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  • Introduction to Robotics: Mechanics and ControlJohn J. Craig

    Preface to the Persian EditionIt was with great pleasure that I learned that this text book would betranslated into the Persian la.nguage. The book is now available in En-glish, Japanese, Korean, Polish, Italian, Chinese, and Persian. I hopethat readers of the Persian la,nguage find my efforts in writing this bookwere worthwhile. My thanks to Prof. Ali Meghdari and his students forthis translation.

    John J. CraigAugust, 1992California, USA

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    plaI .J

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    t3l

    [4]

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    17l

    t8l

    tel

    [10] Y. Koren ,, Robotics for Engineers, McGraw I{ill, lg8b.

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    \A

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    o o \7o o o

    1"](rr-Y)

    /'/

    //

    APz / #p,

    /

    .c-l o.r.l .l+ "& 4 Jtiil s bb1o \ AP,,ltt \ \-YJKJ

  • ,-{ oM;,rtr!.^y tLi?ai' Fo

    oPr =

    t.rlr.f

    (r0- Y )

    V-i,c t #'l orLt-l jJ-t !)',;', * T il

    oPr=ToP, (rP-Y)

    Y-Y Jti s; i;.i5 15)r' k:l JE ,,r-l a5rjl.' a:--il: o.e;.:_1 l:

    tAJiri i..-)' 0- Y.5'- Jr i!r, sScJL P;;5,j'o t:b*'+,jr-a Jti:l ;rLrr,r-.+U, it5tic*J csj!.Jbl:.,!:r"5tJt ,fJkr, !.F1,# j-o f ,,Li;;lJU:t )^Li

    .Jl,r,.5 cJL ,h -,: 4T lt-: ,1-ilst iy .r!:: r i.. Jti:l;.:L .-JL .e: ,"

    .f-l,r/ j* k-rt-rt.33 u:"-l cslt Ct' ,stlil il';. t ,:-1lr,r*-tJ 4lrl t; gi a-,r(^" ,l-+ Cr-l .s!,

    ia; {1} * c,*; 1., {B } *aL6; fT .c*l *51r\t .1,- -n, -; ,,.},-rn Cr-l . \t {B} ;I-l csot.tn 15b-|, .5.u--.n 151,&- qita-/l:J,: AR rs\r oi-5^;.tSu.

    .src/ iLJ b {B} ;L--... o6!-, 1rr, iti t o P"o^c : ;cJ.u. j*.$)./ cd)de A P ,s:t L " P zF fT .c*l *,K .i- ,,-!-ri i/-l .Y

    .$rc,/ dryr4 b o Pr r tSrt k o Pt css.t T .c*l"\,-ri Ft^r .5.,-,rl rJ i/-l .f

    .i- ;r)lrq slS, -71:11 .i- .sti. { rr:a (.-!-{ D 5 r{.73a1\3 r ,st".-;b ,tr. Lt-l jlUU ,.!-+ jl s u z= ^.i il c,,-,u f fu c ,,v:tr\,.1 r;l .c.i-1 .r:.rp jS^,, 6rr3;a--c,JL.# J-+ ;;l.e ai-l ,+t t,,;r:.,*.r .s-.'J.e,:lc*"| .r3,1^t t- -:t* fttr^- al, (i* Jbl t-) \r ;!r, r+F Jl ,u)rq ,,-Jd t t, L ij ;rlrr'r Jbl ,,5

    .tj '^lrlr

    orlr- .l "1,-+;;b

    jl UG ,C;!

    [;']

    :['ii::i]

    ;:Li;, )"6 6r-; Ll ,c*lI

  • f \ ,.F ,l* e p\, crga

    ,s

  • tT arV rolj,l.# r^l'rL I c,t'-l-,, e-tt il,il-r"r''ff: r,p+, "i-lr ti rt:e+ ,{C} : {B} c\xtV J fh c$-; *t b

    I AnEn 'i1,R"Pcone+nP"o^"]Aq_l I I tr\-y)sr l----- -i---- -----lL'oo; \J

    .,Lrr Jrf isul:6li. t ,r-l) ;.-r31 -b .tt,.;-l ,,rl', {A} *X.tkA q &*; d !, { B} vXtV,{A} 1r-:*; {a},;r-; b FJ iyl:b.1,-+ cr-l dU "16.(c-l AT rlri..,::-;lr..;*l f x f #-j,-+ ;:rlr '.-l^. y6u-l yll ,sb..:L ,'jl, .4i .:*r" "of *-;l;,rL ,-it :l ali,r,iT i:,,lr i-t^, ,slt S c*l 61143r, J-+ ,r-;L y-l-2h;L

    .:J,rU- .l Jt-,' ;,c*){, n Pnone : AR:l --i + b "

    Pnonc : "nn-r-rL +tt ,3f '+A ,shF-ljd,il,,:),-sl.f-,--rL 3.r1,:.r) ,-E rjf JtL a +i L: l+l .f-tT

    "on - An' (rY-Y)

    CC J:ai i,.+ {B} *-njy--;lb^Pnone ,(\Y-Y) iJrL;l ,:1.'-...1 t ,.'-u (n P"o*r) = "nn

    o Pnonc * B P,aone (rr-r)

    *\-b +! (rr-r) .J.1t * s!3n

    " Poo^o = -|R n Pnonc = -f nr ^ Paonc (rr-Y)

    L:j tj a.t-re+b uo?. ijd,(FF-Y) r (f Y-Y) ,st^.h!r;l orl.L-l I

    I A*' i,-ARr a Pnoncf'or-[.t,;i t ]

    uoT - aT-, 'L u;.,lir)L; "J t 6 +S +i

    .c*l ;^^ ,1,-+ .,r iyl:'*6 cla ,t:-t-rl=* r ,{ ;l1tt (f O-f 1 iJrL

    ( ro- r)

  • f f ,-F ,!-*,> ,;\1c,!r.r

    0-Y Jb ro.2.r l'o i;l.lh {A} v*tl.36 r c*; e5.u:r, itJ I {B} v*jq \f-Y Jfu'i/-!rfl ...:*l d:jt- Jtiil r-b f tf ;J+tn, *o.-st^l-! 1t,ct1$;-,b:t Z t.tn Jf. .+t=: , Nf .,:*l ,4^* ff

    t-.-J.4 r-o'-;'; &+ {B} =*.,V

    Artl _Br

    Brn _ar

    l' tA??I"rd''

    L"''"

    -r t\?l-'70tA

    o /o

    \

    " th?7

    - o r0o o

    o

    'o oo

    o

    -"rA"o olooo fr"l.1A?7 ofooo fr"lo

    ,ooo o trl .1"

    I\J

    (ff-Y)

    o/0o o o lo o o

    "1[?7 olooo

    o lo o o \7o o o

    t-ig a.*,; ,(f0-T) i.t!;l ,:ti- .l L

    (rv-Y)

    .{A} +.-,.'{B} *nr[ \f-YJSI

  • ,.F rl|'^e; at+l:.rr 2\\.ai f f

    ,L* 6lo+lstr" V- Y;l ':tl .l \ ;rli,g b {D} *akt .,i -/, .tsrc/ jts !2 ,p: \f-r Jfu,

    J12,.at' C-a.J .rJ it" drfk bg,o Jr { ,t+!-+ V;rkL

    'oT = ir Ar ( r^- r)

    (rr-Y)Yr - flr B"r $rr,T .up c*)de tj ut_tpt ,!.t.j iJlt , ,'oT jl \): j;, ,l_l ,Jrl:-/lri ,srt- !

