Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
335
Krzysztof Kozłowski (Ed.)
Robot Motion and Control Recent Developments With 200 Figures
Series Advisory Board
F. Allg¨ower · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis
Editor Professor Dr.-Ing. habil. Krzysztof Kozłowski Poznan University of Technology Institute of Control and Systems, Engineering ul. Piotrowo 3a 60-965 Pozna´n Poland
[email protected] British Library Cataloguing in Publication Data International workshop on Robot Motion and Control (4th : Puszczykowo, Poland : 2004) Robot motion and control : recent developments. (Lecture notes in control and information sciences ; 335) 1.Robots - Control systems - Congresses 2.Robots - Motion - Congresses I.Title II.Kozłowski, Krzysztof 629.8’92 ISBN-13 9781846284045 ISBN-10 184628404X Library of Congress Control Number: 2006923560 Lecture Notes in Control and Information Sciences ISSN 0170-8643 ISBN-10: 1-84628-404-X e-ISBN: 1-84628-405-8 Printed on acid-free paper ISBN-13: 978-1-84628-404-5 © Springer-Verlag London Limited 2006 MATLAB is the registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA. http://www.mathworks.com Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Data conversion by editor. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany (www.ptp-berlin.com) Cover-Design: design & production GmbH, Heidelberg Printed in Germany 987654321 Springer Science+Business Media springer.com
=VIJEGI
Ueb j^fk jlqfs^qflk fk mr_ifpefkd qefp `liib`qflk lc m^mbop fp ql mobpbkq qeb jlpq ob`bkq obpriqp `lk`bokfkd ol_lq jlqflk ^ka `lkqoli ql qeb ol_lqf`p `ljjrkfqv2 Utbkqv "sb lofdfk^i tlohp e^sb _bbk pbib`qba lrq lc 97 m^mbop mobpbkqba ^q qeb Glroqe Jkqbok^qflk^i Xlohpelm lk Sl_lq Nlqflk ^ka Dlkqoli -SlNlDl,48. tef`e t^p ebia fk Qrpw`wvhltl0 Qli^ka0 arofkd clro a^vp colj Krkb 5; ql 640 64482 Uefp Xlohpelm t^p qeb clroqe fk ^ pbofbp lc SlNlDl Xlohpelmp ebia pl c^o -qeb mobsflrp lkbp tbob ebia fk 5===0 64450 ^ka 6446.2 Jq fp ^k fkqbok^qflk^iiv ob`ldkfwba bsbkq0 qb`ekf`^iiv `l1pmlkploba _v qeb JFFF Sl_lqf`p ^ka Brqlj^qflk Tl`fbqv ^ka qeb Qlifpe Tb`qflk lc qeb JFFF Sl_lqf`p ^ka Brqlj^qflk Tl`fbqv De^mqbo2 Jqp 6448 bafqflk t^p ^ipl qb`ekf`^iiv prmmloqba _v qeb qebj^qf` kbqtloh DMBXBS -Difj_fkd ^ka X^ihfkd Sl_lqp.2 Ueb Xlohpelm t^p lod^kfwba _v qeb Jkpqfqrqb lc Dlkqoli ^ka Tvpqbjp Fkdfkbbofkd lc qeb Qlwk^%k Vkfsbopfqv lc Ub`eklildv fk Qli^ka2 Erofkd qeb Xlohpelm qeb jbj_bop lc qeb Jkqbok^qflk^i Qoldo^j Dljjfq1 qbb prddbpqba ql pbib`q qeb jlpq fkqbobpqfkd m^mbop mobpbkqba ^q SlNlDl,482 Ueb ^rqelop tbob ^phba ql