Practical Character Physics for Animators

Transcription

1 Physcs-Based Characters Practcal Character Physcs for Anmators Ar Shapro Insttute for Creatve Technologes Sung-Hee Lee Gwangju Insttute of Scence and Technology R ealsm s mportant n the producton of lve-acton vsual effects when anmated characters occupy the same scene as the lve actors and the lve envronment. In such scenaros, a vrtual character s movements must vsually match the behavor and movements n the lve envronment, or the dscrepancy wll be obvous to the vewer. For example, a character who s jumpng and thus beng brought to the ground by gravty must vsually match an object that s beng thrown n the same scene under the same gravtatonal force. However, the tradtonal tools for creatng 3D character anmaton don t nclude dynamcal nformaton, whch means that the dynamcs of character moton can exst only mplctly n the anmaton framework. Some anmaton systems ncorporate physcal smulaton of rgd and nonrgd solds, fluds, gases, and characters. 4 In addton, techncal anmators often apply dynamc effects durng postprocessng, such as creatng the secondary-moton effects of muscle bulgng and har bouncng. However, the vast majorty of character anmators create most character movement through knematc means. Character anmaton n feature flms often requres a fne-graned level of control over all parts of the character s movement that can t be acheved by current character dynamcs smulaton methods. In addton, a partcular shot s constrants mght requre that an anmated character s moton volate the laws of physcs. For example, ths mght occur when a character needs to move unnaturally to stay n a camera s vew. Also, anmators are typcally traned usng knematc tools and thus develop a hgh level of profcency usng them. Publshed by the IEEE Computer Socety So, conventonal keyframe anmaton has remaned the method of choce for anmaton studos. To generate realstc-lookng moton, professonal anmators typcally combne methods such as keyframng, nverse knematcs, and other tradtonal tools such as curve edtors. Anmators also frequently use reference moton, such as vdeos of people and creatures performng a moton they need to replcate. These references help anmators approxmate the dynamcs of moton, because tradtonal rgs don t nclude dynamcal aspects such as masses and moments of nerta. Thus, Ths system lets anmators anmators must create physcal plausblty wthout the drect n- mprove unrealstc motons n 3D anmaton by vsualzng put of these physcal aspects. We ve developed an nterac- motons physcal propertes tve system that helps anmators such as the center of mass, create more physcally plausble angular momentum, and character motons. To ths end, zero-moment pont, and by t lets anmators vew a charac- comparng the orgnal created ter s or object s moton as f t path to a generated physcsobeyed the laws of physcs. Spe- based path. Anmators then cfcally, our system produces modfy the orgnal path to vsualzatons of dynamcal propmatch the generated path. ertes, such as the center of mass, momentum, and balance. For example, t creates a physcally accurate ballstc moton path alongsde the orgnal knematc path. By comparng the two paths and vewng the addtonal dynamcal nformaton, anmators can adjust the orgnal anmaton to create a more physcally correct anmaton. In addton, our system can automatcally alter the anmaton to account for the dscrepancy between the orgnal anmaton and the physcally correct anmaton. Ths lets nonanmators quckly correct exstng anmaton wthout an anmator s nput //$26. 2 IEEE IEEE Computer Graphcs and Applcatons 45

2 Physcs-Based Characters Related Work n Realstc Anmaton The man artcle focuses on helpng anmators create physcally realstc character anmaton by vsualzng dynamcal propertes of keyframe anmaton. For detals on the mathematcal prelmnares, see A Mathematcal Introducton to Robotc Manpulaton. In contrast, most physcs-based anmaton technques deal wth automatc generaton of anmaton wth mnmal user nputs. Space-tme optmzaton approaches automatcally create physcally plausble anmaton by solvng optmzaton problems subject to physcal and other constrants. 2,3 Another approach develops algorthms that control a character under physcal smulaton. Researchers have constructed such dynamc controllers to let characters perform smple athletc maneuvers, 4 swmmng, 5 stable walkng cycles, 6 reactve motons such as fallng, 7 and other motons such as breathng and graspng. 8,9 Such approaches can work n concert wth knematc anmaton and moton capture. Researchers have also used physcal smulaton to create realstc secondary motons. 