    Xr Ar - Yr 3r ""r (0. -Y)

    .J:r)s-. J=.5;; dl- .S: dtd+ ;l 6l^se.-- \f -YJKJ

  • f 0 .L+ silrdrt,

    ,f,-.r; iJrL n 5 )2p..Jr-+ t d.ffu ,t$-rn '.*l-..,s|,.;tl-irl,.!r., 6trrJrL;l.1r ,.11,,5L, ""f n $+ '.:S 6 uJ[- ,slt b (0"-Y) tL.lr.tj o:l.r-*"1 ,r;l 4-3.bli.-r. b J:*.,!-ri ,r, .f:-rlr J*j,-+ .S,- r .J,-+ iJrt^. .f- ?-, I .r--&, -b

    ,.25T c-r., ,-lLT*

    3r - Yr-' ir Ar "or-' (0 \-Y).c*l o.ui orlr;ti; u-&-t rLi c--r";l ,\0-Y rfS-i.1r

    4 i+, f ;tQ il-:t ,.$\J ,t12eL haL6; ,t+Ifu i15.r) d +{ -.ysrc/ it:; ;t! c4- ..slo.u .rlr ,ril; ,J:1u, ,kt-&-t -. r,r[43'i+. + ! -. .1>,kta, c*.i {D} =nj{-,\f-Y J"fu, t>zsFdi-t; u-S-t a, l:*i -.f,rl1a llJSrs

    ."[u il:L 51;l 6li; \ 0- Y JSJ

  • J-f; {D} +s*; {A} ,-,rt,. ,\O-Y Jfu,r, Oj:-j:jTt r'^'tc*l ,jK ,s-Je ,V pr r;.:", t+t5s 3u.13;^l.r, hr,.rt'6,; 4; ,sb .tf ,gc1- t .jJH. ,k.slrlr,rt! ^it+ .fisT c-)d,r lr rlQ,,sLd,.+ ,-*k;;l 0S- J: j;r \0-r Jfu, l.{g t*l^. lr;rT ;.,t,1: l.ril ,.tu:L ,ratr, ,rtd.rr

    ;l .ulc.,-,Lc {C} rrrt-rkt

    Zr -ir ir-' 3rUrn Urn BrnCl=BACI

    (0r-Y)

    )(or-Y)

    C +A1.,'1TL tli,-r,(or-Y)r (OY-r) t5bAf ;rl:;!ri 62L.-, tri t+:,-l-r,

    xr = Hr 8r Br-, 2r ( or- Y)

    9-l Jt:t tu;gL ;b-i3Jl t eb,{"} .r,rkt Ft t, .5,\P-Y Ji.fu, tt FT ,}-+; +S *tI {S} *rtV,3T

    "1,-+ ro:r.r.r .r.r! frL ,c*l {B} r;t i_t . c^-.i ,r5jtS,

    tU -* u, 3 rf* it^(. J:5.,, J"ci {B} 1-t, d, c----i ,.:*l .}.oi $ -* + I6a;L ,*,: ds r*; Ex ui*+r utS, ,TT .t-rrf Lra;^* 8f .,!-+ '&s+; r-11:

    .+5 +-l-., b ,fV ,r5jl5,ll-i,l ,lu-;; Jr, t, c-l .ul; r) .rti u-l-r: o-rJ.,kil qlStV ,sA6r.\

    tj +L+:-j J.t.t-Ar cdr -.-,f,rh { + l, 6r, *nr[

    TT = +T-' Bsf ZT (oo- Y)

    Ls;+ u,Qti 6tU.tr.&r;G" A-YJi-t,-u c-l,: 66t**.,+-lorlr;LJ r x l;!3r ,r-tl "b:+,t+- f ,lr*.F jj5ub! tU t4l ;-;l.ll 3 ,\,4 p^;: r:t:, t+i .st+r* 15.J c*l ;i jlrr utr*,-rLc*l ;T (oJ'd)) ;l 1,b>.&L.r -* o.t rrtr:..t! 6t{.-Jit, L;tlt:s,st+*,-y[.c-l

  • f V ,rr*{" 4l; i;!-yr!r .,,JU,

    "6 * 6)t.tt 6* 4 t .r.l.: 4,;; JL.,:1t ,5;t5" 6t! \t- YJ'f'J

    J.lj. GVj: cr;Ull .uL,:.t5.- tst+-,_;t) r.2l: f \ ! r! 6rrr tlb ,r_tL 1l:= i: a5.(srl: - \ L

    Jej;:-rc t jl;S ! ;ly.r b 6;;+-.5-- t_T S t-J.e dL Jly cr_l ,r t ,r-nttl t 6U.,[f] ut; t* tr rrtsrtf,1 6t+-Jit, ,Jh t,,'-1,5 Jyt ,*t.d t_ ,t'

    ^S 6ltJ^',tf *--* aa7[ir:Ll ,r-tL.f..,- ily.f ,or--utritS--

    R=(Ir-S)-'(/r*^S) (or- Y)

    e-JU tr 6,(5 = -l' cr--) .r-rh:rrL, .r-yL ..:*l I x I i5.,-,r-;, I7 ;.1 I 3+lcf dd)dJ ;;; (s',.eur sz );!rU, at c.*2,c*l \" x \"

    ' - ltt, ;:" ti"l

    l. Cayley

    (oY- Y )

  • ,-{ eLa; at+.t1.rr t\i4.ai fA

    a.- t6;'.b:t llj,C! f x l;!rr ,r-tL j.S sy,t..u'; 'L =t'**, (6P-Y) 'll',,tJ.$

    "r;-:-;-bL,

    .*U-- ,b R e-tt .uj ;3u1, iJ:L. ,-,* ;ri ,sl.z.rl u. ;rl-irl .R irbs, ,r-tU,-i-S -b t5 j-fuu L ,,5)- Lf_;t, Al-;.,f ,,r_rlr ;t:; l.x,lr:

    *=[* ? 2] (a^- Y )

    -J aJ -r::-..e ,SnrV &- 6l-y3,.. tl: _r, jt,r1- y_l ,,*.r,_r Y- Y ,14 ,t &t?

    ,J:.e--c t':r:{r, !b: J::--a &-Ll l.rrlrri U-l i*.,:-l oli *j Fr-. -rtrt*?,: $lF jll ,r--tV ,;s t :,Jr tt t) +p "r*

    li1 -rlvl -\121 -t

    *.Y-"*.2_.Y.2-"

    (01- Y)

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    ts -,.\ir,-l Ul ,:j 4 ,jtL+ ;)l-i,f L e t l-t' t2n - 6Lby: Jti;l ,[-ca- 6l :l-1: t+-f,(+- iej tr)b & t jL;l iLi-,.l::..c^-J c-J) ilt's s,y,Jrl.u,-*ttlLr c-^"--E'1--. t+Lr, dc*l ;T ,ttf tp-l;r;)tL:. J,-{-.c*l cri,i

    .c*.r jL! 3R fR\ AR 3R A-

    Y-Y JEr r-b tt r4>..1) t"" ;;l-tla, X Jr u-&-tj ,d>.-2) \"" ;;l.tla, 2 Jr,F- ,!, tj ibss s:

    '"'t=32

    -o70o o

    " 1477o

    lo o o

    l" rlrr|"ro"'l'r" "' f ::]

    Rz(Y' ) - (r" -Y)

  • f \ ,r,.ts{e ,-*-Q i:\tt,S, ,rJb

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    t;

    ,.:-,U X-Y-Z sgJt_lg:;-l 64 {IJ} -gr$ 6;14- \;a:--j ts9.Jstjl .,!

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    .t^tJ.(\v-Y Js'i) )s15",.,1:l ,xrSu' rhl ;i clo* Jr t+!:, 6 6.-nrf irt

    [\r"" o olooo "/oool

    Rx(r")= 1.r.." "1A?? -.ro""l (p\-y)L"r"'o oTooo orh??)

    i34 ,J,drtl ,r,_l:tt+ c _---t- ..,br: l) ,?_l S ty,g olal..i.,

    ['r,fV -'tYY 'rtO-lRz(Y.) Rr(f') = l"rO" "zVO -.rftl

    L'r'" o/Ao "rAVJ (PY-Y)

    f'rrtv -'/d" 'r'"-l* R*(Y") Rz(\'" ) = l"rff .rVO -.r0" I

    l"rYO "tlY "rAYJ

    --e s,s-7J. ,l o:ll- .l ,r_jU_rl t+f;, ,'n: -i tSU.lt:,:rls uA* g,_l 1

  • ,# ut|;i,rt+Jg*y t$to; 0o

    .c-ti 6tarrr,-, Jf ufl ) ,6 .Jr.,l \V- YJifu

    l[u_l .r:-1t61 .,rLl ,1i,/ rL-tt-l!J,t61,tJrrSUr;llrl:-7lri,,[[,:l1f y-l;l.r.2lr :.r,-26 i: .9-t 6Lr:1r.2!,i .rt ltj ,:f ],s. lti-l tr +tt

    lr-: ;j,g .tt' J'a + ARxvz(t, g,,o) J:t^. iby *-jL ;r;el c-rdJx-ig pt+l 1-..,, *r,lt+t clt.tr Jf dS*, t+b,

    ',)* t* Jr ,r!:'roll '\.i6,* Jr JL:: yarv .l

    AR"rz(1',0,a)= Rz@) Rv(A) Rxo\

    = [:: -ll :l l': i 'i] [r ;, -;]

    L " ' 'l l_-,0 ,p) L" ,; ,;) (Pr-Y)-r6q ,Jt+lrr: u*;r; c^-b.l .c*l . . . .1 ,sin a t;;;- so rcos a .;,i;- co ;i t, d4J ,j-l r. ,d ,?j ,sJ.^t ,-,rfr l l+1,,r, Jl .trlt c-*^al rt-.-: (Pf-Y) ;, &-rt+--;t ;ttf --,-\.Rt(") f\lr s,Ry(p) r,Rx(f ) l+l 'r/.r-"li J^'

    tre a';.; (21-Y)rr

    lcac1l cosljs^1 - sac1 cas13c11 so's7lAR*rr3,0,a) = lror0 soslsl I cacl sa-sBc1- co.sJ I ttt-tl

    l-''n c|s^r c{tc'1 J

    'L* t t* Jr;!.1r Pitch 'Y

  • 0 \ ,r.6*,{e ,}.2V i}.;,&, ,rJtL,

    k- (f f-Y) iJ:b, .:jLcf Li^L I tCfrr,*,;,tail-2r r-\.'3 clrl i-; ^f +5o.y3,8 1lslt,tn Jr,1 t;lsl+ X,q Jr:r.71:.21-l n -.;t+.-u lt+jl tslb:s cD.

