e + A f e l : 4 E : E l E l Vol.2, No.1 P.48-60
Forward Dynamic ^JE4"f rJg
"l+t e+q :a4q otLl"told
Gomputer Graphic Animation based on Forward Dynamic Simulation
c t r l d
Park, Jihun
* q 4 q r
Hongik University
o!1
3-
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Abstract
This paper present a new technique for doing realistic computer animation. The method is based on forward dynamic simulation and nonlinear problem solving (parameter optimization) technique. Objects are modelled physically and simulated faithfully while satisfying kinematic and geometric constraints. This forward dynamic simr.rlation gives us very realistic motions especially for non-voluntary motions. Then we extend simulation technique to do animation using parameter optiniization. The basic idea is to add motion control ov€r'the entire animation. The motion confrol is finding optimal solutions while satisfying user's animation goals. We provide two different animation technique; one is for rigid body without joint actuators and the other is for rigid body with linear joint actuators. To achieve motion contrcl, we conv€rt single sirnulation to single nonliner fimction evaluation while eitler setting initial conditions as variables for the function or allocating control variables in terms of time. This method is presented with two animation examples: dice-magic and human stand-up.
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Computer Graphics Society 48
Computer Graphlcs Anlmation based on Foilald Dynamic Simulatlon
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49 Computer Graphics Society
Fonrard Dynamic AlEEllol{lE 0ltE! g#El :1rf4 0[Ll40ll{
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Computer Graphics Society
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50
Computer Graphics Animaiion based on Forurard Dynamic Simulation
dynamic simulation ql oJqqE, EE| EJ Hl C€ +
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*-JiE' gaJ",rl zl 4, TlElEi4 J9 model g ^l+
6lq 5-zlel 3g +46JEl. q olB"I 9lq^l dynamic simulation I inverse dynamics *+= 7 '-lq 4 e6I= +=ol o g, forward dynamic simulation r+= C Elqlg d= *^19 7li]El.
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el Fzl 6J+= 3 x| polynomial ol 4.
Suboptimal Control[94] 7l Bg E=el z]g'l
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= dlBg inverse dynamics E,tl$t 4'JEll, University of Pennsylvania ql^l dil6l= "J^i "l El.tl4l =E4gq ol ElBg local optimization I
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5zl9 forward dynamic simulation I ^I+6i;tl ?*
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51 Computer Graphics Societl
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simularion "il +€9= +ag TlElEl ziog "J=
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^ ' J q ^ l = + 4 € s s " J + ^ d € c i i l t E + + E l . sdJ Enloil bounding box € 491^l +E +rlg 4e2l '-l=Eil, ol 4lt.il^l= Fzll9 d4€ol ,tJe.
+58'l=zl a^l6iE aol +,c6'lEl. ol +Ae- ++
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4.
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= 4o14. + 7]dol A +4€ Ultoil €4lalts4 q+= 4l+617.| 96tl^l= B +4€el ^.ilEol EJE
= 4l4lel €E *^1C +E61tr €El. + ol5 qH ,+4e B el 6J Bg .jr6J6)q, B +4€ql sl4 01 +q zl= €Er+ +4014. ia "J+= zl 4ol ^ll E q l E4 €C9= se ^jTllel € E * " i l + 4 l g q , r dg +4e A e-l LI+.l +4li}= aolEl.
3.2 Ealttrl rEg
E l g e Cr]El 4oJ qg{9 + q € + 4 t rl. zB 2 = El'-lel "J4= olilg *Aoil^l forward dynamic simulation + At+= g€?rl. :z B9 u-21 €21 6171 9l6lq "J4= ol:|fl dru[E :afl4. ^lEr.ll ol dE .371 "il= oJxlzl il E d{lg ai46J a'+ 401 €9 B + ileq, f + {rl fl e 4 g 6 J E l . E l + = 6 l + E E + ^ s a r l g q e +c8i9+. oi el 6J simularion g 6lz1 gal€ gzi .I ql6J C4Et 4Sl t-g"de Elgr+ eEl. ?qol zl+EI= Elt}€- y +ol 4 6ln, olal izl "J *6J g x +ol elr 6izl. 9zl9 dz*9 . ol e| a).a +q zl+EE g el rl4, ej+ql^] ++6I= EJg 7 + 7 + x , y + g g f , , f , 4 + 8 ,
i , : - r n g ' f f , , | : f , z l 9 4 . o l u ] +
ODE(ordinary Differential Equation) 9 =zl fiog +El ^l+ql 44 €.E F4 rtlzJg "J5q tEl. ol +I4.I tr14 "Jzl= :aqB 34EJ simulationol € E+. E.E simulation 4:Jol 9lq^l "l+e+el +E ' l 4zi d+e 4+"I^j a49= TlHg ^l+4q 4 q 6 i q o l e 4 .
r g 1 . "J:l ^lg4ol €
o l Xg 4q'X.l^J Ezle dl+qE, 9|4zl g6i= oltrl* SBEJ "Jol "JC6}E+ d'lqol i}4.