2,3 These physcs-based technques pursue a promsng method of anmaton producton, but the ndustry has yet to wdely employ them, for the reasons we menton n the man artcle. Also, physcs-based anmaton s often computatonally heavy, whch prevents ts use n nteractve authorng envronments. In contrast, we developed our approach to mprove conventonal keyframng-based technques physcal realsm by vsualzng knematc anmaton s dynamc propertes. So, the anmaton ndustry can readly employ our system. Our approach vsualzes physcal propertes such as the center of mass, momentum, and the center of pressure (COP). Researchers have explctly used these propertes to ncrease the physcal realsm of exstng anmaton. Usng moton capture data, Anna Majkowska and Petros Faloutsos created flp and backflp motons that obey momentum preservaton laws. 4 Seyoon Tak and Hyeong-Seok Ko 5 and Hyun Joon Shn and hs colleagues 6 enforced a zero moment pont (ZMP) constrant for locomoton or a lnear- or angular-momentum constrant for ballstc motons (COP and ZMP concde for locomoton on the flat ground. 7 ). Adnan Sulejmanpasć and Jovan Popovć produced a physcally vald ballstc moton by adaptng an exstng moton to new constrants. 8 Despte these technques usefulness, many anmators stll prefer to manually edt moton because they need to control varous aspects of t, rangng from knematc constrants to the anmaton s overall style. Our approach dffers from the ones we just descrbed n that, agan, we re nterested prmarly n helpng anmators create physcally plausble anmaton and satsfy varous other requrements, not n actually creatng such motons. Other research has shown that vewers can be senstve to certan types of errors n ballstc moton. 9,2 Our user studes show that anmators often generate anmatons that exceed these thresholds, thus creatng perceptbly mplausble moton. References. R. Murray, Z. L, and S. Sastry, A Mathematcal Introducton to Robotc Manpulaton, CRC Press, A. Wtkn and M. Kass, Spacetme Constrants, Proc. Sggraph, ACM Press, 988, pp C.K. Lu, A. Hertzmann, and Z. Popovć, Learnng Physcs- Based Moton Style wth Nonlnear Inverse Optmzaton, ACM Trans. Graphcs, vol. 24, no. 3, 25, pp J. Hodgns et al., Anmatng Human Athletcs, Proc. Sggraph, ACM Press, 995, pp P.-F. Yang, J. Laszlo, and K. Sngh, Layered Dynamc Control for Interactve Character Swmmng, Proc. 24 ACM Sggraph/ Eurographcs Symp. Computer Anmaton, Eurographcs Assoc., 24, pp J. Laszlo, M. van de Panne, and E. Fume, Lmt Cycle Control and Its Applcaton to the Anmaton of Balancng and Walkng, Proc. Sggraph, ACM Press, 996, pp P. Faloutsos, M. van de Panne, and D. Terzopoulos, Composable Controllers for Physcs-Based Character Anmaton, Proc. Sggraph, ACM Press, 2, pp V.B. Zordan et al., Breathe Easy: Model and Control of Smulated Respraton for Anmaton, Proc. 24 ACM Sggraph/Eurographcs Symp. Computer Anmaton (SCA 4), Eurographcs Assoc., 24, pp N.S. Pollard and V.B. Zordan, Physcally Based Graspng Control from Example, Proc. 25 ACM Sggraph/Eurographcs Symp. Computer Anmaton (SCA 5), ACM Press, 25, pp A. Shapro, F.H. Pghn, and P. Faloutsos, Hybrd Control The system s man purpose s not to generate physcally correct moton automatcally but to nform anmators of the changes necessary to make motons physcally correct. (For a look at some automatc systems, see the Related Work sdebar.) Because the anmator ultmately has complete control over the extent to whch the orgnal anmaton s changed, the system s easly adaptable to a professonal anmator s toolset. Improvng Physcal Realsm We ve found that smply vsualzng the physcal propertes helps anmators create more realstc anmaton. In addton, we ve ntegrated two tools nto keyframe-based anmaton-authorng software. A ballstc-path tool lets anmators easly create or modfy ballstc anmaton. An angularmomentum tool rotates a character s global orentaton to acheve the desred angular momentum. 46 July/August 2

3 for Interactve Character Anmaton, Proc. th Pacfc Conf. Computer Graphcs and Applcatons, IEEE CS Press, 23, pp V.B. Zordan et al., Dynamc Response for Moton Capture Anmaton, ACM Trans. Graphcs, vol. 24, no. 3, 25, pp M. Neff and E. Fume, Modelng Tenson and Relaxaton for Computer Anmaton, Proc. 22 ACM Sggraph/Euro graphcs Symp. Computer Anmaton (SCA 2), ACM Press, 22, pp M. Neff and E. Fume, Methods for Explorng Expressve Stance, Proc. 24 ACM Sggraph/ Eurographcs Symp. Computer Anmaton (SCA 4), Eurographcs Assoc., 24, pp A. Majkowska and P. Faloutsos, Flppng wth Physcs: Moton Edtng for Acrobatcs, Proc. 27 ACM Sggraph/Eurographcs Symp. Computer Anmaton (SCA 7), Euro graphcs Assoc., 27, pp S. Tak and H.