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Animation

When generating motion what matters?

Quality of motion appropriate for rendering style and frame rate
Controllable from UI
Controllable from AI
Skills of the animated character
Personality of the animated character

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How do we animate…

Modifying scene parameters as a function of time

Scripting
Forward Kinematics/Key framing
Inverse Kinematics
Motion capture
Simulation

Behavioral Animation
Physically based�(Dynamics)

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Scripting

Specifying the parameters at every frame

define spinningCube()

rotAngle = pi*frameNumber / 50

define carScript()

carTranslation = 10*(frameNumber / 100)

wheelRotation = pi*frameNumber / 5

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Implications for geometry

Some structure with parameters

Hierarchical �model

Hip

Torso

l.Leg1

r.Leg1

l.Leg2

r.Leg2

Shoulder

Neck

Head

r.Arm1

l.Arm2

r.Arm2

l.Arm1

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How do we animate…

Modifying scene parameters as a function of time

Scripting
Key framing
Inverse Kinematics
Motion capture
Simulation

Behavioral Animation
Physically based�(Dynamics)

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Keyframing

Sometimes we’d like better control over the motion
Thus have an artist do the hard work = Key frames

Disney

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Keyframing

Specify only the important frames,

interpolate the frames in-between

What and how to interpolate is important

interpolating angle

interpolating endpoints

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Keyframe Example

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Interpolation

In order to procedurally generate the in-between frames, use interpolation

Implies we need a point to point correspondence between animation frames

Linear interpolation

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Transformations

What will we be interpolating?

Transforms
(which are? 4x4 matrices)
With respect to a particular frame

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Rotation

How do you interpolate rotations?

You can decompose your rotation into 3 Euler rotations
have problems

Unnatural
Gimbalq lock

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Quaternions

Representation:

w is a scalar
c is a vector (3D)

A rotation by an angle of theta �around an axis of k is:
Interpretation,�a point on the unit sphere in R⁴

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Quaternion

To interpolate, slerp or lerp

Given R0 and R1��
You can interpolate by scaling alpha and computing the quaternion:

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Choosing Animation Properties

Interpolation vs. Approximation

Pass through the point or come close enough?

Computational Complexity

1^st, 2^nd, or 3^rd order polynomials?

Continuity

Allow for kinks or require smoothness?

Global vs. Local Control

Change one point will change all or only small part of the path?

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Interpolating Translations

Interpolate using a cubic B spline

Use an arc length parameterization

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Controlling motion along a path

Speed Control

Given p(t)= the animation path, t is not tied to the function’s arc length and will cause the object to move at different speeds

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Spline-driven Animation

x,y = Q(u)

for u:[0,1]

Equal arc lengths
Equal spacing in u

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Computation of Arc Length

Need to parameterize t in terms of the Arc Length
Thus, we need to know how to compute the arc length of a curve
If we know the length traveled by the object along the path, we can then control the speed at which it traverses this curve.

Can be solved analytically
Or numerically

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Keyframing Interpolation

Inbetweening

1,2,3 4 5 6 7,8,9…

Linear

1,2,3 4 5 6 7,8,9…

Slow in, Slow out

time

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Interpolating Rotations

Objects not only traverse but also rotate, so we must also be able to interpolate different angular poses of an object
First problems becomes one of representation of the angular state of the object.

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Representing Rotations

Fixed Angle Representations

Rotations around the world axes

Euler Angle Representations

Local coordinate system

Axis and angle Representations

Like OpenGL

Quaternions

Euler angles

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Basics of Quaternions

Q=[s,v], where s is a scalar and v is a 3-tuple
Quaternions are very similar to the axis and angle representations, but due to its specific way of being specified, we are able to carry out quaternion mathematics which are quite useful

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Keyframing

Decisions about interpolation

What and how

Fine level of control
Quality of motion depends on skill of animator

Disney

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Implications for geometry

What is being key-framed?

