computer animation · 2001. 11. 30. · pixar 4 3-d and 2-d animation homer 3-d homer 2-d 5 outline...

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1

Computer Animation

Adam Finkelstein

Princeton University

C0S 426, Fall 2001

2Advertisement

3Computer Animation

• What is animation?o Make objects change over time

according to scripted actions

• What is simulation?o Predict how objects change over time

according to physical laws

University of Illinois

Pixar

43-D and 2-D animation

Homer 3-D Homer 2-D

5Outline

• Principles of animation

• Keyframe animation

• Articulated figures

• Kinematics

• Dynamics

Angel Plate 1

6Principles of Traditional Animation

• Squash and stretch• Slow In and out• Anticipation• Exaggeration• Follow through and overlapping action• Timing• Staging• Straight ahead action and pose-to-pose action• Arcs• Secondary action• Appeal

Disney

2


• Squash and stretch

Lasseter `87


• Slow In and Out

Watt Figure 13.5


• Anticipation (and squash & stretch)

Lasseter `87


• Squash and stretch• Slow In and out• Anticipation• Exaggeration• Follow through and overlapping action• Timing• Staging• Straight ahead action and pose-to-pose action• Arcs• Secondary action• Appeal

Disney

11Computer Animation

• Animation pipelineo 3D modelingo Motion specificationo Motion simulationo Shading, lighting, & renderingo Postprocessing

Pixar

12Keyframe Animation

• Define character poses at specific time stepscalled “keyframes”

Lasseter `87

3


• Interpolate variables describing keyframes todetermine poses for character “in-between”

Lasseter `87


• Inbetweening:o Linear interpolation - usually not enough continuity

H&B Figure 16.16

Linear interpolation


• Inbetweening:o Spline interpolation - maybe good enough

H&B Figure 16.11


• Inbetweening:o Cubic spline interpolation - maybe good enough

» May not follow physical laws

Lasseter `87


• Inbetweening:o Cubic spline interpolation - maybe good enough

» May not follow physical laws

Lasseter `87


• Inbetweening:o Inverse kinematics or dynamics

Rose et al. `96

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19Outline




• Kinematics

• Dynamics

Angel Plate 1

20Articulated Figures

• Character poses described by set of rigid bodiesconnected by “joints”

Angel Figures 8.8 & 8.9

Base

Arm

Hand

Scene Graph


Rose et al. `96

• Well-suited for humanoid characters

Root

LHip

LKnee

LAnkle

RHip

RKnee

RAnkle

Chest

LCollar

LShld

LElbow

LWrist

LCollar

LShld

LElbow

LWrist

Neck

Head


Mike Marr, COS 426, Princeton University, 1995

• Joints provide handles for movingarticulated figure


• Inbetweeningo Compute joint angles between keyframes

Watt & Watt

24Example: Walk Cycle

• Articulated figure:

Watt & Watt

5


• Hip joint orientation:

Watt & Watt


• Knee joint orientation:

Watt & Watt


• Ankle joint orientation:

Watt & Watt

28Example: Run Cycle

Mike Marr, COS 426, Princeton University, 1995

29Example: Ice Skating

(Mao Chen, Zaijin Guan, Zhiyan Liu, Xiaohu Qie,CS426, Fall98, Princeton University)

30Example: Horse

(Casey McTaggart, CS426, Fall99)

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31Outline




• Kinematics

• Dynamics

Angel Plate 1

32Kinematics and Dynamics

• Kinematicso Considers only motiono Determined by positions, velocities, accelerations

• Dynamicso Considers underlying forceso Compute motion from initial conditions and physics

33Example: 2-Link Structure

• Two links connected by rotational joints

Θ1

Θ2

X = (x,y)

2

1

(0,0)

“End-Effector”

34Forward Kinematics

• Animator specifies joint angles: Θ1 and Θ2

• Computer finds positions of end-effector: X

))sin(sin),cos(cos( 2121121211 Θ+Θ+ΘΘ+Θ+Θ= llllX

Θ1

Θ2

X = (x,y)

2

1

(0,0)


• Joint motions can be specified by spline curves

Θ1

Θ2

X = (x,y)

2

1

(0,0)

Θ2

Θ1

t


• Joint motions can be specified by initial conditionsand velocities

Θ1

Θ2

X = (x,y)