Cl= 5tr r oJ:l7l Z7l +l^le +El I ElEl €q a *ql czl6lsa+ 6l= 41"+=dg 7}6JEL. ol qe
^ilE =dg uj4 rlzlE 4q"Iol€g 6lz1 93il^]
= "J.l^-l 7j€+, 6l+q ^lE4oldg dl4el il+
d 4 l + o q + a , + + + a + 4 4 ; i l 8 * z l q r J 7l HEg ^l+6}qol €'JEl.
3 . 3 7 l + f t 4 r g g
J.;l €.4l€ simulation 6l=El qq^l €.:lE ol ^lE- +€-a €a5lq 919 4+, zl+6+
(kinematics) "J r-B*E *ls-61'}. oJqls ^l*ol 4+ sgeg gq ^l*el gg -r.rll Eile A+
6 ) a 4 +9l= +€-e €A+El. fq+ 4 +€ol F z l g + i l = F : l € q ( 4 E ) " I = + x l r l t l 4 . ol el -J +de S"Jseriq zlE "t +e 5zl € qel ^I+C r.eJElqdl 6JEl.
3.4 =E r{El
.r.41 E,ll ^lE40ld (sel€€)"il^l c4q 4ee ;59 E-E6l7l 9l 6J tsJHe +7lzls +ts + il=EL r a4= ^sAr+ tsqE ^l+61= * B o l ? , F d 4 = + F e * 3 i + E 6 i l 4 C E . 4 q 4 - 9 . e d 4 6 l = , ' J B o l 4 . ^ s q l r + g r l = ^ l + + o l B e + € E l 7 l = f + + , ^ g " d r + g q e l 6 J . 1 € + 4 oJ qlol oe 4+^l alalol 9_4 aei^1 tsl+4 6l :l *6|El. E6J ol *tse {q+ +C +B 'f 6i 4 * B o l o i q 4 .
4 e * B e c q q 4 s e 4 4 6 1 = " J B " J s l , ol 49 Hahn[9]q] e] 6il 'il.J€ 4ol E]. + crlT]
+E6J q fg r + *ol +4E + *El. q7l^.1 e r l 1 ! + 2 = C " I 1 + 2 = . l 4 . l l r , i , Z + itl 6c+el Eel=, 7 qEi= €iil 4El*6Jg + 4'11q, ; 4El= +4 ?C+"1^l +Z ,l4rl'.l3 L}El.rlq, 7 ntrt= +€.s}= zl dg 'atr.il r{rJ
Computer Graphics Society 52
+4 EqE ulul.llz, e ts *E 4l?€, p E ul
*dl+= +4{4. qzlq^l I dz} + ts *E * sl +48 +4dEl. +q= 15 zlel ulzlel B+ rt C + 6 t o l 4 = E i l , . r . E e J, 1 i , , :,
+ * a +
0 t, 0 z ol4. s+ , ts +EE tlel "Jel el qqolq. 1= -t
* 7 "14. =4-!' ;elg Fzll el €-{zt +E-^l t'lrlq :l;<l f}iQ=. 7}C+E}.
' 3 e 1 a d g a ^ 1 4 e * , : B q l e l adol Cg6l 4 e*gd 4l4l€ Ae 4** alolr Alt}ol tE.
€4. =44 =aol "J+48 ul:zel zlzl g*=El=
zl4g 444J. El:l+el + =dg dzltcl.
. n\ (e-a:J
* , ( E - A ) - J
t -
\ l 9 i - 0 l r t x I
h ( E - A ) : 7 , x ( - i )
( : i + r i x h ) t - ( 1 2 + 0 i x r r 1 . n :
- a ( i i + d "i) .;- (3 + nf xv).trt
"l4e-l + 49 C47l +E^l Elrlq zl zl 9+
g g 44t 4"14.
t ( f + - 6 1
" 7 ) - ( f, * - 6 f
, 7 ) t . i : o t ( f , + 7 I " 7) - ( 3 * e f " 7 ) t . ' i : o e z l 4 ' f : ' I x ; o l t , 1 : i x 1-1x i) qq, i = d.il 4ltt€ +E"*olr, y' = a++
+E*o14. l5 7llel B+ol 44 =9C 4, ot4el a A o l CB4Ezl q"J+El.
t i x l ' 1 x i ) t > p t i . i t
ol adol cgdltr Elllel xl= aol.a, zl;
47'lel 4lte s+6lEe 9lel + 21448 Sotlr ol4e-l + 4C Eil.tl tsts4.