    .o i;l.tla, 2n Jr,_t-;JU;l I Jrt'. c,:t'X-Y-Z 6l_ly c;,,- ,F+r l4l L rk u_l byl'd-U UU'CJ- \Qs-*rr) Jk 6Lrd:L ;l jcr-^..r,-! yrl.. rr-* .p-.rrT a-rr ;!ert{J d+T.r cs-!a rJrL 1,.-* tll-h:-1.5r,rfr) 'r--rv

    tr,srL- bVr-v).+h3 aJ:L t- u,i,-,1:,r;l: t5a5 u!,4!Ar.i;,alrL 1 y-l_2r .r93.Jj;lr Jrar- a-

    AR*rt?Y,0,a) =?"r r

    , Yr

    Trr

    (r0-Y)

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    0 = Atany(-rr,,, ffl + r;.,ya = AtanY(rrl lrl3 , rrrlc0)

    7 = Atan Y (rrl lcp , rrr lc0)\.c*l

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  • ..'5.rrt-;,-rt+J:'Y ) '4?'"i AY

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    .0 = o 1. zr-J'J ,it 6 c*l ;l ,s12.t 6'_1 1s,: slF c* vl:t ,S_,8 - 1. o Jl ,or-r ,_l ts

    {3 - \' ,"ocx = o /o

    1 = AtanY(11r, ?"rr)

    (rv-Y)

    '*l 2-i a)-t'7 +|n .5.-,p - -1"o Jl s

    P - -lo/oo(\ --

    o lo

    ? = -AtanY(r1r, rrr)(P^- Y )

    Z-Y -X -*_st ,s\tst:>j . ;=-; ; -S_t Gt_*7 iliL/ l, {B} -,. .;-;t+t

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    f t il ,{1} cntj Fr. o;r,{ B} S-* oK;, jl cstr. Jr ib:t},,rs.t ,-r_lt>

    c52p Jr .,tr, g aS +J o+y .LU.f *sl ts\lu t .,1:, u ,j-l b* t.J,-rJr iL::,,-,J-l d:"1 * o-.-? !.:.71r f-

    "-F,st+!:r t,-rT jtS.. d t-Jr-r,-,12.-o

    .fEe Z-Y-)I- *-i "s!-l_st,

    ,#;+-,j.,-l U-l ,t-rLt' {;l * , ,Y ,2 csr,"t:ni-;l.r;la".1f-e:..ra:,r, ul_t^j ),;lL-51; jlts>-p:l e.t {B} rstot:^, \A-Y Jifu,,..,f:r *

  • 6f ,rr*{e oZ.; i1\i J1,,rJt!,

    Z6

    Ys

    ?o

    f;R - fr,R. E",R Bo" R (tl-Y).c-l eJ.-3 i! Z-Y-X.tr-:l st-l:, Jt-tn\,c-! c* g;t+bst tl.f-,'l ai -2: .5

    +T.f c*rde tj rt-t'1 {A} du c^.-.i {f } a_t+ ,s;r4-

    f R2,y,*s, - Rz(a) R,(0) ftx(r)

    f." -sd"l lrO 'P'l ft . .l= [: ': ,] l_,, ' ;l [" :: ;i] (v._y)

    kS +l3'.,J"-S"--,r t+--rL =-e tr....1,so = sin e. tce = coso;ir: aft-*,-l

    lcacg casBsl - sc.cl cosBcl * sasTlf Rs,y,1s,(o,, A,l) = lsac/ saspsl * cacl sasBcl - ro"7 |

    l-tB c/s^t c|c^t J(V\-Y),54-t* Jf .r!r, ad o.Ja ,Slt 6 c*l 6laa=l \ )t 'E.i, ,.iy 'a4* AS +5 otft\t E,}.t.JL 1s or[:-- -* c)a b ra.=; ;r1l t:;u. J-; ,-"-fu,.*;/- [ c.,tic*l aJ--,.iti 6t.l.7-9,^,. Jf ib*,* pt+l;l ,-, o.LI cryrrJ .,,-t4 6fa-:c*l

    .Z-Y-X -$l ,st lr; \ A- Y JSJ

  • cl,.t-tn Jr '** 4;t,.r!:: & j,, I f+ I ; I -:j::; T:,.Svz *-;;$bstsT cryr.r (J;! b.JrU. y-l r.t^. J- +,c-l JrL (Pf-Y) lr (Vt-Y).j+L i:t,slt ;tlj,a b Vf -Y) it1! ,F-r ort.ct .c^-J jU ,rLr, ,r-tl ;l Z-Y-Xy'-:l .st-h;

    .r-r -r6t ,; ,frr- iLs:. ,y-tL t ,lt:i Z-Y-X -f,-:l ,sL-lU,J:-,;T c*:r

    ,tf;i,sinB+"Jl

    Z-Y-Z *:l (Jltilt:c*l s; {B} v*rla;;l u-&-' J: t

    ;;r's ra' t " t: :,,1,1?rff" oil y,'I--,:;Y;;r* f*:iu-l ,sfi,t J: j {n) S-* -. r.rltr3 ! q--..; t+!r, u-l ;t-n 6 .45 r-. j;plh,l 2 r,Y ,2 Lsl^1t^- Jr,r!:r - i).rf,.rj.up -*-:lL-:l: il=-rt ,kl

    .€E,l Z-Y-2.{-ll crUl L 0t "i-[ ,,i1; ,c-l or.i,

    r.T .up c-)4 g;3 Jrt jbss ,r-tt ,Jl^; e ,bb;L" i# Jt"r !

    lcacPcl - sasT -cacBsl - sacl cosg]f Rs,y,s,(o,,9,^t) - lsacBcl I co.s1 -sacpsl ! coc1 sosp I tvr-rt

    | -"/rt slsj c0 J:c*l r.uT tj tt,if:r ,r-tl Cpl: I Z-Y-Z ,L-21 6l-11;'+A ,b

    c*l g=:q ibs: ,r_tV

    ftrr rrr rrrlfiRs,y,,,(o, A,l) = l"rr rrr ,rr l (Vr-Y)tt

    Ltt'r ?nrr TrrJ

    (Yr-r)AtanY(f4, + 4,, r.rr)Atan I (rs f sB, rvl sA)Ata,n Y (r1 f sB, - rrt I -s0)

    B_a-

    t-

    It vlr o.2l9 .r ,P Ulr-1r c-,:.. ,L ,>tJ -b tt I Ll ,r,-i,r,. c-). =h tt \ Jl,p = \Ao/oo \0 = "1' $"Syel.Cga*l-."to 1P < \4"/oo,Sbltl,*-

  • 00 ,lr.(;{a r,!l.j 3;!;rr!. ;U,r*t^.,.,1,,b'y t a ,JiV,;l-g-:. t{i,tCL rl_lt:.rr $lri ,-pcs. (yf-f) *l.1ad *; af ,:*l ,;,,-l ,t+JL ,t-l ts vlr.1r.21T c-rg 61 6rlr.rF ,-ir,, .5.- .,:*l

    :r; -ur!i i* hb" ,Js.-ay-lJ) .0 = o 16,J+-t^, 0* ilj,f lt vl3'..i- '6;T ,0 = o," Jl

    0=or"e=ol"

    7 = AtanY(-",'r, rrr)

    (v0- r)

    n-ig c-)d, ur-a =l*.1 ,6;T ,0 = \ A"o Jl ,

    0 = \Ao/o o

    a="/" (VP-y)7 = AtanY(",'t, -rrr)

