Københavns Universitet. Ballet Balance Strategies Pedersen, Camilla; Erleben, Kenny; Sporring, Jon. Published in: Simulation (San Diego, Calif.

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1 univerity of copenhagen København Univeritet Ballet Balance Strategie Peeren, Camilla; Erleben, Kenny; Sporring, Jon Publihe in: Simulation (San Diego, Calif.) Publication ate: 2006 Document Verion Early verion, alo known a pre-print Citation for publihe verion (APA): Peeren, C., Erleben, K., & Sporring, J. (2006). Ballet Balance Strategie. Simulation (San Diego, Calif.), 14(8), Downloa ate: 28. mar

2 Simulation Moelling Practice an Theory 14 (2006) Ballet balance trategie Camilla Peeren a, Kenny Erleben b, Jon Sporring b, * a IT Univerity of Copenhagen, Rue Langgaar Vej 7, DK-2300 Copenhagen, Denmark b Department of Computer Science, Univerity of Copenhagen, Univeritetparken 1, DK-2100 Copenhagen, Denmark Available online 14 November 2006 Abtract Animating phyically realitic human character i challenging, ince human oberver are highly tune to recognize human cue uch a emotion an gener from motion pattern. The main contribution of thi paper i a new moel firmly bae on biomechanic, which i ue to animate balance an baic movement of a ballet ancer. It i upporte by computer imulate experiment an it i in goo agreement with biomechanical meaurement of real-life ancer. Our reult quetion the previou approache in ynamic animation, which only ue the center of gravity trategy, an intea emontrate the viability of the center of preure trategy. Ó 2006 Elevier B.V. All right reerve. Keywor: Biomechanic; Balance trategy; Weight hift trategy; Control mechanim 1. Introuction A long term goal of computer graphic i to increae realim an believability in computer generate animation an picture [1,23,12,4,28,3]. With improve renering technique, the lack of phyical realim an believability i becoming increaingly obviou an annoying to the common oberver. One accompanying long term goal in animation i to increae phyical realim by uing phyic to moel plauible behavior an movement of computer moel. Thi known a phyic-bae animation. Thi paper tuie biomechanical an ballet inpire balance an weight hifting trategie an i bae on original work in [27]. Ballet i a balance art an i a prime focu for learning about human balance an weight hifting trategie. We will ecribe how to make a ynamic animation from the firt of the baic poition in Fig. 1 to obtain a quiet taning on one toe for a 3D humanoi moel. To achieve the goal, the articulate figure move through the four ub-goal hown in Fig. 2: Balancing on two leg, weight hift to the upporting leg, balancing on one leg, an balancing on the toe of one leg. The new moel i firmly bae on biomechanic an i upporte by computer imulate experiment howing goo agreement with biomechanical meaurement of real-life ancer. * Correponing author. are: porring@iku.k (J. Sporring) X/$ - ee front matter Ó 2006 Elevier B.V. All right reerve. oi: /j.impat

3 1136 C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) Fig. 1. Three of the bai ballet poition hown in our imulator: (a) Firt, (b) Secon, an (c) Fifth. The feet are poitione in the frontal plane in all poe. Fig. 2. Image from our imulator illutrating hifting the weight from both leg in firt poe (a) to the left leg (b), to a one-legge tan (c), an finally to a one-toe () quiet taning Survey of pat work The tuy of balance ha been performe in at leat three eparate area of reearch: Biomechanic, Robotic, an Animation. We will in the following highlight ome reearch from thee area. Biomechanic an the tuy of ballet: Ballet i an art-form, where balance play a central role. Claical ballet technique are throughly ecribe in the literature ee, e.g. [11,31]. Biomechanical tuie of ballet ha mainly been tuie through injury cae, e.g. [9]. In the tuying of human balance there are two theorie: Either we balance by controlling the center of ma irectly or inirectly by controlling the center of preure [20]. Empirical invetigation have hown that the velocity of center of ma play a role in balancing [32]. Empirical tuie on real human have been performe on balancing of human veru the poition of the center of ma [26], an hift have been meaure uing a force-platform [24,25]. Finally, a thorough meaurement of the propertie of ma an inertia of human boy part may be foun in [7]. Robotic: Early work on balancing robot may be foun in [30]. Weight hift trategie for walking are often performe through ynamic walking machine, where there i more or le egree of control involve in the walking cycle [21,17,29]. Recently, a number of pectacular an table humanoi robot have been prouce tarting with the Hona robot [15]. Epecially in relation to our work, the center of preure ha recently been introuce a a table control mechanim for balancing robot [14,13]. However pectacular complex motion, uch a hown in Fig. 2 ha not been attempte within thee robot. Animation: A very early mentioning of imulate human motion i given in [2], an early implementation may be foun in [10,22]. The implementation of ancing moel i only carcely icue in the literature, however one exception i [6]. A major inpiration for our work ha been [18,33], where center of ma i ue to control the balance an motion of a humanoi moel of varying complexity. Alternative to center of ma trategie are invere kinematic, e.g. [19], energy moel, e.g. [5], an learning approache, e.g. [8]. 2. The biomechanic of quiet taning A human in quiet taning may be moele by an articulate figure [18,33] coniting of a et of joint an a et of link repreenting boy part. The et of poible joint conit of revolute (1 egr of freeom (DOF)), univeral (2 DOF), an ball-an-ocket (3 DOF) joint. In thi work, we have ue Wooten moel [33], which contain 28 DOF, an ue real meaurement of the ma, m i, center of ma,~r i an moment of inertia of all the boy part [7]. The ankle an hip joint i of particular importance for thi paper, an they are moele by an univeral joint an a ball-an-ocket joint repectively. Traitionally in biomechanic an anatomy motion orientation i ecribe in three plane: The Sagittal (x-axi), the tranvere (y-axi) an the frontal (z-axi) plane. Thee plane are illutrate in Fig. 3(a).

4 C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) Fig. 3. (a) Sagittal, tranvere, an frontal plane together with meial, longituinal an anterior axi. (b) Illutration of biomechanical efinition: Center of ma, center of preure, upport polygon an line of gravity. Meaurement of angle an poition are traitionally alo performe in thee plane [24] by projection onto the repective plane an axe. Typical projection are: The poition of the center of ma, the poition of the center of preure, angle of joint, an the irection of gravity. Working with the projection, rather than the unerlying 3D geometry, allow for comparion with the ubtantial biomechanical literature. Mucle are ue to move an utain poture of the human keleton, an our articulate figure i upplemente by an actuator ytem, which applie joint torque accoring to a imple ampe angular pring moel, ¼ k mucle ðh target h current Þ k mucle _h current : ð1þ In the equation, i the length of the torque vector, h target an h current are target an current angle, h _ current i the current velocity of the angle, an k mucle an k mucle are pring an amping contant. A balance control trategy i a function that etermine upate, Dh, bae on the current tate of the articulate figure, h current, uch that the moel will converge towar a eire tate of quiet balance, i.e. the trategy iteratively etermine new parameter value, h new a, h new ¼ h current þ Dh: ð2þ In the ret of thi article, it will be aume that the moel i place on a planar floor, an the contact between the floor an the feet i repreente by a et of coplanar contact point, ~p j. The upport polygon i efine a the 2D convex hull of all the contact point. The center of ma of the human moel i efine a, X N ~r ¼ 1 m i ~r i ; ð3þ M i where M ¼ P N i m i i the total ma, N i the number of boy part in the moel, m i i the iniviual weight of the boy part, an r i are their location. Note that center of ma i not fixe w.r.t. any location of the boy uring motion of the iniviual boy part. The center of preure i efine a, ~r cp ¼ 1 X K k~n j k~p j ; ð4þ k~nk j where ~n ¼ P K j ~n j i the total normal force acting on the human moel, ~n j i the normal force applie to the human moel at the jth contact point ~p j an K i the number of contact point. For implicity, a pring moel for the floor contact force i ue, where the contact force of the jth contact i, ~ f j ¼ k contact ð~p initial j ~p j Þ k contact _~p j : ð5þ In the equation, ~p initial j i the initial point of contact, ~p j i the current contact point, _~p j i the velocity of the current contact point, an k contact an k contact are pring an amping contant. The vector, ~n j, i calculate