rA 2. +87.J + C44 +E4.1
3.5 er| g*E {el
n4l gr[4 "J4+ gEg e +4= Czl6tll
€El. olag :1 €.dl+ :a +el +a^lololl El+q ol fzl|c}zl 4E.ol4. drJ+= ^lE4lolrd +e ql ul"ll"ldg ElTl q4q +5|g 4zl ,rlzlzl 9lt BIBE gg4El. +E' il4et El+zlzls 4s"J- Erl sgg ^l+g + gq Fqdlg 6l+e 4^i 4 *Eg ^l+e + ilEl. se €-{lz} ${rl {zl
€ +El"il qle 44E Es+ril, 4zlg +4 44
I BaraffI3l.ll el-t dl"J qfi,4. f,g 2 = '{ 4l
el czl +Elgl ciilE !.qt4. 7 = a+ eq i 4 " J 4 e l o l r , 7 = a+ zldg 44t4. * gl - = d+ ,14 i, "l^lel + C4= +El.{r, i
= 4 d+ zldel dgolq. E.r,il7l dzl *EI"|lr Ee C4 +E +.I €"1 ++61s.e
d , > 0 , 7 , > 0 , - J d : 0 .
f B 3 . 4 z l € E a l q q e d l A
I,'i: o -
7 ' i : p l ' t ' i l - l
i, P3
53 Computer Graphics Society
Forward Dynamic Allilol{!t ol=e E#El filq olltl[tol{!
i
motion
:"8 4. El+44
rB 4 E El+;ll4 oI qlEJ ol 6tg Eele4.
4ge al^l€ zJttJ "ld oil4"lEl. "l* 4rl 7 E d+ tr41 $uN;|iEL €alTt *4oJ q= 0 ol E9 AC +=4. E-4l7l 49 d= E^ils +El tsJ
= s l q e j q o , r g . ^ i l z l + E ; s + 4 " J 4 r. 7lA+E+. dz] ; . i € 4^l6le "l*ql * t ?.rr= €C + 9.1=rtl, o1 at"l 0 ol olqg "l +qe 4g'l AEl.
?
t p G . i ) t - , + ,
I h l
4 g a - g t l E ^ l * g A+ +E sgg S5|
czl ol r* ^I+g + 9lq^J €Elq Eri.e 3E| zJ zl E 9l ut sEf4EJg E €S-zl S4. :t44 ol o l H e a + + g E 4 r l € 9 6 J * H o l "lqoe,
*el 444 *BC ^l+6l9El.
4. Forward Dynamic Animation
t +tEl =:t {tol
"JRl4g.E. €:iIEg ^J€-ilel +^Jgile +ts + 9.1=rl, ^Jgiil + F5"J aq 6lzle 6lzl. 5 CEe Elg "J,.J+ + 9l= +*rl5g zlzla !l n, +^dC4lE€ zlFzlzl S4. 4R! 54 7l7jlE I dzlE.I el 6J 6Jg EJ^8dl7l g+ 4E7l= 7l ,l? 9],4. = +4= s= E.^il= 4Fzl€ 714 g rle| zlzlzf 9*g cil.e. E+dJ + ")4. 4*zlzt il= A+= zl"J4"J gFg "J=q t4. :zl4 o l g ^ l 7 l +9 +a+ d+(5El +E)g a+, ul zl"J4"J gtlg 'J=zls. 'Jcl. 44zlzl 9l"E q
! + + 4 E €g gCaiq €,llel CF"ll +CC TltJ + 9l7l 4lEolEl. .rel+ 6'Jg +Eq q= zl t T l z l S = A + q = - s = g 5 o l H l 4 " J 4 o J g E|ol4r 'Jg + 914. q7l.l^l zl*4"J €-E4 Elzl"J4"J +Fel ++g +giJEl, ol3 ulzluf{
Computer Graphics Society
"J gFg €€++rll simulation 49 animation al 71 9l Blq forward dynamic simulation I ^l+6)q olo.! alzl q$ol4. Forward dynamic simulation g " J q + = C q 4 € + g d # q . e 4 l + 6 1 = r + A ol:la g + il4. ol el+ dlBg ol+6lq animation alzl 9st^'l= ^l+zl9 +4.|l *=
simulation C +9 + gl.qB €El. .l el + t+AC 5zl4l EI"J 4lq4a +e.z}. relE computer
graphic animation = computer graphic simulation +
motion controi ol e €4s €Ec + 914.
+^JC'l 4l;J forward dynamic simulation4l gq^l a!+4sE. uJ.g6l= -E= +Ee d4se azl;1o1 eldl r++ g.E_e aTlil= €++ 6lE 'J+=
"JE + Sl4. zlFTl= 7ld r8E4ol Llq
^-i =7lil :+Rl 6lq4 *.zlel +Aql 9q^l +F 7ls "J6lJ €6Jg 'Jrl qtrg, ^lt.ll trl= E+=
^l+6lq F4g 4lq6lq simulation 449 ol ++
animation g EJ + 9lE+. E+ ^lEelol.^Jql a4=
a+ ^lrJg ^ld"l g + d= A+71 F*4. olol t r 1 4 4+4ss a q = ^ l r J g E + + e + 9 4 . .JRl4ss gF Ela^lg ol:l El + *c:l ol os oDE g ^lrJql E++ €.RSE+. 4€!^1<19 E++
6171 9l6lq, 4+ ^lAg El+$ E+s ?t, = S + € E l + * 4 4 q 1 4 d * 5 l q + E € 4 .
