† jCW ehk2jCKPW7n e$ oukvm2† queFE eF™E†zI mvsio Fe ˆJjh N-o I Q† WP[CKPE ehk#qtjCW—7[CKik ehm2†‡j\W7I~e kvo m2sio j\WPERM!que\I o‡I¶Gpsioukvo-Gpoxs7eh o eÏ[loTWiouk eSE qtCj U Pg E‡To ™?] j$n=To U \j W7I m2kWe mvomvsiz† Ifkv=o IqKPE‡ªm IÐG&o qt\j W7I † n=ouk~m2sioxg oTW7qt†QE IpjFO˜E‡†QWio‡IXIvg7eSWPWiorn‰ˆNM Z ¨ Š=¬‚ ÆSZÉ Se W7ºn ÂCZ´]Ш)siTo WG&o qtCj W7Iv† n=ouk~R† W oreNqts jFOmvsio‡IvoYgJoTW7qt†QE Ipm2sio-E †RWio 1 Gpsi† q s †zI . z. z.

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