Vol. xx No. xx,.1 5, 200x 1 Effcent Comutaton of Otmal, Physcally Vald Moton Anthony C. Fang 1 and Nancy S. Pollard 2 1 Deartment of Comuter Scence, Natonal Unversty of Sngaore 2 Robotcs Insttute, Carnege Mellon Unversty 1. Introducton One aealng vson s that a user should be able to desgn a moton trajectory by settng a small number of keyframes and constrants and that the resultng moton should reman otmal n some way. Varatons of ths roblem have aeared n comuter grahcs, bomechancs, and robotcs (e.g., [9] [6] [8] [7] [4]) Deste great nterest and research rogress, however, otmzaton s stll vewed as mractcal when the number of degrees of freedom s hgh; the reutaton s that otmzaton wll be slow, cumbersome, and dffcult to gude toward a desred soluton. To obtan moton for hgh degree of freedom characters usng otmzaton, researchers n comuter grahcs, for examle, have exlored technques rangng from extreme model smlfcaton (e.g., [8]) to enforcng stereotycal momentum atterns [5] to dscardng hyscal constrants entrely and relyng on the anmator to gve the character a sense of weght and balance (e.g., [3]). We have found that hyscally lausble moton can be created for hgh degree of freedom characters by choosng to use only constrants and objectve functons wth dervatves that can be comuted n tme lnear n the number of degrees of freedom of the character. Constrants that fall wthn ths set nclude many of those used to enforce correct hyscs. For examle, the swngng character n Fgure 1 can aly very lttle torque about the bar axs, ground contact forces should fall wthn the frcton cone at the feet, and lnear and angular momentum should be conserved durng the dsmount. As the bass of our algorthm, we resent a anmaton, hyscally based anmaton *1 Block S14 #06-08 3 Scence Drve2, Sngaore, 117543 *2 5000 Forbes Ave., Pttsburgh, PA, 15213, USA Fg. 1 A sngle-fl dsmount from a hgh-bar. (To) Intal guess based on knematc nterolaton of full body moton. (Mddle) Flght duraton s 0.6 seconds; Fl osture s tght and maxmum heght attaned s below hgh-bar. (Bottom) Flght duraton ncreased to 0.8 seconds. Fl osture s relaxed, and maxmum heght attaned exceeds hgh-bar. new formulaton of the equatons of moton that allows frst dervatves of constrants such as these to be comuted n O(D) tme, where D s the number of degrees of freedom (DOF) of the character, vs. the O(D 2 )tme exected from drect analytcal dfferentaton, numercal dfferentaton, or automatc dfferentaton. We add to the exstng body of research ths O(D) algorthm for comutng frst dervatves of a broad range of hyscs constrants for mroved erformance n a otmzaton context (also see [1]). The theoretcal seedu from O(D 2 )too(d) sossble because the ndvdual torques at the character s jonts are not comuted hyscs constrants are formulated based on the aggregate force and torque aled by the character to the envronment. Because ndvdual jont torques are not avalable, our objectve functon should not deend exlctly on those values. Our results suggest, however, that hyscs constrants and a knematc measure of smooth moton are suffcent to cature dynamc effects such as squash-and-stretch and xx xx 1 200x xx
2 Anthony C. Fang Nancy S. Pollard tuckng for faster rotaton, as shown n Fgure 1. 2. Effcent Physcs Constrants We state the otmzaton roblem solved at each stage n the followng form: mn x h(b(t)x) subject to c(t )=0,=1..m, t [t s,t f ] where h s the otmzaton functon; B(t) sasetofba- ss functons; x are the coeffcents, the free arameters of the otmzaton; and c(t ) are the constrants. We use B-slnes as bass functons and follow the standard aroach of enforcng constrants at a fxed number of keyframes. Constrants that enforce hyscal valdty can be formulated as lnear equalty or nequalty constrants on aggregate force. The aggregate force s a reresentaton of all external forces and torques (excludng gravty) that would have