4.1 +95{ll Slt animation
+^85ill= qq+ zl*,7lE 7lz1a- o)21 ?*71 4+ol, simulation .E+ql "iclt zlE4"J flg .fl o l ^ ' l F4g +48 + Sa, d4ee +9 +Aql 96lq F4ol a4€4. ol elEJ A+, simulation I a 4 = d 4 s s 1 7 \ 1 k q a l + € E l . 4 g 4 o l € g ol+6lq .lle"ll.l€9 6,121 916.lq *al{ {i!;j dle+ aol qq4"l€el ^14+El *Ezlzlq 44 g 6l+el ++-? +=qol +4.
r B 5 . 2 l q l 7 1 4 2 r r { 7 1 5 ; 4 q ' I q d e l "J+
Computer Graphics Animation based on Fonvard Dynamic Simulation
- -tx1
-rA 6. 4zll7l-r^.Al 7l Fel forward dynamic animation C 9+
6J+q
"J+
r e 5 = E l i g d 4 € e s + ^ . r € * q E I I
€ n l e + F 4 A o l , J A + 4 € 5 i l 7 l g n , I t + + TIel qol 9lEl. ol 4* "zetlzl 4 a4zl F" +,I zl +=zl. ol gxlol .! j^/ll 7l € 'J4 4 FTll= * dJ + 4r/ll 7l F Slq_E 5-q7lE+ t54r eEl.
ol fl4i^l dl +6J+ (+e +4 ++)= 4.ell 71 4 4 e t l T l + + d + e l 4 4 s l ^ l + . 1 4 , 6 J + e l E + = x, Y, z u3";3:e-99 g5-*ol +. ol A+ ^tl + =a e L4l 71 7l' F"Jql 5-q7loi al4, Fztlg qql + e6l .F +4= ?ol ri. r€ 6 = ol zl+ +zl.l rtl 6J 6J+E ^84^l] "J+= r+4rl=Lll, X *6J9sq c F * € - n 4 ; l ! . y , z * 6 } - q s q e 5 * g 0 . 2 9 ,J4ei- trs+^l 7l tr^.1 6J+Ag e^ldll Bil4. r B4 a"l "*g ++ 44il(local minima)E dJd'J i glqlrl. g5-"1 Hl zlg4"J +€g s6J6l!_s
"++zl "l+ Hl C€4ol q, y+g ++ :il zjil € 7l
^ l 4 z l q l * 6 J + 914. +Edl sBJqE Stsol q
€ 4 1 " J q g f l z l q l = 6 1 7 1 . J = r l l , o l aol +471 € 2++ :=rl= .l++q 6l+o1 4. azl :l ol afruJ Eel"= 7l;lE ari4.J s5-ol ot+ 4_=_11 "J=- + 9l+. ol zl 6J =^J=9 7|rl 6J+el A4dl= * ,l 96lq !l C€ EE-l/l'J zl B4 Simulated Annealing I ^l+6198+.
4.2 gEi{l= 9lt! animation
q7l ^l= 5El =+9 TlTl ^8C41= 9"J ^ll e g o l q E l o l € ( 5 - z l , r l q ) 7 l H g i l ^ l i J 4 . o l * H g 7l =.I 4l ^l g *B.l Hl -ll =+ zl alof laE r|$a|r 9l-s-+t201, ++ zl4loll El]+ Eq+e ^J + + , j B o l Q r E o J t 2 2 l = + € E E a - q + o l q ' ] l . l € 6l9El= dql^l ilol 71 914.561 ol HlBg 4 = y i g " J H E - o l 5 - c " J + 4 9 . I q " t l " i € q l + 4+=J + 9l= 7l B_E_"J.I tsl 6I, Zg l-= ++E .N4oil"l €ql ^I+sJ + 9lEl= aol Q. oletle y*g
oJB=ol ^l*el ,J* 4 ^I*9 €ql Eil+ .]lq"il o l € 7 l H g F " J r J 7 l B s s d l a d J + S q + , o l
* B g 7 1 5 6 1 4 = d d l 4 . d € . i i l = + ^ 8 E 4 l e + 9 4 nJg B + 9l= zlFzl (actuator)g Tlzl n 914.
ol el 6J zi5.zl= CirlE gEg +AdJ + gEl. ol q"J A+, g4l.i tll6J simulation I azl ll g"j olLl el simulation Eiql +F7l "I^l g^J;J €olE ai €6Jg tsJ84. .l ul+ €"Jg HI€6171 96lq
^l ,J.l 4= zlE-71 E dJg *+El 6J g.s7l 914
! L 6 J ^ J E 4 l = E - H - E q E l r l € q z 4 = B 4 "JH]
4 s s d € e l 6 J g "J=q 'll= =+g 7ldEl. .J Rl4qs =+9 ,I+= dJ ^dgnlT} Tlzl = zl+E
!_El ?*El.
