Orientation & Quaternions. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017

Transcription

1 Orienttion & Quternions CSE69: Computer Animtion Instructor: Steve Rotenberg UCSD, Winter 7

2 Orienttion

3 Orienttion We will define orienttion to men n object s instntneous rottionl configurtion Think of it s the rottionl euivlent of position

4 Representing Positions Crtesin coordintes (x,y,z) re n esy nd nturl mens of representing position in 3D spce There re mny other lterntives such s polr nottion (r,θ,φ) nd you cn invent others if you wnt to

5 Representing Orienttions Is there simple mens of representing 3D orienttion? (nlogous to Crtesin coordintes?) Not relly. There re severl populr options though: Euler ngles Rottion vectors (xis/ngle) 3x3 mtrices Quternions nd more

6 Euler s Theorem Euler s Theorem: Any two independent orthonorml coordinte frmes cn be relted by seuence of rottions (not more thn three) bout coordinte xes, where no two successive rottions my be bout the sme xis. Not to be confused with Euler ngles, Euler integrtion, Newton-Euler dynmics, inviscid Euler eutions, Euler chrcteristic Leonrd Euler (77-783)

7 Euler Angles This mens tht we cn represent n orienttion with 3 numbers A seuence of rottions round principle xes is clled n Euler Angle Seuence Assuming we limit ourselves to 3 rottions without successive rottions bout the sme xis, we could use ny of the following seuences: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ

8 Euler Angles This gives us redundnt wys to store n orienttion using Euler ngles Different industries use different conventions for hndling Euler ngles (or no conventions)

9 Euler Angles to Mtrix Conversion To build mtrix from set of Euler ngles, we just multiply seuence of rottion mtrices together: y x y x y z x z y x z x z y x z y z x z y x z x z y x z y c c c s s c s s s c c c s s s s c s s c s c s c c s s c c x x x x y y y y z z z z x y z c s s c c s s c c s s c R R R

10 Euler Angle Order As mtrix multipliction is not commuttive, the order of opertions is importnt Rottions re ssumed to be reltive to fixed world xes, rther thn locl to the object One cn think of them s being locl to the object if the seuence order is reversed

11 Using Euler Angles To use Euler ngles, one must choose which of the representtions they wnt There my be some prcticl differences between them nd the best seuence my depend on wht exctly you re trying to ccomplish

12 Vehicle Orienttion Generlly, for vehicles, it is most convenient to rotte in roll (z), pitch (x), nd then yw (y) In situtions where there is definite ground plne, Euler ngles cn ctully be n intuitive representtion z y front of vehicle x

13 Gimbl Lock One potentil problem tht they cn suffer from is gimbl lock This results when two xes effectively line up, resulting in temporry loss of degree of freedom This is relted to the singulrities in longitude tht you get t the north nd south poles

14 Interpolting Euler Angles One cn simply interpolte between the three vlues independently This will result in the interpoltion following different pth depending on which of the schemes you choose This my or my not be problem, depending on your sitution Interpolting ner the poles cn be problemtic Note: when interpolting ngles, remember to check for crossing the +8/-8 degree boundries

15 Euler Angles Euler ngles re used in lot of pplictions, but they tend to reuire some rther rbitrry decisions They lso do not interpolte in consistent wy (but this isn t lwys bd) They cn suffer from Gimbl lock nd relted problems There is no simple wy to conctente rottions Conversion to/from mtrix reuires severl trigonometry opertions They re compct (reuiring only 3 numbers)

16 Rottion Vectors nd Axis/Angle Euler s Theorem lso shows tht ny two orienttions cn be relted by single rottion bout some xis (not necessrily principle xis) This mens tht we cn represent n rbitrry orienttion s rottion bout some unit xis by some ngle (4 numbers) (Axis/Angle form) Alterntely, we cn scle the xis by the ngle nd compct it down to single 3D vector (Rottion vector)

17 Axis/Angle to Mtrix To generte mtrix s rottion θ round n rbitrry unit xis : ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( z z x z y y z x x z y y y z y x y z x z y x x x c s c s c s c c s c s c s c c

18 Rottion Vectors To convert scled rottion vector to mtrix, one would hve to extrct the mgnitude out of it nd then rotte round the normlized xis Normlly, rottion vector formt is more useful for representing ngulr velocities nd ngulr ccelertions, rther thn ngulr position (orienttion)