-S. Ko, A Physcally Based Moton Retargetng Flter, ACM Trans. Graphcs, vol. 24, no., 25, pp H.J. Shn, L. Kovar, and M. Glecher, Physcal Touch-Up of Human Motons, Proc. th Pacfc Conf. Computer Graphcs and Applcatons, IEEE CS Press, 23, pp M.B. Popovc, A. Goswam, and H. Herr, Ground Reference Ponts n Legged Locomoton: Defntons, Bologcal Trajectores and Control Implcatons, Int l J. Robotcs Research, vol. 24, no. 2, 25, pp A. Sulejmanpasć and J. Popovć, Adaptaton of Performed Ballstc Moton, ACM Trans. Graphcs, vol. 24, no., 25, pp P.S.A. Retsma and N.S. Pollard, Perceptual Metrcs for Character Anmaton: Senstvty to s n Ballstc Moton, ACM Trans. Graphcs, vol. 22, no. 3, 23, pp R. McDonnell, F. Newell, and C. O Sullvan, Smooth Movers: Perceptually Guded Human Moton Smulaton, Proc. 27 ACM Sggraph/ Eurographcs Symp. Computer Anmaton (SCA 7), Eurographcs Assoc., 27, pp Ballstc Paths Tradtonal knematc anmaton systems feature manpulators that let anmators easly create moton paths along the partcular transformaton s drecton. For example, by specfyng two keys along an x-translaton, an anmator can create a straght path n the x-drecton. However, no straghtforward way exsts to create a ballstc path because creatng t requres knowng an object s center of mass and a constrant, such as the startng velocty. If we assume that no external forces affect the mass whle n flght, we can descrbe the pont mass s trajectory r wth respect to tme t as r()= t a+ tb+ 2 Mt g, () 2 n whch a and b are the two parameters determned from a ballstc moton s constrants, such as an orgn and a destnaton, and travelng-tme constrants. M s mass; g s the gravtatonal constant. The system creates ballstc paths n real tme as anmators move around the two endponts and modfy the duraton tme. Fgure shows screen captures of the ballstc paths beng manpulated. To vew multple curves, users can vary the ballstc phase s duraton. Anmators can generate a physcally plausble path by settng startng and endng ponts as constrants n 3D space. Because generatng a ballstc path between two ponts s an underconstraned problem, anmators can also generate multple ballstc trajectores by usng the mnmum and maxmum tme span. Each curve presents a proper ballstc trajectory by ndcatng the path an object must follow to meet the locaton constrants at dfferng tmes. Anmators can also create ballstc paths by specfyng the frst or last frame s poston and velocty. Usng the ballstc paths, anmators can easly correct lnear momentum and thus the center of momentum (COM) trajectory of a character s ballstc moton. An anmator chooses the desred ballstc path, then the system computes the character s COM at each frame and translates ts root node so that the COM concdes wth the ballstc path s current poston. The resultng moton s physcally correct n terms of lnear momentum. Anmators can constran a character s or an object s moton to a ballstc path by retanng the startng and endng locatons and adjustng the tmng to accommodate the ballstc path (see Fgure 2), or manually adjustng the ballstc path untl t mostly matches the orgnal path and then alterng the character s moton by retmng all the moton that occurs durng the ballstc phase (see Fgure 3). When adaptng a ballstc path to the orgnal moton path, the anmator vsually modfes the IEEE Computer Graphcs and Applcatons 47

4 Physcs-Based Characters (a) (b) Fgure. Manpulatng ballstc paths. (a) Our tool generates multple ballstc paths between two locators. (b) Anmators can use t to pregenerate paths between two ponts n a scene nvolvng moton such as jumpng or fallng off a tall structure. Fgure 2. Matchng a ballstc path. The blue curve s the trajectory of the center of mass of the character s (manually created) anmaton. Our system suggests the physcally correct ballstc path (the red curve) that the character s center of mass should follow. The system lets anmators automatcally change the orgnal anmaton to match physcal laws. ballstc path untl he or she fnds a good vsual match. We don t pay partcular attenton to the smoothness of the ballstc trajectory s start and end. Rather, we use the anmator s orgnal preparatory moton durng the preballstc and postballstc phases. Ths doesn t ntroduce vsual artfacts as long as the orgnal moton doesn t dffer much from the corrected moton, whch s usually the case. When the artfacts are vsble, anmators can use conventonal anmaton tools to modfy the frames n the preballstc and postballstc phases durng takeoff and landng. The Angular-Momentum Tool You can change a character s angular momentum n many ways. Our tool changes t by rotatng the character s global orentaton whle