Animation ליאור שפירא גרפיקה ממוחשבת סמסטר א' תש"ע

Transcription

1 Animation ליאור שפירא גרפיקה ממוחשבת סמסטר א' תש"ע מבוסס על שקפים של תומס פנקהאוסר, פרינסטון

2 הצצה לשבוע הבא... רן גל יספר לנו על iwires טל, מהנדס מאלביט יספר לנו על פרויקטים גרפיים מעניינים שהם מפתחים 2

3 Today Animation Simulation Dynamics Kinematics Deformations 3

4 Computer Animation What is animation? Make objects change over time according to scripted actions What is simulation? Predict how objects change over time according to physical laws Pixar University of Illinois

5 Computer Animation Pixar Parody Pixar

6 Outline Keyframe animation Adding inverse kinematics Adding dynamics Lior 09

7 Keyframe Animation Define character poses at specific time steps called keyframes Lasseter `87

8 Keyframe Animation Interpolate variables describing keyframes to determine poses for character in-between Lasseter `87

9 Example: 2-Link Structure Two links connected by rotational joints 2 l 2 End-Effector l 1 X = (x,y) 1 (0,0)

10 Forward Kinematics Animator specifies joint angles: 1 and 2 Computer finds positions of end-effector: X 2 l 2 l 1 X = (x,y) (0,0) 1

11 Forward Kinematics Joint motions can be specified by spline curves 2 l 2 l 1 X = (x,y) (0,0) t

12 Keyframe Animation Inbetweening: Linear interpolation - usually not enough continuity Linear interpolation H&B Figure 16.16

13 Keyframe Animation Inbetweening: Spline interpolation - maybe good enough H&B Figure 16.11

14 Keyframe Animation Inbetweening: Cubic spline interpolation - maybe good enough» May not follow physical laws Lasseter `87

15 Keyframe Animation Inbetweening: Cubic spline interpolation - maybe good enough» May not follow physical laws Lasseter `87

16 Example: Walk Cycle Articulated figure: Watt & Watt

17 Example: Walk Cycle Hip joint orientation: Watt & Watt

18 Example: Walk Cycle Knee joint orientation: Watt & Watt

19 Example: Walk Cycle Ankle joint orientation: Watt & Watt

20 Outline Keyframe animation Adding inverse kinematics Adding dynamics

21 Example: 2-Link Structure What if animator knows position of end-effector 2 l 1 l 2 X = (x,y) End-Effector 1 (0,0)

22 Inverse Kinematics Animator specifies end-effector positions: X Computer finds joint angles: 1 and 2 : 2 l 1 l 2 X = (x,y) (0,0) 1

23 Inverse Kinematics End-effector postions can be specified by spline curves 2 l 2 l 1 X = (x,y) (0,0) 1 x y t

24 Inverse Kinematics Problem for more complex structures System of equations is usually under-defined Multiple solutions 2 l 2 l 3 X = (x,y) l (0,0) Three unknowns: 1, 2, 3 Two equations: x, y

25 Inverse Kinematics Solution for more complex structures: Find best solution (e.g., minimize energy in motion) Non-linear optimization 2 l 2 l 3 X = (x,y) l (0,0)

26 Outline Keyframe animation Adding inverse kinematics Adding dynamics

27 Dynamics Simulation of physics insures realism of motion Lasseter `87

28 Spacetime Constraints Animator specifies constraints: What the character s physical structure is» e.g., articulated figure What the character has to do (keyframes)» e.g., jump from here to there within time t What other physical structures are present» e.g., floor to push off and land How the motion should be performed» e.g., minimize energy

29 Spacetime Constraints Computer finds the best physical motion satisfying constraints Example: particle with jet propulsion x(t) is position of particle at time t f(t) is force of jet propulsion at time t Particle s equation of motion is: Suppose we want to move from a to b within t 0 to t 1 with minimum jet fuel: Minimize subject to x(t 0 )=a and x(t 1 )=b Witkin & Kass `88