5 1138 C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) a the projection of ~ f j onto the contact normal of the floor, an the tangential part i a imple moel of fractional force. Contact force can only be repulive, an attractive contact force, f j, are therefore et to zero. Slipping i obtaine by etting ~p initial j equal to ~p j when the magnitue of the tangential force component excee a multiple of the magnitue of the normal force component, k ~ f friction k > lk~nk. The line of gravity i efine a the line going from the center of ma to the groun in the irection of the gravitational fiel. The point of interection between the line of gravity an the floor i referre to a the projection of the center of ma. Thee concept are illutrate in Fig. 3(b). Balance i efine a an object ability to maintain quiet taning, where quiet taning i obtaine when the projection of the center of ma i kept within the upport polygon [16,18,33]. The implication i that the greater upport polygon, the lower center of ma, the more table the balance an vice vera. The human boy ha a highly place center of ma over a rather mall upport polygon, an a uch the human boy behave a an inverte penulum. In the remainer of thi paper, control trategie for maintaining quiet taning an moving an articulate figure from one poe of quiet taning to another will be icue. 3. Balance trategie: ma center veru preure center In the following, two trategie for balance inpire by biomechanic will be compare: The center of ma trategy an the center of preure trategy. The analyi are one on an articulate figure in 3D bae on [33] taning with parallel. The balance i controlle by the ankle [32]. The center of ma trategy i the traitional balance trategy for ynamic animation, where the angular change i controlle a a function of the projection of the center of ma onto the upport plane [18,33]: Dh ¼ k ðr current r target Þ k v ; ð6þ where Dh i the angular change of the ankle, r current an r target are the projection of the current an target poition of the center of ma onto the tranvere plane, v i the velocity vector of the center of ma an k an k are control parameter. In the center of preure trategy, the moel ue the center of preure to control the center of ma. The goal of thi trategy i to calculate a eire poition of the center of preure, an ue thi for calculating the mucle control: Dr cp ¼ k ðr current r target cp r target Þ k v ; ð7þ ¼ Dr cp þ r current ; ð8þ Dh ¼ k cp ðrcurrent cp r target cp Þ k cp v cp; where Dr cp i the poitional change of the center of preure, r current cp an r target cp ð9þ are the current an the target poition of the center of preure, an k cp an k cp are control parameter. The center of preure theory can be unertoo by looking at the human balance like the balance of an inverte penulum. The center of ma i the top an on the floor i the center of preure. When the poition are right above each other, it i a perfect balance. In orer to change the balance to a new eire poition. The center of ma can tart an acceleration towar the new point by moving the center of preure in the oppoite irection to initiate a falling motion. You o the ame when you balance a match on your finger. Fig. 4 how the reulting ynamic of the projecte center of ma an the center of preure on the anterior axi. The movement of the projection of the center of ma on the articulate figure tart in a poition on the heel to the eire poition on the center of the feet (the center of the upport polygon). The articulate figure manage to balance uing both trategie, but the center of preure trategy require only approximately 1.5 while the center of ma trategy require almot 10 to get a balance in the eire poition. In aition, the center of preure trategy alo ha the mallet amplitue of the ocillation of the center of preure. Thi mean that it ha the bet control over the contact with the groun, becaue movement of the center of preure mean that the weight on the fee are changing. The articulate figure in the balance tet ha only 1 DOF joint an i therefore only balancing by the equation in the agittal plane. Thi i fine