keepng the orgnal keyframe anmaton of each body part unchanged. By dong ths, we can preserve the style of the anmaton that the anmator carefully crafted whle mprovng ts physcal realsm. We provde an effcent algorthm to acheve ths goal. Wth ths algorthm, modfcaton of angular momentum occurs onlne so that the anmators get prompt vsual feedback (see Fgure 4). For notatonal convenence, we use the generalzed notatons for the velocty, nerta, and momentum that provde the combned representaton of the angular and lnear propertes. For the mathematcal prelmnares of the generalzed notatons derved from Le group theory, see the Mathematcal Prelmnares sdebar. Let v, J, and h denote the generalzed velocty, nerta, and momentum of a body part, wth the body part beng the character s root. We can express a velocty v as the sum of the root s velocty and the relatve velocty of to the root: v = v + u, n whch v s the velocty of the root expressed n s body frame. (The left superscrpt ndcates a symbol s reference frame. v, c v, and w v are v expressed n the root frame, the COM frame, and the world frame, respectvely. We don t use the left superscrpt when the symbol s expressed n ts own body frame. For example, v = v.) Lkewse, we can dvde the generalzed momentum nto two parts: h = J v = J ( v + u ) = J v + a, 48 July/August 2

5 n whch a = J u represents the momentum nduced by the velocty of relatve to the root. Then, we compute a character s generalzed momentum n the root frame as h= h = Jv + = a J v + a, : = ˆ Jv + a, (2) n whch the composte rgd-body nerta Ĵ denotes the aggregate nerta of the whole multbody system of a current confguraton. Ĵ and a are a functon of jont angles only and are ndependent of the root s moton. So, they reman constant whle we manpulate the root s translaton and rotaton. Usng Equaton 2, we can effcently compute a character s momentum by calculatng v nstead of recalculatng each body part s velocty as we manpulate the root. We want to determne the confguraton of the root T at a pont n tme at whch the character has the user-specfed momentum c h*. We compute the character s velocty from the confguratons of the current and prevous tme step. We modfy only T at the current tme step; we keep T at the prevous tme step fxed. Because we keep the lnear momentum fxed, the COM frame (a coordnatng frame that s parallel to the world frame wth ts orgn concdng wth the COM) doesn t change whle we rotate the character around the COM. So, gven c h*, ts transformaton wth respect to the world frame w * * c * h = Ad C h s also constant whle we manpulate the character. For convenence, we fnd T such that t creates w h*. The character s current momentum wth respect to the world frame s w h= T Ad * h. (3) * Both Ad - T and h are functons of T, and a closedform soluton doesn t exst. So, we teratvely update T so that w h approaches w h*. Specfcally, we update T by some x whch s defned as = ˆx T d T, and fnd a sutable x that drves w h to w h*. To ths end, we frst relate the change of v wth x. From the defnton ˆv = T T, Fgure 3. A character walkng and jumpng. The ballstc path (red) requres two more frames to complete the trajectory than does the anmated path (blue). The keys for the ballstc path, whch determne the tmng, appear at slghtly dfferent locatons than those n the anmated path. Fgure 4. Vsualzaton of angular momentum. The angular momentum appears as a vector protrudng from the character s center of mass. The yellow arrows ndcate the moton s drecton (usng the rght-hand rule); the vectors sze ndcates the relatve amount of rotaton about that axs. dˆv = ( T dtt ) T + T dt d = xv ˆˆ + T ( T xˆ ) dt ( ) = xv ˆˆ + T T xˆ + Txˆ = xv ˆˆ + vˆ xˆ + xˆ xˆ adˆ v xˆ + h, (4) n whch h s the tme step. Usng Equaton 4 and recallng that Ĵ and a are constant, we express d h n terms of x: d h= ˆJdv x Ĵ ad x+ v h. IEEE Computer Graphcs and Applcatons 49

6 Physcs-Based Characters Mathematcal Prelmnares Gven a homogeneous representaton of a movng body frame T = (R, p) n whch R R 3 3 denotes rotaton and p R 3 translaton, ts generalzed velocty v = [w T, u T ] T expressed n the nstantaneous body frame (hence dubbed body velocty) s ˆv= T T = [ ] ω υ, (A) n whch w and u are, respectvely, the angular and lnear veloctes of T expressed n the body frame. [w] s the skew-symmetrc matrx of w; that s, [w]h = w h for any vector h R 3. We use ˆv for the 4 4 matrx representaton of v. The generalzed momentum h = [k T, l T ] T s expressed as h = Jv, n whch k R 3 and l R 3 represent the angular and lnear momentum (wth respect to the body frame), respectvely. The rgd body s generalzed nerta J