19 Axis/Angle Representtion Storing n orienttion s n xis nd n ngle uses 4 numbers, but Euler s theorem sys tht we only need 3 numbers to represent n orienttion Mthemticlly, this mens tht we re using 4 degrees of freedom to represent 3 degrees of freedom vlue This implies tht there is possibly extr or redundnt informtion in the xis/ngle formt The redundncy mnifests itself in the mgnitude of the xis vector. The mgnitude crries no informtion, nd so it is redundnt. To remove the redundncy, we choose to normlize the xis, thus constrining the extr degree of freedom

20 Mtrix Representtion We cn use 3x3 mtrix to represent n orienttion s well This mens we now hve 9 numbers insted of 3, nd therefore, we hve 6 extr degrees of freedom NOTE: We don t use 4x4 mtrices here, s those re minly useful becuse they give us the bility to combine trnsltions. We will not be concerned with trnsltion tody, so we will just think of 3x3 mtrices.

21 Mtrix Representtion Those extr 6 DOFs mnifest themselves s 3 scles (x, y, nd z) nd 3 shers (xy, xz, nd yz) If we ssume the mtrix represents rigid trnsform (orthonorml), then we cn constrin the extr 6 DOFs b c bc b c c b

22 Mtrix Representtion Mtrices re usully the most computtionlly efficient wy to pply rottions to geometric dt, nd so most orienttion representtions ultimtely need to be converted into mtrix in order to do nything useful (trnsform verts ) Why then, shouldn t we just lwys use mtrices? Numericl issues Storge issues User interction issues Interpoltion issues

23 Quternions

24 Complex Numbers In lgebr, we study complex numbers of the form: + bi where i = (or i = )

25 Product of Complex Numbers If we multiply two complex numbers together, we get: + bi c + di = c + bci + di + bdi = c bd + bc + d i = α + βi

26 Polr Coordintes We cn think of complex number s point in the complex plne, where nd b re the Crtesin coordintes of the point We cn lso define polr coordintes r (distnce or mgnitude) nd θ (ngle) where r = + b θ = tn b,

27 Euler s Formul Remember Euler s Formul from lgebr? e iθ = cos θ + i sin θ This llows us to write complex number in polr form: re iθ = r cos θ + ir sin θ The product of two complex numbers in polr form is: r e iθ r e iθ = r r e i θ +θ

28 Product of Complex Numbers If we multiply two complex numbers c nd c together, the mgnitude of the product will eul the product of the mgnitudes of the originl two complex numbers The ngle θ of the product will eul the sum of the ngles of the two originl numbers Therefore, if we use complex numbers with mgnitudes of., we cn use them to represent rottions in the complex plne

29 Quternions Quternions re n interesting mthemticl concept with deep reltionship with the foundtions of lgebr nd number theory Invented by W.R.Hmilton in 843 In prctice, they re most useful to us s mens of representing orienttions A uternion hs 4 components 3

30 Quternions (Imginry Spce) Quternions re ctully n extension to complex numbers Of the 4 components, one is rel sclr number, nd the other 3 form vector in imginry ijk spce! i i j k jk ij i j k3 j k ijk kj ki ik ji

31 Product of Quternions If we multiply two uternions p nd together, we get: p = p + ip + jp + kp 3 + i + j + k 3 = p + i p + p + j p + p + k p 3 + p 3 + ij p p + ik p 3 p 3 + jk p 3 p 3 + i p + j p + k p 3 3

32 Product of Quternions = p + i p + p + j p + p + k p 3 + p 3 + ij p p + ik p 3 p 3 + jk p 3 p 3 + i p + j p + k p 3 3 = p p p p i p + p + p 3 p 3 + j p + p p 3 + p 3 + k p 3 + p 3 + p p

33 Quternions (Sclr/Vector) Sometimes, they re written s the combintion of sclr vlue s nd vector vlue v s, v where s v 3

34 Quternion Multipliction We cn perform multipliction on uternions if we expnd them into their complex number form i j k3 ss i j k i j k v v, sv 3 sv v v 3

35 Quternion Multipliction Note tht two unit uternions multiplied together will result in nother unit uternion This corresponds to the sme property of complex numbers Remember tht multipliction by complex numbers cn be thought of s rottion in the complex plne As uternions hve 3 imginry components, they cn effectively represent rottions in 3 plnes Quternions extend the plnr rottions of complex numbers to 3D rottions in spce