6 C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) Fig. 4. Comparion of the ynamic of the ankle for the two trategie: (a) Center of ma trategy an (b) Center of preure trategy. r cp i the center of preure an r i the center of ma. becaue the poition of the projection of the center of ma i the ame a the poition of the center of preure on the meial axi from tart. In all our other tet the articulate figure i both balancing in agittal an in frontal plane. We have one the balance tet on another moel [27] than Wooten [33] an obtaine imilar reult. 4. Dynamic of a ballet ancer Quiet taning on the toe of one leg i central in all ballet training. Thi i emaning, ince the ancer ha to balance on a very mall upport polygon while at the ame time looking at eae. We have evelope trategie for obtaining a balance on the left leg with the right foot by the left knee a illutrate in Fig. 2. The motion to balance i obtaine through a number of balance controller an mucle action hown in Fig. 5(a), an the etail of which will be icue in the following. The one leg poe i ue, when ancer turn in a pirouette, an it i perhap the mot baic of all balance in ballet training. Both in quiet balance an in mot of the baic ballet exercie, the leg are trictly eparate into the working leg (oing the exercie) an the upporting leg. Shifting the weight between the leg i baic, an it houl preferably be one without rawing the attention of the auience. In our tet the left leg i the upporting leg, an the right leg i the working leg Weight hifting trategie The weight hifting trategy ecribe in the following wa inpire by the analyi of real ancer preente in the literature [24]. They howe that angle change in the hip an ankle are nearly ientical uring a weight hift, an that the center of preure tart moving towar the working leg an en up being on the upporting leg. The lat reult i only explainable from the center of preure balance trategy. To hift the weight, we eignate the left ankle to be the controlling joint uing (9). The eire poition of the projection of the center of ma where on the center of the upporting leg foot. The angular change in the hip joint where calculate from the angular change in the left ankle. The right ankle controlle the poition of the right foot center of preure. Fig. 5. (a) The moel, motion, an controller ue to obtaine balance on one leg. (b) A rotation in the ankle with traight leg implie a rotation in the hip, an a rotation in the pine i require in orer to keep the upper boy vertical. (c) Plot howing poition of the center of preure an center of ma projecte onto the meial axi.

7 1140 C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) Ballet aethetic require that the upper boy i kept parallel to the line of gravity uring a weight hift in the frontal plane with both feet fixe on the floor. It ha been claime [24] that ancer keep their upper boy vertical by a counter rotation in the hip joint, however thi i only phyically poible when the leg are parallel a illutrate in Fig. 5(b). Ballet ancer compenate for the hip rotation by a counter rotation in the lumbar region of the lower back [11], an therefore a control function in the pine i require. Dancer control the boy center by the tomach mucle. Experience ha hown that thee tomach mucle are extremely important for aethetic motion of the articulate figure [27], thu a control function in the pelvi i ue to inhibit rotation in the Sagittal plane. Both control function are moele uing a pring law imilar to (1). The final weight hift of the articulate figure i hown in Fig. 2. The reulting articulate figure agree with meaurement performe on real ancer [24] a follow: The meaure angle change are nearly ientical uring the weight hift, an the center of preure tart to move towar the right foot an en in a poition on the left foot. The lat reult are howe in Fig. 5(c). On the in-axi zero i between the center of the leg, the negative number are the foot of the working/right leg an the poitive are foot of the upporting/left leg Movement to quiet taning on one toe The final movement to obtain a one-toe quiet taning i achieve by lifting the non-upporting leg, an hifting from a foot tan to a toe tan, ee Fig. 2. It i not ifficult to raie the leg, however the major challenge i to keep balance on a very mall upport polygon. Two trategie have been evelope: A trategy for raiing the right leg, an a trategy for making a weight hift to the toe. Similarly to the weight hifting balance trategy, both are bae on the center of preure