R 6 6 has ths structure: J = I m[ r] T m[ r ] m, n whch m s the mass, I R 3 3 s the rotatonal nerta matrx, r R 3 s the poston of the center of mass, and R 3 3 s the dentty matrx. Equaton B s coordnatenvarant (t holds wth respect to any coordnate frame). Gven a coordnate frame T and a generalzed velocty g = [w T, u T ] T, the adjont mappng Ad T s represented as Ad Tgˆ = TgT ˆ, or n matrx form as R Ad Tg = [ pr ] R ω υ. We use the adjont mappng n the coordnate transformaton of the generalzed velocty. The correspondng dual adjont mappng that performs the coordnate transformaton of the generalzed momentum s Ad * T ; t has the form of the transpose of Ad T; that s, Ad * T = Ad T T. For example, the generalzed velocty, momentum, and nerta wth respect to the world frame ( w v, w h, w J, respectvely, wth the left superscrpt w mplyng the world reference frame) are w v = Ad Tv w * h= T Ad h (B) w * J= Ad T JAd T. We can easly verfy that Ad T = Ad T and AdTAdT2 = AdTT 2. Assumng that lnk of a multbody system s the root lnk, the confguraton T of the body frame {} of wth respect to the world frame s T = T G, n whch T s the confguraton of the root and G denotes the relatve confguraton of {} wth respect to the root. Substtutng Equaton C nto Equaton A, we can decompose the body velocty v : v = v + u, n whch v = Ad G v s the velocty of the root expressed n the body frame {} and u = u u T ω, υ T s the relatve velocty of to the root: ωu υ [ ] u = G Ġ. (C) The left superscrpt denotes the reference frame. In the man artcle, we use the transformatons of the generalzed velocty, momentum, and nerta of a lnk to the root: v = Ad G v, h = Ad G * h, and * J = Ad J G Ad G. The Le bracket ad g s another mappng for the generalzed velocty, defned as adˆ ggˆ2 = gˆgˆ2 gˆ2g ˆ or, n matrx form, ad g g = [ ω ] 2 [ υ] [ ω] ω υ 2 2. * The dual ad g for the generalzed momentum s ts transpose adg * = ad T g. Note that ad gv = -ad vg and ad g + ad v = ad g+v. The Le bracket occurs when Ad s dfferentated. For example, f Ad T s dfferentated wth respect to tme t, d dt Ad = Ad ad, T T v n whch v s the body velocty of T. For the proof, see Newton-Type Algorthms for Dynamcs-Based Robot Movement Optmzaton. Reference. S.-H. Lee et al., Newton-Type Algorthms for Dynamcs- Based Robot Movement Optmzaton, IEEE Trans. Robotcs, vol. 2, no. 4, 25, pp July/August 2

7 Fnally, we compute the change of w h due to x: d w * * h= d( Ad ) h+ Ad d h T T = Ad ad h+ Ad * T * x * T d = Ad ad + Ad * * T h x * T d * * Ad ad + ˆ T h Jadv + ˆ J x, (5) h h h * n whch ad h x and Ĵad v x account for the effect of the coordnate change of T due to x on h and v, respectvely; h Ĵx s the added momentum due to x. By solvng Equaton 5 for x, we can compute the x that wll create the desred d w h. Based on the relatons we just derved, here s the algorthm to compute T gven c h*: w * * c *. h = Ad C h, n whch C s the COM frame wth respect to the world frame. 2. Compute Ĵ and a (see Equaton 2). 3. Compute w h (see Equaton 3). 4. whle d w h = w h* - w h s above a threshold, do 5. Compute x by solvng Equaton T Te gˆ x, n whch < g. 7. Update v, h (see Equaton 2), and w h. g controls the dstance for the next teraton. We change only the angular part of c h* and keep the lnear momentum fxed. However, both the angular and lnear parts of w h* change, owng to the coordnate transformaton. Keyframe Anmatons Physcal Accuracy Here, we nvestgate the physcal accuracy of professonal anmators keyframe anmatons. Specfcally, we compute the ballstc motons center of mass, lnear momentum, and angular momentum and nvestgate how accurately these propertes follow Newton s laws. For walkng and runnng anmatons, we compute the center of pressure (COP) and verfy whether t s actually n the support polygon. Fgure 5 shows snapshots n whch COM and COP are vsualzed. Ballstc Motons We collected 3 ballstc motons of human-lke characters, such as jumpng and fallng motons. Each moton lasted from 4 to 27 frames. We examned 5 frames total. In the ballstc phase, the path of the COM that s projected to the horzontal plane should form a straght lne. Fgure 6a s the hstogram of errors n the ballstc motons horzontal plane. (a) Fgure 5. The center of mass projected on the ground (red crcle), the support polygon (red polygon), and the center of pressure (COP; the purple crcle) for keyframe anmaton of (a) a humanod character and (b) a nonhumanod character. The nonhumanod character s horzontal (b) shape causes greater nstablty n the COP calculaton than the humanod character s shape does. Here, the error s the dstance from the COM s horzontal projecton to the lne