4.2.1 =qel Bg +a E-aE El+71 e'lEJ *E
oJg =+"I -Jol 4+6J oi, ol e "Jalq €"1 ++ +€.I ErTl zl+al=Zl = Zltalqol +4.
.l ag =+9 ^l zld(oriein)4 €d(insertion)9 oJ ol^i, t+el 6Jel ill 71 7l +qA 4, ol 6Jol zl€- 6l= EJ6}S "J + 9lq. s€E +dql ^l =+ol z+
+"I= ^l 4rt^14 dlElql =+ol zl+6}-= *-Jq gq BlEl= cross product 6lE, 6J =+"1 96I ul il = +€E trr<i9 "J + 9lt;}. ol Ag forward dynamic simulation "il trJ;il+E €El. t q7l^l = + E d J e 9 l .I ^ l 9 ^ l ,J 9 ; J + s " J q g q z l = t l ol -'., *99 *6Je: .tirl'tl ol € F_+.I ^Jcnlg € i l . l t r ] " 4 € e E J 4 . "l zie'il+ i9oJ 7Nt'og,
MUSCLE ORIGIN
g
--'t
')i \ ',
- - ,
a A 7 J+ soteusql 4 6J moment arm
att
II
55 Computer Graphics Society
Forward Dynamic Alte[o]{!t 01tt g#q refE ollLl[fl01{!
tleelF9 r-E'ol ef,6l2l ^l*.el 6} +€ol B + 9 ] = E 3 " I = 4 r { < l o l 9 . l t } 7 € e ; } q . l ! " I " 1 d61= d'l Hl -l €dEl zl^llB'l ^I?-el *€ e.z
= R^lg + 9l=L{, rag ^}a},ol B + 9l= +
€el E3= -r ^l*E €il €t.l rlzl Bzlzl zl q E o l 4. rls9 :B€ 44+ "J ^lol ql 9il;l=
soleus J$9 +El'JEl.
4.2.2 ^l,Jel il+E ^JC4zl qEJ *B
ol tsJH€- interpolationol 96lq q€5J ^lq
E_ ^l 'J9 flf+s. S€g + tl4l 6l= ,iol rl. olell q el= ^84-i szl. zl5-:19 g4 "Jg ol]qElol .€
"J4a "Jq. zl5-zlel ?*9 m ol j1, {alE .'Jg f(x) ol El. ^l uld?l 0 doil^l+El, a7l +E7l 0 oJ 'll, -gdzl= z|tszl= 2 iA gd'J4. "l A+ a 4:, ^l ,Je ]1A9flEl. €.E zl5-il= /l++ ?J+
e- 6-l + cl El
r I + E t + x t ) = + + + l t E l . I f o , f t , l z
€ rlzl9 4lq E+e "J=r, ol rt-E-ol tr6IE A 4 4 - q E - X r ) E E s J q . ^ l ) z l H + 4 = e l E l = +qzl 96l= qi-e E4g slE + 9.14= ag F +El.
ol *9 el€ oJRl4 ^l 7lzl. Se]zl (n+l) zll9
^Iqdg ^l+61_il., 44el ^riqd9 o, r, ..., n ol 4 '. +zzl. t-d 4lol dg ^J46il Hzl. reJrtl + 4 = + + g q e l z i l ^ l + 6 1 = C i l " l r I 6 J q q " I ol dg BIEE, 4 =:+6J"I r-l]ull ^.ls EIC 4qg zt6}qo} 6J+. ol el 6J +49 ol +71 96ll^.i= 6lLl E ^ l q d . i l 4 4 3 + 6 J g g 6 l q q e l 4 g 4 €
4 z l = T l z l = q q = ^ l + e E l . "l 4El= Z 4 +szl. !E'J +471 olqEilol ,gql ,J4= ^l ,Jg B ^ l 6 + 4 9 * C + 9 l o o - 4 , 3 + 4 4 = ^ l ,J g H + e = t r } . r q E ^ l + 9 = + t r + = E l c r + 4 4 . oleliJ {r H+ EEI€ i =+
"l^1.
a : ( +ii...zi>'
o l r l t o l q ' I o l € E ^ l = H I C € d J + 4 B g
" l + 6 l q E . q , ; = a7la|ol ?9114. "l =zl r l e ^ I + z l . I e l 6 l a a g = a o l 4 ,
- i , ,
i : 0 , I , " ' , n E ^ i + 6 l q u ( D E ^ J d + E l . 8 1 q € 9 € C . J "rl, +rtlg *BE 120,22)+y 94 ,rlqdg ^l ,J41 4dll "J€6J A4 s . a e * e € . e 2 1 g q - ; r l q d , J q A 4 g "Jels
€49 + *eq, "1 'zJ4 zlillE B+s "J-E- +
? t 4 . 4 t ^ r l q d g 7 J 4 € e 1 il e I "l gqot .J 4. ol ag =j+zJ_e-.e- 'J4= ^l ,Jg Ols} ;Jee
^q ^l ag normalize ^l 71 7l qBol 4.
