CMSC 425: Lecture 9 Basics of Skeletal Animation and Kinematics

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1 CMSC 425: Lecture 9 Basics f Skeletal Animatin and Kinematics Reading: Chapt f Gregr, Game Engine Architecture. The material n kinematics is a simplificatin f similar cncepts develped in the field f rbtics, knwn as the Denavit-Hartenberg parameters. Game Animatin: Mst cmputer games invlve characters that mve arund in a fluid and cntinuus manner. Unlike bjects that mve accrding t the basic laws f phsics (e.g., balls, prjectiles, vehicles, water, smke), the animatin f skeletal structures ma be subject t man cmple issues, such as phsilgical cnstraints (e.g., hw t jump nt a platfrm), athletic technique (e.g., hw a ftball quarterback thrws a pass), stlistic chices (e.g., hw a dancer mves), r simpl the arbitrar cnventins f everda life (e.g., hw t hld chpsticks). Prducing natural lking animatin is the tpic f ur net few lectures. Skeletal Mdel and Tree Structure: The mst cmmn frm f character animatin used in high-end -dimensinal games is thrugh the use f skeletal animatin. A character is mdeled as skin surface stretched ver a skeletal framewrk which cnsists f mving jints cnnected b segments called bnes (see Fig. (a) and (b)). Nte that skin refers t the mdel s surface, which tpicall includes nt nl skin but the clthing the mdel is wearing (see Fig. (c)). Animatin is perfrmed b mdifing the relatinships between pairs f adjacent jints, fr eample, b altering jint angles and defrming the skin accrdingl. We will discuss this prcess etensivel later. bnes jints (a) (b) (c) Fig. : (a) and (b) skeletal mdel and (c) the bind (r reference) pse. A skeletal mdel is based n a hierarchical representatin where peripheral elements (e.g., hands and feet) are linked as children f mre central elements (e.g., legs, arms, trs, etc.). Clearl, a skeletal mdel can be represented internall as a multi-wa rted tree, in which each nde represents a single jint. The bnes f the tree are nt eplicitl represented, since (as we shall see) the d nt pla a significant rle in the animatin r rendering prcess. Lecture 9 Spring 28

2 We will discuss later hw the skin is represented. Fr nw, let us cnsider hw the jints are represented. We assume that the tree is represented as an standard (rted, unrdered) multi-wa tree. Fr eample, fr an nde it shuld be pssible t enumerate all the children f this nde, t test whether the nde is the rt, and if it is nt the rt t determine its parent nde. In this lecture, we will dente each jint b an integer inde, sa j, and we will let p(j) dente j s parent. If we cnsider just the parent links, the result is an inverted tree structure, where all paths lead t the rt (see Fig. 2(b)). We can make the assumptin that the rt is identified b a special inde, sa j =. In additin, each nde will stre sme internal infrmatin, as will be discussed belw. Bind Pse: Befre discussing animatin n skeletal structures, it is useful t first sa a bit abut the ntin f a pse. In humanid and animal skeletns, jints mve b means f rtatins (as ppsed sa t translatin, which arises with sme rbts). Assigning angles t the varius jints f a skeletn uniquel specifies the skeletn s eact gemetric structure, called its pse. When a designer defines the initial laut f the mdel s skin, the designer des s relative t a default pse, which is called the reference pse r the bind pse. 2 Fr human skeletns, the bind pse is tpicall ne where the character is standing upright with arms etended straight ut t the left and right (similar t Fig. (b) abve). Jint Internal Infrmatin: Each jint can be thught f as defining its wn jint crdinate frame (see Fig. 2(a)). Recall that in affine gemetr, a crdinate frame cnsists f a pint (the rigin f the frame) and three mutuall rthgnal unit vectrs (the,, and z aes f the frame). Given the skeletn s inverted tree structure (see Fig. 2(b)), rtating a jint can be achieved b appling a suitable rtatin transfrmatin t its assciated crdinate frame. Each frame f the hierarch is understd t be psitined relative t its parent s frame. In this wa, when the shulder jint is rtated, the descendants jints (elbw, hand, fingers, etc.) als mve as a result (see Fig. 2(c)). Change-f-Crdinates Transfrmatin: In rder t determine the mtin f the varius bnes that result frm sme jint rtatin, we need t knw the relatinships between the varius jints f the skeletn. There is a ver general and elegant wa f ding this thrugh the applicatin f affine gemetr. Given an tw crdinate frames in d-dimensinal space, it is pssible t cnvert a pint (r free vectr) represented in ne crdinate frame t its representatin in the ther frame b multipling the pint (given as a (d + )-dimensinal vectr in hmgeneus crdinates) times an suitable (d + ) (d + ) matri. The resulting affine transfrmatin is called a change-f-crdinates transfrmatin. Cnstructing such transfrmatins is an eercise in linear algebra. Fr the sake f cmpleteness, let us cnsider the prcess in a simple 2-dimensinal eample. Suppse we have tw It is rather interesting t think abut hw this happens fr ur wn jints. Fr eample, ur shulder jint has tw degrees f freedm, since it can pint ur upper arm in an directin it likes. Yur elbw als has tw degrees f freedm. One degree cmes b fleing and etending ur frearm. The ther can be seen when u turn ur wrist, as in turning a dr knb. Yur neck has (at least) three degrees f freedm, since, like ur shulder, u can pint the tp f ur head in an directin, and, like ur elbw, u can als turn it clckwise and cunterclckwise. 2 I suspect that the name bind pse arises because designers attach r bind the skin t the mdel relative t this initial pse. Lecture 9 2 Spring 28