mobm^ob buqbkaba sbopflkp lc qebfo m^mbop rpfkd lkiv rm ql 69* lc qeb lofdfk^i `lkqbkq mr_ifpeba fk qeb IebVXXW\aZf bY g[X ?bheg[ BagXeaTg\baT_ Nbe^f[bc ba JbUbg Fbg\ba TaW TYPXL_TNLWWd EPO`YOLY_ @LYT[`WL_Z] LWZYR L C]P^N]TMPO :PZXP_]TN CL_S F\ebf'_Tj @T_\V^\ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 56= lrqmrq cbba_^`h obdri^qflk0 ^pvjmqlqf` pq^_fifw^qflk lc ^ "uba1mlfkq ^ka0 jlob dbkbo^iiv0 lc ^ajfppf_ib klk1pq^qflk^ov qo^gb`qlofbp0 mo^`qf`^i pq^_fifw^qflk lc dbkbo^i qo^gb`qlofbp2 B klq^_ib bu`bmqflk fp qeb cTg[ Yb__bj\aZ mol_ibj0 tef`e tfii klq _b `lkpfaboba ebob _b`^rpb fq fp pifdeqiv af!bobkq fk k^qrob2 Uefp mol_ibj fp klkbqebibpp fjmloq^kq clo ^mmif`^qflkp0 ^ka tb obcbo qeb ob^abo ql b2d2 Z;0 6=\ clo obi^qba `lkqoli abpfdk obpriqp2 Ueb jbqelap obsfbtba fk qeb m^mbo `lsbo ^ i^odb o^kdb lc ^mmif`^qflkp> mlpfqflk `lkqoli lc sbef`ibp -b2d2 `^o1mi^qllkfkd0 [`orfpfkd jlab) `lkqoli.0 mlpfqflk ^ka lofbkq^qflk `lkqoli -b2d2 m^ohfkd0 pq^_fifw^qflk lc mob1mi^kkba obcbobk`b qo^gb`qlofbp0 qo^`hfkd lc jlsfkd q^odbqp.2 Glo `lkqolii^_ib ifkb^o K. Kozłowski (Ed.): Robot Motion and Control, LNCIS 335, pp. 3–23, 2006. © Springer-Verlag London Limited 2006
4
P. Morin and C. Samson
pvpqbjp0 ifkb^o pq^qb cbba_^`hp molsfab pfjmib0 bÆ`fbkq0 ^ka ol_rpq `lkqoli plirqflkp2 Cv `lkqo^pq0 clo klkelilkljf` pvpqbjp0 af!bobkq qvmbp lc cbba_^`h i^tp e^sb _bbk molmlpba0 b^`e lkb `^oovfkd fqp pmb`f"` ^as^kq^dbp ^ka ifjfq^qflkp2 Uefp afsbopfqv fp m^oqiv grpqf"ba _v pbsbo^i qeblobqf`^i obpriqp0 ob`^iiba croqebo fk qeb m^mbo0 tef`e ^``lrkq clo qeb afÆ`riqv3fjmlppf_fifqv lc abofsfkd cbba_^`h i^tp bkaltba tfqe ^ii qeb dlla molmboqfbp lc ifkb^o cbba_^`hp clo ifkb^o `lkqoli pvpqbjp2 Bp ^ `lkpbnrbk`b0 qeb `elf`b lc ^ `lkqoli ^mmol^`e clo ^ dfsbk ^mmif`^qflk fp ^ j^qqbo lc `ljmoljfpb0 abmbkafkd lk qeb pvpqbj `e^o^`qbofpqf`p ^ka qeb mbocloj^k`b obnrfobjbkqp2 Bq qefp mlfkq0 pfjri^qflkp `^k molsfab rpbcri `ljmibjbkq^ov drfabifkbp clo qeb `elf`b lc qeb `lkqoli i^t2 Erb ql pm^`b ifjfq^qflkp0 tb ^ob klq ^_ib ql fk`irab pfjri^qflk obpriqp fk qefp m^mbo0 _rq tb obcbo qeb fkqbobpqba ob^abo ql Z67\ tebob ^ abq^fiba pfjri^qflk pqrav clo ^ `^o ifhb1sbef`ib0 _^pba lk qeb `lkqoli i^tp ebob molmlpba0 fp dfsbk2 Xefib qeb mobpbkq m^mbo obsfbtp jlpq lc qeb `i^ppf`^i qo^gb`qlov qo^`hfkd mol_ibjp clo klkelilkljf` sbef`ibp0 fq fp _v kl jb^kp ^ `ljmibqb prosbv lc bufpqfkd `lkqoli jbqelap2 Cbpfabp m^mbo pfwb `lkpfabo^qflkp0 qelpb ebob afp`rppba ^ob mofj^ofiv _^pba lk lro ltk bumbofbk`b0 ^ka ob#b`q lro mobcbobk`bp2 Glo prosbv1ifhb bumlpfqflkp0 tb obcbo qeb ob^abo ql b2d2 Z;056\2 Ueb