2

1

(0,0)

1.02.1

250)0(60)0(

21

21

−=Θ

=Θ

=Θ=Θ

dt

d

dt

d

oo

7

37Example: 2-Link Structure

• What if animator knows position of “end-effector”

Θ1

Θ2

X = (x,y)

2

1

(0,0)

“End-Effector”

38Inverse Kinematics

• Animator specifies end-effector positions: X

• Computer finds joint angles: Θ1 and Θ2:

xllyl

yllxl

))cos(())sin((

))cos(()sin((

22122

221221 Θ++Θ

Θ++Θ−=Θ

Θ1

Θ2

X = (x,y)2

1

(0,0)

−−+=Θ −

21

22

21

221

2 cosll

llxx

2


• End-effector postions can be specified by splinecurves

Θ1

Θ2

X = (x,y)

2

1

(0,0)

y

x

t


• Problem for more complex structureso System of equations is usually under-definedo Multiple solutions

Θ1

Θ2

2

1

(0,0)

X = (x,y)

3

Θ3

Three unknowns: Θ1, Θ2 , Θ3Two equations: x, y


• Solution for more complex structures:o Find best solution (e.g., minimize energy in motion)o Non-linear optimization

Θ1

Θ2

2

1

(0,0)

X = (x,y)

3

Θ3


Fujito, Milliron, Ngan, & SanockiPrinceton University

“Ballboy”

8

43Summary of Kinematics

• Forward kinematicso Specify conditions (joint angles)o Compute positions of end-effectors

• Inverse kinematicso “Goal-directed” motiono Specify goal positions of end effectorso Compute conditions required to achieve goals

Inverse kinematics provides easierspecification for many animation tasks,but it is computationally more difficult

Inverse kinematics provides easierspecification for many animation tasks,but it is computationally more difficult

44Overview

• Kinematicso Considers only motiono Determined by positions, velocities, accelerations

• Dynamicso Considers underlying forceso Compute motion from initial conditions and physics

45Dynamics

• Simulation of physics insures realism of motion

Lasseter `87

46Spacetime Constraints

• Animator specifies constraints:o What the character’s physical structure is

» e.g., articulated figureo What the character has to do

» e.g., jump from here to there within time to What other physical structures are present

» e.g., floor to push off and lando How the motion should be performed

» e.g., minimize energy


• Computer finds the “best” physical motionsatisfying constraints

• Example: particle with jet propulsiono x(t) is position of particle at time to f(t) is force of jet propulsion at time to Particle’s equation of motion is:

o Suppose we want to move from a to b within t0 to t1with minimum jet fuel:

0'' =−− mgfmx

dttft

t∫1

0

2)(Minimize subject to x(t0)=a and x(t1)=bWitkin & Kass `88


• Discretize time steps:

02

''2

11 =−−

+−= −+ mgf

h

xxxxm i

iiii

2∑i

ifhMinimize subject to x0=a and x1=b

211

1

2''

'

h

xxxx

h

xxx

iiii

iii

−+

−

+−=

−=

Witkin & Kass `88

9


• Solve withiterativeoptimizationmethods

Witkin & Kass `88


• Advantages:o Free animator from having to specify details of

physically realistic motion with spline curveso Easy to vary motions due to new parameters

and/or new constraints

• Challenges:o Specifying constraints and objective functionso Avoiding local minima during optimization


• Adapting motion:

Witkin & Kass `88

Heavier Base

Original Jump



Witkin & Kass `88

Hurdle



Witkin & Kass `88

Ski Jump


• Editing motion:

Li et al. `99

10


• Morphing motion:

Gleicher `98


• Advantages:o Free animator from having to specify details of

physically realistic motion with spline curveso Easy to vary motions due to new parameters

and/or new constraints

• Challenges:o Specifying constraints and objective functionso Avoiding local minima during optimization

57Dynamics

• Other physical simulations:o Rigid bodieso Soft bodieso Clotho Liquidso Gaseso etc.

Hot Gases(Foster & Metaxas `97)

Cloth(Baraff & Witkin `98)

58Summary




• Kinematicso Forward kinematicso Inverse kinematics

• Dynamicso Space-time constraintso Also other physical simulations

computer animation · 2001. 11. 30. · pixar 4 3-d and 2-d animation homer 3-d homer 2-d 5 outline...

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