Rigid Body Dynamics. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018

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1 Rg Bo Dnmcs CSE169: Compute Anmton nstucto: Steve Roteneg UCSD, Wnte 2018

2 Coss Pouct k j

3 Popetes of the Coss Pouct

4 Coss Pouct c c c c

5 Coss Pouct c c c c c c

6 Coss Pouct ˆ ˆ c c c

7 Ht Opeto We ve ntouce the ht opeto whch convets vecto nto skew-smmetc mt ( T = ) Ths llows us to tun coss pouct of two vectos nto ot pouct of mt n vecto Ths s mnl fo lgec convenence, s the ot pouct s ssoctve (lthough stll not commuttve) = (non commuttve) መ c = መ c (ssoctve)

8 Devtve of Rottng Vecto Let s s tht vecto s ottng oun the ogn, mntnng fe stnce At n nstnt, t hs n ngul veloct of

9 Devtve of Rottng Mt f mt A s g 33 mt ottng wth ngul veloct Ths mples tht the,, n c es must e ottng oun The evtves of ech s e,, n c, n so the evtve of the ente mt s: A A ˆ A

10 Pouct Rule The pouct ule efnes the evtve of poucts c c c c

11 Pouct Rule t cn e etene to vecto n mt poucts s well B A B A B A

12 Egenvlue Equton

13 Smmetc Mt

14 Smmetc Mt Dgonlton

15 Dnmcs of Ptcles

16 Knemtcs of Ptcle poston v veloc t v 2 2 ccelet on

17 Mss, Momentum, n Foce m mss p mv momentum f p m foce

18 Moment of Momentum The moment of momentum s vecto L p Also known s ngul momentum (the two tems men scll the sme thng, ut e use n slghtl ffeent stutons) Angul momentum hs pllel popetes wth lne momentum n ptcul, lke the lne momentum, ngul momentum s conseve n mechncl sstem t s tpcll epesente wth cptl L, whch s unfotuntel nconsstent wth ou stn of usng lowecse fo vectos

19 Moment of Momentum L s the sme fo ll thee of these ptcles p 3 p 2 p L p 1

20 Moment of Momentum L s ffeent fo ll of these ptcles L p p p 2 1 p 3

21 Moment of Foce (Toque) The moment of foce (o toque) out pont s the te of chnge of the moment of momentum out tht pont τ L

22 Moment of Foce (Toque) f τ f v v τ f p v τ p p L τ p L m

23 Rottonl net L=p s genel epesson fo the moment of momentum of ptcle n cse whee we hve ptcle ottng oun the ogn whle keepng fe stnce, we cn e-epess the moment of momentum n tems of t s ngul veloct

24 Rottonl net L L L v v L p L ˆ ˆ ˆ ˆ m m m m m m

25 Rottonl net ˆ ˆ m m m

26 Rottonl net L m m m m m m m m m

27 Rottonl net The ottonl net mt s 33 mt tht s essentll the ottonl equvlent of mss t eltes the ngul momentum of sstem to ts ngul veloct the equton L Ths s sml to how mss eltes lne momentum to lne veloct, ut otton s tonl complet p mv

28 Sstems of Ptcles

29 Sstems of Ptcles m totl n 1 m totl mss of ll ptcles cm m m poston of cente of mss p cm p m v totl momentum

30 Veloct of Cente of Mss cm totl cm totl cm cm cm cm cm m m m m m m m m v p p v v v v

31 Foce on Ptcle The chnge n momentum of the cente of mss s equl to the sum of ll of the foces on the nvul ptcles Ths mens tht the esultng chnge n the totl momentum s nepenent of the locton of the pple foce p cm p p cm p p f

32 Sstems of Ptcles The totl moment of momentum oun the cente of mss s: L cm p L cm cm p

33 Toque n Sstem of Ptcles L cm p τ cm L cm p τ cm p τ cm f

34 Sstems of Ptcles We cn see tht sstem of ptcles ehves lot lke ptcle tself t hs mss, poston (cente of mss), momentum, veloct, cceleton, n t espons to foces f f cm We cn lso efne t s ngul momentum n elte chnge n sstem ngul momentum to foce pple to n nvul ptcle τ cm f

35 ntenl Foces f foces e genete wthn the ptcle sstem (s fom gvt, o spngs connectng ptcles) the must oe Newton s Th Lw (eve cton hs n equl n opposte ecton) Ths mens tht ntenl foces wll lnce out n hve no net effect on the totl momentum of the sstem As those opposte foces ct long the sme lne of cton, the toques on the cente of mss cncel out s well

36 Dnmcs of Rg Boes

37 Knemtcs of Rg Bo Fo the poston of the cente of mss of the g o: v cm cm cm v cm cm 2 cm 2

38 Knemtcs of Rg Bo

39 Rg Boes We tet g o s sstem of ptcles, whee the stnce etween n two ptcles s fe We wll ssume tht ntenl foces e genete to hol the eltve postons fe. These ntenl foces e ll lnce out wth Newton s th lw, so tht the ll cncel out n hve no effect on the totl momentum o ngul momentum The g o cn ctull hve n nfnte nume of ptcles, spe out ove fnte volume nste of mss eng concentte t scete ponts, we wll conse the enst s eng vle ove the volume

40 Rg Bo Mss Wth sstem of ptcles, we efne the totl mss s: m n 1 m Fo g o, we wll efne t s the ntegl of the enst ρ ove some volumetc omn Ω m

41 Rg Bo Cente of Mss The cente of mss s: cm

42 Rottonl net of Ptcle

43 Rg Bo Rottonl net

44 Rg Bo Rottonl net The g o ottonl net s 33 smmetc mt tht encoes the stuton of mss oun the cente of mss t s clculte clcultng the ntegls on the pevous sle ntegtng ove the volume of the g o whee nctes the vecto fom the cente of mss to the poston of the volume ntegton element n ρ epesents the enst t tht locton These ntegls cn e clculte wth nltcl fomuls fo smple shpes lke sphees, clnes, n oes Thee lso ests n nltcl technque fo computng them on tngle meshes s well (Mtch-Eel lgothm)

45 Rottonl net Dgonlton

46 Dgonlton of Rottonl net

47 Rottonl net of Bo

48 Rottonl net of Sphee

49 Rottonl net

50 Devtve of Rottonl netl A A A A A A A A A A A A A A ˆ ˆ ˆ ˆ T T T T T T T T T

51 Devtve of Angul Momentum τ τ τ L τ L ˆ ˆ

52 Newton-Eule Equtons f m τ

53 Apple Foces & Toques f cg f τ cg f 1 m f 1 τ

54 Popetes of Rg Boes m A v p mv L f m τ f

55 Rg Bo Smulton RgBo { vo Upe(flot tme); vo ApplFoce(Vecto3 &f,vecto3 &pos); pvte: // constnts flot Mss; Vecto3 Rotnet; //,, & fom gonl net // vles Mt34 Mt; // contns poston & oentton Vecto3 Momentum,AngMomentum; }; // ccumultos Vecto3 Foce,Toque;

56 Rg Bo Smulton RgBo::ApplFoce(Vecto3 &f,vecto3 &pos) { Foce += f; Toque += (pos-mt.) f }

57 Rg Bo Smulton RgBo::Upe(flot tme) { // Upe poston Momentum += Foce * tme; Mt. += (Momentum/Mss) * tme; // Mt. = poston } // Upe oentton AngMomentum += Toque * tme; Mt33 = Mt 0 Mt T // A 0 A T Vecto3 = -1 L flot ngle = * tme; // mgntue of Vecto3 s = ; s.nomle(); Mt.RotteUntAs(s,ngle);

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