m^mbo,p p`lmb fp ^ipl ifjfqba ql [`i^ppf`^i) klkelilkljf` sbef`ibp0 clo tef`e qeb [e^oa) klkifkb^ofqfbp ^ofpb bu`irpfsbiv colj qeb hfkbj^qf`p2 Nlob dbkbo^i klkelilkljf` jb`e^kf`^i pvpqbjp0 fk qeb pbkpb lc b2d2 Z8\0 ^ob klq `lkpfaboba ebob2 Ueb m^mbo fp lod^kfwba ^p cliiltp2 Tbsbo^i jlabip ^ob fkqolar`ba fk Tb`qflk 5260 tfqe pljb lc qebfo molmboqfbp _bfkd ob`^iiba2 Tb`qflk 527 fp qeb `lob lc qeb m^mbo> qeb j^fk qo^gb`qlov qo^`hfkd mol_ibjp ^ob obsfbtba colj _lqe qeb ^mmif`^qflk ^ka `lkqoli abpfdk sfbtmlfkqp2 Jk m^oqf`ri^o0 ^as^kq^dbp ^ka ifjfq^qflkp fkebobkq ql pmb`f"` qvmbp lc cbba_^`h `lkqoliibop ^ob afp`rppba2 Gfk^iiv0 pljb `lk`irafkd obj^ohp ^ob molsfaba2
&$' <TIJQNSL TK DJMNHQJX" :NSJRFYNHX Jk qefp pb`qflk0 pljb ^pmb`qp lc qeb jlabifkd lc klkelilkljf` sbef`ibp ^ob ob`^iiba ^ka fiirpqo^qba fk qeb `^pb lc rkf`v`ib ^ka `^o1ifhb sbef`ibp2 Ueb molmboqfbp obsfbtba fk qefp pb`qflk ^mmiv -lo buqbka. ql jlpq tebbiba sbef`ibp rpba fk ob^i1ifcb ^mmif`^qflkp2
*(+(* >TYPXL_TN^ b(](_( LY
-525.
/&
tebob qeb eA ,p abklqb [cobb) s^of^_ibp0 qeb NA ,p ^ob pjllqe sb`qlo "biap -s2c2. loqeldlk^i ql qeb #B ,p0 ^ka ^ 6 afj-H.2 B pq^qb pm^`b obar`qflk ql ^ pr_j^kfclia F lc qeb jb`e^kf`^i `lk"dro^qflk pm^`b H -pbb b2d2 Z:\ clo jlob abq^fip. vfbiap ^ `lkqoli jlabi fk qeb p^jb cloj0 tfqe qeb cliiltfkd molmboqfbp tef`e tfii _b ^pprjba ql elia colj klt lk2
C]Z[P]_TP^3 C(* ^ 6 _ >@ afj-F .0 C(+ qeb NA ,p ^ob ifkb^oiv fkabmbkabkq ^q ^kv a 0 F 0 C(, qeb NA ,p p^qfpcv qeb Mfb Bidb_o^ S^kh Dlkafqflk lk F 0 f2b2 clo ^kv a 0 F 0 pm^kANA -a .5 ZNA 5 NB \-a .5 ZNA 5 ZNB 5 NC \\-a .5 4 4 4B @ <E 4 Sb`^ii qe^q Qolmboqv C(, bkprobp qe^q Tvpqbj -525. fp il`^iiv `lkqolii^_ib ^q
^kv mlfkq0 ^ka dil_^iiv `lkqolii^_ib fc F fp `lkkb`qba -pbb b2d2 Z690 Qolm2 7259\.2 B _^pf` bu^jmib fp qeb rkf`v`ib1ifhb ol_lq lc Gfd2 5250 telpb hfkbj^qf` jlabi tfqe obpmb`q ql qeb fkboqf^i co^jb 7 @ -45 Z 5 [ . fp dfsbk _v> %
3
a] @ e
`lp )
%
4
)
@ pfk A /
%
3 4
e
4
@4A
-526. 4 5 @ -h5 i5 ).% 0 e qeb pfdkba ilkdfqrafk^i sbil`fqv lc qeb sbef`ib,p _lav0 fqp ^kdri^o sbil`fqv2
tfqe a ^ka e
&
'
&
'
j0
y
0
j0
ϕ P1
θ
P0
P0
y
x
i0
0
θ
x
i0
@][. 0.0. Znk {toiirk 2r3 gtj igx 2x36roqk |knoirky B pb`lka bu^jmib fp qeb `^o1ifhb sbef`ib lc Gfd2 5252 B hfkbj^qf` jlabi clo qefp pvpqbj fp 4 3 3 4 `lp ) 4 B pfk ) C B 4C C C a] @ e B -527. B q^k 3 C / e B @ 4A A @ Q 5 4 &
'
6
P. Morin and C. Samson
tfqe a @ -h5 i5 )5 3.% 0 3 qeb pqbbofkd tebbi ^kdib0 ^ka Q qeb afpq^k`b _bqtbbk G ^ka G 2 Bk bnrfs^ibkq0 _rq pifdeqiv pfjmibo0 jlabi fp dfsbk _v %
&
3
a] @ e
&
3 4
4
`lp ) B pfk ) C B C
'
@
4
A/
e
4
B4C B C ' @4A
-528.