to be aled to the character root to exlan the character s moton. For examle, when the character s swngng on a hgh bar or monkey bars, the amount of torque that can be aled about the bar axs s constraned. Let aggregate force f 0 be reresented as [ ] f 0 = f a 0 f b 0 1 where f a 0 s lnear force and fb 0 s torque about the world orgn. Aggregate force s translated to a constrant ont c as follows: [ ] [ ] f c = f a c f a 0 f b = c f b 0 c0 f a 0 2 where c 0 s the world vector from the orgn to c. The bar contact constrant can then be exressed as τ max < s bar f b c <τ max 3 where τ max s the scalar torque lmt, s bar s the bar axs, and s bar f b c s a rojecton oeraton that results n torque about the bar axs. When constrants are formulated as functons of aggregate force, they can be evaluated effcently usng any effcent nverse dynamcs algorthm. However, at each stage of the otmzaton, dervatves of constrants and objectve functons may also be requred. For examle, the sequental quadratc rogrammng algorthm used n [9] makes use of frst dervatves of the constrants and all arameters are affected by q Fg. 2 The effect of arameter q s roagated u the tree wth veloctes v and back down the tree wth momentum terms. Comutng 0/ q requres O(D) tme and results n an O(D 2 ) algorthm for comutng the momentum Jacoban. both frst and second dervatves of the objectve functon. Gven free arameters x, the B-slne coeffcents defnng the character s trajectory, frst dervatves of constrants are smle functons of frst dervatves of aggregate force, whch s exressed as follows: q v 0 f 0 x = f 0 q q x + f 0 q q x + f 0 q q x 4 At any tme t, character oston q, velocty q, andacceleraton q are known, and terms q/ x, q/ x, and q/ x are avalable trvally from the defnton of B- slnes. The term f 0 / q, whch we wll refer to as the force Jacoban, s the most dffcult term n ths exresson, and the trck s to comute ths term n tme lnear n the degrees of freedom of the character. 2. 1 Lnear Tme Momentum Jacoban Effcently comutng f 0 / q, the force Jacoban, requres effcently comutng 0 / q, the momentum Jacoban, because aggregate force f 0 s the tme dervatve of aggregate momentum 0. The usual way to comute aggregate momentum s to formulate the followng recurson: v =X 1 v 1 +s q =X +1 +1 +I v 5 6 where 0 s the desred result. Note that we use satal vector notaton as n [2], and so both lnear and angular momentum terms are reresented n Equatons 5 and 6. Veloctes v are roagated from base to leaf, and momentum s roagated from leaf to base. Fgure 2 shows ths rocess. Parameter q aears n the coordnate transforms X +1 and X+1, and so every v j for j > deends on q, and every j for j 0deends on q. Unrollng the recurson to collect terms for 0 / q requres O(D) tme. There are D terms q, JRSJ Vol. xx No. xx 2 xx, 200x
Effcent Comutaton of Otmal, Physcally Vald Moton 3 Fg. 3 q arameters from lnk to the base are affected by q * I* The effect of rewrtng the recurson s to lmt the effect of q to arameters collected at jonts between and 0. Terms requred for the momentum Jacoban are accumulated n a sngle ass from leaf to base, and the momentum Jacoban can be comuted n lnear tme. and ths aroach wll lead to an O(D 2 ) comutaton for the momentum Jacoban. There s no clever way to smlfy the calculaton by aggregatng terms when t s resented n ths form. We observe that rewrtng the recurson solves ths dlemma: I 0 =X +1 I +1 X+1 +I 7 =X +1 +1 +I v 8 0 = 0 9 Thekeythngtonotceheresthat s exressed as a functon of v, whch s a local varable at lnk. As a result, only roagaton from leaf to base s requred, and each arameter q j does not affect terms comuted for jonts j + 1 and beyond (Fgure 3). Also note that s n general not equal to f 0. Atermsuerscrted wth an asterx should be treated only as an ntermedary quantty, unless ts subscrt s zero n whch case t s the desred aggregate result. A lnear tme exresson for the momentum Jacoban can be derved n a straghtforward manner based on ths form of the recurson (see [1]). Note that we are not smlfyng or changng the outcome of the dynamcs comutaton, only changng the order n whch terms are comuted. Aggregate momentum 0 and the momentum Jacoban are exactly the same n both formulatons. 