    L'_lg '+4.^t f.1t cttArlr./li.st-b, ,c^,[.iX-Y-z 6LJy:J-3 o:lr cr cr3-++ (E 6b.:l:r]ri t- ,+'.K6tg!.-1ru)L (iJ ! i[, o- {!,tul:-1!ri a2l;l.5._ }t>.Z-Y-Ty':l ut_l.U ),Z-y-X;L_rl6.x--..r ,si4lj YY 't-st-",-;l ,;L{JL ,t'lr.7f u-l .t-J lq;,l ,*l sgla,,^. Jr-Jr ifrr + Ll-,.:l:.1!f \Y ,as5^,.-. t_r_l tt ,r-*J1,,.'+,|; U-l "r;-r4;" 19ll:l:-rll.-rl; ,st-l:.,i ,j:t jX d5+Jb.y.ly'_rl .rt-lt + Ltyri-_r :lr-1f \ y j,c-1,. 6l-r.,r.,bd :l-r-,pfi \ Y t+- r:,-JJrr ,.ljl.raci .5,- 1 .r*-r Cla;-5J,2 Gy Sr,/_:l .St-lr; ,ta Sr(!_l t .:ylr:5.-e ;l-l csl.lt:n Jr Jl* tsLpfrrll orLf*.| ! i!:: ,-r-;r, ut=: .s!rglrrlr-rlri.jjj*. il3x-;; jr? Ll ,srl'r 6;r. c-&_ry trsl:-2!ri y-l;l .5,_ e" *^lGl

    .c*l ,-ui .r[5,,,1-!.lr ((r-.,D d-3^r, 1; c*rqi J12-a*. tf--;t ,t-l .sl

    ,sti-tlj: x a* ct.'-t,s.,ir.is, L...^tL 6-;r+-i: i;!rit,;)l,r:l"qsl: : t* tt13,e6. 6;+ u,a_t^i jl Jt, tr_l .ry,g ,J:4 a-r: \"o ! JrL-ro ,(6.,- tsbt*l il *

    ":*) t-J ,lj opr ,;+ tt ct-*.?L" .c*l Jrll

    {B} =*th ;l J-i .-a:.ai.r-2el ,*ra, ,*t:, L-.1b s un'al.*1s\ ili,r b 6;+,l"-.& -bt lt

  • ,F,16,-rt*rr tl*i 67

    Jr l, {B} ,-,+ .# A;,a-l ,j.[:- {A} fb *-rtt[ t E e.-nj.16 .;l..r+: jLrr c-! c-: ;)tB * 0 L-el;;;l.lL, Ak ,lt"

    {A} t d-*; {B},r56;+.vlr,g,r.rL:,.r!5: Jrt,, t* r}61k.,ltt'.r;li J'L L-sb s tr ,r-f L ;rT :,:l.r ;L:; R*(e) L-AR(k,q 't^; [t ;rl-ig L-r) bl: )t orl-* j:1. "l;ilsll2-i,r--;.,l;t*;lU,3r

    L L;l oR .,lttc,L-.;h,t b rt o-(- ,st*|, UU .*5.f A-i b rl-r ;-rl.Clr -,, ? ,-sb .t.*,g $J .b.,|:ra;r1l .FtsT cryi& c;+ J.e;cs!, r x \ 6ylrr, E Cc w-,- 0 ;lt>i;lrl

    .(+:r !, \1-Y Jfu) rrnr.r u'!t^i (,)ti i:+) K \ l),JrL, ilts>,r-tL,:; vt ;;l {A} -L.l c|t-tnjl ,ri-,jLrr yn 6 r,K:^

    .r,i .up t) tj 6ll.,.,i--,st+fr, ,st*l '23e6

    Rx(o) - cos dsin 0

    .l-.in B I

    .o, A-l(vv-Y)

    ,Jrb, \sl: t .t* 'AE \ 1- Y Jfuc.,til *-f,-:lLi,r.,lrr.e,'.1 0L-1l:lRrltr.{e}cc^#{B}6}.}-lrsltd.-lLu-l .\

    .[f] ,i,r .+.[ ,r!rt ].[-sl\*i ),rl*l ,+--,

    Zn

    ?B

    ?o

  • f cosd " sin0lnye)-l \ . I tv^-Y)[- sin 0 ' cos 0Jfcos0 -sind "l

    RzQ)- lsin0 cosd "l (vr-y)L" \J.5 (r+r lt 7-Y u-r) rb ;E; ;-tl-ie,.r.:L ,p t "::;:t;;{ ,irr,r!;,

    I k,k,uo + ro k,krao - k"so k"k,uo + kusdlRK(0) - lf"fuu| + k"s0 krkra? 1c0 krk,u0 - k"s0 | (A"-y)

    Lk,k,u0 - krs| kuk,u0 I k,s0 k,k,a0 + c0 l..:*l ok =lk, ks krl's,n0 = \ -cos0,s0 = sin0,c0 = cos0;lr:^S,ll oK jt1.,,':, 6L-lr r> o*z c*:*t dJt 1> c*,!c*-r i,,b il 0 c.,)!

    .r-i.f Cd)de ;-J,gaS +5 +_i .;Se ,.J,--rr i!r, ,r_tL ;_t; + lt *-sl: ) r-rtu ;-l (4. - Y) iJ:L,c-rl Jrt,., ibst ,;-,,,t.5.- ,jtL+ relj,-l,ol.eiJr ,-i;n L-j:: ulr5> rr^. jl

    'f-ulU-l uS€'.(Y-Y s 7-\

    (A \-Y)

    '6aT

    Ct) jl0 S K 'r*l^t "rl'-aL,T t-:.2r ,r[^r y-l e-H ;l u.z;.

    f.rr ?nrr r,,..l1"r,, rrr ,r.lL'r' rrr 'rrj

    .5t+-t) c*lr,-itS,l U ;rbst ,r-tlJt .trl c*t o

    AR*(d) =

    g=A.or( frr- \+Y

    rrr * rrr

    ["r, - t,rlR=*l',,'-",, 1

    Lttt - t''tJ(^ Y- Y)

  • ) J-rtu et / ,Sl:l ,..JS."*U^' e-.-2s \ h' ) jp :,'-r:;;j, "6;+ jt^.J.;1.,, dr.1lr rrt: ( -nk,-0) J5; + u-S-t c$,(o k,q J:t'.. i-e!,rsbst*,?-f q,.r!r: ,r-tljl

    "J',-.r;.rr,.r-!rQ..:*l ;t1: ,r-tl it" q,L2"i-2r

    ,slr-sl: tst+f:r-r: dc*l ;T tai-l.rr -r.-+ ile..- .C*Hl l, J- rr u-lrl .F- +t:cs t,.r!rrrl.ti..+Lt d.:*l Fu..

    =1 i* t rLti Jrt* ,-r!:r Jt^'''S*S

    ,(AY-Y) ilaI 4:-:*,-u.:lrl J- .rr.r.tp;r{&l 3l9y^. tr,-r;,rS ,I- -*-.(4.-.: t Y-Y U-;) rlr.up cd)l, ,glr,d = \A'o [- s0 = "o Su;t5,o ts

    A-Y JEr rJr ,r3 ,c-l *:- {A} t l+l { B) :s}u. '*i r12-o Cr-l r {B} *rrl"g?.srrl o:lr;!er 0;;lxl+$rSe irr;l ^t)

    r R - ["rV.Y orvoV "r"lr jrt.rt1T &-)& ;Su. J:at {A} * c*; I {B} *-n.,V d!, "I+ ,r-t\.+l.f rld)t ,t:5;n'1 ,r-tl Glr:t d-*; ,(l'-Y) il.it tt isf i/--S-ti t,

    .r91 $lF [. " o]Tt r,! rtS,rl:r,,c-l Ei!, Jtt' Jtill ,*r,rkt i-r.-rr;r,'

    4ry1 -gt

    "/1ff " /"7Y" /"py ./1ff

    -.7\'0f " 7\'Afo ro o ro

    "71,0f o lo

    -"f l"0f " /"" 1A7? o loo ro \/'

    (^r-Y)

    I

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    .'s b | "1,-+

    l-Y Jhr r,l',r Jr ',?.e

    ,d-l & {A\ t l+l { B} ,'-;.,C t*j i"+ {B} -. r,rL3;btr06;l.sla,,rtS,lAP = f\t" I t" Yl"liLii;l^SAk : ['rVo!'rV"V " t"fr

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  • 01 ,l;*{e r1l; ir!;r!r ,rJU,

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    {A} q c*; t+T irr, Ll ,r,:....r {B} I tA} t, ?r a,;15- ,sl+-;r?,-s!,,1, ri)j; bljd,c*l rJ,i f l: ,rb.1, t* Gil J djt- JtiJl

    ft'" o to o to t'" I

    A nt I "r" \/" o to tr. | (Ar_y)A'r = | "r. o ro \/" fl,

    I

    L"r' olo olo \l'J

    ,-*,t J: i ;t:1 {B'} a, c*i {B} vntkt ,.*,, i,:-r +

    [\l' oro oro -\r'lB,n- l'r" \/" olo -rr'lE'T =

    l";. o to \/. -y1,. | ,Ao-Y)

    L'r" oro olo \l"J

    .:l:;!er {A'} 1r;*; t {B'} -. r,rt+a bli,r,hr,rh ,K-s G;lt& -U tt JL;l orLi-,| \ folj.rl r:r-lJ-;r :oj,t Jy^a ttS,g i+r;l aS 612r,. Jf ib1o U-l