ol *9 ulB.I 4dfl ^JEil {+ : e+ 6J ;iq d i . i ^ j e l E E I i , - . +&+ e+. + q7l^l
6 + 4 e ) 4 E = + E l q l j l , x , y , z = o l q ' l d l . g rltlgel zl = =l-Ezlel 4,21, a, B, 7 z 7li 4 -s.Zlel HJ6JS +4d4.
i , : ( fv fz; "' l-i)'
l : ( x y z a B y 0 t 0 z . . . 0 , ) '
5. it4 6lt= *7t+lt1 gH
".r?,tl zl + r.Bll zl F" 5:rlql^-l Hl+EL+ (S g +zl -J+)= zlzlzl 4 4ei7l F +drJel 71 4 9 ^ I + . 1 q , d J i e l E+= X, y, Z HJ6Josg g 5 - * . 1 4 . "l a+ ^lE eag 4zll)l7l F.Jql 5 ol Tlot 6lq, F7il9 4.]l +?61 .t 6J+= aol El.
f{tlr ___lr-
i -/-- =\\
| (r.\ i1J
L r - - . . J
rg B zlt-il7l "J zl dg r]4 *zl "l ts oil Ll uil ol .9
E_ Fzl. ol dl + €"iel EEI = 5J+ Ll=ril s ^I+
sl }J dJ -3,- =ol zl!-- ri ol 4. ol ol el 4l E ol q EI ol C el +4ql ulzl qel Tlzl Bl ++iE "J5+ ?14.
rel r qzj ql 4l oJ3rJg 7l8J + il=Lll, ol ;I"+
= z J € - . i L l ' i l o l € . 9 + 4 € . E d 6 . l 7 l e l 'J ' i o s _-- ^l+614Ll, ceLI= 5-zl-3 EJ=-rl 9+ +4e s_,F_ ^l+"J + 9,lEl. :E6J ol rle zIE 3ag +s kinematic "i rJ6lq €41 !i3 + 9l:- 549 7l
^l ,1 ?lEl. + ol qql ^l :- 3l + ezl zl dol ^l zlol
^ J 5-El I m €44 91s4, Czle qlg ilel +F_
r;, g ol r|. 7l.ll Ll "J 4 q -:- ^i ,J.l rl6J € I EJ+
E_ "J:,tr}. dJ+= ^JC6l7l 96lq ;rlol dc F=
'tl, 4zl ^l: 4+ ^l 4, +rJ, g9 .'ll ;(l d.J ^l A
* 0 a , l i , 2 i q l E 4 . . l ^ l q d E o l F - - g t q 3 o . g 0 , 1,2 +jl " l E - 3 - € . 9 4 . , / , r l z l q d i o l l { l e l 6 J e l 4 l 'l d } o - f ! , s i , s i , . i ' , r } ! r t l 4 - q - a 4 l q i l i q l ^ l e l i l e l 9 i l , + E , z l + F - z l F 7 l . " J i : 0 , 1 , 2 " l c l . f , A t g C € H t 6lH "l + p€716J 6J+ol 7l = 6l+ €+4"J ^l ,Jql
q /l ql ^l dl +aJ+=
7 2 , ,
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Computer Graphics Society
fg 6 = "lq.J +4"1 EI+ *+el ^J?^I g+
E +4lil=Sl, X ts*6}9_eel g€."tg r.46lr. y, z *i}e-eel +5*e 0.2e1 +4ea trEl^l7lB
^l 6J+rl9 a^14 H9t4. rg4 +ol ge ++
44il(local minima)E ga!} + 919.4. +Fol uf zlgzjoJ +59 g+6lsz.*+71 ol+ tsl C
€ 4 o 1 4 , F + e + + 3 4 i l = T l ; ( l = a g q " J e + 914. +Eq s.JqE gtol qB4 "Jqtsflzl q+El7l El=El, olaol 94zl €€zJ9 =4= ol ++el 6l+014. STlilol =?+ HEI= 7l6llE a r+4ql €-Eol ol? 4sil "J= + q4. olel+ F c=g 7la B+el 43ai= ,*71 g6lq Hl cB
rE-:a4oJ zlB4 Simulated Annealing I ^l+6) 9El.
5 . 1 + g E H l d € = = : ; l l " J 7 l E Hl d€ Hl+ ++"1 Hl dB 4lq+6J+E TlTl +41"ll Efl6J 34.11= *71 gdlq cRG2[r3]= ^]
86iHEl. rE+ cRGz e+ ag El d€ package
= tlEl+,/ng +e ++ 44iE Tlzl a 9= 6}
+e-l +414 44rfg *zl *d4. olel+ E^il=
4aE'l7l +l6lq dld€ ue_:1*41,\] ++ 44il E "J*g q 4 e+q rt=g dl+q^t ++ g=
g ^ l E n t E l . c= ol *Bg "l+6lq +44 4zi atg 4= !-+e fl4Rl s+zJol+.
r g I = + + 3 u l i = E q ' l z l e - l + + A
€ Hl CB s-Er4Bg 9-1oj f,,r]E€ +E+{E}.