30 Spacetime Constraints Solve with iterative optimization methods Witkin & Kass `88

31 Spacetime Constraints Advantages: Free animator from having to specify details of physically realistic motion with spline curves Easy to vary motions due to new parameters and/or new constraints Challenges: Specifying constraints and objective functions Avoiding local minima during optimization

32 Spacetime Constraints Adapting motion: Original Jump Heavier Base Witkin & Kass `88

33 Spacetime Constraints Adapting motion: Hurdle Witkin & Kass `88

34 Spacetime Constraints Adapting motion: Ski Jump Witkin & Kass `88

35 Spacetime Constraints Advantages: Free animator from having to specify details of physically realistic motion with spline curves Easy to vary motions due to new parameters and/or new constraints Challenges: Specifying constraints and objective functions Avoiding local minima during optimization

36 Example: Manipulation of Sims. Interactive Manipulation of Rigid Body Simulations. Popovic et al Siggraph Popovic

37 Summary Keyframe animation Poses specified at key times In-betweening to fill in the rest Incorporating inverse kinematics Makes keyframes easier to specify Incorporating dynamics Makes animation easier to adapt

38 38

39 Simulation Dynamics Considers underlying forces Compute motion from initial conditions and physics Kinematics Considers only motion Determined by positions, velocities, accelerations

40 Dynamics Hodgins

41 Passive Dynamics No muscles or motors Smoke Water Cloth Fire Fireworks Dice McAllister

42 Passive Dynamics Physical laws Newton s laws Hook s law Etc. Physical phenomena Gravity Momentum Friction Collisions Elasticity Fracture McAllister

43 Fun with Bunny Porous Flow in Particle-Based Fluid Simulations Lenaerts et al, SIGGRAPH 2007 Fire with cellular patterns Ron Fedkiw A Finite Element Method for Animating Large Viscoplastic Flow Bargteil et al SIGGRAPH

44 Particle Systems A particle is a point mass Mass Position Velocity Forces Color Lifetime p = (x,y,z) Use lots of particles to model complex phenomena Keep array of particles Newton s laws v

45 Our Particle enum ParticleType { Create,Update }; struct Vector2d{ double x; double y; v p = (x,y) }; struct Particle { Vector2d Pos; //Position of the particle Vector2d Vel; //Velocity of the particle int age; //Current age of the particle int LifeSpan; //Age after which the particle dies int color; int size; }; 45

46 Particle Systems For each frame: Create new particles and assign attributes Delete any expired particles Update particles based on attributes and physics Render particles

47 Our Particle void Init(long num_part,long num_forces, Vector2d Forces[], ParticleType part_type,vector2d vel, Vector2d pos1,vector2d pos2, int lifespan, int color, int size){ Particles = new Particle[num_part]; ParticleNum=num_part; for(int i=0;i<num_forces;i++){ TotForce.x+=Forces[i].x; TotForce.y+=Forces[i].y; } } Particle_Type=part_type; Vel=vel; Pos1=pos1; Pos2=pos2; LifeSpan=lifespan; Color=color; Size=size; randomize(); InitParticles(); 47

48 Creating Particles Where to create particles? Predefined source Surface of shape Where particle density is low etc. McAllister

49 Creating Particles Where to create particles? Predefined source Surface of shape Where particle density is low etc. Reeves

50 Deleting Particles When to delete particles? Predefined sink Surface of shape Where density is high Life span Random McAllister

51 Rendering Particles Rendering styles Points Polygons Shapes Trails etc. McAllister

52 Rendering Particles Rendering styles Points Polygons Shapes Trails etc. McAllister

53 Rendering Particles Rendering styles Points Polygons Shapes Trails etc. McAllister

54 Rendering Particles Rendering styles Points Polygons Shapes Trails etc. McAllister

55 Particle Systems For each frame: Create new particles and assign attributes Delete any expired particles Update particles based on attributes and physics Render particles McAllister

56 Equations of Motion Newton s Law for a point mass f = ma Computing particle motion requires solving second-order differential equation Add variable v to form coupled first-order differential equations