5
tfqe a @ -h5 i5 )5 ' .% ^ka ' >@ -q^k 3.7Q2 Olqb qe^q qeb `lkqoli s^of^_ib -528. af!bop colj qeb lkb fk -527. _v ^ c^`qlo -5 / q^k 3.7Q2 '
e
'
fk
*(+(+ >TYPXL_TN^ b(](_( L @ZaTYR 9]LXP B dbkbof` molmboqv lc sbef`ibp fp qeb fks^of^k`b -lo pvjjbqov. tfqe obpmb`q ql qeb Mfb dolrm lc ofdfa jlqflkp fk qeb mi^kb2 Nlob mob`fpbiv0 cliiltfkd Z8\0 qeb pq^qb pm^`b F `^k rpr^iiv _b ab`ljmlpba ^p ^ molar`q F @ @ # J 0 tebob @ @ < # = ) J> -6. fp ^ppl`f^qba tfqe qeb sbef`ib,p _lav `lk"dro^qflk -mlpfqflk ^ka lofbkq^qflk. fk qeb mi^kb0 ^ka J fp ^ppl`f^qba tfqe [fkqbok^i) pq^qb s^of^_ibp lc qeb sbef`ib2 Xfqe qefp ab`ljmlpfqflk0 lkb e^p '
&
) *
a @ Xc 5
X @ -h5 i5 ).% 0 @5 c 0 J
-529.
^ka Tvpqbj -525. `^k _b tofqqbk ^p
X] @ c] @
D F A
D F A
/&
eA NA? -X5 c.5
-52:.
eA NAJ -c.4
/&
Glo bu^jmib0 J @ 5 clo qeb rkf`v`ib1ifhb sbef`ib tebob^p0 fk qeb `^pb lc qeb `^o1ifhb sbef`ib0 J @ = tfqe c @ 3 clo Tvpqbj -527.0 ^ka J @ < tfqe c @ ' clo Tvpqbj -528.2 @ fp bkaltba tfqe qeb dolrm molar`q &
3)
XX &
'
>@ @
h i
*
&
&
/ I-) . &
)
&
/)
)
h i
*4
'
'
A
-52;.
'
tfqe XA @ -hA 5 iA 5 )A .% 0 @ -Z @ 55 6.0 ^ka I-). qeb olq^qflk j^qofu lc ^kdib )2 Glo ^kv "uba c qeb [sb`qlo "biap) NA? -X5 c. ^ob ibcq1fks^of^kq t2o2q2 qefp dolrm molar`q0 f2b2 clo ^kv "uba X 0 @ ^ka ^kv plirqflk d 2!+ -X -d.5 c-d.. ql -52:.0 d 2!+ -X X -d.5 c-d.. fp ^ipl ^ plirqflk ql -52:.0 ^ppl`f^qba tfqe qeb p^jb `lkqoli fkmrq2 Cv rpfkd qefp fks^of^k`b molmboqv0 fq fp pfjmib ql pelt qe^q qeb hfkbj^qf`p t2o2q2 ^ jlsfkd co^jb 7I @ -4I 5 ZI 5 [I . -pbb Gfd2 526. fp dfsbk _v %
%
: Zxgpkizux Zxgiqotm lux Tutnurutusoi \knoirky
D F
X] = @
A
D F
eA NA? -X= 5 c. ! I'-X= 5 XI .X] I 5
-52@ XI! X @ @I-!)I . i ! iI ) ! )I
*4
3
)
!i= ' -X= 5 XI . >@ @I-!)I . h= A ^ka I
&
4
j0
jr ye
θe
5
*4 A
4
-52=.