B=9 ^lE+ q qel ++el *6J 9l rzle ^lE 4=El, "Jg -H-+ +e 6J+ 44*q gdqE r q 6 i + q = E+ 459 4q6lq, r zldol^i 7ll +6lq 446il= *ol ++q.
: a g 9 . + 4 € H I C €
r E-:A4 EJ
"l 4dg Eq +EJ6}7l 9]6tl pVM 5lr+ Ae H€ilq zlBg ^l+6lB 4l.|l ^lAg B+q ol+
Ar+4olE+. pVM g ^l*altr 3El nr1 (t n I tr+ 4+) 9**9 e.e-^1,r'1= ^l+6lq nzl9 Ee 4l^]"tl= 6J+E "l+ttg AltElE+ 6lr. +Elzl E'l+el gE4l^l= olSg +46141 ;J4. ol A+, ++ 4lt ^laol zJEl El+ E44oJEl, 7llt! ^l a g 4+q .J+ 4l+ ^l+ol fs^l^ *4rl zJ "J + r3-€El. o.| ^lag El+e] ee^tl^l= ^l+g 4 aq= {{ol n*(a}u}el ;*+ 4lA} ^l a)*(Hl .3
€ * c 4 q R I + q + ) q l
" r " i l * a E e t a t t ^ t
A"J rs€4.
5.2 Simulated Annealing
ul C€ oe-:24"J 7l Bel fgt tlde t"l +/j1g +e ++ 44^l= zlzlr 91ts ++el + a l z l 4 4 { r C * z l * + 4 = d o l q . € - E o l s l + +4'|= +'-l qdg aolEl. Simulated Annealing g ol el ;J C,I= + + g=El, rag F4l4ql ,l
= = E a o l 4 . "l "*Hql^l= tr+ac oJele + ddlq -Jialg "Jdl E=El, "JE € aJ+ aq cl dsEl +e 6J+ *g Tlzl= E+ <lE-g 6Jtl Pl ol=ol:1, "J.J d|+ ri5-"1 qdjlq *5lEJ4F_ + E ++= ol+6lq "J+ Eol €ioJosr4, zl+' ;l 4ol "lU aes r1-=-r11 6lq ++ 44Alql ^l e qg + 9141 dll =El. 499 Simulated Annealing I 7le Algorithm g +4{ ,i.l 4.
(1) +g ^149s 7 el "Jel I ^l zl B+ al
= 1, e dq'J4.
T < - T o , i r - 7 o
Q) ^14 E+ al=g ^l+6lq 6J+a\ fl.i)
= 4l-tFJ4.
E <- fl;;)
G) k = 1, ... , k, el {}+q1 rl;il T 7l + e q = q
1u1 ,,.NE-S Ef rt5 7 = tq+o (b) r"l "ilEJ"}= il+<fg 4lt}6la
E'<- fl;')
(c) 4€.6Ji€ +6J^l A Ar+"l] tr+4 x 1 . - x ' , E < - E
(d) gE T A ,JL ^l7Jq.
T \ k + l ) < - c x f l i ) , 0 ( c ( 1
Computer Graphics Society Initial Guessinc
via User lnterfac-e
57
Forward Dynamic AlE40lf,E 0ltEt d#El f.ilEl 0llLl0ll0l{!
@) x 71 44 d+ tt5"1 j1, E zl :t 424 El+ <i.l 4.
Simulated Annealing9 ^l+;J ol E^fl dg + :ll zj 3,1 4tlg *=rll eel = ^l ,Jol q+ 94 ,J { E } = a . 1 4 . E e E + a € q o l d n l + 6 J ^ l a .rl gg €6Jg +=rll, ol al + E^IdE9 _=9;lzl 95lq EJi Zltl ^l ,J, + ^l g4ol € ^l Zg €ol E+ 6lir, E+ ^l*zlzl .J+E ee,l 6l9gss + n l 4 ; 1 4 i l 9 g j r . X l 4 . r 4 3 l q o l { 1 9 * = +
^ l r s r e g + g ^ l 7 J 4 . r E 6 J q l q ' l . l €ql 9lq
^ l = 7 l * 3 1 4 < I t g , s + a o l o l q n * e . a l o l
;l 4^l .ll Tlrl+ 4+, ++61 +g dis4l Eol€"J
+ 9l+.