57 Solving the Equations of Motion Initial value problem Know p(0), v(0), a(0) Can compute force at any time and position Compute p(t) by forward integration f p(0) p(t) Hodgins

58 Solving the Equations of Motion Euler integration p(t+t)=p(t) + t v(t) v(t+t)=v(t) + t f(x,t)/m Hodgins

59 Solving the Equations of Motion Euler integration p(t+t)=p(t) + t v(t) v(t+t)=v(t) + t f(p(t),t)/m Problem: Accuracy decreases as t gets bigger Hodgins

60 Solving the Equations of Motion Midpoint method (2 nd order Runge-Kutta) Compute an Euler step Evalute f at the midpoint Take an Euler step using midpoint force» v(t+t)=v(t) + t f( p(t) + 0.5*t v(t),t) Hodgins

61 Solving the Equations of Motion Adapting step size Compute p a by taking one step of size h Compute p b by taking 2 steps of size h/2 Error = p a - p b Multiply step size by factor (constant/error) p b error p a

62 Particle System Forces Force fields Gravity, wind, pressure Viscosity/damping Liquids, drag Collisions Environment Other particles Other particles Springs between neighboring particles (mesh) Useful for cloth

63 Particle System Forces Witkin

64 Example: Gravity McAllister

65 Example: Fire

66 Example: Bouncing Off Wall Requires Collision detection Collision response (dynamic forces) Witkin

67 Example: Bouncing Off Wall Witkin

68 Example: Bouncing Off Wall Witkin

69 Example: Bouncing Off Wall Witkin

70 Example: Bouncing Off Wall Witkin

71 Example: Bouncing Off Particles

72 Advancing our Particles void Run(){ while(!kbhit()){ delay(10); cleardevice(); //delay(10); for(int i=0;i<particlenum;i++){ if(particles[i].age >= 0){ setfillstyle(1,particles[i].color); setcolor(particles[i].color); //circle((int)particles[i].pos.x,(int)particles[i].pos.y,particles[i].size); fillellipse( ); Particles[i].Vel.x+=TotForce.x; Particles[i].Vel.y+=TotForce.y; Particles[i].Pos.x+=Particles[i].Vel.x; Particles[i].Pos.y+=Particles[i].Vel.y; Particles[i].age++; if(particles[i].age > Particles[i].LifeSpan){ if(particle_type == Create) { InitParticle(i); } else { Particles[i].age = -1; } } } } } } 72

73 Example: Cloth Spring-mass mesh Hooke s law f = force k s = spring constant d = p - q s = resting length Hodgins

74 Example: Cloth Spring-mass mesh Hodgins

75 Example: Cloth Animating Developable Surfaces Simulating Knitted Cloth at the Yarn Level

76 Example: Flocks & Herds

77 Summary Particle systems Lots of particles Simple physics Interesting behaviors Waterfalls Smoke Cloth Flocks Solving motion equations Simplest method is Euler integration Better to use adaptive step sizes

78 Passive vs. Active Dynamics Hodgins

79 Active Dynamics Motions Physics Controllers Learning Behaviors States Cognition Planning Funge99

80 Motion Example 1: how do worms move? Grzeszczuk95

81 Snake Motion Grzeszczuk95

82 Worm Biomechanical Model Grzeszczuk95

83 Worm Physics f a X dl k( L I) D f = force along spring direction dt f / m 1 m fdtdt k = spring force constant D = damping force I = current spring length L = minimum energy spring length plus forces due to friction with ground. Miller88

84 Her Majesty s Secret Serpent Miller89

85 Time permitting OTHER TOPICS 85

86 Other topics in Animation Motion Capture Skinning Deformations Differential representation Cage deformations Deformation transfer 86

87 What do we expect from surface deformation? Smooth effect on the large scale As-rigid-as-possible effect on the small scale (preserves details)