36 Unit Quternions For convenience, we will use only unit length uternions, s they will be sufficient for our purposes nd mke things little esier 3 These correspond to the set of vectors tht form the surfce of 4D hypersphere of rdius The surfce is ctully 3D volume in 4D spce, but it cn sometimes be visulized s n extension to the concept of D surfce on 3D sphere

37 Quternions s Rottions A uternion cn represent rottion by n ngle θ round unit xis : or cos x cos, sin sin sin If is unit length, then will be lso y z sin

38 Quternions s Rottions sin cos sin cos sin cos sin sin sin cos 3 z y x z y x

39 Quternion Negtion We see tht uternion cn be represented s rottion round unit xis This leds to potentil redundncy if we negte both the xis nd the rottion ngle This corresponds to negting ll 4 components of the uternion This leds to the sme orienttion in 3D spce! This is n importnt issue to remember: for every orienttion (3x3 orthonorml mtrix), we cn ctully produce opposite uternions tht mp to the sme orienttion

40 Quternion to Mtrix To convert uternion to rottion mtrix:

41 Mtrix to Quternion Mtrix to uternion is little more complex nd reuires nlyzing multiple cses to get the best numericl precision See Sm Buss s book 3D Computer Grphics (p.35) for description of the lgorithm

42 Mtrix to Quternion void Quternion::FromMtrix(const Mtrix44& mtx) { flot trce=mtx..x+mtx.b.y+mtx.c.z; if(trce>=mtx..x && trce>=mtx.b.y && trce>=mtx.c.z) { s=.5f*srtf(trce+.f); flot tmp=.5f/s; x=tmp*(mtx.b.z-mtx.c.y); y=tmp*(mtx.c.x-mtx..z); z=tmp*(mtx..y-mtx.b.x); } else if(mtx..x>mtx.b.y && mtx..x>mtx.c.z) { x=.5f*srtf(.f*mtx..x-trce+.f); flot tmp=.5f/x; s=tmp*(mtx.b.z-mtx.c.y); y=tmp*(mtx.b.x+mtx..y); z=tmp*(mtx..z+mtx.c.x); } else if(mtx.b.y>mtx.c.z) { y=.5f*srtf(.f*mtx.b.y-trce+.f); flot tmp=.5f/y; s=tmp*(mtx.c.x-mtx..z); x=tmp*(mtx.b.x+mtx..y); z=tmp*(mtx.c.y+mtx.b.z); } else { z=.5f*srtf(.f*mtx.c.z-trce+.f); flot tmp=.5f/z; s=tmp*(mtx..y-mtx.b.x); x=tmp*(mtx..z+mtx.c.x); y=tmp*(mtx.c.y+mtx.b.z); } }

43 Product of Quternions A uternion cn be used to represent n orienttion The product of two uternions represents new orienttion tht is orienttion rotted by orienttion If we used mtrices to represent the orienttions insted, we would hve MM In other words: toqut(m) * toqut(m) = ± toqut(m *M)

44 Quternion Dot Products The dot product of two uternions works in the sme wy s the dot product of two vectors: p p p p p33 p cos The ngle between two uternions in 4D spce is hlf the ngle one would need to rotte from one orienttion to the other in 3D spce

45 Spheres Think of person stnding on the surfce of big sphere (like plnet) From the person s point of view, they cn move in long two orthogonl xes (front/bck) nd (left/right) There is no perception of ny fixed poles or longitude/ltitude, becuse no mtter which direction they fce, they lwys hve two orthogonl wys to go From their point of view, they might s well be moving on infinite D plne, however if they go too fr in one direction, they will come bck to where they strted!

46 Hyperspheres Now extend this concept to moving in the hypersphere of unit uternions The person now hs three orthogonl directions to go No mtter how they re oriented in this spce, they cn lwys go some combintion of forwrd/bckwrd, left/right nd up/down If they go too fr in ny one direction, they will come bck to where they strted

47 Hyperspheres Now consider tht person s loction on this hypersphere represents n orienttion Any incrementl movement long one of the orthogonl xes in curved spce corresponds to n incrementl rottion long n xis in rel spce (distnces long the hypersphere correspond to ngles in 3D spce) Moving in some rbitrry direction corresponds to rotting round some rbitrry xis If you move too fr in one direction, you come bck to where you strted (corresponding to rotting 36 degrees round ny one xis)

48 Hyperspheres A distnce of x long the surfce of the hypersphere corresponds to rottion of ngle x rdins This mens tht moving long 9 degree rc on the hypersphere corresponds to rotting n object by 8 degrees Trveling 8 degrees corresponds to 36 degree rottion, thus getting you bck to where you strted This implies tht nd - correspond to the sme orienttion