segment connectng the horzontal projecton of the COMs of the frst and last frames. The errors are normalzed by the lne segment s length. The fgure shows that the error s less than. for more than 9 percent of the frames. Ths relatvely low error seems due to how anmators create keyframes for ballstc motons. Because anmators usually create a straght lne from the root node s startng and endng poston and adjust the heghts of the n-between keyframes, the COM trajectory s horzontal projecton shows a mostly straght lne unless a character often changes ts pose. Fgures 6b and 6c show the hstogram of normalzed errors of the lnear momentum n the horzontal (x-z) plane. Here, the error s the dfference between the current momentum and the average momentum, normalzed by the average momentum s magntude. When comparng the error of the lnear momentum wth that of the COM, we can see that the keyframe s locaton n the lne s less correct from the perspectve of the lnear momentum, even though the n-between keyframes make a lne that s mostly straght. Fgures 6d, 6e, and 6f show the hstogram of normalzed errors of the angular momentum n the x, y, and z drectons. Here, we defne the error n the same way as n the lnear momentum. The error n the angular momentum s much greater than that n the lnear momentum. One reason for ths mght be that angular momentum s less perceptble to human eyes than lnear momentum s. Except for flppng jumps, durng whch IEEE Computer Graphcs and Applcatons 5

8 Physcs-Based Characters Frequency (%) Frequency (%) Frequency (%) (a) (c) (e) Infnty (b) (d) Infnty (f) Frequency (%) Frequency (%) Frequency (%) Infnty Infnty Infnty Fgure 6. Measurng the physcal correctness of ballstc motons created by professonal anmators usng conventonal keyframng. The normalzed error of the (a) center of momentum (COM) n the horzontal plane, (b) lnear momentum n the x drecton, (c) lnear momentum n the z drecton, (d) angular momentum n the x drecton, (e) angular momentum n the y drecton, and (f) angular momentum n the z drecton. the whole body s rotaton s explct, the ballstc moton s angular momentum s hard to perceve wthout a vsualzaton tool such as ours. By fttng Equaton to the COM trajectory, we estmate a scene s gravty. Table I shows ths gravty relatve to earth gravty (9.8 m/s 2 ) for each scene. The gravty devates consderably among scenes rangng from.5 g to 3.3 g. The medan of the gravtes s.95 g, whch s close to real gravty. Ths shows that the keyframe anmatons create ballstc motons under true gravty on average, but that large devatons exst among scenes. Research has shown that observers can detect devatons n horzontal or vertcal acceleratons. 5 Our data demonstrates that hand-anmated moton often exceeds such thresholds. Another nterpretaton of the varaton n estmated gravty s that bg dfferences exst n anmators notons of a character s sze. If that s true, lower gravty n the data ndcates the anmator consd- Table. Gravty n the test scenes. Scene g* *g: measured gravty/ July/August 2

9 7 ers the character bgger than t actually s; hgher gravty means the anmator thnks the character s smaller than t s. Other research has suggested that preparatory moton mght also be an mportant consderaton n the perceved vsual qualty of ballstc moton. 6 Our method doesn t address the preparatory or recovery phases; t leaves such tasks to anmators. Ambulatory Motons We collected fve scenes of walkng motons (65 frames) and fve scenes of runnng motons (3 frames) of bped characters and examned whether the COP s n the support polygon. Fgure 7 shows the hstogram of the COP s normalzed dstance from the support polygon. The dstance s postve when the COP s outsde the support polygon; t s normalzed by the support polygon s mean radus (computed as the permeter dvded by 2p). For walkng, the COP s n the support polygon n more than 6 percent of the frames and s wthn the normalzed dstance of n 85 percent of frames. The experment shows that the keyframe anmaton has relatvely hgh accuracy n terms of the COP. Ths suggests that the COP s an mportant ndcator of ambulatory motons realsm. Somewhat naturally, the error grows as a character s speed ncreases. For runnng, the COP s n the support polygon n only 33 percent of frames and s wthn the normalzed dstance of n 7 percent of frames. Dscusson Here, we dscuss the advantages and lmtatons of our system and observatons made when the system was used by anmators. Effectveness and Impact Many anmators have found our system useful for mprovng keyframe anmaton s realsm. The manually created anmatons n Fgure 2 and the accompanyng vdeo (see org/.9/mcg.2.22) are