88 Several approaches FFD (space deformation) Lattice-based (Sederberg & Parry 86, Coquillart 90, ) Curve-/handle-based (Singh & Fiume 98, Botsch et al. 05, ) Cage-based (Ju et al. 05, Joshi et al. 07, Kopf et al. 07) Pros: efficiency almost independent of the surface resolution possible reuse Cons: space warp, so can t precisely control surface properties images taken from [Sederberg and Parry 86] and [Ju et al. 05]

89 Several approaches Surface-based approaches Multiresolution modeling Zorin et al. 97, Kobbelt et al. 98, Lee 98, Guskov et al. 99, Botsch and Kobbelt 04, Differential coordinates linear optimization Lipman et al. 04, Sorkine et al. 04, Yu et al. 04, Lipman et al. 05, Zayer et al. 05, Botsch et al. 06, Fu et al. 06, Non-linear global optimization approaches Kraevoy & Sheffer 04, Sumner et al. 05, Hunag et al. 06, Au et al. 06, Botsch et al. 06, Shi et al. 07, images taken from PriMo, Botsch et al. 06

90 Surface-based approaches Pros: direct interaction with the surface control over surface properties Cons: linear optimization suffers from artifacts (e.g. translation insensitivity) non-linear optimization is more expensive and non-trivial to implement

91 Sorkine et al 2004 LAPLACIAN SURFACE EDITING 91

92 Differential Coordinates Differential coordinates are defined by the discrete Laplacian operator: 1 δ L( v ) v v i i i j di j N () i For highly irregular meshes: cotangent weights [Desbrun et al. 99] average of the neighbors

93 Why differential coordinates? They represent the local detail / local shape description The direction approximates the normal The size approximates the mean curvature 1 d δ v v v v i i v N() i len( ) 0 i 1 len( ) v 1 lim v vds H( v ) n len( ) i i i v i ds

94 Why differential coordinates? Local detail representation enables detail preservation through various modeling tasks Representation with sparse matrices Efficient linear surface reconstruction

95 Overall framework Compute differential representation LV ( ) Pose modeling constraints v u, ic i i Reconstruct the surface in least-squares sense V arg min L( V ) vi ui V ic 2 2

96 Overall framework ROI is bounded by a belt (static anchors) Manipulation through handle(s)

97 Problem: invariance to transformations The basic Laplacian operator is translation-invariant, but not rotation- and scale-invariant Reconstruction attempts to preserve the original global orientation of the details

98 Implicit definition of transformations The idea: solve for local transformations AND the edited surface simultaneously! V n 2 arg min L( vi ) Ti ( δi) vj u V i1 jc j 2 Transformation of the local frame

99 Defining the transformations T i How to formulate T i? Based on the local (1-ring) neighborhood Linear dependence on the unknown v i k k i i i i i i i i i T T T v v v v v v arg min ( ) ( ) n i i j V j i i j C L V T δ v v u Members of the 1-ring of i-th vertex T i

100 Defining the transformations T i First attempt: define T i simply by solving T i arg min T i k vi j1 j T v i i j 2 T i v i v 1 i v 2 ik vi v 1 i v 2 ik

101 Defining the transformations T i Plug the expressions for T i into the least-squares reconstruction formula: arg min ( ) n i j i i i j j V C V T L v v δ u Linear combination of the unknown v i

102 Constraining T i Trivial solution for T i will result in membrane surface reconstruction To preserve the shape of the details we constrain T i to rotations, uniform scales and translations t11 t12 t13 t14 t21 t22 t23 t24 Ti t31 t32 t33 t34 t41 t42 t43 t44 Linear constraints on t lm so that T i is rotation+scale+translation??

103 Constraining T i 3D case Not linear in 3D: rotation + s exp H si H uniform scale T hh H is 33 skew-symmetric, Hx hx Linearize by dropping the quadratic term

104 Adjusting T i Due to linearization, T i scale the space along the h axis by cos When is large, this causes anisotropy Possible correction: Compute T i, remove the scaling component and reconstruct the surface again from the corrected i Apply our technique from [Lipman et al. 04] first, and then the current technique with small.

105 Some results

106 Some results

107 Some results