2. 2 Lnear Tme Force Jacoban In a tradtonal nverse dynamcs formulaton, acceleratons and forces are exressed as the tme dervatves of Equatons 5 and 6: a =X 1 a 1 +s q +v ˆ s q f =X +1 f +1 +I a +v ˆ I v 10 11 where the symbol ˆ s the cross roduct oerator for satal vectors. As wth momentum, ths form results n an exresson for the force Jacoban that requres O(D 2 ) tme to comute. For fast comutaton, we nstead take the tme dervatve of Equaton 8, whch results n f =X +1 f +1 +v ˆ +I a + İ v 12 Ths equaton has the roertes we are lookng for. Velocty v and acceleraton a are local to lnk, and terms are roagated from leaf to base only. Note that as wth aggregate momentum, f s n general dfferent from the actual jont force f f 0. Dfferentatng Equaton 12 and accumulatng the coeffcents of dervatve elements results n a comact exresson for analytcal dervatves, whch are gven n [1]. Each artal dervatve of the aggregate force wth resect to jont ostons, veloctes, and acceleratons may be obtaned n constant tme, and the full Jacoban may be obtaned n lnear tme. 3. Results Otmal Motons. Fgures 1 and 4 show a samlng of our results. Fgure 1 shows a dsmount. From to to bottom: ntal moton, results wth a flght tme of 0.6s, and results wth a flght tme of 0.8s. Note the looser tuck and the hgher flght trajectory n the 0.8s moton. The ntal moton aears very unstable at landng. The character would fall over. Ths effect s elmnated n the otmzaton by enforcng the hyscs constrants of ground contact. Ths otmzaton requred 26 seconds on a 750MHz Pentum 3 comuter. Fgure 4 shows ntal and fnal moton for a monkey bars examle. Ths otmzaton requred 2.4 mnutes on a 750MHz comuter. No touch-u was done on the results. In artcular, the geometry of the monkey bars was not modeled. In ths examle, notce the swngng of the legs and arms, as well as body roll, tch, and yaw. All of these effects are obtaned as a result of the otmzaton rocess. In ths examle, the ntal moton s rgd translaton of the entre character. Tmng. To emrcally test the advantage of our method for fast dervatve comutaton, we ran a test examle 5 tmes, each tme wth the dentcal setu excet that a dfferent technque was used to comute all requred frst dervatves. Fgure 5 summarzes the results. The dfferentaton technques tested were: xx xx 3 200x xx
4 Anthony C. Fang Nancy S. Pollard Fg. 4 Intal and otmzed moton for a monkey bar examle. Technque Tme er teraton Average % error Our method 0.11s 0 Drect method 0.62s 0 NR1 0.97s 0.10 NR2 1.92s 1.0e-04 NR3 5.73s 1.5e-06 Fg. 5 Tme requred for one teraton usng a varety of dfferentaton technques. Our method. Analytcal gradent comutaton usng our aroach. Drect method. Analytcal gradent obtaned by drect dfferentaton of the equatons of moton. NR1. Numercal dfferentaton by ordnary forward dfferences. NR2. Numercal dfferentaton by central dfferences. NR3. Rchardson-extraolaton of order 6. 4. Dscusson Ths aer contrbutes to hyscally based otmzaton by defnng and exlorng a restrcted class of otmzaton roblems where hyscs constrants are ncluded and frst dervatves of constrants and objectve functons can be comuted n lnear tme. The fact that frst dervatves can be comuted n lnear tme nstead of quadratc tme suggests that our roblem s smler than revous hyscally based aroaches and smlar n comlexty to very successful knematc aroaches such as mnmzng dstance to a reference moton. We susect that our soluton landscae wll be smoother than revous hyscally based otmzaton aroaches, makng t feasble to handle more comlex characters. For actvtes where jont torque