the results of substantal tme and effort devoted by professonal anmators. So, some of these anmatons are already qute realstc and leave less room for mprovement usng our tool. In some sense, ths shows that physcal realsm s a key factor for hgh-qualty anmaton. Other examples show that our system mproves the physcal realsm sgnfcantly. Also, when our system s used from the begnnng of the anmaton process, t mght ncrease that process s effcency. Our vsualzaton tools can also serve as a gavel that s, confrmaton that anmaton s ndeed realstc. Indvdual anmators mght have Frequency (%) Frequency (%) (a) (b) Infnty Dstance between the zero-moment pont and the support polygon Infnty Dstance between the zero-moment pont and the support polygon Fgure 7. Hstograms of the normalzed dstance between the COP and the support polygon for anmatons of (a) walkng and (b) runnng created by professonal anmators usng conventonal keyframng. The COPs of keyframe anmaton turned out to be farly accurate, beng mostly wthn the normalzed dstance of from the support polygon. ther own sense of physcal correctness. Ths can cause debate over how a character should move when collaboratng anmators have a dfferent sense of physcal ntuton. We ve observed that our tool helps anmators agree on physcally correct anmaton by quantfyng the moton s dscrepancy. For example, the ballstc path can ndcate the exact number of frames n whch an anmaton should be slowed down or sped up. We determned that n a character-heavy lveacton flm, between and 6 percent of the shots usng anmated characters have ballstc moton such as fallng or jumpng. Our system could mprove many of these shots. Of course, many more shots beneft from COM and COP vsualzaton. Our system s most effectve when the anmator understands how to change the physcal curves to create better anmatons. So, t was useful to create vdeos that showed, for example, the proper locaton of the COM and COP (or zero-moment pont) durng walkng, runnng, or other motons. IEEE Computer Graphcs and Applcatons 53

10 Physcs-Based Characters Scalng Large and Small Characters The characters scale can dramatcally mpact the resultng anmaton. Other research has provded detaled explanatons for scalng dynamc control systems on the bass of tme, postons, and veloctes. 7 We relate such scalng strateges to the applcaton of knematcally based anmaton of large and small characters. For example, large creatures appear to move more slowly under proper physcal condtons, whereas small creatures appear to move more quckly. The tme t requred for an object to travel to a maxmum heght h s Fgure 8. Usng moton capture data as a tranng tool for anmators. The COM s outsde the support polygon whle the character turns. Fgure 9. How scale affects the appearance of moton. The fgure shows a normal-szed man and a man tmes larger. A ball s placed n moton around each of them. The ball movng around the larger man wll take 3.9 tmes as long as the one movng around the normal-szed man. Such tranng vdeos can effectvely teach anmators how to nterpret our system, as n Fgure 8. Of course, the anmator chooses whether to make the character act human-lke. Use of COM and COP The COM and COP gudelnes explctly expressed the notons of physcal correctness n anmatons. For example, a humanod s COM tends to follow a path from about the contact foot to the other contact foot at a later tme. Our system was able to serve as an nstructonal ad for creatng smoother moton because, for example, users could drectly see the dea of carryng the weght from one foot to another n the COM s projecton. t = 2 h g, (22) n whch g s the force of gravty. So, gravtyrelated movements such as walkng (n whch gravty pulls the character s swngng leg to the ground) or throwng objects wll appear slower or faster f we consder two characters of dfferng szes, h and h 2, and ther respectve tmes, t and t 2, to complete such actons. The rato of tme requred for that acton to occur would be t t2 = h h2. Our observatons have shown that anmators often make large errors when the characters they re anmatng are ether much larger or much smaller than normal-szed objects. For example, gant men wll move too quckly for ther relatve sze, and small men wll move much too slowly for ther real sze. We hypothesze that ths s due to the famlarty of seeng characters of normal sze move and the unfamlarty of seeng gants or mnature people move (see Fgure 9). Lmtatons and Future Work Our method for modfyng angular momentum changes only the character s global orentaton. It doesn t change the character s pose. There are many other ways to correct