ir θr
yr
0r
θ
y
xe 0
x
xr
i0
@][. 0.1. Qotksgzoiy }7x7z7 g su|otm lxgsk Olqb qe^q Tvpqbj -52 qo^gb`qlofbp obar`ba ql "uba mlfkqp -eI @ 4.0 `loobpmlkafkd ql m^ohfkd1ifhb ^mmif`^qflkp0 ^ka qo^gb`qlofbp clo tef`e eI albp klq `lksbodb ql wbol2 72 Qo^`qf`^i pq^_fifw^qflk2 Ueb fab^ fp ql obi^u qeb ^pvjmqlqf` pq^_fifw^qflk l_1 gb`qfsb2 Glo j^kv ^mmif`^qflkp0 mo^`qf`^i pq^_fifw^qflk vfbiafkd riqfj^qbiv _lrkaba ^ka pj^ii qo^`hfkd boolop fp prÆ`fbkq2 Olq lkiv qeb qeblobqf`^i l_pqor`qflk obsb^iba _v Ueblobj 5 kl ilkdbo eliap fk qefp `^pb0 _rq ^ipl ^kv qo^gb`qlov0 klq kb`bpp^ofiv ^ajfppf_ib0 `^k _b pq^_fifwba fk qefp t^v2 Xb klt obsfbt fk jlob abq^fip qebpb af!bobkq qo^gb`qlov qo^`hfkd mol_ibjp ^ka pqo^qbdfbp2
*(,(* B`_[`_ 9PPOMLNV 6ZY_]ZW Bp mlfkqba lrq fk Tb`qflk 526280 qeb ifkb^ofw^qflk lc ^ `lkqoli pvpqbj -525. p^qfpcvfkd Qolmboqv C(*0 ^q ^kv bnrfif_ofrj -a5 e. @ -a 5 4.0 fp kbfqebo `lkqolii^_ib klo ^pvjmqlqf`^iiv pq^_fifw^_ib2 Uefp ^``lrkqp clo qeb afÆ`riqv lc `lkqoliifkd pr`e ^ pvpqbj2 Jk pljb ^mmif`^qflkp0 eltbsbo0 fq fp klq kb`bpp^ov ql `lkqoli qeb crii pq^qb a 0 _rq lkiv ^ sb`qlo crk`qflk Y >@ -Y 5 4 4 4 5 YG .% lc a ^ka0 mlppf_iv0 d2 Jk m^oqf`ri^o0 fc ` ^0 qebk qeb j^mmfkd %
*
D 9Y 9Y F e N -a . @ N -a .e a ]@ e 2!+ 9Y A A 9a 9a 9a A
&
-525=.
/&
tfqe N -a . @ -N -a . " " " ND -a ..0 j^v _b lkql0 ^p ^ obpriq lc qeb crii o^khbakbpp lc qeb j^qofu 3@ 3H N -a.2 Jc qefp fp qeb `^pb0 Y `^k _b b^pfiv `lkqoliiba sf^ fqp qfjb1abofs^qfsb2 Ul fiirpqo^qb qefp c^`q0 `lkpfabo qeb `^o1mi^qllkfkd mol_ibj0 &
*
G lk} 2}kgq3 {toluxsoz gyy{svzouty }7x7z7 {f gxk gryu xkw{oxkj ot znk jk"tozout ul gysvzuzoi yzghoroz 2ykk a:@ N -W -#.. " " " ND -W -#.. 9# 9# &
&
G
-526=.