49 Hyperspheres Consider wht would hppen if this ws not the cse, nd if 8 degrees long the hypersphere corresponded to 8 degree rottion This would men tht there is exctly one orienttion tht is 8 opposite to reference orienttion In relity, there is continuum of possible orienttions tht re 8 wy from reference They cn be found on the eutor reltive to ny point on the hypersphere

50 Hyperspheres Also consider wht hppens if you rotte book 8 round x, then 8 round y, nd then 8 round z You end up bck where you strted This corresponds to trveling long tringle on the hypersphere where ech edge is 9 degree rc, orthogonl to ech other edge

51 Quternion Joints One cn crete skeleton using uternion joints One possibility is to simply llow uternion joint type nd provide locl mtrix function tht tkes uternion Another possibility is to lso compute the world mtrices s uternion multiplictions. This involves little less mth thn mtrices, but my not prove to be significntly fster. Also, one would still hve to hndle the joint offsets with mtrix mth

52 Quternions in the Pose Vector Using uternions in the skeleton dds some complictions, s they cn t simply be treted s 4 independent DOFs through the rig The reson is tht the 4 numbers re not independent, nd so n nimtion system would hve to hndle them specificlly s uternion To del with this, one might hve to extend the concept of the pose vector s contining n rry of sclrs nd n rry of uternions When higher level nimtion code blends nd mnipultes poses, it will hve to tret uternions specilly

53 Quternion Interpoltion

54 Liner Interpoltion If we wnt to do liner interpoltion between two points nd b in norml spce Lerp(t,,b) = (-t) + (t)b where t rnges from to Note tht the Lerp opertion cn be thought of s weighted verge (convex) We could lso write it in it s dditive blend form: Lerp(t,,b) = + t(b-)

55 Sphericl Liner Interpoltion If we wnt to interpolte between two points on sphere (or hypersphere), we don t just wnt to Lerp between them Insted, we will trvel cross the surfce of the sphere by following gret rc

56 Sphericl Liner Interpoltion We define the sphericl liner interpoltion of two unit vectors in n-dimensionl spce s: Slerp( t,, b) sin t sin t sin sin b where : cos b

57 Quternion Interpoltion Remember tht there re two redundnt vectors in uternion spce for every uniue orienttion in 3D spce Wht is the difference between: Slerp(t,,b) nd Slerp(t,-,b)? One of these will trvel less thn 9 degrees while the other will trvel more thn 9 degrees cross the sphere This corresponds to rotting the short wy or the long wy Usully, we wnt to tke the short wy, so we negte one of them if their dot product is <

58 Bezier Curves in D & 3D Spce Bezier curves cn be thought of s higher order extension of liner p interpoltion p p p p 3 p p p p

59 de Cstleju Algorithm p Find the point x on the curve s function of prmeter t: p p p 3

60 de Cstleju Algorithm p Lerp Lerp Lerp t, p t, p t, p, p, p, p 3 p p p 3

61 de Cstleju Algorithm r r r Lerp Lerp t, t,,, r

62 de Cstleju Algorithm x Lerp t, r r, r x r

63 de Cstleju Algorithm x

64 de Cstleju Algorithm 3,,,,,, p p p p p p t Lerp t Lerp t Lerp,,,, r r t Lerp t Lerp,, r r x t Lerp 3 p p p p

65 Bezier Curves in Quternion Spce We cn construct Bezier curves on the 4D hypersphere by following the exct sme procedure using Slerp insted of Lerp It s good ide to flip (negte) the input uternions s necessry in order to mke it go the short wy There re other, more sophisticted curve interpoltion lgorithms tht cn be pplied to hypersphere Interpolte severl key poses Additionl control over ngulr velocity, ngulr ccelertion, smoothness

66 Quternion Summry Quternions re 4D vectors tht cn represent 3D rigid body orienttions We choose to force them to be unit length Key nimtion functions: Quternion-to-mtrix / mtrix-to-uternion Quternion multipliction: fster thn mtrix multipliction Slerp: interpolte between rbitrry orienttions Sphericl curves: de Cstleju lgorithm for cubic Bezier curves on the hypersphere

67 Quternion References Animting Rottion with Quternion Curves, Ken Shoemke, SIGGRAPH 985 Quternions nd Rottion Seuences, Kuipers