lmts are mortant, torque nformaton must be taken nto account to roduce good results. An extreme examle of ths stuaton s the assve swng of a mult-lnk chan. Mnmzng acceleratons whle mantanng hyscs constrants would roduce a result that was vald for the body as a whole but would requre non-zero torques at the jonts no whng moton would be seen. Mnmzng sum squared torques would roduce the desred results. (Of course, truly assve moton can be created much more easly usng forward dynamc smulaton.) More commonly, a lmted set of torques or energy terms may be mortant. For examle, the monkey bars moton aears to requre very hgh torque at the wast. When hyscal arameters at certan jonts are dentfed as mortant, our method can be extended to rovde and dfferentate these arameters for any K jonts wth runnng tmes of O(KD), reachng the exected bound of O(D 2 ) when all jont torques are requred. An nterestng research roblem s to determne automatcally when torques at a gven jont should be consdered. Fnally, we would lke to emhasze that the man advantage of our aroach may be as art of a more comlete anmaton system. Our vson s that the ablty to enforce hyscs constrants effcently should be just one of the tools avalable to the anmator. Detals of the desred moton could be fleshed out usng moton cature data, rocedural technques, keyframes, and/or objectve functons arorate to the secfc task. We have shown that hyscs constrants can be enforced n an JRSJ Vol. xx No. xx 4 xx, 200x
effcent manner. Incororatng hyscs constrants nto tradtonally knematc anmaton aroaches s one drecton of future work. Acknowledgements Ths work was suorted n art by NSF awards CCR-0093072 and IIS-0205224. References [ 1 ] A. C. Fang and N. S. Pollard. Effcent synthess of hyscally vald human moton. In SIGGRAPH 03 Proceedngs, 2003. [ 2 ] R. Featherstone. Robot Dynamcs Algorthms. Kluwer Academc Publshers, Boston, MA, 1987. [ 3 ] M. Glecher. Moton edtng wth sacetme constrants. In Proceedngs of the 1997 Symosum on Interactve 3D Grahcs, ages 139 148, Provdence, RI, Arl 1997. [ 4 ] M. Hardt, J. W. Helton, and K. Kreutz-Delgado. Otmal bed walkng wth a comlete dynamcal model. In Proceedngs of the 38th IEEE Conference on Decson and Control, 1999. [5] C.K.LuandZ.Poovć. Synthess of comlex dynamc character moton from smle anmatons. In SIGGRAPH 02 Proceedngs, 2002. [ 6 ] Z. Lu, S. J. Gortler, and M. F. Cohen. Herarchcal sacetme control. In SIGGRAPH 94 Proceedngs, Annual Conference Seres, ages 35 42. ACM SIGGRAPH, ACM Press, July 1994. [ 7 ] M. G. Pandy and F. C. Anderson. Dynamc smulaton of human movement usng large-scale models of the body. In Proc. IEEE Intl. Conference on Robotcs and Automaton, 2000. [8] Z. Poovć and A. Wtkn. Physcally-based moton transformaton. In SIGGRAPH 99 Proceedngs, Annual Conference Seres. ACM SIGGRAPH, ACM Press, August 1999. [ 9 ] A. Wtkn and M. Kass. Sacetme constrants. In J. Dll, edtor, Comuter Grahcs (SIGGRAPH 88 Proceedngs), volume 22, ages 159 168, August 1988. Effcent Comutaton of Otmal, Physcally Vald Moton 5 Anthony C. Fang Anthony Fang s an Assstant Professor n the Deartment of Comuter Scence at the Natonal Unversty of Sngaore. He receved hs PhD n Comuter Scence at Brown Unversty n 2003. Hs rmary research nterest s n the synthess of hyscally based anmaton of humanlke characters. Nancy S. Pollard Nancy Pollard s an Assstant Professor n the Robotcs Insttute and the Comuter Scence Deartment at Carnege Mellon Unversty. She receved her PhD n Electrcal Engneerng and Comuter Scence from the MIT Artfcal Intellgence Laboratory n 1994, where she erformed research on gras lannng for artculated robot hands. Before jonng CMU, Nancy was an Assstant Professor and art of the Comuter Grahcs Grou at Brown Unversty. Her rmary research objectve s to understand how to create natural moton for anmated human characters and humanod robots. xx xx 5 200x xx