3 j j j 4 j 5 j 2 j j 4 j 5 j j 2 j j 2 j j j 4 j j j 5 (a) (b) (c) Fig. 2: (a) Skeletal mdel, (b) inverted tree structure, and (c) rtating a frame prpagates t the descendants. crdinate frames, F and G (see Fig. ). Let F., F., and F. dente F s rigin pint, and its tw basis vectrs. Define G., G. and G. similarl. F p G G. [F ] = (2,, ) G. [F ] = (, 2, ) G. [F ] = (4, 2, ) F p G F. [G] = 2 5, 5, F. [G] = 5, 2 5, F. [G] = ( 2,, ) Fig. : Change-f-crdinates transfrmatin. Given an pint in space, it can be represented either with respect t F s crdinate sstem r G s. Fr an pint p, define p [F ] t be p s hmgeneus crdinates relative t frame F, and define p [G] similarl fr frame G. We can d the same fr an vectr v. In rder t define the change-f-crdinates transfrmatin, we need t knw first what G s basis elements are relative t F. In the abve eample, it is eas t verif that G. [F ] = (2,, ), G. [F ] = (, 2, ), and G. [F ] = (4, 2, ). (Recall that we are using affine hmgeneus crdinates, where the last cmpnent is t dente a vectr r t dente a pint.) Als, it is eas t verif that ( ) ( 2 F. [G] = 5, 5,, F. [G] = 5, 2 ) 5,, and F. [G] = ( 2,, ). T btain the change-f-crdinates transfrmatins frm G t F, define T F G t be the transfrmatin that maps a pint given in G s crdinate sstem t its representatin in F s crdinate sstem. This transfrmatin can be represented as a matri whse clumns are Lecture 9 Spring 28

4 G. [F ], G. [F ], and G. [F ] : T [F G] = Cnversel, t cnvert the ther directin, define T G F t be the transfrmatin that maps a pint given in F s crdinate sstem t its representatin in G s crdinate sstem. This transfrmatin can be represented as a matri whse clumns are F. [G], F. [G], and F. [G] : T [G F ] = 2/5 /5 2 /5 2/5 (See an standard reference n linear algebra fr a prf.) Change-f-Crdinates Eample: T test this, let s cnsider the pint p and vectr v in Fig. 4. F p G v p [F ] = (,, ) v [F ] = (,, ) F p G v p [G] = (,, ) v [G] = (,, ) Fig. 4: Eample f appling the change-f-crdinates transfrmatin. Clearl, p [F ] = (,, ) and p [G] = (,, ). Appling the abve transfrmatins, we btain the epected results 2 4 T [F G] p [G] = 2 2 p [F ], and T [G F ] p [F ] = 2/5 /5 2 /5 2/5 p [G], Net, cnsider v. We have v [F ] = (,, ) and v [G] = (,, ). Again, appling the abve transfrmatins, we have 2 4 T [F G] v [G] = 2 2 v [F ], and T [G F ] v [F ] = 2/5 /5 2 /5 2/5 v [G]. Lecture 9 4 Spring 28