6 . g g q l
6.1 Elg +^l9l
o l q l = r e 10 ql + 44+ 9=rll, +^19
€ - " J 4 q g F d i l i l + 4 d q l = q 7 1 9 , ? ^ 1 9 z l ' 1 ' E g 9 s E l 4 t i e q , + 4 d " 1 E q T l + z l "J-3 4zl "J +qlE ' 1 '
E g B q o t € E l = a o l E l . o l aol !L6l+9 ^lEd.l 9q g}E "24e1 7 1 , 4tll7lE" qle+el zlol de ol a+ Hl + EJ+7I t El+/ag :l4 'Je 'J€6J zl 9€ zI4 + 914=
' r l o l t r } . o l z l " J €gg dl d€ rsr4"J 7l Hg ^l 8+ tll€ tt= 'JBg qfirtl {}E}. ol EalEg €
,l I610l Simulated Annealing 7l B ol ^I+q fi o
Ll, +eEJ Bl d€ o_a:zrll "J 7l HE g+61 tszl aJ+.
rB 10. "l€i^19 ollLlEilol .€
6 . 2 A l g E g 0 { ^ {
q l + o l r l E l l o l €
ol ql= rB u ql +4+ 9l=EJl, ^l+q +el 6l+ +aql + 16 7tl el +s+ =+9 Tlrl tl+"rl ^1, =+9 6Jg ^ilq B+.a Tlrl a+E "tl ul ulol ,!ol r|. ol +alEg 571 96lq El C€ "J A ^ l ; I B " l ^ l + 9 i s q , a 7 l 9 Z { l 5 + e ^ } + zlTl "Jq+4. BE ^l+E +d6l ^l 9]= trEll=
,1 "+ dJ+e €n, ol el 6J E-+g R]+^l 7l = =+
.J9 3l "l 6J+= Hl t€ *c4 .l^l *.1 .I= a
" l E , t . o l e l .J . l q ' l o l del Hl + d J + € z l t 6 l = E l ts Sparc Classic 4l r.l + 2 Bol 4.9€tr1. + :l,c
^ l ag ^l+4E "Jq"l "l+ "*.1 =l*€4. r?9
=+g ^!ol €tlql 44 tr61=Lll, a4!"1 ^-l 9g
^l .l €++ Bg *+6Jg .lln 91*9 *+tr}.
7 . e E
Forward dynamic simulation9 e+El rellE otl Ll Elol €ql ^l+61= 7l-^J =ezl .J ol += d+
+ = " l q o l o l € g 7 l ; 6 } / l r I i z l 4 E " l 4 . + 6 1 B l z l g 4 S l g 5 g " J r l 4 S l 7 l + + + = 9 " I q ' t l ol €g ^l+6lq tEq rlzl "il= €rJol +7'l g=
4. :lzl 4 tsq+C ^l+dlq forward dynamic simularion 9 6J a+ ^l g4ol ^d ^l Aol 9-4 AA 4. "l ag ol+ *g ^lael +gs Fzlel Erl=
n e J 6 l T i 4 E . l 4 . a 4 4 q s + g = F z l g r " J B EIg ^l A"J tr5171 q.J"J + 9.lsss, =81 tsl zlg4"J 5lzl9 .lul otl ol €q ilq^l 7l +49 ol +"J .ilq'lold sEl= €{l €+ol g + 9.1= a ol tr}. IL6J S5-g 4lq6l= 7l Bol €Eq zl = ol += ++"rl ++ 41 4dlol y*71 419o1 Q. :.zlu|
El zlg4"J g*'ol gt.J€ "ilq'lol €9 6171 96ll llE forward dynamic simulation ol 9i4..l 4.
Forward dynamic simulation9 ^l+,-lq zlF 7lc 714. jZnl €-nlg olqltl ol€6J uli, q! +=
.l,{l ^Iq= alZTlTl +3-614. = =+ ^]49 zl +6lqt20l 6Jg "J5ol '1= ^lol 7l E os g A+,
3 + t l a " J e g C " J e l 'J 4 o l + , = + E q + e
" J r l O D E E € q o l 6 l E s u l C 6 J o l € 4 . . l , l g
, .):
. 4 . : t . ,
Computer Graphics Society 58
++ 'l +€ 716l+9 rq6lE trJ+ El dl ri€oJ
*€ g= 3|ol forward dynamic simulation 4l oJ
e s l q 4 l E l . c € + e t r 3 i++= 4lol?g ^l+
;lq 'Jgq forward dynamic simulationB 4+122) g q o l e E l H l C € o l E l . " l * B e I i z l z l * B I + ? J € E I E E j 1 9 l + . o J 4 " l d l t € " J + +
a 4 c e E l + + = . q g a o l + + + 4 4 i l = Y + o l zlzl_r. Xlcl. *sEq €+ *6*? El+:- 919
forward dynamic simulation g ^l+i'J oNrluNol ! 7l B ^lzlzl= ^-l s Eli6lq ol € *B"l EJ+ E
c 4 " J z l E + g 8 - l = a o l 4 .
8. gr E^d
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59 Computer Graphics Society
Forward Dynamic {tailol{58 olt* =#El frfiE oilq0|101{!
Realtime Articulated Figure Animation using Multiple Kinematic Constraints, Computer Graphics, vol 24, no 2, pp
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Spatial Constraints, MS-CIS Universify of Pennsylvania, 1989.
ao-no January,
lzsl
t30l 1261
t27l
t35l
t36l t31l
Computer Graphics Society 60