  • ,# ,eV; ,-rt+lr''..l J \i -4i I o

    J: j,,4rt ,tb:t c*i.,*5^*b {A'} ,. c.*; L {B'} jLrr,(A.-Y) il+llia. &-l a +-i t, .r.,rl cdra (A'-f) Qb., ;tt-S u-6-h t uly.f l, wnrl9'

    t-rcl 4 s c*l [. " .]" iT ;t5,rl:, ,.r,1_; Jti:l

    A' ,t-B,r =

    Aq-pt=

    '7 tl'l ' t" ?Y. /,?y "/tff

    -'lfOF " /f 0fo

    /o o

    /o

    "ll'0f 'f'l-"/fOt "r"

    I

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    "7f0f-'7\'0f

    " 147?o

    lo o o

    (^^-Y)

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    AT _ I,T },,T f,,T (Av- Y )

    gi 6rr.c )bL S

    '/1ff " /"7Y" /" ?Y . /tff

    -"7!'0f .7\"0fo

    /o o o o

    /o o o

    c*l ;.:a

    -\/\r-l\/\rl

    t;::]

    ,s\a oxT c*)4 ,s;+ 'L- q ! ;tS,r-i.5.,- ,trj3u; ir.,;| ^S 6sn Jr ;tbst t,Jlop: {B'} : {A'} a5+J +j .t}lap :t-*,-l ,i.r.- il ,x1Ky^. Jr iLs;&JLs ,=-tir$ st ct-l 6;+.:B;;l .t-J rlt ib': u* ts, r t+T 1+,. c*l ,i51;; ! \ Y-Y ,r-- .s-r,g pt-+l '15-J, 7 -e ibs: tp 6il, t+T irr ;t(, vl;c;l ;I '+:1

    *_ll .St..t.L!gta;;., l,'t* ,,[a, :.rs ,V :l ;Tyr a5 r_1lr )rs E ,s-&_t:lr-2!l lsrJ;-6- ,r_U ,sU.ultUl,c*l *,lJy_l ;:e.r-.. ;l gju ,rt, y_l_rr,J,5c^n e Jl .t-rt,rl;- .l*-ll

    .rj ''^llr ;t, rl:;!l i/-l ,f tL J--*.>--; ,*,d J:t-. L-sl: s k = lk, k, k,l' Jrl'. Jttu, *J. y'-:l ,sl"r-ft*

    r-r,g0(t=krslnT; 0(r = ky Sln i

    00(r=kzslnii €r=cos-(Ar-Y)

  • t \ 6iS&E ut-Li i]9 fi, dt!,

    c--t rli, il '&\ l+l oo J -\:*; F..- J"-.l(,_;l qJ;tq u-l S ,>-l t:,ef+ef+e[+e[-\ (r. -Y)

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    tFg dLtj) 0:+ ,c^-l ,L_51 ,56.n

    fr-r.;-y.lR,= lf(.,ey{eye1)

    lr(er€s - €y€1)

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    Y(e1e1 - €r(r) Y(.,e1' + eref)l\ - Ye { - Y.l Y(e 1e; - .,,.r) I (1\-y)Y(eyes*erer) \ - Y.l - r.ll

    f,F tL:' ,r-;u JlTrr - Ttrc,

    -

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    et=iy'\+rrr*rrt*rrr

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    ..lL r,.rp./t t-\, \l ;-;!-2r o.,l.e-e tre;.5 c--l;;y,(At-y).st++r. ++,-rrlo.r.i ^t :ti S er_1e ,b :l d Ck ;-p

    c,!r.r tr jl rrl.r- ,l \ ;rl-i,l lrl+-;r+-r t+&

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    jl ,s;J+",slt,lW t,-t& U_lJI,JLro.,:.c*l orL aL;l a;lra*>l.o;^> Cr-l tr i#

    ..u Joli fJer,r -rl=* ,- "=- ,J:., 16{ 6;+ '*j

    rli sta;lr; ,Lrr 1-Yya 3 ur- q5te-1lrr1 ..rii 15LCl--i y: .6-ll: -rSs-,rt(, dt^_rl:, ! .lo ,JL.oi r;l;r..llc,;ti, U ,l ,4tr.1lrra 4T lt-:,c-l c,eLt l-o.2b2 i/-l ,.1,-{ .#r+tr ,rtt -o bt aL;1\:r,.5: .f-Frl;16 olti,'larlr; (,r;-rlr") ,(:*" :6tb,g- &tS..2rLp; c*l gS-. r.!. ,lrr, j: .r-i[. ;L-S,-b{ .-e 5.0;\s\.;r..-,L_: { sr.\r.f,-S"-.+-5.. -b I I Y \ - Y J"fu, .2r oJ-1, orlr lt l; ,Jtr lltra ^- >{:, .*! ^*B d)l;, ;lJ,_S"_ ! t ;_r. tL. ,r,,.trlr ;L*! !-{a r ,rl, ;;lsl ,)t j'.L-: yl:r1 u ,}-l

    .it!

    v2 vJ

    ///

    .1112 c,-c- 6tr.2l:r1 Y \ - Y JS,J

  • ?Y gl g;b.1ly upe,.,

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    'fu,t+T jrr .t{.-- e [.r-11:r1 ,y. L.L ,U)'q..ff i .L..- ,bt.;. tb, s:+s

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    ,Jr.i ib Ji.rr5,- c-l OS-, ^S ,,g\4;tstrgJ ;l u-ll U st i-,; I tj tt:c^-lrr f+lf

    -)l b t ,c^t? e ,;l.ll 2 .3)r 2b t -r .tQ i 3i ;l S c,-l 6rlsr; u!; rtry,1lr p-, jT Ul e U-t .;l E a.5y rlrt /.st"jl UG .-ut, a:-,:l: ,r(:-, E tlr;-

    .c*l "rL;

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  • ,-{ eV;,-rkrr.ry t\L.j ?l

    .lrvr "1,-.tr

    YY-YJSJ

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    ort - 'l td+il ii+.-,(fO-Y)r (f t-Y).st".l+!-2: o.,r-l o:lr !l^i.i JS- t'1r-. d,t-t^. .trdr$ tt+i-

    .l-.4,g cir-rlrr,. f x f q5L+--rL it; b)lt t n:-, wt';l 3;lu. 1.q;,1Jt*ut+!r, ,'Jc. .r1lSu. ilrlt;U i,t-t-.. f J,,,I,-ri .sL+--rL v-e 4j

    t'-r31 -b tt tj ut-t-l b rlt;^ 4 (4.1r.a)

    op=AR in $n op ( 1o- Y)

    rtV I #R ,e-tV b rJ +-*J,.rS"-r: b .r[: sl.t--r[ lt:.,| a5c*l ;T ,-ir,' .5.,-L_i ,--tn.rr-1

    op=iR op (tt-Y)

    .up 3i ,k Y? ),-'r,> J^t Of q;! ,;l6r.e: Jg- a- 6, illR.lr.2eic-ra,? s u-J J'^t t 1.1 !:- 3 $?-Y) ;rl=s t5 itt i e-tV v-e1t+l ..-1,

  • ?6 e.V

    F,Lc YY 2,r2.b ,j* rY il* r.rr. d,q-b-, ,[5:l.r-- ,d1,;--r, .c*l g.a .,|.s.a:*l

    ! t+,-,-;U 9 ,rrtj' ,l cful:l'rt-r. +, ,b-,. ,j-+ ,tt. jl ,,;,-rt ,tlt ,s4* Jt t l,f-,d :.1 l3 c,f-L^.. .rt,F-"!-

    op =ARop=ARAP=ARAP _AP

    in "rR DPin C Pnp

    (1v-Y)

    ;t--t-..:l.u;. ; -: )ll5 u1,,_,F J^r \ A s --p J^r YV t+-,;t,.L,:l.u;,.[.c*l Jsl Gy tt

    ;l urL-; rtr; ,.+li e ,IJJL qU BR s ,3R,AR c*l gS^, tCt ril r5;Li:r, !, iT ,'r/ r*t^, ,t i- I $ R c--l -i4 ,;t-f ,-r-l tr.,-J .J,-+ A P; ! b too Pt

    .+3:1t ], \ ?-Y u_r .f-tt6+ t-s& sl$:t^(; qJS,s!r

    \"_y JL, r-# v*.rkk .tl.u; ;1.r, d.r."r: tf l ,AR 3R,,rfr) ,r-tL 2t t-tyb ,sltb .ia

    ..r;! \ A :l -# g.1 ,Jtek :l-t'; 3 YV ;lc-.tr; llicl,-r-L v-* kV ,r_tV jrt u e , r,3R,s4* L, +4

    e,_lR i.,E,_frR L,Cr:CrxC,

    ( 1^- Y)

    t .c*.1 \0 g.1 rgt+Lc :l.u;e,Yf dy i,t-L, 11l;l ,slt:-e,5t4lc :lu

    6vB. Noble, Applied Linear Algebrv, Prentice-Hall, 1969.