5 Abstractin Revisited: We have seen the use f hmgeneus matrices befre fr the purpse f perfrming affine transfrmatins (that is, fr mving bjects arund in space). We are using the same mechanism here, but the meaning is quite different. Here the bjects are nt being mved, rather we are simpl translating the names f pints and vectrs frm ne crdinate sstem t anther. The gemetric bjects are themselves nt mving. Yu might wnder, What s the difference in hw u lk at it? Recall that in affine gemetr we defined pints and (free) vectrs as different abstract bjects, that empl the same representatin (hmgeneus vectrs). Here, we are distinguishing tw different tpes f peratins (affine transfrmatins versus change-f-crdinates transfrmatins), but using the same representatin (hmgeneus matrices) fr bth. Even thugh the same representatin is being used, these shuld be cnceptualized as tw ver different peratins. Jint Transfrmatins: Returning t the prblem f skeletal sstems, let us assume that we are wrking in -dimensinal space, and cnsider the skeletn in its bind pse. Fr an tw jints j and k, define T [k j] t be the change-f-crdinates transfrmatin that maps a pint in jint j s crdinate sstem t its representatin in k s crdinate sstem. (At this pint we are nt cnsidering jint rtatins.) That is, if v is a clumn vectr in hmgeneus crdinates representing f a pint relative t j s crdinate sstem, then v = T [k j] v is eactl the same pint in space, but it is epressed in crdinates relative t k s crdinate frame. (In previus lectures I have used p fr pints and v fr free vectrs. Since we are using p fr parent, I ll refer t pints b the letter v, but dn t be cnfused.) Given an nn-rt jint j, define the lcal-pse transfrmatin, dented T [p(j) j], t be an affine transfrmatin that cnverts a pint in j s crdinate frame t its representatin in its parent s (p(j)) crdinate frame. Define the inverse lcal-pse transfrmatin, dented T [j p(j)], t be the inverse f this transfrmatin. That is, it cnverts a pint epressed relative t j s parent s frame back t j s frame. (Recall that these transfrmatins d nt change the psitin f a pint. The simpl translate the same pint frm its representatin in ne frame t anther.) v k v [k] = (2,, ) v [j] = (,, ) i j v [i] = (,, ) (a) (b) Fig. 5: Three jints i, j, and k in a (rather nnstandard) bind pse. The pint v is represented in hmgeneus crdinates relative each frame. Cnsider three jints i, j, and k, where i = p(j) and j = p(k) (see Fig. 5(a)). The lcal-pse transfrmatin fr k, T [p(k) k], can be epressed mre succinctl as T [j k]. Given a pint v [k] Lecture 9 5 Spring 28

6 epressed relative t k s frame, we can epress it relative t j s frame as v [j] = T [j k] v [k]. Similarl, a pint v [j] epressed relative t j s frame can be epressed relative t i s frame as v [i] = T [i j] v [j]. Cmbining these, we can epress a pint in k s frame relative t i s frame b taking the prduct f these tw matrices v [i] = T [i j] T [j k] v [k] = T [i k] v [k], where T [i k] = T [i j] T [j k]. Clearl, b multipling apprpriate chains f the lcal-pse transfrmatins and their inverses, we can walk up and dwn the paths f the tree allwing us t cnvert a pint relative t an ne jint int its representatin relative t an ther jint. An Eample: T make this a bit mre cncrete, let us cnsider an eample in 2-dimensinal space. Cnsider the pse shwn in Fig. 5(a). (This is nt a nrmal bind pse, since the elbw shuld nt be bent, but it makes fr a mre interesting case.) Let i dente the shulder jint, j the elbw jint, and k the hand jint. Cnsider a pint v that lies tw units bend the mdel s inde finger. Its hmgeneus crdinates relative t the hand frame are (2,, ). (Since the -ais pints up.) Its crdinates relative t the elbw frame are (,, ), and its crdinates relative t the shulder frame are (,, ) (see Fig. 5(b)). That is, v [k] = 2 v [j] = v [i] = Because k s crdinate frame lies 4 units alng the -ais relative t j s crdinate frame, the lcal pse transfrmatin T [j k] (which maps a pint in k s crdinate frame t j s crdinate frame) clearl increases the -crdinate b 4 units. Thus: T [j k] = 4 We can easil check this, since just as we epected. T [j k] v [k] = 4 2 v [j], Net, cnsider hw t map a pint in j s crdinate frame t its parent s frame i. Observe that the -crdinate f the transfrmed pint (its vertical distance) is the -crdinate f the riginal pint. Thus, the middle rw f the matri is (,, ). The -crdinate f the Lecture 9 Spring 28