Uo^kpsbopb crk`qflkp ^iilt ql rpb #] 5 4 4 4 5 #] G ^p ^aafqflk^i -sfoqr^i. `lkqoli fkmrqp2 Uefp fp obi^qba ql qeb fab^ lc `lkqoliiba lp`fii^qlo fk Z54\2 Glo qeb `^o jlabi -528.0 qeb rpbcrikbpp lc qebpb `ljmibjbkq^ov fkmrqp fp bumif`fqba fk qeb cliiltfkd ibjj^2 &
: Zxgpkizux Zxgiqotm lux Tutnurutusoi \knoirky
:B
?PXXL *( >G !+ < #= #< WXabgX T f`bbg[ YhaVg\ba. =X"aX g[X RaX\Z[Ube( fgTgX j Ul ) *4 3) * h= ! I-)= ! W . W B i= W C C -5274. j >@ B A @ )= ! W '
&
&
(
'
(
' !W TaW g[X ThZ`XagXW Vbageb_ iXVgbe e' >@ -e 5 e 5 !#] 5 4 4 4 4 4 4 5 !#] G .% . L[Xa, j] @ :-j5 W . -A -#.'e / ; -j .G -a= 5 d. / e < -j .. -5275. j\g[ A -#. g[X `Tge\k fcXV\"XW \a eX_Tg\ba -526=., / ' ' ( (0 ) * I -!j . 4 I -j . I-j . !>> 4 :-j5 W . >@ 5 ; -j . >@ 4 B 4 B TaW < -j . >@ -45 45 j 5 4.% . Jc W fk -5274. fp ^ qo^kpsbopb crk`qflk0 qebk :-j5 W .A -#. fp ^ crii1o^kh j^qofu )
&
'
&
&
(
' &
(
(
'
'
)
^ka fq fp pfjmib ql rpb -5275. fk loabo ql abofsb ^ `lkqoli tef`e ^pvjmqlqf`^iiv pq^_fifwbp j @ 42 Tr`e ^ `lkqoli i^t fp mlfkqba lrq _v qeb cliiltfkd molmlpfqflk2
C]Z[Z^T_TZY /( N\g[ g[X abgTg\baf bY EX``T 0, Tffh`X g[Tg W \f T geTafiXefX YhaVg\ba Ybe g[X VTe `bWX_ -528., TaW Vbaf\WXe g[X WlaT`\V YXXWUTV^ _Tj ! " e' >@ A 6 -#. !; -j .G -a= 5 d. ! -:-j5 W ..! P -j . -5276. &
j\g[ A 6 -#. T e\Z[g-\aiXefX bY
A -#. TaW P -j . >@ -\ j 5 \ j 5 6\ q^k-j 76.5 \ h .% -\A 8 4.4 L[Xa, Ybe Tal eXYXeXaVX geT]XVgbel XI fhV[ g[Tg X] I \f UbhaWXW, 0. j @ 4 \f XkcbaXag\T__l fgTU_X Ybe g[X Vbageb__XW flfgX` -5275., 1. Tal geT]XVgbel a= bY g[X Vbageb__XW geTV^\aZ Xeebe `bWX_ -5254. VbaiXeZXf gb W ->G . TaW, j\g[ Ta TWXdhTgX V[b\VX bY #-4., g[\f fXg \f XkcbaXag\T__l &
&
'
'
(
(
fgTU_X.
)
)
Uefp obpriq peltp qe^q fq fp mlppf_ib ql pq^_fifwb qeb qo^`hfkd boolo a= ql qeb pbq W ->G . clo Tal obcbobk`b qo^gb`qlov XI 2 Tfk`b >G fp `ljm^`q ^ka W fp pjllqe0 W ->G . fp _lrkaba2 Jk m^oqf`ri^o0 fc W ->G . fp `lkq^fkba fk ^ pj^ii kbfde_loella lc qeb lofdfk0 qebk qeb qo^`hfkd boolo a= fp riqfj^qbiv pj^ii0 j[TgXiXe qeb qo^gb`qlov XI 2 Uefp ib^sbp rp tfqe qeb mol_ibj lc abqbojfkfkd qo^kpsbopb crk`qflkp2 Jk Z66\0 ^ dbkbo^i clojri^ fp molmlpba clo aofcqibpp pvpqbjp2 B c^jfiv lc qo^kpsbopb crk`qflkp0 `ljmrqba fk m^oq colj qefp clojri^0 fp mlfkqba lrq _bilt2 Uebpb crk`qflkp ^ob ab"kba lk > 0 f2b2 qebv abmbka lk qtl s^of^_ibp # 5 # 2 Jq fp `ib^o qe^q clo qeb `^o jlabi0 qtl fp qeb pj^iibpq krj_bo lc s^of^_ibp clo tef`e -526]oZg\^] Oh[hmb\l )F@>O*5 ;99k\abo^ _hk OZmbhgZe J^\aZgb\l Zg] >gZerlbl5 |ur7 :