    D. Ballard, and C. Brown, Computer Vision, Prentice-Hall, 1982.

    O. Bottema, and B. Roth, Theoretical Kinematics, North Holland,Amsterdam., 1979.

    t1l

    12)

    t3l

  • t4l R.p. paul, .Robo t Maniputators,MlT ,r"rrlnl: "*" '\i*i

    77

    t5] I. Shames, Engineering Mechanics, 2nd edition, Prentice-Hall, 1967.

    16] Symon, Mechonics,3rd edition, Addison-Wesley, 1971.

    l7l B. Gorla, and M. Renaud, Robots Manipulateurs, Cepadues-Editions, Toulouse, 1984.

    t{at*.ts 6 s;llla, Xe Jr tr, rd>..2) 0 tilsl+ 2o Jr l+l .P.rl:, [\O] \-ypt+l 'u f t *t ! !1 t+L:, U-l nt +51+, c/f:, ',-_tl

    .t-rct.:lr a!3r.JA)

    a-.;r fA;;l.rh *o Jr .-,? )r4>.-1) l" ;11-r;la, Yo Jr l+l oP;lry [\0] Y-Y

    1tz.,l 'u f t 4t t: I Lpb, U-l E +51+ Glr:t,r_tV .:-*e orl: at-1r.Jer

    .c*l ,P {A} t l-r:,1 {B} ,,*l {JJ jt il t.,12-aa {B) *nrqe [\ t] r- Y,-,. atlt J F^rc/ ;fe: a-.Jr 0 e;lsla' 2" Jr l' {B} v-nrv w6 Gbtt,r-tV c.*l ,-rlL.r*fng *tt 6 6;l-tla, Xs Jr bPV

    .$rc/ -J o P a. B P :l b l.r-,,1r2 ,J: j.c*l J-Lr {A) t l+l {B} '.-l dJrlt tj;'u-1 {B} =*ja3 t\t] f-YJf b J-S =n'Y'-': forcf ;fer a-_2: t"" ;;l.sla, 2u Jf 1., ;rT ,-,rtr.2l:2 *i.ei 3 ,sb::,r-tU c-l ,-rlL.gfng a-yr f0 ti$la, Xs

    .$rc/ -JAP ,, BP;ll.(i - \/) r,*_-rlr) ,>-l e-oi ) "*o',\ .'-:y_rh [, f x Y .r_tl f ,B [\r]

    5c....r- \ .i:rl.rl ! )c,ARt'-:rltt,6_* f:*.r--r:l d-)de t (A'-Y) iJ:t^. [Y\]

    ,r-tV.5-- J:h, L-sl: s rt*, 3 xS 6i-1tl;t l- J*; l, ,;-r&l [Yf]x-;;;:-L Ll ,-u,J Oi(t Y-Y) ijrb, Jvjbt,t_l ,slt.r.rr c-:qlJ abs'

    t-J,g ttt -,: L 0 : \A'o J 0 = oo rfE.-rtCL U p-t fut.SJs,.at- bbss r-tV ,-'t-tn Jlt Ls;;+-;-[; d+-,-i ug,lr..l2t-|u IYf]

    :45;Gi dr+ JtS-[ ,1,;* b -\..rar r*j; J:L i-1! : t*

    0-Y

    ,-lv-Y

    A-Y

  • 7Y Wt-

    Procedure RMTOAA (VAR R: mat33; VAR K: vec3; VARtheta: real);

    Jy.at, JrL L-slj str Jt:- Jlys;:+- ul,-[; .S.p,-r: .s-S-t LE-r.t-)Jlr *i; ilt,rt ,f-tL

    Procedure AATORM (VAR K: vec3; VAR theta: real: VARR: mat33);

    4S.ii-- .+J !r"l r:-,p3,e .r:3.rt ! t+i J 4.l) ,lt pr (-i, t )[ Lbr,y_; 3:t+J it^-,|U L-,,!+?:r, .:JL r> .d t- .ll;u- 6tf t 6\D,,-rti. L-l aS

    .+s.r!F,c-[i 6trr5r... Jr,,:t-J.6,,,J:,,r tsL-lr; 6DL A-Y ir,-; [YV] 1-Y.+5r1.$ Z-Y-Z,L-51 6l_ly 6lrlJ A-Y y_; [YY] \"-Y

    d,t.il) ,rlt r,_b*Q c _^..-E ,.rL:, ib,: ,;--tl,9s ,-.ri" ,rU,_t; ttr [\"] \ \-Y:(c--4

    ',;)ul ,rbt - t);+:oa,?r'lry [\f] \Y-Y

    BV=

    €\ ^::l' Jl

    Arnpl=

    "14?? -or0oo olooo \\f"or0oo "14?? 6looo -Y/oo/ooo oorooo \7ooo t/"

    .+5 q-t^, L nV

    \0-Y ,f5.-i;: tji.l:.;t,Jlc$ orl: sr--rt r:"-[ l dI il csl*th [y \] \f-y!, Ef u,?..l=.lr iU; b t+l ,yJ ,lt ,4; E ,*+r +J 1*t,++r

    .ui a*tr...

    '14?? -ol0"o olooo \\2"o16"" "14?? olooo -\/'o/ooo oorooo \7ooo A/'

    li.;"1

    Xr-

  • ,f ,lA;.rkJcr/ 5\)r..oi 7A

    Brn -aI

    \7""o olooo olooo olo

    olooo .1h?7 -o7foo \o/o

    oorooo .rott

    t"r^r, -trr"

    "1h?7 -o16oo olooo -Yt'.7f\'\' '/V0' -or0o" -flo'/Y0" "7fl'\' "147? l'7'3r-

    , {/} l.lnl ,r,4,rT €-rr, n c,JL I AT'^*l^.60 d J-r.j [f t]AP'rLL J K .tj,t abst0;;llla, R Jr {B} u ; '.rlJ:k t't {B}

    .('.,.rf"1 {A} vrrkr i+, J R ,-{ .:JL el ,25u.ir+ tB) .ll;;ti,!-s"- L. 6;11--r: t+- {B} r {A} -rtta.- e: [rf]k '^*-rlt1- Jr iL-r! 0 s;l.lla u+ t,c*l {A} r JrLt lll :4T,r, c^-).r

    -$-t o.rVr, .l5r'. i!r:

    \f-Y

    @AR _ ARK(0)

    ,>-l ,$y *'^\b +or iLl;

    f R - erce

    ;T-1: 'f

    r5-: c.:t3; lj ut-tot ,.rlrr ,-p-JU a- 1J*5t 4[ c;-rlry [YY] \t-Y

    AP -ARZnconDe

    K-l i' r;:r

    i]

    d ** t' -rt lt ob:t ,r-tl u l.ril '5 c*l ,i ,5 ir,-l f t+l ,slt ,tv &-r,-T c*r" il;tV p f Rl;

    A p = f n o p

  • tt Wr-

    &,{ ?t'.rfrr tsLp,_ tL tt c^rr t b rltt dc*l ;ri -&.t ,bAP =AR\n$nDrAP =ARSRCPA p = An u pAP _AP

    .pJ.1-L&"To.^r t ;bnPr-t ,tSu.-j-il \.. .L[- 6.o P bfrr-rb ,O=.3L ,rrLo 1* , ,rrr.i; JL :r j] ibss ,r-tl u ,-&t 6-r jl,,t+l ulr ,ju U-:+..-rr.tu, rh-l i,e \". .uL- \rgRs,8R,dl? ++

    . '=rr rlf l ,S,l tt-... ! g-" t ?-f .Sbk rl-ro- .5 !., ;1.-t^,al-,.:; ot(:-, ,l :.1 ,: .t;j .S,_ ;rlr ;[:; ,slr. rs$t,-sL-i Jfu t\t] \V-Yu4r-/..:*l r$ c:lr ;t:; ,t(- t ;.,,-l ,roi^. a- Yl'-Y J*,.2r .c-l 15lilFl,i.st*trr r i;l.ll*,f\s ua5,'.5-ur,r, .rt-:; xyi^i.a y \ ,1a2 0cfi5: at o:;- .JSs. Lf;.- | xy Z^)a 7 7l;;tl U z ua*v.+[s; Jti;l