the angular momentum that nvolve changng parts of the character s body whle leavng other parts the same. For example, a person can change hs or her angular momentum by wndmllng hs or her arms n a crcular manner. We don t provde a tool to let anmators explore all these possbltes. Instead, we focus on ease of use and automaton. It sn t clear that a sngle useful correcton method exsts that wll yeld better anmaton due to anmaton constrants, such as requrng a character s feet to be n contact wth the ground durng landng. So, t s the anmator s role to manually correct the character s posture to resolve ths dscrepancy. In the prevous example, the angular momentum durng flght can be automatcally adjusted but would 54 July/August 2

11 need to be manually smoothed wth the landng posture to obtan correctness. Also, here we computed the COP under the assumpton that the ground s flat. However, characters often walk or run on uneven ground. When the contact ponts between the character and the ground aren t coplanar, our system can t defne the support polygon. For such cases, we ll need to extend the support polygon to 3D space. The algorthms we use to generate physcal vsualzaton are straghtforward to mplement and compatble wth most knematc anmaton systems. In addton, the system doesn t requre anmators to change the methods by whch they generate anmaton, thus leveragng ther exstng sklls. Also, the generated moton s qualty s addtve t doesn t replace the underlyng anmaton. The anmator may choose to use the system only f t enhances the moton s realsm. Although we desgned our system for lve-acton vsual effects, the technques work for almost any anmaton purpose that requres or desres better physcal realsm for characters, such as systems for fully 3D envronments or prebaked anmatons for vdeo games. Our system can also help enforce consstency of character moton across an anmaton studo. Typcally, several anmators wll create anmatons of a partcular character for dfferent scenes. A move requrng heavy vsual effects mght requre coordnatng dozens of anmators to produce hundreds of anmatons for a small number of characters. Wth our system, dfferent anmators anmatons of the same character tend to be more consstent wth each other. Ths s because the anmators don t have to rely solely on ther ndvdual senses of tmng and space; they can use our system s nteractve vsual feedback. Acknowledgments Ths materal s based mostly on research we completed whle at Rhythm & Hues Studos. Both of us are correspondng authors. We thank the anonymous revewers for ther helpful comments, whch mproved the artcle. Sung-Hee Lee was supported partly by the Global Fronter R&D Program of the Natonal Research Foundaton, Korea (NRF- MAXA ). A Practcal Dynamcs System, Proc. 23 ACM Sggraph/Eurographcs Symp. Computer Anmaton, Eurographcs Assoc., 23, pp Massve, Massve Software, 2; com. 3. Havok Behavor, Havok Inc., 2; com/ndex.php?page=havok-behavor. 4. Endorphn 2.7, NaturalMoton Ltd., 2; www. naturalmoton.com/endorphn. Although we desgned our system for lveacton vsual effects, the technques work for almost any anmaton purpose that requres or desres better physcal realsm. 5. P.S.A. Retsma and N.S. Pollard, Perceptual Metrcs for Character Anmaton: Senstvty to s n Ballstc Moton, ACM Trans. Graphcs, vol. 22, no. 3, 23, pp P.S.A. Retsma, J. Andrews, and N.S. Pollard, Effect of Character Anmacy and Preparatory Moton on Perceptual Magntude of s n Ballstc Moton, Computer Graphcs Forum, vol. 27, no. 2, 28, pp N. Pollard, Smple Machnes for Scalng Human Moton, Proc. Computer Anmaton and Smulaton 99, Sprnger, 999, pp. 3. Ar Shapro s a research scentst at the Insttute for Creatve Technologes. He prevously worked as a graphcs scentst at Rhythm & Hues Studos, and an R&D engneer at Industral Lght and Magc. Hs research nterests nclude realstc character moton, physcs-based anmaton, and anmaton tools. Shapro has a PhD n computer scence from the Unversty of Calforna, Los Angeles. Contact hm at shapro@ct.usc.edu. Sung-Hee Lee s a lecturer at the Gwangju Insttute of Scence and Technology s School of Informaton and Communcatons. Hs research nterests nclude human modelng and anmaton, physcs-based anmaton, multbody dynamcs, and humanod robotcs. He prevously was a postdoctoral researcher at Honda Research Insttute USA. Lee has a PhD n computer scence from the Unversty of Calforna, Los Angeles. He s a member of IEEE and ACM Sggraph. Contact hm at shl@gst.ac.kr. References. Z. Kačć-Alesć, M. Nordenstam, and D. Bullock, Selected CS artcles and columns are also avalable for free at IEEE Computer Graphcs and Applcatons 55