7 new pint (its distance t the right f the shulder) is minus the ld -crdinate. Thus, the first rw f the matri is (,, ). Therefre, the transfrmatin that achieves this change f crdinates is T [i j] = Again, we can check this, since as we epected. T [i j] v [j] = v [i], Nw, t btain the transfrmatin T [k i], we multipl these tw matrices (translating frm k t j, then j t i) 4 T [i k] = T [i j] T [j k] = 4 Finall, t check this we have T [i k] v [k] = 4 Again, this is just what we epect t happen. 2 v [i]. Of curse, appling this in -dimensinal space will invlve handling 4 4 matrices and the assciated rtatin and translatin matrices. While this wuld be much harder t d b hand, it can be dne in b a similar prcess that is purel mechanical (and hence, eas t prgram). Frward Kinematics: Net, suppse that in additin t knwing the lcal-pse transfrmatins and their inverses, we als knw the rtatin transfrmatins assciated with the individual jints f the sstem. Kinematics (als called frward kinematics) is the prblem f determining where a pint is transfrmed as a result f these rtatins. We can appl ur knwledge f rtatin transfrmatins and the lcal-pse transfrmatins and their inverses t slve this prblem. Fr eample, recall the pint v in ur earlier eample (see Fig. (a)) Suppse that the elbw is rtated b cunterclckwise b (see Fig. (b)), and then the shulder is rtated clckwise b 45, that is, cunterclckwise b 45 (see Fig. (c)). The questin that we want t cnsider, where is the pint v mapped t as a result f these tw rtatins? Let v be its psitin after the elbw rtatin and let v be its psitin after bth rtatins. Befre getting t the answer, recall frm ur earlier lecture n affine gemetr the rtatin transfrmatins in hmgeneus crdinates: cs sin /2 /2 Rt( ) = sin cs /2 /2 Lecture 9 7 Spring 28

8 i v k j v k i j i 45 j k v (a) (b) Fig. : Frward kinematics eample. (c) and Rt( 45 ) = cs 45 sin 45 sin 45 cs 45 / 2 / 2 / 2 / 2 We need t decide which frame t use as ur reference frame. Let s use the shulder jint i, since it is the mst glbal. (Generall, we wuld select the rt f ur skeletn tree.) We saw alread hw t cmpute v [i]. Using this as a starting pint, let s first cnsider the effect f the elbw rtatin. Because the elbw rtatin ccurs abut the elbw s crdinate frame, we first need t translate v int its representatin with respect t j s frame (b multipling b the inverse lcal-pse transfrmatin T [j i] ). We then appl the rtatin abut the elbw jint. Finall, we cnvert this representatin back t the shulder frame (b appling the lcal-pse transfrmatin T [i j] ). Thus, we have v [i] = T [i j] Rt( ) T [j i] v [i]. This ields a representatin f v relative t the shulder frame. Since the secnd rtatin is perfrmed abut the shulder frame, we d nt need t perfrm an change-f-crdinate transfrmatin. We can just appl the rtatin transfrmatin directl. This ields: Putting bth steps tgether, we have v [i] = Rt( 45 ) v [i]. v [i] = Rt( 45 ) T [i j] Rt( ) T [j i] v [i]. Tp-dwn r Bttm-up? Yu might wnder wh we did the elbw rtatin first fllwed b the shulder transfrmatin. Des the rder reall matter? The issue is that ur lcal-pse transfrmatins have been built under the assumptin that the mdel is in the bind pse, that is, nne f the jints are rtated. If we were t have perfrmed the shulder rtatin first, and then attempted t appl the inverse lcal-pse transfrmatin T [j i] t cnvert the result frm the shulder s frame t the elbw frame, we wuld discver that this transfrmatin is Lecture 9 8 Spring 28

9 n lnger crrect. The reasn is that the entire arm (and the elbw jint in particular) has mved int a new psitin, but T [j i] was defined based n its riginal psitin. T avid this prblem, the transfrmatins shuld be applied in a bttm-up manner, first rtating the descendant ndes (e.g., wrist) and then wrking up t their ancestrs (e.g., elbw and then shulder). Take-Awa Lessn: I must acknwledge that implementing this b b hand wuld be a mess (especiall in -space), but hpefull u get the idea. B using ur lcal-pse transfrmatins (and pssibl their inverses), we can change t the crdinate frame where the rtatin takes place, then appl the rtatin, then translate back. While it wuld be mess t write dwn all the transfrmatins, if we have precmputed the lcal pse transfrmatins and their inverses, this can all be prgrammed in a straightfrward manner b traversing the tree (in pstrder) and perfrming simple matri multiplicatins. Lecture 9 9 Spring 28