    .j-r:T c-rar z 5,r,0 6ltl;;l;Lo:; pJl, ^ P 'aLjI

    .sgliclt:-l ;,Lc.;... Yf- Y JSJ

  • ,t2.1616a15-1r.ti.,,- .-r,-j rsb; rs.&-) 6rnd- ff:,;;6Yf-Y .$ y: ,[L, tt-l r s,p ,a aaL/ & .c-l 65tS 'il.-:;.. rtLtg& LErl3 d-- t$ J:4 JliLr. b 0 : a (J;t^4-Jb .c*,1 r.r*: ob ;Lijrlx,l ; uAi iL-U , ,;-,- 1r & .t; ,-l s-rcl J^f l' :l -rr d*J t n P ili; .;16r alui,- .t-.J.,C P .b s12,, 'alj; o tSc*l usl*

    .I-r:1 ar* 15 ,B ,a Lsif al-r-c)a-b51s r--i't* Jr t-,+t$ailrla;42 X yzuJr l+l tfrtttf @,s -l:l .st-|r., 3-r!-r, ,6 ;UtL + +i ! .rr.f crl: ;125r rl: qlrlu,r.-\

    ^iU" .'-;..-t' J:zj R*(6)Rr(rD qV ! s* s-14 6;14-.5,'-{,J13,.e* ,4 *uJJu.;t.+l c.:U t:r. v.;Jke .stor.l'^. Jf LCtrt u-lL-: -b q tF,-/t =-* -t

    *tur -h *,..,/-!rq 'r.d Rv?b)n*@)f tfu r, d rri! ^-t, -b I c-*l ;g' Ll .r-,;l: & 6t* ;1:e Srr,z'

    ,.rfr).-:ti .-. fJqe r, r.]---.i -- ,Sr,ru,-,. fJh )l q-tn $- p-.q a!9:

    rrrr-,/ 1lW|b ilR*(ilR"(/)n;'(d)

    e12,ea,,,ti il"! r.-r. cs-,t +J:l R*(il v-,- t', ,[\] .*U ]-r; "t-l

    .c:f oL.;;- Yf - YJfu

  • Y\ W)-

    ,* t+-,-;t vt' t*y r-.rC .b.. ;1.2r r,f ,s--J. ,l a.o)L .y_t+ -r[-,L_51 6l-121 '^Er-*, t, Jrt., ib1o tr_tt Jtfu, ,,61, 621 a uj L .c-l ,r.l

    .(c*l rr..l rrlr (VY-Y) \| 1.4g:;) 4.pl c-r r. b h : .0 ,a) Z-Y-Z --\J-l- Q' .l,-+ ,l:r7 ;1T tl t S | 0 qlsl^; k ,lrr. Jr e ,lrt ;rb-, ty. ] C,

    't--& -b t;_i€Q' = R*(O)Q

    c*-:r, ,\*c./ *t t-j ttry,-or, 6b,&.:tst JyS,(1. -Y)ilaf ;lr:tr- I Lr_-pi

    e,= ecoso+sino(K xe)+ (\ _cosr)(K.ilK

    ,cosd = \ ,sin 0 - 0 t; r-ht ll;.$q*.:) 0+-f .rl:r sL-l:,,,Jt [\O] y\-y.{.rrl c-)4 5 k 'lr;,lt t* Jr 0 jbst ! Jrt, i;,s> ,r-1,L ,(dt = o r

    .r.:5 rrl.d .l (4" -f ) .lrl;lc -^.--l; J.-SJl: .jt:rr: dr+r ;t:;,Y \-Y e_r'aa; ;lc:ti- .l t, [f " J yy-y

    .(:rl.l c_--.rl t+t:r ft+il \+/ ,r-,-) .lrlr .,.,-t++tr,u P\ 'ALi;u 'G)tt *J. , iT -*rV ef .r.-,-; s.lJ,fll [yO] yf-y

    :-r;l;:L Lli; U-l \y t5 tj al5; .S i_r. u p7 5 ,u p1

    .'rl' ,lt {A} *nrl11 ir., r ,P1 .\.c-l {A} jl X .t-t* + ih r uP, .Y

    .:.11r -7!ri {A} J XY '^^;- tt ? s,x. €,,}.t u} +.S--rj u p, .Y.+5..:tjl ,- .uLitSl .st+*rt 6bb,#J:"j [fO] yf-y

    -r) d,e-oi ),€oi,\11 llc,-21..;fr, ,r--rVtaj_sy-rti, d+^, j[.: [f.J y0-y.i-\F ii

    ulrr tsl+-- jU'4S rto ,s! ilrLr' lt.[-tl,st-l:, il ^.c5-^-.J" +5c^,[ [ff] lr-l-sr1161 ir!.

    .t-r-lc#)d, t ofJhi. ,YO-Y ,[5*: ,, qi 1., f tof @

    .4ts-l c*r4 I, tT JbL,f O-Y JS- q ^"i q [\A] yA-y

    .x-uT c*rde ,8T JIJL ,YO-Y ,fS-: a, a-.r; l [\0] yi-yrO-r JS- * .", ! [\O] io-y

  • ,{ ,C6,rt+{.y tt$*i VY

    'J,*- a6":f { ort k:,s\.nt[ Y0-YJKJ

    .r-rrl ca)t b Af -rbi,,Y?-l Jf- q ^"-t ! [\0]

    .I-rrl cd)4J t tT rlli, ,Y?-Y JS- t ."-t ! [\0]

    .{rri c*rt b 3f Jl.r;, ,Y?-Y JS- q a-.r; t [\0]

    .t_u-l d*:a, ,"^T JIJL,Y?-l ,[5,:.,."-t t: [\01.,.:*l r-b l.r )t orl--.e,ifr) *-tl * ,,tQt> +5 -B [Y " ]

    r\ -YrY-Y

    rr-Y

    r2I

    t< s ----l

    T2I

    l*-s-*l,J,s-,sbc-S q J-> ,s49.rV Yt-YJS."i

  • lY w;s ,tstbl Lls a- l54 -J,.-(c;.L:Y ,JL;i) b,itp t, $ 4- s* fft] @f.-t *rr ;tl.i.1lr ,S5bTL1t ,,:.:,r5rn L ggl;i p Sr-t -L f.*

    .,:*l 6sl;TLrs i(lrt + /r) ,stl, ,u.rt N ..sU; y 5.,2,:.c*l o.li orlry-; .1,-+ t\O](@

    AqtBl=

    "rY0 "tYY "tA? 6/"'lAV -" ,6" o lo o -f /"

    .rtt .rto -.ro' \'/"

    .o-.rsT d*:a, "oT tt t (r, \ r-)V+.;rjlor-i J-, --L- f--: t I $ tF b u, s or ii_yl:r, er [Y0] fA-Ytltt st U ,gb L_:lt ,S ib:: dt 7 a1 ,..L 1a

    "y_l a5.rpL cUl:

    url43; -8, J^r .S,_ .* r-t ,r!r, ,-S_t orVt) :uU -rap cn0 JtqU (.t; g-, L- rLg; -r:) a-o)li t*+ ,c).--aV ,;,-l1l o:te---,1 tr .(c*t u*-sljbbss ,r-tL ,ir.:-^^r ,d! ,1, ai;rt+!i ! l-! ,r[,

    :-0, ] iub.t+5

    , .-l.uk tS,_ 'r-l\.5-_ ;tfrh .5 (JtS-q ,U t 6lo.lr; Jt-tay>ti) +r*,_:.: ,;-rrflt 1rv1 rl-Y

    .+j;GI (1\-Y) L,t J.r5+t-., lr err- ibst ,r-tt r" t, )lL,L-51 6l-111 .5 (JtLt, ,t,, t ,5lr,.lr1 ,-rt'^.i!) +-4i ,r"l-rr3l 1ff1 f"-Y+D ((r-,)) cl*-9^,, ,, .rJ a*l-.' l-, e:* jbst ,f-tt ,o ! )12 Z-X-Z

    .+J.-.,tj ,st-l:, .5 (JtLt, ;t\:+ lgla.trq t-t12aa. )l:.) +--t ,;_rr(Jt 1ff1 f \-Y+V {r9)) c*-;,, ,, .rJ a*1.- t ,}r.*, bb1o e_tt r" t, }Iz X-Y-X

    '+5

    ( Yc-.-.'i) uaSt&bl d-r-U-l 3 +--i 6la'Lr21;,c*l oti.; i-s; AtanY eb LJ t:*6 ihiL ,t Jt .\

    .xt i_J- ! eb.5..,- ti A-^;e J, t4-;:f- €lre,t-*6 t, lt";rl csjlt;:J,:L i;i^, .la:rd?rt\yb t+r,ri.g? uKrl:rae (y x y ibs,-,r-tt,sb+) 0 \:l:t, ! -. r,rt.6; .5--,'-;l: -b tt,-&-t orV{ .r+, jt-,:j (Y x \,r-tV.i- .sb*)+r.ai JtS-t. ,1.,; q 6lo.t1.r;; .{