Hierarchical model?
Skeleton+skin?
Also ‘path’ must be represented

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More about skeletons+skin

Two part representation

Skeleton/bones/rig
Mesh/skin

Bones in green

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skinning

Rigid skinning associates one bone per vertex
Smooth skinning weights vertices

To interpolate transforms of n neighboring bones

http://www.okino.com/conv/skinning.htm

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skinning

Skinning can be accomplished in a vertex shader
http://graphics.cs.cmu.edu/projects/sma/jamesTwigg_SMA_medium.avi

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How do we animate…

Modifying scene parameters as a function of time

Scripting
Key framing
Inverse Kinematics
Motion capture
Simulation

Behavioral Animation
Physically based�(Dynamics)

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Kinematics

The study or specification of motion,

independent of the underlying physics

that created the motion

Articulated Figure

A figure made up of a series of links (bones)

connected at joints

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Kinematics

Degrees of Freedom (DOF)

End Effector

State Vector

The vector of independent variables

describing the state of the articulated figure

Number of variables in the state vector

The pose of the end of a chain of links

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Forward Kinematics

Given the state vector,

calculate the pose of the end effector

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kinematics

Forward kinematics is the application of motion to the hierarchical model given constraints on the motion

Pose each joint individually

Maya user guide

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Inverse kinematics

Given

an initial pose
A final pose
Joint �constraints

Solve for the �intermediate �frames

Maya user guide

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Inverse Kinematics

Given the pose of the end-effector,

calculate the state vector

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Inverse Kinematics

The kinematic function is usually highly nonlinear

and hard to invert

Consider it as locally linear

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Inverse Kinematics

Calculating the Jacobian

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Implications for geometry

Kinematics applied to?

Hierarchical model?
Skeleton+skin?
(no path but state and time)

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How do we animate…

Modifying scene parameters as a function of time

Scripting
Key framing
Inverse Kinematics
Motion capture
Simulation

Behavioral Animation
Physically based�(Dynamics)

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Motion Capture

Natural-looking motion
Hard to generalize motions

Registration is difficult
“Weightless” according to professional animators

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Motion Capture�Images courtesy Microsoft Motion Capture Group

Feb 13, Spring 2002

CS 4455

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Implications for geometry

Acquisition (registration)
Mapping to existing geometry (bones+skin)

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How do we animate…

Modifying scene parameters as a function of time

Scripting
Key framing
Inverse Kinematics
Motion capture
Simulation

Behavioral Animation
Physically based�(Dynamics)

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Simulation (Broadly Defined)

Physics is hard to simulate
Pseudo-physics is somewhat hard
Control is very hard
Gives Generalization + Interactivity

User/AI

Desired

Behavior

Control

Forces and Torques

Model

Numerical Integrator

Graphics

State

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Example – shark animation

K. Blizard

User/AI

Desired

Behavior

Control

Forces and Torques

Model

Numerical Integrator

Graphics

State

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Behavioral Animation

Animating by describing an actor’s behavior

An actor’s behavior defines how the actor

interacts with other actors and the environment

TRex()

if(player is close)

eatPlayer()

else if(can see player)

chasePlayer()

else

wander()

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Behavioral Animation

Boids (birds) [Reynolds87]

Collision Avoidance

avoid collisions with �nearby flockmates

Velocity Matching

match velocity with nearby flockmates

Flock Centering

stay near centroid of nearby flockmates

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Behavioral Animation

Useful for crowd animations

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Group Behaviors

Lots of background characters to create a feeling of motion
Make area appear interesting, rich, populated

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Flocking -- (HalfLife, Unreal)

What might go wrong?

Simple version:

Compute trajectory to head towards centroid

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Group Behaviors

Reaction to neighbors

Craig Reynolds

SIGGRAPH 1987

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Particles

Flocking quickly becomes a sub-problem of a particle system simulation….
Can be used generally to represent fluid (smoke, fire works, etc.)
Each particle contains local state

Position
Velocity
Age
Lifespan
Rendering properties…

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Dynamics

Particle Systems [Reeves83]

Create new particles

Assign initial attributes

Remove dead particles

Update particles

Render particles

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Physically based animation/Dynamics

Using “physics” to define the animation

Can use “augmented” laws of physics

For example

(1)

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ODE basics

To do things right, its worth looking at foundations…
Physically based modeling (Baraff and Witkin)

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