trategy. To raie the leg, the left ankle i kept a the controlling joint, an the center of preure trategy i ue to keep the balance on the center of the left foot. To hift the weight to one toe, the center of preure trategy i ue in two tep: Firtly, for the controlling ankle joint, an econly to control the toe joint, when the ankle ha been traightene. When the upport polygon i getting maller the human boy mall eviation from a tiff penulum make it neceary to have a control of the center of ma at the left hip joint to maintain the balance. Lifting the right leg to an aethetically pleaing poe i performe uing a pring law. The pine an pelvi are controlle a explaine in the previou ection. The analyi of the reult how a motion of the center of preure, when the right foot let go of the floor contact. Thi i not entirely in agreement with meaurement on real ance. Otherwie the reult how a table line. 5. Dicuion From a ballet point of view, controlling the center of preure i alo a way of controlling the contact of the feet with the groun. Ballet ancer are very much aware of the relation between their feet an the groun, ince it trongly influence their balance, their tance, an the auience impreion of the ancer boy. Balance an weight hifting are the mot baic technique in ballet. Thi paper ha hown how you can ue reult from biomechanic for eveloping trategie for ynamic animation of an articulate figure of a human. The focu ha been on balancing trategie, where two trategie have been compare: center of ma trategy, an the center of preure trategy. The latter howe far the bet reult, an our reult emontrate that uing the center of preure trategy make it poible to evelop trategie for an articulate figure of a human to balance in a very complicate poition. The center of preure trategy i connecte with the theory of the balance of the inverte penulum. We have ue it in a imple form jut working on the ankle joint, an we emontrate that it i an effective way of balancing for quiet taning among other poition. However, the inverte penulum i a too coare implification of the human boy, when the working leg i lifte balance i obtaine on one leg. Thi movement require motion of joint in the whole boy. We ha to rely on the center of ma trategy at the hip joint to tabilize the balance on one leg. The concluion i that human balance i complex, an that human poibly ue both the center of preure trategy an the center of ma trategy.

8 The center of preure trategy a preente in thi article i for fine tuning the balance. That i, the trategy i only ueful, when the projection of the center of ma i inie the upport polygon: In the cae of a fall, the natural human repone woul be to prouce counter movement uch a moving a leg, which in fact move an increae the ize of the upport polygon in orer to regain balance; the preure trategy coul be ue to etimate ueful upport polygon in orer to calculate a ueful leg movement, but uch algorithm have not been invetigate in thi paper. Our reult are in goo agreement with reult from biomechanic. We attribute the minor eviation oberve a implification in our moel uch a the implifie keleton an the mucle ue. From a eigner point of view, the ue of virtual pring for moeling actuator force an contact force are attractive ue to their implicity, but they require a lot of parameter tuning, an thee parameter are rather enitive to global moel change. Learning may be incorporate to avoi exceive parameter tuning, but reult are not guarantie to look aethetically pleaing. Reearch in fining a better metho for fining parameter coul make phyical animation of human much more ueful alo in area outie reearch. Future tep in our reearch will be to evelop trategie for exercie on one leg. Reference C. Peeren et al. / Simulation Moelling Practice an Theory 14 (2006) [1] William W. Armtrong, Mark W. Green, The ynamic of articulate rigi boie for purpoe of animation, The Viual Comput. 1 (4) (1985) [2] Norman I. Baler, Stephen W. Smoliar, Digital repreentation of human movement, ACM Comput. Surv. 11 (1) (1979) [3] B.A. Barky, N. Baler, D. Zeltzer (E.), Making Them Move: Mechanic Control an Animation of Articulate Figure, The Morgan Kaufmann Serie in Computer Graphic an Geometric Moeling, Morgan Kaufman Publiher Inc, [4] R. Barzel, A.H. Barr, A moeling ytem bae on ynamic contraint, In Comput. 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