    "-;* ,lS,}* (r,y,,0) ;-,^ cr_Uo*

  • .L*_JU JS- t (r,y,0) ,j-+; ,slt ..rrr,,l+l r t, :::;;:

    Yf

    +$;UT ,* t, uLttj

    Procedure UTOI (VAR uform: vec3; VAR iform: frame);

    Jtf :l .1,_+) "User form TO Internal form " .jr.ii- UTOI aT.2, ofdbL 2i:l J:l r;u ,61 -elt,.tt.c-l (r-- tL w*.,V r-,f at r-u:Sorll--.1IV ,11.j+t .-l .r-tl Jy*at. *rtrkt fr;l 1r: 1# ),(r,y,,0);t.!t lt*nt$lylr.f .c*l Y x Y;te:'rrt-.i-r Y x \;tS,.2lrr1 ..5,_Z,s-&-tLlt --j.o-l [' " \] f- L;;T1t o5.r+, il:; r x f 6,3-.r;ls

    ,45;GT * b aT g ,r: f t+jl t $l Ll;r.r-) ,k ;- d +.-,-r

    Procedure ITOU (VAR iform: framel VAR uform: vec3);

    4,5':fi-. .l * ,Fb;-r;l .15 *-*F"qr, L ,.,!-.rr rr 6+:*,-i 6la.lr12-; .f

    Procedure TMULT (VAR brela, crelb, crela: frame);

    U-l u.;l d+J u3' .c*l Lflf f-'- -/-rL, t,tstt)t Jrl rb r: ,d:, rD,i.(\brela - AT '>U.) ,;L.f uQ^d rJa:,r-r 1t+l -br 31,, ,srS,l.h.:;.

    +J;UT t) J.t-tn+lt;.,ir,{L; ay|'b "1,-+.S- af+-,-i seluL2li .f

    Procedure TINVERT (VAR brela, arelb: frame);

    ,t^;L 6,,_l u,Ll .J +J a--.3; .r;*l tA)f lJ) r+ J cr)1;3 Jtl .& ,o-f -,-lt,tt.(brela - AT y:.) +L;cr LF.L rJAr,rr ,,9:l r.b-r: 6t., ,s.,6

    .15 ,rli- 'l iJ=-:t .s\)) Jp'-att hf,rtla y-l .Jlo..\J o:l: ,r-1 ,shrtrt+a .0t-*S q,(,:-l d>.y) r.zr...>/ 0 i\ ,t.t) [o y 0];-t d-b.- ! (Y y-r,^,.1r)

    d t*+.5 +J 1..J ,+ J) r) U|,(\O-Y) J't' $L r5rlr; .:5.3,rr o:l:J TMULT 6tra,.[y * .xl*. d +.-,-i Lsla,lrt..re: ;E^; I hpt+; U-l 4J,c*l pl 3f a:.;ji c-r+ 6udrl, *J^ ! (frl ,st*--r.r:) TINVERT

    {A} * €-# ! tB} -.r,rtl,t ,brela.},-rr,.y-!r[.c*l (;) relative,,rL,,* brelarr rel .\'l''S't "':'i

  • VA WP

    .xiJL;-l

    XT - [, y e] - [\ \/o - \/o \".r.]

    XT - l, y 4 = l" r' Yt. f0r"]7,7 - l, y el- [-lr " - Y/' - \'"/"]

    ..r:1L.j v4 ,atr r, F,L: '^b:+,(.+y ,r,,[a--) .r5

  • JA t, .,S1', tfu tJiLAi t{ 6J L.;:-,u

    rn.ri,o \ -ldtL ;l l.rc:S rt7-l 6bri dJ .b-rr .u* b .Sr rJ c-l ,5f rt' .-(.-lt .-.st^ =ii;l ft\ At 6ki'.' , i-r,3 ,.rLJ tLe2-,;tL ,&t^+ ftr ir9Ja,..-2r .JSu..S*-t^l- ,4tu_4 ;-rcr tgJJ (r(, g;t.-i:, L- ;r -rn t- iL.; r, c.;) ;ti&|.t-Je trtb,Sr ,)Lj r.qbJ Lr-x.r uL#1, i-lL


IKO - bankowość mobilna PKO BP

IKO - bankowość mobilna PKO BP

Wichtige Informationen! iko / corratec Bayerncup Seriedienstagsradler.com/.../04/Iko-Corratec-Bayern-Cup-2011.pdf · 2011. 4. 6. · 23.07.2011 iko / corratec Hill–Climb in Kiefersfelden

Wichtige Informationen! iko / corratec Bayerncup Seriedienstagsradler.com/.../04/Iko-Corratec-Bayern-Cup-2011.pdf · 2011. 4. 6. · 23.07.2011 iko / corratec Hill–Climb in Kiefersfelden

Wersja demonstracyjna aplikacji IKO · 2013-12-13 · Wersja demonstracyjna aplikacji IKO Korzystanie z IKO jest bezpłatne, proste i bezpieczne. Wystarczy konto w PKO Banku Polskim

Wersja demonstracyjna aplikacji IKO · 2013-12-13 · Wersja demonstracyjna aplikacji IKO Korzystanie z IKO jest bezpłatne, proste i bezpieczne. Wystarczy konto w PKO Banku Polskim

2/2021 - IKO

2/2021 - IKO

Makalah Iko Zat Warna

Makalah Iko Zat Warna

IKO Enerfoil - Summary Brochure

IKO Enerfoil - Summary Brochure

IKO Ener-Air - Summary Brochure

IKO Ener-Air - Summary Brochure

چگونه یک فروشگاه اینترنتی راه اندازی کنید؟

چگونه یک فروشگاه اینترنتی راه اندازی کنید؟

IKO System - Automatiser sa prospection

IKO System - Automatiser sa prospection

SA DE/X - IKO International

SA DE/X - IKO International

IKO Appli Guide Ro Web

IKO Appli Guide Ro Web

چگونه یک فروشگاه اینترنتی بهتر داشته باشیم

چگونه یک فروشگاه اینترنتی بهتر داشته باشیم

Proyecto de Grado Iko Postres&Aperitivos

Proyecto de Grado Iko Postres&Aperitivos

DIRECCIÓN TR FoQ Hospital San Joan de Déu. Barcelona.

DIRECCIÓN TR FoQ Hospital San Joan de Déu. Barcelona.

AISLAR EFICIENTEMENTE€¦ · IKO enertherm, sistemas de aislamiento de alta eficacia 2 3 IKO Insulations 4-5 Un mundo sostenible 6-7 IKO enertherm 8 IKO enerherm rooftop 9-11 Nuestros

AISLAR EFICIENTEMENTE€¦ · IKO enertherm, sistemas de aislamiento de alta eficacia 2 3 IKO Insulations 4-5 Un mundo sostenible 6-7 IKO enertherm 8 IKO enerherm rooftop 9-11 Nuestros

IKO ADVA NTパンフ 1124 - ikont.co.jp

IKO ADVA NTパンフ 1124 - ikont.co.jp

IKO: GUIDE LINEARI · 6 7 GUIDE LINEARI IKO C-Lube è il più avanzato sistema autolubrificante esclusivamente dedicato alle guide lineari IKO. Questa soluzione garantisce ai pattini

IKO: GUIDE LINEARI · 6 7 GUIDE LINEARI IKO C-Lube è il più avanzato sistema autolubrificante esclusivamente dedicato alle guide lineari IKO. Questa soluzione garantisce ai pattini

ALLINONE DAKSSEEN - hdobv.nl · IKO carrara,de witte bitumen dakbaan voor het ECO dak 10 - 12 IKO compact roof, massief dakpakket voor extra waterdichtheid en zekerheid 13 IKO green

ALLINONE DAKSSEEN - hdobv.nl · IKO carrara,de witte bitumen dakbaan voor het ECO dak 10 - 12 IKO compact roof, massief dakpakket voor extra waterdichtheid en zekerheid 13 IKO green

VOCABULARY - فروشگاه در راه ترجمهdl.alijafarnode.ir/file/sample.Vocabulary.F.T.C.B.S.workbook.pdf · VOCABULARYFORTHE COLLEGE-BOUNDSTUDENT ByHAROLDLEVINE ChairmanofEnglish

VOCABULARY - فروشگاه در راه ترجمهdl.alijafarnode.ir/file/sample.Vocabulary.F.T.C.B.S.workbook.pdf · VOCABULARYFORTHE COLLEGE-BOUNDSTUDENT ByHAROLDLEVINE ChairmanofEnglish

Voor elke toepassing een oplossing - plastiekvw.be · iko insulations iko insulations maakt deel uit van de internationale iko group een toonaangevende producent van dakdichting,

Voor elke toepassing een oplossing - plastiekvw.be · iko insulations iko insulations maakt deel uit van de internationale iko group een toonaangevende producent van dakdichting,